Sphere in a simple shear flow

Example 52 On the basis of the kinetic theory, which is used to model collision-dominated gas-solid flows, derive a general expression of solid stresses of elastic spheres in a simple shear flow. 

Figure 7-2. Illustration of the decomposition of the problem of a freely rotating sphere in a simple shear flow as the sum of three simpler problems (a) a sphere rotating in a fluid that is stationary at infinity, (b) a sphere held stationary in a uniform flow, and (c) a nonrotating sphere in a simple shear flow that is zero at the center of the sphere. The angular velocity Cl in (a) is the same as the angular velocity of the sphere in the original problem. The translation velocity in (b) is equal to the undisturbed fluid velocity evaluated at the position of the center of the sphere. The shear rate in (c) is equal to the shear rate in the original problem.

This is the velocity field for a stationary, nonrotating sphere in a simple shear flow. A sketch of the fluid pathlines in the xix2 plane for this case is reproduced in Fig. 8 2(a). If the sphere is allowed to rotate with some angular velocity 12, the corresponding velocity field... 

Thus the sphere rotates with an angular velocity that is just 1 /2 the vorticity of the undisturbed flow. The velocity field for a freely rotating sphere in a simple shear flow is identical to Eq. (8-51), except that... 

Similar methods were employed by Schonberg et al. (1986) to investigate the multiparticle motions of a finite collection of neutrally buoyant spheres suspended in a Poiseuille flow. They were also used by Ansell and Dickinson (1986) to simulate the fragmentation of a large colloidal floe in a simple shear flow. 

The Lift on a Sphere That is Rotating in a Simple Shear Flow... 

The composite velocity field for a sphere rotating with the angular velocity (8-52) in a simple shear flow, Eq. (8 46), is thus the sum of (8-51) and (8-54). The fluid pathlines in the X X2 plane for this case are shown in Fig. 8-2(b). The corresponding torque on the sphere is... 

Clearly, for either a nonrotating or freely rotating sphere, the velocity field in a simple shear flow is fully 3D, so that the solution methods of earlier sections would necessarily fail. 

A. Acrivos, Heat transfer at high Peclet number from a small sphere freely rotating in a simple shear flow, J. Fluid Mech. 46, 233-40 (1971). 

The constitutive equation, (2-60), for the stress, on the other hand, will be modified for all fluids in the presence of a mean motion in which the velocity gradient Vu is nonzero. To see that this must be true, we can again consider the simplest possible model system of a hard-sphere or billiard-ball gas, which we may assume to be undergoing a simple shear flow,... 

Problem 7-8. Sphere in a Linear Flow. A rigid sphere is translating with velocity U and rotating with angular velocity ft in an unbounded, incompressible Newtonian fluid. The position of the sphere center is denoted as xp (that is, xf, is the position vector). At large distances from the sphere, the fluid is undergoing a simple shear flow (this is the undisturbed velocity field). We may denote this flow in the form... 

In spite of the fact that there are actually quite a large number of axisymmetric problems, however, there are many important and apparently simple-sounding problems that are not axisymmetric. For example, we could obtain a solution for the sedimentation of any axisymmetric body in the direction parallel to its axis of symmetry, but we could not solve for the translational motion in any other direction (e.g., an ellipsoid of revolution that is oriented so that its axis of rotational symmetry is oriented perpendicular to the direction of motion). Another example is the motion of a sphere in a simple linear shear flow. Although the undisturbed flow is 2D and the body is axisymmetric, the resulting flow field is fully 3D. Clearly, it is extremely important to develop a more general solution procedure that can be applied to fully 3D creeping-flow problems. 

Figure 9-7. A schematic representation of a sphere of radius a and a constant surface temperature Tq in a fluid of ambient temperature Trxj that is undergoing a simple shear flow n, = y vi. Because the sphere is assumed to rotate freely in the ambient flow, it rotates with an angular velocity f2 = (y/2)k about the z axis.

The direct boundary integral formulation was used to simulate suspended spheres in simple shear flow. The viscosity was then calculated by integration of the surface tractions on the moving wall. Figure 10.28 shows a typical mesh for the domain and spheres for these simulations in this mesh, the box has dimensions of 1 x 1 x 1 (Length units)3 and 40 spheres of radius of 0.05 length units. 

Deformation of a Sphere in Various Types of Flows A spherical liquid particle of radius 0.5 in is placed in a liquid medium of identical physical properties. Plot the shape of the particle (a) after 1 s and 2 s in simple shear flow with y 2s1 (b) after 1 s and 2 s in steady elongational flow with e = 1 s 1. (c) In each case, the ratio of the surface area of the deformed particle to the initial one can be calculated. What does this ratio represent ... 

Finally, the lift on a stationary sphere in simple shear flow can again be shown to be zero because of symmetry. If, for example, the lift were nonzero in the positive y direction (see Fig. 7-2), then it would also have to be nonzero in the negative y direction because the symmetry of the flow problem is such that there is no distinction between v > 0 and y < 0. Thus the lift must be zero. 

Figure 8-1. The undisturbed flow and spherical coordinate system for simple shear flow, u = yy, in the vicinity of a sphere.

Figure 8-2. Fluid pathlines in the x x2 plane (see Fig. 8-1) for simple shear flow past (a) nonrotating and (b) rotating spheres. The nonrotating case was obtained from Eq. (8-51), and the rotating case was obtained by adding (8-54) to (8-51).

We have previously obtained solutions by other techniques for the problems of a rigid sphere immersed in axisymmetric straining or simple shear flow of an unbounded fluid. In this subsection, it is shown that those two problems also can be solved very simply by means of a superposition of fundamental singularities at the center of the sphere. 

F. Heat Transfer From a Sphere in Simple Shear Flow at Low Peclet Numbers... 

F. HEAT TRANSFER FROM A SPHERE IN SIMPLE SHEAR FLOW AT LOW PECLET NUMBERS... 

Thus in this section, we consider the special case of heat transfer from a rigid sphere when the undisturbed fluid motion, relative to axes that translate with the sphere, is a simple, linear shear flow,... 

Figure 9-8. Correlation between the Nusselt number and Peclet number for heat transfer at low Peclet number from a sphere with constant surface temperature in (a) uniform streaming flow and (b) simple shear flow.

The first term in this expression is, of course, the familiar result for pure conduction from a heated sphere and is the same for all flows. The second term represents the first contribution of convection and should be compared with the second term in Eq. (9 60), which is the Nusselt number for forced convection heat transfer at low Pe when the flow is uniform streaming. The most important observation is that the dependence on Pe is different in the two cases, being O(Pe) in the uniform streaming flow and 0(Pe1/2) in simple shear flow. The two results, (9-60) and (9-191), are plotted in Fig. 9-8. Evidently, the Nusselt number for simple shear flow exceeds the value for uniform streaming flow for Pe< 1 where the two asymptotic predictions are valid. Although the numerical difference between the two results is small, the most important conclusion from the analysis is not the numerical magnitude of corrections to the conduction heat flux but rather the fact that the asymptotic form of the convection contribution clearly depends on flow type. In general, heat transfer correlations developed for one type offlow will not carry over to some other type offlow. 

The question at hand is whether circumstances exist for this rather simple situation in which the conditions (1) and (2) are satisfied so that boundary-layer analysis can be applied. So far as the first condition is concerned, the only flows of (9-266) that have open streamlines are those with X > 0 (which includes simple shear flow). On the other hand, there is a nonzero hydrodynamic torque on the sphere that causes it to rotate for all flows in this subgroup except X = 1. Thus, for a sphere in the general linear 2D flow, given by (9-266), there are only two cases that satisfy the conditions for applicability of boundary-layer theory ... 

Before concluding the discussion of high-Peclet-number heat transfer in low-Reynolds-number flows across regions of closed streamlines (or stream surfaces), let us return briefly to the problem of heat transfer from a sphere in simple shear flow. This problem is qualitatively similar to the 2D problem that we have just analyzed, and the physical phenomena are essentially identical. However, the details are much more complicated. The problem has been solved by Acrivos,24 and the interested reader may wish to refer to his paper for a complete description of the analysis. Here, only the solution and a few comments are offered. The primary difficulty is that an integral condition, similar to (9-320), which can be derived for the net heat transfer across an arbitrary isothermal stream surface, does not lead to any useful quantitative results for the temperature distribution because, in contrast with the 2D case in which the isotherms correspond to streamlines, the location of these stream surfaces is a priori unknown. To resolve this problem, Acrivos shows that the more general steady-state condition,... 

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). 

Any flow exerts a frictional or viscous stress 17 [see Eq. (5.1)] onto the wall of the vessel in which the liquid is flowing or onto particles in the liquid. This has several consequences, the simplest one being that particles move with the liquid some others are illustrated in Figure 5.3, which applies to simple shear flow. A solid sphere rotates, as mentioned. Solid anisometric particles also rotate, but not at a completely constant rate. An elongated particle will rotate slower when it is oriented in the direction of flow than when in a perpendicular direction. This means that such particles show on average a certain preference for orientation in the direction of flow, although they keep rotating. 

In elongational flow of the same velocity gradient, the viscous stresses are greater see Eq. (5.2). Anisometric solid particles become aligned in the direction of flow flexible particles become extended, i.e., anisometric, and aligned. Particles that are close to each other become separated from each other, while they can stay together in simple shear flow a doublet of spheres then continues rotating as one dumbbell-shaped particle. 

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