Two dimensional bubbles and drops in a yield stress fluid.

摘要

We have implemented a finite-element code for a two-phase system consisting of incompressible two-dimensional Newtonian bubbles or droplets moving in a continuous Bingham fluid. The code employs a level-set method to track the deformable interface and a continuously differentiable viscosity function for the continuous phase that approaches the discontinuous Bingham model as a regularization parameter goes to zero.;Converged solutions for the slow gravitational rising (falling) of single bubbles (droplets) through a Bingham fluid in a large container show the presence of unyielded "ears" on the equatorial axis adjacent to the bubble (droplet) surface. The container boundaries are beyond the outer yield surface, and the flow is independent of the container length scale.;Collinear bubble or droplet pairs in a gravitational field interact in a way that is qualitatively similar to the interactions in a Newtonian outer fluid. Fore-aft symmetry is broken for two collinearly rising bubbles, with the formation of a cap on the upper bubble and an inverted teardrop shape on the lower, forming a "shade tree" following coalescence. The dynamics of shape development in the Bingham material differ from those for a Newtonian continuum, however. There is an unyielded region that initially extends between the two equatorial planes, and there is a recirculating flow that causes flattening of the trailing edges of bubbles. The trailing bubble evolves to a teardrop shape from an intermediate "peanut" shape. With decreasing Bond number (ratio of gravitational to interfacial stresses) the trailing bubble or droplet develops an unusual cusped "fishtail" shape.;Bubble pairs can rise under conditions where the buoyant force on a single bubble does not exceed the integrated yield stress. A collinear configuration is the most effective in terms of overcoming yield stress and a side-by-side configuration is the least effective, with an off-center configuration intermediate. The calculations also show that the off-centered bubbles tend to align when they rise, increasing the coalescence efficiency in a bubbly fluid.;It is also observed that the evolution of three bubbles, and possibly more, can be understood by considering the interactions between each pair of bubbles.