Sphere in Linear Flows Axisymmetric Extensional Flow and Simple Shear

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. 

We begin with the problem of a solid sphere centered at the stagnation point of an axisymmetric straining flow in which the undisturbed velocity at infinity takes the dimensionless form 

In this technique, we seek the disturbance velocity field, 

Thus we suggest trying to construct a solution that is a superposition of a stresslet and a potential quadrapole ud4 both located at the center of the sphere, that is, 

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