Poiseuille flow with stochastically perturbed boundary

We investigate, for the first time, the influence of stochastically perturbed boundaries on the flow of an incompressible Newtonian fluid. Two canonical configurations are examined: planar channel flow and axisymmetric pipe flow. The boundary perturbations are modeled as zero-mean stochastic processes, incorporated via a deterministic framework in which the characteristic spatial frequency of the boundary roughness is assumed to be asymptotically large compared to the intrinsic length scales of the flow. Asymptotic expansions for the pressure field are derived under these assumptions, and the implications of the stochastic boundary forcing on the global discharge are analyzed.