Equilibrium conditions and symmetries for foams in contact with solid surfaces

The patterns formed by liquid foams in contact with solid surfaces resemble 2D foams. We derive the equilibrium equations for such 2D foams when the solid surface is curved and smooth, generalising the standard case for flat Hele-Shaw cells. The equilibrium conditions at the vertices in 2D are invariant under conformal transformations. We establish the transformation rules for films under conformal maps. Even if conformal invariance does not hold in general, by considering foams confined between two closely spaced non-parallel plates, we show that an exponential profile gives a pattern that resembles the image of a regular hexagonal foam under a complex logarithm map, to lowest order.