Fluids (liquids and gases) cannot withstand shear stress — they flow under any applied tangential force. This chapter explores pressure, buoyancy, fluid dynamics, viscosity, and surface tension.
Pressure in Fluids
Pressure = Force / Area (unit: Pascal, Pa = N/m2).
Pressure at a depth h in a static fluid: P = P0 + rho g h
where P0 is atmospheric pressure, rho is fluid density, g is acceleration due to gravity.
Pascal's Law: Pressure applied to an enclosed fluid is transmitted undiminished to every part of the fluid and the walls of the container. (Basis of hydraulic machines.)
Atmospheric pressure ~ 1.013 x 105 Pa (one atmosphere). Measured by a mercury barometer.
Archimedes' Principle and Buoyancy
Buoyant force = weight of fluid displaced = rhofluid x Vsubmerged x g
Archimedes' Principle: When a body is partially or fully submerged in a fluid, it experiences an upward buoyant force equal to the weight of the fluid displaced.
Condition for floating: rhobody ≤ rhofluid (object floats if average density is less than or equal to fluid density).
Fluid Dynamics: Equation of Continuity
For streamlined (ideal) flow: A1 v1 = A2 v2 (Equation of Continuity)
where A is cross-sectional area and v is fluid velocity. This is conservation of mass for incompressible flow.
Bernoulli's Theorem
For an ideal (non-viscous, incompressible) fluid in steady flow:
P + (1/2) rho v2 + rho g h = constant
- This is energy conservation per unit volume. Applications:
- Venturimeter: measures flow speed
- Torricelli's theorem: efflux speed from a tank: v = √(2gh)
- Magnus effect: spinning ball curves in air
- Lift on aircraft wings
Viscosity
Real fluids have internal friction called viscosity. Coefficient of viscosity (eta):
F = eta A (dv/dx), unit: Pa.s (or Poise in CGS)
Stokes' Law: Viscous drag on a sphere: F = 6 pi eta r v
Terminal velocity: vt = 2r2(rho - sigma)g / (9 eta)
where rho = density of sphere, sigma = density of fluid.
- Reynolds Number: Re = rho v D / eta
- Re < 1000: Laminar (streamline) flow
- Re > 2000: Turbulent flow
Surface Tension
Surface tension (T) = Force per unit length on the liquid surface, unit: N/m.
Excess pressure inside a liquid drop: deltaP = 2T/r
Excess pressure inside a soap bubble: deltaP = 4T/r (two surfaces)
Capillarity: Rise of liquid in a narrow tube: h = 2T cos(theta) / (rho g r)
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A hydraulic lift has pistons of area 5 cm2 and 500 cm2. A force of 10 N on the small piston. What load does the large piston support?
By Pascal's law: F2/A2 = F1/A1 => F2 = 10 x (500/5) = 1000 N
A tank of height 5 m is filled with water (rho = 1000 kg/m3). Find efflux speed from a hole at the bottom.
v = √(2gh) = √(2 x 10 x 5) = 10 m/s
A steel ball (radius 2 mm, density 8000 kg/m3) falls in glycerine (density 1300 kg/m3, eta = 1.5 Pa.s). Find terminal velocity.
vt = 2r2(rho - sigma)g / (9 eta) = 2 x (2x10-3)2 x (8000-1300) x 10 / (9 x 1.5)
= 2 x 4x10-6 x 6700 x 10 / 13.5 = 0.536 / 13.5 = 0.0397 m/s
Water flows in a pipe; diameter narrows from 4 cm to 2 cm. If speed at wide end is 1 m/s, find speed at narrow end.
A1 v1 = A2 v2. pi(0.02)2 x 1 = pi(0.01)2 x v2. v2 = 4/1 = 4 m/s
A soap bubble of radius 3 cm. T = 0.03 N/m. Find excess pressure inside.
deltaP = 4T/r = 4 x 0.03 / 0.03 = 4 Pa
Water rises 5 cm in a capillary tube (contact angle = 0, rho = 1000 kg/m3, g = 10 m/s2). Find radius if T = 0.072 N/m.
h = 2T cos(theta)/(rho g r) => r = 2T/(rho g h) = 2 x 0.072/(1000 x 10 x 0.05) = 0.144/500 = 2.88 x 10-4 m
At what speed will flow in a pipe of diameter 2 cm change from laminar to turbulent? (rho = 1000 kg/m3, eta = 10-3 Pa.s, Re = 2000)
v = Re x eta / (rho D) = 2000 x 10-3 / (1000 x 0.02) = 2/20 = 0.1 m/s
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Common mistakes
- Soap bubble has two surfaces — excess pressure = 4T/r, not 2T/r. A liquid drop has only one surface: 2T/r.
- In Bernoulli's equation, ensure the flow is steady and the fluid is ideal (non-viscous, incompressible).
- Pascal's law applies to static fluids — do not apply it directly to flowing fluids.
Summary
Fluid pressure increases with depth. Pascal's law underlies hydraulic machines. Buoyancy explains floating and sinking. Bernoulli's theorem links pressure, speed, and height in ideal flow. Viscosity and surface tension govern real fluid behaviour.