Circular Cylindrical Coordinates

To obtain governing equations and boundary conditions, we adopt the circular cylindrical coordinate system that is shown, along with a top view of the eccentric cylinder system, in Fig. 5-1. In this system, the origin of the coordinate system is chosen to be coincident with the central axis of the inner cylinder, which is assumed to have a radius a and an angular velocity 2 in the direction shown. The radius of the outer cylinder is assumed to be a( 1 + e), and its center is offset along the 0 = 0 axis relative to the center of the inner cylinder by an amount ask. In the journal-bearing problem, the outer cylinder does not rotate ( 2 = 0). The surface of the inner cylinder is thus... 

The solution of these equations by means of standard eigenfunction expansions can be carried out for any curvilinear, orthogonal coordinate system for which the Laplacian operator V2 is separable. Of course, the most appropriate coordinate system for a particular application will depend on the boundary geometry. In this section we briefly consider the most common cases for 2D flows of Cartesian and circular cylindrical coordinates. 

Starting with (7-51) and (7-52), we can also obtain a general solution for xj/ in terms of a circular cylindrical coordinate system, and this solution is more immediately applicable to real problems. The governing equations in polar cylindrical coordinates are... 

For 2D body shapes, h2 = 1. In addition either h2/hx = 1 (for example elliptical cylindrical, bipolar, or parabolic cylindrical coordinate22), or h2/hx = 1 + 0(Pe xli) (for circular cylindrical coordinates assuming that r = 1 is the surface of the cylinder). Hence (9 254) simplifies to the universal form, at least for all 2D geometries for which there is a known analytic coordinate system ... 

In the prolate case, the ellipsoidal coordinates (f, //) are related to circular cylindrical coordinates (r, z) by the transformation z = c cosh cos y and r —c sinh sin q. The coordinate surface / = /o (constant) defines the surface of a prolate spheroid, with major and minor semiaxes ao and bo given by... 

Figure B.4 Diagram (a) defining r, , and in the spherical polar coordinate system and (b) defining / , 0, and Z in the circular cylindrical coordinate system.

If/(r) is cylindrically symmetric, it is convenient to adopt the circular cylindrical coordinates (JR, O, Z) with the Z axis coincident with the symmetry axis and R lying in the plane perpendicular to it (see Figure B.4b)./(r) is then a function of R and Z only. The transform F(s) is then also a function of two variables, sR and sz, where sz is the component of s in the symmetry axis direction and sr is the component in the direction perpendicular to it. When the x axis direction is chosen so that the vector s is in the XOZ plane, we have... 

A circular waveguide with inner radius a is shown in Fig. 4.14. Here the axis of the waveguide is aligned with the z axis of a circular-cylindrical coordinate system, where p and are the radial and azimuthal coordinates, respectively. If the walls are perfectly conducting and the dielectric material is lossless, the equations for the TE , modes are... 

T able 1.2. NavierStokes formulation in a circular cylindrical coordinate system for an incompressible fluid... 

Figure 17.2. Definitions ofithe position ofia point M and the velocity vector in a circular cylindrical coordinate system ofi axis Oz...

A circular cylindrical coordinate system (r, 0, z) is employed with origin at the center of the bottom plate and positive z-axis pointing to the top plate. The plates have a (common) radius R and separation (gap) H. The top plate is driven at an angular velocity of W t) such that its angular displacement is given by... 

A circular cylindrical coordinate system (r, 6, z) is taken at the mid-plane as shown in Fig. 9. 

Consider now the problem of steady motion in an infinitely long cylindrical tube of circular cross-section and radius a, and let (r,2) denote cylindrical coordinates about the tube axis. Since satisfies... 

Equation (8.12) is a form of the convective dijfusion equation. More general forms can be found in any good textbook on transport phenomena, but Equation (8.12) is sufficient for many practical situations. It assumes constant diffusivity and constant density. It is written in cylindrical coordinates since we are primarily concerned with reactors that have circular cross sections, but Section 8.4 gives a rectangular-coordinate version applicable to flow between flat plates. 

The charge density of dust transported through ducts and the resultant electric fields at the duct Inner walls was monitored by a Monroe Electronics Inc., Model 171 electric fieldmeter. All the electrostatic sampling In the field was performed In circular cross-section ducts. Thus, the electrostatic field Intensity, for this geometry, can be determined from Poisson s equation using the cylindrical coordinate system. 

Solution. The tangential velocity, v, of an interstitial traveling along a circular path of radius R in Fig. 3.8 will be proportional to the force Pi = —Hi VP exerted by the dislocation. In cylindrical coordinates, P is proportional to sin6/r, so... 

For simplicity, it is assumed that the impact is a Hertzian collision. Thus, no kinetic energy loss occurs during the impact. The problem of conductive heat transfer due to the elastic collision of solid spheres was defined and solved by Sun and Chen (1988). In this problem, considering the heat conduction through the contact surface as shown in Fig. 4.1, the change of the contact area or radius of the circular area of contact with respect to time is given by Eq. (2.139) or by Fig. 2.16. In cylindrical coordinates, the heat conduction between the colliding solids can be written by... 

Fully developed flow in a pipe, i.e., a duct with a circular cross-sectional shape, will first be considered [l],[2],[3]. The analysis is, of course, carried out using the governing equations written in cylindrical coordinates. The z-axis is chosen to lie along the center line of the pipe and the velocity components are defined in the same way that they were in Chapter 2, i.e., as shown in Fig. 4.3. 

To describe the flow between the inner and outer cylinders, we use cylindrical coordinates (r, 6, z) and note that the fluid moves in a circular motion the velocities in the radial the axial directions are zero = 0, Vz = 0, and due to symmetry... 

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 ... 

There are three possibilities corresponding to the dimension of the distribution. The first is a ID concentration distribution (d = 1), in which the diffusing species spreads evenly in the z directions from an initial line pulse at z = 0 on the xz plane. In this case, the variable r in (6-37) is the Cartesian variable z. The second case is a circularly symmetric distribution for c (d = 2), which evolves by diffusion on a plane from an initial compact planar pulse. In this case, r in (6 37) is the radial component of a polar (or cylindrical) coordinate system that lies in the diffusion plane. The third case is a spherically symmetric distribution corresponding to d = 3, which evolves at long times from a compact 3D pulse that diffuses outward into the frill 3D space. In this case, r is the radial variable of a spherical coordinate system. To obtain the long-time form of the distribution we must solve (6-37), but subject to the integral constraint that the total amount of the diffusing species is constant, independent of time ... 

To illustrate this fact, we may consider the 2D heat transfer problem of uniform flow past a heated circular cylinder with uniform surface temperature. In this case, if we look for a solution in the asymptotic form (9-15) for low Peclet numbers, the nondimensional governing equation and boundary conditions for do are again (9-16) and (9-17), but this time are expressed in cylindrical coordinates, namely,... 

Figure 9-5. A sketch showing the cylindrical coordinate system used for analysis of streaming flow past a circular cylinder.

A second, equally dramatic difference between the potential-flow theory and experimental observation was that the flow patterns were often completely different. In the case of streaming flow past a circular cylinder, for example, the potential-flow solution is fore aft symmetric with no indication of a wake downstream of the body. To show this we simply solve the potential-flow equation in cylindrical coordinates,... 

To illustrate this latter point, we first derive equations for the inner boundary-layer region for the specific problem of streaming flow past a circular cylinder, starting from the equations of motion expressed in a cylindrical coordinate system. These are... 

First, however, it is important to recognize that the form of equations (10 28), (10-30), and (10 32) is independent of the geometry of the body (i.e., independent of the cross-sectional shape for any 2D body). Although we started our analysis with the specific problem of flow past a circular cylinder, and thus with the equations of motion in cylindrical coordinates, the equations for the leading-order approximation in the inner (boundary-layer) region reduce to a local, Cartesian form with Y being normal to the body surface and x... 

Let us consider a thin liquid film of thickness h flowing by gravity on the surface of a vertical circular cylinder of radius a. In the cylindrical coordinates TZ, ip, Z, the only nonzero component of the liquid velocity satisfies the equation... 

Transient flow in tubes under instantaneously applied pressure gradient. The problem on a transient laminar viscous flow in an infinite circular tube under a constant pressure gradient instantaneously applied at t = 0 is considered in [449]. The equation of motion in the cylindrical coordinates has the form... 

Pulsating laminar flow in a circular tube. Let us give an exact solution of yet another problem without initial conditions. Consider the problem about a laminar viscous flow in an infinite circular tube with the axial pressure gradient varying according to a harmonic law. Since the problem is axisymmetric, in the cylindrical coordinates 1Z, Z it can be represented in the form... 

Statement of the problem. Let us consider laminar steady-state fluid flow in a circular tube of radius a with Poiseuille velocity profile (see Section 1.5). We introduce cylindrical coordinates 1Z, Z with the Z-axis in the direction of flow. We assume that for Z > 0 the temperature on the wall is equal to the constant T2. In the entry area Z < 0, the temperature on the wall is also constant but takes another value T. ... 

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... 

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