Solid body rotation

Solid-Body Rotation When a body of fluid rotates in a sohd-body mode, the tangential or circumferential velocity is linearly proportional to radius ... 

The Giesekus criterion for local flow character, defined as

extensional flow, 0 in simple shear flow and — 1 in solid body rotation [126]. The mapping of J> across the flow domain provides probably the best description of flow field homogeneity current calculations in that direction are being performed in the authors laboratory. 

An alternative to the rotating disk method in a quiescent fluid is a stationary disk placed in a rotating fluid. This method, like the rotating disk, is based on fluid mechanics principles and has been studied using benzoic acid dissolving into water [30], Khoury et al. [31] applied the stationary disk method to the study of the mass transport of steroids into dilute polymer solutions. Since this method assumes that the rotating fluid near the disk obeys solid body rotation, the stirring device and the distance of the stirrer from the disk become important considerations when it is used. A similar device was developed by Braun and Parrott [32], who used stationary spherical tablets in a stirred liquid to study the effect of various parameters on the mass transport of benzoic acid. 

For a uniform angular velocity ( > = constant, i.e., a solid body rotation ), n = — 1, whereas for a uniform tangential velocity ( plug flow ) n = 0, and for inviscid free vortex flow co = c/r2, i.e., n = 1. Empirically, the exponent n has been found to be typically between 0.5 and 0.9. The maximum value of Ve occurs in the vicinity of the outlet or exit duct (vortex finder) at r = De/2. 

When solid body rotation is assumed in the CE (model B), the degree of differential rotation at its base is too low to trigger efficient shear-induced turbulence between the outer part of the hydrogen burning shell (HBS) and the CE (solid lines in Fig. la). On the contrary, in our model C the differential rotation... 

Great care has to be given to the physics of rotation and to the treatment of its interaction with mass loss. For differentially rotating stars, the structure equations need to be written differently [9] than for solid body rotation. For the transport of the chemical elements and angular momentum, we consider the effects of shear mixing, meridional circulation, horizontal turbulence and in the advanced stages the dynamical shear is also included. Caution has to be given that advection and diffusion are not the same physical effect. 

Equation 26 is accurate only when the liquids rotate at the same angular velocity as the bowl. As the liquids move radially inward or outward these must be accelerated or decelerated as needed to maintain solid-body rotation. The radius of the interface, r, is also affected by the radial height of the liquid crest as it passes over the dischaige dams, and these crests must be considered at higher flow rates. 

The AO = (w/r)dt terms represent the solid-body rotation due to the circumferential velocity w. So, even if there were no shearing (i.e., da = df8 = 0) there would still be a... 

One must be aware of the practical limitations that are inherent in the one-dimensional analysis of a problem like this one. If the analysis is carried to an infinite amount of time, the solution of Eq. 4.119 would predict that the fluid rotation is induced in an infinite extent of space surrounding the shaft. It is obvious that while such a shaft can produce motion in the nearby fluid, it cannot ultimately bring the entire atmosphere up to a solid-body rotation. After a certain amount of time, as the rotating fluid expands outward, the one-dimensionality must be interruped by encounter with surfaces or by fluid instability. 

The flow field in Eq. (Al-7) is really just a solid-body rotation which rotates, but does not deform, the fluid element. As a result, the rate-of-strain tensor D is the zero tensor, and the Finger strain tensor is the unit tensor. 

Slipping The slipping regime occurs when the granular bed undergoes solid body rotation and then... 

In fact as long as each of the stencils sums to 1, the matrix will always have a unit eigenvalue with an eigencolumn of Is 20Can we do this Yes, because each stencil defines an affine combination, and affine combinations are invariant under translations. In fact they are invariant under solid body rotations, scalings and affine transforms too. 

In turbulent mixing unwanted phenomena such as solid body rotation and central surface vortices may occur. In solid body rotation the fluid rotates as if it was a solid mass, and as a result little mixing takes place. At high impeller rotational speeds the centrifugal force of the impeller moves the fluid out to the walls creating a surface vortex. This vortex may even reach down to the impeller resulting in air entrainment into the fluid [87]. 

The characteristic solid body rotation is the primary flow pattern in un-bafiled tanks. To avoid these phenomena, baffles are installed in the tank. On wall baffles are sketched in Fig 7.1. Generally, baffles are placed in the tank to modify the flow and surface destroy vortices. Baffles mounted at the tank wall are most common, but also bottom baffles, floating surface baffles and disk baffles at the impeller shaft are possible. Often tank wall baffles are mounted a certain distance from the wall, as illustrated in Fig 7.3. This creates a different flow pattern in the tank. The purpose of installing baffles away from the wall is to avoid dead zones where liquid is seldom exchanged and where impurities accumulate. Experiments have confirmed that the flow patterns in baffled agitated tanks are different from the flow patterns in unbaffled agitated tanks. In baffled tanks the discharge flow dissipates partly in the bulk... 

The solvent is assumed to be in solid body rotation at an angular speed (o, and the solute is assumed to move circumferentially with the solvent. A single solute is considered, that is, a binary mixture, and a cylindrical coordinate system rotating with the angular speed (o is adopted. The solute concentration is then a function only of the time t and radial distance r from the rotation axis. The continuity (diffusion) equation (Eq. 3.3.15) can therefore be written... 

Solid-body rotation (sometimes called forced vortex). 

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