Abstract: A local nonlinear stability analysis using a eight-mode
expansion is performed in arriving at the coupled amplitude equations
for Rayleigh-Bénard-Brinkman convection (RBBC) in the presence
of LTNE effects. Streamlines and isotherms are obtained in the
two-dimensional unsteady finite-amplitude convection regime. The
parameters’ influence on heat transport is found to be more
pronounced at small time than at long times. Results of the
Rayleigh-Bénard convection is obtained as a particular case of
the present study. Additional modes are shown not to significantly
influence the heat transport thus leading us to infer that five minimal
modes are sufficient to make a study of RBBC. The present problem
that uses rolls as a pattern of manifestation of instability is a needed
first step in the direction of making a very general non-local study of
two-dimensional unsteady convection. The results may be useful in
determining the preferred range of parameters’ values while making
rheometric measurements in fluids to ascertain fluid properties such
as viscosity. The results of LTE are obtained as a limiting case of
the results of LTNE obtained in the paper.