Rotating fluid

Consider an open bucket of water resting on a turntable that is rotating at an angular velocity on (see Fig. 4-5). The (inward) radial acceleration due to the rotation is o r, which results in a corresponding radial pressure gradient at all points in the fluid, in addition to the vertical pressure gradient due to gravity. Thus the pressure differential between any two points within the fluid separated by dr and dz is 

Just like the accelerating tank, the shape of the free surface can be determined from the fact that the pressure is constant at the surface, i.e., 

This can be integrated to give an equation for the shape of the surface  

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Results of test work with this filter, producing cakes of 1 mm thickness using a 3 mm clearance, have been pubUshed (33,34). The cake formed on the medium was generally stable, giving high filtration rates over long periods of time, and the precoat type cake did not blind with time. There was no evidence of any size selectivity of the process the only exception was conventional filter aids which were preferentially picked up by the rotating fluid. This... 

As the Reynolds number rises above about 40, the wake begins to display periodic instabiUties, and the standing eddies themselves begin to oscillate laterally and to shed some rotating fluid every half cycle. These still laminar vortices are convected downstream as a vortex street. The frequency at which they are shed is normally expressed as a dimensionless Strouhal number which, for Reynolds numbers in excess of 300, is roughly constant ... 

Mechanical Gleaning. A cleaner is a hydrocyclone device utilizing fluid pressure to create rotational fluid motion (20). Pulp is introduced tangentially near the top of the cleaner. Contaminants denser than water such as chemically treated toner inks and sand migrate toward the outer wall of the cleaner and exit in a separate (reject) stream. For most forward cleaners, optimal ink removal efficiency is obtained at a pulp consistency of 0.2—0.3%. Most forward cleaners deinking efficiency declines at pulp feed consistencies greater than 0.4%. However, a cleaner said to be efficient at 1.2% pulp consistency has been reported (39). 

Theoretically, for a particle of a given size that moves in the highly rotating fluid flow in a cyclone, a particular radial orbit position may be found in every horizontal plane of the cyclone where the outward centrifugal force is just balanced by the drag exerted on the particle by the radial inward fluid flow. If Stokes s law (13.16) is assumed, then the position of the equilibrium orbit on each horizontal plane of the cyclone may be obtained and is given by... 

An appropriate model of the Reynolds stress tensor is vital for an accurate prediction of the fluid flow in cyclones, and this also affects the particle flow simulations. This is because the highly rotating fluid flow produces a. strong nonisotropy in the turbulent structure that causes some of the most popular turbulence models, such as the standard k-e turbulence model, to produce inaccurate predictions of the fluid flow. The Reynolds stress models (RSMs) perform much better, but one of the major drawbacks of these methods is their very complex formulation, which often makes it difficult to both implement the method and obtain convergence. The renormalization group (RNG) turbulence model has been employed by some researchers for the fluid flow in cyclones, and some reasonably good predictions have been obtained for the fluid flow. 

Power is the external measure of the mixer performance. The power put into the system must be absorbed through friction in viscous and turbulent shear stresses and dissipated as heat The power requirement of a system is a function of the impeller shape, size, speed of rotation, fluid density and viscosity, vessel dimensions and internal attachments, and posidon of the impeller in this enclosed system. 

In his work on the equilibrium form of a rotating fluid mass, Poincar6 indicated a powerful method for dealing with transient situations by means of a parameter (not to be confused with the parameter p previously mentioned) in the differential equation itself, which leads to the following definition ... 

Greenspan, H.R, The Theory of Rotating Fluids, Cambridge University Press, Cambridge, 1969, pp. 1-5. 

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. 

K Smith, C Colton. Mass transfer to a rotating fluid, Part II. Transport from the base of an agitated cylindrical tank. AIChE J 18 958, 1972. 

N Khoury, J Mauger, S Howard. Dissolution rate studies from a stationary disk/ rotating fluid system. Pharm Res 5 495, 1988. 

This is known as the Euler turbine equation, because it applies directly to turbines and all rotating fluid machinery. We will find it useful later in the analysis of the performance of centrifugal pumps. 

Rotational fluid velocities are calculated since horizontal (rotational) flow prevails in the hydrodynamic regime within the dissolution vessels. Thus, the overall hydrodynamics and hence dissolution is dominated by the substantially higher rotational (tangential) fluid velocities. 

Golub, J. P., and H. L. Swinney. 1975. Onset of turbulence in a rotating fluid. Physical Review Letters 35 927. 

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. 

Stationary disc in uniformly rotating fluid PV6CJ-1/2 0.761 fl (r) 85... 

P. Hillion, T. Takabayasi, and J. P. Vigier, Relativistic hydrodynamics of rotating fluid masses moving with the velocity of light, Acta Physica Polonica XIX (1960). 

J. P. Vigier, F. Halbwachs, and P. Hillion, Lagrangian formalism in relativistic hydrodynamics of rotating fluid masses, Nuovo Cimento 90(58), 818 (1958). 

The practical consequence of Eq. (2) for rotational fluid flow can be quite dramatic, as demonstrated by the high wind speeds that may be generated near the center of free vortex flows—e.g., tornadoes and typhoons. 

While the viscous model for the evolution of protoplanetary disks has had some success in matching some of the general properties of protoplanetary disks, such as the observed mass accretion rates and effective temperatures, the exact source of the viscosity remains the subject of ongoing studies. Currently, the most popular candidates for driving the mass transport in protoplanetary disks are the magneto-rotational instability (MRI) and gravitational instability. A third candidate, shear instability, has also been proposed based on laboratory experiments of rotating fluids (Richard Zahn 1999), but questions remain as to whether these results can be extended to the scale of protoplanetary disks. 

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