36 research outputs found
Doubly-periodic array of bubbles in a Hele-Shaw cell
Exact solutions are presented for a doubly-periodic array of steadily moving
bubbles in a Hele-Shaw cell when surface tension is neglected. It is assumed
that the bubbles either are symmetrical with respect to the channel centreline
or have fore-and-aft symmetry, or both, so that the relevant flow domain can be
reduced to a simply connected region. By using conformal mapping techniques, a
general solution with any number of bubbles per unit cell is obtained in
integral form. Several examples are given, including solutions for multi-file
arrays of bubbles in the channel geometry and doubly-periodic solutions in an
unbounded cell.Comment: 15 pages, 12 figure
Time-evolving bubbles in two-dimensional stokes flow
A general class of exact solutions is presented for a time evolving bubble in a two-dimensional slow viscous flow in the presence of surface tension. These solutions can describe a bubble in a linear shear flow as well as an expanding or contracting bubble in an otherwise quiescent flow. In the case of expanding bubbles, the solutions have a simple behavior in the sense that for essentially arbitrary initial shapes the bubble will asymptote an expanding circle. Contracting bubbles, on the other hand, can develop narrow structures ('near-cusps') on the interface and may undergo 'break up' before all the bubble-fluid is completely removed. The mathematical structure underlying the existence of these exact solutions is also investigated