Slender body approximation

Consequently analytical methods are mostly confined to creeping flows. Roughly, there are two types of problems that can be solved. The first of these deals with interfaces that show small deviations from simple geometric forms, as for instance the case of a slightly deformed sphere settling in an infinite fluid. The second type constitutes cases where interfacial position changes, but only very slowly. Then its variation can be neglected to the first approximation and the lubrication theory approximation or the slender body approximation applied. It should be noted that both the above methods yield approximate solutions. 

The cases of a sphere and slightly deformed sphere in a uniform flow field are considered first in Sections 4 and 5. The mathematical method used conventionally in these problems is the regular asymptotic expansion. The reader is introduced to this method. In Section 6, the dip coating problem under the lubrication theory approximation is examined. (The closely related slender body approximation is outlined in Problem 7.5.) A more sophisticated method of matched asymptotic expansions is used to solve this problem and its main features... 

Figure 4.8 shows that Eq. (4-27) gives a good approximation for the drag on a cylinder with motion normal to the axis for the range in which experimental results are available. The curve obtained from the exact results for spheroids can be used to estimate for very small or large E. The slender-body result, Eq. (4-37), appears to be applicable for E > 3. 

In the remainder of this section, we outline results for two types of problems for which internal distributions of singularities have been used to advantage. The first, originally pursued by Chwang and Wu,13 considers bodies of very simple shape - spheres, prolate ellipsoids of revolution (spheroids), and similar cases for which exact solutions can be obtained for some flows either by a point or line distribution involving only a few singularities. The second class of problems is for very slender bodies for which an approximate solution can be obtained by means of a distribution of stokeslets along the particle centerline.14... 

Approximate Solutions of the Creeping-Flow Equations by Means of Slender-Body Theory... 

Eq. (8-124). However, for a slender body, the magnitude ofthe position vector that connects points at the body surface and a point on the centerline is very small (for the same xi), and thus the stokeslet distribution at the body surface can be accurately approximated by use of only a stokeslet distribution on the centerline. This qualitative discussion can be formalized in a straightforward manner. 

With u°° (x) specified, the algebraic equation (8-187) can thus be solved to determine a first approximation to the stokeslet density distribution in the slender-body limit, 0. 

Tuck established a mathematical expression for squat with a slender body theory. The slender body theory assumes that the beam, draft, and water depth are very small relative to ship length. This theory uses potential flow where the continuity equation becomes Laplace s equation. The flow is taken to be inviscid and incompressible and is steady and irrotational. In restricted water, the problem is divided into the inner and the outer problems, following a technique of matched asymptotic expansions to construct an approximate solution. The inner problem deals with flow very close to the ship. The potential is only a function of y and 2 in the Cartesian coordinate system. In the cross-flow sections, the potential function... 

In devolatilizing systems, however, Ca 1 and the bubbles deform into slender S-shaped bodies, as shown in Fig. 8.12. Hinch and Acrivos (35) solved the problem of large droplet deformation in Newtonian fluids. They assumed that the cross section of the drop is circular, of radius a, and showed that the dimensionless bubble surface area, A, defined as the ratio of the surface area of the deformed bubble A to the surface area of a spherical bubble of the same volume, is approximated by (36) ... 

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