# Core Principles of Fluid Mechanics: Pressure, Viscosity, and Bernoulli's Principle > A first course in fluid mechanics, built from four ideas and derived rather than asserted. We weigh the column of water above a point to get pressure at depth, and read the straight pressure line off a graph. We shear a thin layer of fluid between two plates to define shear stress, the velocity gradient and viscosity, then compare air, water, oil and honey. We watch the same pipe run laminar and then turbulent, and let the Reynolds number decide which. Finally we follow a slug of water into a narrowing throat, balance the work done on it against the energy it gains, and arrive at Bernoulli's equation and the pressure drop a venturi makes. Assumes only basic mechanics and the idea of a rate of change. - Canonical watch page: [Core Principles of Fluid Mechanics: Pressure, Viscosity, and Bernoulli's Principle](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle) - Publisher: [Academa, Inc.](https://academa.ai) - Subject: Physics - Published: 2026-10-05T00:58:48.418Z - Updated: 2026-10-05T00:58:48.418Z - Duration: PT615S (10 minutes 15 seconds) - Chapters: 4 - Views: 0 - Language: en-US - Access: Free - Video stream: [HLS content](https://academa.ai/media/l/01M44QVH6A57GW6WAWAZF9M63J/0/dark/master.m3u8) - Embed: [Player](https://academa.ai/embed/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle) - Audiovisual record: [Semantic JSON](https://academa.ai/media/l/01M44QVH6A57GW6WAWAZF9M63J/0/semantic.json) - Thumbnail: [Image](https://academa.ai/media/l/01M44QVH6A57GW6WAWAZF9M63J/0/dark/poster.jpg) ## Description How fluids push, resist, and flow: pressure with depth, viscosity, laminar and turbulent flow, and Bernoulli's equation. ## Chapters - [00:00–02:35.923 · Pressure and Depth](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=0) - [02:35.923–04:46.876 · Viscosity](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=155.92262592592593) - [04:46.876–07:6.135 · Laminar and Turbulent Flow](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=286.87633425925924) - [07:6.135–10:15 · Bernoulli's Equation](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=426.13498042681715) ## Transcript ### [00:00 · Pressure and Depth](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=0) Fluid mechanics gets a remarkable amount of mileage out of four ideas, and this lecture builds them one at a time. Pressure grows with depth. Viscosity is a fluid's resistance to being sheared. Flow comes in two characters, laminar and turbulent. And the last idea, Bernoulli's equation, ties speed to pressure. So start with the easiest case there is. Here is a tank of water, standing perfectly still and open to the air at the top. Nothing is flowing yet. Pick a point somewhere inside, and call it P. It sits a depth h below the surface. What presses down on P is everything above it. This column of water, h tall, standing on a small area A. Pressure at a point is not only a downward push, though. It presses equally hard in every direction, up, down and sideways, and that is what lets us talk about the pressure at a point rather than the pressure on a face. So weigh the column. Its weight is its mass times g, its mass is the density times the volume, and the volume is the area times the height. Pressure is that force divided by the area it acts on. The area cancels straight out, and what is left is the density, times g, times the depth. Now go deeper. The column grows taller, so it weighs more, and the pressure at P climbs with it. Come back up and it falls again. Nothing but the depth changed. One term is still missing. The air above the surface is pressing down too, so the total pressure at depth h is atmospheric pressure plus rho g h. Put that on a graph. Depth across, in metres. Pressure up, in kilopascals. At the surface, where the depth is zero, the reading is just the atmosphere, about one hundred and one kilopascals. Put a marker two metres down, and then take it to ten metres. There the pressure is about two hundred kilopascals, roughly twice what the air alone gives you. Twenty metres down it is close to three hundred. The line is dead straight, and its slope is rho g, about ten kilopascals for every metre of water. And notice what never appears. Not the width of the tank, not its shape, not how much water is in it. Depth is what sets the pressure. ### [02:35.923 · Viscosity](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=155.92262592592593) A fluid at rest only pushes. To get it to resist, you have to make it slide over itself. So here are two flat plates with a thin layer of fluid between them. The lower one is bolted down. A real fluid sticks to a solid surface. Right at the lower plate the fluid does not move at all, and that is the no slip condition. It holds at every wall, in every flow. Now drag the upper plate sideways at a steady speed U. The fluid touching it is carried along at exactly that speed, for the same reason. In between, every layer slides over the one below it. The speed climbs evenly from zero at the bottom to U at the top, so the profile is a straight line. Here is the thing worth noticing. To keep that plate moving you have to keep pushing it. The push, divided by the plate's area, is the shear stress, tau. And the shear stress is proportional to how quickly the speed changes as you go up, which is the velocity gradient. The constant in front is the viscosity, mu. For a straight profile the gradient is easy. It is just U over h, the plate speed divided by the gap. Watch what happens when you drag the plate faster. The profile tips over, the gradient steepens, and the stress you have to supply goes up in proportion. Ease off, and it relaxes again. Viscosity is a property of the fluid itself, measured in pascal seconds. And fluids differ by an almost absurd margin. Air sits at about two hundred thousandths of a pascal second. Water is one thousandth, roughly fifty times more. Olive oil is near a tenth. And honey is up around ten, which is half a million times stickier than air. But it is one law for all of them. Stress equals viscosity times gradient. Only the number in front changes, and that single number decides whether a flow stays orderly or breaks up. ### [04:46.876 · Laminar and Turbulent Flow](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=286.87633425925924) Set a fluid moving along a pipe and it does not have to be tidy about it. Here is the tidy case: water running gently, with its layers gliding along in parallel. Inject a thread of red dye into the middle of it and the thread simply stays a thread, all the way down the pipe. Neighbouring layers never trade places. That orderly case is called laminar flow. Now turn the speed up, and the very same pipe does something completely different. The paths no longer stay in lane. They tumble over each other, and the dye is shredded across the whole pipe within a few diameters. That is turbulent flow. So what decides which one you get? A tug of war. Inertia carries a parcel of fluid onward in whatever direction it already had, and viscosity drags it back into line with its neighbours. That tug of war has a number attached to it, and it is called the Reynolds number. Density, speed and pipe diameter on top, standing for inertia. Viscosity underneath. Every unit cancels, so what comes out is a bare count with no units at all. Take one centimetre of water pipe. At three centimetres a second, Reynolds comes to about three hundred. Speed it up to ten centimetres a second and we reach one thousand. Still firmly laminar, and the layers hold. Push on to about twenty three centimetres a second, and we arrive at two thousand three hundred. For a pipe this is the critical value, where laminar flow starts to lose its grip. From there up to around four thousand the flow flickers between the two characters. At fifty centimetres a second, Reynolds five thousand, it is reliably turbulent. Think about what did not change there. The same pipe, the same water, the same viscosity. Only the speed. And the difference costs you. Turbulent flow mixes beautifully, which is sometimes what you want, but it also drags harder and eats far more pressure along a pipe than laminar flow does. ### [07:6.135 · Bernoulli's Equation](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=426.13498042681715) Here is a pipe that narrows in the middle, with a steady stream of water running through it from left to right. Nothing is piling up inside, and water is hard to squash. So whatever volume passes this wide section each second has to pass the throat as well. Area times speed is the same at both. Halve the area and the speed has to double. The flow runs faster through the throat, every time, and that is purely bookkeeping. Now the pressure, and for that we follow a small slug of fluid from the wide part into the throat. Fluid behind pushes it forward, fluid ahead pushes back, and the net work done on it is the pressure difference times its volume. That work has to go somewhere. The slug speeds up, so its kinetic energy rises by one half rho V times the change in v squared. And if the pipe also climbs, some of the work goes into lifting the slug, which is the change in its potential energy. Work in equals energy gained. That single line is the whole argument: the pressure drop pays for the extra speed and for the extra height. Divide the whole thing through by the volume and collect the two stations on opposite sides. Pressure, plus one half rho v squared, plus rho g y, is the same everywhere along the stream. Every term is now an energy per unit volume. The first is the pressure itself. The second is the kinetic term, which grows with the square of the speed. The third is the height term. Our pipe is horizontal, so the height is the same at both stations and that term drops out. What is left relates pressure to speed, and nothing else. Here is the answer, then. The speed in the throat is larger, so the kinetic term there is larger, and the pressure there has to be smaller to keep the total fixed. Stand a tube up out of each section and you can watch it happen. The water climbs high where the pipe is wide and slow, and sits lower where it is narrow and fast. Put numbers on it. Water at two metres a second in the wide part, doubling to four in the throat. One half of a thousand, times sixteen minus four, is six thousand pascals, so the throat sits six kilopascals lower. A narrowing like this is called a venturi, and that pressure drop is useful rather than a nuisance. Measure the drop between the two tubes and you have measured the flow rate. Four ideas, then. At rest, pressure builds with depth and with nothing else. In shear, stress is viscosity times the velocity gradient. In a pipe, the Reynolds number tells you whether the layers hold or break up. And along a stream, faster always means lower pressure. Those four will carry you a long way, because almost everything else in fluid mechanics is one of them applied somewhere new. ## About Academa, Inc. Academa makes technical knowledge easier to understand through visual lectures and lets learners request new lecture videos on the topics they need. ## Complete audiovisual record Immutable source: [semantic.json](https://academa.ai/media/l/01M44QVH6A57GW6WAWAZF9M63J/0/semantic.json) Record version: 1. Render attempt: 0. ### How to read this timeline Each scene owns its object identifiers. A beat's board is the complete board when listed, empty when marked empty, and unchanged from the nearest earlier listed board in the same scene when marked unchanged. Action times are absolute positions in the published video. ### Scene 1: [Pressure and Depth](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=0) Span: 00:00–02:35.923 (0s–155.92262592592593s). #### Objects - column: a Polygon \[yellow\] drawn in tank (vertices=((2.6, (4.2 - depth)), (3.4, (4.2 - depth)), (3.4, 4.2), (2.6, …, fill\_opacity=0.35) - depth: a VariableNumber (initial\_value=1.0, format\_spec='.1f') - drop: a Line \[yellow\] labelled "h" drawn in tank (start=(1.9, 4.2), end=(1.9, (4.2 - depth)), dashed=True) - floor: a Line \[gray\] drawn in tank (start=(0.6, 0.4), end=(5.4, 0.4)) - graph: an Axes (x\_range=(0.0, 20.0), y\_range=(0.0, 320.0), x\_ticks\_every=5.0) - h\_read: a VariableNumber (initial\_value=2.0, format\_spec='.0f') - head\_read: a Heading that says "Reading the Numbers" - law: a Math \[text\] that says "$p = p\_0 + rho g h$" - left\_wall: a Line \[gray\] drawn in tank (start=(0.6, 4.7), end=(0.6, 0.4)) - line: a FunctionPlot \[blue\] drawn in graph (function=\, x\_range=(0.0, 20.0)) - line\_2: a Line \[yellow\] drawn in graph (start=(0.0, 101.3), end=(0.0, 0.0), dashed=True) - p\_read: a VariableNumber (initial\_value=120.9, format\_spec='.0f') - point: a Point \[yellow\] drawn in graph (location=(0.0, 101.3)) - point\_2: a Point \[yellow\] drawn in graph (location=(15.0, 248.5)) - probe: a PlotPoint \[yellow\] labelled "121 thin upright("kPa")" drawn in graph (target='line', x=\) - push\_down: an Arrow \[green\] drawn in tank (start=(4.6, 3.05), end=(4.6, 2.55)) - push\_left: an Arrow \[green\] drawn in tank (start=(5.25, 2.4), end=(4.75, 2.4)) - push\_right: an Arrow \[green\] drawn in tank (start=(3.95, 2.4), end=(4.45, 2.4)) - push\_up: an Arrow \[green\] drawn in tank (start=(4.6, 1.75), end=(4.6, 2.25)) - question: a Panel that says "Why does a fluid push harder the deeper you go, and how hard does it push?" - right\_wall: a Line \[gray\] drawn in tank (start=(5.4, 0.4), end=(5.4, 4.7)) - roadmap: a Block \[text\] that says "Pressure grows with depth. Viscosity resists shearing. Flow is laminar or turbulent. Bernoulli links speed and pressure." - slope: a Math \[text\] that says "$rho g approx 9.8 thin frac(upright("kPa"), upright("m"))$" - spot: a Point \[magenta\] labelled "P" drawn in tank (location=(3.0, (4.2 - depth))) - surface: a Line \[blue\] drawn in tank (start=(0.6, 4.2), end=(5.4, 4.2)) - tank: a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 5.0), aspect=(6.0, 5.0)) - water: a Polygon \[blue\] drawn in tank (vertices=((0.6, 0.4), (5.4, 0.4), (5.4, 4.2), (0.6, 4.2)), fill\_opacity=0.22) - work: a Derivation \[text\] that says "$W &= m g \\ &= rho V g \\ &= rho A h g \\ p &= frac(W, A) \\ &= rho g h$" #### Beats ##### [00:00](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=0) Narration: Fluid mechanics gets a remarkable amount of mileage out of four ideas, and this lecture builds them one at a time. Board: Empty. Actions: - [00:00](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=0): question is shown on the screen, written out. - [00:2.81](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=2.81): roadmap is shown on the screen, written out. ##### [00:6.927](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=6.927499999999999) Narration: Pressure grows with depth. Viscosity is a fluid's resistance to being sheared. Flow comes in two characters, laminar and turbulent. And the last idea, Bernoulli's equation, ties speed to pressure. Board: roadmap — a Block \[text\] that says "Pressure grows with depth. Viscosity resists shearing. Flow is laminar or turbulent. Bernoulli links speed and pressure."; question — a Panel that says "Why does a fluid push harder the deeper you go, and how hard does it push?" Actions: - [00:7.276](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=7.276): roadmap (the "Pressure" part) is emphasized. - [00:9.307](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=9.307): roadmap (the "Pressure" part) is no longer emphasized. - [00:9.307](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=9.307): roadmap (the "Viscosity" part) is emphasized. - [00:15.414](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=15.413999999999998): roadmap (the "Viscosity" part) is no longer emphasized. - [00:15.414](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=15.413999999999998): roadmap (the "laminar or turbulent" part) is emphasized. - [00:19.907](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=19.906999999999996): roadmap (the "Bernoulli" part) is emphasized. - [00:19.907](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=19.906999999999996): roadmap (the "laminar or turbulent" part) is no longer emphasized. - [00:21.718](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=21.718): roadmap is hidden from the screen — left the board. - [00:21.718](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=21.718): roadmap (the "Bernoulli" part) is no longer emphasized. ##### [00:22.918](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=22.918) Narration: So start with the easiest case there is. Here is a tank of water, standing perfectly still and open to the air at the top. Nothing is flowing yet. Board: question — a Panel that says "Why does a fluid push harder the deeper you go, and how hard does it push?" Actions: - [00:22.918](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=22.918): tank is shown on the screen, written out. - [00:26.506](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=26.506): left\_wall is shown on the screen, written out. - [00:26.506](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=26.506): floor is shown on the screen, written out. - [00:26.506](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=26.506): right\_wall is shown on the screen, written out. - [00:26.982](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=26.982000000000003): water is shown on the screen, written out. - [00:29.745](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=29.745): surface is shown on the screen, written out. ##### [00:33.573](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=33.573) Narration: Pick a point somewhere inside, and call it P. It sits a depth h below the surface. Board: question — a Panel that says "Why does a fluid push harder the deeper you go, and how hard does it push?"; tank — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 5.0), aspect=(6.0, 5.0)); left\_wall — a Line \[gray\] drawn in tank (start=(0.6, 4.7), end=(0.6, 0.4)); floor — a Line \[gray\] drawn in tank (start=(0.6, 0.4), end=(5.4, 0.4)); right\_wall — a Line \[gray\] drawn in tank (start=(5.4, 0.4), end=(5.4, 4.7)); water — a Polygon \[blue\] drawn in tank (vertices=((0.6, 0.4), (5.4, 0.4), (5.4, 4.2), (0.6, 4.2)), fill\_opacity=0.22); surface — a Line \[blue\] drawn in tank (start=(0.6, 4.2), end=(5.4, 4.2)) Actions: - [00:36.011](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=36.010999999999996): spot is shown on the screen, written out. - [00:38.043](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=38.043000000000006): drop is shown on the screen, written out. ##### [00:40.744](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=40.744) Narration: What presses down on P is everything above it. This column of water, h tall, standing on a small area A. Board: question — a Panel that says "Why does a fluid push harder the deeper you go, and how hard does it push?"; tank — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 5.0), aspect=(6.0, 5.0)); left\_wall — a Line \[gray\] drawn in tank (start=(0.6, 4.7), end=(0.6, 0.4)); floor — a Line \[gray\] drawn in tank (start=(0.6, 0.4), end=(5.4, 0.4)); right\_wall — a Line \[gray\] drawn in tank (start=(5.4, 0.4), end=(5.4, 4.7)); water — a Polygon \[blue\] drawn in tank (vertices=((0.6, 0.4), (5.4, 0.4), (5.4, 4.2), (0.6, 4.2)), fill\_opacity=0.22); surface — a Line \[blue\] drawn in tank (start=(0.6, 4.2), end=(5.4, 4.2)); spot — a Point \[magenta\] labelled "P" drawn in tank (location=(3.0, (4.2 - depth))); drop — a Line \[yellow\] labelled "h" drawn in tank (start=(1.9, 4.2), end=(1.9, (4.2 - depth)), dashed=True) Actions: - [00:44.622](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=44.622): column is shown on the screen, written out. ##### [00:49.622](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=49.622) Narration: Pressure at a point is not only a downward push, though. It presses equally hard in every direction, up, down and sideways, and that is what lets us talk about the pressure at a point rather than the pressure on a face. Board: question — a Panel that says "Why does a fluid push harder the deeper you go, and how hard does it push?"; tank — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 5.0), aspect=(6.0, 5.0)); left\_wall — a Line \[gray\] drawn in tank (start=(0.6, 4.7), end=(0.6, 0.4)); floor — a Line \[gray\] drawn in tank (start=(0.6, 0.4), end=(5.4, 0.4)); right\_wall — a Line \[gray\] drawn in tank (start=(5.4, 0.4), end=(5.4, 4.7)); water — a Polygon \[blue\] drawn in tank (vertices=((0.6, 0.4), (5.4, 0.4), (5.4, 4.2), (0.6, 4.2)), fill\_opacity=0.22); surface — a Line \[blue\] drawn in tank (start=(0.6, 4.2), end=(5.4, 4.2)); spot — a Point \[magenta\] labelled "P" drawn in tank (location=(3.0, (4.2 - depth))); drop — a Line \[yellow\] labelled "h" drawn in tank (start=(1.9, 4.2), end=(1.9, (4.2 - depth)), dashed=True); column — a Polygon \[yellow\] drawn in tank (vertices=((2.6, (4.2 - depth)), (3.4, (4.2 - depth)), (3.4, 4.2), (2.6, …, fill\_opacity=0.35) Actions: - [00:55.358](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=55.358): push\_down is shown on the screen, written out. - [00:55.484](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=55.48399853617733): push\_up is shown on the screen, written out. - [00:55.61](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=55.60999707235466): push\_right is shown on the screen, written out. - [00:55.764](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=55.76433089996578): push\_left is shown on the screen, written out. ##### [01:3.771](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=63.771) Narration: So weigh the column. Its weight is its mass times g, its mass is the density times the volume, and the volume is the area times the height. Board: question — a Panel that says "Why does a fluid push harder the deeper you go, and how hard does it push?"; tank — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 5.0), aspect=(6.0, 5.0)); left\_wall — a Line \[gray\] drawn in tank (start=(0.6, 4.7), end=(0.6, 0.4)); floor — a Line \[gray\] drawn in tank (start=(0.6, 0.4), end=(5.4, 0.4)); right\_wall — a Line \[gray\] drawn in tank (start=(5.4, 0.4), end=(5.4, 4.7)); water — a Polygon \[blue\] drawn in tank (vertices=((0.6, 0.4), (5.4, 0.4), (5.4, 4.2), (0.6, 4.2)), fill\_opacity=0.22); surface — a Line \[blue\] drawn in tank (start=(0.6, 4.2), end=(5.4, 4.2)); spot — a Point \[magenta\] labelled "P" drawn in tank (location=(3.0, (4.2 - depth))); drop — a Line \[yellow\] labelled "h" drawn in tank (start=(1.9, 4.2), end=(1.9, (4.2 - depth)), dashed=True); column — a Polygon \[yellow\] drawn in tank (vertices=((2.6, (4.2 - depth)), (3.4, (4.2 - depth)), (3.4, 4.2), (2.6, …, fill\_opacity=0.35); push\_down — an Arrow \[green\] drawn in tank (start=(4.6, 3.05), end=(4.6, 2.55)); push\_up — an Arrow \[green\] drawn in tank (start=(4.6, 1.75), end=(4.6, 2.25)); push\_right — an Arrow \[green\] drawn in tank (start=(3.95, 2.4), end=(4.45, 2.4)); push\_left — an Arrow \[green\] drawn in tank (start=(5.25, 2.4), end=(4.75, 2.4)) Actions: - [01:3.771](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=63.771): push\_down is hidden from the screen. - [01:3.771](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=63.771): push\_up is hidden from the screen. - [01:3.771](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=63.771): push\_right is hidden from the screen. - [01:3.771](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=63.771): push\_left is hidden from the screen. - [01:5.919](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=65.919): tank moves to a new place on the board. - [01:5.919](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=65.919): work is shown on the screen, written out. - [01:10.47](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=70.47): work is shown on the screen, written out. - [01:12.943](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=72.943): work is shown on the screen, written out. ##### [01:15.088](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=75.0875) Narration: Pressure is that force divided by the area it acts on. The area cancels straight out, and what is left is the density, times g, times the depth. Board: question — a Panel that says "Why does a fluid push harder the deeper you go, and how hard does it push?"; tank — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 5.0), aspect=(6.0, 5.0)); left\_wall — a Line \[gray\] drawn in tank (start=(0.6, 4.7), end=(0.6, 0.4)); floor — a Line \[gray\] drawn in tank (start=(0.6, 0.4), end=(5.4, 0.4)); right\_wall — a Line \[gray\] drawn in tank (start=(5.4, 0.4), end=(5.4, 4.7)); water — a Polygon \[blue\] drawn in tank (vertices=((0.6, 0.4), (5.4, 0.4), (5.4, 4.2), (0.6, 4.2)), fill\_opacity=0.22); surface — a Line \[blue\] drawn in tank (start=(0.6, 4.2), end=(5.4, 4.2)); spot — a Point \[magenta\] labelled "P" drawn in tank (location=(3.0, (4.2 - depth))); drop — a Line \[yellow\] labelled "h" drawn in tank (start=(1.9, 4.2), end=(1.9, (4.2 - depth)), dashed=True); column — a Polygon \[yellow\] drawn in tank (vertices=((2.6, (4.2 - depth)), (3.4, (4.2 - depth)), (3.4, 4.2), (2.6, …, fill\_opacity=0.35) Actions: - [01:16.631](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=76.631): work is shown on the screen, written out. - [01:19.743](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=79.74300000000001): work is shown on the screen, written out. ##### [01:26.264](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=86.26400000000001) Narration: Now go deeper. The column grows taller, so it weighs more, and the pressure at P climbs with it. Come back up and it falls again. Nothing but the depth changed. Board: Unchanged from the preceding beat in this scene. Actions: - [01:27.286](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=87.28599999999999): spot is redrawn as the numbers it depends on change. - [01:27.286](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=87.28599999999999): drop is redrawn as the numbers it depends on change. - [01:27.286](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=87.28599999999999): column is redrawn as the numbers it depends on change. - [01:27.286](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=87.28599999999999): depth ticks to 3.4. - [01:33.335](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=93.335): spot is redrawn as the numbers it depends on change. - [01:33.335](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=93.335): drop is redrawn as the numbers it depends on change. - [01:33.335](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=93.335): column is redrawn as the numbers it depends on change. - [01:33.335](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=93.335): depth ticks to 1.8. ##### [01:38.323](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=98.32300000000001) Narration: One term is still missing. The air above the surface is pressing down too, so the total pressure at depth h is atmospheric pressure plus rho g h. Board: Unchanged from the preceding beat in this scene. Actions: - [01:43.664](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=103.664): law is shown on the screen, written out. - [01:45.464](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=105.464): A box is drawn around law. - [01:48.111](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=108.1105): law moves to a new place on the board. - [01:48.111](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=108.1105): question is hidden from the screen — left the board. - [01:48.111](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=108.1105): tank is hidden from the screen — left the board. - [01:48.111](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=108.1105): left\_wall is hidden from the screen — tank left the board. - [01:48.111](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=108.1105): floor is hidden from the screen — tank left the board. - [01:48.111](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=108.1105): right\_wall is hidden from the screen — tank left the board. - [01:48.111](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=108.1105): water is hidden from the screen — tank left the board. - [01:48.111](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=108.1105): surface is hidden from the screen — tank left the board. - [01:48.111](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=108.1105): spot is hidden from the screen — tank left the board. - [01:48.111](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=108.1105): drop is hidden from the screen — tank left the board. - [01:48.111](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=108.1105): column is hidden from the screen — tank left the board. - [01:48.111](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=108.1105): work is hidden from the screen — left the board. ##### [01:49.311](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=109.3105) Narration: Put that on a graph. Depth across, in metres. Pressure up, in kilopascals. At the surface, where the depth is zero, the reading is just the atmosphere, about one hundred and one kilopascals. Board: law — a Math \[text\] that says "$p = p\_0 + rho g h$" Actions: - [01:49.311](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=109.3105): graph is shown on the screen, written out. - [01:51.412](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=111.41199999999999): line is shown on the screen, written out. - [01:57.194](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=117.19399999999999): point is shown on the screen, grown. - [01:58.529](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=118.52899999999998): line\_2 is shown on the screen, drawn. - [02:0.882](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=120.88239626618487): point is hidden from the screen. - [02:0.882](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=120.88239626618487): line\_2 is hidden from the screen. ##### [02:4.667](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=124.667) Narration: Put a marker two metres down, and then take it to ten metres. There the pressure is about two hundred kilopascals, roughly twice what the air alone gives you. Board: law — a Math \[text\] that says "$p = p\_0 + rho g h$"; graph — an Axes (x\_range=(0.0, 20.0), y\_range=(0.0, 320.0), x\_ticks\_every=5.0); line — a FunctionPlot \[blue\] drawn in graph (function=\, x\_range=(0.0, 20.0)) Actions: - [02:5.352](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=125.35199999999998): probe is shown on the screen, written out. - [02:7.546](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=127.54599999999998): probe is redrawn as the numbers it depends on change. - [02:7.546](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=127.54599999999998): h\_read ticks to 10.0. - [02:7.546](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=127.54599999999998): p\_read ticks to 199.4. ##### [02:15.113](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=135.1125) Narration: Twenty metres down it is close to three hundred. The line is dead straight, and its slope is rho g, about ten kilopascals for every metre of water. Board: law — a Math \[text\] that says "$p = p\_0 + rho g h$"; graph — an Axes (x\_range=(0.0, 20.0), y\_range=(0.0, 320.0), x\_ticks\_every=5.0); line — a FunctionPlot \[blue\] drawn in graph (function=\, x\_range=(0.0, 20.0)); probe — a PlotPoint \[yellow\] labelled "121 thin upright("kPa")" drawn in graph (target='line', x=\) Actions: - [02:15.519](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=135.51899999999998): probe is redrawn as the numbers it depends on change. - [02:15.519](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=135.51899999999998): h\_read ticks to 20.0. - [02:15.519](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=135.51899999999998): p\_read ticks to 297.5. - [02:20](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=139.99999999999997): slope is shown on the screen, written out. - [02:20.476](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=140.47599999999997): law (the "rho g" part) is emphasized. - [02:24.076](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=144.0755): law (the "rho g" part) is no longer emphasized. ##### [02:24.675](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=144.6755) Narration: And notice what never appears. Not the width of the tank, not its shape, not how much water is in it. Depth is what sets the pressure. Board: law — a Math \[text\] that says "$p = p\_0 + rho g h$"; slope — a Math \[text\] that says "$rho g approx 9.8 thin frac(upright("kPa"), upright("m"))$"; graph — an Axes (x\_range=(0.0, 20.0), y\_range=(0.0, 320.0), x\_ticks\_every=5.0); line — a FunctionPlot \[blue\] drawn in graph (function=\, x\_range=(0.0, 20.0)); probe — a PlotPoint \[yellow\] labelled "121 thin upright("kPa")" drawn in graph (target='line', x=\) Actions: - [02:32.198](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=152.19799999999998): law (the "h" part) is emphasized. - [02:32.907](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=152.90699999999998): point\_2 is shown on the screen, grown. - [02:34.032](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=154.03183333333337): law (the "h" part) is no longer emphasized. - [02:34.282](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=154.28183333333337): graph is hidden from the screen — left the board. - [02:34.282](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=154.28183333333337): line is hidden from the screen — graph left the board. - [02:34.282](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=154.28183333333337): probe is hidden from the screen — graph left the board. - [02:34.282](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=154.28183333333337): law is hidden from the screen — left the board. - [02:34.282](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=154.28183333333337): slope is hidden from the screen — left the board. - [02:34.881](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=154.8809592592593): point\_2 is hidden from the screen. ### Scene 2: [Viscosity](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=155.92262592592593) Span: 02:35.923–04:46.876 (155.92262592592593s–286.87633425925924s). #### Objects - bottom\_plate: a Line \[gray\] drawn in plates (start=(0.4, 0.5), end=(5.6, 0.5)) - head\_numbers: a Heading that says "How Sticky, in Numbers" - head\_shear: a Heading that says "Shearing a Fluid" - layer\_1: an Arrow \[green\] drawn in plates (start=(1.0, 0.95), end=((1.0 + (0.42 \* u\_top)), 0.95)) - layer\_2: an Arrow \[green\] drawn in plates (start=(1.0, 1.4), end=((1.0 + (0.85 \* u\_top)), 1.4)) - layer\_3: an Arrow \[green\] drawn in plates (start=(1.0, 1.85), end=((1.0 + (1.27 \* u\_top)), 1.85)) - layer\_4: an Arrow \[green\] drawn in plates (start=(1.0, 2.3), end=((1.0 + (1.7 \* u\_top)), 2.3)) - mu\_units: a Math \[text\] that says "$\[mu\] = upright("Pa") dot.op upright("s")$" - noslip\_def: a Panel that says "A real fluid sticks to a solid surface. At the wall, the fluid moves with the wall and not past it." - plate\_pull: an Arrow \[red\] labelled "U" drawn in plates (start=(3.2, 2.62), end=((3.2 + (1.2 \* u\_top)), 2.62)) - plates: a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 3.2), aspect=(6.0, 3.2)) - profile: a Line \[yellow\] labelled "u(y)" drawn in plates (start=(1.0, 0.5), end=((1.0 + (1.7 \* u\_top)), 2.3), dashed=True) - shear: a Derivation \[text\] that says "$tau &= frac(F, A) \\ &= mu frac(dif u, dif y) \\ &= mu frac(U, h)$" - stuck: a Point \[green\] labelled "u = 0" drawn in plates (location=(1.0, 0.5)) - table: a Table \[text\] that says "Fluid $mu$ in Pa s air $1.8 times 10^(-5)$ water $1.0 times 10^(-3)$ olive oil $8 times 10^(-2)$ honey $approx 10$" (rows=(('Fluid', '$mu$ in Pa s'), ('air', '$1.8 times 10^(-5)$'), ('w…, header=True) - top\_plate: a Line \[gray\] drawn in plates (start=(0.4, 2.3), end=(5.6, 2.3)) - u\_top: a VariableNumber (initial\_value=1.0, format\_spec='.1f') #### Beats ##### [02:35.923](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=155.92262592592593) Narration: A fluid at rest only pushes. To get it to resist, you have to make it slide over itself. So here are two flat plates with a thin layer of fluid between them. The lower one is bolted down. Board: Empty. Actions: - [02:35.923](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=155.92262592592593): head\_shear is shown on the screen, written out. - [02:43.922](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=163.92162592592592): plates is shown on the screen, written out. - [02:46.964](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=166.96362592592592): bottom\_plate is shown on the screen, written out. - [02:46.964](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=166.96362592592592): top\_plate is shown on the screen, written out. ##### [02:49.131](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=169.13112592592591) Narration: A real fluid sticks to a solid surface. Right at the lower plate the fluid does not move at all, and that is the no slip condition. It holds at every wall, in every flow. Board: plates — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 3.2), aspect=(6.0, 3.2)); head\_shear — a Heading that says "Shearing a Fluid"; bottom\_plate — a Line \[gray\] drawn in plates (start=(0.4, 0.5), end=(5.6, 0.5)); top\_plate — a Line \[gray\] drawn in plates (start=(0.4, 2.3), end=(5.6, 2.3)) Actions: - [02:54.669](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=174.66862592592594): stuck is shown on the screen, written out. - [02:56.956](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=176.95562592592591): plates moves to a new place on the board. - [02:56.956](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=176.95562592592591): noslip\_def is shown on the screen, written out. ##### [03:1.828](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=181.82812592592592) Narration: Now drag the upper plate sideways at a steady speed U. The fluid touching it is carried along at exactly that speed, for the same reason. Board: noslip\_def — a Panel that says "A real fluid sticks to a solid surface. At the wall, the fluid moves with the wall and not past it."; plates — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 3.2), aspect=(6.0, 3.2)); head\_shear — a Heading that says "Shearing a Fluid"; bottom\_plate — a Line \[gray\] drawn in plates (start=(0.4, 0.5), end=(5.6, 0.5)); top\_plate — a Line \[gray\] drawn in plates (start=(0.4, 2.3), end=(5.6, 2.3)); stuck — a Point \[green\] labelled "u = 0" drawn in plates (location=(1.0, 0.5)) Actions: - [03:4.452](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=184.45162592592592): plate\_pull is shown on the screen, written out. - [03:7.181](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=187.18062592592594): layer\_4 is shown on the screen, written out. ##### [03:11.345](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=191.34462592592593) Narration: In between, every layer slides over the one below it. The speed climbs evenly from zero at the bottom to U at the top, so the profile is a straight line. Board: noslip\_def — a Panel that says "A real fluid sticks to a solid surface. At the wall, the fluid moves with the wall and not past it."; plates — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 3.2), aspect=(6.0, 3.2)); head\_shear — a Heading that says "Shearing a Fluid"; bottom\_plate — a Line \[gray\] drawn in plates (start=(0.4, 0.5), end=(5.6, 0.5)); top\_plate — a Line \[gray\] drawn in plates (start=(0.4, 2.3), end=(5.6, 2.3)); stuck — a Point \[green\] labelled "u = 0" drawn in plates (location=(1.0, 0.5)); plate\_pull — an Arrow \[red\] labelled "U" drawn in plates (start=(3.2, 2.62), end=((3.2 + (1.2 \* u\_top)), 2.62)); layer\_4 — an Arrow \[green\] drawn in plates (start=(1.0, 2.3), end=((1.0 + (1.7 \* u\_top)), 2.3)) Actions: - [03:13.133](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=193.13262592592594): layer\_3 is shown on the screen, written out. - [03:13.259](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=193.2586583478593): layer\_2 is shown on the screen, written out. - [03:13.385](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=193.38469076979266): layer\_1 is shown on the screen, written out. - [03:20.505](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=200.50462592592592): profile is shown on the screen, written out. ##### [03:22.138](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=202.13812592592592) Narration: Here is the thing worth noticing. To keep that plate moving you have to keep pushing it. The push, divided by the plate's area, is the shear stress, tau. Board: noslip\_def — a Panel that says "A real fluid sticks to a solid surface. At the wall, the fluid moves with the wall and not past it."; plates — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 3.2), aspect=(6.0, 3.2)); head\_shear — a Heading that says "Shearing a Fluid"; bottom\_plate — a Line \[gray\] drawn in plates (start=(0.4, 0.5), end=(5.6, 0.5)); top\_plate — a Line \[gray\] drawn in plates (start=(0.4, 2.3), end=(5.6, 2.3)); stuck — a Point \[green\] labelled "u = 0" drawn in plates (location=(1.0, 0.5)); plate\_pull — an Arrow \[red\] labelled "U" drawn in plates (start=(3.2, 2.62), end=((3.2 + (1.2 \* u\_top)), 2.62)); layer\_4 — an Arrow \[green\] drawn in plates (start=(1.0, 2.3), end=((1.0 + (1.7 \* u\_top)), 2.3)); layer\_3 — an Arrow \[green\] drawn in plates (start=(1.0, 1.85), end=((1.0 + (1.27 \* u\_top)), 1.85)); layer\_2 — an Arrow \[green\] drawn in plates (start=(1.0, 1.4), end=((1.0 + (0.85 \* u\_top)), 1.4)); layer\_1 — an Arrow \[green\] drawn in plates (start=(1.0, 0.95), end=((1.0 + (0.42 \* u\_top)), 0.95)); profile — a Line \[yellow\] labelled "u(y)" drawn in plates (start=(1.0, 0.5), end=((1.0 + (1.7 \* u\_top)), 2.3), dashed=True) Actions: - [03:31.368](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=211.36762592592592): shear is shown on the screen, written out. ##### [03:33.512](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=213.51162592592593) Narration: And the shear stress is proportional to how quickly the speed changes as you go up, which is the velocity gradient. The constant in front is the viscosity, mu. Board: Unchanged from the preceding beat in this scene. Actions: - [03:39.282](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=219.28162592592594): shear is shown on the screen, written out. - [03:42.428](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=222.42762592592592): shear (the "mu" part) is emphasized. - [03:44.146](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=224.14612592592593): shear (the "mu" part) is no longer emphasized. ##### [03:44.746](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=224.74612592592592) Narration: For a straight profile the gradient is easy. It is just U over h, the plate speed divided by the gap. Board: Unchanged from the preceding beat in this scene. Actions: - [03:46.488](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=226.48762592592593): shear is shown on the screen, written out. ##### [03:52.73](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=232.72962592592592) Narration: Watch what happens when you drag the plate faster. The profile tips over, the gradient steepens, and the stress you have to supply goes up in proportion. Ease off, and it relaxes again. Board: Unchanged from the preceding beat in this scene. Actions: - [03:55.215](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=235.21462592592593): plate\_pull is redrawn as the numbers it depends on change. - [03:55.215](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=235.21462592592593): layer\_4 is redrawn as the numbers it depends on change. - [03:55.215](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=235.21462592592593): layer\_3 is redrawn as the numbers it depends on change. - [03:55.215](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=235.21462592592593): layer\_2 is redrawn as the numbers it depends on change. - [03:55.215](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=235.21462592592593): layer\_1 is redrawn as the numbers it depends on change. - [03:55.215](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=235.21462592592593): profile is redrawn as the numbers it depends on change. - [03:55.215](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=235.21462592592593): u\_top ticks to 1.8. - [04:3.226](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=243.2256259259259): plate\_pull is redrawn as the numbers it depends on change. - [04:3.226](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=243.2256259259259): layer\_4 is redrawn as the numbers it depends on change. - [04:3.226](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=243.2256259259259): layer\_3 is redrawn as the numbers it depends on change. - [04:3.226](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=243.2256259259259): layer\_2 is redrawn as the numbers it depends on change. - [04:3.226](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=243.2256259259259): layer\_1 is redrawn as the numbers it depends on change. - [04:3.226](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=243.2256259259259): profile is redrawn as the numbers it depends on change. - [04:3.226](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=243.2256259259259): u\_top ticks to 0.6. - [04:5.71](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=245.71012592592592): shear moves to a new place on the board. - [04:5.71](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=245.71012592592592): head\_shear is hidden from the screen — left the board. - [04:5.71](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=245.71012592592592): noslip\_def is hidden from the screen — left the board. - [04:5.71](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=245.71012592592592): plates is hidden from the screen — left the board. - [04:5.71](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=245.71012592592592): bottom\_plate is hidden from the screen — plates left the board. - [04:5.71](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=245.71012592592592): top\_plate is hidden from the screen — plates left the board. - [04:5.71](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=245.71012592592592): stuck is hidden from the screen — plates left the board. - [04:5.71](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=245.71012592592592): plate\_pull is hidden from the screen — plates left the board. - [04:5.71](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=245.71012592592592): layer\_4 is hidden from the screen — plates left the board. - [04:5.71](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=245.71012592592592): layer\_3 is hidden from the screen — plates left the board. - [04:5.71](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=245.71012592592592): layer\_2 is hidden from the screen — plates left the board. - [04:5.71](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=245.71012592592592): layer\_1 is hidden from the screen — plates left the board. - [04:5.71](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=245.71012592592592): profile is hidden from the screen — plates left the board. ##### [04:6.91](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=246.9101259259259) Narration: Viscosity is a property of the fluid itself, measured in pascal seconds. And fluids differ by an almost absurd margin. Board: Empty. Actions: - [04:6.91](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=246.9101259259259): head\_numbers is shown on the screen, written out. - [04:10.672](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=250.6716259259259): mu\_units is shown on the screen, written out. - [04:13.47](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=253.4696259259259): table is shown on the screen, written out. ##### [04:16.659](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=256.6591259259259) Narration: Air sits at about two hundred thousandths of a pascal second. Water is one thousandth, roughly fifty times more. Board: mu\_units — a Math \[text\] that says "$\[mu\] = upright("Pa") dot.op upright("s")$"; head\_numbers — a Heading that says "How Sticky, in Numbers" Actions: - [04:17.008](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=257.0076259259259): table is shown on the screen, written out. - [04:21.152](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=261.1516259259259): table is shown on the screen, written out. - [04:23.022](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=263.0216259259259): table is indicated — a transient flash. ##### [04:25.026](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=265.0261259259259) Narration: Olive oil is near a tenth. And honey is up around ten, which is half a million times stickier than air. Board: Unchanged from the preceding beat in this scene. Actions: - [04:25.375](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=265.3746259259259): table is shown on the screen, written out. - [04:27.848](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=267.8476259259259): table is shown on the screen, written out. - [04:30.146](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=270.1456259259259): table is indicated — a transient flash. ##### [04:32.975](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=272.9751259259259) Narration: But it is one law for all of them. Stress equals viscosity times gradient. Only the number in front changes, and that single number decides whether a flow stays orderly or breaks up. Board: Unchanged from the preceding beat in this scene. Actions: - [04:36.156](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=276.1556259259259): shear (the "mu frac(dif u, dif y)" part) is emphasized. - [04:39.976](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=279.9756259259259): table (the "column=2" part) is emphasized. - [04:42.577](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=282.5766259259259): shear (the "mu frac(dif u, dif y)" part) is no longer emphasized. - [04:45.585](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=285.5846675925926): table (the "column=2" part) is no longer emphasized. - [04:45.835](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=285.8346675925926): head\_numbers is hidden from the screen — left the board. - [04:45.835](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=285.8346675925926): mu\_units is hidden from the screen — left the board. - [04:45.835](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=285.8346675925926): shear is hidden from the screen — left the board. - [04:45.835](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=285.8346675925926): table is hidden from the screen — left the board. ### Scene 3: [Laminar and Turbulent Flow](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=286.87633425925924) Span: 04:46.876–07:6.135 (286.87633425925924s–426.13498042681715s). #### Objects - cap\_lam: a Tex \[text\] that says "Laminar: the layers slide, and the dye stays a thread." - cap\_turb: a Tex \[text\] that says "Turbulent: the layers mix, and the dye is torn apart." - dye: a Line \[red\] drawn in laminar (start=(0.7, 1.5), end=(5.3, 1.5)) - head\_re: a Heading that says "One Number Decides" - head\_regimes: a Heading that says "Two Ways to Flow" - lam\_1: an Arrow \[blue\] drawn in laminar (start=(0.7, 0.85), end=(5.0, 0.85)) - lam\_2: an Arrow \[blue\] drawn in laminar (start=(0.7, 1.2), end=(5.2, 1.2)) - lam\_3: an Arrow \[blue\] drawn in laminar (start=(0.7, 1.8), end=(5.2, 1.8)) - lam\_4: an Arrow \[blue\] drawn in laminar (start=(0.7, 2.15), end=(5.0, 2.15)) - lam\_bottom: a Line \[gray\] drawn in laminar (start=(0.4, 0.4), end=(5.6, 0.4)) - lam\_top: a Line \[gray\] drawn in laminar (start=(0.4, 2.6), end=(5.6, 2.6)) - laminar: a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 3.0), aspect=(6.0, 3.0)) - note\_high: a Tex \[text\] that says "Above about $4000$: turbulent." - note\_low: a Tex \[text\] that says "Below about $2300$ in a pipe: laminar." - note\_mid: a Tex \[text\] that says "$2300$ to $4000$: transition, and unsteady." - point: a Point \[yellow\] drawn in scale (location=(2300.0, 0.0)) - point\_2: a Point \[yellow\] drawn in scale (location=(5500.0, 0.0)) - point\_3: a Point \[yellow\] drawn in scale (location=(700.0, 0.0)) - pointer: a Point \[yellow\] labelled "300" drawn in scale (location=(\, 0.0)) - re\_def: a Math \[text\] that says "$upright("Re") = frac(rho v D, mu)$" - re\_val: a VariableNumber (initial\_value=300.0, format\_spec='.0f') - scale: a NumberLine labelled "upright("Re")" (x\_range=(0.0, 6000.0), include\_numbers=True, ticks\_every=1000.0) - turb\_1: a ParametricCurve \[blue\] drawn in turbulent (function=\, t\_range=(0.6, 5.4)) - turb\_2: a ParametricCurve \[blue\] drawn in turbulent (function=\, t\_range=(0.6, 5.4)) - turb\_bottom: a Line \[gray\] drawn in turbulent (start=(0.4, 0.4), end=(5.6, 0.4)) - turb\_dye: a ParametricCurve \[red\] drawn in turbulent (function=\, t\_range=(0.6, 5.4)) - turb\_top: a Line \[gray\] drawn in turbulent (start=(0.4, 2.6), end=(5.6, 2.6)) - turbulent: a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 3.0), aspect=(6.0, 3.0)) #### Beats ##### [04:46.876](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=286.87633425925924) Narration: Set a fluid moving along a pipe and it does not have to be tidy about it. Here is the tidy case: water running gently, with its layers gliding along in parallel. Board: Empty. Actions: - [04:46.876](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=286.87633425925924): head\_regimes is shown on the screen, written out. - [04:48.258](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=288.25833425925924): lam\_bottom is shown on the screen, written out. - [04:48.258](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=288.25833425925924): lam\_top is shown on the screen, written out. - [04:50.034](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=290.03433425925925): laminar is shown on the screen, written out. - [04:56.013](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=296.01333425925924): lam\_1 is shown on the screen, written out. - [04:56.13](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=296.1301825153917): lam\_2 is shown on the screen, written out. - [04:56.247](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=296.2470307715241): lam\_3 is shown on the screen, written out. - [04:56.364](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=296.3638790276566): lam\_4 is shown on the screen, written out. ##### [04:59.04](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=299.03983425925924) Narration: Inject a thread of red dye into the middle of it and the thread simply stays a thread, all the way down the pipe. Neighbouring layers never trade places. Board: laminar — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 3.0), aspect=(6.0, 3.0)); head\_regimes — a Heading that says "Two Ways to Flow"; lam\_bottom — a Line \[gray\] drawn in laminar (start=(0.4, 0.4), end=(5.6, 0.4)); lam\_top — a Line \[gray\] drawn in laminar (start=(0.4, 2.6), end=(5.6, 2.6)); lam\_1 — an Arrow \[blue\] drawn in laminar (start=(0.7, 0.85), end=(5.0, 0.85)); lam\_2 — an Arrow \[blue\] drawn in laminar (start=(0.7, 1.2), end=(5.2, 1.2)); lam\_3 — an Arrow \[blue\] drawn in laminar (start=(0.7, 1.8), end=(5.2, 1.8)); lam\_4 — an Arrow \[blue\] drawn in laminar (start=(0.7, 2.15), end=(5.0, 2.15)) Actions: - [05:0.038](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=300.0383342592592): dye is shown on the screen, written out. - [05:7.781](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=307.7813342592592): dye is indicated — a transient flash. ##### [05:9.891](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=309.8913342592592) Narration: That orderly case is called laminar flow. Now turn the speed up, and the very same pipe does something completely different. Board: laminar — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 3.0), aspect=(6.0, 3.0)); head\_regimes — a Heading that says "Two Ways to Flow"; lam\_bottom — a Line \[gray\] drawn in laminar (start=(0.4, 0.4), end=(5.6, 0.4)); lam\_top — a Line \[gray\] drawn in laminar (start=(0.4, 2.6), end=(5.6, 2.6)); lam\_1 — an Arrow \[blue\] drawn in laminar (start=(0.7, 0.85), end=(5.0, 0.85)); lam\_2 — an Arrow \[blue\] drawn in laminar (start=(0.7, 1.2), end=(5.2, 1.2)); lam\_3 — an Arrow \[blue\] drawn in laminar (start=(0.7, 1.8), end=(5.2, 1.8)); lam\_4 — an Arrow \[blue\] drawn in laminar (start=(0.7, 2.15), end=(5.0, 2.15)); dye — a Line \[red\] drawn in laminar (start=(0.7, 1.5), end=(5.3, 1.5)) Actions: - [05:11.783](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=311.7833342592592): laminar moves to a new place on the board. - [05:11.783](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=311.7833342592592): cap\_lam is shown on the screen, written out. - [05:13.617](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=313.6173342592592): turbulent is shown on the screen, written out. - [05:13.617](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=313.6173342592592): turb\_bottom is shown on the screen, written out. - [05:13.617](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=313.6173342592592): turb\_top is shown on the screen, written out. - [05:17.321](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=317.32133425925923): turb\_1 is shown on the screen, written out. - [05:17.541](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=317.5408903166203): turb\_2 is shown on the screen, written out. ##### [05:18.676](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=318.67583425925926) Narration: The paths no longer stay in lane. They tumble over each other, and the dye is shredded across the whole pipe within a few diameters. That is turbulent flow. Board: cap\_lam — a Tex \[text\] that says "Laminar: the layers slide, and the dye stays a thread."; laminar — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 3.0), aspect=(6.0, 3.0)); turbulent — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 3.0), aspect=(6.0, 3.0)); head\_regimes — a Heading that says "Two Ways to Flow"; lam\_bottom — a Line \[gray\] drawn in laminar (start=(0.4, 0.4), end=(5.6, 0.4)); lam\_top — a Line \[gray\] drawn in laminar (start=(0.4, 2.6), end=(5.6, 2.6)); lam\_1 — an Arrow \[blue\] drawn in laminar (start=(0.7, 0.85), end=(5.0, 0.85)); lam\_2 — an Arrow \[blue\] drawn in laminar (start=(0.7, 1.2), end=(5.2, 1.2)); lam\_3 — an Arrow \[blue\] drawn in laminar (start=(0.7, 1.8), end=(5.2, 1.8)); lam\_4 — an Arrow \[blue\] drawn in laminar (start=(0.7, 2.15), end=(5.0, 2.15)); dye — a Line \[red\] drawn in laminar (start=(0.7, 1.5), end=(5.3, 1.5)); turb\_bottom — a Line \[gray\] drawn in turbulent (start=(0.4, 0.4), end=(5.6, 0.4)); turb\_top — a Line \[gray\] drawn in turbulent (start=(0.4, 2.6), end=(5.6, 2.6)); turb\_1 — a ParametricCurve \[blue\] drawn in turbulent (function=\, t\_range=(0.6, 5.4)); turb\_2 — a ParametricCurve \[blue\] drawn in turbulent (function=\, t\_range=(0.6, 5.4)) Actions: - [05:23.644](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=323.64433425925927): turb\_dye is shown on the screen, written out. - [05:27.232](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=327.23233425925923): cap\_turb is shown on the screen, written out. ##### [05:29.202](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=329.2018342592593) Narration: So what decides which one you get? A tug of war. Inertia carries a parcel of fluid onward in whatever direction it already had, and viscosity drags it back into line with its neighbours. Board: cap\_lam — a Tex \[text\] that says "Laminar: the layers slide, and the dye stays a thread."; laminar — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 3.0), aspect=(6.0, 3.0)); cap\_turb — a Tex \[text\] that says "Turbulent: the layers mix, and the dye is torn apart."; turbulent — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 3.0), aspect=(6.0, 3.0)); head\_regimes — a Heading that says "Two Ways to Flow"; lam\_bottom — a Line \[gray\] drawn in laminar (start=(0.4, 0.4), end=(5.6, 0.4)); lam\_top — a Line \[gray\] drawn in laminar (start=(0.4, 2.6), end=(5.6, 2.6)); lam\_1 — an Arrow \[blue\] drawn in laminar (start=(0.7, 0.85), end=(5.0, 0.85)); lam\_2 — an Arrow \[blue\] drawn in laminar (start=(0.7, 1.2), end=(5.2, 1.2)); lam\_3 — an Arrow \[blue\] drawn in laminar (start=(0.7, 1.8), end=(5.2, 1.8)); lam\_4 — an Arrow \[blue\] drawn in laminar (start=(0.7, 2.15), end=(5.0, 2.15)); dye — a Line \[red\] drawn in laminar (start=(0.7, 1.5), end=(5.3, 1.5)); turb\_bottom — a Line \[gray\] drawn in turbulent (start=(0.4, 0.4), end=(5.6, 0.4)); turb\_top — a Line \[gray\] drawn in turbulent (start=(0.4, 2.6), end=(5.6, 2.6)); turb\_1 — a ParametricCurve \[blue\] drawn in turbulent (function=\, t\_range=(0.6, 5.4)); turb\_2 — a ParametricCurve \[blue\] drawn in turbulent (function=\, t\_range=(0.6, 5.4)); turb\_dye — a ParametricCurve \[red\] drawn in turbulent (function=\, t\_range=(0.6, 5.4)) Actions: - [05:34.031](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=334.03133425925927): turb\_dye is indicated — a transient flash. - [05:39.105](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=339.1053342592592): dye is indicated — a transient flash. - [05:42.415](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=342.41483425925924): cap\_lam is hidden from the screen — left the board. - [05:42.415](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=342.41483425925924): cap\_turb is hidden from the screen — left the board. - [05:42.415](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=342.41483425925924): head\_regimes is hidden from the screen — left the board. - [05:42.415](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=342.41483425925924): laminar is hidden from the screen — left the board. - [05:42.415](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=342.41483425925924): lam\_bottom is hidden from the screen — laminar left the board. - [05:42.415](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=342.41483425925924): lam\_top is hidden from the screen — laminar left the board. - [05:42.415](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=342.41483425925924): lam\_1 is hidden from the screen — laminar left the board. - [05:42.415](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=342.41483425925924): lam\_2 is hidden from the screen — laminar left the board. - [05:42.415](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=342.41483425925924): lam\_3 is hidden from the screen — laminar left the board. - [05:42.415](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=342.41483425925924): lam\_4 is hidden from the screen — laminar left the board. - [05:42.415](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=342.41483425925924): dye is hidden from the screen — laminar left the board. - [05:42.415](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=342.41483425925924): turbulent is hidden from the screen — left the board. - [05:42.415](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=342.41483425925924): turb\_bottom is hidden from the screen — turbulent left the board. - [05:42.415](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=342.41483425925924): turb\_top is hidden from the screen — turbulent left the board. - [05:42.415](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=342.41483425925924): turb\_1 is hidden from the screen — turbulent left the board. - [05:42.415](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=342.41483425925924): turb\_2 is hidden from the screen — turbulent left the board. - [05:42.415](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=342.41483425925924): turb\_dye is hidden from the screen — turbulent left the board. ##### [05:43.615](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=343.6148342592592) Narration: That tug of war has a number attached to it, and it is called the Reynolds number. Density, speed and pipe diameter on top, standing for inertia. Viscosity underneath. Board: Empty. Actions: - [05:43.615](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=343.6148342592592): head\_re is shown on the screen, written out. - [05:43.615](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=343.6148342592592): scale is shown on the screen, written out. - [05:47.643](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=347.64333425925923): scale moves to a new place on the board. - [05:47.643](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=347.64333425925923): re\_def is shown on the screen, written out. - [05:53.251](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=353.25133425925924): re\_def (the "rho v D" part) is emphasized. - [05:54.563](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=354.56333425925925): re\_def (the "mu" part) is emphasized. - [05:54.563](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=354.56333425925925): re\_def (the "rho v D" part) is no longer emphasized. - [05:56.792](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=356.79183425925925): re\_def (the "mu" part) is no longer emphasized. ##### [05:57.392](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=357.3918342592592) Narration: Every unit cancels, so what comes out is a bare count with no units at all. Take one centimetre of water pipe. At three centimetres a second, Reynolds comes to about three hundred. Board: re\_def — a Math \[text\] that says "$upright("Re") = frac(rho v D, mu)$"; scale — a NumberLine labelled "upright("Re")" (x\_range=(0.0, 6000.0), include\_numbers=True, ticks\_every=1000.0); head\_re — a Heading that says "One Number Decides" Actions: - [06:8.259](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=368.2593342592593): pointer is shown on the screen, written out. ##### [06:9.951](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=369.95083425925924) Narration: Speed it up to ten centimetres a second and we reach one thousand. Still firmly laminar, and the layers hold. Board: re\_def — a Math \[text\] that says "$upright("Re") = frac(rho v D, mu)$"; scale — a NumberLine labelled "upright("Re")" (x\_range=(0.0, 6000.0), include\_numbers=True, ticks\_every=1000.0); head\_re — a Heading that says "One Number Decides"; pointer — a Point \[yellow\] labelled "300" drawn in scale (location=(\, 0.0)) Actions: - [06:13.457](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=373.45733425925926): pointer is redrawn as the numbers it depends on change. - [06:13.457](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=373.45733425925926): re\_val ticks to 1000.0. - [06:15.57](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=375.57033425925925): note\_low is shown on the screen, written out. ##### [06:18.48](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=378.4803342592592) Narration: Push on to about twenty three centimetres a second, and we arrive at two thousand three hundred. For a pipe this is the critical value, where laminar flow starts to lose its grip. Board: re\_def — a Math \[text\] that says "$upright("Re") = frac(rho v D, mu)$"; note\_low — a Tex \[text\] that says "Below about $2300$ in a pipe: laminar."; scale — a NumberLine labelled "upright("Re")" (x\_range=(0.0, 6000.0), include\_numbers=True, ticks\_every=1000.0); head\_re — a Heading that says "One Number Decides"; pointer — a Point \[yellow\] labelled "300" drawn in scale (location=(\, 0.0)) Actions: - [06:22.892](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=382.89233425925926): pointer is redrawn as the numbers it depends on change. - [06:22.892](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=382.89233425925926): re\_val ticks to 2300.0. - [06:26.305](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=386.30533425925927): point is shown on the screen, grown. - [06:28.572](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=388.57243241876847): point is hidden from the screen. - [06:29.533](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=389.5333342592593): note\_mid is shown on the screen, written out. ##### [06:31.05](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=391.0503342592592) Narration: From there up to around four thousand the flow flickers between the two characters. At fifty centimetres a second, Reynolds five thousand, it is reliably turbulent. Board: re\_def — a Math \[text\] that says "$upright("Re") = frac(rho v D, mu)$"; note\_low — a Tex \[text\] that says "Below about $2300$ in a pipe: laminar."; note\_mid — a Tex \[text\] that says "$2300$ to $4000$: transition, and unsteady."; scale — a NumberLine labelled "upright("Re")" (x\_range=(0.0, 6000.0), include\_numbers=True, ticks\_every=1000.0); head\_re — a Heading that says "One Number Decides"; pointer — a Point \[yellow\] labelled "300" drawn in scale (location=(\, 0.0)) Actions: - [06:36.588](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=396.5883342592592): pointer is redrawn as the numbers it depends on change. - [06:36.588](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=396.5883342592592): re\_val ticks to 5000.0. - [06:40.802](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=400.8023342592592): note\_high is shown on the screen, written out. ##### [06:42.633](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=402.63333425925924) Narration: Think about what did not change there. The same pipe, the same water, the same viscosity. Only the speed. Board: re\_def — a Math \[text\] that says "$upright("Re") = frac(rho v D, mu)$"; note\_low — a Tex \[text\] that says "Below about $2300$ in a pipe: laminar."; note\_mid — a Tex \[text\] that says "$2300$ to $4000$: transition, and unsteady."; note\_high — a Tex \[text\] that says "Above about $4000$: turbulent."; scale — a NumberLine labelled "upright("Re")" (x\_range=(0.0, 6000.0), include\_numbers=True, ticks\_every=1000.0); head\_re — a Heading that says "One Number Decides"; pointer — a Point \[yellow\] labelled "300" drawn in scale (location=(\, 0.0)) Actions: - [06:50.737](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=410.7373342592592): re\_def (the "v" part) is emphasized. - [06:51.62](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=411.6198342592593): re\_def (the "v" part) is no longer emphasized. ##### [06:52.22](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=412.21983425925924) Narration: And the difference costs you. Turbulent flow mixes beautifully, which is sometimes what you want, but it also drags harder and eats far more pressure along a pipe than laminar flow does. Board: Unchanged from the preceding beat in this scene. Actions: - [06:53.299](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=413.29933425925924): point\_2 is shown on the screen, grown. - [06:56.62](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=416.6201373310721): point\_2 is hidden from the screen. - [07:2.97](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=422.97033425925923): point\_3 is shown on the screen, grown. - [07:4.495](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=424.4945009259259): head\_re is hidden from the screen — left the board. - [07:4.495](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=424.4945009259259): note\_high is hidden from the screen — left the board. - [07:4.495](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=424.4945009259259): note\_low is hidden from the screen — left the board. - [07:4.495](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=424.4945009259259): note\_mid is hidden from the screen — left the board. - [07:4.495](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=424.4945009259259): re\_def is hidden from the screen — left the board. - [07:4.495](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=424.4945009259259): scale is hidden from the screen — left the board. - [07:4.495](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=424.4945009259259): pointer is hidden from the screen — scale left the board. - [07:5.093](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=425.0933137601505): point\_3 is hidden from the screen. ### Scene 4: [Bernoulli's Equation](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=426.13498042681715) Span: 07:6.135–10:14.735 (426.13498042681715s–614.7352095934839s). #### Objects - area\_throat: a Line \[yellow\] labelled "A\_2" drawn in pipe (start=(3.85, 1.41), end=(3.85, 2.29), dashed=True) - area\_wide: a Line \[yellow\] labelled "A\_1" drawn in pipe (start=(1.55, 0.9), end=(1.55, 2.8), dashed=True) - bottom\_wall: a ParametricCurve \[gray\] drawn in pipe (function=\, t\_range=(0.5, 7.5)) - c0: a Math \[text\] that says "$A\_1 v\_1 = A\_2 v\_2$" - c1: a Math \[text\] that says "$v\_2 = frac(A\_1, A\_2) v\_1$" - const: a Math \[text\] that says "$p + frac(1, 2) rho v^2 + rho g y = upright("constant")$" - energy: a Derivation \[text\] that says "$W &= (p\_1 - p\_2) V \\ Delta K &= frac(1, 2) rho V (v\_2^2 - v\_1^2) \\ Delta U &= rho V g (y\_2 - y\_1) \\ (p\_1 - p\_2) V &= Delta K + Delta U$" - example: a Math \[text\] that says "$Delta p = frac(1, 2) rho (v\_2^2 - v\_1^2) = 6 thin upright("kPa")$" - head\_energy: a Heading that says "Where the Energy Goes" - head\_law: a Heading that says "Bernoulli's Equation" - head\_recap: a Heading that says "What to Carry Away" - horiz: a Math \[text\] that says "$p\_1 + frac(1, 2) rho v\_1^2 = p\_2 + frac(1, 2) rho v\_2^2$" - pipe: a Figure (x\_range=(0.0, 8.0), y\_range=(0.0, 5.0), aspect=(8.0, 5.0)) - question: a Panel that says "Where this pipe narrows, what happens to the speed, and what happens to the pressure?" - recap\_1: a Text \[text\] that says "1. Pressure at rest: $p = p\_0 + rho g h$." - recap\_2: a Text \[text\] that says "2. Viscosity: $tau = mu thin dif u slash dif y$." - recap\_3: a Text \[text\] that says "3. Reynolds number picks laminar or turbulent." - recap\_4: a Text \[text\] that says "4. Bernoulli: faster flow, lower pressure." - speed\_throat: an Arrow \[green\] labelled "v\_2" drawn in pipe (start=(4.3, 1.85), end=(5.9, 1.85)) - speed\_wide: an Arrow \[green\] labelled "v\_1" drawn in pipe (start=(1.05, 1.85), end=(1.95, 1.85)) - top\_wall: a ParametricCurve \[gray\] drawn in pipe (function=\, t\_range=(0.5, 7.5)) - tube\_throat: a Line \[blue\] labelled "p\_2" drawn in pipe (start=(4.15, 2.29), end=(4.15, 3.3)) - tube\_wide: a Line \[blue\] labelled "p\_1" drawn in pipe (start=(2.3, 2.76), end=(2.3, 4.35)) #### Beats ##### [07:6.135](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=426.13498042681715) Narration: Here is a pipe that narrows in the middle, with a steady stream of water running through it from left to right. Board: Empty. Actions: - [07:6.135](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=426.13498042681715): question is shown on the screen, written out. - [07:6.599](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=426.59898042681715): pipe is shown on the screen, written out. - [07:7.064](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=427.0639804268171): bottom\_wall is shown on the screen, written out. - [07:7.064](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=427.0639804268171): top\_wall is shown on the screen, written out. ##### [07:11.936](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=431.9364804268171) Narration: Nothing is piling up inside, and water is hard to squash. So whatever volume passes this wide section each second has to pass the throat as well. Area times speed is the same at both. Board: pipe — a Figure (x\_range=(0.0, 8.0), y\_range=(0.0, 5.0), aspect=(8.0, 5.0)); question — a Panel that says "Where this pipe narrows, what happens to the speed, and what happens to the pressure?"; bottom\_wall — a ParametricCurve \[gray\] drawn in pipe (function=\, t\_range=(0.5, 7.5)); top\_wall — a ParametricCurve \[gray\] drawn in pipe (function=\, t\_range=(0.5, 7.5)) Actions: - [07:17.846](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=437.84598042681716): area\_wide is shown on the screen, written out. - [07:20.144](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=440.14398042681717): area\_throat is shown on the screen, written out. - [07:23.139](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=443.13898042681717): pipe moves to a new place on the board. - [07:23.139](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=443.13898042681717): c0 is shown on the screen, written out. ##### [07:24.819](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=444.81948042681717) Narration: Halve the area and the speed has to double. The flow runs faster through the throat, every time, and that is purely bookkeeping. Board: c0 — a Math \[text\] that says "$A\_1 v\_1 = A\_2 v\_2$"; pipe — a Figure (x\_range=(0.0, 8.0), y\_range=(0.0, 5.0), aspect=(8.0, 5.0)); question — a Panel that says "Where this pipe narrows, what happens to the speed, and what happens to the pressure?"; bottom\_wall — a ParametricCurve \[gray\] drawn in pipe (function=\, t\_range=(0.5, 7.5)); top\_wall — a ParametricCurve \[gray\] drawn in pipe (function=\, t\_range=(0.5, 7.5)); area\_wide — a Line \[yellow\] labelled "A\_1" drawn in pipe (start=(1.55, 0.9), end=(1.55, 2.8), dashed=True); area\_throat — a Line \[yellow\] labelled "A\_2" drawn in pipe (start=(3.85, 1.41), end=(3.85, 2.29), dashed=True) Actions: - [07:25.481](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=445.48098042681715): speed\_wide is shown on the screen, written out. - [07:27.211](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=447.2109804268172): c1 is shown on the screen, written out. - [07:29.742](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=449.7419804268171): speed\_throat is shown on the screen, written out. - [07:33.225](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=453.2249804268171): pipe moves to a new place on the board. - [07:33.225](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=453.2249804268171): c0 is hidden from the screen — left the board. - [07:33.225](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=453.2249804268171): c1 is hidden from the screen — left the board. - [07:33.225](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=453.2249804268171): question is hidden from the screen — left the board. ##### [07:33.825](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=453.82498042681715) Narration: Now the pressure, and for that we follow a small slug of fluid from the wide part into the throat. Fluid behind pushes it forward, fluid ahead pushes back, and the net work done on it is the pressure difference times its volume. Board: pipe — a Figure (x\_range=(0.0, 8.0), y\_range=(0.0, 5.0), aspect=(8.0, 5.0)); bottom\_wall — a ParametricCurve \[gray\] drawn in pipe (function=\, t\_range=(0.5, 7.5)); top\_wall — a ParametricCurve \[gray\] drawn in pipe (function=\, t\_range=(0.5, 7.5)); area\_wide — a Line \[yellow\] labelled "A\_1" drawn in pipe (start=(1.55, 0.9), end=(1.55, 2.8), dashed=True); area\_throat — a Line \[yellow\] labelled "A\_2" drawn in pipe (start=(3.85, 1.41), end=(3.85, 2.29), dashed=True); speed\_wide — an Arrow \[green\] labelled "v\_1" drawn in pipe (start=(1.05, 1.85), end=(1.95, 1.85)); speed\_throat — an Arrow \[green\] labelled "v\_2" drawn in pipe (start=(4.3, 1.85), end=(5.9, 1.85)) Actions: - [07:33.825](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=453.82498042681715): head\_energy is shown on the screen, written out. - [07:44.832](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=464.83198042681715): energy is shown on the screen, written out. ##### [07:49.054](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=469.05398042681713) Narration: That work has to go somewhere. The slug speeds up, so its kinetic energy rises by one half rho V times the change in v squared. Board: pipe — a Figure (x\_range=(0.0, 8.0), y\_range=(0.0, 5.0), aspect=(8.0, 5.0)); bottom\_wall — a ParametricCurve \[gray\] drawn in pipe (function=\, t\_range=(0.5, 7.5)); top\_wall — a ParametricCurve \[gray\] drawn in pipe (function=\, t\_range=(0.5, 7.5)); area\_wide — a Line \[yellow\] labelled "A\_1" drawn in pipe (start=(1.55, 0.9), end=(1.55, 2.8), dashed=True); area\_throat — a Line \[yellow\] labelled "A\_2" drawn in pipe (start=(3.85, 1.41), end=(3.85, 2.29), dashed=True); speed\_wide — an Arrow \[green\] labelled "v\_1" drawn in pipe (start=(1.05, 1.85), end=(1.95, 1.85)); speed\_throat — an Arrow \[green\] labelled "v\_2" drawn in pipe (start=(4.3, 1.85), end=(5.9, 1.85)); head\_energy — a Heading that says "Where the Energy Goes" Actions: - [07:53.709](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=473.70898042681716): energy is shown on the screen, written out. ##### [07:59.348](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=479.3479804268171) Narration: And if the pipe also climbs, some of the work goes into lifting the slug, which is the change in its potential energy. Board: Unchanged from the preceding beat in this scene. Actions: - [08:4.99](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=484.98998042681717): energy is shown on the screen, written out. ##### [08:6.972](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=486.97198042681714) Narration: Work in equals energy gained. That single line is the whole argument: the pressure drop pays for the extra speed and for the extra height. Board: Unchanged from the preceding beat in this scene. Actions: - [08:7.982](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=487.98198042681713): energy is shown on the screen, written out. - [08:13.415](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=493.4149804268171): energy (the "Delta K + Delta U" part) is emphasized. - [08:16.701](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=496.7009804268172): energy is hidden from the screen — left the board. - [08:16.701](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=496.7009804268172): head\_energy is hidden from the screen — left the board. - [08:16.701](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=496.7009804268172): energy (the "Delta K + Delta U" part) is no longer emphasized. ##### [08:17.901](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=497.9009804268171) Narration: Divide the whole thing through by the volume and collect the two stations on opposite sides. Pressure, plus one half rho v squared, plus rho g y, is the same everywhere along the stream. Board: pipe — a Figure (x\_range=(0.0, 8.0), y\_range=(0.0, 5.0), aspect=(8.0, 5.0)); bottom\_wall — a ParametricCurve \[gray\] drawn in pipe (function=\, t\_range=(0.5, 7.5)); top\_wall — a ParametricCurve \[gray\] drawn in pipe (function=\, t\_range=(0.5, 7.5)); area\_wide — a Line \[yellow\] labelled "A\_1" drawn in pipe (start=(1.55, 0.9), end=(1.55, 2.8), dashed=True); area\_throat — a Line \[yellow\] labelled "A\_2" drawn in pipe (start=(3.85, 1.41), end=(3.85, 2.29), dashed=True); speed\_wide — an Arrow \[green\] labelled "v\_1" drawn in pipe (start=(1.05, 1.85), end=(1.95, 1.85)); speed\_throat — an Arrow \[green\] labelled "v\_2" drawn in pipe (start=(4.3, 1.85), end=(5.9, 1.85)) Actions: - [08:17.901](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=497.9009804268171): head\_law is shown on the screen, written out. - [08:24.089](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=504.08898042681716): const is shown on the screen, written out. ##### [08:32.119](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=512.1194804268172) Narration: Every term is now an energy per unit volume. The first is the pressure itself. The second is the kinetic term, which grows with the square of the speed. The third is the height term. Board: pipe — a Figure (x\_range=(0.0, 8.0), y\_range=(0.0, 5.0), aspect=(8.0, 5.0)); bottom\_wall — a ParametricCurve \[gray\] drawn in pipe (function=\, t\_range=(0.5, 7.5)); top\_wall — a ParametricCurve \[gray\] drawn in pipe (function=\, t\_range=(0.5, 7.5)); area\_wide — a Line \[yellow\] labelled "A\_1" drawn in pipe (start=(1.55, 0.9), end=(1.55, 2.8), dashed=True); area\_throat — a Line \[yellow\] labelled "A\_2" drawn in pipe (start=(3.85, 1.41), end=(3.85, 2.29), dashed=True); speed\_wide — an Arrow \[green\] labelled "v\_1" drawn in pipe (start=(1.05, 1.85), end=(1.95, 1.85)); speed\_throat — an Arrow \[green\] labelled "v\_2" drawn in pipe (start=(4.3, 1.85), end=(5.9, 1.85)); const — a Math \[text\] that says "$p + frac(1, 2) rho v^2 + rho g y = upright("constant")$"; head\_law — a Heading that says "Bernoulli's Equation" Actions: - [08:35.753](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=515.7529804268172): const (the "p" part) is emphasized. - [08:39.039](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=519.0389804268171): const (the "frac(1, 2) rho v^2" part) is emphasized. - [08:39.039](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=519.0389804268171): const (the "p" part) is no longer emphasized. - [08:43.393](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=523.3929804268172): const (the "frac(1, 2) rho v^2" part) is no longer emphasized. - [08:43.393](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=523.3929804268172): const (the "rho g y" part) is emphasized. - [08:44.403](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=524.4029804268172): const (the "rho g y" part) is no longer emphasized. ##### [08:45.003](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=525.0029804268172) Narration: Our pipe is horizontal, so the height is the same at both stations and that term drops out. What is left relates pressure to speed, and nothing else. Board: Unchanged from the preceding beat in this scene. Actions: - [08:50.831](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=530.8309804268172): horiz is shown on the screen, written out. ##### [08:54.717](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=534.7169804268171) Narration: Here is the answer, then. The speed in the throat is larger, so the kinetic term there is larger, and the pressure there has to be smaller to keep the total fixed. Board: pipe — a Figure (x\_range=(0.0, 8.0), y\_range=(0.0, 5.0), aspect=(8.0, 5.0)); bottom\_wall — a ParametricCurve \[gray\] drawn in pipe (function=\, t\_range=(0.5, 7.5)); top\_wall — a ParametricCurve \[gray\] drawn in pipe (function=\, t\_range=(0.5, 7.5)); area\_wide — a Line \[yellow\] labelled "A\_1" drawn in pipe (start=(1.55, 0.9), end=(1.55, 2.8), dashed=True); area\_throat — a Line \[yellow\] labelled "A\_2" drawn in pipe (start=(3.85, 1.41), end=(3.85, 2.29), dashed=True); speed\_wide — an Arrow \[green\] labelled "v\_1" drawn in pipe (start=(1.05, 1.85), end=(1.95, 1.85)); speed\_throat — an Arrow \[green\] labelled "v\_2" drawn in pipe (start=(4.3, 1.85), end=(5.9, 1.85)); const — a Math \[text\] that says "$p + frac(1, 2) rho v^2 + rho g y = upright("constant")$"; horiz — a Math \[text\] that says "$p\_1 + frac(1, 2) rho v\_1^2 = p\_2 + frac(1, 2) rho v\_2^2$"; head\_law — a Heading that says "Bernoulli's Equation" Actions: - [08:57.62](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=537.6199804268172): speed\_throat is emphasized. - [08:59.64](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=539.6399804268171): horiz (the "frac(1, 2) rho v\_2^2" part) is emphasized. - [09:1.857](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=541.8569804268171): horiz (the "frac(1, 2) rho v\_2^2" part) is no longer emphasized. - [09:1.857](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=541.8569804268171): horiz (the "p\_2" part) is emphasized. - [09:3.831](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=543.8309804268172): speed\_throat is no longer emphasized. - [09:5.247](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=545.2474804268171): horiz (the "p\_2" part) is no longer emphasized. ##### [09:5.847](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=545.8474804268171) Narration: Stand a tube up out of each section and you can watch it happen. The water climbs high where the pipe is wide and slow, and sits lower where it is narrow and fast. Board: Unchanged from the preceding beat in this scene. Actions: - [09:10.886](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=550.8859804268171): tube\_wide is shown on the screen, written out. - [09:13.707](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=553.7069804268172): tube\_throat is shown on the screen, written out. ##### [09:16.478](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=556.4784804268172) Narration: Put numbers on it. Water at two metres a second in the wide part, doubling to four in the throat. One half of a thousand, times sixteen minus four, is six thousand pascals, so the throat sits six kilopascals lower. Board: pipe — a Figure (x\_range=(0.0, 8.0), y\_range=(0.0, 5.0), aspect=(8.0, 5.0)); bottom\_wall — a ParametricCurve \[gray\] drawn in pipe (function=\, t\_range=(0.5, 7.5)); top\_wall — a ParametricCurve \[gray\] drawn in pipe (function=\, t\_range=(0.5, 7.5)); area\_wide — a Line \[yellow\] labelled "A\_1" drawn in pipe (start=(1.55, 0.9), end=(1.55, 2.8), dashed=True); area\_throat — a Line \[yellow\] labelled "A\_2" drawn in pipe (start=(3.85, 1.41), end=(3.85, 2.29), dashed=True); speed\_wide — an Arrow \[green\] labelled "v\_1" drawn in pipe (start=(1.05, 1.85), end=(1.95, 1.85)); speed\_throat — an Arrow \[green\] labelled "v\_2" drawn in pipe (start=(4.3, 1.85), end=(5.9, 1.85)); const — a Math \[text\] that says "$p + frac(1, 2) rho v^2 + rho g y = upright("constant")$"; horiz — a Math \[text\] that says "$p\_1 + frac(1, 2) rho v\_1^2 = p\_2 + frac(1, 2) rho v\_2^2$"; head\_law — a Heading that says "Bernoulli's Equation"; tube\_wide — a Line \[blue\] labelled "p\_1" drawn in pipe (start=(2.3, 2.76), end=(2.3, 4.35)); tube\_throat — a Line \[blue\] labelled "p\_2" drawn in pipe (start=(4.15, 2.29), end=(4.15, 3.3)) Actions: - [09:17.094](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=557.0939804268171): example is shown on the screen, written out. - [09:31.107](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=571.1069804268172): A box is drawn around example. ##### [09:32.519](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=572.5194804268172) Narration: A narrowing like this is called a venturi, and that pressure drop is useful rather than a nuisance. Measure the drop between the two tubes and you have measured the flow rate. Board: pipe — a Figure (x\_range=(0.0, 8.0), y\_range=(0.0, 5.0), aspect=(8.0, 5.0)); bottom\_wall — a ParametricCurve \[gray\] drawn in pipe (function=\, t\_range=(0.5, 7.5)); top\_wall — a ParametricCurve \[gray\] drawn in pipe (function=\, t\_range=(0.5, 7.5)); area\_wide — a Line \[yellow\] labelled "A\_1" drawn in pipe (start=(1.55, 0.9), end=(1.55, 2.8), dashed=True); area\_throat — a Line \[yellow\] labelled "A\_2" drawn in pipe (start=(3.85, 1.41), end=(3.85, 2.29), dashed=True); speed\_wide — an Arrow \[green\] labelled "v\_1" drawn in pipe (start=(1.05, 1.85), end=(1.95, 1.85)); speed\_throat — an Arrow \[green\] labelled "v\_2" drawn in pipe (start=(4.3, 1.85), end=(5.9, 1.85)); const — a Math \[text\] that says "$p + frac(1, 2) rho v^2 + rho g y = upright("constant")$"; horiz — a Math \[text\] that says "$p\_1 + frac(1, 2) rho v\_1^2 = p\_2 + frac(1, 2) rho v\_2^2$"; example — a Math \[text\] that says "$Delta p = frac(1, 2) rho (v\_2^2 - v\_1^2) = 6 thin upright("kPa")$"; head\_law — a Heading that says "Bernoulli's Equation"; tube\_wide — a Line \[blue\] labelled "p\_1" drawn in pipe (start=(2.3, 2.76), end=(2.3, 4.35)); tube\_throat — a Line \[blue\] labelled "p\_2" drawn in pipe (start=(4.15, 2.29), end=(4.15, 3.3)) Actions: - [09:34.644](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=574.6439804268172): area\_throat is indicated — a transient flash. - [09:40.844](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=580.8439804268172): tube\_throat is indicated — a transient flash. - [09:43.304](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=583.3044804268171): const is hidden from the screen — left the board. - [09:43.304](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=583.3044804268171): example is hidden from the screen — left the board. - [09:43.304](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=583.3044804268171): head\_law is hidden from the screen — left the board. - [09:43.304](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=583.3044804268171): horiz is hidden from the screen — left the board. - [09:43.304](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=583.3044804268171): pipe is hidden from the screen — left the board. - [09:43.304](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=583.3044804268171): bottom\_wall is hidden from the screen — pipe left the board. - [09:43.304](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=583.3044804268171): top\_wall is hidden from the screen — pipe left the board. - [09:43.304](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=583.3044804268171): area\_wide is hidden from the screen — pipe left the board. - [09:43.304](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=583.3044804268171): area\_throat is hidden from the screen — pipe left the board. - [09:43.304](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=583.3044804268171): speed\_wide is hidden from the screen — pipe left the board. - [09:43.304](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=583.3044804268171): speed\_throat is hidden from the screen — pipe left the board. - [09:43.304](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=583.3044804268171): tube\_wide is hidden from the screen — pipe left the board. - [09:43.304](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=583.3044804268171): tube\_throat is hidden from the screen — pipe left the board. ##### [09:44.504](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=584.5044804268172) Narration: Four ideas, then. At rest, pressure builds with depth and with nothing else. In shear, stress is viscosity times the velocity gradient. Board: Empty. Actions: - [09:44.504](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=584.5044804268172): head\_recap is shown on the screen, written out. - [09:46.745](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=586.7449804268172): recap\_1 is shown on the screen, written out. - [09:51.03](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=591.0299804268172): recap\_2 is shown on the screen, written out. ##### [09:55.554](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=595.5539804268171) Narration: In a pipe, the Reynolds number tells you whether the layers hold or break up. And along a stream, faster always means lower pressure. Board: recap\_1 — a Text \[text\] that says "1. Pressure at rest: $p = p\_0 + rho g h$."; recap\_2 — a Text \[text\] that says "2. Viscosity: $tau = mu thin dif u slash dif y$."; head\_recap — a Heading that says "What to Carry Away" Actions: - [09:57.086](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=597.0859804268172): recap\_3 is shown on the screen, written out. - [10:2.392](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=602.3919804268172): recap\_4 is shown on the screen, written out. ##### [10:6.181](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=606.1809804268172) Narration: Those four will carry you a long way, because almost everything else in fluid mechanics is one of them applied somewhere new. Board: recap\_1 — a Text \[text\] that says "1. Pressure at rest: $p = p\_0 + rho g h$."; recap\_2 — a Text \[text\] that says "2. Viscosity: $tau = mu thin dif u slash dif y$."; recap\_3 — a Text \[text\] that says "3. Reynolds number picks laminar or turbulent."; recap\_4 — a Text \[text\] that says "4. Bernoulli: faster flow, lower pressure."; head\_recap — a Heading that says "What to Carry Away" Actions: - [10:13.694](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=613.6935429268171): head\_recap is hidden from the screen — left the board. - [10:13.694](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=613.6935429268171): recap\_1 is hidden from the screen — left the board. - [10:13.694](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=613.6935429268171): recap\_2 is hidden from the screen — left the board. - [10:13.694](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=613.6935429268171): recap\_3 is hidden from the screen — left the board. - [10:13.694](https://academa.ai/@sina/lectures/core-principles-of-fluid-mechanics-pressure-viscosity-and-bernoulli-s-principle?t=613.6935429268171): recap\_4 is hidden from the screen — left the board.