Rayleigh-Benard convection, heating from below

BENARD CONVECTION CELLS. When a layer of liquid is heated from below, the onset of convection is marked by the appearance of a regular array of hexagonal cells, the liquid rising in the center and falling near the wall of each cell. The criterion for the appearance of the cells is that the Rayleigh number should exceed 1700 (for rigid boundaries). 

F. NATURAL CONVECTION IN A HORIZONTAL FLUID LAYER HEATED FROM BELOW-THE RAYLEIGH-BENARD PROBLEM... 

In this section, we consider the classic problem of a fluid layer of depth d, with an upper surface that is an interface with air that is maintained at an ambient temperature 7o. The fluid layer is heated from below, and we shall assume that the lower fluid boundary is isothermal with temperature T (> To). This problem sounds exactly like the Rayleigh-Benard problem with a free upper surface. However, we consider the fluid layer to be very thin (i.e., d small) so that the Rayleigh number, which depends on d3, is less than the critical value for this configuration. Nevertheless, as previously suggested, the fluid layer may still undergo a convective motion that is due to Marangoni instability. 

The most well-known example of pattern formation is Rayleigh-Benard convection which appears when a fluid layer is uniformly heated from below... 

Let us first recall briefly the classical Benard-Rayleigh problem of thermal convection in an isotropic liquid. When a horizontal layer of isotropic liquid bounded between two plane parallel plates spaced d apart is heated from below, a steady convective flow is observed when the temperature difference between the plates exceeds a critical value A 7. The flow has a stationary cellular character with a spatial periodicity of about 2d. The mechanism for the onset of convection may be looked upon as follows. A fluctuation T in temperature creates warmer and cooler regions, and due lO buoyancy effects the former tends to move upwards and the latter downwards. When AT < AT, the fluctuation dies out in time because of viscous effects and heat loss due to conductivity. At the threshold the energy loss is balanced exactly and beyond it instability develops. Assuming a one-dimensional model in which T and the velocity (normal to the layer) vary as exp (i j,y) with x ji/rf, the threshold is given by the dimensionless Rayleigh number... 

Secondly, heat is transferred towards the surface, within the 0.4 mm mixed con-duction/convection layer, via a very large temperature gradient of the order of 5000-10,000 K/m, by a relatively weak thermal process. With a high thermal impedance, the process consists of a static thermal conductance enhanced about 1.5-2.5 times by penetration of some of the intermittent convection from the Rayleigh/Benard convection below. 

If either the monomer or the polymer, or both, are liquid natural convection, caused by the heat liberated by the exothermic reaction, can occur. Consider first the case when the monomer is liquid and the polymer is solid (cf. Section 1). We will discuss separately upward and downward propagating fronts. If the front propagates upward, then the chemical reaction heats the monomer from below which reminds of the classical Rayleigh-Benard problem. If the Rayleigh number is sufficiently large, then the planar front loses its stability and stationary natural convection above the front occurs. For descending planar fronts there is no such convective instability. An approximate analytical approach allows one to find stability conditions for the propagating reaction front and to determine the modes which appear when the front loses stability [22]. 

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