Mechanical Properties of Fluids: Quick Revision

1. Pressure and Pascal's Law

P = \dfrac{F}{A}

P = P_0 + \rho g h

Pressure is a scalar; depth alone sets it, not shape or volume.

Hydraulic lift

F_2 = F_1\dfrac{A_2}{A_1}

Pascal’s law: pressure applied anywhere in an enclosed fluid is transmitted undiminished everywhere. Work done is equal on both pistons.

2. Continuity and Bernoulli

Equation of continuity

A_1 v_1 = A_2 v_2

Volume flow rate $Av$ stays fixed along a pipe; narrower section ⟹ faster flow.

Bernoulli's equation

P + \tfrac12\rho v^2 + \rho g h = \text{constant along a streamline}

Valid for steady, incompressible, non-viscous flow. Horizontal pipe: faster flow ⟹ lower pressure.

3. Venturi Meter

Venturi pressure drop

P_1 - P_2 = \tfrac12\rho\left(v_2^2-v_1^2\right) = \tfrac12\rho\,v_1^2\left[\left(\dfrac{A_1}{A_2}\right)^{2}-1\right]

Throat (narrow section) always has the lowest manometer column — lowest pressure, highest speed.

4. Viscosity, Stokes' Law and Terminal Velocity

\dfrac{F}{A} = \eta\,\dfrac{dv}{dz}

Gas viscosity rises with temperature; liquid viscosity falls.

Stokes' law

F = 6\pi\eta r v

Terminal velocity

v_t = \dfrac{2r^2(\rho-\sigma)g}{9\eta}

$\rho$ = sphere density, $\sigma$ = fluid density. $v_t \propto r^2$ — bigger particles settle faster.

5. Reynolds Number

Re = \dfrac{\rho v D}{\eta}

Laminar: $Re \lesssim 1000\text{–}2000$
Turbulent: $Re$ above that range

Large $Re$ : inertial forces dominate over viscous damping.

6. Surface Tension — Excess Pressure

S = \dfrac{F}{l} = \dfrac{\text{Energy}}{\text{Area}}

Liquid drop (1 surface): $\Delta P = 2S/r$
Soap bubble (2 surfaces): $\Delta P = 4S/r$

7. Capillary Rise

Capillary rise

h = \dfrac{2S\cos\theta}{\rho g r}

Water wets glass ( $\theta$ small, $\cos\theta>0$ ): rises. Mercury ( $\theta>90^\circ$ , $\cos\theta<0$ ): depressed. Rise is inversely proportional to tube radius $r$ .

Full derivations and worked examples: detailed notes →