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