Probabilistic computation of Poiseuille flow velocity fields

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 B 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.

Food transport in theC. eleganspharynx

“…Equation·13 was solved numerically by simulated diffusion on a lattice. The method is based on that described by Hunt et al (1995), but we made one important change. Hunt et al (1995) calculated the average number of steps that a particle placed in the interior of the lumen takes to reach the wall.…”

“…The method is based on that described by Hunt et al (1995), but we made one important change. Hunt et al (1995) calculated the average number of steps that a particle placed in the interior of the lumen takes to reach the wall. We instead counted the average number of times a diffusing particle hit each lattice point.…”

Anomalous diffusion of fluid momentum and Darcy-like law for laminar flow in media with fractal porosity

“…A viscous fluid flow through porous medium is accompanied by diffusion of the fluid momentum [10,[23][24][25]. In this regard, it is easy to understand that the hydrodynamic properties of different types of fractal materials may be quite different [9].…”

Exact solutions and physical analogies for unidirectional flows

“…In 1871, Boussinesq [13] recognized that de Saint-Venant's theory of torsion is mathematically equivalent to Stokes' theory of Poiseuille flow. Ten years later Heaviside [14] noted the equivalence of beam torsion and the magnetic self-induction of an electrical wire, which was eventually recognized as another analogy for pipe flow [15], along with the "membrane analogy" identified by Prandtl [16] in 1903.…”