Couple stress Rayleigh–Bénard convection in a square cavity

Consistent skew-symmetric couple stress theory, which introduces an intrinsic characteristic length scale parameter l into the formulation, is applied here to model the Rayleigh–Bénard convection problem. The focus is on the potential effect of size-dependency on the stability of Rayleigh–Bénard convection. We consider a two-dimensional flow and study the effects of couple-stresses by developing and then applying a stream function-vorticity-temperature computational fluid dynamics formulation. Details are provided on the governing equations for size-dependent flow, based on the Boussinesq approximation. Afterwards, the formulation is applied to the Rayleigh–Bénard convection problem in a square cavity to examine numerically the stability of the flow as a function of the length scale parameter l. The investigation covers a range of Rayleigh numbers, and includes an evaluation of the critical value beyond which convective instabilities start to appear, based upon an overall kinetic energy norm. The additional boundary conditions associated with consistent couple stress theory are found to play an important role in determining this critical Rayleigh number and also affect the flow patterns. Furthermore, the developments here may encourage the study of size-dependent multiphysics flow phenomena in general, which in turn may reveal new features and mechanisms of energy dissipation and storage at the smallest continuum scales.