Thermal stability of radiating fluids: The Bénard problem

The Bénard problem of the radiating nongray fluids is examined in terms of the Eddington approximation. The nongrayness of radiation is prescribed by the ratio and product of the Planck and Rosseland means of the absorption coefficient, ${\mathrm{\eta \hspace{0.17em}=\hspace{0.17em}(\alpha }}_P{\mathrm{/\alpha }}_R{)}^{\text{1/2}}$ and ${\alpha }_M{\mathrm{\hspace{0.17em}=\hspace{0.17em}(\alpha }}_P{\alpha }_R{)}^{\text{1/2}}$ , respectively. Effects of radiation on the classical problem are then characterized by four parameters: the Planck number, ${\mathcal{P}}_0$ (the ratio of conduction to radiation), optical thickness, ${\mathrm{\tau \hspace{0.17em}=\hspace{0.17em}\alpha }}_{M}d$ ($d$ being the distance between the plates) nongrayness of the fluid $\eta$ and the emissivity of boundaries ${\epsilon }_0$ and ${\epsilon }_1$ , respectively. The radiation in general has a stabilizing effect; decreasing ${\mathcal{P}}_0$ , increasing degree of nongrayness for $\text{η\hspace{0.17em}>\hspace{0.17em}1}$ , changing color of boundaries from black to mirror all delay the onset of instability. The boundary color and nongrayness of gas are responsible for the extrema observed in stability curves. Accuracy of the Eddington approximation is checked with the exact solution and the convergence of the approximate solution is studied in terms of the first and second approximations. Results are given for black‐black, mirror‐mirror, and black‐mirror boundaries.