Probabilistic computation of Poiseuille flow velocity fields

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Abstract
Velocity fields for Poiseuille flow through tubes having general cross section are calculated using a path integral method involving the first‐passage times of random walks in the interior of the cross sectional domain D of the pipe. This method is applied to a number of examples where exact results are available and to more complicated geometries of practical interest. These examples include a tube with ‘‘fractal’’ cross section and open channel flows. The calculations demonstrate the feasibility of the probabilistic method for pipe flow and other applications having an equivalent mathematical description (e.g., torsional rigidity of rods, membrane deflection). The example of flow through a fractal pipe shows an extended region of diminished flow velocity near the rough boundary which is similar to the suppressed vibration observed near the boundaries of fractal drums.

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