Axisymmetric shear flow

Drops and bubbles. Axisymmetric shear flow past a drop was studied in [474,475], We denote the dynamic viscosities of the fluid outside and inside the drop by p and p.2- Far from the drop, the stream function satisfies (2.5.3) just as in the case of a solid particle. Therefore, we must retain only the terms with n = 3 in the general solution (2.1.5). We find the unknown constants from the boundary conditions (2.2.6)-(2.2.10) and obtain... 

Axisymmetric shear flow past a gas bubble was studied numerically in [472],... 

Spherical particle. First, we consider an axisymmetric shear flow, where the dimensional fluid velocity components remote from the particle have the following form in the Cartesian coordinates X, X2, Xy... 

Spherical bubble. The problem of mass transfer to a spherical bubble in an axisymmetric shear flow at low Reynolds numbers was solved numerically in the entire range of Peclet numbers in [251], The results for the mean Sherwood number can be approximated by formula (4.7.10), where the corresponding value from the second row in Table 4.4 at (3 = 0 must be substituted into the right-hand side. Thus, we obtain the formula... 

For high Peclet numbers (Pe > 100), the mean Sherwood number for a drop in an axisymmetric shear flow is well approximated by the positive root of the cubic equation... 

Bubble Axisymmetric shear flow at high Reynolds numbers Analytical, DBLA 0 [359]... 

Consider a single, freely suspended axisymmetric particle in a homogeneous shear flow held of an incompressible Newtonian liquid. The free suspension condition implies that the net instantaneous force and torque on the particle vanish. There is, however, a finite net force along the axis that one half of the particle exerts on the other, as shown schematically in Fig. 7.25. 

In retrospect, the effect of the change of variables has been to deform the velocity field from axisymmetric stagnation flow over a sphere to linear shear flow along a flat plate. The main advantage of the new coordinates is that the coefficients of the derivatives in (32) are independent of X, and consequently Duhamel s theorem can be applied. Thus, the following procedure can be used (1) solve equation (32) subject to a uniform surface concentration (2) extend this solution to one valid for an arbitrary, nonuniform surface concentration by applying Duhamel s theorem (3) select the surface concentration which satisfies (33a). 

Note that G is time-periodic a non-Brownian axisymmetric particle rotates indefinitely in a shearing flow. This rotation is called a Jeffery orbit (Jeffery 1922). The period P required for a rotation of tt in a Jeffery orbit is ... 

In spite of the fact that there are actually quite a large number of axisymmetric problems, however, there are many important and apparently simple-sounding problems that are not axisymmetric. For example, we could obtain a solution for the sedimentation of any axisymmetric body in the direction parallel to its axis of symmetry, but we could not solve for the translational motion in any other direction (e.g., an ellipsoid of revolution that is oriented so that its axis of rotational symmetry is oriented perpendicular to the direction of motion). Another example is the motion of a sphere in a simple linear shear flow. Although the undisturbed flow is 2D and the body is axisymmetric, the resulting flow field is fully 3D. Clearly, it is extremely important to develop a more general solution procedure that can be applied to fully 3D creeping-flow problems. 

The expressions (8-44) and (8 45) represent a complete, exact solution of the creeping-flow equations for a completely arbitrary linear flow. Among the linear flows of special interest are axisymmetric pure strain, which was solved by means of the eigenfunction expansion for axisymmetric flows in the previous chapter, and simple shear flow, for which... 

Sphere in Linear Flows Axisymmetric Extensional Flow and Simple Shear... 

We have previously obtained solutions by other techniques for the problems of a rigid sphere immersed in axisymmetric straining or simple shear flow of an unbounded fluid. In this subsection, it is shown that those two problems also can be solved very simply by means of a superposition of fundamental singularities at the center of the sphere. 

The closely related problem of a rigid sphere in a linear shear flow is very easy to solve now by analogy to the solution for an axisymmetric straining flow. We consider the problem in the form... 

Problem 8-16. Axisymmetric Prolate Spheroid Held Fixed in a Shear Flow. Let us consider an axisymmetric, prolate ellipsoid whose surface is given by... 

The general problem of the stability of various types of shear flow has occupied a great deal of research for nearly a century. There are two basic classes of problem. First, is the instability of a parallel shear flow such as simple shear flow or either 2D or axisymmetric Poiseuille flow, in which the problem of stability is a very subtle balance of viscous and inertia effects. Although the basic flow is very simple in these cases, the analysis of instability is difficult and involved, so much so that complete books have been written on this subject alone.21 At the end of this chapter, we will return to a very brief discussion of this class of problems. 

In the problem of linear shear flow past a spherical drop (bubble), the Stokes equations (2.1.1) and the boundary conditions at infinity (2.5.1) must be completed by the boundary conditions on the interface and the condition that the solution is bounded inside the drop. In particular, in the axisymmetric case, the boundary conditions (2.2.6)-(2.2.10) are used. 

Solid particles. In the case of axisymmetric straining shear flow, the boundary conditions (2.5.1) remote from the particle have the form... 

Note that in the case of axisymmetric linear shear flow to the surface of a spherical particle (drop or bubble), there are two isolated stagnation points 9 = 0 and 9 = it, as well as the stagnation line 9 = it/2. 

For the axisymmetric and plane shear flows (see Table 4.4), the error of formula (4.8.1) does not exceed 1%. 

Let us consider mass transfer for a translational flow past a solid spherical particle, where the flow field remote from the particle is the superposition of a translational flow with velocity U and an axisymmetric straining shear flow, the translational flow being directed along the axis of the straining flow. The dimensional fluid velocity components in the Cartesian coordinates relative to the center of the particle have the form... 

Drop, bubble Axisymmetric shear Stokes flow 2 1... 

Axisymmetric simple shear flow occurs in a straight cylindrical tube this so-called Poiseuille flow is depicted in Figure 5.4a, further on. The figure also illustrates another point. The liquid velocity v equals zero at the wall of the tube and is at maximum in the center, whereas the velocity gradient is zero in the center and is at maximum at the wall. This is more or less the case in many kinds of flow. The flow velocity at the wall of a vessel always equals zero, at least for a Newtonian liquid (explained below). 

Figure 2.4 Critical Weber number for break-up of drops in various types of flow. Singledrop experiments in two-dimensional simple shear (a — 0), hyperbolic flow (a = l) and intermediate types/-as well as a theoretical result for axisymmetrical extensional flow (ASE)." The hatched area rrfers to apparent We values obtained in a colloid mill" ...

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