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Viscosity

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). The mean relative speed differs from the mean speed.

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Viscosity

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  1. Viscosity

  2. 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).

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

  4. 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 Collision Rate

  5. The mean time t 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 l combines the mean time and velocity. Mean Free Path

  6. 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 b a

  7. 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 l can be related to the cross section. Relative Flux

  8. 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 h Pzz Pzx ux z x

  9. 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 h. Momentum Transport

  10. 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

  11. 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 Jz Flux proportional to gradient The proportionality is the coefficient of self-diffusion D. Self-Diffusion

  12. Diffusion Equation • The particles are conserved in the layer Dz. • 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.

  13. 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. Diffusion Coefficient

  14. 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

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