The Horizontal Shallow Cavity with a Free Surface

The Horizontal Shallow Cavity with a Free Surface 

A problem that is closely related to the enclosed-cavity problem of the preceding subsection is that sketched in Fig. 6-7(b). The primary difference is that the upper boundary is now assumed to be an interface that can deform as a consequence of the flow within the cavity. As a consequence, the flow will not be unidirectional in the core. 

The boundary conditions atz = 0 are also unchanged and are given by (6 124b). Finally, because of the presence of the impermeable end walls, the zero-flux condition, (6 125), is preserved, though now in the slightly more general form 

The primary new feature is that the boundary conditions (6-124a) are now replaced with boundary conditions for an interface. We assume that the fluid above the interface is air (or some other gas). Hence these boundary conditions can be adopted from Eqs. (6-9) (6-21). The kinematic condition, (6-9), (for a steady interface shape) becomes, 

We assume, for simplicity, that the interface is pinned at the comers of the cavity, i.e., 

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