Core Principles of Fluid Mechanics: Pressure, Viscosity, and Bernoulli's Principle

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.

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.

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.

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.