Exact Solutions of the Stokes Equations

An extensive discussion and bibliography of further references to conventional wall-effects will be found in Happel and Brenner (H9). To this list should be added the theoretical study by Dean and O Neill (D4) of the rotation of a sphere about an axis parallel to a nearby plane wall in an otherwise unbounded fluid, and a companion study by O Neill (02a) of the translation of a sphere parallel to a plane wall. Bipolar coordinates were employed to obtain exact solutions of the Stokes equations. These studies are particularly interesting in view of the fact that, due to the asymmetry of the flow, the rotating sphere experiences a force parallel to the wall whereas the translating sphere experiences a torque about the sphere center parallel to the wall. According to the remarks made in the second paragraph of Section II,C,3, these cross effects may be expressed in terms of the coupling dyadic in Eqs. (38) and (39) (where 0 refers to the sphere center). The Dean-O Neill in-... 

Although the Hadamard-Rybcynski result is an exact solution of the Stokes equation, it is inconsistent with many experimental results for small bubbles or drops. Bond [4] and Bond and Newton [5] showed that sufficiently small bubbles or drops behave like rigid spheres. This phenomenon was eventually explained by Frumkin and Levich [6] (see Ref. 7) as being due to... 

Using asymptotic analysis, Sadhal and Johnson [9] rigorously established the conditions under which an immobilized surfactant cap would form on a bubble or drop in Strokes flow. They also found an exact solution of the Stokes equation for a fluid sphere with an immobilized surfactant cap. 

In the work of Hasimoto [8] the exact solution of the Stokes equation for the fluid flow in fibrous porous medium in the form of the infinite series is used to predict the permeability of fibrous porous media. He found the theoretical prediction of the permeability of fibrous porous media using only the terms of lowest order of this series ... 

The three exact solutions of Navier-Stokes equations presented in this chapter have allowed rrs to familiarize orrrselves with Navier-Stokes equations and to rrse the concept of stress for a Newtonian flirid. They also illustrate simple properties regarding pressure distributiorr withirr the flows. 

The exact solution of the convection-diffusion equations is very complicated, since the theoretical treatments involve solving a hydrodynamic problem, i.e., the determination of the solution flow velocity profile by using the continuity equation or -> Navier-Stokes equation. For the calculation of a velocity profile the solution viscosity, densities, rotation rate or stirring rate, as well as the shape of the electrode should be considered. 

A small number of exact solutions of the nonlinear Navier Stokes equations have been discovered more or less by accident. A discussion of some of these solutions can be found in other textbooks.3 However, they are all special cases that do not lead to solutions for a broader class of problems, nor do they generally provide physical insights that can be transferred to other problems. If we approach the question of exact analytic solutions from a more pragmatic or systematic point of view, it is evident that the most important class of problems for which we should expect exact solutions is those for which the nonlinear terms in the equations are identically equal to zero, i.e.,... 

Boundary layer approximation. The Landau problem, which was described above, is an example of an exact solution of the Navier-Stokes equations. Schlichting [427] proposed another approach to the jet-source problem, which gives an approximate solution and is based on the boundary layer theory (see Section 1.7). The main idea of this method is to neglect the gradients of normal stresses in the equations of motion. In the cylindrical coordinates (71, ip, Z), with regard to the axial symmetry (Vv = 0) and in the absence of rotational motion in the flow (d/dip = 0), the system of boundary layer equations has the form... 

Note that in fact the plane Poiseuille flow (1.17) is also an exact solution of the full Navier-Stokes equation. However, it was shown by linear stability analysis that this becomes unstable to small perturbations at a critical Reynolds number of 5772. In fact, the transition to turbulence is observed experimentally at even lower values of Re around 1000. 

Carruthers (12) and Milson and Pamplin (13) have discussed the implications of the resulting oscillations on crystal growth. In this section we shall examine exact solutions of the Navier-Stokes equations for a two-dimensional simplified model of a molten zone which is in the form of a cavity or slot of liquid of depth d, supported on the bottom, but with a free surface on top. The zone is heated over the length, - < x < , by a flux, q, and cooled on its ends at x +L, where L > . We shall study the core region inside - < x < for which a similarity solution exists. Thus we are neglecting end-effects. 

As was briefly pointed out in section 18.3.1, the Carman-Kozeny equation does not work well towards the limit of e = 1 Carman himself stated that the equation should not be used for e > 0.8. Several researchers have attempted to derive a model with a more realistic and general outcome. Perhaps the most significant attempt is that by Rudnick who used a ffee-surface cell model by Happel in which each particle is assumed to be a sphere at the centre of a cell, the volume of which is such that the porosity of each cell is the same as that of the bed. If the tangential stresses at the boundaries of adjoining cells are set to zero, an exact solution of the general Navier-Stokes equations exists, assuming that the inertial terms are negligible. 

The Navier-Stokes equation is one of the basic governing equations for study of fluid flow related to various disciplines of engineering and sciences. It is a partial differential equation whose integration leads to the appearance of some constants. These constants need to be evaluated for exact solutions of the flow field, which are obtained by imposing suitable boundary conditions. These boundary conditions have been proposed based on physical observation or theoretical analysis. One of the important boundary conditions is the no-slip condition, which states that the velocity of the fluid at the boundary is the same as that of the boundary. Accordingly, the velocity of the fluid adjacent to the wall is zero if the boundary surface is stationary and it is equal to the velocity of the surface if the surface is moving. This boundary condition is successful... 

A second aspect of irreversibility concerns the approach to equilibrium. Since this question arises only for an isolated system, we will suppose the external forces to be turned off at some time, say < = 0, the system then being isolated for later times. Now the Navier-Stokes equations imply an approach to equilibrium, with monotonically increasing entropy. However, for times t > 0 the ensemble (51) is an exact solution to the Liouville equation for an isolated system, and as is well known the quantity S of Eq. (85) must then be time-independent and cannot approach its equilibrium value. The resolution of this paradox lies in the approximate nature of the Navier-Stokes equations, which hold only for slowly varying processes. The exact transport relations, taking into account processes of arbitrarily rapid variation, are evidently such as to maintain S constant. One may say that the low frequency contributions to S increase at the expense of the high frequency contributions, and measurements of a sufficiently coarse nature will show an apparent approach to equilibrium. Thus the approach to equilibrium is obtained as a natural consequence of an approximation method suited to slowly varying processes. 

Wang, C.Y., 1989. Exact solutions of the unsteady Navier-Stokes equations. Appl. Mech. Rev. 42, S269-S282. 

Wang, C.Y, 1991. Exact solutions of the steady-state Navier-Stokes equations. Annu. Rev. Fluid Mech. 23, 159-177. 

A conclusion of the above discussion is that exptessing an exact solution to Navier-Stokes equations is not sufficient in itself. One also needs to be able to demonstrate, with all necessary rigor, that this solution is stable under the conditions considered. This requires mastering the theory of instability in fluid flows. ... 

The Rankine model is of great practical interest, although it is not an exact solution of the Navier-Stokes equations. In the presence of viscosity, the vorticity discontinuity at r = o is removed. The radius a of the vortex widens in time if a secondary flow does not counteract the effect of viscosity. 

However, an exact solution to the problem of convective diffusion to a solid surface requires first the solution of the hydrodynamic equations of motion of the fluid (the Navier-Stokes equations) for boundary conditions appropriate to the mainstream velocity of flow and the geometry of the system. This solution specifies the velocity of the flrrid at any point and at any time in both tube and yam assembly. It is then necessary to substitute the appropriate values for the local fluid velocities in the convective diffusion equation, which must be solved for boundary cortditiorts related to the shape of the package, the mainstream concentration of dye and the adsorptions at the solid surface. This is a very difficrrlt procedure even for steady flow through a package of simple shape. " ... 

Exact Solutions to the Navier-Stokes Equations. As was tme for the inviscid flow equations, exact solutions to the Navier-Stokes equations are limited to fairly simple configurations that aHow for considerable simplification both in the equation and in the boundary conditions. For the important situation of steady, fully developed, laminar, Newtonian flow in a circular tube, for example, the Navier-Stokes equations reduce to... 

The key problem in using Eq. (3.1) is the specification of p. We ask whether we can derive an expression for p. The velocity components u, v, and w, although random, are related through conservation of mass and momentum for the flow, that is, they are governed by the stochastic Navier-Stokes and continuity equations. In general, as we have noted, an exact solution for u, v, and w is unobtainable. We can, however, consider an idealized situation in which the statistical properties of u, v, and w are specified a priori. Then, in so doing, we wish to see if we can obatin an exact solution of Eq. (2.4) from which p can be obtained through Eq. (2.6). 

In principle, one can write down all of these forces and formulate the Newtonian equations of motion for the fluid this yields a complicated differential equation known as the Navier-Stokes equation [1-3]. A complete solution of the Navier-Stokes equation gives the exact trajectory and velocity of each fluid element. In practice, the calculations are often difficult because one must simultaneously account for all fluid elements and the interactions between these elements caused by the viscous drag forces. (The simultaneous motion of many interacting fluid elements is analogous to the simultaneous motion of many interacting mechanical objects, the latter being so complicated that it is described as the many body problem. ) However, in certain cases, the Navier-Stokes equation is reduced to a tractable form by the existence of steady low-velocity flow and high symmetry in the flow conduit (e.g., capillary tubes of circular cross section). We will examine such simple cases shortly. 

More complicated 3D effects were studied in Refs. 6 and 7 with the help of 3D Monte Carlo digital simulation performed with a rather powerful computer (RISK System/6000). Sedimentation FFF with different breadth-to-width channel ratios and both codirected and counterdirected rotation and flow were studied. Secondary flow forming vortexes in the y-z plane is generated in the sedimentation FFF channel, both due to its curvature, and the Coriolis force caused by the centrifuge rotation. The exact structure of the secondary flow was calculated by the numerical solution of the Navier-Stokes equations and was used in the Monte Carlo simulation of the movement of solute molecules. 

Although the full Navier Stokes equations are nonlinear, we have studied a number of problems in Chap. 3 in which the flow was either unidirectional so that the nonlinear terms u Vu were identically equal to zero or else appeared only in an equation for the crossstream pressure gradient, which was decoupled from the primary linear flow equation, as in the ID analog of circular Couette flow. This class of flow problems is unusual in the sense that exact solutions could be obtained by use of standard methods of analysis for linear PDEs. In virtually all circumstances besides the special class of flows described in Chap. 3, we must utilize the original, nonlinear Navier Stokes equations. In such cases, the analytic methods of the preceding chapter do not apply because they rely explicitly on the so-called superposition principle, according to which a sum of solutions of linear equations is still a solution. In fact, no generally applicable analytic method exists for the exact solution of nonlinear PDEs. 

Arbitrary three-dimensional straining shear flows past a porous particle were considered in [524], The flow outside the particle was described by using the Stokes equations (2.1.1). It was assumed that the percolation of the outer liquid into the particle obeys Darcy s law (2.2.24). The boundary conditions (2.5.1) remote from the particles and the conditions at the boundary of the particle described in Section 2.2 were satisfied. An exact closed solution for the fluid velocities and pressure inside and outside the porous particle was obtained. 

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