Mechanical Properties of Fluids: Class 11 Physics · Chapter 9

9.1 Introduction

Tip a steel tumbler of water into a glass and the water takes the glass's shape at once. When a pressure cooker whistles, steam rushes out of its vent. Water and steam look nothing alike, yet they share one ability: both flow. Any substance that can flow is called a fluid (तरल), so the word covers liquids and gases alike.

Air, the blood your heart pumps (about five litres a minute at rest), the sap in a neem tree and the monsoon clouds over the Western Ghats are all fluids on the move.

What sets a fluid apart from a solid? A solid holds its own shape; a fluid has none and settles into whatever holds it. Liquids and gases then part ways over volume. A litre of milk is still a litre in a jug or in a bottle, but a gas spreads to fill every corner of its container. In Chapter 8 you saw that pressure changes the volume of any material. For solids and liquids the change is tiny, while a gas gives way easily: gases are far more compressible.

The decisive difference is how a material answers a shearing force, one that acts along a surface. Glue an eraser to your desk and push its top sideways: it leans a little and then holds, because the solid develops a shear stress that balances your push. Push the top of a layer of liquid and it keeps sliding, however gentle the push. A fluid at rest can support essentially no shear stress, and that is exactly why it flows (Fig. 9.1).

We first study fluids at rest: pressure, how it grows with depth and the machines that use it. Then we set fluids moving and use energy conservation to reach Bernoulli's principle. Two properties of real fluids close the chapter: viscosity, a kind of internal friction, and surface tension, which makes a liquid surface behave like a stretched skin.

9.2 Pressure

Press a drawing pin into a soft board with your thumb. The point goes in, yet the flat head does not hurt you, although the same force acts at both ends. A heavy truck spreads its load over ten or more tyres, and a camel's broad padded feet keep it from sinking into desert sand. In each case the effect is decided by the force divided by the area that carries it.

Now lower a stone into a bucket of still water. The water pushes on every part of the stone and of the bucket, and each push is perpendicular to the surface it acts on. Why? A fluid at rest cannot hold a shear stress (Section 9.1). If the water pressed on a wall at a slant, it would feel a slanted reaction whose part along the wall is a shear, and it would slide. Still water can only push straight on (Fig. 9.3, first step).

To measure the push at a point, picture a tiny sensor: a sealed, airless can closed by a light piston of area $A$, held out by a calibrated spring. Placed in the fluid, the piston moves in until the spring's force balances the fluid's normal force $F$ on it. The ratio gives the average push per unit area over the piston:

Shrinking the piston towards a point defines the pressure (दाब) there:

The SI unit of pressure, $1\mathrm{N}{\mathrm{m}}^{-2}$, is called the pascal (Pa) after the French scientist Blaise Pascal (1623–1662), whose work on fluids we meet next; larger pressures are often quoted in standard atmospheres, $1\mathrm{a}\mathrm{t}\mathrm{m}=1.013\times {10}^5\mathrm{P}\mathrm{a}$, close to the air pressure at sea level. The dimensions of pressure are $[\mathrm{M}{\mathrm{L}}^{-1}{\mathrm{T}}^{-2}]$. Notice that pressure carries no direction: (9.2) uses only the force normal to the little surface, and that surface may face any way. Pressure is a scalar.

Because a fluid has no fixed shape, we describe how much matter it packs into a space by its density, the mass per unit volume:

Density has dimensions $[\mathrm{M}{\mathrm{L}}^{-3}]$ and unit $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$. Dividing by the density of water at 4 °C, $1.000\times {10}^3\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$, gives the relative density, a pure number. Mercury's is about 13.6, so a litre of mercury has a mass of 13.6 kg (Table 9.1). Squeezing hardly changes a liquid's density, while a gas becomes denser in proportion to the pressure on it.

9.2.1 Pascal's Law

Float a metre under the surface of a pool and turn your head slowly. Your eardrums feel the same squeeze whichever way they face. Experiments of this kind led Pascal to two conclusions about a fluid at rest, now called Pascal's law: the pressure at a point acts equally in every direction, and it is the same at all points on one horizontal level. Both follow from Newton's laws.

For the second part of the law, take a thin horizontal rod of the fluid with flat ends of area $A$. Gravity acts across the rod, so along it only the end pushes $P_{L}A$ and $P_{R}A$ act. The rod stays at rest, so $P_{L}A=P_{R}A$ and $P_L=P_R$. Any two points on one level, joined through the same fluid, share one pressure; if they did not, fluid would flow from the high-pressure end towards the other.

9.2.2 Variation of Pressure with Depth

Dive towards the bottom of a deep pool and your ears start to hurt. The deeper you go, the more water lies above you, and that water must be held up by the pressure beneath it. The argument below turns this into a formula.

Look at what (9.7) leaves out. The area $A$ cancelled, and nothing in the result describes the vessel. The pressure at a point depends only on how far below the free surface it lies, not on the width or shape of the container or on how much liquid it holds. A tall thin pipe of water and a broad tank filled to the same height press equally hard on each square centimetre of their bases. This surprising result is called the hydrostatic paradox (Fig. 9.4, last step).

Join vessels of different shapes at the bottom and pour in water: it settles to one level in all of them, since only a common level gives each height in the joining pipe a single pressure. Masons use exactly this when they mark a level line on two walls with a clear plastic pipe full of water; the water surfaces at the two ends always lie at the same height.

9.2.3 Atmospheric Pressure and Gauge Pressure

We live at the bottom of an ocean of air. Every square metre of ground carries the weight of all the air piled above it, right up to where the atmosphere fades out, and that weight per unit area is the atmospheric pressure. Near sea level its value is about $1.013\times {10}^5\mathrm{P}\mathrm{a}$: the air above each square metre weighs as much as a loaded 10-tonne truck. We are not crushed because our body fluids push outward almost as hard.

Evangelista Torricelli (1608–1647), who had worked as Galileo's assistant, found how to measure it in 1643. He filled a glass tube about a metre long, sealed at one end, with mercury, covered the open end, upended it in a dish of mercury and then uncovered it. The mercury sank until its column stood about 76 cm above the level in the dish, leaving an almost empty space at the top. This instrument is the mercury barometer (Fig. 9.5).

Why does the column stop there? The space above it holds only a trace of mercury vapour, whose pressure is negligible. A point in the tube at the level of the dish's surface and a point on that open surface lie on one level in the same mercury, so they share one pressure, which is $P_a$ at the open surface. The column of height $h$ above the first point must supply it:

Here $\rho$ is the density of mercury. With $\rho =1.36\times {10}^4\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$ and $h=0.760\mathrm{m}$, (9.8) gives $1.013\times {10}^5\mathrm{P}\mathrm{a}$; a water barometer would need a tube over 10 m tall. Doctors still quote blood pressure in millimetres of mercury; the pressure of a 1 mm column of mercury, about $133\mathrm{P}\mathrm{a}$, is called one torr in Torricelli's honour. Weather scientists use the bar (${10}^5\mathrm{P}\mathrm{a}$) and the millibar; weather charts mark pressure in hectopascals, and $1\mathrm{h}\mathrm{P}\mathrm{a}$ is one millibar.

Sea-level pressure rises and falls by a few percent with the weather; a steady fall of the barometer warns of an approaching low-pressure system, such as a Bay of Bengal cyclone.

To measure a pressure difference we use an open-tube manometer: a U-tube partly filled with a liquid, one arm connected to the gas being tested and the other open to the air (Fig. 9.5, last step). Points at one level in the liquid share one pressure, so the gas pressure $P$ exceeds $P_a$ by the extra liquid column $h$ in the open arm: $P-P_a=\rho gh$. A manometer therefore reads gauge pressure directly. For small differences a light liquid such as water or oil gives a readable height; for large ones, dense mercury keeps the tube short.

9.2.4 Hydraulic Machines

Squeeze the bottom of a closed toothpaste tube and paste appears at the nozzle; poke a water balloon and it bulges everywhere at once. A push on one part of an enclosed fluid is felt throughout it. Try it with a plastic bottle filled to the brim with water, capped, and pierced with pinholes at several places: squeeze it gently and every jet grows stronger at the same moment, whichever way the hole faces. The extra pressure from your hand has reached all points of the water equally.

A hydraulic lift puts this to work (Fig. 9.6). Two pistons, a small one of area $A_1$ and a large one of area $A_2$, close the two ends of a sealed, liquid-filled system. A force $F_1$ on the small piston adds a pressure $F_1\mathrm{/}{A}_1$, which reaches the large piston unchanged and pushes it up with a force

The force is multiplied by $A_2\mathrm{/}{A}_1$, the mechanical advantage. Energy is not created, though. The liquid is practically incompressible, so the volume pushed out of the small cylinder, $A_1{d}_1$, equals the volume added under the large piston, $A_2{d}_2$. The large piston moves a distance smaller by the same factor, and $F_1{d}_1=F_2{d}_2$: for frictionless pistons, the work put in equals the work got out.

Car brakes use the same idea. At each wheel, a piston in a wheel cylinder presses the brake pads or shoes against a spinning disc or drum. All four wheel cylinders are joined by narrow pipes full of brake fluid to a single master cylinder under the pedal, whose piston is much smaller. A modest push on the pedal therefore becomes a large force at every wheel, and because the fluid carries one pressure throughout, the four wheels are braked equally, which keeps the car from slewing sideways. Excavator arms, a barber's adjustable chair and a mechanic's bottle jack are other hydraulic machines.

9.3 Streamline Flow

Light an agarbatti in a still room and watch its smoke. For the first few centimetres it rises as a smooth, unbroken thread; higher up the thread wavers and breaks into curls and puffs. The same air behaves in two quite different ways. The study of fluids in motion, fluid dynamics, starts by describing the smooth kind precisely.

To describe a flow we can watch fixed places rather than individual particles. Imagine a camera fixed on one spot in a canal, recording the velocity of whatever water passes that spot. If the reading never changes, at that spot and at every other, the flow is steady. Steady does not mean uniform: water may run faster through a narrow stretch than a wide one, so a given particle can speed up as it travels. What stays fixed is the velocity at each place.

In steady flow the pattern of streamlines is a fixed map of the motion. Two streamlines can never cross: at the crossing a particle would have two possible velocities, and the flow could not be steady. A bundle of neighbouring streamlines forms a tube of flow; fluid enters it at one end and leaves at the other, but never crosses its sides.

The product $Av$ is the volume flow rate, the volume passing each second, in ${\mathrm{m}}^3{\mathrm{s}}^{-1}$. The equation of continuity says it is the same all along a pipe; it is the law of conservation of mass for an incompressible fluid. Where a pipe narrows the fluid must speed up, and where it widens the fluid slows down. On a streamline picture, crowded lines mean fast flow. Partly closing the end of a garden hose with your thumb uses this: if the open area falls to a fifth while the flow rate holds steady, the water leaves five times faster and reaches much farther.

Steady flow survives only at low enough speeds. Above a critical speed, which depends on the fluid, the pipe and any obstacles, the flow becomes turbulent (विक्षुब्ध): the velocity at a point changes erratically and the fluid churns into eddies of many sizes. The breaking agarbatti smoke, the foaming rapids of the Ganga near Rishikesh and the swirling wake behind a speeding bus are all turbulent. Smooth flow in which neighbouring layers slide past one another without mixing is called laminar flow.

9.4 Bernoulli's Principle

Water entering the narrow part of a pipe speeds up, so something must push it forward. Only the surrounding fluid can, through a pressure that is higher behind the water than ahead of it. Faster flow should therefore go with lower pressure. In his book Hydrodynamica (1738) the Swiss physicist Daniel Bernoulli turned this idea into an exact relation by applying energy conservation to a moving fluid.

Each term of Bernoulli's equation is measured in $\mathrm{J}{\mathrm{m}}^{-3}$, which is the same as a pascal, so pressure itself can be read as a store of energy per unit volume. Put $v=0$ everywhere and (9.13) gives back $P_1-P_2=\rho g(h_2-h_1)$, the hydrostatic result (9.6): fluid statics is a special case.

The equation is only as good as its assumptions. It needs steady flow: in turbulent flow the speed and pressure at a point fluctuate and it fails. It treats the fluid as incompressible, ignoring energy stored by squeezing it, so a gas qualifies only when its density barely changes. And it assumes no viscosity (Section 9.5); in a real fluid some mechanical energy turns into heat, and the pressure downstream comes out lower than (9.12) predicts. For water and for air at modest speeds it is nonetheless remarkably useful.

A scent atomiser or a paint spray gun blows air fast across the top of a thin tube dipping into the liquid; the pressure there drops, so the liquid is pushed up and torn into droplets. A Bunsen burner draws in air at its base for the same reason.

9.4.1 Speed of Efflux: Torricelli's Law

Efflux means outflow. Torricelli noticed that liquid spurting from a hole in a tank leaves with the speed it would gain by falling freely from the level of the liquid's surface down to the hole. Bernoulli's equation shows why.

This is exactly the speed of a stone dropped from rest through a height $h$, and for the same reason: potential energy $\rho gh$ per unit volume becomes kinetic energy $\frac{1}{2}\rho v^2$. A drum of water with a tap $0.80\mathrm{m}$ below its surface delivers water at $\sqrt{2(9.8)(0.80)}\approx 4.0\mathrm{m}{\mathrm{s}}^{-1}$. In the opposite extreme, when the gas above the liquid is at a pressure far above atmospheric, the $2gh$ term hardly matters and the container pressure alone sets the jet speed. Fire extinguishers, spray cans and, in a more complicated way, rocket engines work in this regime.

9.4.2 Dynamic Lift

A fluid streaming past a moving body can push it sideways, at right angles to its motion. This dynamic lift keeps a soaring eagle aloft with barely a wingbeat, holds up aircraft and bends the path of a spinning ball. Bernoulli's principle explains at least part of it.

Start with a ball that only moves forward. Seen from the ball, air streams past it, and the pattern of streamlines above the ball is a mirror image of the pattern below (Fig. 9.11). Equal speeds mean equal pressures, so the air pushes neither up nor down; it only drags the ball back.

Spin breaks the symmetry. A spinning ball carries a thin skin of air round with it. With backspin, that skin moves along with the oncoming air over the top of the ball and against it underneath, so the air speeds up above and slows below; the streamlines crowd together on top and spread out underneath. Bernoulli's principle then gives a lower pressure above than below, and the ball is pushed upward. Lift caused by spin is called the Magnus effect. A spin bowler's drift and a footballer's curling free kick show it. (A fast bowler's swing has a different cause: the seam makes air cling differently to the two sides.)

An aircraft wing has a special cross-section, an aerofoil: rounded at the front, curved on top and tapering to a sharp rear edge. Tilted slightly up into the oncoming air, it makes the air over its upper surface flow faster than the air beneath. The lower pressure on top gives an upward lift that, in level flight, balances the aircraft's weight. Seen another way, the wing deflects air downwards and, by Newton's third law, the air pushes the wing up; both pictures describe the same flow.

9.5 Viscosity

Stir a jar of honey and a glass of water with the same spoon at the same speed: the honey fights back far more. Real fluids resist the sliding of one layer over another, a kind of internal friction called viscosity (श्यानता). It acts only while layers move relative to each other; a fluid at rest feels none.

The cleanest way to see viscosity is to shear a thin film of liquid steadily. Spread oil between two flat plates a small distance $l$ apart, keep one plate fixed and slide the other parallel to it at a constant speed $v$ (Fig. 9.12). The liquid clings to both plates (the oil touching each plate moves with it), so across the gap the velocity climbs in a straight line from zero to $v$. Keeping the plate going needs a steady force $F$, because each layer drags on its neighbours. Swap the oil for honey and $F$ shoots up for the same $v$: honey is more viscous. Smooth, layered flow of this kind is laminar.

The same clinging shapes flow in a pipe. The liquid at the wall stays put, so the velocity profile is a rounded curve, zero at the wall and largest on the axis: the liquid moves in nested cylindrical sleeves, each sliding inside the next (last step of Fig. 9.12).

A solid under a steady shear stress takes up a fixed strain. A fluid cannot: a small block of the liquid that starts out rectangular is sheared into a parallelogram that leans further and further for as long as the plate moves. In a time $\mathrm{\Delta }t$ its top shifts by $\mathrm{\Delta }x=v\mathrm{\Delta }t$ relative to its bottom, so the shear strain $\mathrm{\Delta }x\mathrm{/}l$ keeps growing. For water, oils and many other liquids, experiment shows that the stress $F\mathrm{/}A$ on a plate of area $A$ is proportional not to the strain but to its rate, $\mathrm{\Delta }x\mathrm{/}(l\mathrm{\Delta }t)=v\mathrm{/}l$. The constant of proportionality defines the viscosity:

Viscosity has dimensions $[\mathrm{M}{\mathrm{L}}^{-1}{\mathrm{T}}^{-1}]$. Runny liquids such as water, petrol and alcohol have low viscosity; glycerine, honey, engine oil and tar have high viscosity. Blood, at a few millipascal seconds, is thicker than water. Table 9.2 gives some values.

Temperature matters a great deal. Heat a liquid and its molecules break free of their neighbours more easily, so its viscosity falls: warm ghee pours where cold ghee barely moves, and water at 100 °C is between three and four times less viscous than at 20 °C. In a gas the trend reverses. Gas molecules hardly attract one another; a gas's viscosity comes from molecules darting between layers and carrying momentum with them. Heating makes them faster, so a gas becomes more viscous when hot.

9.5.1 Stokes' Law

A marble sinking through a bottle of shampoo, a dust grain settling in still air and a food pellet drifting down through aquarium water each drag the nearby fluid along, so the fluid holds them back. For a small sphere moving slowly, George Gabriel Stokes (1819–1903) showed in 1851 that this drag is proportional to the sphere's speed, its radius and the fluid's viscosity:

We shall not derive it, but its dimensions check: $[\eta ][a][v]=[\mathrm{M}{\mathrm{L}}^{-1}{\mathrm{T}}^{-1}][\mathrm{L}][\mathrm{L}{\mathrm{T}}^{-1}]=[\mathrm{M}\mathrm{L}{\mathrm{T}}^{-2}]$, a force. The law holds while the flow round the sphere stays smooth, which means small, slow spheres or very viscous fluids. In Chapter 5 you calculated the work done by resistive forces; Stokes' law tells us what such a force is for a small sphere.