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 )