yy yy xx yy zz pressure stresses due to linear compressibility rate of strain v

# Yy yy xx yy zz pressure stresses due to linear

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yy yy xx yy zz pressure stresses due to linear compressibility rate of strain v p p V y σ µε λ ε ε ε µ λ = − + + + + = − + + JG 2 ( ) zz zz xx yy zz p σ µε λ ε ε ε = − + + + + 2 xy yx xy v u x y τ τ µε µ = = = + 2 xz zx xz u w z x τ τ µε µ = = = + 2 yz zy yz w v y z τ τ µε µ = = = +
( ) . j i ij ij ij ij ij j i u u p V p x x τ δ µ δ λ δ τ = − + + + = − + JG thermodynamic pressure viscous stresses Note the inclusion of pressure because if velocity vanishes normal stress = - pressure (hydrostatic) Fluid at rest , 0 , , ij i j p i j τ = = 0 B p ρ −∇ = JG Stoke ‘s hypothesis 2 3 λ µ = − (1845) For air & most gas mixtures For liquids . 0 V = JG
. . 3 2 3 xx yy zz V V u v w u v w p x y z x y z σ σ σ µ λ + + = − + + + + + + JG JG ±²²³²²´ ±²²³²²´ 0 3 (2 3 ) . p V µ λ = = − + + JG ±²³²´ Define: mean (mechanical) pressure , p ( ) 1 3 xx yy zz p σ σ σ = − + + Mean pressure in a deforming viscous fluid is not equal to the thermodynamic pressure but distinction is rarely important 2 . 3 normal viscous stresses p p V λ µ = + JG ±²²³²²´ usually small in typical flow problems controversial subject