NUMERICAL-SOLUTIONS FOR THE DEFORMATION OF A BUBBLE RISING IN DILUTE POLYMERIC FLUIDS

Abstract

The steady-state deformation of a bubble rising in polymeric liquid has been investigated using a general numerical technique for the solution of free-boundary problems in non-Newtonian fluid mechanics. The technique is based on a finite-difference solution of the governing equations on an orthogonal curvilinear coordinate system, which is constructed numerically and adjusted to fit the boundary shape exactly at any time. The problem was analyzed based on the constitutive equation proposed by Chilcott and Rallison [J. Non-Newtonian Fluid Mech. 29, 381 (1988)], which models a dilute polymer solution as a suspension of dumbbells with finite extensibility. Computations were carried out to investigate the effect of conformation change of polymer chains on the bubble deformation for various values of the Deborah number, maximum chain extensibility (i.e., roughly proportional to molecular weight), capillary number, and the Reynolds number. Numerical results show good agreement with existing experimental findings reported elsewhere. Especially, the tendency of transition from a prolate shape to a cusped shape has been observed in the creeping flow limit.