Domain perturbations

E. FLOW IN A WAVY-WALL CHANNEL - DOMAIN PERTURBATION METHOD ... 

E. Flow in a Wavy-Wall Channel - 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. 

It will be noted that the dimensionless problem is characterized by two dimensionless parameters, the amplitude of the corrugation e and the ratio of the channel width to the wavelength of the corrugations d/L. The domain perturbation method that we use to solve... 

The procedure just outlined can be extended to calculate higher-order terms in the expansion (4-103). We calculate only one more term at 0(e2). The governing equation and the symmetry condition for w2 are identical to (4-110) and (4-111). The boundary condition at y = 1/2 from the domain perturbation procedure can be obtained from (4-105c) ... 

For this limiting case, the boundary conditions, (4-126), can also be approximated in terms of asymptotically equivalent boundary conditions applied at y = 1 /2. The details follow those of the previous subsection entirely and are not repeated here. We again use the symmetry condition at y = 0 and condition (4-126) at y = 1 /2. The results of applying the domain perturbation analysis to both u and v are... 

We have not chosen to carry out the calculation to higher order in s. At 0(e), there is no change in the volumetric flow rate. However, if we examine the boundary condition for ih at T = 1/2 from the domain perturbation expression [(4-104), with u replacing w], we see that there will be two terms, one proportional to cos(47tx) and the other independent of x. It is thus clear that the volumetric flow rate will change at 0(e2). 

One complication is that the boundary conditions (4-264)-(4-266) must be applied at the bubble surface, which is both unknown [that is, specified in terms of functions R(t) and fn(9,tangent unit vectors n and t, that appear in the boundary conditions are also functions of the bubble shape. In this analysis, we use the small-deformation limit s 1 to simplify the problem by using the method of domain perturbations that was introduced earlier in this chapter. First, we note that the unit normal and tangent vectors can be approximated for small e in the forms... 

To avoid this, we use domain perturbation theory (see Section E) to transform from the exact boundary conditions applied at rs = R + sf to asymptotically equivalent boundary conditions applied at the spherical surface rs = R(t). For example, instead of a condition on ur at r = R(t) + ef from the kinematic condition, we can obtain an asymptotically equivalent condition at r = R by means of the Taylor series approximation... 

The method of domain perturbations was used for many years before its formal rationalization by D. D. Joseph D. D. Joseph, Parameter and domain dependence of eigenvalues of elliptic partial differential equations, Arch. Ration. Mech. Anal. 24, 325-351 (1967). See also Ref. 3f. The method has been used for analysis of a number of different problems in fluid mechanics A. Beris, R. C. Armstrong and R. A. Brown, Perturbation theory for viscoelastic fluids between eccentric rotating cylinders, J. Non-Newtonian Fluid Mech. 13, 109-48 (1983) R. G. Cox, The deformation of a drop in a general time-dependent fluid flow, J. Fluid Mech. 37, 601-623 (1969) ... 

D. D. Joseph, Domain perturbations The higher order theory of infinitesimal water waves, Arch. Ration. Mech. Anal. 51,295-303 (1975) D. D. Joseph and R. Fosdick, The free surface on a liquid between cylinders rotating at different speeds, Arch. Ration. Mech. Anal. 49, 321-81 (1973). 

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. 

The method of domain perturbations was first introduced in Chap. 4. However, the main ideas are repeated here. We begin with the condition (6 139). Introducing the expansions (6-159b), we find that this becomes... 

Hence the velocity and pressure distributions at 0(1) can be completely determined within the core region (away from the end walls), to within an arbitrary constant for p 0). In fact, the flow is a simple unidirectional flow, as is appropriate for traction-driven flow between two plane surfaces. The turning flow that must occur near the ends of the cavity influences the core flow only in the sense that the presence of impermeable end walls requires a pressure gradient in the opposite direction to the boundary motion in order to satisfy the zero-mass-flux constraint. But now, a remarkable feature of the domain perturbation procedure is that we can use our knowledge of the unidirectional flow that is appropriate for an undeformed interface at 0(1) to directly determine the 0(5) contribution to the interface shape function in (6-159a) without having to determine any other feature of the solution at 0(5). 

The first two terms match the result obtained earlier by means of the expansion (6-149) and (6 151) applied to Eq. (6-148). The latter is, however, computed only to terms of 0(Ca/s3), and thus does not contain the third term in (6 185). The effort involved in obtaining (6 185) by means of the domain perturbation technique is, however, greater than the analysis to obtain the result (6-148) by means of the thin-film approach, and the latter does not make any a priori restriction on the shape function h. These observations suggest that the thin-film approach is both simpler and more powerful for this particular class of problems. It should be emphasized, however, that the domain perturbation technique can sometimes yield results when no other approach will work, and it has proven to be an invaluable tool in obtaining analytic solutions for a wide variety of free-boundary problems, both in fluid mechanics and other subjects. 

We may recall from the preceding example that 5 determines the magnitude of the interface deformation. We assume, in what follows, that 5 is no larger than 0(1). In fact, we follow the example of the previous section and consider two distinct cases. First, we use the conventional thin-film analysis to obtain the leading-order term (only) in the thin-film parameter e, but with the possibility of variations in the dimensionless film thickness functions h(x) of 0(1), corresponding to 5 = 0(1). Second, we employ the domain perturbation technique to consider the case of 5 = 0(e) second order in e, thus incorporating the O(sRe) and 0(ePe) terms as indicated in (6 216). 

If we were to extend the solution to include corrections at 0(e), we could use the method of domain perturbations about the 0(1) estimate of the interface shape to obtain interface boundary conditions at 0(e). For example, we can express h x) in the form of an expansion in e,... 

The domain perturbation anaylsis largely follows that from the preceding problem. We assume that... 

As in the domain perturbation example of the preceding problem, we see that 9/ (0)/9x is independent of x, as one would expect in the absence of interface deformation at this order of approximation, and again... 

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 ... 

In the present section, we consider the translational motion of a drop through a quiescent fluid for Ca <very small, and this facilitates the derivation of an approximate analytic solution by the method of domain perturbations. 

The first approximation of the deviation from sphericity is contained in the function f>(, t), and we have anticipated, from the preceding arguments, that the magnitude of term is proportional to Ca. The presumption that the expansion for /(x) proceeds in whose powers of Ca is based on the fact that domain perturbation techniques almost always lead to a regular (rather than singular) asymptotic structure. 

It remains to determine the four sets of constants, An, Bn, Cn, and Dn, and the function fo that describes the 0(Ca) correction to the shape of the drop. For this, we still have the five independent boundary conditions, (7-207)-(7-210). It can be shown that the conditions (7-207), (7-208), and (7-209) are sufficient to completely determine the unknown coefficients in (7-213) and (7-215). Indeed, for any given (or prescribed) drop shape, the four conditions of no-slip, tangential-stress continuity and the kinematic condition are sufficient along with the far-field condition to completely determine the velocity and pressure fields in the two fluids. The normal-stress condition, (7-210), can then be used to determine the leading-order shape function /0. Specifically, we can use the now known solutions for the leading-order approximations for the velocity components and the pressure to evaluate the left-hand side of (7 210), which then becomes a second-order PDE for the function /(). The important point to note is that we can determine the 0(Ca) contribution to the unknown shape knowing only the 0(1) contributions to the velocities and the pressures. This illustrates a universal feature of the domain perturbation technique for this class of problems. If we solve for the 0(Cam) contributions to the velocity and pressure, we can... 

Derive dimensionless equations and boundary conditions whose solution would be sufficient to determine the drop velocity (and shape) to 0(8). Use the method of domain perturbations to express all boundary conditions at the deformed drop interface in terms of equivalent conditions at the spherical surface of the undeformed drop. Show that 5 = Ca. 

Note To apply the reciprocal theorem, the shape of the drop for the complementary problem generally would have to be exactly the same as the shape in the problem of interest. However, because the original problem can be reduced by means of domain perturbations to an equivalent problem with the boundary conditions applied at the spherical surface, r = 1, we may also conveniently choose the drop to be spherical for this complementary problem. 

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