A viscous oil of shear viscosity fills the gap between two flat plates a distance apart

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Kinematics, conservation laws and ideal flow · Viscous flow and dynamical similarity

Fluid Dynamics, Ch Kinematics in three dimensions and Cauchy's equation of motion

Problem

A viscous oil of shear viscosity

\mu = 0.50\,\mathrm{Pa\,s}

fills the gap between two flat plates a distance

h = 2.0\,\mathrm{mm}

apart. The lower plate is fixed and the upper plate moves at constant speed

U = 0.40\,\mathrm{m\,s^{-1}}

in its own plane; the flow between the plates is steady simple shear,

\mathbf{u} = (Sy, 0, 0)

with

S = U/h

, and the fluid is incompressible. ((a)) For an incompressible Newtonian fluid the stress is

\sigma_{ij} = -p\delta_{ij} + 2\mu e_{ij}

, where

e_{ij}

is the rate-of-strain tensor. Using the fact that

e_{ij}

is a real symmetric tensor with an orthogonal set of eigenvectors, show that every eigenvector of

e_{ij}

is also an eigenvector of

\sigma_{ij}

, so the principal axes of stress and of rate of strain coincide. ((b)) For the shear flow between the plates, work out the components of

e_{ij}

, hence the dissipation function

\Phi = 2\mu\,e_{ij}e_{ij}

(uniform across the gap), and integrate

\Phi

over unit area of the gap to get the dissipation rate per unit plate area. Compare this with the rate at which the moving plate does work per unit area on the fluid, and state whether the two rates agree.

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