Cylindrical coordinate system

In an axisymmetric flow regime all of the field variables remain constant in the circumferential direction around an axis of symmetry. Therefore the governing flow equations in axisymmetric systems can be analytically integrated with respect to this direction to reduce the model to a two-dimensional form. In order to illustrate this procedure we consider the three-dimensional continuity equation for an incompressible fluid written in a cylindrical (r, 9, 2) coordinate system as... 

Now we intend to derive nonpenetration conditions for plates and shells with cracks. Let a domain Q, d B with the smooth boundary T coincide with a mid-surface of a shallow shell. Let L, be an unclosed curve in fl perhaps intersecting L (see Fig.1.2). We assume that F, is described by a smooth function X2 = i ixi). Denoting = fl T we obtain the description of the shell (or the plate) with the crack. This means that the crack surface is a cylindrical surface in R, i.e. it can be described as X2 = i ixi), —h < z < h, where xi,X2,z) is the orthogonal coordinate system, and 2h is the thickness of the shell. Let us choose the unit normal vector V = 1, 2) at F,, ... 

Note that 0" < A< 60". The invariants A , and form a cylindrical coordinate system relative to the principal coordinates, with axial coordinate / A equally inclined to the principal coordinate axes, with radial coordinate /3t, and with angular coordinate The plane A" = 0 is called the II plane. Because the principal values can be ordered arbitrarily, the representation of A through its invariants in n plane coordinates has six-fold symmetry. 

As with other types of rotating machinery, an axial compressor can be described by a cylindrical coordinate system. The Z axis is taken as running the length of the compressor shaft, the radius r is measured outward from the shaft, and the angle of rotation 6 is the angle turned by the blades in Figure 7-2. This coordinate system will be used throughout this discussion of axial-flow compressors. 

A similar mathematical model to that just described for bench slot exhausts can again be used, but in this case the Laplace equation should be employed in a cylindrical coordinate system (see Fig. 10.83), namely,... 

It is convenient to employ two sets of coordinate systems. Spherical polar coordinates r, Q, A) are defined with the origin at the vertex of the cone the axis is 0=0, the surface of the conical portion of the cyclone is the cone 0 = 0% and the azimuthal coordinate is A. Using the same origin, cylindrical polar coordinates (R, A, Z) are defined, where R = r sin 0 and the Z-axis coincides with the axis 0=0. 

In addition to the Cartesian and normal/tangential coordinate systems, the cylindrical (Figure 2-11) and spherical (Figure 2-12) coordinate systems are often used. 

Figure 2-11. Equations of motion in a cylindrical coordinate system,...

We now formalize the definition of piston flow. Denote position in the reactor using a cylindrical coordinate system (r, 6, z) so that the concentration at a point is denoted as a(r, 9, z) For the reactor to be a piston flow reactor (also called plug flow reactor, slug flow reactor, or ideal tubular reactor), three conditions must be satisfied ... 

Homogeneous difference schemes for stationary equations in spherical and cylindrical coordinate systems have been designed in Chapter 3. 

A model that employs a two-dimensional cylindrical coordinate system and assumes axial symmetry with respect to r- and z-axes is developed. Figure 3.2.2 shows the coordinate system, computing region, and... 

Although the foregoing example in Sec. 4.2.1 is based on a linear coordinate system, the methods apply equally to other systems, represented by cylindrical and spherical coordinates. An example of diffusion in a spherical coordinate system is provided by simulation example BEAD. Here the only additional complication in the basic modelling approach is the need to describe the geometry of the system, in terms of the changing area for diffusional flow through the bead. 

Let us consider the SHV mode of the TCP flow as the base state of a dynamical system described by three velocity components Vr, V0 and Vz relative to the fixed cylindrical co-ordinate system depicted in Figure 4.4.7(b). This dynamical system is described mathematically by the equations of motion of the particle trajectories in the 3D (r, 0, z) coordinate system ... 

A generalized mass balance equation in other coordinate systems is sometimes useful. In the cylindrical system, Eq. (19) becomes... 

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. 

In general, the velocity profile will be curved but as equation 1.33 contains only the local velocity gradient it can be applied in these cases also. An example is shown in Figure 1.13. Clearly, as the velocity profile is curved, the velocity gradient is different at different values of y and by equation 1.32 the shear stress r must vary withy. Flows generated by the application of a pressure difference, for example over the length of a pipe, have curved velocity profiles. In the case of flow in a pipe or tube it is natural to use a cylindrical coordinate system as shown in Figure 1.14. 

Continuity and rate equations can also be written in a cylindrical coordinate system for the two-dimensional annular chromatography. Assuming steady state and neglecting velocity and concentration variations in the radial direction, the above-mentioned equations may then be written as... 

Fig. 15.1 A sketch of a cylindrical geometry coordinate system for a cylindrical resonant cavity. The dimensions are determined by the range of frequencies to be used in the studies...

As mentioned in the introduction to this chapter this is a necessary condition when approximating the cylindrical screw in the Cartesian coordinate system. The screw rotation theory, New Theory line, predicts that the rate should constantly increase as the channel gets deeper. When a fixed positive pressure occurs for the screw rotation model, the New Theory with Pressure line, the predictions fits the data very well for all H/Ws. Thus for modern screw designs with deeper channels, reduced energy dissipation, and lower discharge temperatures, the screw rotation model would be expected to provide a good first estimation of the performance of the extruder regardless of the channel depth for Newtonian polymers. 

For linear molecules, convention dictates that the high-symmetry axis be the z axis and then the Hessian of II(p) is diagonal in this coordinate system. Moreover, the expansion of Eq. (5.114) can be reduced to a two-dimensional one by using spherical polar coordinates to exploit the cylindrical symmetry. The expansion can be written as [355]... 

The cylindrical coordinate system and cylindrical control volume are illustrated in Figure 2.6. There are some differences in the development of a mass balance equation on a cylindrical control volume. Primarily, the rdOdx side of the control volume increases in area as r increases. For the control volume of Figure 2.6, the area normal to the r-coordinate would be... 

Wood Kirkwood (Ref 36a) assumed a curved shock front leading a zone which is cylindrically symmetric. Their coordinates were x, coincident with the axis of the cylindrical chge, and r, the radial distance from the axis. The vector mass velocitytf has an axial component u and a radial component >. Fig 3Oof Ref 66, p 157 is a sketch of the flow in a coordinate system which moves with.the deton wave. Here = space coordinate within reaction wave ... 

Fig 30 Schematic diagram of cylindrically symmetric flow In a detonation wave, with coordinate system at rest in the detonation front... 

The flow lines in a coordinate system at rest in the unreacted expl are shown in Fig 32a. A spherical deton is not steady, since the radius of curvature increases with time. For an instantaneously steady spherical segment of shock front moving in the direction of axis of a cylindrical chge, the flow lines betw the front and the C-J plane in a coordinate system at rest in the shock front will diverge, as shown in Fig 32b... 

Figure 8.2 Cylindrical polar coordinate system. The z axis lies along the axis of the infinite cylinder.

In the same spirit, a dyadic Green function in a cylindrical coordinate system is written as... 

---