Method of reflections

A number of authors from Ladenburg (LI) to Happel and Byrne (H4) have derived such correction factors for the movement of a fluid past a rigid sphere held on the axis of symmetry of the cylindrical container. In a recent article, Brenner (B8) has generalized the usual method of reflections. The Navier-Stokes equations of motion around a rigid sphere, with use of an added reflection flow, gives an approximate solution for the ratio of sphere velocity in an infinite space to that in a tower of diameter Dr ... 

Boundary effects on the electrophoretic mobility of spherical particles have been studied extensively over the past two decades. Keh and Anderson [8] applied a method of reflections to investigate the boundary effects on electrophoresis of a spherical dielectric particle. Considered cases include particle motions normal to a conducting wall, parallel to a dielectric plane, along the centerline in a slit (two parallel nonconducting plates), and along the axis of a long cylindrical pore. The double layer is assumed to be infinitely thin... 

FIG. 3. Dimensionless electrophoretic velocity U/Uq of a sphere vs. distance parameter A for particle s motion normal to a conducting plane. Solid line is the exact numerical results and dashed line represents the approximate results from the method of reflections. 

An analytical study using the method of reflections was conducted by Chen and Keh [9] to investigate the electrophoretic motion of two freely suspended nonconducting spherical particles with infinitely thin double layer. The particles may differ in size and zeta potential, and they are oriented arbitrarily with respect to the imposed electric field. The resulting translational and angular electrophoretic velocities are given by... 

Ennis and White [56] employed the method of reflections to investigate the electrophoresis of two spherical particles with equilibrium double layers of arbitrary thickness. Their analysis assumes that the zeta potential of the particle is small and the double layers do not overlap significantly. One interesting finding from their study is that the particles with equal zeta potential do interact with each other when the double layer thickness is finite, unlike the no-interaction result for the case of infinitely thin ion cloud. The leading order interaction between two particles is still but the... 

The exact numerical results for s at various ajlat were obtained by Keh and Yang [16] using the collocation results and are presented in Table 2. The corresponding approximate results from (93) are also shown in the same Table for comparison. It can be found that the results of s predicted by the method of reflections have significant errors when the ratio ajlat is small. For equilibrium double layer of finite thickness, an approximate analytic expression for atj was obtained by Ennis and White [56] using the reflection results, but no exact solution is available as yet. 

In a series of papers, Felderhof has devised various methods to solve anew one- and two-sphere Stokes flow problems. First, the classical method of reflections (Happel and Brenner, 1965) was modified and employed to examine two-sphere interactions with mixed slip-stick boundary conditions (Felderhof, 1977 Renland et al, 1978). A novel feature of the latter approach is the use of superposition of forces rather than of velocities as such, the mobility matrix (rather than its inverse, the grand resistance matrix) was derived. Calculations based thereon proved easier, and convergence was more rapid explicit results through terms of 0(/T7) were derived, where p is the nondimensional center-to-center distance between spheres. In a related work, Schmitz and Felderhof (1978) solved Stokes equations around a sphere by the so-called Cartesian ansatz method, avoiding the use of spherical coordinates. They also devised a second method (Schmitz and Felderhof, 1982a), in which... 

Hydrodynamic Interactions Between Widely Separated Particles - The Method of Reflections... 

The best available analytic technique is known as the method of reflections. It is an iterative scheme that applies when the separation distance between the particles or between the particle and wall is large relative to the characteristic dimension of the particle(s).27 The basic idea is to approximate the solution as a series of terms that satisfy the creeping-flow... 

However, determination of functions simultaneously satisfying boundary conditions at all shells requires a solution of a system of 2n equations with 2n unknowns. For this reason a method of reflections is suggested which can be used for the inductive excitation... 

The method of reflections, described above, is applied for every harmonic. The total electric field is expressed through the integral, and its integrand defines a reaction of a medium due to action of corresponding harmonic of the primary field. For this reason for electric field in the internal area we have the following expression ... 

Fortunately, in most practical investigations, such complete information is unnecessary. Rather, it usually suffices to know only certain components of these dyadics, and then only in limiting cases. If ajl represents a characteristic particle-to-wall dimension ratio, these limiting cases correspond to the extreme cases where ajl is either very small or very near unity. In the former case the method of reflections (cf. H9) provides a useful technique for obtaining the wall correction. In the latter case, corresponding to the situation where the particle is extremely close to the wall, lubrication-theory type approximations (B7, B29, Cll, D7, G5d, H15, K8, M9, MIO, S8) normally suffice to obtain the required correction. Intermediate cases are rarely of interest. 

In the translational case the method of reflection yields (see B8, H3, and references to earlier papers given therein), to the lowest degree of approximation,... 

There are two ways to solve this problem. The first method, known as the method of reflections, is an approximate method. It is similar to the method of successive approximations, and its application was pioneered by Smoluchowski for a system of n solid particles located relatively far from each other. Let us set forth the general idea of this method for the case of two spherical particles undergoing translational motion in the xz plane with the velocities Ua and Uy (Fig. 8.1). 

In the Ref [14], the method of reflections was applied to calculations of three-particle and four-particle interactions. It was shown that, as compared to pair interactions, three- and four-particle interactions introduce corrections of the order 0(l/r" ) and 0(l/r ) to the corresponding velocity perturbations, where r is the characteristic distance between particles. A generalization for the N-particle case was made in [15]. The velocity perturbation is found to be of the order 0(l/r + ). In the same work, expressions for the mobility functions are derived up to the terms of order 0(l/r ). It should be kept in mind that the corresponding expressions are power series in 1/r, so to calculate the velocities at small clearances between particles (it is this case has presents the greatest interest), one has to take into account many terms in the series, or to repeat the procedure of reflection many times. In addition to analytical solutions, numerical solutions of a similar problem are available, for example, in [16]. At small clearances between particles, the application of numerical methods is complicated by the need to increase the number of elements into which particle surfaces are divided in order to achieve acceptable accuracy of the solution. 

To increase the accuracy of the definition of factors fi in cases when the particles are located relatively far apart, one should take advantage of the method of reflections, which allows one to obtain an approximate solution in the form of a power series, r. In this way, terms of higher order are taken into account in Eqs. (12.49). Omitting simple but bulky calculations, accurate to terms of r , we obtain ... 

We have introduced new designations JI2 = Fi Fe Fi = FiIFc-The factors of hydrodynamic resistance hy depend on the relative distance s between the drops. Paper [43] solves the problem of slow central motion of two drops in a liquid where the two drops and the liquid all have different viscosities. The factors hy are found in the infinite series form. The appoximate solution of a similar problem is obtained by method of reflection in [46], and hy are found as power series in ratios Ri/r and R2/T which may be considered as asymptotic expressions for factors hy at s 2. An asymptotic expression for h is obtained in [39] for small values of the gap between the drops at s 2 ... 

TV. Meglinsky, A.N. Bashkatov, E.A. Genina, D.Yu. Churmakov, and V.V. Tuchin, Study of the Possibility of Increasing the Probing Depth by the Method of Reflection Confocal Microscopy upon Immersion Clearing of Near-Surface Human Skin Layers, Laser Physics, vol. 13, no.l, 2003, pp. 

The constitution of the received compounds 2-6 is confirmed by NMR H spectrums. Identification IR the spectrums received by a method of reflectance in a solid phase, frequency of phenolic hydroxyl is in area of 3100 cm that can be a consequence of intermolecular interaction of a proton group OH bunches with the functional group The similar effect is considered in work [11]. 

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