Translational-shear flow

Note that since the problem of Stokes flow is linear, one can find the velocity and pressure fields in translational-shear flows as the superposition of solutions describing the translational flow considered in Section 2.2 and shear flows considered in the present section. 

Mass Transfer in a Translational-Shear Flow and in a Flow With Parabolic Profile... 

Mass exchange between a spherical particle and the translational-shear flow (4.9.1) at high Peclet numbers was studied in [175]. For the mean Sherwood number depending on the parameters... 

The mean Sherwood number for the translational-shear flow (4.9.1) past a spherical drop under limiting resistance of the continuous phase at high Peclet numbers can be calculated by the formulas [164]... 

To construct approximate formulas for the Sherwood number in the case of a translational shear flow past particles and bubbles in the entire range of Peclet numbers, one can use formulas (4.7.9) and (4.7.10) where Shpoo and Shboo must be replaced by the right-hand sides of Eqs. (4.9.3), (4.9.4) and (4.9.5), (4.9.6), respectively, with 0 = 0. 

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... 

Here, the sphere center is instantaneously situated at point 0 the sphere center translates with velocity U, while it rotates with angular velocity (a r is measured relative to 0 its magnitude r is denoted by r. Moreover, f = r/r is a unit radial vector. The latter solution is derivable in a variety of ways e.g., from Lamb s (1932) general solution (Brenner, 1970). [Equation (2.12) represents a superposition (Brenner, 1958) of three physically distinct solutions, corresponding, respectively, to (i) translation of a sphere through a fluid at rest at infinity (ii) rotation of a sphere in a fluid at rest at infinity (iii) motion of a neutrally buoyant sphere suspended in a linear shear flow. The latter was first obtained by Einstein (1906, 1911 cf. Einstein, 1956) in connection with his classic calculation of the viscosity of a dilute suspension of spheres, which formed part of his 1905 Ph.D. thesis.]... 

Flowability If we re considering particles, powders, and other products that are intended to flow, then this is a very important consideration. These materials need to easily flow from bins, hoppers, and out of boxes for consumer products. Powder flowability is a measure-able characteristic using rotational shear cells (Peschl) or translational shear cells (lenike) in which the powder is consolidated under various normal loads, and then the shear force is measured, enabling a complete yield locus curve to be constructed. This can be done at various powder moistures to create a curve of flowability versus moisture content. Some minimal value is necessary to ensure free flow. Additional information on these devices and this measure can be found in Sec. 21, Sohd-Solid Operations and Processing. ... 

The aerodynamic diameter is one of the most common equivalent diameters. It can be defined as the diameter of a unit den.sity sphere with the same terminal settling velocity as the particle being measured. The aerodynamic diameter is commonly used to describe the mt)lioii of particles in collection devices such as cyclone separators and impactors. However, in shear flows, the motion of irregular particles may not be characterized accurately by the equivalent diameter alone because of the complex rotational and translational motion of inegular particles compared with spheres. That is, the path of the irregular particle may not follow that of a particle of the same aerodynamic diameter. It is of course possible that there may be a. sphere of a certain diameter and unit density that deposits at the same point this could be an average point of deposition because of the effects of turbulence or the. stochastic behavior of irregular particles. 

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.

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. 

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,... 

For viscous flows around particles whose size is much less than the characteristic size of flow inhomogeneities, the velocity distribution (1.1.15) can be viewed as the velocity field remote from the particle. The special case Gkm = 0 corresponds to uniform translational flow. For Vj.(0) = 0, Eq. (1.1.15) describes the velocity field in an arbitrary linear shear flow. 

In practice, one often deals with the case in which particles are completely entrained by the flow and convective transfer due to the shear flow plays the main role. In studying the corresponding diffusion processes, it is convenient to attach the frame of reference to the particle center of gravity then this frame moves translationally at the particle velocity, and the particle itself can freely rotate around the origin. For linear shear flow, the fluid velocity components have the form... 

Let us consider mass transfer for a translational flow past a solid spherical particle, where the flow field remote from the particle is the superposition of a translational flow with velocity U and an axisymmetric straining shear flow, the translational flow being directed along the axis of the straining flow. The dimensional fluid velocity components in the Cartesian coordinates relative to the center of the particle have the form... 

Mass and Heat Transfer Between Cylinders and Translational or Shear Flows... 

Transient Mass Transfer in Steady-State Translational and Shear Flows... 

In this chapter, some problems of mass and heat transfer with various complicating factors are discussed. The effect of surface and volume chemical reactions of any order on the convective mass exchange between particles or drops and a translational or shear flow is investigated. Linear and nonlinear nonstationary problems of mass transfer with volume chemical reaction are studied. Universal formulas are given which can be used for estimating the intensity of the mass transfer process for arbitrary kinetics of the surface or volume reaction and various types of flow. 

It was shown in [ 166,351 ] that Eq. (5.1.5) provides several valid initial terms of the asymptotic expansion of the Sherwood number as Pe —> 0 for any kinetics of the surface chemical reaction. (Specifically, one obtains three valid terms for the translational Stokes flow and four valid terms for an arbitrary shear flow.)... 

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