  We analyse a model for thermal convection in a class of generalized
Navier-Stokes equations containing fourth order spatial derivatives of the
velocity and of the temperature. The work generalises the isothermal model of
A. Musesti. We derive critical Rayleigh and wavenumbers for the onset of
convective fluid motion paying careful attention to the variation of
coefficients of the highest derivatives. In addition to linear instability
theory we include an analysis of fully nonlinear stability theory. The theory
analysed possesses a bi-Laplacian term for the velocity field and also for the
temperature field. It was pointed out by E. Fried and M. Gurtin that higher
order terms represent micro-length effects and these phenomena are very
important in flows in microfluidic situations. We introduce temperature into
the theory via a Boussinesq approximation where the density of the body force
term is allowed to depend upon temperature to account for buoyancy effects
which arise due to expansion of the fluid when this is heated. We analyse a
meaningful set of boundary conditions which are introduced by Fried and Gurtin
as conditions of strong adherence, and these are crucial to understand the
effect of the higher order derivatives upon convective motion in a microfluidic
scenario where micro-length effects are paramount. The basic steady state is
the one of zero velocity, but in contrast to the classical theory the
temperature field is nonlinear in the vertical coordinate. This requires care
especially dealing with nonlinear theory and also leads to some novel effects.
