Computation of steady flow past a sphere in a tube using a PTT integral model

Abstract

A Phan-Thien- Tanner (PTT) integral model which is exactly equivalent to the usual differential form is used with a new numerical scheme to simulate the Poiseuille flow of PTT fluids and the steady motion of a sphere along the axis of a cylinder. The exact solution for the Poiseuille flow of a PTT fluid presented here can be used to determine the rheological parameters, as well as to test numerical schemes and codes. With this scheme, convergent simulation results for flow past sphere are obtained to W i, (Weissenberg number) = 0.5, when ϵ =0.02, and up to W i=2.9, when ϵ =0.25, for the finest mesh used (M4). The tube diameter is twice the sphere diameter in all cases. The mesh refinement results exhibit good convergence properties. All of the curves of the dimensionless drag force versus W i for three different meshes are close to each other right up to where the numerical simulation diverges. There is no upturn in the drag for these results. The numerical simulation results for the dimensionless drag forces are in good agreement with those obtained by other methods using the differential form of the PTT model. One of the interesting features of the non-Newtonian stress field is the minimum value of τ zz, which is negative, appearing in the downstream region near the rear stagnation point, rather than at the upstream region near the front stagnation point as usually happens for UCM fluids. For the PTT fluid, above some level of viscoelasticity, a negative value of velocity component V z appears downstream near the rear stagnation point, which tells us there must be a vortex behind the sphere, although it is very weak at the Weissenberg numbers we have reached. We also find that the numerical method, solving the non-Newtonian extra stress based on the strain history calculation along the streamline, is an accurate scheme, which captures well the thin stress boundary layers and high stress regions The stress calculated by streamline integration is often very sensitive to small changes in kinematics, which poses a big challenge to computational stability