Conjectures on certain boundary-layer equations and natural coordinates

Abstract

The problem of exclusion of steady downward unsaturated seepage from underground cavities is reducible to a linear convection—diffusion equation with a no normal-flux condition at the cavity surface. Various exact solutions indicate that a roof boundary-layer analysis centred on the upstream stagnation point, and neglecting peripheral variation, gives to remarkable accuracy the quantity θmax, the crucial dimensionless potential determining whether or not water enters the cavity. The great accuracy of this analysis is attributed to the use of curvilinear coordinates natural to the cavity configuration. Global information (such as up to three separate characteristic lengthscales) is injected into the localized boundary-layer formulation via the metric coefficient of the natural coordinates. These are essential to the boundary-layer analysis. Cartesian coordinates, on the other hand, invariably suggest that no boundary layer exists! Definition of the natural coordinates is discussed and means of constructing them about arbitrary cavities are developed. Results for smooth cavities support the conjecture that roof geometry near the upstream stagnation point largely determines θmax, with downstream details unimportant. Comparison of solutions for flat-roofed rectangular and cylindrical cavities with those for strips and discs indicate, however, that the conjecture applies only in weak form to cavities of polygonal cross-section.