Stokes equations exact solutions

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). 

The fundamental physical laws governing motion of and transfer to particles immersed in fluids are Newton s second law, the principle of conservation of mass, and the first law of thermodynamics. Application of these laws to an infinitesimal element of material or to an infinitesimal control volume leads to the Navier-Stokes, continuity, and energy equations. Exact analytical solutions to these equations have been derived only under restricted conditions. More usually, it is necessary to solve the equations numerically or to resort to approximate techniques where certain terms are omitted or modified in favor of those which are known to be more important. In other cases, the governing equations can do no more than suggest relevant dimensionless groups with which to correlate experimental data. Boundary conditions must also be specified carefully to solve the equations and these conditions are discussed below together with the equations themselves. 

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. 

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. 

After Navier- Stokes equation has been written down in the first half of nineteenth century, few exact solutions were obtained for few fluid flows. In one such case, Stokes compared theoretical prediction with available experimental data for pipe flow and found no agreement whatsoever. Now we know that the theoretical solution of Stokes corresponded to undisturbed laminar flow, while the experimental data given to him corresponded to a turbulent flow. This problem was seized upon by Osborne Reynolds, who explained the reason for such mismatches by his famous pipe flow experiments (Reynolds, 1883). It was shown that the basic flow obtained as a... 

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. 

Even with these simplifications, however, it is rarely possible to obtain analytic solutions for fluid mechanics or heat transfer problems. The Navier Stokes equation for an isothermal fluid is still nonlinear, as can be seen by examination of either (2 89) or (2 91). The Bousi-nesq equations involve a coupling between u and 6, introducing additional nonlinearities. It will be noted, however, that, provided the density can be taken as constant in the body-force term (thus neglecting any natural convection), the fluid mechanics problem is decoupled from the thermal problem in the sense that the equations of motion, (2 89) or (2-91), and continuity, (2-20), do not involve the temperature 0. The thermal energy equation, (2-93), is actually a linear equation in the unknown 6, once the Boussinesq approximation has been introduced. In that case, the only nonlinear term is dissipation, but this involves the product E E and can be treated simply as a source term that will be known once Eqs. (2-89) or (2 91) and (2 20) have been solved to determine the velocity. In spite of being linear, however, the velocity u appears as a coefficient (in the convective derivative term). Even when the form of u is known (either exactly or approximately), it is normally quite a complicated function, and this makes it extremely difficult to obtain analytic solutions for 0 even though the governing equation is linear. 

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.,... 

We have seen several examples of unidirectional and ID flows for which the Navier-Stokes equations simplify to a linear form so that exact analytical solutions can be obtained. The closest analogy would be problems for which u V0 = 0. Of course, this is just the limit of pure conduction (or pure diffusion) such as the problem considered in the previous... 

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. 

If the boundaries of the flow domain are not parallel, the magnitude of the primary velocity component must vary as a function of distance in the flow direction. This not only introduces a number of new physical phenomena, as we shall see, but it also means that the Navier-Stokes equations cannot be simplified following the unidirectional flow assumptions of Chap. 3, and exact analytical solutions are no longer possible. In this chapter, we thus consider only a special limiting case, known as the thin-gap limit, in which the distance between the boundaries is small compared with the lateral gap width. In this case, we shall see that we can obtain approximate analytical solutions by using the asymptotic and scaling techniques that were introduced in the preceding chapter. 

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. 

Problem 7-14. The Young-Goldstein-Block Problem Revisited. Let us reconsider the Young-Goldstein Block problem, but in this case we directly seek the solution for the case in which the temperature gradient has the value that causes the velocity of the bubble, U, to be exactly equal to zero. Starting with the governing Stokes equations and boundary conditions, nondimensionalize and re-solve for this particular case. It may be useful to remember that this solution is valid when the hydrodynamic force on the bubble is exactly equal to its buoyant force. 

Statement of the problem. In this section we describe one of the few cases in which a nonlinear boundary value problem for the Navier-Stokes equations admits an exact closed-form solution. 

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... 

Let us consider a semi-infinite fluid bounded by a by a rigid plane -oo < X < oo, Y > 0. Two exact solutions are known for the transient flow near a plate [427], These solutions correspond to rather simple flows governed by linear equations of motion. However, in these cases the Navier-Stokes equations are linearized because the nonlinear convective terms are identically zero (Vx dVx /dX = 0) rather than because we neglect these terms. The equation of motion has the form... 

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. 

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. 

The equations of motions (Navier-Stokes equations) are deterministic. This means that solutions with exactly the same boundary and initial conditions will produce the same results. In very small and simple geometries, it is possible to solve the equations of motions in a turbulent regime using a method similar to the false-transient method discussed in Chapter 16. Practical engineering problems require far too much computer storage and time for this to be feasible. 

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-... 

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. 

The starting point is a Mathematical Model, i.e. the set of equations and boundary conditions, which covers the physics of the flow most suitable. For some problems the governing equations are known accurately (e.g. the Navier-Stokes equations for incompressible Newtonian fluids). However for many phenomena (e.g. turbulence or multiphase flow) and especially for the description of ceramic materials or wall slip phenomena the exact equations are either not available or a numerical solution is not feasible. 

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