Stream function axisymmetric creeping flow

Shortcut Methods for Axisymmetric Creeping Flow in Spherical Coordinates. All the previous results can be obtained rather quickly with assistance from information in Happel and Brenner (1965, pp. 133-138). For example, the general solution for the stream function for creeping viscous flow is... 

The system considered in this chapter is a rigid or fluid spherical particle of radius a moving relative to a fluid of infinite extent with a steady velocity U. The Reynolds number is sufficiently low that there is no wake at the rear of the particle. Since the flow is axisymmetric, it is convenient to work in terms of the Stokes stream function ij/ (see Chapter 1). The starting point for the discussion is the creeping flow approximation, which leads to Eq. (1-36). It was noted in Chapter 1 that Eq. (1-36) implies that the flow field is reversible, so that the flow field around a particle with fore-and-aft symmetry is also symmetric. Extensions to the creeping flow solutions which lack fore-and-aft symmetry are considered in Sections II, E and F. 

Hence, two-dimensional axisymmetric potential flow in spherical coordinates is described by = 0 for the scalar velocity potential and = 0 for the stream function. Recall that two-dimensional axisymmetric creeping viscous flow in spherical coordinates is described by E E ir) = 0. This implies that potential flow solutions represent a subset of creeping viscous flow solutions for two-dimensional axisymmetric problems in spherical coordinates. Also, recall from the boundary condition far from submerged objects that sin 0 is the appropriate Legendre polynomial for the E operator in spherical coordinates. The methodology presented on pages 186 through 188 is employed to postulate the functional form for xlr. 

Calculate the stream function for axisymmetric fully developed creeping viscous flow of an incompressible Newtonian fluid in the annular region between two concentric tubes. This problem is analogous to axial flow on the shell side of a double-pipe heat exchanger. It is not necessary to solve algebraically for all the integration constants. However, you must include all the boundary conditions that allow one to determine a unique solution for i/f. Express your answer for the stream function in terms of ... 

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