Cylindrical Navier-Stokes Equations

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.) ... 

The first numerical study on the transient flow of a single liquid droplet impinging onto a flat surface, into a shallow or deep pool was performed by Harlow and Shannon)397 In their work, the full Navier-Stokes equations were solved numerically in cylindrical... 

The hydrodynamic equation of motion (Navier-Stokes equation) for the stationary axial velocity, vfr), of an incompressible fluid in a cylindrical pore under the influence of a pressure gradient, dP /dz, and an axial electric field, E is... 

A number of authors from Ladenburg (LI) to Happel and Byrne (H4) have derived such correction factors for the movement of a fluid past a rigid sphere held on the axis of symmetry of the cylindrical container. In a recent article, Brenner (B8) has generalized the usual method of reflections. The Navier-Stokes equations of motion around a rigid sphere, with use of an added reflection flow, gives an approximate solution for the ratio of sphere velocity in an infinite space to that in a tower of diameter Dr ... 

Develop the dimensionless Navier-Stokes equations for cylindrical coordinates. 

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... 

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

Instead of starting with the governing differential equation itself, Eq. 4.7, go back to the momentum balances on a cylindrical element. When the Navier-Stokes equations were... 

Based on a differential cylindrical control volume, derive steady-state momentum balances for the axial and circumferential directions, i.e., the Navier-Stokes equations. 

Consider the steady flow inside a cylindrical channel, which is described by the two-dimensional axisymmetric continuity and Navier-Stokes equations (as summarized in Section 3.12.2). Assume the Stokes hypothesis to relate the two viscosities, low-speed flow, a perfect gas, and no body forces. The boundary-layer derivation begins at the same starting point as with axisymmetric stagnation flow, Section 6.2. Assuming no circumferential velocity component, the following is a general statement of the Navier-Stokes equations ... 

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... 

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

Description of the hydrodynamics in the cylindrical capillary experimental design is fairly simple. Considering only electrostatic and fluid frictional forces acting upon the suspended particles, apparent particle mobility at a given location r across the diameter of the capillary may be represented by a solution to the Navier-Stokes equation in the scheme of a coordinate system with the origin in the center of the capillary by... 

A drawback to the rectangular cell design is that the hydrodynamic description of fluid flow is much more complicated than in the cylindrical cell. Though it is beyond the intent of this discussion to go into full detail, analytical solutions to the Navier-Stokes equation for steady laminar fluid flow have been derived that can be used for calculation of electro-osmosis at flat plates in the rectangular cell configuration where electro-osmosis may differ at the upper, lower, and side chamber walls [ 15). 

Taylor bubbles (gas plugs) in a vertical tube move under the influence of surface tension, inertia, gravitation, and viscous effects. For a Newtonian fluid with constant viscosity and density these phenomena can be described by the Navier-Stokes equations for circular geometry using cylindrical coordinates ... 

In engineering, not only cylindrical pipes are used, but also pipelines with elliptic, square, or triangular cross sections [380], The area D and the cross section shape are taken constant along the longitudinal flow direction Ox. It is known for this problem that the normal flow velocity components Uy and U- are equal to zero, and the only non-zero component is the longitudinal velocity U = U(y, z) that depends on two coordinates (y, z)) in D. Therefore, the complete Navier—Stokes equations are reduced to... 

Equation (3-10), which we have derived from the Navier-Stokes equations, governs the unknown scalar velocity function for all unidirectional flows, i.e., for any flow of the form (3-1). However, instead of Cartesian coordinates (x, y, z), it is evident that we could have derived (3-10) by using any cylindrical coordinate system (q, 1/2, z) with the direction of motion coincident with the axial coordinate z. In this case,... 

The continuity and Navier-Stokes equations in cylindrical and spherical coordinate systems are given in Supplement 5. 

Let us use the cylindrical coordinate system TZ, ip, Z, where the coordinate Z is measured from the disk surface along the rotation axis. Taking account of the problem symmetry (the unknown variables are independent of the angular coordinate Navier-Stokes equations in... 

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... 

The electrolyte fluid flow is assumed to be well represented by the incompressible Navier-Stokes equations (1,6), which in cylindrical coordinates can be expressed as (6),... 

Booth and Hirst [10] examined the squeeze film problem for two rigid circular parallel plates of radius separated by an oil film of thickness fi (h<Starting with the Navier-Stokes equations in cylindrical coordinates, they obtained the relations... 

The Navier-Stokes equations are the differential momentum balances for a three-dimensional flow, subject to the assumptions that the flow is laminar and of a constant-density newtonian fluid and that the stress deformation behavior of such a fluid is analogous to the stress deformation behavior of a perfectly elastic isotropic solid. These equations are useful in setting up momentum balances for three-dimensional flows, particularly in cylindrical or spherical geometries. 

The cylindrical coordinate form of the Navier-Stokes equations is shown in many textbooks, e.g., Bird et al. [8]. Starting with that form, derive the Poiseuille equation. 

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