Motion induced between parallel plates with offset centers of radial stretching and shrinking

The flow between parallel plates separated by distance $h$ is investigated where the upper and lower plates respectively stretch and shrink at the same rate $a$ and the centers of stretching and shrinking are horizontally separated by distance $2\,l$. A reduction of the Navier-Stokes equation yields two ordinary differential equations dependent on a Reynolds number $R = ah^2/\nu$. In addition a free parameter $\gamma$ appears which corresponds to a uniform pressure gradient acting along the line connecting the stretching/shrinking centers. We consider three cases: $\gamma = 0$, $\gamma = O(1)$ and $\gamma = O(R)$. The flow is described by two functions of the plate-normal coordinate $\eta = z/h$: the first $f(\eta)$ has an analytical solution while the second $g(\eta)$ must be resolved numerically. The small-$R$ solutions are found and the large-$R$ asymptotic behaviors of the wall shear stresses and the centerline velocities are obtained by matching the viscous boundary layer flows to the interior inviscid motion.