Plane shear flow

The motion of a free spherical particle in an arbitrary plane shear flow was studied in [342, 343]. 

Figure 2.11. Linear shear flow past a freely rotating circular cylinder in the 7Z -plane (the limit streamlines P = Ps are marked bold) (a) simple shear flow ( STg = 1) (b) general case of plane shear flow (0 < n < 1)...

For the axisymmetric and plane shear flows (see Table 4.4), the error of formula (4.8.1) does not exceed 1%. 

Simple Shear and Arbitrary Plane Shear Flows... 

By virtue of the no-slip condition on the surface, a sphere freely suspended in a plane shear flow will rotate at a constant angular velocity fl equal to the flow rotation velocity at infinity. The solution of the corresponding three-dimensional hydrodynamic problem on a particle in a Stokes flow is given in [343]. 

To describe the solution of the mass transfer problem for an arbitrary plane shear flow at high Peclet numbers, we introduce dimensionless quantities by the formulas ... 

Stone. P.A.. Roy. A.. Larson, R.G., Waleffe, F., and Graham, M. (2004) Polymer drag reduction in exact coherent structures of plane shear flow. Phys. Fluids. 16, 3470-3482. 

Waleffe, F. (1998) Three-dimensional coherent states in plane shear flows. Phys. Rev. Lett., 81, 4140-A143. 

Da Cruz, E, S. Emem, M. Prochnow, J.-N. Roux, and R Chevoir. 2005. Rheophysics of dense granular materials Discrete simulation of plane shear flows. Phys. Rev. E 72, 021309. 

E. Dubois-Violette, E. Guyon, I. Janossy, P. Pieranski and P. Manneville, Theory and experiments on plane shear flow instabilities in nematics, J. Mecanique, 16, 733-767 (1977). 

Figure A3.1.5. Steady state shear flow, illustrating the flow of momentum aeross a plane at a height z.

Liquids are able to flow. Complicated stream patterns arise, dependent on geometric shape of the surrounding of the liquid and of the initial conditions. Physicists tend to simplify things by considering well-defined situations. What could be the simplest configurations where flow occurs Suppose we had two parallel plates and a liquid drop squeezed in between. Let us keep the lower plate at rest and move the upper plate at constant velocity in a parallel direction, so that the plate separation distance keeps constant. Near each of the plates, the velocities of the liquid and the plate are equal due to the friction between plate and liquid. Hence a velocity field that describes the stream builds up, (Fig. 15). In the simplest case the velocity is linear in the spatial coordinate perpendicular to the plates. It is a shear flow, as different planes of liquid slide over each other. This is true for a simple as well as for a complex fluid. But what will happen to the mesoscopic structure of a complex fluid How is it affected Is it destroyed or can it even be built up For a review of theories and experiments, see Ref. 122. Let us look into some recent works. 

A straightforward estimate of the maximum hardness increment can be made in terms of the strain associated with mixing Br and Cl ions. The fractional difference in the interionic distances in KC1 vs. KBr is about five percent (Pauling, 1960). The elastic constants of the pure crystals are similar, and average values are Cu = 37.5 GPa, C12 = 6 GPa, and C44 = 5.6 GPa. On the glide plane (110) the appropriate shear constant is C = (Cu - C12)/2 = 15.8 GPa. The increment in hardness shown in Figure 9.5 is 14 GPa. This corresponds to a shear flow stress of about 2.3 GPa. which is about 17 percent of the shear modulus, or about C l2n. 

For convenience, a mathematically simple arrangement is considered. It consists of a fluid layer of finite constant thickness, confined by two rigid parallel planes of infinite extension. Steady laminar shear flow is created in this layer by fixing one plane in space and moving the other one with constant speed in a direction parallel to both planes. In this way, a truly uniform and time independent shear rate q is created in the liquid. The magnitude of this shear rate is simply given by the ratio of the said speed to the mutual distance of the planes. Experimentally such an arrangement is approximated e.g. by the use of two coaxial cylinders. When the gap between the inner surfaces of these cylinders is made small compared with their radii, the above mentioned situation can be realized to a sufficient extent. 

The state of stress in a flowing liquid is assumed to be describable in the same way as in a solid, viz. by means of a stress-ellipsoid. As is well-known, the axes of this ellipsoid coincide with directions perpendicular to special material planes on which no shear stresses act. From this characterization it follows that e.g. the direction perpendicular to the shearing planes cannot coincide with one of the axes of the stress-ellipsoid. A laboratory coordinate system is chosen, as shown in Fig. 1.1. The x- (or 1-) direction is chosen parallel with the stream lines, the y- (or 2-) direction perpendicular to the shearing planes. The third direction (z- or 3-direction) completes a right-handed Cartesian coordinate system. Only this third (or neutral) direction coincides with one of the principal axes of stress, as in a plane perpendicular to this axis no shear stress is applied. Although the other two principal axes do not coincide with the x- and y-directions, they must lie in the same plane which is sometimes called the plane of flow, or the 1—2 plane. As a consequence, the transformation of tensor components from the principal axes to the axes of the laboratory system becomes a simple two-dimensional one. When the first principal axis is... 

The strategy is as follows. We start by rewriting the equations in cylindrical coordinates (r, ,z). The variables we consider are the layer displacement u (now in the radial direction) from the cylindrical state, the director n, and the fluid velocity v. The central part of the cylinder, r < Ri, containing a line defect, is not included. It is not expected to be relevant for the shear-induced instability. We write down linearized equations for layer displacement, director, and velocity perturbations for a multilamellar (smectic) cylinder oriented in the flow direction (z axis). We are interested in perturbations with the wave vector in the z direction as this is the relevant direction for the hypothetical break-up of the cylinder into onions. The unperturbed configuration in the presence of shear flow (the ground state) depends on r and 0 and is determined numerically. The perturbations, of course, depend on all three coordinates. We take into account translational symmetry of the ground state in the z direction and use a plane wave ansatz in that direction. Thus, our ansatze for the perturbed variables are... 

Using this concept, Erwin [9] demonstrated that the upper bound for the ideal mixer is found in a mixer that applies a plane strain extensional flow or pure shear flow to the fluid and where the surfaces are maintained ideally oriented during the whole process this occurs when N = 00 and each time an infinitesimal amount of shear is applied. In such a system the growth of the interfacial areas follows the relation given by... 

Now consider a case of collision of a fixed single sphere with a cloud of particles as shown in Fig. 5.8(b). The sphere of radius r is subjected to a shear flow of a particle cloud with a velocity gradient of dUp/dy. The particle number density is denoted as n, and the mass of a particle is m. Select the velocity Up as zero in the center plane of the sphere. The relative velocity is estimated by... 

The experimental arrangement analyzed in this section is shown in Figure 2.11. Here a shear flow is applied in the (x, y) plane, with the flow in the x direction. Due to the symmetry of the flow, the principal axes of the refractive index tensor will have 3 oriented... 

Oblique incidence of light through a shear flow. The axes defining the components n j and n2 lie in the (JC, y) plane. 

It is instructive to present the solution to equation (7.106) for the case of simple shear flow. For a spheroid oriented with its symmetry axis defined by the polar angle 9 relative to the z axis, and azimuthal axis <(> measured in the (x, y) plane relative to the x axis, equation (7.106) produces the following two equations for a simple shear flow of the form, v = G (0, x, 0) ... 

The velocity field between the cone and the plate is visualized as that of liquid cones described by 0-constant planes, rotating rigidly about the cone axis with an angular velocity that increases from zero at the stationary plate to 0 at the rotating cone surface (23). The resulting flow is a unidirectional shear flow. Moreover, because of the very small i//0 (about 1°—4°), locally (at fixed r) the flow can be considered to be like a torsional flow between parallel plates (i.e., the liquid cones become disks). Thus... 

Figure E7.2 compares a stepwise increase in interfacial area in simple shear flow with optimal initial orientation, and simple shear flow where, at the beginning of each step, the interfacial area element is placed 45° to the direction of shear. The figure shows that, whereas in the former case the area ratio after four shear units is 4.1, in the latter case the ratio is 6.1, with a theoretical value of 7.3 when the 45° between the plane and direction of shear is maintained at all times. We note, however, that it is quite difficult to generate steady extensional flows for times sufficiently long to attain the required total elongational strain. This is why a mixing protocol of stepwise stretching and folding (bakers transformation) is so efficient. Not only does it impose elongational stretching, but it also distributes the surface area elements over the volume.

Fig. E7.2 Schematic representation of interfacial area increase in simple shear flow, (a) With initial orientation of 45° to the direction of shear, and after each shear unit, the plane is rotated hack to 45° orientation, (b) With optimal initial orientation and no rotation.

The research of Roy Jackson combines theory and experiment in a distinctive fashion. First, the theory incorporates, in a simple manner, inertial collisions through relations based on kinetic theory, contact friction via the classical treatment of Coulomb, and, in some cases, momentum exchange with the gas. The critical feature is a conservation equation for the pseudo-thermal temperature, the microscopic variable characterizing the state of the particle phase. Second, each of the basic flows relevant to processes or laboratory tests, such as plane shear, chutes, standpipes, hoppers, and transport lines, is addressed and the flow regimes and multiple steady states arising from the nonlinearities (Fig. 6) are explored in detail. Third, the experiments are scaled to explore appropriate ranges of parameter space and observe the multiple steady states (Fig. 7). One of the more striking results is the... 

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