Analytic results for viscoelastic flow in a porous tube

We consider the problem of an upper-convected Maxwell fluid that is injected into or sucked from a cylindrical tube with porous walls. Many salient features of the flow are revealed by solving for the stresses assuming Newtonian kinematics. It is thereby found that the stress components have eigensolutions that in suction are eliminated by boundary conditions imposed on the centerline, which is the upstream boundary for suction. For injection, the centerline is the downstream boundary; boundary conditions applied at the centerline then do not eliminate the eigensolutions. As a result, numerical integration of the differential equations starting at the centerline is stable for suction, unstable for injection. This conclusion applies whether or not Newtonian kinematics are assumed. It is shown here that the eigensolution in injection is eliminated, and the integration stabilized, if integration is started at a position of zero radial flow outside the tube wall, in the region of “pre-history”, and carried out in the direction of flow, towards the centerline of the tube. The solution obtained in this way shows steep stress gradients that are driven by a biaxial extensional near-singularity in the region of pre-history.