Benard problem

Eidson, T. M. (1985). Numerical simulation of the turbulent Rayleigh-Benard problem using subgrid modelling. Journal of Fluid Mechanics 158, 245-268. 

Previous computations (189) show that the critical value of Rat for non-Boussinesq conditions is approximately the same as that for a Boussinesq fluid in a box heated from below, at least when H2 is the carrier gas. Thus, results from the stability analysis of the classical Rayleigh-Benard problem of a two-dimensional fluid layer heated from below (see reference 190 for a review) may be used to indicate the type of behavior to be expected in a horizontal reactor with insulated side walls. As anticipated from this analysis, an increase in the reactor height from 2 to 4 cm raises the value of Rat to 4768, which is beyond the stability limit, Rat critical = 2056, for a box of aspect ratio 2 (188). The trajectories show the development of buoyancy-driven axial rolls that are symmetric about the midplane and rotating inward. For larger values of Rat (>6000), transitions to three-dimensional or time-de-... 

HYDRODYNAMIC AND HYDROMAGNETIC STABILITY. S. Chandrasekhar. Lucid examination of the Rayleigh-Benard problem clear coverage of the theory of instabilities causing convection. 704pp. 5b x 8b. 64071-X Pa. 12.95... 

M. Renardy and Y. Renardy, Pattern selection in the Benard problem for a viscoelastic fluid, Z. Angew. Math. Mech., 43 (1992) 154-180. 

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

The boundary conditions depend, of course, on the nature of the flat surfaces at z = 0 and 1. We shall return to conditions for the velocity components shortly. It may be noted that the Rayleigh Benard problem has historically been stated in a slightly different form, which is equivalent to rescaling 0 to 0 = O/Pr. The difference is that (12 181) and (12 184) then become... 

Figure 12-5. Stability criteria for the Rayleigh-Benard problem. The two curves shown are the neutral stability curves for the modes n = 1 and n = 2. The region above the curve for n = I is unstable, whereas that below is stable. The critical Rayleigh number is 657.511 at a critical wave number of 2.221...

It is worthwhile illustrating the proof of the principle of exchange of stabilities for the Rayleigh-Benard problem. Not only will this allow us to discuss the derivation of instability criteria for the case of no-slip boundaries, but the approach to proving this principle can also be applied to other problems. 

As in the case of the Rayleigh-Benard problem, there is a steady-state solution of these equations. 

As in the Rayleigh Benard problem, the boundary conditions for two free surfaces are... 

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 governing equations for the linear stability theory are the same as for the Rayleigh-Benard problem, namely (12-215), except that it is customary to drop the buoyancy terms because these are of secondary importance for very thin fluid layers where Marangoni instabilities are present but Ra <neutral state. Assuming that... 

The boundary conditions at the lower solid boundary are the same as for the Rayleigh Benard problem,... 

Problem 12-11. Marangoni Instability (The Principle of Exchange of Stabilities). Following the procedure that was outlined in Section F for the Rayleigh-Benard problem, prove that the principle of exchange of stabilities is valid for the Marangoni instability problem (Section H). 

Problem 12-17. Buoyancy-Driven Instability of a Fluid Layer in a Porous Medium Based on Darcy s Law. We consider the classical Rayleigh-Benard problem of a fluid layer that is heated from below, except in this case, the fluid is within a porous medium so that the equations of motion are replaced with the Darcy equations, which were discussed in Subsection Cl of this chapter. Hence the averaged velocity within the porous medium is given by Darcy s law... 

Analyze the linear stability of the fluid layer. The analysis is similar to the classical Rayleigh-Benard problem, but simpler because the Navier-Stokes equations are replaced with Darcy s law. It can be solved analytically. In place of the Rayleigh number, you should find that the stability depends on a modified Rayleigh number,... 

As noted above, these authors also proved the general validity, for the Rayleigh problem, of the principle of exchange of stabilities. Further, by formulating the problem in terms of a variational principle, Pellew and South-well devised a technique which led to a very rapid and accurate approximation for the critical Rayleigh number. Later, a second variational principle was presented by Chandrasekhar (C3). A review by Reid and Harris (R2) also includes other approximate methods for handling the Benard problem with solid boundaries. 

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]. 

KAPRAL - What is the nature of the microscopic boundary condition used in the nonequilibrium molecular dynamics simulation of the Benard problem in order to obtain the temperature gradient How far into the fluid does the temperature boundary layer extend ... 

MARESCHAL - In the Benard problem, the thermal boundaries are simulated along the ways developped in non-equilibrium molecular dynamics, using stochastic boundary conditions (see G. Ciccotti). The boundary layer does not extend over more than a mean free path in the system and can hardly be seen in our measurements. 

Let us consider a plane sandwich cell into which the cathode injects electrons that negatively charge neutral molecules. Due to an excess negative charge density Q(z) near the cathode, a force QE is directed to the anode and tries to shift the charged layer of a liquid to the right (Fig. 29a). Since the cell is sealed and the liquid is incompressible, a circular convective flow occurs in order to reduce the internal pressure (Fig. 29b, c). This case resembles the well-known Benard problem in the thermoconvection of liquids. 

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