The motion of a piezoviscous fluid under a surface load

•We investigate the motion of a piezoviscous fluid under a surface load.•We derive an analytical solution for surface load problem for Navier–Stokes fluid.•We solve the full problem numerically using a variant of spectral collocation method.

Using a variant of a spectral collocation method we numerically solve the problem of the motion of a highly viscous fluid with pressure dependent viscosity under a surface load, which is a problem relevant in many applications, in particular in geophysics and polymer melts processing. We compare the results with the results obtained by the classical Navier–Stokes fluid (constant viscosity). It turns out that for a realistic parameter values the two models give substantially different predictions concerning the motion of the free surface and the velocity and the pressure fields beneath the free surface.As a byproduct of the effort to test the numerical scheme we obtain an analytical solution—for the classical Navier–Stokes fluid—of the surface load problem in a layer of finite depth.