Two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability

Since the transport of biological fluids through contracting or expanding vessels is characterized by low seepage Reynolds numbers, the current study focuses on the viscous flow driven by small wall contractions and expansions of two weakly permeable walls. The scope is limited to two-dimensional symmetrical solutions inside a simulated channel with moving porous walls. In seeking an exact solution, similarity transformations are used in both space and time. The problem is first reduced to a nonlinear differential equation that is later solved both numerically and analytically. The analytical procedure is based on double perturbations in the permeation Reynolds number R and the wall expansion ratio α. Results are correlated and compared via variations in R and α. Under the auspices of small R and α, the analytical result constitutes a practical equivalent to the numerical solution. We find that, when suction is coupled with wall contraction, rapid flow turning is precipitated near the wall where the boundary layer is formed. Conversely, when injection is paired with wall expansion, the flow adjacent to the wall is delayed. In this case, the viscous boundary layer thickens as injection or expansion rates are reduced. Furthermore, the pressure drop along the plane of symmetry increases when the rate of contraction is increased and when either the rate of expansion or permeation is reduced. As nonlinearity is retained, our solutions are valid from a large cross-section down to the state of a completely collapsed system.