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

Abstract

1964). Thermal instability in a horizontal fluid layer: effect of boundary conditions and nonlinear temperature profile.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.

Penetrative ferroconvection via internal heating in a saturated porous layer with constant heat flux at the lower boundary

“…Sparrow et al [39] Present study increase in M 1 leads to either increase in destabilizing magnetic force or decrease in stabilizing viscous force on the system and hence it has a destabilizing effect on the system. A closer inspection of the figure further reveals that the magnetic force is to reinforce together with buoyancy force and to hasten the onset of ferroconvection when compared to their effect in isolation.…”

“…To validate the numerical procedure used in the present study, the critical Rayleigh number R tc and the corresponding critical wave number a c obtained under the limiting case of M 1 ¼0, Da À 1 ¼0 and N s ¼0 (ordinary viscous fluid layer) for different values of Biot number Bi are compared with those of Sparrow et al [39] in Table 1. We note that the agreement is good and thus verify the accuracy of the numerical procedure employed.…”

Electrothermoconvective instability in a heat generating dielectric fluid layer

“…To validate the solution procedure, computations are carried out first under the limiting conditions of R e = 0 and also when N s = 0. The critical Rayleigh number R tc obtained for R e = 0 and for various values of N s are compared with the results of Sparrow et al [10] in Table 1, while R tc obtained for N s = 0 and for different values of R e are compared with those of Roberts [2] in Table 2. From the tables we note that the agreement is excellent and thus verifies the accuracy of the method employed.…”

“…This can aptly be modeled as volumetric heat generation within the dielectric fluid causing basic temperature distribution to be nonlinear which in turn has a profound effect on the stability of the system. The onset of thermal instability in a horizontal ordinary viscous fluid layer, subject to an internal heat generation, has been analyzed by Sparrow et al [10] and Shivakumara and Suma [11]. In the latter paper an additional effect of vertical throughflow is also considered.…”

The Sun’s Supergranulation

“…Even more interestingly, fixed heat flux boundary conditions naturally favour marginally stable convection cells with infinite horizontal extent compared to the layer depth, or convection cells with a very large but finite horizontal extent when a weak modulation of the heat flux is allowed for (Sparrow et al, 1964;Hurle et al, 1967;Van der Borght, 1974;Busse and Riahi, 1978;Chapman and Proctor, 1980;Depassier and Spiegel, 1981). This case is therefore very different from the standard Rayleigh-Bénard case with fixed temperature boundary conditions, which gives rise to cells with aspect ratio of order unity.…”