Equations in Cylindrical Coordinates

Torgunakov V.G. et al. Two-level system for thermographic monitoring of industrial thermal units. Proc. of VTI Intern. S-T conference. Cherepovets, Russia, pp. 45-46, 1997. 2. Solovyov A.V., Solovyova Ye.V. et al. The method of Dirichlet cells for solution of gas-dynamic equations in cylindrical coordinates, M., 1986, 32 p. 

A first principle mathematical model of the extruder barrel and temperature control system was developed using time dependent partial differential equations in cylindrical coordinates in two spatial dimensions (r and z). There was no angular dependence in the temperature function (3T/30=O). The equation for this model is (from standard texts, i.e. 1-2) ... 

Difference schemes for a stationary equation in cylindrical coordinates. 

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.) ... 

We solve this problem by using a generalized mass balance equation in cylindrical coordinates, centered on the disk ... 

The Modified Bessel Functions. By an argument similar to that employed in 1 we can readily show that Laplace s equation in cylindrical coordinates d2yi, 1 dy> 1 d tp d2yi... 

The diffusive flux rates would be treated similarly. The area of the control volume changing with radius is the reason the mass transport equation in cylindrical coordinates, given below - with similar assumptions as equation (2.18) - looks somewhat different than in Cartesian coordinates. 

Develop the heat transport equation in cylindrical coordinates for temperature. 

Any flow with a nonuniform velocity profile will, when spatial mean velocity and concentration are taken, result in dispersion of the chemical. For laminar flow, the well-described velocity profile means that we can describe dispersion analytically for some flows. Beginning with the diffusion equation in cylindrical coordinates (laminar flow typically occurs in small tubes) ... 

The general solution of the diffusion equation in cylindrical coordinates is a = ai lnr+ a2, and using the boundary conditions above to determine the constants ai and <12,... 

Solution. Starting with the diffusion equation in cylindrical coordinates (see Eq. 5.8) and using the scaling parameter to change variables, the diffusion equation in 77-space becomes... 

The strategy is as follows. We start by rewriting the equations in cylindrical coordinates (r, ,z). The variables we consider are the layer displacement u (now in the radial direction) from the cylindrical state, the director n, and the fluid velocity v. The central part of the cylinder, r < Ri, containing a line defect, is not included. It is not expected to be relevant for the shear-induced instability. We write down linearized equations for layer displacement, director, and velocity perturbations for a multilamellar (smectic) cylinder oriented in the flow direction (z axis). We are interested in perturbations with the wave vector in the z direction as this is the relevant direction for the hypothetical break-up of the cylinder into onions. The unperturbed configuration in the presence of shear flow (the ground state) depends on r and 0 and is determined numerically. The perturbations, of course, depend on all three coordinates. We take into account translational symmetry of the ground state in the z direction and use a plane wave ansatz in that direction. Thus, our ansatze for the perturbed variables are... 

We now transform the governing equations in cylindrical coordinates into polar coordinates. Since the motion is axisymmetric, the transformation from (r, z) to (R, 6), as shown in Fig. 3.1, is analogous to the transformation from Cartesian coordinates (x, y) to cylindrical coordinates (r, 0) in a two-dimensional domain. The stream function is related to the velocity components in polar coordinates by... 

The diffusion equations in cylindrical coordinates for the redox species O and R, with no chemical reactions occurring in solution are given below ... 

The theoretical model assumes a line heat source dissipating heat radially into an infinite solid, initially at uniform temperature. The fundamental heat conduction equation in cylindrical coordinates, assuming uniform radial heat transfer, is [3] ... 

The equations of continuity, momentum and energy are summarized in Tables 6.1, 6.2 and 6.3, respectively. The vectorial forms in Tables 6.1 and 6.3 are provided for scholars of heat transfer that would like to go to three dimensional applications. The equations in cylindrical coordinates may be obtained from the rectangular coordinate equations by the use of the appropriate transformations. 

Consider a cylindrical elemental control volume of dimensions Ar, rA0, and Az in the r, 0 and z directions, respectively. Derive the continuity equation in cylindrical coordinates. 

Derive the two-dimensional energy equation in cylindrical coordinates using the control volume shown in Fig. P2.1. 

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. 

Beginning with the three-dimensional heat-conduction equation in cartesian coordinates [Eq. (l-3a)], obtain the general heat-conduction equation in cylindrical coordinates [Eq. (1-36)]. 

Writing the vector Helmholtz equation in cylindrical coordinates, using our trial function for u, and invoking the Ansatz that ij/ is a slowly varying function of z in order to drop the term in d ip/dz, we obtain an equation for P and q, namely. 

The scalar Helmholtz equation in cylindrical coordinates has the form... 

Upon writing the scalar Helmholtz equation in cylindrical coordinates and dropping the d g/dz term as discussed in Section III, we find the following equation for... 

The general heat conduction equation in cylindrical coordinates can be obtained from an energy balance on a volume element in cylindrical coordinates, shown in Fig. 2-23, by following the steps just outlined. It can also be obtained directly from Eq. 2-38 by coordinate transformation u ing the... 

Starting with an energy balance on a ring-shaped volume element, derive the two-dimensional steady heat conduction equation in cylindrical coordinates for T(r, z) for the caf.se of constant thermal conductivity and no heat generation./... 

As an aside, it is worth mentioning, that the technique described earlier can also be used for solving partial differential equations in cylindrical coordinates. For example, consider the Graetz problem, [1]... 

Laplace Transform Technique for Parabolic Partial Differential Equations in Cylindrical Coordinates... 

Often inversion to time domain solution is not trivial and the time domain involves an infinite series. In section 8.1.4 short time solution for parabolic partial differential equations was obtained by converting the solution obtained in the Laplace domain to an infinite series, in which each term can easily inverted to time domain. This short time solution is very useful for predicting the behavior at short time and medium times. For long times, a long term solution was obtained in section 8.1.5 using Heaviside expansion theorem. This solution is analogous to the separation of variables solution obtained in chapter 7. In section 8.1.6, the Heaviside expansion theorem was used for parabolic partial differential equations in which the solution obtained has multiple roots. In section 8.1.7, the Laplace transform technique was extended to parabolic partial differential equations in cylindrical coordinates. In section 8.1.8, the convolution theorem was used to solve the linear parabolic partial differential equations with complicated time dependent boundary conditions. For time dependent boundary conditions the Laplace transform technique was shown to be advantageous compared to the separation of variables technique. A total of fifteen examples were presented in this chapter. 

Hansen [34] gives an informative derivation of the momentum equations in cylindrical coordinates employing an inertial frame. 

The gradient of v is part of the momentum equations. In cylindrical coordinates it is expressed as ... 

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