Creeping flow general solutions

Figure 11.8 shows typical curves for Re/Rex as functions of t and M, calculated from Eqs. (11-31) to (11-36) for 7 = 2.65. Even for low Rex (curve 2), the velocity approaches the terminal value more rapidly than predicted by the creeping flow solution. At higher Rex the steady terminal velocity is approached more rapidly, but the value required to achieve a given fraction of Rcx increases with Rex- The trajectory is generally more sensitive to Rex than to 7 as shown by Fig. 11.9, where we have plotted the t and required to... 

First, and most important, nonlinear dynamics provides an intellectual framework to pursue the consequences of nonlinear behavior of transport systems, which is simply not possible in an intellectual environment that is based upon a linear mentality, characterized by well-behaved, regular solutions of idealized problems. One example that illustrates the point is the phenomenon of hydrodynamic dispersion in creeping flows of nondilute suspensions. It is well known that Stokes flows are exactly reversible in the sense that the particle trajectories are precisely retraced when the direction of the mean flow is reversed. Nevertheless, the lack of reversibility that characterizes hydrodynamic dispersion in such suspensions has been recently measured experimentally [17] and simulated numerically [18], Although this was initially attributed to the influence of nonhydrodynamic interactions among the particles [17], the numerical simulation [18] specifically excludes such effects. A more general view is that the dispersion observed is a consequence of (1) deterministic chaos that causes infinitesimal uncertainties in particle position (due to arbitrarily weak disturbances of any kind—... 

We have seen that the Navier-Stokes and continuity equations reduce, in the creeping-motion limit, to a set of coupled but linear, PDEs for the velocity and pressure, u andp. Because of the linearity of these equations, a number of the classical solution methods can be utilized. In the next three sections we consider the general class of 2D and axisymmetric creeping flows. For this class of flows, it is possible to achieve a considerable simplification of the mathematical problem by combining the creeping-flow and continuity equations to produce a single higher-order DE. 

The general solution (7-71) can be applied to examine 2D flows in the region between two plane boundaries that intersect at a sharp corner. This class of creeping motion problems was considered in a classic paper by Moffatt,11 and our discussion is similar to that given by Moffatt. A typical configuration is shown in Fig. 7-5 for the case in which one boundary at 6 = 0 is moving with constant velocity U in its own plane and the other at 6 = a is stationary. 

Now, we have expressed the general streamfunction, (7-149), and the disturbance flow contribution in (7-150) and (7-151), in terms of spherical coordinates. However, we have not yet specified a body shape. Thus the linear decrease of the disturbance flow with distance from the body must clearly represent a property of creeping-flows that has nothing to do with specific coordinate systems. Indeed, this is the case, and the velocity field (7-151) plays a very special and fundamental role in creeping-flow theory. It is commonly known as the Stokeslet velocity field and represents the motion induced in a fluid at Re = 0 by a point force at the origin (expressed here in spherical coordinates).17 We shall see later that the Stokeslet solution plays an important role in many aspects of creeping-flow theory. 

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. 

A second, even more important, fact, is that a general solution of the creeping- (dimensional) flow equations is... 

Now, let us see how the preliminary concepts in this section can be put together to achieve a general representation procedure for the solution of general classes of creeping-flow problems. We begin our discussion with the simplest case of spherical particles. 

We shall see that the stokeslet solution plays a fundamental role in creeping flow theory. We have already seen in Section E of Chap. 7 that it describes the disturbance velocity far away from a body of any shape that exerts a nonzero force on an unbounded fluid. Indeed, when nondimensionalized and expressed in spherical coordinates, it is identical to the velocity field, (7 151). In the next section we use the stokeslet solution to derive a general integral representation for solutions of the creeping-flow equations. 

To obtain a general integral representation for solutions of the creeping-flow equations, it is necessary first to derive a general integral theorem reminiscent of the Green s theorem from vector calculus. 

The preceding sections have been concerned primarily with direct solution techniques for problems in creeping-flow theory. Here, we discuss several general topics that evolve directly from these developments. The first two involve application of the so-called reciprocal theorem of low-Reynolds-number hydrodynamics. 

The reciprocal theorem is derived directly from the general integral formula, (8-111). For this purpose, we identify u and u, as well as T and T, as the solutions of two creeping-flow problems for flow past the same body but with different boundary conditions on the body surface 3 D. 

Use the general representation of solutions for creeping flows in terms of vector harmonic functions to solve for the velocity and pressure fields in the two fluids, as well as the deformation and surfactant concentration distribution functions, at steady state. You should find... 

It is important to recognize that the analysis presented in this section is generally applicable to any high-Peclet-number heat transfer process that takes place across a region of closed-streamline flow. In particular, the limitation to small Reynolds number is not an intrinsic requirement for any of the development from Eq. (9-309) to Eq. (9-334). It is only in the specification of a particular form for the function V( ) that we require an analytic solution for f and thus restrict our attention to the creeping-flow limit. Indeed all of (9-309)-(9-334) apply for any closed-streamline flow at any Reynolds number, provided only... 

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