The Disturbance Equations and Boundary Conditions

The Rayleigh Benard configuration is sketched in Fig. 12 4. We assume that the fluid is between two infinite plane surfaces that are separated by a distance d. The lower surface is at a constant temperature i and the upper one is at a lower temperature T0. Our starting point is the time-dependent Boussinesq equations, (12-160), (12 169), and (12-170), but now in dimensionless form, 

Here d is the characteristic length scale, the characteristic velocity scale is denoted as uc but is left unspecified for the moment, the characteristic time scale is d uc, and 

If we examine (12 174) more closely, we see that there are several possible choices for the characteristic velocity. An apparently convenient choice is to designate uc = vo/d as this appears to reduce the number of parameters to the greatest extent. This choice of characteristic velocity is really saying that the characteristic time scale is the diffusion time d2/v0. The governing equations, (12-173) and (12 174), then become 

The new dimensionless parameter that appears in these equations is known as the Grashof number, 

The first thing to notice is that there is a steady-state solution for arbitrary values of the dimensionless parameters, namely, 

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