Moderate‐to‐large injection and suction driven channel flows with expanding or contracting walls

Abstract

AbstractWe consider in this paper the incompressible laminar flow in a porous channel with expanding or contracting walls. While the head‐end is closed by a compliant membrane, the downstream end is left unobstructed. For symmetric injection or suction along the uniformly expanding porous walls, the Navier‐Stokes equations are reduced to a single, nonlinear, ordinary differential equation. The latter is obtained via similarity transformations in both time and space. The resulting equation is then solved both numerically and asymptotically, using perturbations in the crossflow Reynolds number R. Two separate approaches are presented for each of the injection and suction cases, respectively. For the large injection case, the governing equation is first integrated and the resulting third‐order differential equation is solved using the method of variation of parameters. For the large suction case, the governing equation is first simplified near the wall and then solved using successive approximations. Results are then correlated and compared for variations in R and the dimensionless wall expansion rate α. For injection‐induced flow, the asymptotic solution becomes more accurate when R/α is increased. Its deviation from the classic sinusoidal profile arising in nonexpanding channels becomes less significant with successive increases in R. For suction‐induced flows, faster wall contractions increase the effective Reynolds number ‐(α+R), thus leading to more precise approximations. For the same absolute value of R, the suction‐flow approximation tends to be the most accurate of the two and the least sensitive to variations in α. As ‐(α+R) is increased, the suction profile approaches the linear form anticipated in nonexpanding channels. By comparison with the injection‐induced flow, suction is characterized by improved accuracy, sharper flow turning, and larger shear.