Velocity fields, cylindrically symmetri

Because of the cylindrical symmetry of the total electric field in the column plasma, in two-term approximation the expansion of the velocity distribution can be represented by the expression f U,v/v,r) = 2n) mg/2f [fQ(U,r)+f U,r)v /v+f.(U,r)v,/v]. This expansion includes, in addition to the isotropic distribution fo(U, r), a radial component fr(U, r) and an axial component f (U,r) of the vectorial anisotropic part of the velocity distribution. In particular, this radial distribution component allows the particle and energy current density of the electrons in the radial direction to be described and thus reveals significant aspects of the electron confinement by the radial... 

When a nondeformable object is implanted in the flow field and the streamlines and equipotentials are distorted, the nature of the interface does not affect the potential flow velocity profiles. However, the results should not be used with confidence near high-shear no-slip solid-liquid interfaces because the theory neglects viscous shear stress and predicts no hydrodynamic drag force. In the absence of accurate momentum boundary layer solutions adjacent to gas-liquid interfaces, potential flow results provide a reasonable estimate for liquid-phase velocity profiles in Ihe laminar flow regime. Hence, potential flow around gas bubbles has some validity, even though an exact treatment of gas-Uquid interfaces reveals that normal viscous stress is important (i.e., see equation 8-190). Unfortunately, there are no naturally occurring zero-shear perfect-slip interfaces with cylindrical symmetry. 

The complete velocity field inside the spinning rotor of the gas centrifuge (assuming the fluid to be a single component ideal gas) is determined by the equations of motion. These are shown in full on pp. 83,85, and 319 of Ref. 19. In cylindrical coordinates with axial symmetry and at steady... 

At an axi-symmetric boundary Neuman conditions are used for all the fields, except for the normal velocity component which is zero because the flow direction turns at this point. The assumption of cylindrical axi-symmetry in the computations prevents lateral motion of the dispersed gas phase and leads to an unrealistic radial phase distribution [73, 66[. Krishna and van Baten [73] obtained better agreement with experiments when a 3D model was applied. However, experience showed that it is very difficult to obtain reasonable time averaged radial void profiles even in 3D simulations. 

It is desired to estimate the electrophoretic velocity U of a long, nonconducting, charged cylindrical particle of length L and radius a and with a low surface potential I, as a result of the application of an electric field parallel to the symmetry axis. The Debye length is arbitrary but finite, and the flow is a low Reynolds number, inertia free one. 

Potential Flow Transverse to a Long Cylinder Via the Scalar Velocity Potential. The same methodology from earlier sections is employed here when a long cylindrical object of radius R is placed within the flow field of an incompressible ideal fluid. The presence of the cylinder induces Vr and vg within its vicinity, but there is no axis of symmetry. The scalar velocity potential for this two-dimensional planar flow problem in cylindrical coordinates must satisfy Laplace s equation in the following form ... 

The symmetry is further broken, and the effect of the rotation translated into a poloidal field, through the combined action of circulation and turbulence. An initially axisymmetric field is sheared by differential rotation, and if it is initially cylindrical (Bz) or poloidal Br, Bg), then an azimuthal field (B ) results. Here r and 9 are the radius and latitude, respectively. A poloidal field results from a toroidal potential field. Bp = A x A, so that the toroidal magnetic field results from a distortion of the poloidal field. Finally, to convert the toroidal field back into a toroidal potential, some additional symmetry breaking is required. Turbulence in a rotating medium has vor-ticity, or handedness, which is parallel to the local angular-velocity vector and neither radial nor even hemispherically symmetric. 

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