The Stability Of Bénard System And The Orthogonality Of Eigenfunction

The standard Bénard system refers to an infinite horizontal layer of incompressible fluid in which an adverse temperature gradient is maintained by heating from below. Because of the inhibition of viscosity and gravitation of fluid, the fluid remains at rest (i.e. basic flow) when the temperature difference between the bottom and top of the layer is small. When the temperature difference surpasses a critical value, the basic flow will become unstable and the thermal convection sets in. The perturbation equations of the basic state are the Oberbeck-Boussinesq equations which are nonlinear partial differential equations.Most mathematical models describing the flow are nonlinear partial differential equations, and the traditional starting point of an investigation of hydrodynamic stability problem is the eigenvalue analysis, i.e. the method of linearized stability. For some flows, notably those with instability driven by thermal or centrifugal forces, the predictions of...