(95c) Stability of the Shape of a Cusped Bubble Rising in a Viscoelastic Fluid

We present the results of a numerical study of the shape deformation of a bubble rising through a straight tube filled with an immiscible viscoelastic fluid. The finite volume method is used to solve the axisymmetric incompressible flow equations subject to the free-surface boundary conditions in an orthogonal curvilinear coordinate system. The FENE-CR constitutive equations are adopted for modeling the stress terms, and the artificial compressibility factor is introduced in the continuity equation to effectively increase the convergence rate of the computations. The interface shape is updated using the dynamic and kinematic conditions, and a boundary-fitted grid conforming to the deforming bubble shape is generated at each time step by solving the requisite elliptic partial differential equations. The effects of bubble size and rheological properties of the suspending fluid on the steady shape and mobility of the bubble are examined. Under certain conditions (e.g. high Deborah and capillary numbers), a sharp axisymmetric trailing edge develops into a three-dimensional cusp with a "knife-edge" shape. A temporal three-dimensional stability analysis of the axisymmetric base shape is performed to elucidate the underlying physics of this phenomenon. The mode with azimuthal wavenumer n = 2 is found to be the only unstable mode. The energy budget of the disturbances is investigated to determine the production and dissipation terms affecting the growth of this instability.