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Weakly NonLinear Stability of Magnetized Ferrofluids at Large Critical Wavenumbers
 Blennerhassett and P.J. Stiles
"... The structure of linear and weakly nonlinear stability eigenfunctions in a thin layer of magnetized ferrofluid heated from above is examined in the limit as the critical wavenumber a of the disturbance becomes large. Two cases are examined. The first, using idealized boundary conditions, allows an ..."
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The structure of linear and weakly nonlinear stability eigenfunctions in a thin layer of magnetized ferrofluid heated from above is examined in the limit as the critical wavenumber a of the disturbance becomes large. Two cases are examined. The first, using idealized boundary conditions, allows an exact solution to be found and the second, using physical boundary conditions, reveals a coreboundary layer structure in which the core solution is asymptotic to the exact solution in the large critical wavenumber limit and boundary layers of thickness O(a \Gamma4=3 ) develop. The critical value of the instability parameter in this case is found to have an expansion in inverse powers of a 2=3 . Asymptotic expressions for the Nusselt number are also presented, showing that the presence of boundary layers increases the heat transfer across the fluid layer. The reason for developing these solutions is that they form the basis for an analysis of the strongly nonlinear vortices that develop...
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 t ..."
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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 \Gamma4=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 . Wea...