title: Multiple steadily translating bubbles in a Hele-Shaw channel.
abstract: Analytical solutions are constructed for an assembly of any finite number of bubbles in steady motion in a Hele-Shaw channel. The solutions are given in the form of a conformal mapping from a bounded multiply connected circular domain to the flow region exterior to the bubbles. The mapping is written as the sum of two analytic functions-corresponding to the complex potentials in the laboratory and co-moving frames-that map the circular domain onto respective degenerate polygonal domains. These functions are obtained using the generalized Schwarz-Christoffel formula for multiply connected domains in terms of the Schottky-Klein prime function. Our solutions are very general in that no symmetry assumption concerning the geometrical disposition of the bubbles is made. Several examples for various bubble configurations are discussed.
authors.affiliation: Department of Mathematics , Imperial College London , 180 Queen's Gate, London SW7 2AZ, UK.
authors.forename: Christopher C
authors.initials: CC
authors.lastname: Green
authors.affiliation: Department of Mathematics , Imperial College London , 180 Queen's Gate, London SW7 2AZ, UK ; Departamento de Física , Universidade Federal de Pernambuco , Recife 50670-901, Brazil.
authors.forename: Giovani L
authors.initials: GL
authors.lastname: Vasconcelos
keywords: Hele-Shaw flows; Schottky–Klein prime function; Schwarz–Christoffel formula; bubble dynamics; free boundary problems