Sphere general linear flows

In this section, we consider the problem of a nonrotating sphere in a general linear flow of an unbounded fluid, namely,... 

To obtain a solution for the complete class of flows given by (8 59), we again construct a solution of the creeping-motion equations by means of the superposition of vector harmonic functions. The development of a general form for the pressure and velocity fields in the fluid exterior to the drop follows exactly the arguments of the preceding solution for a solid sphere in a linear flow, and the solution therefore takes the same general form [see Eq. (8-39) and (8 11)], that is,... 

Problem 8-4. Consider a sphere suspended in the general linear shear flow , = T,-yXy under creeping-flow conditions. If the sphere is free to rotate, what is its angular velocity... 

Before we leave the present problem, the reader s attention is called to several generalizations of the predicted relationship (9-161) between Nu and Pe for Pe <asymptotic method to provide insight into the form of correlations between dimensionless parameters, with a minimum of detailed analysis. The first is due to Batchelor14 and Acrivos,15 who showed that the correlation (9-191), first derived for a sphere in linear shear flow, could be generalized easily and extended to the much more general case of a rigid, heated sphere in an arbitrary linear flow... 

The question at hand is whether circumstances exist for this rather simple situation in which the conditions (1) and (2) are satisfied so that boundary-layer analysis can be applied. So far as the first condition is concerned, the only flows of (9-266) that have open streamlines are those with X > 0 (which includes simple shear flow). On the other hand, there is a nonzero hydrodynamic torque on the sphere that causes it to rotate for all flows in this subgroup except X = 1. Thus, for a sphere in the general linear 2D flow, given by (9-266), there are only two cases that satisfy the conditions for applicability of boundary-layer theory ... 

Here, the sphere center is instantaneously situated at point 0 the sphere center translates with velocity U, while it rotates with angular velocity (a r is measured relative to 0 its magnitude r is denoted by r. Moreover, f = r/r is a unit radial vector. The latter solution is derivable in a variety of ways e.g., from Lamb s (1932) general solution (Brenner, 1970). [Equation (2.12) represents a superposition (Brenner, 1958) of three physically distinct solutions, corresponding, respectively, to (i) translation of a sphere through a fluid at rest at infinity (ii) rotation of a sphere in a fluid at rest at infinity (iii) motion of a neutrally buoyant sphere suspended in a linear shear flow. The latter was first obtained by Einstein (1906, 1911 cf. Einstein, 1956) in connection with his classic calculation of the viscosity of a dilute suspension of spheres, which formed part of his 1905 Ph.D. thesis.]... 

Problem 7-26. An Alternative Derivation of the Solution for Stokes Flow. In Subsection B.4, we showed that the force acting on a sphere that translates through a fluid at low Reynolds number can be expressed in terms of a resistance tensor A in the form F = A U. A generalization of this idea is that the pressure and velocity fields around the sphere must also be a linear function of the vector U, and thus expressible in terms of a vector pressure and a tensor velocity in the form... 

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. 

In many physical processes, however, the particle has a density very nearly equal to that of the fluid, and the particle then has no (or negligible) translational velocity relative to the fluid. Provided that the linear dimensions of the particle are small compared with distances over which the velocity gradient in the ambient flow field changes significantly, the flow near the particle is then effectively due to a force-free particle in an ambient velocity field that can be approximated as varying linearly with position. An important question is whether the rate of heat transfer from a heated sphere in such a flow is still given by a correlation of the form (9 159) for Pe 1. More generally, of course, we may ask whether the type... 

---