E Shallow-Cavity Flows

Three examples of shallow-cavity flows that we consider in this section are sketched in Fig. 6-7. At the top is the case in which all four boundaries are solid walls, the fluid is assumed to be isothermal, and the motion is driven by tangential motion of the lower horizontal boundary. In the middle, a generalization of this problem is sketched in which the fluid is still assumed to be isothermal and driven by motion of the lower horizontal boundary, but the upper boundary is an interface with air that can deform in response to the flow within the cavity. Finally, the lower sketch shows the case in which fluid in the shallow cavity is assumed to have an imposed horizontal temperature gradient, produced by holding the end walls at different, constant temperatures, and the motion is then driven by Marangoni stresses on the upper interface. In the latter case, there will also be density gradients that can produce motion that is due to natural convection, but this contribution is neglected here (however, see Problem 6-13.) 

The last of the three problems in Fig. 6-7 is qualitatively related to the thermocapillary flows thatare important in the processing of single crystals for microelectronics applications. A typical configuration is sketched in Fig. 6 8, in which a cylindrical solid passes through a heating coil (a furnace), is melted, and then resolidifies into a single crystal of high quality. 

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Equation (6-148), plus the boundary conditions (6 142) and the integral constraint (6 143), is sufficient to determine h(x). We should note that we do not necessarily expect Eq. (6-148) to hold all the way to the end walls atx = 0 andx = 1, for it was derived by means of the governing equation, (6-119), (6-120) and (6-137), and these are valid only for the core region of the shallow cavity. Nevertheless, we will at least temporarily ignore this fact and integrate (6-148) over the whole domain, with the promise to return to this issue later. Qualitatively, we can see that the interface deformation is determined by a balance between the nonuniform pressure associated with the flow in the cavity, e g., Eq. (6 145), which tends to deform the interface, and the effects of capillary and gravitational forces, both of which tend to maintain the interface in its flat, undeformed state, i.e., h = 1. 

Problem 6-13. Buoyancy-Driven Circulation in a Shallow Cavity. A variation on the problems discussed in Section E is to consider the circulation of fluid within a shallow cavity driven by buoyancy effects when there is a temperature difference between the two ends of the cavity. This has been used as a model problem for the recirculation flows within an estuary. 

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