Laminar Flow in a Uniformly Porous Channel with Large Injection

Abstract

SummaryThe author has previously discussed the flow in a two-dimensional channel with porous walls through which fluid is uniformly injected or extracted. Yuan has given a solution for high injection in which certain derivatives of the axial velocity become infinite at the centre of the channel. Physically this means that there must be a viscous layer at the centre of the channel and that Yuan’s solution is ignoring the shear layer. In this paper the complete solution for large injection is obtained by the method of inner and outer expansions and therefore includes the viscous layer. The resulting series solutions are confirmed by numerical results.

Asymptotic solutions for laminar flow based on blood circulation through a uniformlyporous channel with retractable walls and an applied transverse magnetic field

“…However, here the unboundedness does not seem to occur in practical application. Later, the existence of a viscous shear layer, pointed out by Terrill [40], would result in the appearance of the unboundedness. To eliminate the unboundedness in f ′′′ , Zhou and Majdalani [38] obtained a uniformly valid composite solution using matched-asymptotic expansions with logarithmic corrections.…”

Internal Flows Driven by Wall-Normal Injection

“…This of course signaled the presence of a thin boundary layer that necessitated special treatment. The corresponding boundary layer would later be captured by Terrill (1965) who also described an insightful technique to solve this problem numerically. In the interim, Berman (1958a) published his second work in which he extended the original planar problem to various geometric settings.…”

“…Despite the accuracy of their technique, their power series depended on two arbitrary constants that could only be determined numerically through a trial and error procedure. According to Terrill (1964), their method could be viewed as suitable for intermediate values of Re (15 ≤ Re ≤ 35). Otherwise, a transformation of the governing equation could be more effective at achieving direct numerical integration.…”

“…Nonetheless, it remained somewhat unique in its ability to provide a single analytical approximation that applied over the entire range of Re, a feat that standard perturbation methods failed to accomplish. Along similar lines, Terrill (Terrill, 1964;1965) compiled a comprehensive and detailed résumé of the perturbation solutions of this problem over all ranges of the Reynolds number. Therein, he derived and discussed several limiting cases such as Re = 0, |Re|≪1, Re → +∞,R e→ −∞, and compared the various solutions with numerical simulations based on Runge-Kutta integration.…”

“…As for the case of large suction, Berman (1953) first remarks that the limit of the reduced ODE cannot be used to obtain a solution owing to the reduction in order of the governing equation. Later, Sellars (1955) and Terrill (1964) invoke a procedure that permits the extraction of a closed-form analytical approximation for the large Re case by implementing a coordinate transformation that takes into account the spatial relocation of the boundary layer to the sidewall region. It is widely believed that Berman (1953) was among the earliest to examine the problem of laminar viscous flow bounded by porous surfaces (see Dauenhauer & Majdalani, 2003;Zhou & Majdalani, 2002).…”