The general linear isotropic relation between the viscous stress and the velocity…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Fluids & Continuum Physics · Solid continuum mechanics and complex fluids

Fluid Dynamics, Ch The stress tensor completed: constitutive law and fluid properties

Problem

The general linear isotropic relation between the viscous stress and the velocity gradient is

\tau_{ij} = A_{ijkl}\,\partial u_k/\partial x_l

with

A_{ijkl} = a\,\delta_{ij}\delta_{kl} + b\,\delta_{ik}\delta_{jl} + c\,\delta_{il}\delta_{jk}

, and

\partial u_k/\partial x_l = e_{kl}+\Omega_{kl}

splits into the symmetric rate of strain

e_{kl}

and the antisymmetric part

\Omega_{kl}

. ((a)) Show that the

b

and

c

terms of

\tau_{ij}

depend on

\Omega_{kl}

only through the combination

b-c

, and use the requirement that rigid rotation carries no viscous stress to show

b=c\equiv\mu

. ((b)) With

b=c=\mu

and

\mu_b \equiv a+\tfrac23\mu

, the stress reduces to

\tau_{ij} = 2\mu\big(e_{ij}-\tfrac13 e_{kk}\delta_{ij}\big)+\mu_b e_{kk}\delta_{ij}

. Show that the dissipation function

\Phi=\tau_{ij}e_{ij}

reduces to a sum of two non-negative squared terms, and deduce the sign that

\mu

and

\mu_b

must each have.

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