JEE Physics Mechanical Properties Of Fluids Notes Set 02

Fluid Dynamics

For steady flow of an ideal fluid, the equation of continuity is \( A_1v_1=A_2v_2 \) (conservation of mass), so the speed is greater in the narrow section, with \( v\propto 1/r^2 \). Bernoulli's theorem expresses conservation of energy: \( P+\tfrac12\rho v^2+\rho gh=\text{constant} \), where the first term is the pressure head \( P/\rho g \).

Applications: the speed of efflux from a hole at depth h is \( \sqrt{2gh} \) (Torricelli), and the horizontal range from a hole at depth h in a tank of height H is \( 2\sqrt{h(H-h)} \). The dynamic lift on an aeroplane wing is \( \tfrac12\rho(v_u^2-v_l^2)A \), and sprayers, atomisers and the Bunsen burner all rely on low pressure where the speed is high. A venturimeter measures flow speed.

Viscosity

Viscous force between layers is \( F=-\eta A\dfrac{dv}{dx} \) (Newton's law). The CGS unit of \( \eta \) is the poise, and the SI unit is Pa s (1 Pa s = 10 poise). The viscosity of liquids decreases and of gases increases with temperature. Stokes' law gives the viscous drag on a sphere as \( F=6\pi\eta rv \). The terminal velocity is \( v_t=\dfrac{2r^2(\rho-\sigma)g}{9\eta} \), proportional to \( r^2 \), and it is independent of the height of fall.

Poiseuille's formula for the volume flow through a capillary is \( Q=\dfrac{\pi Pr^4}{8\eta l} \), so the flow is proportional to \( r^4 \). The Reynolds number \( Re=\dfrac{\rho vD}{\eta} \) tells whether flow is laminar (below about 2000) or turbulent (above about 3000).