Someone argues: "The viscous stress must be a linear

PHYSICS · Subjective · 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

Someone argues: "The viscous stress must be a linear, isotropic function of the velocity gradient

\partial u_i/\partial x_j = e_{ij}+\Omega_{ij}

, where

e_{ij}

is the symmetric rate of strain and

\Omega_{ij}

is the antisymmetric rotation tensor. Because both

e_{ij}

and

\Omega_{ij}

are perfectly good tensors, isotropy permits a term built from each, so

\tau_{ij} = 2\mu e_{ij} + \lambda\Omega_{ij}

for some constant

\lambda

, and a fluid element spinning fast enough would feel an extra viscous stress from the

\lambda\Omega_{ij}

term." Find the error in this argument. State which of the four assumptions behind the isotropic constitutive law rules out the

\lambda\Omega_{ij}

term, and use the rigid rotation

\mathbf{u} = (-\omega y, \omega x, 0)

as the test case.

Browse the problem library