Cylindrical coordinates determination

It is tempting to say that the velocity vectors in uniaxial extension are all in the axial direction. Consider a homogeneous, incompressible cylindrical rod of initial radius Ro and length lo that is suspended vertically with one end fixed at X = 0. The other end of the rod is extended in the axial (x) direction until / = 21o. Using conventional cylindrical coordinates, determine the final position of a point that was located at r = Ro, x — lo/2 prior to extension. 

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.) ... 

To calculate d4/d , we need to evaluate partial derivatives, such as U->4/7) , which measures the rate of change in energy with the order parameter. To do so we need to define generalized coordinates of the form ( , qi, , qN-1). Classical examples are spherical coordinates (r, 6, o), cylindrical coordinates (r, 0, z) or polar coordinates in 2D. Those coordinates are necessary to form a full set that determines... 

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. 

Thus the determinant must vanish, det (T — A.1) = 0, which expanded, for example, in cylindrical coordinates, is written as... 

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,... 

The chain conformation it was treated mathematically in terms of atoms regularly spaced along helices, an approach equivalent to defining the contents of the unit cell in terms of cylindrical coordinates. Packing of the chains was characterized by defining the distances between molecules as determined from the equatorial reflections on an x-ray fiber pattern. j ... 

Using a heat transfer limited model based on Fourier s Law in cylindrical coordinates (developed at the Colorado School of Mines), hydrate dissociation under field conditions provides an order of magnitude (higher) prediction, which is acceptable in the industrial setting (see Chapter 8 and Appendix B for more details, including a description of CSMPlug to determine dissociation dimensions and times). 

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... 

The passage of electrons or other particles with charge q and mass m through an electrostatic lens system is governed by their motion under the action of the electric field. In the case considered here, cylindrical symmetry around the optical axis (z-axis) and paraxial rays will be assumed. Of the cylindrical coordinates only the transverse radial coordinate p and the distance coordinate z are of relevance, and the electrostatic potential of the lens is given by q>(p, z). As shown in Section 10.3.1, in the paraxial approximation the potential q>(p, z) is fully determined by the potential symmetry axis. Hence, the equations of motion and the fundamental differential equation of an electrostatic lens depend only on this potential. The fundamental lens equation is given by (see equ. (10.38))... 

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. 

The origin of the cylindrical coordinates on the center-line is determined by the inter-center distance, d = h + R + R2, between the spheres and the distances, d and 2, of the center of each sphere to the origin. We have d = and... 

In this section we explain how to determine V and the spatial derivatives of the unit vectors in cylindrical coordinates. 

For the cylindrical coordinates the metric coefficients can be determined from (C.75) ... 

The pressure gradient is one of the gradients of a scalar variable that are part of the equations in question. In cylindrical coordinates it is determined as ... 

To evaluate the double integral in Eq. (5.161) it is advantageous to change variables according to ri,r2 — ri,r = T — and to employ cylindrical coordinates such that dr = dxdj/dz — det Jd dpdz where the determinant of the Jacobian for this transformation det J = p. Moreover, we reedize from Eq. (5.150) that we need to compute the difference Z2 — Zi to calculate the second virial coefficient of the confined quantum gas. Hence, by immediately carrying out the trivial integrations over. ti, yi, and we obtain... 

To see whether the condition (9-320) provides the added input that is required for determining 0o(VO hi the closed-streamline region, it is convenient to express it in terms of the independent variables f and rj (which we choose as lines orthogonal to cylindrical coordinates (r, < />). To do this, we need only introduce the appropriate scale or metric factors for the (coordinate system, h,/, and hri. The reader is reminded that these scale factors are defined such that the length of a differential line element, expressed in the (f, >]) system, takes the form... 

Suppose that in the spherical (or cylindrical) coordinates, the surface of a particle (drop, bubble) is described by the equation r = R(9), where r is the dimensionless (referred to the characteristic length) radial coordinate and 9 is the angular coordinate. Then the velocity field near the interface is determined by the dimensionless stream function rp = [r-R(9)]mf(9), and the value F(k, k+1) in (4.6.22) is calculated by the following formulas [166] in the axisymmetric case, 0 < 8 < rr and... 

Before we enter into a more detailed discussion on the determination of the molecular electric quadrupole moments and on additivity rules for atom susceptibilities, we will draw some general conclusions from the theoretical expressions for the g- and -values given in Eqs. (1.2) and (1.4), respectively. We first restate that the perturbation sums are necessarily zero if the total electronic wavefunction (for simplicity we may tliink of a Slater determinant) has cylindrical symmetry with respect to the rotational axis in consideration. To see this, we recall that in cylindrical coordinates with a as the sjmimetry axis ... 

Pore-water profiles were used together with solid-phase dissolution rates in diagenetic models to determine first-order anoxic precipitation rate constants for both Mn and Fe. A two-dimensional cylindrical coordinate model was employed to account for the effects of biogenic irrigation of burrows on pore-water Mn " distributions. Two-dimensional diffusion can result in a decrease in Mn " with depth that would be interpreted as evidence for precipitation and cause overestimation of precipitation rates in a one-dimensional model. 

Figure 3. Left Analyte is moved stepwise around the CSP with a grid that is determined by cylindrical coordinates. Right Dimensions of an imaginary elliptical cylinder surrounding the analyte molecule. Figures provided by K.D. Bartle.

In problem 2.4 the movement of charged colloidal particles in water contained in an annular gap between two oppositely charged infinitely long cylinders was considered. The potential distribution across the gap d(j)ldr was assumed constant, where (j) is the potential and r is the radial cylindrical coordinate. It is desired here to take the analysis one step further and determine the potential and concentration distribution within the gap, neglecting any electrode effects. 

Consider the diffusion of solute A from the surface of a cylinder of radius R into a homogeneous tissue (Figure 3.4b). For example, the cylinder might represent the external surface of a capillary that contains a high concentration of a drug. The concentration within the tissue, in the region r > R, can be determined by solving Equation 3-31 in cylindrical coordinates ... 

This condition far from the snbmerged object is used to determine the functional form of the scalar velocity potential. For example, in cylindrical coordinates. 

Cylindrical coordinates are a generalization of polar coordinates to three dimensions, obtained by augmenting r and 9 with the Cartesian z coordinate. (Alternative notations you might encounter are r or p for the radial coordinate and 9 or 4> for the azimuthal coordinate.) The 3x3 Jacobian determinant is... 

Integration in Eqn (10.2) is performed over V area occupied by the current sources. As (r, (p, z) one can mark cylindrical coordinates of a free point in space, and as (p,6,Q—cylindrical coordinates of a point that belongs to V area. The magnetic induction vector is determined from the equality B = rotA.. Magnetic carriers (separate nanoparticles or their aggregates) can be considered as magnetic dipoles. In general, the dipole is influenced by the force F and mechanical moment M which are defined by the known formulas ... 

The relevance of contact resistances in SPS process has been simulated and confirmed, with the simulation shown in Fig. 6.15 as an example [4]. The system simulated is a Model 1050-Sumitomo SPS, where a solid graphitic cylinder is inserted into the die. The 2D cylindrical coordinate system of coupled thermal and electrical problems is numerically solved by using Abaqus (FEM). The heat losses due to radiation from all exposed surfaces, except those on the ends of the rams, have been considered, where a constant temperature of 25 °C is used for the simulation. Thermophysical parameters of all materials are available in that study. A proportional feedback controller based on the outer surface temperature of the die is modeled, in order to determine the voltage drop applied at two ends of the rams. This controller is used to imitate the actual proportional integral derivative (PID), which is observed in real SPS facilities. It is used to apply electric power input to the system when experiments are conducted in terms of temperature controlling. 

To determine the velocities in the cylindrical coordinate system (CCS), the momentum equation has to be expressed in cylindrical coordinates. Again, the flow is assumed to be in steady state inertia, centrifugal, and gravitational forces are assumed negligible and the fluid viscosity constant. For the angular component this can be written as ... 

The axisymmetric meniscus under a conical surface is chosen as the subject, as shown in Fig. 12. The cylindrical coordinates r and z are taken to be the radial and horizontal directions, respectively. If some relations of differential geometry are inserted into the radii of curvature in Laplace equation (2), the profile of the axisymmetric meniscus can be determined by the following differential equation [31,62]. 

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