Abstract: An analytical study is conducted to examine the
coupling between penetrative convection and Rayleigh-Bénard-Marangoni (RBM)
convection in a shallow pool of liquid whose density exhibits a maximum at
temperature The liquid layer sits on a plate of
low thermal conductance and underlies an air layer. The free surface that
emerges in the process is assumed non-deformable. The liquid, whose density is
assumed to have a quadratic dependence on temperature with an extremum at is heated from below. A linear
stability analysis is conducted in the parameter space of the problem consisting
of the Rayleigh number, the Marangoni number and a density extremum number. The
main purpose of the present study is to isolate the influence of the non-linear
equation of state of the fluid on the threshold values of the Rayleigh and
Marangoni numbers and associated flow patterns at the onset of convection.
Keywords and phrases: fluid dynamics, hydrodynamic stability, thin fluid films.
