Cylindrically symmetric velocity field

We now discuss that in the jetting regime radius selection is dominated by simple hydrodynamics [46]. We assume that the shear stress and the velocity difference across the membrane can be neglected. For the analysis of the hydrodynamic flow profile, we can hence focus on the fluid motion inside and outside of the tube. In the experiments shown in Figure 11.10, the outer silicate solution is confined to a glass cylinder of radius J cyi(l-1 cm)- Along its central axis, buoyant cupric sulfate solution ascends as a cyHndrical jet of radius R. The cylindrically symmetric velocity fields v(r), which solve the Navier-Stokes equations, are... 

It has been shown by Svendsen et al. [161], among others, that the time averaged experimental data on the flow pattern in cylindrical bubble columns is close to axi-symmetric. Fair agreement between experimental- and simulated results are generally obtained for the steady velocity fields in both phases, whereas the steady phase distribution is still aproblem. Therefore, it was anticipated that the 2D axi-symmetric simulations capture the pertinent time-averaged flow pattern that is needed for the analysis of many (not aU) mechanisms of interest for chemical engineers. Sanyal et al. [137] and Krishna and van Baten [81] for example stated that the 2D models... 

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. 

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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