Nonlinear Sciences – Pattern Formation and Solitons
Scientific paper
2011-01-19
Prog. Theor. Phys. 125 (2011) 1
Nonlinear Sciences
Pattern Formation and Solitons
46 pages; 4 figures
Scientific paper
10.1143/PTP.125.1
The Rosensweig instability has a special character among the frequently discussed instabilities. One distinct property is the necessary presence of a deformable surface, and another very important fact is, that the driving force acts purely via the surface and shows no bulk effect. These properties make it rather difficult to give a correct weakly nonlinear analysis. In this paper we give a detailed derivation of the appropriate amplitude equation based on the hydrodynamic equations emphasizing the conceptually new procedures necessary to deal with the distinct properties mentioned above. First the deformable surface requires a fully dynamic treatment of the instability and the observed stationary case can be interpreted as the limiting case of a frozen-in characteristic mode. Second, the fact that the driving force is manifest in the boundary conditions, only, requires a considerable change in the formalism of weakly nonlinear bifurcation theory. To obtain the amplitude equations a combination of solubility conditions and (normal stress) boundary conditions has to be invoked in all orders of the expansions.
Bohlius Stefan
Brand Helmut R.
Pleiner Harald
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