Creeping flow Stokes’ problem

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—... 

A very important consequence of approximating the Navier-Stokes equations by the creeping-flow equations is that the classical methods of linear analysis can be used to obtain exact solutions. Equally important, but less well known, is the fact that many important qualitative conclusions can be reached on the basis of linearity alone, without the necessity of obtaining detailed solutions. This, in fact, will be true of any physical problem that can be represented, or at least approximated, by a system of linear equations. In this section we illustrate some qualitative conclusions that are possible for creeping flows. 

The reader may find the result (7-16) surprising. As already noted, it is well known that a rotating and translating sphere in a stationary fluid will often experience a sideways force (that is, lift) that will cause it to travel in a curved path-think, for example, of a curve ball in baseball or an errant slice or hook in golf. The difference between these familiar examples and the problem previously analyzed is that the Reynolds numbers are not small and the governing equations are the full, nonlinear Navier-Stokes equations rather than the linear creeping-flow approximation. Thus the decomposition to a set of simpler component problems cannot be used, and it is not possible to deduce anything about the forces on the... 

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. 

As an example of the application of (7-131), we consider creeping flow past an arbitrary axisymmetric body with a uniform streaming motion at infinity. For the case of a solid sphere, this is known as Stokes problem. In the present case, we begin by allowing the geometry of the body to be arbitrary (and unspecified) except for the requirement that the symmetry axis be parallel to the direction of the uniform flow at infinity so that the velocity field will be axisymmetric. A sketch of the flow configuration is shown in Fig. 7 11. We measure the polar angle 9 from the axis of symmetry on the downstream side of the body. Thus ij = I on this axis, and ij = — 1 on the axis of symmetry upstream of the body. 

A second problem, closely related to Stokes problem, is the steady, buoyancy-driven motion of a bubble or drop through a quiescent fluid. There are many circumstances in which the buoyancy-driven motions of bubbles or drops are of special concern to chemical engineers. Of course, bubble and drop motions may occur over a broad spectrum of Reynolds numbers, not only the creeping-flow limit that is the focus of this chapter. Nevertheless, many problems involving small bubbles or drops in viscous fluids do fall into this class.23... 

To complete the specification of the problem for 9, we must specify a particular velocity field u. In the case of Re <creeping-flow solutions of Chaps. 7 and 8, and it is again convenient to focus our attention on the case of a sphere in a uniform streaming flow, in which a first approximation to the velocity field is given by the Stokes solution, Eq. (7-158), from which we can calculate the velocity components by means of (7-102). 

But before the virtues of the results and the approach are extolled, the method must be described in detail. Let us therefore return to a systematic development of the ideas necessary to solve transport (heat or mass transfer) problems (and ultimately also fluid flow problems) in the strong-convection limit. To do this, we begin again with the already-familiar problem of heat transfer from a solid sphere in a uniform streaming flow at sufficiently low Reynolds number that the velocity field in the domain of interest can be approximated adequately by Stokes solution of the creeping-flow problem. In the present case we consider the limit Pe I. The resulting analysis will introduce us to the main ideas of thermal (or mass transfer) boundary-layer theory. 

Since diffusion processes in fluids are characterized by very large values of the Schmidt number, we must point out that in problems of convective mass transfer in fluid media, the Peclet number is also large even for low Reynolds numbers at which the Stokes flow law applies (for a creeping flow). 

At very low Reynolds numbers below about 1, the term creeping flow is used to describe flow at very low velocities. This type of flow applies for the fall or settling of small particles through a fluid. Stokes law is derived using this type of flow in problems of settling and sedimentation. 

This chapter starts by considering a very simple situation where the above approximation is made and a creeping solution of the Navier-Stokes equation obtained. In situations with simple symmetry, however, it is often easier, under creeping flow conditions, to look at the problem starting with the defining equation for Newtonian viscosity (equation [4.6]) and some examples of this type are examined. A discussion of the viscosity of simple liquids in terms of the molecular behaviour follows and the chapter ends with a brief account of non-Newtonian liquids and of methods for measuring liquid viscosity. 

As a very simple example of the use of the Navier-Stokes equation under creeping flow conditions, consider the flow of a Newtonian liquid in a parallel-sided, semi-infinite channel in the absence of body forces. Let the boundaries be planes located at y = /z, each lying in an xz plane and let the liquid flow be in the z direction. The motion is assumed to be so slow that a creeping solution is obtained, the liquid is assumed to be incompressible and r is treated as a constant. From the symmetry of the problem the liquid flow velocity is a function of y only, so that the three components of the Navier-Stokes equation are ... 

Given its efficiency, BEM has been applied to a variety of problems involving large deformations of a free surface. Several solutions have been developed for problems related to the nonlinear evolution of water waves, [9-11] and for problems related to nonlinear deformations of both viscous and inviscid drops [12,13]. BEM has been applied to several applications of creeping (Stokes) flows in liquid columns [14-16] and in annular layers [16]. Inviscid solutions have also been obtained for both infinite [16] and finite-length [13] liquid jet problems, as well as for dripping flows [13], fountains [13], and fluid sloshing problems [17]. 

Analytical solutions to the bed expansion problem, without resort to empiricism, are almost entirely limited to spheres in creeping (i.e., viscous or Stokes) flow, Reo < 0.2. Many of these are discussed by Happel and Brenner (1965) and later by Jean and Fan (1989). 

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