Asymptotic methods domain perturbation method

We could obtain a general solution of Eq. (4-94). However, there is no obvious way to apply the boundary condition at the channel wall, at least in the general form (4-95) or (4-96). The method of domain perturbations provides an approximate way to solve this problem for e << 1. The basic idea is to replace the exact boundary condition, (4-96), with an approximate boundary condition that is asymptotically equivalent for e 1 but now applied at the coordinate surface y = d/2. The method of domain perturbations leads to a regular perturbation expansion in the small parameter s. 

The interface boundary conditions, (6-139)-(6-141), are applied atz = h. However, because we now assume that h takes the asymptotic form, (6 159a), it is convenient to use the method of domain perturbations to transform these conditions into asymptotically equivalent boundary conditions applied at the undeformed surface, z = 1. 

Inside the drop, we require that the velocity and pressure fields be bounded at the origin [which is a singular point for the spherical coordinate system that we will use to solve (7 199)]. Finally, at the drop surface, we must apply the general boundary conditions at a fluid interface from Section L of Chap. 2. However, a complication in using these boundary conditions is that the drop shape is actually unknown (and, thus, so too are the unit normal and tangent vectors n and t and the interface curvature V n). As already noted, we can expect to solve this problem analytically only in circumstances when the shape of the drop is approximately (or exactly) spherical, and, in this case, we can use the method of domain perturbations that was first introduced in Chap. 4. In this procedure, we assume that the shape is nearly spherical, and develop an asymptotic solution that has the solution for a sphere as the first approximation. An obvious question in this case is this When may we expect the shape to actually be approximately spherical ... 

Ca <very small, and we shall see that we can obtain an approximate analytic solution using the asymptotic method of domain perturbations. 

There are a number of variations for the perturbation technique. Among them, the method of matched asymptotic expansion is the easiest to apply. The method is useful for obtaining expansions from separate domains of validity. 

In this chapter, we will consider the boundary layer function method (or, in short, boundary function method), which gives the possibility of constructing a uniform asymptotic approximation (or, briefly, uniform asymptotics) for the solution u e) of the singularly perturbed problem (i.e., an asymptotic approximation in the whole domain D). 

Beginning with this section, we will now consider partial differential equations. Let us consider a general approach for constructing asymptotic expansions of the solutions of singularly perturbed linear partial differential equations, which was proposed in the well-known fundamental work of Vishik and Lyusternik [27]. We will illustrate the idea of this approach (known in the literature as the method of Vishik-Lyusternik) on a simple example of an elliptic equation in a bounded domain with smooth boundary. 

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