09Class 11 Physics

Mechanical Properties of Fluids formulas

Pressure, flow and surface tension formulas for Mechanical Properties of Fluids. Most numericals here use Bernoulli and continuity together.

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Pressure and buoyancy

Pressure at depth: $\displaystyle p = p_0 + \rho g h$
Pascal's law (hydraulic lift): $\displaystyle \frac{F_1}{A_1} = \frac{F_2}{A_2}$
Archimedes' principle: $\displaystyle F_b = \rho_{\text{fluid}}\, V_{\text{displaced}}\, g$

Flowing fluids

Continuity equation: $\displaystyle A_1v_1 = A_2v_2$
Bernoulli's equation: $\displaystyle p + \tfrac12\rho v^2 + \rho g h = \text{constant}$
For steady, incompressible, non-viscous flow.
Torricelli's law: $\displaystyle v = \sqrt{2gh}$

Viscosity

Viscous force: $\displaystyle F = \eta A\frac{dv}{dx}$
Stokes' law: $\displaystyle F = 6\pi\eta r v$
Terminal velocity: $\displaystyle v_t = \frac{2r^2(\rho - \sigma)g}{9\eta}$
ρ is the sphere's density and σ the fluid's.
Reynolds number: $\displaystyle R_e = \frac{\rho v d}{\eta}$

Surface tension

Surface tension: $\displaystyle S = \frac{F}{L}$
Excess pressure in a drop: $\displaystyle \Delta p = \frac{2S}{r}$
Excess pressure in a soap bubble: $\displaystyle \Delta p = \frac{4S}{r}$
A bubble has two surfaces.
Capillary rise: $\displaystyle h = \frac{2S\cos\theta}{\rho g r}$

Where students slip up

Using 2S/r for a soap bubble. It has two surfaces, so it's 4S/r.
Applying Bernoulli to viscous flow, or between points on different streamlines.
Forgetting gauge pressure leaves out atmospheric pressure, while absolute pressure includes it.

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