Rayleigh problem

There are some small differences between conventions for spherical harmonics Yim(6,(p) in different texts (following most chemists, we use the so-called "Condon and Shortley" [10] convention). Note These functions are also the solutions to the Rayleigh problem of the normal modes of waves on a flooded planet (s, p, d,f functions), and they also occur in the study of earthquakes. 

Figure 3-11. (a) The velocity profiles at various increasing times above an infinite flat plate that suddenly begins moving in its own plane at t = 0 with a constant velocity U (the Rayleigh problem), (b) The selfsimilar velocity profile for the same problem obtained by the rescaling of the distance from the plate y with the diffusion length scale Jvt for all t > 0. 

Thus the Rayleigh problem is reduced to the solution of the ODE, (3-126), subject to the two boundary conditions, (3-127) and (3-128). When a similarity transformation works, this reduction from a PDE to an ODE is the typical outcome. Although this is a definite simplification in the present problem, the original PDE was already linear, and the existence of a similarity transformation is not essential to its solution. When similarity transformations exist for more complicated, nonlinear PDEs, however, the reduction to an ODE is often a critical simplification in the solution process. 

Given the equation and boundary conditions, (3 126), (3 127), and (3 128), the solution of the Rayleigh problem is very simple. Integrating (3 126) twice with respect to i], we obtain... 

Necessary conditions for the existence of a self-similar solution are that (1) the governing PDE must reduce to an ODE for F as a function ot// alone, and (2) the original boundary and initial conditions must reduce to a number of equivalent conditions for F that are consistent with the order of the ODE. Of course, a proof of sufficient conditions for existence of a selfsimilar solution would require a proof of existence of a solution to the ODE and boundary conditions that are derived for F. In general, however, the problems of interest will be nonlinear, and we shall be content to derive a self-consistent set of equations and boundary conditions and attempt to solve this latter problem numerically rather than seeking a rigorous existence proof. Let us see how the systematic solution scheme based on the general form (3-135) works for the Rayleigh problem. 

This is precisely the self-similar solution of the Rayleigh problem, (3-134), which was obtained in the previous section. Notice that the length scale d drops completely out of this limiting form of the solution (3-158). This is consistent with our earlier observation that the presence of the upper boundary should have no influence on the velocity field for sufficiently small times 7 <[Pg.151]

Thus, in the limit e —> 0, the present problem reduces to the Rayleigh problem, with governing equation and boundary/initial conditions given by (3 119) and (3-120b). 

We may start with Eq. (3-175) for the solid phase, expressed (for convenience) in terms of 0S as defined as in (3-178). This equation is, in fact, identical to the governing equation, (3-119), for the Rayleigh problem of Section E. The boundary conditions, also expressed in terms of 6, are... 

In the infinitesimal solid layer at t = 0, the temperature is also the freezing point 6m. Now, in view of the analysis of (3-119) for the Rayleigh problem, it is evident that a general solution of (3-175), expressed in self-similar form, is... 

This form for the solution is consistent with the idea, borrowed from the Rayleigh problem, that the temperature imposed at the base of the container will propagate up the cylinder a distance s/icj by conduction in a time t. Hence this should also be characteristic of the distance that the interface has propagated up into the cylinder in time t. 

Problem 3-10. The Rayleigh Problem with Oscillating Boundary Motion. Consider an incompressible, Newtonian fluid that occupies the region above a single, infinite plane boundary. Beginning at t = 0, this boundary oscillates back and forth in its own plane with a velocity ux = U sin o>t(t > 0). 

Problem 3-18. Rayleigh Problem with Constant Shear Stress. Consider a semi-infinite body of fluid that is bounded below, at y = 0, by an infinite plane, solid boundary. For times l < 0, the fluid and the boundary are motionless. However, for / > 0, the boundary moves in its own plane under the action of a constant force per unit area, F. Assume that the boundary has zero mass so that its inertia can be neglected. 

Problem 3-20. Generalization of the Rayleigh Problem. A plate bounding a fluid initially... 

Problem 3-21. The Rayleigh Problem with Prescribed Boundary Acceleration. An... 

Problem 10-9. Translating Flat Plate. Consider the high-Reynolds-number laminar boundary-layer flow over a semi-infinite flat plate that is moving parallel to its surface at a constant speed (7 in an otherwise quiescent fluid. Obtain the boundary-layer equations and the similarity transformation for f (r ). Is the solution the same as for uniform flow past a semi-infinite stationary plate Why or why not Obtain the solution for f (this must be done numerically). If the plate were truly semi-infinite, would there be a steady solution at any finite time (Hint. If you go far downstream from the leading edge of the flat plate, the problem looks like the Rayleigh problem from Chap. 3). For an arbitrarily chosen time T, what is the regime of validity of the boundary-layer solution ... 

The analysis with disturbance quantities of the form of Eqs. (10.6.15) indicates a periodic structure in the x, z plane but the shape of the cells associated with the solution is not specified and higher order nonlinear theory is required to define a particular cellular structure. Palm (1960) has shown that in the parallel Rayleigh problem for steady buoyancy driven convection of a liquid film heated from below, the cells approach a hexagonal form as a consequence of the variation of the kinematic viscosity with temperature. 

Summary of Boundary Conditions Used in the Rayleigh Problem... 

In 1940, Pellew and Southwell (P3) resolved the Rayleigh problem for free-free, solid-free, and solid-solid boundaries and obtained good... 

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. 

In a recent paper. Sparrow, Goldstein, and Jonsson (Sll) presented the solution to the Rayleigh problem with a radiation boundary condition of the type of Eq. (38) at the upper free surface. Some of their results, for both a constant flux and a constant temperature bottom, are shown in Table IV. Of special interest is their solution for the limiting case of a constant-flux upper surface and a constant-flux bottom, for which the critical Rayleigh number was found to be only 320. 

In Pearson s analysis, the fluid is assumed to be supporting an adverse linear temperature gradient p, and all the fluid properties except for the surface tension a are taken to be independent of temperature. Thus, the governing equations for the present system are identical to those of the Rayleigh problem, except for the absence of the gravitational term, so that, in place of Eq. (29), there results... 

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... 

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