Thermal Convection in a Sixth-Order Generalized Navier–Stokes Fluid

In this work, we deal with a problem of thermal convection for a fluid satisfying Navier-Stokes equation containing the spatial derivatives of the velocity field of sixth order, with the introduction of a tri-Laplacian term. It was pointed out by several authors, for example, Fried and Gurtin, that contributions of higher order take into account microlength effects; these phenomena are relevant in microfluidic flows. In particular, we follow the isothermal model of Musesti, using a Boussinesq approximation, so that the density in the body force term depends on the temperature to consider buoyancy effects that occur when the fluid is heated and it expands. We discuss different meaningful boundary conditions that have a key role to understand the effects of higher-order derivatives in microfluidic scenarios with convection. We carry out the complete study of linear and nonlinear stability for the flow. In addition, we complete the treatment with the analysis of critical wavenumbers and Rayleigh numbers for convection in the fluid.