Notes Viscosity
! Thursday: class will begin late, 11:30am ! In reality, nearby molecules travelling at different velocities occasionally bump into each other, transferring energy • Differences in velocity reduced (damping) • Measure this by strain rate (time derivative of strain, or how far velocity field is from rigid motion) • Add terms to our constitutive law
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Strain rate Viscous stress
! At any instant in time, measure how fast chunk ! As with linear elasticity, end up with two parameters if we of material is deforming from its current state want isotropy: • Not from its original state viscous ! ij = 2µ"˙ ij + #"˙ kk$ij • So we"re looking at infinitesimal, incremental strain • µ and ! are coefficients of viscosity (first and second) updates • These are not the Lame coefficients! Just use the same symbols • Can use linear Cauchy strain! • ! damps only compression/expansion ! So the strain rate tensor is " Usually !!-2/3µ (exact for monatomic gases) " So end up with # & 1 "ui "u j !˙ ij = 2 % + ( % ( % ( "u "u j 2 "u "x j "xi viscous i k $ ' ! ij = µ' + # $ij * & "x j "xi 3 "xk )
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