Size-Dependent Fluid Mechanics: Theory and Application
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Some physical experiments exhibit size-dependency in fluid flow at small scales. This necessitates the introduction of couple stresses in the corresponding continuum theory. This in turn requires the vorticity to be considered as an additional degree of freedom associated with the angular momentum equation. Subsequently, the formulation accounts not only for stretches of the fluid elements, but also their bending deformations. The resulting size-dependent couple-stress fluid mechanics then can be used to explore the flow behavior at small sizes, such as micro- and nano-scales, and also to bridge between atomistic and classical continuum theories. The work here concentrates on two-dimensional flow and examines the effects of couple-stresses by developing and then applying a stream function-vorticity computational fluid dynamics formulation. Details are provided both on the governing equations for size-dependent flow and on the corresponding numerical implementation. Afterwards, the formulation is applied to the much studied lid-driven cavity problem to investigate the behavior of the flow as a function of the length scale parameter l . The investigation covers a range of Reynolds numbers, and includes an evaluation of the critical value beyond which a stationary response is no longer possible. This provides us with different unexpected sets of results for different boundary conditions when accounting for couple-stresses. These in turn might explain different chaotic behaviors of fluid flow. Since problems of thermoviscous flows are of prime importance for many physical processes, the classical Boussinesq equations for the Rayleigh-Bénard convection problem are modified by including couple stresses, which account for size-dependency. Then the stability of natural convection in a square cavity is studied numerically and the onset of convective instability within a range of Rayleigh and Prandtl numbers is investigated.