Rayleigh–Bénard Convection in a Radiating Fluid

Abstract

Linear and weakly nonlinear stability analyses of Rayleigh–Bénard convection (RBC) in a radiating Newtonian fluid are studied in the paper. The optical properties of the Newtonian fluid are considered to be independent of the wavelength of radiation. A gray medium thus assumed allows us to consider two asymptotic cases: (a) optically thin fluid medium (transparent) and (b) optically thick fluid medium (opaque). Using the solution in terms of a truncated Fourier series representation, we arrive at the analytical expression for the Rayleigh number and examine the thermal radiation properties. A modified Lorenz model, which has in it the influence of the radiation parameters, is derived. The analytically intractable three-dimensional Lorenz model is then projected into the one-dimensional Stuart–Landau equation. The analytical solution of the Stuart–Landau equation is used to quantify the heat transport. It is shown that the radiation inhibits the primary instability of convection in both transparent and opaque media. However, the delay of convection is more in the opaque medium compared to that in the transparent medium. Inclusion of a transparent medium creates a “heat-sink-like situation,” whereas the opaque medium leads to an “enhanced-thermal-diffusivity situation.” Both these situations result in diminished heat transport in the RBC system. The analytical expression of the Hopf–Rayleigh number is obtained by linearizing the modified Lorenz model around one of its postonset critical points. This number provides information about the onset of chaos in the dynamical system. The impact of the radiation effect is to delay the appearance of chaos.