Size-Dependent Couple Stress Fluid Mechanics: The Influence of Boundary Conditions

In size-dependent couple stress fluid mechanics, which involves a length parameter $l$, the corresponding modified Navier-Stokes equations are $\rho \frac{D{\rm {\bf v}}}{Dt}=-\nabla p+\mu \;\nabla^{2}{\rm {\bf v}}-\mu l^{2}\;\nabla^{2}\nabla^{2}{\rm {\bf v}}$. The term involving $l$ is of fourth order, which then requires the prescription of additional boundary conditions compared to the classical case. Therefore, the boundary conditions in the size-dependent theory must include specification of either the tangential component of rotations ${\rm {\bf \omega }}$ on the boundary or the tangential moment-tractions ${\rm {\bf m}}^{\left( n \right)}$. Here we concentrate on two-dimensional flows and explore the consequences of prescribing different boundary conditions in size-dependent couple-stresses fluid mechanics by using computational fluid dynamics. We investigate the characteristics of flow for the cavity problem based upon the equation above and the Boussinesq approximation for the Rayleigh-Benard problem. This provides us with interesting, unexpected results for various boundary conditions, when accounting for couple-stresses. These in turn might explain different mechanisms for energy dissipation, as well as for chaotic behaviors of fluid flow.