On bubble rising in a Hele-Shaw cell filled with a non-Newtonian fluid

Abstract

The problem of a bubble rising due to buoyancy in a Hele-Shaw cell filled with a viscous fluid is a classical free-boundary problem first posed and solved by Saffman & Taylor. In fact, due to linearity of the flow equations the problem is reduced to that of a bubble transported by uniform fluid flow. Saffman and Taylor provided explicit expressions for the bubble shape. Steady propagation of bubbles and fingers in a Hele-Shaw cell filled with a nonlinearly-viscous fluid was studied by Alexandrou & Entov. In Alexandrou & Entov, it was shown that for a nonlinearly viscous fluid the problem of a rising bubble cannot be reduced to that of a steadily transported bubble, and should be treated separately. This note presents a solution of the problem following the general framework suggested in Alexandrou & Entov. The hodograph transform is used in combination with finite-difference and collocation techniques to solve the problem. Results are presented for the cases of a Bingham and power-law fluids.