We consider viscous flow and solute transport in a tube with a corrugated surface, whose topology cannot be accurately described due to either a scarcity of data or measurement errors (or, more frequently, both). This uncertainty can be addressed by modeling the geometry of the roughness as a stationary random field. Consequently, flow and transport equations become stochastic. To solve these equations, the irregular boundary of the flow domain is mapped onto a smooth one (no roughness), with the impact of the original geometry encapsulated in the components of the contravariant tensor associated with the mapping. A perturbation expansion is then employed to compute the statistics of the velocity field, which is used to derive the effective viscosity and quantify the diffusion mechanism of an advected passive solute. It is observed that the effective viscosity depends on the degree of disorder in the roughness geometry and, therefore, cannot be regarded solely as a property of the fluid. Instead, the concentration is shown to satisfy a diffusion-type equation with time-dependent effective parameters. In particular, the effective diffusion coefficient converges to a constant (asymptotic) value after a transitional period, which depends on the correlation scale of the random signal describing the surface topology. The analytical nature of the solutions allows for the treatment of surfaces with, among other characteristics, short correlation lengths—an issue that cannot be effectively addressed using Monte Carlo simulations.

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• Laminar flows
• Stokesian dynamics