Properties of Matter — Flow, Surface Tension & Viscosity (Medium)

Quick Recap — Fluid Dynamics, Surface Tension & Viscosity

• Continuity: $A v=$ const, so $v\propto\dfrac{1}{A}\propto\dfrac{1}{r^2}$. Bernoulli: $P+\tfrac12\rho v^2+\rho g h=$ const (fast flow $\Rightarrow$ low pressure).
• Efflux (Torricelli): $v=\sqrt{2gh}$.
• Surface tension: excess pressure in a drop $\dfrac{2T}{r}$, in a soap bubble $\dfrac{4T}{r}$. Capillary rise $h=\dfrac{2T\cos\theta}{r\rho g}$ (so $h\propto\dfrac1r$).
• Surface energy $=T\times$area; work to blow a soap bubble $=8\pi r^2 T$ (two surfaces).
• Viscosity: Stokes drag $F=6\pi\eta r v$; terminal velocity $v_t=\dfrac{2r^2(\rho-\sigma)g}{9\eta}$ (so $v_t\propto r^2$).

Worked mini-example. Water ($T=0.072$ N/m) rises in a capillary of radius $0.2$ mm: $h=\dfrac{2T}{r\rho g}=\dfrac{2\times0.072}{(2\times10^{-4})(1000)(10)}=0.072$ m $=7.2$ cm.