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Boundary Layers
Published in Alan W. Bush, Perturbation Methods for Engineers and Scientists, 2018
Mathematically the occurrence of boundary layers is associated with the presence of a small parameter multiplying the highest derivative in the governing equaton of a process. A straightforward perturbation expansion using an asymptotic sequence in the small parameter leads to differential equations of lower order than the original governing equation. In consequence not all of the boundary or initial conditions can be satisfied by the perturbation expansion. This is an example of what is commonly referred to as a singular perturbation problem. The technique for overcoming the difficulty is to combine the straightforward expansion, which is valid away from the boundary where a condition is not satisfied, with an expansion which is valid within a layer adjacent to this boundary. The straightforward expansion is referred to as the outer expansion. The inner expansion associated with the boundary layer region is expressed in terms of a stretched variable, rather than the original independent variable, which takes due account of the scale of certain derivative terms. The inner and outer expansions are matched over a region located at the edge of the boundary layer. The technique is called the method of matched asymptotic expansions.
Chapter 3: Methods for Second-Order Nonlinear Differential Equations
Published in Andrei D. Polyanin, Valentin F. Zaitsev, Handbook of Ordinary Differential Equations, 2017
Andrei D. Polyanin, Valentin F. Zaitsev
Remark 3.25. The method of matched asymptotic expansions is successfully applied for the solution of various problems in mathematical physics that are described by partial differential equations; in particular, it plays an important role in the theory of heat and mass transfer and in hydrodynamics.
Thermophoresis at small but finite Péclet numbers
Published in Aerosol Science and Technology, 2018
The derivation of Equation (1) is based on the assumption that the Péclet number is identically zero (and the Reynolds number is also zero). For large aerosol particles undergoing thermophoresis in high temperature gradients, values of would be of the order 0.1 and the convective transport of the heat may not be neglected relative to the conduction in the fluid. Even if heat convection plays only a minor role in determining the instantaneous thermophoretic motion of an aerosol particle at small Péclet numbers, it may result in an appreciable cumulative effect over a period of time (Khair 2013). Also, the Péclet number in thermophoresis can never be zero in reality, and the possibility of singular behavior at should be examined to insure that even small values of will not significantly alter the result of Equation (1). These motivate an investigation on the effect of heat convection in the fluid on thermophoresis. The objective of the present work is to analyze the thermophoretic motion of an aerosol sphere when the Péclet number is small but nonzero. A perturbation method of matched asymptotic expansions (Van Dyke 1975; Bender and Orszag 1978; Nayfeh 1981; Leal 1992) will be employed to solve the problem and an approximate expression for the particle velocity up to as a function of the relevant parameters is given by Equation (27) together with Equations (28), (32), (45), and (54).
Two-dimensional shear waves scattering by a large rigidity strip with interface crack
Published in Waves in Random and Complex Media, 2018
Using the method of matched asymptotic expansions with the thickness-to-length ratio as the perturbation parameter, analogously to results obtained previously (see, e.g. [23]), it can be shown that the field satisfies in the domain the Helmholtz equation and the following effective boundary conditions on the interval