Thermal Instability in Differentially Heated Inclined Fluid Layers

In an inclined adversely heated fluid layer confined between two rigid boundaries in a slot of large aspect ratio it is found that the unicellular base flow in the conduction regime becomes unstable with the formation of stationary secondary rolls with their axes along the line of inclination (x-rolls) for large Prandtl number fluids and axes perpendicular to the line of inclination (y-rolls) for small Prandtl number fluids. However, for angles near the vertical, the curve of the critical Rayleigh number versus inclination for x-rolls rises above that for y-rolls even for large Prandtl number fluids so that in a vertical fluid layer only cross rolls (y-rolls) could develop. The stability equations, as well as the results, reduce to those available for the horizontal fluid layer for which x-rolls are as likely to occur as y-rolls. It is seen that even a small inclination to the horizontal is enough to assign a definite direction for these two-dimensional cells, this direction depending on the Prandtl number. It is hoped that this basic information will be of help in the determination of the magnitude of the secondary cells in the postinstability regime and the heat transfer characteristics of the thin fluid layer.