Steady channel flow of a Bingham fluid with non-flat walls: Analytical solution up to first-order perturbation terms

This work investigates the steady flow of an incompressible Bingham fluid in a planar channel with non-flat walls. By exploiting a small-aspect-ratio assumption, we construct an asymptotic expansion in the geometric parameter ɛ and derive the full solution up to first order. At leading order, the analysis recovers the classical structure with a continuous rigid plug, whose velocity is determined through a nonlinear algebraic equation derived from the momentum balance in the unyielded region. The first-order problem is shown to be fully linear, with all coefficients depending solely on the zeroth-order solution. Explicit formulas for the first-order corrections to the velocity field, pressure gradient, and yield surface are obtained. Numerical tests demonstrate that these corrections are typically negligible but may reach up to 10% under specific parameter combinations. The results clarify the robustness and limitations of lubrication-type approximations for viscoplastic flows in channels of variable width.