Navier-Stokes equations coordinates

The governing equations for mass flow, energy flow, and contaminant flow in a room will be the continuity equation, Navier-Stokes equations (one in each coordinate direction), the energy equation, and the mass transport equation, respectively. 

For steady, incompressible fluid flow in a cyclone separator, the governing Navier-Stokes equations of motion are given, in a Cartesian coordinate system, by ... 

Next, we substitute these dimensionless variables into the incompressible Navier-Stokes equations (equation 9.16). In Cartesian coordinates, the T component of the first equation reads... 

We use computational solution of the steady Navier-Stokes equations in cylindrical coordinates to determine the optimal operating conditions.Fortunately in most CVD processes the active gases that lead to deposition are present in only trace amounts in a carrier gas. Since the active gases are present in such small amounts, their presence has a negligible effect on the flow of the carrier. Thus, for the purposes of determining the effects of buoyancy and confinement, the simulations can model the carrier gas alone (or with simplified chemical reaction models) - an enormous reduction in the problem size. This approach to CVD modeling has been used extensively by Jensen and his coworkers (cf. Houtman, et al.) ... 

Equation A.22 is the Navier-Stokes equation for the x-component of motion in rectangular Cartesian coordinates. The corresponding equations for they and z components are obvious. 

Alternative methods of analysis have been examined and evaluated. Shokoohi and Elrod[533] solved the Navier-Stokes equations numerically in the axisymmetric form. Bogy15271 used the Cosserat theory developed by Green.[534] Ibrahim and Linl535 conducted a weakly nonlinear instability analysis. The method of strained coordinates was also examined. In spite of the mathematical or computational elegance, all of these methods suffer from inherent complexity. Lee15361 developed a 1 -D, nonlinear direct-simulation technique that proved to be a simple and practical method for investigating the nonlinear instability of a liquid j et. Lee s direct-simulation approach formed the... 

Equations (if.4) and (ff.S) are solved, along with the continuity equation (which does not change upon nondimensionalization), in a Cartesian coordinate system using the Fourier-Galerkin (spectral) technique under periodic boundary conditions in all three space dimensions. The scheme is similar to that used by Orszag [8] for direct solution of the incompressible Navier-Stokes equations. More details can be found in [9] and [7], and the scheme may be considered to be pseudospectral. ... 

These are called the incompressible Navier-Stokes equations in Cartesian coordinates. 

Develop the dimensionless Navier-Stokes equations for cylindrical coordinates. 

Since velocity is a vector quantity, it is usually necessary to identify the component of the velocity, as was done for the rectangular Cartesian coordinate system in Eq. (1). The value of the integral as it differs from zero may be employed as a measure of the accuracy with which average characteristics (Kl) of the stream may be used to describe the macroscopic aspects of turbulence. Such methods do not yield results of practical significance when applied to the solution of the Navier-Stokes equations. 

Stokes Postulates Stokes s postulates provide the theory to relate the strain-rate to the stress. As a result the forces may be related to the velocity field, leading to viscous-force terms in the Navier-Stokes equations that are functions of the velocity field. Working in the principal coordinates facilitates the development of the Stokes postulates. 

We have discussed stresses and strain rates. A critical objective is to relate the two, leading to equations of motion governing how fluid packets are accelerated by the forces acting on them. Generally, we are working toward a differential-equation description of a momentum balance, F = ma. The approach is to represent both the forces and the accelerations as functions of the velocity field. The result will be a system of differential equations in which velocities are the dependent variables and the spatial coordinates and time are the independent variables (i.e., the Navier-Stokes equations). 

The final objective of this chapter was to develop quantitative relationships between a fluid s strain-rate and stress fields. Expressions for the strain rates were developed in terms of velocities and velocity gradients. Then, using Stokes s postulates, the stress field was found to be proportional to the strain rates and a physical property of the fluid called viscosity. The fact that the stress tensor and strain-rate tensor share the same principal coordinates is an important factor in applying Stokes s postulates. The stress-strain-rate relationships are fundamental to the Navier-Stokes equations, which describe conservation of momentum in fluids. 

Working in cylindrical coordinates, and substituting the force-per-unit-volume expressions that stem from the stress tensor (Eqs. 2.137, 2.138, and 2.139), the Navier-Stokes equations can be written as... 

Notice that in these equations the terms on the right-hand side are written with a certain resemblance to the rows of the stress tensor, Eq. 2.180. The pressure gradients have been written as a separate terms. In the r and 9 equations, the final term collects some of the left-overs in going from Eqs. 3.57 and 3.58, yet maintaining the other terms in a form analogous to the stress tensor. The z equation has no left-over terms, which is also the case for the Navier-Stokes equations in cartesian coordinates. 

Expanded into cylindrical coordinates, the constant-viscosity Navier-Stokes equations are given as... 

The term (ui V) V, which is called vortex stretching, originates from the acceleration terms (2.3.5) in the Navier-Stokes equations, and not the viscous terms. In two-dimensional flow, the vorticity vector is orthogonal to the velocity vector. Thus, in cartesian coordinates (planar flow), the vortex-stretching term must vanish. In noncartesian or three-dimensional flows, vortex stretching can substantially alter the vorticity field. 

As with the axisymmetric stagnation-flow case, deriving the tubular stagnation-flow equations begins with the steady-state three-dimensional Navier-Stokes equations (Eqs. 3.58, 3.59, and 3.60). The approach depends on essentially the same assumptions as the axial stagnation flows described earlier, albeit with the similarity requiring no variation in the axial coordinate. The velocity field is presumed to be described in terms of a stream function that has the form... 

The most general equations for the laminar flow of a viscous incompressible fluid of constant physical properties are the Navier-Stokes equations. In terms of the rectangular coordinates x, y, z, these may be written ... 

When treating this simple case of electro-osmosis mathematically we immediately realize that the y component of the Navier-Stokes equation disappears. All y derivatives are zero because no quantity can change with y due to the symmetry. We further assume that the liquid flows only parallel to the x coordinate that is parallel to the applied field. Then vz 0 and vy =0. As a consequence all derivatives of vy and vz are zero. From the equation of... 

In all the above derivations in this section, the influence of viscosity is neglected so that analytical solutions for velocity and pressure profiles can be obtained. When the viscosity of fluid is taken into account, it is difficult to obtain any analytical solution. Kuts and Dolgushev [35] solved numerically the flow field in the impingement of two axial round jets of a viscous impressible liquid ejected at the same velocity from conduits with the same diameter and located very close to each other. The mathematical formulation incorporated the complete Navier-Stokes equations transformed into stream and velocity functions in cylindrical coordinates r and z, with the assumption that the velocity profiles at the entrance and the exit of the conduit were parabolic. The continuity equation is given by Eq. (1.22) and the equations for motion in dimensionless form are ... 

First, consider the case where the flow is parallel to the cylinders. It is assumed that the fluid is moving through the annular space between the cylinder of radius a and the fluid envelope of equivalent radius b, as shown in Fig. 7.14. Assume that the fluid motion is in the creeping flow regime so that inertia terms can be omitted from the Navier-Stokes equations. Thus, in cylindrical coordinates, we have... 

The symbol defined as V2 is called the Laplacian. Table 2.4 lists the components of the Navier-Stokes equation in the various coordinate systems. 

TABLE 2.4 The Navier-Stokes Equation in Several Coordinate Systems... 

As was mentioned above, the Navier-Stokes equations are obtained by the appli-cation of the conservation of momentum principle to the fluid flow. The same control volume that was introduced above in the discussion of the continuity equation is considered and the conservation of momentum in each of the three coordinate directions is separately considered. The net force acting on the control volume in any of these directions is then set equal to the difference between the rate at which momentum leaves the control volume in this direction and the rate at which it enters in this direction. The net force arises from the pressure forces and the shearing forces acting on the faces of the control volume. The viscous shearing forces for two-dimensional flow (see later) are shown in Fig. 2.3. They are expressed in terms of the velocity field by assuming the fluid to be Newtonian and are then given by [4],[5] ... 

In expressing the Navier-Stokes equations in these coordinates it is convenient to define ... 

Using this notation, the Navier-Stokes equations become in cylindrical coordinates ... 

In this theory, equilibrium flow is obtained using thin shear layer (TSL) approximation of the governing Navier- Stokes equation. However, to investigate the stability of the fluid dynamical system the disturbance equations are obtained from the full time dependent Navier- Stokes equations, with the equilibrium condition defined by the steady laminar flow. We obtain these in Cartesian coordinate system given by. 

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