Brinkman Electro-Osmotic Crossflow in a Microtube with a Hydrophobic and Randomly Corrugated Wall

A second-order domain perturbation analysis is developed for electro-osmotic transport through a long, rigid microtube containing a Newtonian electrolyte and a homogeneous porous matrix. The wall radius varies along the tube axis, forming axisymmetric circumferential corrugations represented by a zero-mean stationary random process. Momentum transport is described by the Darcy–Brinkman model, electrostatic screening by the linearized Poisson–Boltzmann equation, and tangential wall mobility by a traction-based slip condition. Spectral expressions are obtained for the mean pressure-gradient, flow-rate and streaming-potential corrections. The matched-screening case α = k is treated using an alternative finite spatial basis. Sinusoidal and trapezoidal profiles illustrate how roughness amplitude and harmonic content affect the normalized transport corrections; comparisons distinguish equal peak amplitude from equal root-mean-square displacement. The smooth, clear-fluid, no-slip limit recovers the Rice–Whitehead solution. For a selected sinusoidal case with finite porous resistance and slip, comparison with an independent finite difference solution gives a mean pressure-gradient discrepancy of approximately 0.0791% at ϵ = 0.10 under prescribed electrical forcing. The formulation provides asymptotic transport predictions for sufficiently small wall displacement and slope, low wall potential and creeping flow. Its prospective use is the assessment of roughness effects in porous electro-osmotic liquid-delivery passages within these restrictions.