The dynamics of simple fluids in steady circular shear

An isotropic simple fluid of constant density ρ \rho is confined between two infinite horizontal planes which rotate steadily about separate vertical ( z ) \left ( z \right ) axes with common angular velocity ω \omega . We show that at least one solution to the exact equations of motion is determined by the differential equation \[ d d z ( η d u d z ) − i ρ ω u = 0 \frac {d}{{dz}}\left ( {\eta \frac {{du}}{{dz}}} \right ) - i\rho \omega u = 0 \] where u ( z ) u\left ( z \right ) is a complex variable representing the horizontal velocity and η ( | d u / d z | , ω ) \eta \left ( {\left | {du/dz} \right |,\omega } \right ) is a complex shear modulus. This equation represents the extension to nonlinear viscoelasticity of the previous works of Berker and of Abbott and Walters on linear viscoelastic fluids, for which η \eta reduces to the usual dynamic viscosity η ∗ ( ω ) \eta *\left ( \omega \right )