Asymmetrical suction and injection in laminar channels with porous walls: a fixed point approach

The problem of laminar flow in a rectangular channel with a pair of porous walls is considered. The porous walls allow fluid to be injected into or sucked out of the channel at constant velocities normal to the walls; the velocities at each wall are not necessarily of equal magnitude nor symmetrical in direction. In this article, a unique solution to this problem is shown to exist for sufficiently low Reynolds numbers through the application of Banach's fixed point theorem. This serves to further the discussion about the uniqueness of solutions for this problem, whilst also demonstrating the suitability of a fixed point approach to this family of fluid dynamics problems. References S. S. Almuthaybiri and C. C. Tisdell. Laminar flow in channels with porous walls: Advancing the existence, uniqueness and approximation of solutions via fixed point approaches. J. Fixed Point Theory Appl. 24.3, 55 (2022). doi: 10.1007/s11784-022-00971-8 A. S. Berman. Laminar flow in channels with porous walls. J. Appl. Phys. 24.9 (1953), pp. 1232–1235. doi: 10.1063/1.1721476 H. Guo, C. Gui, P. Lin, and M. Zhao. Multiple solutions and their asymptotics for laminar flows through a porous channel with different permeabilities. IMA J. Appl. Math. 85.2 (2020), pp. 280–308. doi: 10.1093/imamat/hxaa006 A. J. Jerri. Introduction to Integral Equations with Applications. New York: Dekker Inc., 1985, p. 254 R. M. Terrill and G. M. Shrestha. Laminar flow through parallel and uniformly porous walls of different permeability. Z. Angew. Math. Phys. 16.4 (1965), pp. 470–482. doi: 10.1007/BF01593923 F. M. White, B. F. Barfield, and M. J. Goglia. Laminar flow in a uniformly porous channel. J. Appl. Mech. 25.4 (1958), pp. 613–617. doi: 10.1115/1.4011881