Solution of the adjoint problem for instabilities with a deformable surface: Rosensweig and Marangoni instability

We present a method to find the adjoint system of equations and the corresponding boundary conditions for free deformable surfaces. Motivated by the nonlinear discussion of the Rosensweig instability in ferrogels using the energy method, we treat the surface as dynamic and take the stationary limit only in the very end. We analyze the adjoint system of dynamic equations together with its corresponding boundary conditions and present as a solution the adjoint eigenvectors for the Rosensweig instability. The method is also applied to pure surface tension driven convection (Marangoni convection).