Fluid Dynamics

Instructor: Zhang Hepeng
中文版：流体动力学
Back to: Soft Matter Physics Course Notes

Core Picture

Soft matter usually moves in water, oil, or complex fluids. At micron scales, inertia is often weak and viscosity dominates. When force is removed, motion stops almost immediately.

Continuum Description

Fluid dynamics describes a fluid with velocity field v(r,t), pressure p(r,t), and density rho.

material D=$\frac{D}{Dt}=\partial_t+\mathbf v\cdot\nabla$

continuity=$\partial_t\rho+\nabla\cdot(\rho\mathbf v)=0$

incompressible=$\nabla\cdot\mathbf v=0$

The material derivative follows a moving fluid element. Incompressibility is a good approximation for water and many soft-matter experiments.

Viscosity and Newtonian Fluid

For a Newtonian fluid, shear stress is proportional to shear rate.

shear=$\tau=\eta\dot\gamma$

stress=$\sigma_{ij}=-p\delta_{ij}+\eta(\partial_i v_j+\partial_j v_i)$

viscosity=$\eta$ measures momentum diffusion

Navier-Stokes and Low-Re Limit

Navier-Stokes is momentum conservation. In soft matter, the low-Reynolds-number limit often applies, giving the Stokes equation.

NS=$\rho\frac{D\mathbf v}{Dt}=-\nabla p+\eta\nabla^2\mathbf v+\mathbf f$

Re=$\rho U L/\eta$

Stokes=$0=-\nabla p+\eta\nabla^2\mathbf v+\mathbf f$

constraint=$\nabla\cdot\mathbf v=0$

Small Re means viscous forces dominate and inertial memory is negligible. This is the basis of microfluidics, colloidal hydrodynamics, and cellular-scale motion.

Drag on a Sphere

A sphere of radius R moving with speed U through a viscous fluid experiences Stokes drag.

drag=$F=6\pi\eta R U$

mobility=$U=\mu F$

mu=$\mu=\frac{1}{6\pi\eta R}$

This formula connects directly to the Stokes-Einstein relation in Brownian motion.

Poiseuille Flow

Flow through a narrow tube is extremely sensitive to tube radius.

Poiseuille=$Q=\frac{\pi a^4\Delta p}{8\eta L}$

resistance=$\frac{\Delta p}{Q}=\frac{8\eta L}{\pi a^4}$

Exam Cue

For micron-scale motion in water, estimate Re first. If Re is small, use Stokes equation, Stokes drag, mobility, and then connect to Stokes-Einstein diffusion when thermal noise is included.