Supercritical analysis of strongly nonlinear vortices in magnetized ferrofluids

The structure of two-dimensional vortices in a thin layer of magnetized ferrofluid heated from above is examined in the limit as the critical wavenumber, a, of the roll cells becomes large. In particular, we present a nonlinear asymptotic description of the vortex pattern that occurs directly above the critical point in parameter space where instability first sets in. Two cases are examined. First, an idealized case where the fluid layer has boundary conditions appropriate for a free surface and second, a more physical situation where the fluid is confined between rigid horizontal magnetic pole pieces. The idealized problem has a relatively simple solution structure, which is a leading-order approximation to the solution of the physical problem. As the critical wavenumber increases, boundary layers of thickness O(a^-4/3) develop at the walls in the physical problem and the critical value of the instability parameter has an asymptotic expansion in inverse powers of a^2/3. Weakly nonlinear solutions are extended into the unstable region of parameter space where the convection is described by fully nonlinear equations and the perturbations to the basic state have finite amplitude. This analysis is new since similar large-wavenumber investigations in other problems do not apply at critical conditions. As the instability parameter is increased above its critical value, a hierarchy of bifurcations in the equations governing the temperature perturbation occurs, necessitating the inclusion of progressively more harmonics in the solution. Numerical solutions for the first few of these temperature equations are presented, and asymptotic expressions for the heat transfer across the fluid layer for both sets of boundary conditions are derived. Although the conduction state still dominates the heat transfer, the solution structure is fundamentally different from previous weakly nonlinear theories. Finally, some numerical solutions of the full steady-state governing equations are presented and compared with the asymptotic structures. These results verify that the solutions obtained with both sets of boundary conditions are asymptotic to one another in the large-wavenumber limit and that the asymptotic solution is a useful approximation in this limit.