Azimuthal velocity

The azimuthal component of the fluid velocity, v, is identical to vx and the local fluid angular velocity is co = v /q. This azimuthal velocity is y sin0oo(0) = qoo(0) and the local shear rate, y, is -sin0 which, for no slip boundary conditions, is Q/a for small a. Under uniform shear with no slip, it may be shown that dvx/dy 0 and dvx/dy y[2, 17]. 

For large values of 6, Manohar [42], and Banks [4] solved the boundary layer Eqs. (2)-(5) numerically with a finite difference method. Manohar s results for the meridional and azimuthal velocity gradients on the spherical surface have been curve-fitted by Newman [45] and Chin [18] to follow the following equations in the regime of 0 < 9 < 7t/2 ... 

For turbulent flow on a rotating sphere or hemisphere, Sawatzki [53] and Chin [22] have analyzed the governing equations using the Karman-Pohlhausen momentum integral method. The turbulent boundary layer was assumed to originate at the pole of rotation, and the meridional and azimuthal velocity profiles were approximated with the one-seventh power law. Their results can be summarized by the... 

The Chilton-Colburn analogy can be also used to estimate the local mass transfer rate in laminar flow where the wall shear stress is related to the azimuthal velocity gradient by... 

With a view to any simulation, a few important items have to be addressed. First of all, it has to be decided whether the flow to be simulated is 2-D, 21/2-D, or 3-D. When the flow is, e.g., axis-symmetrical and steady, a 2-D simulation may suffice. For a flow field in which all variables, including the azimuthal velocity component, may not depend on the azimuthal coordinate, a 21/2-D simulation may be most appropriate. Most other cases may require a full 3-D simulation. It is tempting to reduce the computational job by casting the 3-D flow field into a 2-D mode. The experience, however, is that in 2-D simulations the turbulent viscosity tends to be overestimated in this way, the flow... 

Reynolds number reverse osmosis coefficient Schmidt number Sherwood number filament surface area of spacer specific surface area of spacer spiral wound module bubble rising velocity mean azimuthal velocity time... 

Velocity maps were measured with How-compensated spin-echo sequences [Rof l], The expefimental data were analysed in terms of a power law for the shear stress a[2 which solves the constitutive equation (T 2 = f)(dy/d/)" for the shear-rate-dependent viscosity rj and the applied shear strain y. The azimuthal velocity at radius r and rotation speed wr of the inner cylinder can be expressed as... 

The second group, with 0 = 0, and 0 = 0 for which the spectral function is denoted L(z), comprises all the V-particles. They precess with the constant azimuthal velocity at the frozen polar angle 0 = 0, in which the effective potential is minimal. Particles of this group, unlike the librators, do not occupy the flat part of the hat curved potential well. 

A number of commercially available rotating disk dynamic membrane systems have been tested for different applications. The DMF module (Pall Corp., New York) consists of several disks mounted on the same shaft, such that each can rotate between two annular membranes with maximum rotation speed of 3450 rpm, corresponding to an azimuthal velocity of 20 m/s at the disk tip. Pali s lab-scale system has been tested for protein separation [64] and filtration of recombinant yeast cells [65]. The studies of the application of the rotating disk dynamic membrane indicated that high-shear-enhanced filtration is much less sensitive to the solid concentration. 

The number of particles of charge q and mass m in a ring of major radius R (see Fig. 1) is obtained as follows. The circulating ion current 1 = (Ni/27rR) q V0 V0 = R is the mean azimuthal velocity and 0. is the mean rotational frequency. In general, e... 

The bulk flow is given by the incompressible Navier-Stokes equations which follow. Note that eventhough we assume that the flow variables are independent of 9, we will allow for a non-zero azimuthal velocity u(r,. 2, t). This can arise from a solid body rotation, for example and can be used to model the effects of centrifugal forces on capillary instability. The momentum and continuity equations (in component) form are ... 

At this end, to demonstrate superiority of our classical algorithms, we show some sample results of our most recent FDF simulation of the Sandia/Sydney swirl burner [35]. This configuration is selected as it is one of the most challenging turbulent flames for prediction. Figure 3 shows the contours of the azimuthal velocity field as predicted by our FDF. The simulated results agree with experimental data better than any other classical methods currently available [36]. But the computational time requirements are excessive. As another example. Fig. 4 shows the contour of filtered temperature field for the symbolic Taylor-Green vortex flow as obtained via FDF coupled with a discontinuous Galerkin flow solver [37]. Quantum computation may potentially provide a much more efficient means for such simulations. 

The system of balance equations adopted for the charged species is given in Table 8.1. In this table, the electrons have an axial velocity and an azimuthal velocity g the... 

A detailed consideration of relative gas motion is necessary to determine the angles of incidence and the amount of the relative gas velocity. Therefore, Fig. 5.2 shows the motion of the jet and the gas phase. According to the sketch, the relative motion consists of three parts. There is a relative movement due to the axial velocity of the jet w and because of the impressed gas flow with the velocity Vg. Additionally, as a result of the rotation of the system the jet is exposed to a relative gas motion in the azimuthal direction. While the local azimuthal velocity of the system is given by Vrot = am, the velocity of the gas phase in the azimuthal direction has to be modeled. According to measurements of Mescher and Walzel [36] the azimuthal gas velocity depends on radial distance to the atomizer and is between... 

Fig. 7 Azimuthal velocity of binary fluids in a rotating cylinder with fio = O.O / bF/moa yi. The viscous fluids with particle mass m and mo are located dXr

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