Plane polar coordinates definition

From the definition of a particle used in this book, it follows that the motion of the surrounding continuous phase is inherently three-dimensional. An important class of particle flows possesses axial symmetry. For axisymmetric flows of incompressible fluids, we define a stream function, ij/, called Stokes s stream function. The value of Imj/ at any point is the volumetric flow rate of fluid crossing any continuous surface whose outer boundary is a circle centered on the axis of symmetry and passing through the point in question. Clearly ij/ = 0 on the axis of symmetry. Stream surfaces are surfaces of constant ij/ and are parallel to the velocity vector, u, at every point. The intersection of a stream surface with a plane containing the axis of symmetry may be referred to as a streamline. The velocity components, and Uq, are related to ij/ in spherical-polar coordinates by... 

Figure A10.4 (a) The coordinate system and definition of an infinitesimal volume c/r for cylindrical polar coordinates, the factor u enters dx because of the definition of an arc length using angles measured in radians, (b) A slice through the H—H bond showing the charge density on a plane perpendicular to the H2 Cco axis, (c) The geometry used to obtain the distances r and /-j from each nuclear centre these are the varieJales used in the s-orbital basis functions, since they define the nuclear-electron separations.

Coordinates of the centres of mass and orientations of the two acetylene molecules (1 and 2) in the (2x2) unit cell on MgO(lOO). The values obtained from LEED-IV data [24], neutron diffraction (ND) [26], and semi-empirical potential calculations [24,25] are tabulated. The definitions of x, y, and O are given in Fig. 4. 9 is the polar tilt of the molecule away from the surface plane, and z is with respect to the surface MgO layer. The two calculated z values are for effective charges of 1.2 and 2, respectively. 

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