A viscous fluid does not slip past a fractal free surface

Two-dimensional channel flow past the surface of an extremely polydisperse foam can behave as if the foam surface is a smooth plane subjected to the no-slip boundary condition, even when the flowing fluid in fact experiences zero tangential stress at the surface. This limiting behavior obtains whenever (1) The actual foam surface approximates a "fat'' fractal, a set characterized by Hausdorff dimension between 1 and 2. And (2) The channel height is large compared with surface roughness. And (3) The flow near the boundary is a slow viscous flow.

Two powerful mathematical results, the measurable Riemann mapping theorem and Perron's solution of Dirichlet's problem, guarantee that a solution of the fractal boundary problem exists and is unique. Fourier series representation of the worst case bound shows that the disturbances introduced by the foam surface invariably decay exponentially away from the surface. As a result, except within a thin "boundary layer'', a linear shear velocity profile across the channel obtains. Had the foam surface been smooth and planar, as usual in textbook channel flow, the solution would be a parabolic velocity profile. Not a linear shear.