Singular perturbation methods, such as the method of multiple scales and the method of matched asymptotic expansions, give series in a small parameter e which are asymptotic but (usually) divergent. In this survey, we use a plethora of examples to illustrate the cause of the divergence, and explain how this knowledge can be exploited to generate a 'hyperasymptotic' approximation. This adds a second asymptotic expansion, with different scaling assumptions about the size of various terms in the problem, to achieve a minimum error much smaller than the best possible with the original asymptotic series. (This rescale-and-add process can be repeated further.) Weakly nonlocal solitary waves are used as an illustration.

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The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series

An exact solution to the Navier–Stokes equations for the flow in a channel or tube with an accelerating surface velocity is presented. By means of a similarity transformation the equations of motion are reduced to a single ordinary differential equation for the similarity function which is solved numerically. For the two-dimensional flow in a channel, a single solution is found to exist when the Reynolds number R is less than 310. When R exceeds 310, two additional solutions appear and form a closed branch connecting two different asymptotic states at infinite R. The large R structure of the solutions consists of an inviscid fluid core plus an O(R−1) thin boundary layer adjacent to the moving wall. Matched-asymptotic-expansion techniques are used to construct asymptotic series that are consistent with each of the numerical solutions.For the axisymmetric non-swirling flow in a tube, however, the situation is quite different. For R [Lt ] 10[sdot ]25, two solutions exist which form a closed branch. Beyond 10[sdot ]25, no similarity solutions exist within the range 10[sdot ]25 0. These solutions, however, do not evolve from the R = 0 state nor do they bifurcate from the non-swirling solutions at any finite value of R.

Steady flow in a channel or tube with an accelerating surface velocity. An exact solution to the Navier—Stokes equations with reverse flow

Techniques for computational fluid dynamics modelling of flow in membrane channels

http://majdalani.eng.auburn.edu/publications/pdf/2002%20-%20Majdalani%20-%20JB%20-%20Two-dimensional%20viscous%20flow%20between%20slowly%20expanding%20or%20contracting%20walls%20with%20weak%20permeability.pdf

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 a: Results are correlated and compared via variations in R and a: Under the auspices of small jRj and jaj; 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. r 2002 Elsevier Science Ltd. All rights reserved.

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

Numerous investigations ([1] and [4–9]) have been made of laminar flow in a uniformly porous circular pipe with constant suction or injection applied at the wall. The object of this paper is to give a complete analysis of the numerical and theoretical solutions of this problem. It is shown that two solutions exist for all values of injection as well as the dual solutions for suction which had been noted by previous investigators. Analytical solutions are derived for large suction and injection; for large suction a viscous layer occurs at the wall while for large injection one solution has a viscous layer at the centre of the channel and the other has no viscous layer anywhere. Approximate analytic solutions are also given for small values of suction and injection.

On laminar flow through a uniformly porous pipe

The problem of laminar flow through a porous annulus with constant velocity of suction at the walls and with swirl is reduced to the solution of four non-linear differential equations. The significance of each of these equations is discussed. By taking the swirl to be zero series solutions are obtained for (i) small suction or blowing (ii) when the total flow into the channel through the walls is small. Finally the asymptotic behaviour of the flow for large suction or blowing is discussed.

Flow through a porous annulus

The steady laminar flow of a viscous incompressible fluid through a two-dimensional channel, having fluid sucked or injected with different velocities through its uniformly porous parallel walls is considered. A solution for small suction Reynolds number has been given by the authors in a previous paper. The purpose of this paper is to present a solution valid for large Reynolds numbers for the cases of (i) suction at both walls, and (ii) suction at one wall and injection at the other. A technique of matching outer and inner expansions is used to obtain an asymptotic solution for both of these cases. Further a perturbation solution for the case of suction at one wall and injection at the other is obtained by choosing the difference between two wall velocities as the perturbation parameter. Both asymptotic and perturbation solutions are confirmed by exact numerical solutions. As expected, the resulting solutions show the presence of the usual suction boundary layers in both types of flow considered in this paper.

Laminar flow through a channel with uniformly porous walls of different permeability

The problem of heat transfer in a two-dimensional porous channel has been discussed by Terrill [6] for small suction at the walls. In [6] the heat transfer problem of a discontinuous change in wall temperature was solved. In the present paper the solution of Terrill for small suction at the walls is revised and the whole problem is extended to the cases of large suction and large injection at the walls. It is found that, for all values of the Reynolds number R, the limiting Nusselt number Nu
∞ increases with increasing R.

R. M. Terrill

Papers

On laminar flow through a uniformly porous pipe

Flow through a porous annulus

Laminar flow through a channel with uniformly porous walls of different permeability

Heat and mass transfer in laminar flow between parallel porous plates