Coordinates cylindrical polar

The change in the coordinate system affects the strain components and, after some manipulation, one obtains 

The various stress components are shown in Fig. 4.14 and the shear components can be shown to be symmetric. The equilibrium equations have a different form in the new coordinate system. This leads to new relationships between the stresses and the Airy stress function. Using the same approach as that outlined in the last section, the following revised versions of Eqs. (4.23) and (4.24) are obtained 

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Figure Bl.9.5. Geometrical relations between the Cartesian coordmates in real space, the spherical polar coordinates and the cylindrical polar coordinates.

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. 

Thus, the velocity components of the fluid may be determined, in cylindrical polar coordinates, as follows ... 

However, more accurate predictions for the spin velocity may be obtained with allowance made for the effect of viscosity in the governing equation for the spin velocity. According to the experimental data of Keisall," which indicates that the spin velocity in a cyclone is a function of R only, the A component of Eq. (13.1) in the cylindrical polar coordinate sy stem may reduce to... 

To take a particular case, consider the cylindrical polar coordinates (o, [Pg.3]

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

The convective-diffusion equation in cylindrical polar coordinates in... 

That T is a series of Bessel rather than trigonometric functions is merely a consequence of using cylindrical polar coordinates (r, j, cz ) for atoms in real space and (R, iji, i/a for points in reciprocal space. Not only is this a convenient framework for describing a helical molecule, but it can lead to economies in computing T. For helices, only Bessel terms with... 

Figure 1. Reciprocal space coordinates (R, i/<, 1) are the cylindrical polar coordinates...

It is convenient to use spherical polar coordinates (D,a,i i) at some stages of the calculation and cylindrical polar coordinates at others to define position in reciprocal... 

This is a two-dimensional problem handled in cylindrical polar coordinates in which the longitudinal z axis is a symmetry axis for the system. The relevant coordinates are (r, 0) and the gradient is applied along the polar axis direction (i.e., across a diameter). The relaxing boundary is at a radial distance r = a from the cylinder center ... 

Note the r dr in the integrand, which is necessary because cylindrical polar coordinates are used. The relationship between the average velocity and the pressure gradient can be obtained by integrating Eq. (9.7) using Eq. (9.8) this is the Hagen-Poiseuille law. The peak velocity is twice the average velocity in pipe flow ... 

A sketch of the physical problem and a definition of the cylindrical polar coordinates (r, creeping-flow limit, the governing equations and boundary conditions are... 

Cylindrical polar coordinates p, Cartesian coordinates by the equations of transformation... 

Problem 17-1. The equation for the free particle is separable in many coordinate systems. Using cylindrical polar coordinates, set up and separate the wave equation, obtain the solutions in

recursion formula for the coefficients in the series solution of the p equation. Hint In applying the polynomial method, omit the step of finding the asymptotic solution. 

Problem 17-2. Calculate p) for a harmonic oscillator in a state represented by fn mrH of Equation 17-27. Shew that p, is zero in the same state. Hint Transform — into cylindrical polar coordinates. 

Notation. As used for isotope separation, the gas centrifuge is a cylinder of radius a and length L, rotating about a vertical axis with angular velocity u rad/s. Cylindrical polar coordinates are used, with the following notation for position and velocity components ... 

Differential enrichment equation. Under steady-state conditions, the differential equation for conservation of light component, in cylindrical polar coordinates, is... 

The cylindrical polar coordinate system is another three-dimensional coordinate system. It uses the variables p, 4>, and z, already defined and shown in Fig. 2.11. The equations needed to transform from Cartesian coordinates to cylindrical polar coordinates are Eqs. (2.39) and (2.40). The third coordinate, z, is the same in both Cartesian and cylindrical polar coordinates. Equations (2.37) and (2.38) are used for the reverse transformation. 

EXAMPLE 2.6 Find the cylindrical polar coordinates of the point whose Cartesian coordinates are (l.(K)0, —4.000, —2.000). 

The magnitude of the position vector in cylindrical polar coordinates is given... 

A surface is represented in cylindrical polar coordinates by the equation z = p. Describe the shape of the surface. 

EXERCiSE7.l9 [ Find the Jacobian for the transformation from Cartesian to cylindrical polar coordinates. Q... 

Coordinate systems such as spherical polar or cylindrical polar coordinates are called orthogonal coordinates, because an infinitesimal displacement produced by changing only one of the coordinates is perpendicular (orthogonal) to a displacement produced by an infinitesimal change in any one of the other coordinates. 

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