Classical Benard Problem Research Articles

Convection in multi-component rotating fluid layers via the auxiliary system method

The onset of thermal convection in a uniformly rotating horizontal layer filled by a Navier–Stokes multi-component fluid mixture, heated from below and salted partly from above and partly from below, is investigated via the new approach named auxiliary system method Rionero (Rend Lincei Mat Appl 25:1–44, 2014). In the free–free case, via the generalization of the Rionero Linearization Principle: “Decay of linear energy for any initial data implies decay of nonlinear energy at any instant” [given in Rionero (Rend Lincei Mat Appl 25:1–44, 2014) in the absence of rotation], it is shown that conditions guaranteeing linear stability of thermal conduction solution guarantee also absence of subcritical instabilities and global exponential asymptotic nonlinear stability. The classical Benard problem is investigated via a procedure different from the celebrated one given in Chandrasekhar (Hydrodynamic and hydromagnetic stability, 1981).

Stability of lid-driven shallow cavity heated from below

Hydrodynamic and thermal stability of combined thermal buoyancy and lid-driven shear flow in a shallow cavity is analyzed by means of linearized perturbation theory. The analysis considers a cavity heated from below and cooled at the upper moving lid. A numerical procedure, which has generality with respect to boundary conditions, Reynolds, and Prandtl numbers, is described for solution of the linearized model equations. A direct numerical integration (Runge-Kutta with Newton-Raphson) method is used to solve the differential conservation equations. This method gives an exact result for the classical Benard problem where the flow becomes unstable at a critical Rayleigh number, Ra c = 1707.76. The numerical results show the existence of two critical wave numbers depending on whether the dominant force driving the flow is due to buoyancy or shear. For Pr ⩽ 0.1 the instability is due to the buoyancy force for constant heat flux boundary conditions, while for Pr = 1 the instability is due to the shear force. Increasing the Reynolds number stabilizes the flow, and reducing the Prandtl number makes the flow more unstable.

A Benard convection problem with a perturbed lower wall

The classical Benard problem of fluid confined between two horizontal boundaries and heated from below is perturbed by making the elevation of the lower wall a small and slowly varying function of the horizontal coordinate. Modified amplitude equations, both linear and nonlinear, are found, and modified critical Rayleigh numbers are calculated. For the type of solution found it appears that the horizontal wavenumber is uniquely determined.