The dynamics of simple fluids in steady circular shear

An isotropic simple fluid of constant density $\rho$ is confined between two infinite horizontal planes which rotate steadily about separate vertical $\left ( z \right )$ axes with common angular velocity $\omega$. We show that at least one solution to the exact equations of motion is determined by the differential equation \[ \frac {d}{{dz}}\left ( {\eta \frac {{du}}{{dz}}} \right ) - i\rho \omega u = 0\] where $u\left ( z \right )$ is a complex variable representing the horizontal velocity and $\eta \left ( {\left | {du/dz} \right |,\omega } \right )$ 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 $\eta$ reduces to the usual dynamic viscosity $\eta *\left ( \omega \right )$