Three-Dimensional Shear Flows

A method for solving three-dimensional problems on the diffusion boundary layer based on a three-dimensional analog of the stream function, was proposed in [348, 350]. In [27, 166, 353], this method was used for studying mass exchange between spherical particles, drops, and bubbles and three-dimensional shear flow. 

David.son, L, Large eddy simulation A dynamic one-equation subgrid model for three-dimensional recirculating flow. In llth Int. Symp. on Turbulent Shear Flow, vol. 3, pp. 26.1-26.6, Grenoble, 1997. 

A three dimensional turbulent flow field in unbaffled tank with turbine stirrer or 6-paddle stirrer was numerically simulated by the method of finite volume elements [80], whereas in the case of free surface the vortex profile was also determined using iterative techniques. The prediction of the velocity and turbulence fields in the whole tank and the stirrer power was compared with literature data and their own results. Of the two simulation techniques used, turbulent eddy-viscosity/zc-e turbulence model and the DS model (differential 2. order shear stress), only the latter produced satisfactory results. In particular it proved that fluctuating Coriolis forces have to be taken into account by source terms in the transport equation for the Reynolds shear stress. 

Table 3.9 shows many various rheological models used to categorize and model fluid systems. It is written in terms of the three-dimensional form (where terms are discussed in Macosko (1994)) and the simple two-dimensional shear-flow relationship (where terms have been defined here). 

Purely viscous constitutive equations, which account for some of the nonlinearity in shear but not for any of the history dependence, are commonly used in process models when the deformation is such that the history dependence is expected to be unimportant. The stress in an incompressible, purely viscous liquid is of the form given in equation 2, but the viscosity is a function of one or more invariant measures of the strength of the deformation rate tensor, [Vy - - (Vy) ]. [An invariant of a tensor is a quantity that has the same value regardless of the coordinate system that is used. The second invariant of the deformation rate tensor, often denoted IId, is a three-dimensional generalization of 2(dy/dy), where dy/dy is the strain rate in a one-dimensional shear flow, and so the viscosity is often taken to be a specific function-a power law, for example-of (illu). ]... 

Becu L, Anache D, ManneviUe S, Cohn A (2007) Evidence for three-dimensional unstable flows in shear-banding wotmUke miceUes. Phys Rev E 76(1) 011503... 

Solutions of methylceUuloses are pseudoplastic below the gel point and approach Newtonian flow behavior at low shear rates. Above the gel point, solutions are very thixotropic because of the formation of three-dimensional gel stmcture. Solutions are stable between pH 3 and 11 pH extremes wiU cause irreversible degradation. The high substitution levels of most methylceUuloses result in relatively good resistance to enzymatic degradation (16). 

The Prandtl mixing length concept is useful for shear flows parallel to walls, but is inadequate for more general three-dimensional flows. A more complicated semiempirical model commonly used in numerical computations, and found in most commercial software for computational fluid dynamics (CFD see the following subsection), is the A — model described by Launder and Spaulding (Lectures in Mathematical Models of Turbulence, Academic, London, 1972). In this model the eddy viscosity is assumed proportional to the ratio /cVe. 

This fitted the data well up to volume fractions of 0.55 and was so successful that theoretical considerations were tested against it. However, as the volume fraction increased further, particle-particle contacts increased until the suspension became immobile, giving three-dimensional contact throughout the system flow became impossible and the viscosity tended to infinity (Fig. 2). The point at which this occurs is the maximum packing fraction, w, which varies according to the shear rate and the different types of packings. An empirical equation that takes the above situation into account is given by [23] ... 

The concepts of interface rheology are derived from the rheology of three-dimensional phases. Characteristic for the interface rheology is the coupling of the motions of an interface with the flow processes in the bulk close to the interface. Thus, in interface rheology the shear and dilatational stresses of the interface are in equilibrium with the corresponding shear stress in the bulk. An important feature is the compressibility of the adsorption layer of an interface in contrast, the flow elements of the bulk are incompressible. As a result, compression or dilatation of the adsorption layer of a soluble surfactant is associated with desorption and adsorption processes by which the interface tends to reinstate the adsorption equilibrium with the bulk phase. 

In pseudoplastic substances shear thinning depends mainly on the particle or molecular orientation or alignement in the direction of flow, this orientation is lost or regained at the same speed. Additionally many dispersions show this potential for particle or molecule interactions, this leads to bonds creating a three-dimensional network structure. They are often build-up from relatively weak hydrogen or ionic bonds. When the network is disturbed. 

The calculation method and equations presented in the previous sections are for Newtonian fluids such that the flow due to screw rotation and the downstream pressure gradient can be solved independently, that is, via the principle of superposition. Since most resins are highly non-Newtonian, the rotational flow and pressure-driven flow in principle cannot be separated using superposition. That is, the shear dependency of the viscosity couples the equations such that they cannot be solved independently. Potente [50] states that the flows and pressure gradients should only be calculated using three-dimensional (3-D) numerical methods because of the limitations of the Newtonian model. 

Runnels and Eyman [41] report a tribological analysis of CMP in which a fluid-flow-induced stress distribution across the entire wafer surface is examined. Fundamentally, the model seeks to determine if hydroplaning of the wafer occurs by consideration of the fluid film between wafer and pad, in this case on a wafer scale. The thickness of the (slurry) fluid film is a key parameter, and depends on wafer curvature, slurry viscosity, and rotation speed. The traditional Preston equation R = KPV, where R is removal rate, P is pressure, and V is relative velocity, is modified to R = k ar, where a and T are the magnitudes of normal and shear stress, respectively. Fluid mechanic calculations are undertaken to determine contributions to these stresses based on how the slurry flows macroscopically, and how pressure is distributed across the entire wafer. Navier-Stokes equations for incompressible Newtonian flow (constant viscosity) are solved on a three-dimensional mesh ... 

In order to model this type of flow field geometrically, Beltrami found that it was necessary to consider a three-dimensional circular axisymmetric flow in which the velocity and vorticity field lines described a helical pattern. This helicoidal flow field was unique in that the pitch of the circular helices decreased as the radius from the central axis increased. This produces a specialized shear effect between the field lines of successively larger cylindrical tubes constituting the respective helices. In the limit of such a field, the central axis of the flow also serves as a field line (see Fig. 3). 

Altobelli et al., used a more elaborate three-dimensional MRI technique to study the flow of suspended particles (44) and granular flows (27), also studied by Ng et al. (45) for pellet-sized pills under load, while being sheared in a nonmagnetic shear box, similar to the Jenicke cell (22). This technique holds great potential for detecting details of particulate movements and deformations of three-dimensional particulate assemblies, but is currently limited to very low shearing velocities. 

A similar analysis can be performed for three-dimensional lattices subjected to the same flow. The corresponding maximum concentration curve in three dimensions is shown in Fig. 3 as a function of the flow parameter X. This curve displays a discontinuous dependence on X in the neighborhood of X = 0, revealing a very special feature of simple shear flow. The saw-tooth property characterizing hyperbolic flows (X > 0) is derived from the best estimates... 

Fig. 4. The two possible configurations for a three-dimensional lattice in a simple shear flow. The velocity field is given by u = Gz e is the projection of I3 onto the x-y plane (a) slide flow, (b) tube flow.

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