Viscosity

Average Speed The Maxwell-Boltzmann distribution is a function of the particle speed. The average speed follows from integration.  Spherical shell in velocity- space The relative velocity between particles is reduced by sqrt(2).

Relative Speed The mean relative speed differs from the mean speed.  Each molecule contributes

Collision Rate A reduced version of transport theory can found from a simple model of collisions.  Probability P(t)  Collision rate w The probability distribution in time is exponential.  Normalized to 1 at t = 0  Differential probability p

Mean Free Path The mean time  between collisions comes from the probability distribution.  Integrate by parts The kinetic energy of a gas can be characterized by the mean particle speed. The mean free path combines the mean time and velocity.

Scattering Scattering cross-section depends on the relative size of particles and their relative velocity.  Identical particles  Relative velocity v ’, mass m, radius a. Hard spheres have cross sections independent of velocity. a a b

Relative Flux A particle in a small volume experiences a relative flux.  Incidence based on relative speed The total scattered is the flux times the cross section.  Collision rate The mean free path can be related to the cross section.

Shear Stress A stress is a force per unit area.  Normal stress perpendicular to area  Shear stress perpendicular A fluid in motion static can support a shear stress.  Velocity gradient  Coefficient of viscosity  z uxux P zx P zz x

Momentum Transport One sixth of the particles will cross a plane in a given direction at a time. The stress is related to the net momentum change. Relate this to the gradient to get the viscosity coefficient .

Viscosity The viscosity is a retarding force due to motion in the fluid.  Friction or drag The viscosity depends on the material and temperature, not on the density.  Assumed low density – single collisions  High enough density to primarily collide with particles, not walls

Self-Diffusion Assume a fluid that is non- uniform in one dimension.  Number density n(z ) Identify a plane with a flux.  Plane area A  Perpendicular flux J z  Flux proportional to gradient The proportionality is the coefficient of self-diffusion D.

Diffusion Equation The particles are conserved in the layer  z.  Relates number to flux  Partial differential equation in t, z Use the assumed gradient to get a pde in n only. This is Fick’s diffusion equation.

Diffusion Coefficient One sixth of the particles will cross a plane in a given direction at a time. Find the flux from the mean velocity. Relate this to the gradient to get D.

Thermal Transport Consider the flow of heat through a plane.  Temperature gradient  Coefficient of thermal conductivity Find the coefficient by using the mean energy transfer.  Relate to specific heat