This article describes a self-similarity solution of the Navier-Stokes equations for a laminar, incompressible, and time-dependent flow that develops within a channel possessing permeable, moving walls. The case considered here pertains to a channel that exhibits either injection or suction across two opposing porous walls while undergoing uniform expansion or contraction. Instances of direct application include the modeling of pulsating diaphragms, sweat cooling or heating, isotope separation, filtration, paper manufacturing, irrigation, and the grain regression during solid propellant combustion. To start, the stream function and the vorticity equation are used in concert to yield a partial differential equation that lends itself to a similarity transformation. Following this similarity transformation, the original problem is reduced to solving a fourth-order differential equation in one similarity variable that combines both space and time dimensions. Since two of the four auxiliary conditions are of the boundary value type, a numerical solution becomes dependent upon two initial guesses. In order to achieve convergence, the governing equation is first transformed into a function of three variables: The two guesses and. At the outset, a suitable numerical algorithm is applied by solving the resulting set of twelve first-order ordinary differential equations with two unspecified start-up conditions. In seeking the two unknown initial guesses, the rapidly converging inverse Jacobian method is applied in an iterative fashion. Numerical results are later used to ascertain a deeper understanding of the flow character. The numerical scheme enables us to extend the solution range to physical settings not considered in previous studies. Moreover, the numerical approach broadens the scope to cover both suction and injection cases occurring with simultaneous wall motion.

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We 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

This paper focuses on the theoretical treatment of the laminar, incompressible, and time-dependent flow of a viscous fluid in a porous channel with orthogonally moving walls. Assuming uniform injection or suction at the porous walls, two cases are considered for which the opposing walls undergo either uniform or nonuniform motions. For the first case, we follow Dauenhauer and Majdalani ͓Phys. Fluids 15, 1485 ͑2003͔͒ by taking the wall expansion ratio ␣ to be time invariant and then proceed to reduce the Navier-Stokes equations into a fourth order ordinary differential equation with four boundary conditions. Using the homotopy analysis method ͑HAM͒, an optimized analytical procedure is developed that enables us to obtain highly accurate series approximations for each of the multiple solutions associated with this problem. By exploring wide ranges of the control parameters, our procedure allows us to identify dual or triple solutions that correspond to those reported by Zaturska et al. ͓Fluid Dyn. Res. 4, 151 ͑1988͔͒. Specifically, two new profiles are captured that are complementary to the type I solutions explored by Dauenhauer and Majdalani. In comparison to the type I motion, the so-called types II and III profiles involve steeper flow turning streamline curvatures and internal flow recirculation. The second and more general case that we consider allows the wall expansion ratio to vary with time. Under this assumption, the Navier-Stokes equations are transformed into an exact nonlinear partial differential equation that is solved analytically using the HAM procedure. In the process, both algebraic and exponential models are considered to describe the evolution of ␣͑t͒ from an initial ␣ 0 to a final state ␣ 1. In either case, we find the time-dependent solutions to decay very rapidly to the extent of recovering the steady state behavior associated with the use of a constant wall expansion ratio. We then conclude that the time-dependent variation of the wall expansion ratio plays a secondary role that may be justifiably ignored.