Viscous flows down an inclined plane from point and line sources

Abstract

The flow of a viscous fluid from a point or line source on an inclined plane is analysed using the equations of lubrication theory in which surface tension is neglected. At short times, when the gradient of the interfacial thickness is much greater than that of the plane, the fluid is shown to spread symmetrically from the source, as on a horizontal plane. At long times, the flow is predominantly downslope, with some cross-slope spreading for the case of a point source. Similarity solutions for the long-time behaviour of the governing nonlinear partial differential equations are found for the case in which the volume of fluid increases with time like tα, where α is a constant. The two-dimensional equations appropriate to a line source are hyperbolic in the self-similar regime and the similarity profile is found analytically to end abruptly at a downslope position which increases like t(2α+1)/3. Inclusion of higher-order terms in the analysis resolves this frontal shock into a boundary-layer structure of width comparable to the thickness of the current. Owing to the term representing cross-slope spreading, the mathematical structure of the equations is considerably more complex for flow from a point source and the similarity form is found numerically in this case. Though the downslope and cross-slope extents of the current again increase with time according to a power-law if α > 0, they also depend on a power of In t if α = 0. The leading-order near-source structure is shown to be that of steady flow from a constant-flux source of strength given by the instantaneous flow rate. For sources with α > 1, the contact line advances at all points on the perimeter of the flow and the entire plane is eventually covered by the flow; for sources with 0 < α < 1, only a portion of the contact line is advancing at any time and only that part of the plane with |y| [les ] cx3α/(4α+3) is eventually covered, where x and y are the downslope and cross-slope coordinates and c is a constant. The theoretical spreading relationships and planforms are found to be in good agreement with experimental measurements of constant-volume and constant-flux flows of viscous fluids from a point source on a plane. At very long times, however, the experimental flows are observed to be unstable to the formation of a capillary rivulet at the nose of the current.