Gradients of Potential Fields on Rough Surfaces: Perturbative Calculation of f(a) for Small Surface Dimension

Kausik Sarkar & Charles Meneveau
Department of Mechanical Engineering
The Johns Hopkins University | Baltimore, MD 21218

ABSTRACT: An exact solution to the Laplace equation with Dirichlet boundary condition on a simple boundary is used in an iterative fashion to study the case of a stochastic rough surface with fractal dimension D=2+e. For small e, an analytic expression is derived for the spectrum of singularities f(a) of the gradient of the potential normal to the boundary, using the random multiplier approach. A binomial approximation to the multiplicative process is shown to severely underestimate f(a) for low values of a.

Phys. Rev. E, 47, pp. 957-966

DOI: 10.1103/PhysRevE.47.957 | Full PDF
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