Navier-Stokes equations in cylindrical coordinates

The equations of motion (F-r, Fv, and Fz are the components of the external volume force) are 

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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.) ... 

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 general time-dependent governing equations of fluid flow in a straight cylindrical tube are given by the continuity and the Navier-Stokes equations in cylindrical coordinates. 

Navier-Stokes equations in cylindrical coordinate system (r, (p, z) are... 

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

Write out the continuity, Navier-Stokes, and energy equations in cylindrical coordinates for steady, laminar flow with constant fluid properties. The dissipation term in the energy equation can be ignored. Using this set of equations, investigate the parameters that determine the conditions under which similar" velocity and temperature fields will exist when the flow over a series of axisymmetrie bodies of the same geometrical shape but with different physical sizes is considered. 

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... 

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... 

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... 

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 mass and momentum equations, that is, the Navier-Stokes approximation expressed in cylindrical coordinates with axisymmetry assumption, are [50, 51] ... 

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

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,... 

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),... 

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. 

The transport equations describing the instantaneous behavior of turbulent liquid flow are three Navier-Stokes equations (transport of momentum corresponding to the three spatial coordinates r, z, in a cylindrical polar coordinate system) and a continuity equation. The instantaneous velocity components and the pressure can be replaced by the sum of a time-averaged mean component and a root-mean-square fluctuation component according to Reynolds. The resulting Reynolds equations and the continuity equation are summarized below ... 

Problem By writing the Stokes equation (Navier-Stokes equation without the inertial term) in cylindrical coordinates, show that the velocity profile in the tube is parabolic.Designate by G the pressure gradient responsible for the transport of fluid (in the present problem, G is generated by the Laplace underpressure existing at the upstream interface), and set the velocity at the solid/liquid interface equal to zero. Under these conditions, deduce the average velocity in the tube (Poiseuille s law) and, from there, the viscous force F that opposes the progress of the fluid [equation (5.40)]. 

T able 17.1. Navier-Stokes equations, incompressibility, and velocity curl expressed in a cylindrical coordinate system with axis Oz... 

For cylindrical coordinates, each velocity component is written as, and v, respectively. The melt film on the disk is moved by friction between the film and the disk, and is transferred outward by centrifugal force. The velocity of the fluid changes in the boundary layer, with a thickness of 6 near the disk surface. The thickness of the boundary layer decreases when the angular velocity of a disk increases. Since the angular velocity is constant in this study, the thickness of the boundary layer at the surface of the disk should also remain unchanged. Under the conditions that the velocity profile is steady-state and that the edge effect of the disk can be negligible, Navier-Stokes equations (momentum conservation law) can be described as follows ... 

By solving the Navier-Stokes equations of Newtonian fluid, one can obtain the following equation, which describes the pressure distribution under the chip in the cylindrical coordinate system ... 

This completes the derivation of the basic equations for swirling flows from the Navier-Stokes equations. When deriving flow equations, particularly in cylindrical coordinates, this method is safer than using heuristic arguments. 

In order to formulate the flow equations for a fluid, for instance, for the gas in the cyclone or swirl tube, we must balance both mass and momentum. The mass balance leads to the equation of continuity the momentum balance to the Navier-Stokes equations for an incompressible Newtonian fluid. When balancing momentum, we have to balance the x-, y- and -momentum separately. The fluid viscosity plays the role of the diffusivity. Books on transport phenomena (e.g. Bird et ah, 2002 Slattery, 1999) will give the full flow equations both in Cartesian, cylindrical and spherical coordinates. 

To analyze the linear stability of a Couette flow, we begin with the Navier Stokes and continuity equations in a cylindrical coordinate system. The frill equations in dimensional form can be found in Appendix A. We wish to consider the fate of an arbitrary infinitesimal disturbance to the base flow and pressure distributions (12 114) and (12 116). Hence we consider a linear perturbation of the form... 

The equations can also be written using cylindrical and spherical coordinates. The solutions to the equations are called velocity fields or flow fields. The equations were developed by Claude-Louis Navier (1785-1836) in 1822, and developed further by George Stokes (1819-1903), and find many applications including the study of the flow of fluids in pipes and over surfaces. 

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