Skip to main content Accessibility help

Thermal instability in a horizontal fluid layer: effect of boundary conditions and non-linear temperature profile

  • E. M. Sparrow (a1), R. J. Goldstein (a1) and V. K. Jonsson (a1)


An investigation is carried out to determine the conditions marking the onset of convective motion in a horizontal fluid layer in which a negative temperature gradient occurs somewhere within the layer. In such cases, fluid of greater density is situated above fluid of lesser density. Consideration is given to a variety of thermal and hydrodynamic boundary conditions at the surfaces which bound the fluid layer. The thermal conditions include fixed temperature and fixed heat flux at the lower bounding surface, and a general convective-radiative exchange at the upper surface which includes fixed temperature and fixed heat flux as special cases. The hydrodynamic boundary conditions include both rigid and free upper surfaces with a rigid lower bounding surface. It is found that the Rayleigh number marking the onset of motion is greatest for the boundary condition of fixed temperature and decreases monotonically as the condition of fixed heat flux is approached. Non-linear temperature distributions in the fluid layer may result from internal heat generation. With increasing departures from the linear temperature profile, it is found that the fluid layer becomes more prone to instability, that is, the critical Rayleigh number decreases.



Hide All
Bénard, H. 1900 Tourbillions cellulaires dan une nappe liquids. Rev. gén. Sci. pur. appl. 11, 126171, 130928.
Block, M. J. 1956 Surface tension as the cause of Bénard cells and surface deformation in a liquid film. Nature, Lond., 178, 650651.
Goldstein, A. W. 1959 Stability of a horizontal fluid layer with unsteady heating from below and time dependent body forces. NASA Tech. Rep. no. R-4.
Jeffreys, H. 1926 The stability of a layer of fluid heated below. Phil. Mag., 2, 83344.
Jeffreys, H. 1928 Some cases of instability in fluid motion. Proc. Roy. Soc. A, 118, 195208.
Lin, C. C. 1958 The Theory of Hydrodynamic Stability, pp. 1069. Cambridge University Press.
Low, A. R. 1929 On the criterion for stability of a layer of viscous fluid heated from below. Proc. Roy. Soc. A, 125, 18095.
Ostrach, S. 1957 Convection phenomena in fluids heated from below. Trans. A.S.M.E., 79, 299305.
Pearson, J. R. A. 1958 On convection cells induced by surface tension. J. Fluid Mech., 4, 489500.
Pellew, A. R. & Southwell, R. V. 1940 On maintained convective motion in a fluid heated from below. Proc. Roy. Soc. A, 176, 31243.
Rayleigh, Lord 1916 On convection currents in a horizontal layer of fluid when the higher temperature is on the under side. Phil. Mag. 32, 52946.
Sani, R. 1963 Convective instability. Ph.D. Thesis in Chemical Engineering, University of Minnesota.
Thomson, J. 1982 On a changing tessellated structure in certain liquids. Proc. Glasgow Phil. Soc. 13, 469.
MathJax is a JavaScript display engine for mathematics. For more information see


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed