Multiple steady bubbles in a Hele-Shaw channel

We construct analytical solutions, in the form of conformal mappings, solving the free boundary problem for the shapes of any finite number of steadily translating bubbles in a Hele-Shaw channel. These solutions can be decomposed into the sum of two analytic functions - corresponding to the complex potentials in the laboratory and co-travelling frames - which conformally map a bounded multiply connected circular domain onto respective degenerate polygonal domains (infinite strips with interior slits of finite-length which are either horizontally or vertically aligned). These functions are obtained using the generalised Schwarz-Christoffel formula for multiply connected domains in terms of the Schottky-Klein prime function. The solutions we have found are very general and make no assumptions on the geometrical arrangement of the bubbles.