Chapter 9 ยท Introduction

Chapter 9 โ€” Mechanical Properties of Fluids

Liquids and gases flow, press on every surface they touch, cling to some walls and shy away from others. This chapter explains pressure in a fluid at rest, the energy balance of a moving fluid, the internal friction called viscosity, and the skin-like surface of a liquid.

A diver at the bottom of a five-metre pool feels the water squeezing her ears, yet a hydraulic jack lets a mechanic lift a car with one hand, and the same law explains both. An aircraft weighing four hundred tonnes is held up by air flowing a little faster over its wings than under them. A raindrop falling from a cloud a kilometre up does not hit you at the $140\ \text{m/s}$ that free fall would give, but at a gentle few metres per second. And water creeps up a thin glass tube, while mercury in the same tube sinks below the level outside.

All of these belong to the mechanics of fluids โ€” substances that flow, liquids and gases alike. In Chapter 8 you saw how a solid resists being stretched, squeezed or sheared. A fluid cannot resist a shear at all when it is at rest; it simply flows. That one property shapes the whole chapter. You will start with fluids at rest: pressure, how it grows with depth, atmospheric and gauge pressure, and Pascal's law behind hydraulic lifts and brakes. Then you will set the fluid moving and use conservation of mass and of energy to reach the equation of continuity and Bernoulli's principle, with the speed of a jet from a tank, the Venturi meter and the lift on a wing. Real fluids are sticky: viscosity, Stokes' law and terminal velocity explain the raindrop. Finally the surface itself: surface tension and surface energy, the pressure inside drops and bubbles, and capillary rise.

Pressure $p=\dfrac{F}{A}$โ†’$p=p_a+\rho gh$โ†’Pascal's lawโ†’$A_1v_1=A_2v_2$โ†’Bernoulli's principleโ†’Viscosityโ†’Terminal velocityโ†’Surface tensionโ†’Capillary rise
๐ŸŽฏ By the end of this chapter you canโ€ฆ
  • define pressure and density, explain why pressure at a point in a fluid at rest acts equally in all directions, and use $p=p_a+\rho gh$ with atmospheric, gauge and absolute pressure, barometers and U-tube manometers;
  • state Pascal's law and analyse hydraulic lifts, presses and brakes, $\dfrac{F_1}{A_1}=\dfrac{F_2}{A_2}$, including the work done;
  • distinguish streamline and turbulent flow, and use the equation of continuity $A_1v_1=A_2v_2$;
  • derive Bernoulli's principle $p+\tfrac12\rho v^2+\rho gh=\text{constant}$ from the workโ€“energy theorem and apply it to the speed of efflux $\sqrt{2gh}$, the Venturi meter, blood flow and dynamic lift;
  • use the coefficient of viscosity $\eta$, the critical velocity and Reynolds number $Re=\dfrac{\rho vd}{\eta}$, Stokes' law $F=6\pi\eta rv$ and the terminal velocity $v_t=\dfrac{2r^2(\rho-\sigma)g}{9\eta}$;
  • relate surface tension to surface energy, use the angle of contact, the excess pressures $\dfrac{2S}{r}$ (drop) and $\dfrac{4S}{r}$ (soap bubble), and the capillary rise $h=\dfrac{2S\cos\theta}{\rho gr}$;
  • tackle Board, JEE Main, JEE Advanced and NEET-style problems on fluids.
Teaser โ€” Three Vessels, One Pressure

Three vessels have the same circular base (radius $5.0\ \text{cm}$) and are filled with water to the same depth $h$. The wide one holds far more water than the narrow one. Before you look at the readouts, guess: which base is pushed hardest by the water? Drag the depth slider or press Play to fill them together.

40 cm
gauge pressure on each baseโ€”
force on each baseโ€”
weight of water (wide)โ€”
weight of water (narrow)โ€”

Sections in this chapter

9.1

Pressure, Pascal's Law and Hydraulic Machines

Pressure and density, variation of pressure with depth, atmospheric and gauge pressure, barometers, manometers, and hydraulic lifts and brakes.

$p=p_a+\rho gh$
Open section โ†’9.2

Streamline Flow and Bernoulli's Principle

Streamlines, the equation of continuity, Bernoulli's principle, speed of efflux, the Venturi meter, blood flow and the lift on a wing.

$p+\tfrac12\rho v^2+\rho gh=\text{const}$
Open section โ†’9.3

Viscosity, Stokes' Law and Terminal Velocity

Internal friction in fluids, the coefficient of viscosity, the drag on a small sphere and why falling drops reach a steady speed.

$v_t=\dfrac{2r^2(\rho-\sigma)g}{9\eta}$
Open section โ†’9.4

Surface Tension and Capillarity

Surface tension and surface energy, angle of contact, excess pressure in drops and bubbles, and the rise of liquids in thin tubes.

$h=\dfrac{2S\cos\theta}{\rho gr}$
Open section โ†’
โœ… Before you start
You will need Newton's laws and the idea of equilibrium of forces, including free-body diagrams (Class 11 Physics, Chapter 4); work, kinetic energy, gravitational potential energy $mgh$, the workโ€“energy theorem and conservation of energy (Chapter 5, Sections 5.1โ€“5.2); and the idea of stress as force per unit area, with the bulk modulus, from Chapter 8 (Mechanical Properties of Solids). From Class 10 Science you should remember density, buoyancy and Archimedes' principle in outline. From mathematics you need rearranging formulae, square roots, areas and volumes of circles, cylinders and spheres, and significant figures and dimensions (Chapter 1).
โ† 8.2 Elastic Moduli and Poisson's Ratio9.1 Pressure, Pascal's Law and Hydraulic Machines โ†’