The Energy Method in Hydrodynamic Stability
Dr. Brian Straughan
Department of Mathematical Sciences
University of Durham, UK
September 6, 2002
We review the state of the art concerning recent developments where unconditional nonlinear stability is proved in some problems in fluid dynamics. Open questions to fundamental problems are included and we detail several recent advances yielding unconditional stability results via new Lyapunov functionals.
Attention is focused on the linear instability and nonlinear stability of a model for convection induced Rayleigh numbers very close to the critical ones of linear stability theory, and this demonstrates that linear theory effectively captures the physics of the onset of convection. Realistic boundary conditions appropriate to fixed surfaces are analyzed, and these reflect the experiments performed by Krishnamurti. The results obtained here lend much credence to the use of the model of Krishnamurti for convection in a fluid layer stably stratified when the heat source depends on the concentration of the stratifying agent.
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