Viscosity

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle to start. Drop a small steel ball into a tall jar of glycerine. Gravity pulls on it the whole time, so it should keep speeding up, right? But it does not! After a moment, it sinks at a steady speed. So who is holding it back? By the end of this lesson, you will even calculate that speed. Today: viscosity, the friction inside a fluid. It comes up often in JEE, so let's begin.

0:31Newton's law of viscosity

So, think of honey poured from a spoon. Picture the flowing fluid as thin layers, sliding over one another like the pages of a book. Neighbouring layers drag on each other with a tangential force: along the layers, not across them. Okay? Newton's law of viscosity gives this force. F equals eta, times capital A, times d v by d z. Capital A is the area of the layers, and d v by d z is the velocity gradient: how fast the speed changes as we move across the layers, along z. Eta, the Greek letter, is the coefficient of viscosity. With me? Now its unit. Eta is force over area, times d z over d v: newton per square metre, times seconds. So eta is in pascal second, and one pascal second equals ten poise. Careful, students often flip this: one pascal second is ten poise, not one tenth of a poise. Right? Quick one. What are the dimensions of eta? The square brackets mean the dimensions of, with M for mass, L for length, T for time. Force, MLT to the power minus two, divided by area, L squared, and by the velocity gradient, T to the power minus one. That gives M L to the power minus one, T to the power minus one. Got it?

1:58Heat it up: liquids and gases differ

Now, a kitchen question. Warm a jar of honey. Does it flow faster, or slower? Faster! Correct! For liquids, viscosity falls as temperature rises. It comes from the attraction between molecules, and heating weakens that attraction. Clear? Now heat some air instead. Does its viscosity fall too? No, no, careful! For gases it is the opposite: viscosity rises with temperature. Here it comes from momentum carried between the layers by moving molecules, and heating increases this.

2:34Stokes' law and terminal velocity

Okay? Now, back to our steel ball. For a sphere moving slowly through a fluid, Stokes' law gives the viscous drag: F equals six pi eta r v. Pi is the Greek letter pi, r is the radius, and v is the speed. Only for slow motion, okay? On the right is a graph of the ball's velocity against time; we will come to it in a moment. The ball picks up speed. What happens to this drag? Yes, it grows, because the drag is proportional to v! Look at the graph: the velocity rises, then flattens out at v t, the terminal velocity, with a small t as subscript. There, the weight m g, mass times the acceleration due to gravity, equals the upthrust, capital F with subscript B, plus the drag. No net force, constant speed. With me? Let's open this up. The weight is the volume, four over three pi r cubed, times rho, the density of the sphere, times g. The upthrust is the same volume times sigma times g, and the drag is still six pi eta r v t. Careful: sigma here is the density of the fluid, not surface tension! Right? Now solve for v t: two r squared, times rho minus sigma, times g, over nine eta. See the r squared? Double the radius, and v t becomes four times. Clear? Now, what if rho is less than sigma? The angled sign means less than: a body lighter than the fluid. What happens then? Then rho minus sigma is negative, and so is v t: the body moves up! That is why air bubbles rise through a liquid. Got it?

4:18Poiseuille's equation and a topper's trick

Now, the last formula, for flow through a narrow tube: Poiseuille's equation. Q equals pi P r to the power four, over eight eta l. Capital Q is the volume flowing per second, capital P the pressure difference across the tube, r the radius, and small l the length. Okay? So tell me. Halve the radius, and keep the same pressure difference and length. Does the flow become half? No, no, careful! Q is proportional to r to the power four; the curly symbol means proportional to. The double arrow means so: halve r, and the flow becomes one half to the power four, just one sixteenth! Right? Now a trick toppers use. Capital R, the viscous resistance, is eight eta l over pi r to the power four. Then Q equals P over R, just like current equals potential difference over resistance! Clear? So tubes combine just like electrical resistances: in series, the same Q flows through each, and the resistances add. Got it?

5:22Example 1: force to drag a plate over a 1 mm oil layer

Now, Example one. Take a plate of area zero point one square metre, pulled at zero point five metres per second over a one millimetre layer of oil on a fixed surface. Eta is zero point one pascal second. Find the force needed. Pause the video, and try it first. Done? Okay, let's check. The oil touching the plate moves with it, at zero point five metres per second; the oil on the fixed surface stays at rest. So across the layer, the speed drops from zero point five to zero. Okay? So the velocity gradient is v over d, with v the plate's speed and d the thickness of the layer. Zero point five over one, right? No, no, wait! d is one millimetre, that is ten to the power minus three metres. So v over d is zero point five over ten to the power minus three: five hundred per second; s to the power minus one means per second. Got it? Now, F equals eta, times capital A, times v over d: zero point one, times zero point one, times five hundred. Five newtons! Capital N stands for newtons. Forget the millimetres, and you are a thousand times too small. Clear?

6:43Example 2: terminal velocity of a steel ball in glycerine

Now, Example two: our steel ball from the start! Radius one millimetre, density seven thousand eight hundred kilograms per cubic metre. It falls in glycerine, with sigma one thousand two hundred sixty kilograms per cubic metre and eta one point five pascal second. Take g as nine point eight, and find the terminal velocity. Pause the video and try it. Done? Let's see. Rho is seven thousand eight hundred, so that goes straight into the formula, right? Oh no, careful! The formula has rho minus sigma, because the glycerine pushes up with its upthrust. Seven thousand eight hundred minus one thousand two hundred sixty: six thousand five hundred forty kilograms per cubic metre. Okay? Next, the radius in metres. One millimetre is ten to the power minus three metres, so r squared is ten to the power minus six square metres. Right? Now substitute: v t equals two times ten to the power minus six, times six thousand five hundred forty, times nine point eight, over nine times one point five. The top is about zero point one two eight; the bottom is thirteen point five. With me? Divide: v t is about nine point five times ten to the power minus three metres per second; the wavy equals sign means approximately. That is nine point five millimetres per second, slow and steady, like our ball in the jar. Puzzle solved!

8:19Quick revision: Newton's law, Stokes' law and Poiseuille

So, quick revision. Newton's law: F equals eta, capital A, d v by d z, and one pascal second is ten poise. Heated, a liquid's viscosity falls and a gas's rises. Stokes, for a slow sphere: F equals six pi eta r v, and v t equals two r squared, rho minus sigma, g, over nine eta. Poiseuille: Q equals pi P r to the power four, over eight eta l, so Q goes as r to the power four, and resistances combine like electrical ones. Now try the practice problems on this page. All the best!