Simplifications to the Navier-Stokes equations

Simplifications to the Navier-Stokes equations are produced when the Reynolds number is very small or very large, Re — 0 or Re — oo. These limiting cases are never reached in reality but they represent asymptotic solutions and are better approximations the larger or smaller the Reynolds number is. We will investigate these limiting cases in the following. 

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An alternative approach to seeking simplifications to the Navier-Stokes equations is to accept the full set of equations, but approximate each term in the equation with a simpler form that permits solutions to be developed. Although the resulting equations are only approximately correct, the advent of modern digital computers has allowed them to be written with great fineness, so that highly accurate solutions are achieved. These techniques are called CFD. 

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... 

From now on, the permeation in (16) is neglected as it is several orders of magnitude smaller than the advection due to the radial component of the velocity vr (now playing the role of vz in the planar case). As far as the velocity perturbation is concerned, our aim is to describe its principal effect-the radial motion of smectic layers, i.e., instead of diffusion (permeation) we now have advective transport. In this spirit we make several simplifications to keep the model tractable. The backflow-flow generation due to director reorientation-is neglected, as well as the effect of anisotropic viscosity (third and fourth line of (19)). Thereby (19) is reduced to the Navier-Stokes equation for the velocity perturbation, which upon linearization takes the form... 

One of the main approaches to the analysis and simplification of the Navier-Stokes equations is as follows. One assumes that the nonlinear inertia term (V V) V is small compared with the linear viscous term vAY and hence can be neglected altogether or taken into account in some special way. This method is well-founded for Re = LUjv -C 1 and is widely used for studying the motion of... 

J. Boundary-layer equations. When laminar flow is occurring in a boundary layer, certain terms in the Navier-Stokes equations become negligible and can be neglected. The thickness of the boundary layer <5 is arbitrarily taken as the distance away from the surface where the velocity reaches 99% of the free stream velocity. The concept of a relatively thin boundary layer leads to some important simplifications of the Navier-Stokes equations. 

Re is a dimensionless parameter. Its definition constitutes a powerful tool for the transfer of information from experiments performed at the laboratory scale on various hydrodynamic phenomena to very large (e.g. airplanes) or very small (particles) scales. Via eq 1.36 Re also provides the possibilty of further simplification of the Navier-Stokes equation if the value of Re is very large or very small. 

There is no general solution of the Navier-Stokes equations, which is due in part to the non-linear inertial terms. Analytical solutions are possible in cases when several of the terms vanish or are negligible. The skill in obtaining analytical solutions of the Navier-Stokes equations lies in recognizing simplifications that can be made for the particular flow being analysed. Use of the continuity equation is usually essential. 

The Navier-Stokes equations have been derived and written in a form that exposes V-V explicitly. In large measure this is done in anticipation of the simplifications that accrue for incompressible flow where V-V = 0. It is also important to recognize situations in which compressible fluids (i.e., gases) behave as though they were incompressible, thus permitting the incompressible-flow simplifications. 

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. 

Solution of the Navier-Stokes equations of fluid How is possible using numerical methods. In porous media, considerable simplification is of ten possible because velocities are generally low and flow is confined to the small spaces between... 

If one is interested in properties that vary on very long distance and time scales it is possible that a drastic simplification of the molecular dynamics will still provide a faithful representation of these properties. Hydrodynamic flows are a good example. As long as the dynamics preserves the basic conservation laws of mass, momentum and energy, on sufficiently long scales the system will be described by the Navier-Stokes equations. This observation is the basis for the construction of a variety of particle-based methods for simulating hydrodynamic flows and reaction-diffusion dynamics. (There are other phase space methods that are widely used to simulate hydrodynamic flows which are not particle-based, e.g. the lattice Boltzmann method [125], which fall outside the scope of this account of MD simulation.)... 

The Reynolds number, which characterizes the importance of inertial forces compared to viscous forces is around unity. This implies that the non-linear terms in the Navier-Stokes equations are weak, easing the task of solving these equations. For some systems, the assumption of Stokes flow may be reasonable, i.e., the inertial terms are set equal to zero this affords a significant simplification of the fluid flow problem [160]. The Reynolds number is independent of pressure, when everything else is held constant. 

Ludwig Prandtl, the father of boundary-layer theory, after making the conceptual division of the flow discussed in Sec. 10.1, set out to calculate the flow in the boundary layer. He chose as his starting point the Navier-Stokes equations (Sec. 7.9) and simplified them by dropping the terms he considered unimportant. His simplifications are as follows ... 

There are several major simplifications that can be used to tackle these equations. First of all, fluid flow is almost always neglected altogether. Then the Navier-Stokes equation is not needed at all, v is no longer a variable and we only need to solve the director equation (7) which will now be... 

This process provides the mapping only. The equation describing the physics (e.g., the Navier-Stokes equation) must be transformed into (, ri) coordinates and solved. For instance, in reservoir simulation, Darcy s pressure equation must be expressed in terms of, t and solved. Thompson s simplification lies not in the transformed equation, which may contain mixed derivatives and variable coefficients, but in the computational domain itself, because it takes on a rectangular form amenable to simple numerical solution. 

For a basic analysis of the problem, we can use the Reynolds equations, which are the time-averaged form of the Navier-Stokes equations (see Section 2-5 and Chapter 5). The major problem is to simplify the equations and obtain additional relations between the unknowns. One idea to provide simplification is to assume that turbulent fluctuations are random in nature and can therefore be treated by... 

Boundary-layer behavior is one of several potential simplifications that facilitate channel-flow modeling. Others include plug flow or one-dimensional axial flow. The boundary-layer equations, however, are the ones that require the most insight and effort to derive and to establish the ranges of validity. The boundary-layer equations retain a full two-dimensional representation of all the field variables as well as all the nonlinear behavior of Navier-Stokes equations. Nevertheless, when applicable, they provide a very significant simplification that can be used to great benefit in modeling. 

Modeling a disk by solving the full three-dimensional Navier-Stokes equations is a complicated task. Moreover, it is still not fully understood what is the cause of frictional forces in the disk. Molecular viscosity is by orders of magnitude too small to cause any appreciable accretion. Instead, the most widely accepted view is that instabilities within the disk drive turbulence that increases the effective viscosity of the gas (see Section 3.2.5). A powerful simplification of the problem is (a) to assume a parameterization of the viscosity, the so-called a-viscosity (Shakura Syunyaev 1973) ((3-viscosity in the case of shear instabilities, Richard Zahn 1999) and (b) to split the disk into annuli, each of which constitutes an independent one-dimensional (ID) vertical disk structure problem. This then constitutes a 1+1D model a series of ID vertical models glued together in radial direction. Many models go even one step further in the simplification by considering only the vertically integrated or representative quantities such as the surface density X(r) = p(r, z.)dz... 

No simplification can be used for the problem of the backward facing penetrable step but the full Navier—Stokes equations. Therefore, no solution is available to validate the numerical algorithm. To be aware of it, the numerical algorithm shortly described in the previous section was tested over the whole range of the above-mentioned problems. In this case, the outlet boundary condition (3 = which is associated with the steady flow in an infinite duct, was used. The results of two numerical performances for the flow regime Re = 100 and EPR dimensions h = 0.3 and L x = 1, are shown in Fig. 3.16 the halves of flows in each case are symmetric. Let us analyze them. 

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. 

For 2D and axisymmetric flows, however, the general representation results, (7-31) and (7 32), do lead to a very significant simplification, both for creeping flows and for flows at finite Reynolds numbers where we must retain the ftdl Navier Stokes equations. The reason for this simplification is that the vector potential A can be represented in terms of a single... 

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