A spectral collocation scheme for the flow of a piezo-viscous fluid in ducts with slip conditions

Abstract

In this paper we present a numerical scheme based on spectral collocation methods to investigate the flow of a piezo-viscous fluid, i.e., a fluid in which the rheological parameters depend on the pressure. In particular, we consider an incompressible Navier–Stokes fluid with pressure dependent viscosity flowing in: (i) a two-dimensional non-symmetric planar channel; (ii) a three-dimensional axisymmetric non-straight conduit. For both cases we impose the Navier slip boundary conditions that can be reduced to the classical no-slip condition for a proper choice of the slip parameter. We assume that the dependence of the viscosity on the pressure is of exponential type (Barus law), even though the model can be replaced by any other viscosity function. We write the mathematical problem (stress based formulation) and discretize the governing equations through a spectral collocation scheme. The advantage of this numerical procedure, which to the authors’ knowledge has never been used before for this class of fluids, lies in in the ease of implementation and in the accuracy of the solution. To validate our model we compare the numerical solution with the one that can be obtained in the case of small aspect ratio, i.e., the leading order lubrication solution. We perform some numerical simulation to investigate the effects of the pressure-dependent viscosity on the flow. We consider different wall functions to gain insight also on the role played by the channel/duct geometry. In both cases (i), (ii) we find that the increase of the coefficient appearing in the viscosity function results in a global reduction of the flow, as physically expected.