Steady rotational flow

Figure 5,16. It is assumed that by using an exactly symmetric cone a shear rate distribution, which is very nearly uniform, within the equilibrium (i.e. steady state) flow held can be generated (Tanner, 1985). Therefore in this type of viscometry the applied torque required for the steady rotation of the cone is related to the uniform shearing stress on its surface by a simplihed theoretical equation given as...

These are the most successful types of reactors presently available. The Internal reciprocating plunger types, for example, that of Nelles in Jankowski et al (1978), do not provide a steady uniform flow. Of those operating with rotating blowers or turbines, the best known are those of Garanin et al (1967), Brown and Bennett (1972), Livbjerg and Villadsen (1971). These and that of Rbmer and Luft (1974) are shown on Figures 2.4.2 a-d. 

This is a statement of Bernoulli s theorem the quantity v2l2+Plp+gh is constant throughout the fluid for steady, irrotational flow. Equation A.33 is the same as equation 1.11. It will be recalled that, for rotational flow with friction, the engineering form of Bernoulli s equation applies only along a streamline and allowance must be made for frictional losses. 

The first mode may occur when a droplet is subjected to aerodynamic pressures or viscous stresses in a parallel or rotating flow. A droplet may experience the second type of breakup when exposed to a plane hyperbolic or Couette flow. The third type of breakup may occur when a droplet is in irregular flow patterns. In addition, the actual breakup modes also depend on whether a droplet is subjected to steady acceleration, or suddenly exposed to a high-velocity gas stream.[2701[2751... 

Just as there arc many types of fluids, so there arc. partly as a result, many types of fluid flow. Uniform flow is steady in lime, or the same at all points in space. Steady flow is flow of which the velocity at a point fixed with respect to a fixed system of coordinates is independent of lime. Many common types of flow can be made steady by a suitable choice of coordinates. Rotational flows have appreciable vorticily, and they cannot he described mathematically by a velocity potential function. Turbulent flow is flow in which the fluid velocity at a fixed point fluctuates with lime in a nearly random way. The motion is essentially rotational, and is... 

Application of the steady state condition in such cases shows that any rotational flow is possible, i.e., the intensity of flow is indeterminate. This fact has puzzled chemists and physicists for many years. The first explicit statement concerning this puzzle known to the author is by Wegscheider in a paper of 1902 (26). 

Yamane, K., Nakagawa, M., Altobelli, S., Tanaka, T., and Tsuji, Y. (1998), Steady particulate flows in a horizontal rotating cylinder, Phys. Fluids, 10,1419-1427. 

For solutions of nonspherical particles the situation is more complicated and the physical picture can be described qualitatively as follows for a system of particles in a fluid one can define a distribution function, F (Peterlin, 1938), which specifies the relative number of particles with their axes pointed in a particular direction. Under the influence of an applied shearing stress a gradient of the distribution function, dFfdt, is set up and the particles tend to rotate at rates which depend upon their orientation, so that they remain longer with their major axes in position parallel to the flow than perpendicular to it. This preferred orientation is however opposed by the rotary Brownian motion of the particles which tends to level out the distribution or orientations and lead the particles back toward a more random distribution. The intensity of the Brownian motion can be characterized by a rotary diffusion coefficient 0. Mathematically one can write for a laminar, steady-state flow ... 

Water is poured in steadily from the top. If the flow rate is too slow, the top cups never fill up enough to overcome friction, so the wheel remains motionless. For faster inflow, the top cup gets heavy enough to start the wheel turning (Figure 9.1.1a). Eventually the wheel settles into a steady rotation in one direction or the other (Figure 9.1.1b). By symmetry, rotation in either direction is equally possible the outcome depends on the initial conditions. 

By increasing the flow rate still further, we can destabilize the steady rotation. Then the motion becomes chaotic the wheel rotates one way for a few turns, then some of the cups get too full and the wheel doesn t have enough inertia to carry them over the top, so the wheel slows down and may even reverse its direction (Figure 9.1.1c). Then it spins the other way for a while. The wheel keeps changing direction erratically. Spectators have been known to place bets (small ones, of course) on which way it will be turning after a minute. 

A very interesting series of studies of the influence of end effects in the rotating concentric cylinder problem has been published by Mullin and co-workers T. Mullin, Mutations of steady cellular flows in the Taylor experiment,J. Fluid Mech. 121, 207-18 (1982) T. B. Benjamin and T. Mullin, Notes on the multiplicity of flows in the Taylor experiment, J. Fluid Mech. 121, 219-30 (1982) K. A. Cliff and T. Mullin, A numerical and expwerimental study of anomalous modes in the Taylor experiment, J. Fluid Mech. 153, 243-58 (1985) G. Pfister, H. Schmidt, K. A. Cliffe and T. Mullin, Bifurcation phenomena in Taylor-Couette flow in a very short annulus, J. Fluid Mech. 191, 1-18 (1988) K. A. Cliffe, 1.1. Kobine, and T. Mullin, The role of anomalous modes in Taylor-Couette flow, Proc. R. Soc. London Ser. A 439, 341-57 (1992) T. Mullin, Y. Toya, and S. I. Tavener, Symmetry breaking and multiplicity of states in small aspect ratio Taylor-Couette flow, Phys. Fluids 14, 2778-87 (2002). 

Figure 1. Hydrophoretic separation principle, (a) Schematic showing a hydrophoretic device with anisotropic microfluidic obstacles, (b and c) Simulated streamlines in the device. The slanted groove patterns on the channel generate rotational flows by using a steady axial pressure gradient, (d) Different particle ordering according to particle size by steric hindrance mechanism. (Reproduced with permission from Ref [20] Copyright 2009, American Chemical Society.)...

Figure 2.4 Streamlines of the steady cellular flow composed of an array of counter-rotating vortices (top row) and the spreading in time of a weakly diffusing passive concentration field. Time increases from top to bottom.

This equation, often referred to as the moment-of-momentum equation is one of the basic tools in the analysis of rotating fluid machines, turbines, pumps, and other devices [5]. In the steady-state flow dLldt) is zero and equals m p.so we have... 

The stream function is very useful because its physical significance is that in steady flow lines defined by ip= constant are streamlines which are the actual curves traced out by the particles of fluid. A stream function exists for all two-dimensional, steady, incompressible flow whether viscous or inviscid and whether rotational or irrotational. 

There are two basic types of extruders continuous and discontinuous or batch type extruders. Continuous extruders are capable of developing a steady, continuous flow of material, whereas batch extruders operate in a cyclic fashion. Continuous extruders utilize a rotating member for transport of the material. Batch extruders generally have a reciprocating member to cause transport of the material. 

Weissenberg Camera. A camera for X-ray diffraction analysis of crystal structures. The crystal is rotated, in the X-ray beam. The film is rotated and moved parallel to the axis of rotation. Weissenberg Rheogoniometer. In essence a cone and plate viscometer (q.v.), but with accessories and control systems to allow it to measure, as well as viscosity, elasticity and a wide range of flow properties such as dilatancy, thixotrophy, relaxation phenomena, etc in steady rotation or in torsional oscillation, over wide but controlled ranges of temperature and shear. 

For creep of viscoelastic liquids, one of the rotational geometries is usually chosen. A constant torque is suddenly applied (actually, within a small time interval t ) and after lapse of a period somewhat longer than ti the angular displacement is followed as a function of time. After steady-state flow has been achieved, it is desirable to remove the torque and follow the creep recovery as described in Chapter 1, Section E. Absence of perceptible friction is essential. For temperature control, which is important because of the strong dependence of viscoelastic properties on temperature, there must be no heat leakage through the device used to apply the torque connecting members of low thermal conductivity are useful. 

With a suitable mechanical drive, small angular oscillating deformations can be imposed on steady rotation at various shear rates, using either coaxial cylinder or cone and plate geometry, and the dynamic mechanical properties can be measured under conditions of non-Newtonian flow. By use of annular pumping geometry with steady rotation of the outer cylinder, the viscoelastic properties under conditions of non-Newtonian flow perpendicular to the oscillatory motion can be measured. ... 

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