Cylindrical co-ordinates

Ur, Ur, and u- are the cartesian results Ur and u- are the results in cylindrical co-ordinates. Note u is in the direction of positive P, pressing into the half plane for compressive loading. The tangential displacements of the free surface are towards the origin in agreement with our intuition. 

For the annular element of radius, r, in Fig. 4.63 it is more convenient to use cylindrical co-ordinates so this equation may be rewritten as... 

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 ... 

Initial modelling on predicting velocity profiles in pultrusion dies was carried out by Gorthala et al. (1994). Here a two-dimensional mathematical model in cylindrical co-ordinates with a control-volume-based finite-difference method was developed for resin fiow, cure and heat transfer associated with the pultrusion process. Raper et al. 

Taking cylindrical co-ordinates (r, , z) with FjFa as z-axis, and its mid-point as origin, we can write the equations of these surfaces of revolution... 

Then they choose numerical solution of continuum turbulent flow equations with effective viscosity coefficient p, = Pr + Pm (Pm - coefficient of molecular viscosity) by the use of K-e turbulence model with method of finite elements on irregular computational grid [274]. These equations in cylindrical co-ordinates are as following ... 

The second characteristic of MO distinguishing their behaviom in the saddle region r w 0 concerns the (A, saddle point A 1,/z 0 we have... 

Motion in a capillary is described using cylindrical co-ordinates r, z and 6 L and R are the length and radius of the capillary tube and Pq — is the pressure drop along the tube. This is illustrated in Figure 3.16. Only shear thinning fluids are considered in this section. 

Only the z component of the stress balance equation in cylindrical co-ordinates is required to describe the motion. In the absence of gravity a momentum balance gives (Bird et al, 1960) ... 

Figure 3.16. Side and end views of capillary defining (a) cylindrical co-ordinates r, z and 9 and (b) radius, R, length, L, and pressure, Pq and l.

Equation (68) gives the electrostatic potential at the surface of a solute ion in a pure dielectric solvent. To find the electrostatic potential, Tg, for electrostatic charge, e, in a dielectric solution requires that, Tg, be a solution of the Poisson— Boltzmann equation [21, 25]. Owing to the spherical and planar geometry of a charge in an interfacial system, Onsager and Samaras chose cylindrical co-ordinates (, X) for the Poisson—Boltzmann equation. 

Corpuscle rule we will set in its cylindrical co-ordinates (r (p z). Speed of a corpuscle it is definable three cocTaBiiammHMHit/— tangential, — radially extending W—axial speeds. 

The capture ratio is thus computed as follows. We will choose a cylindrical co-ordinate system with the centre, which has been had in the center of the big drop, and the radial co-ordinate y, perpendicular to a heading of its falling. 

Here x, z and %, c, are cylindrical co-ordinates in real and reciprocal space, respectively (see FIGURE 2). Po is the average value (scaled here to the electron density) which C approaches at long distances, while Jo is the usto ord Bessel function. 

---