Journal of Fluid Mechanics

A similarity solution for slow viscous flow down an inclined plane

Peter C.  Smith a1
a1 Department of Meteorology, Massachusetts Institute of Technology

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A similarity solution is found for the distribution of layer thickness in viscous source flow down an inclined plane. With this soldion a t lowest order, an asymptotic expansion in inverse powers of the downstream co-ordinate x allows a small correction for upstream influence. A simple experiment confirms the major features of the similarity solution: (i) a parabolic cross-stream variation in the layer thickness; (ii) spreading of the flow according to an $x^{\frac{3}{7}}$ power law; (iii) thin-ning of the layer along streamlines like $x^{-\frac{1}{7}}$; and (iv) surface velocities which vary as the square of the layer thickness. Deviations of the layer-thickness measurements from the parabolic profile follow the trend predicted by the first-order corrections, whereas systematically high measured values are explained qualitatively in terms of waves at the free surface.

(Published Online March 29 2006)
(Received February 16 1972)
(Revised October 3 1972)