Reynolds number shear

A general study of the streamlines for a circular cylinder in simple shear flow can he found in the following papers C. R. Robertson and A. Acrivos, Low Reynolds number shear flow past a rotating circular cylinder, Part I, Momentum Transfer, J. Fluid Mech. 40, 685-704 (1970) R. G. Cox, I. Y. Z. Zia, and S. G. Mason, Particle motions in sheared suspensions, 15. Streamlines around cylinders and spheres, J. Colloid Interface Sci. 27, 7-18 (1968). 

Flow Past Bodies. A fluid moving past a surface of a soHd exerts a drag force on the soHd. This force is usually manifested as a drop in pressure in the fluid. Locally, at the surface, the pressure loss stems from the stresses exerted by the fluid on the surface and the equal and opposite stresses exerted by the surface on the fluid. Both shear stresses and normal stresses can contribute their relative importance depends on the shape of the body and the relationship of fluid inertia to the viscous stresses, commonly expressed as a dimensionless number called the Reynolds number (R ), EHp/]1. The character of the flow affects the drag as well as the heat and mass transfer to the surface. Flows around bodies and their associated pressure changes are important. 

In order to select the pipe size, the pressure loss is calculated and velocity limitations are estabHshed. The most important equations for calculation of pressure drop for single-phase (Hquid or vapor) Newtonian fluids (viscosity independent of the rate of shear) are those for the deterrnination of the Reynolds number, and the head loss, (16—18). 

Radial-flow impellers include the flat-blade disc turbine, Fig. 18-4, which is labeled an RlOO. This generates a radial flow pattern at all Reynolds numbers. Figure 18-17 is the diagram of Reynolds num-ber/power number curve, which allows one to calculate the power knowing the speed and diameter of the impeller. The impeller shown in Fig. 18-4 typically gives high shear rates and relatively low pumping capacity. 

Other scale-up factors are shear, mixing time, Reynolds number, momentum, and the mixing provided by rising bubbles. Shear is maximum at the tip of the impeller and may be estimated from Eq. (24-5), where the subscripts s and I stand for small and large and Di is impeller diameter [R. Steel and W. D. Maxon, Biotechnm. Bioengn, 4, 231 (1962)]. 

Other seale-up faetors are mixing time, Reynolds number, and shear. Shear is maximum at the tip of the impeller and may be determined by Equation 11-101 [17] ... 

Process results, 316, 323, 324 Pumping number, 400 Radial flow, 291 Reynolds number, 299, 303 Scale up, 312-318 Shear rate, 315... 

Chain degradation in turbulent flow has been frequently reported in conjunction with drag reduction and in simple shear flow at high Reynolds numbers [187], Using poly(decyl methacrylate) under conditions of turbulent flow in a capillary tube, Muller and Klein observed that the hydrodynamic volume, [r ] M, is the determining factor for the degradation rate in various solvents and at various polymer concentrations [188], The initial MWD of the polymers used in their experiments are, however, too broad (Mw/Iiln = 5 ) to allow for a precise... 

The ratio u/d represents the velocity gradient in the fluid, and thus the group (pu/d) is proportional to the shear stress in the fluid, so that (pu2)/(pu/d) = (dup/p) = Re is proportional to the ratio of the inertia forces to the viscous forces. This is an important physical interpretation of the Reynolds number. 

Many materials of practical interest (such as polymer solutions and melts, foodstuffs, and biological fluids) exhibit viscoelastic characteristics they have some ability to store and recover shear energy and therefore show some of the properties of both a solid and a liquid. Thus a solid may be subject to creep and a fluid may exhibit elastic properties. Several phenomena ascribed to fluid elasticity including die swell, rod climbing (Weissenberg effect), the tubeless siphon, bouncing of a sphere, and the development of secondary flow patterns at low Reynolds numbers, have recently been illustrated in an excellent photographic study(18). Two common and easily observable examples of viscoelastic behaviour in a liquid are ... 

As indicated in Section 3.7.9, this definition of ReMR may be used to determine the limit of stable streamline flow. The transition value (R ur)c is approximately the same as for a Newtonian fluid, but there is some evidence that, for moderately shear-thinning fluids, streamline flow may persist to somewhat higher values. Putting n = 1 in equation 3,140 leads to the standard definition of the Reynolds number. 

As indicated earlier, non-Newtonian characteristics have a much stronger influence on flow in the streamline flow region where viscous effects dominate than in turbulent flow where inertial forces are of prime importance. Furthermore, there is substantial evidence to the effect that for shear-thinning fluids, the standard friction chart tends to over-predict pressure drop if the Metzner and Reed Reynolds number Re R is used. Furthermore, laminar flow can persist for slightly higher Reynolds numbers than for Newtonian fluids. Overall, therefore, there is a factor of safety involved in treating the fluid as Newtonian when flow is expected to be turbulent. 

Equation 5.2 is found to hold well for non-Newtonian shear-thinning suspensions as well, provided that the liquid flow is turbulent. However, for laminar flow of the liquid, equation 5.2 considerably overpredicts the liquid hold-up e/,. The extent of overprediction increases as the degree of shear-thinning increases and as the liquid Reynolds number becomes progressively less. A modified parameter X has therefore been defined 16 171 for a power-law fluid (Chapter 3) in such a way that it reduces to X both at the superficial velocity uL equal to the transitional velocity (m )f from streamline to turbulent flow and when the liquid exhibits Newtonian properties. The parameter X is defined by the relation... 

Because concentrated flocculated suspensions generally have high apparent viscosities at the shear rates existing in pipelines, they are frequently transported under laminar flow conditions. Pressure drops are then readily calculated from their rheology, as described in Chapter 3. When the flow is turbulent, the pressure drop is difficult to predict accurately and will generally be somewhat less than that calculated assuming Newtonian behaviour. As the Reynolds number becomes greater, the effects of non-Newtonian behaviour become... 

The average Nusselt number, Nu, is presented in Fig. 4.10a,b versus the shear Reynolds number, RCsh- This dependence is qualitatively similar to water behavior for all surfactant solutions used. At a given value of Reynolds number, RCsh, the Nusselt number, Nu, increases with an increase in the shear viscosity. As discussed in Chap. 3, the use of shear viscosity for the determination of drag reduction is not a good choice. The heat transfer results also illustrate the need for a more appropriate physical parameter. In particular. Fig. 4.10a shows different behavior of the Nusselt number for water and surfactants. Figure 4.10b shows the dependence of the Nusselt number on the Peclet number. The Nusselt numbers of all solutions are in agreement with heat transfer enhancement presented in Fig. 4.8. The data in Fig. 4.10b show... 

Shear Reynolds number, based on the shear viscosity Radius... 

With turbulent channel flow the shear rate near the wall is even higher than with laminar flow. Thus, for example, (du/dy) ju = 0.0395 Re u/D is vaHd for turbulent pipe flow with a hydraulically smooth wall. The conditions in this case are even less favourable for uniform stress on particles, as the layer flowing near the wall (boundary layer thickness 6), in which a substantial change in velocity occurs, decreases with increasing Reynolds number according to 6/D = 25 Re", and is very small. Considering that the channel has to be large in comparison with the particles D >dp,so that there is no interference with flow, e.g. at Re = 2300 and D = 10 dp the related boundary layer thickness becomes only approx. 29% of the particle diameter. It shows that even at Re = 2300 no defined stress can be exerted and therefore channels are not suitable model reactors. 

In order to calculate the shear stress in the chamber several parameter have to be fulfilled To achieve a two-dimensional flow the ratio of width to height of the flow channel is 5 1 [43]. To maintain a laminar flow the Reynolds number, given as... 

This is most easily achieved by rotating the inner cylinder and keeping the outer fixed in the laboratory frame. Note, however, that this geometry leads to the formation of Taylor vortex motion if inertial effects become important (Reynolds number Re 1). Most rheo-NMR experiments are actually performed at low Re. In the cylindrical Couette, the natural coordinates are cylindrical polar (q, <(>, z) so the shear stress is denoted and is radially dependent as q 2. The strain rate across the gap is given by [2]... 

In equation 5.3, and when calculating the Reynolds number for use with Figure 5.7, the fluid viscosity and density are taken to be constant. This will be true for Newtonian liquids but not for non-Newtonian liquids, where the apparent viscosity will be a function of the shear stress. 

The three basic types of impeller which are used at high Reynolds numbers (low viscosity) are shown in Figures 10.55a, b, c. They can be classified according to the predominant direction of flow leaving the impeller. The flat-bladed (Rushton) turbines are essentially radial-flow devices, suitable for processes controlled by turbulent mixing (shear controlled processes). The propeller and pitched-bladed turbines are essentially axial-flow devices, suitable for bulk fluid mixing. 

The Weber-Reynolds number (Re/We) is defined as the ratio of surf ace tension of a bubble to viscous shear on the bubble surface due to bubble motion ... 

Annular flow. In annular flow, as mentioned in Section 3.4.6.1, modeling of the interfacial shear remains empirical. For adiabatic two-phase flow, Asali et al. (1985) suggested that the friction factor, fjfs, is dependent on a dimensionless group for the film thickness, 8+, as defined in Eq. (3-136), and the gas Reynolds number, Rec ... 

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