Linear shear flow

Bentley, B. J., and Leal, L. G., A computer-controlled four-roll mill for investigations of particle and drop dynamics in two-dimensional linear shear flows. J. Fluid Mech. 167, 219-240 (1986). 

Assuming that the flow above the wall is linear shear flow, the velocity field can be modeled, from Eq. (6), as Vx = jy, where the velocity increases linearly away from the wall, with slope j. Near the wall, a reflection of the weighting function, similar to the reflection of the Green s function [Eq. (36)], must be applied,... 

Sign Convention of the Stress Tensor x Consider a linear shear flow and examine... 

In retrospect, the effect of the change of variables has been to deform the velocity field from axisymmetric stagnation flow over a sphere to linear shear flow along a flat plate. The main advantage of the new coordinates is that the coefficients of the derivatives in (32) are independent of X, and consequently Duhamel s theorem can be applied. Thus, the following procedure can be used (1) solve equation (32) subject to a uniform surface concentration (2) extend this solution to one valid for an arbitrary, nonuniform surface concentration by applying Duhamel s theorem (3) select the surface concentration which satisfies (33a). 

The vast majority of work on particle-surface electrostatic interactions has neglected any effects due to particle motion. However, both theoretical [31,32] and experimental work [33-35] have been done on the problem of a charged particle interacting with a charged wall in a linear shear flow. In the theoretical treatment, it is assumed that the double layer thickness is small compared to both the particle diameter and the surface-to-particle gap. Hence, changes in the pressure and potential profiles in the gap caused by motion can be written as small perturbations to their equilibrium profiles. In the region outside the small double layers, the fluid velocity v and perturbation pressure dp are governed by Stokes equations... 

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.]... 

In the present rheological context, lattice deformation may be regarded as arising from the transport of neutrally buoyant lattice points suspended within a macroscopically homogeneous linear shear flow. The local vector velocity field v at a general (interstitial or particle interior) point R of such a spatially periodic suspension can be shown to be of the form... 

The tests were also performed on computer-generated data in which additional uniform or non-uniform motion was added, to study how far the CG algorithm could be pushed beyond its original design parameters. For uniform motion, CG tracking was as successful as in the quiescent case for small drifts but failed for drifts of the order of half the particle-particle separation. For non-uniform (linear shear) flows with small strains between frames the identification worked correctly, but large non-uniform displacements caused major tracking errors. 

Kariyasaki [70] studied bubbles, drops, and solid particles in linear shear flow experimentally, and showed that the lift force on a deformable particle is opposite to that on a rigid sphere. For particle Reynolds numbers between 10 and 8 the drag coefficient could be estimated by Stokes law. The terminal velocity was determined to be equal to that of a particle moving in a quiescent... 

Davis, M.H., Two Unequal Spheres in a Slow Linear Shear Flow, Rept. NCAR-TN/STR-64, National Center for Atmospheric Research, Boulder, CO, 1971. 

In the intermediate case, in which neither of the limiting cases (3-26) pertains, neither choice for uc is preferable over the other. In this case, the velocity profile is somewhere between the linear shear flow of Gd2 /p U and the parabolic profile of Gd2 hi U. Either of the dimensionless solution forms, (3-24) or (3-29), is satisfactory. Both show that the form of the velocity profile depends on the dimensionless ratio Gd2 / iiU. Indeed, when converted back to dimensional variables, the two dimensionless solution forms are identical [recall that u = u/U in (3-24), but u = u/(Gd2/p) in (3-29)] ... 

This dimensional form of the solution emphasizes the fact that the flow is a linear combination of plane Poiseuille flow and linear shear flow, with the relative magnitude determined by the ratio of Gd2/yu to U. 

Hence, to achieve the best possible approximation to a linear shear flow, the Couette device must have a very thin gap relative to the cylinder radius. 

On the other hand, because the circular Couette flow is generally adopted as a convenient substitute for a simple shear flow, it maybe tempting to analyze the experimental data as though we exactly had simple shear between two plane boundaries. This would mean dividing rrg r=a with an estimated velocity gradient given by the velocity difference of the two walls divided by the gap width as would be exactly correct for a linear shear flow. The latter is simply... 

If we were to estimate the viscosity by using only the ratio of the measured shear stress, (3-80), to the actual velocity gradient, (3 82), at r = a (instead of the full strain rate E ) as would be valid for a linear shear flow, we would obtain... 

J. A. Schonberg and E. J. Hinch, Inertial migration of a sphere in Poiseuille flow, J. Fluid Mech. 203, 517-524 (1989) E. S. Asmolov, The inertial lift on a small particle in a weak-shear parabolic flow, Phys. of Fluids 14, 15-28 (2002) P. Cherukat, J. B. McLaughlin, and D. S. Dandy, A computational study of the inertial lift on a sphere in a linear shear flow field, Int. J. of Multiphase Flow, 25, 15-33 (1999). 

S. Wakiya, Application of bipolar co-ordinates to the two-dimensional creeping-motion of a liquid. I. Flow over a projection or a depression on a wall, J. Phys. Soc. Jpn. 39, 111 3-20 (1975) M. E. O Neill, On the separation of a slow linear shear flow from a cylindrical ridge or trough in a plane, Z. Angew. Math. Phys. 28,438-48 (1977) A. M. J. Davis and M. E. O Neill, Separation in a Stokes flow past a phase with a cylindrical ridge or trough, Q. J. Mech. Appl. Math. 30, 355-68 (1977). 

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 8-4. Shape of a viscous drop in (a) simple, linear shear flow and (b) uniaxial extension for X = 1 and Ca = 0.2.

The closely related problem of a rigid sphere in a linear shear flow is very easy to solve now by analogy to the solution for an axisymmetric straining flow. We consider the problem in the form... 

Problem 8-4. Consider a sphere suspended in the general linear shear flow , = T,-yXy under creeping-flow conditions. If the sphere is free to rotate, what is its angular velocity... 

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,... 

The governing DE and boundary conditions for the temperature field are again (9 1) and (9-2), and the dimensionless equation and boundary conditions are (9-7) and (9-8), with the same definition for the dimensionless temperature (9-3) and the sphere radius as a characteristic length scale. Only the form of the velocity field, u, and the choice of characteristic velocity are different for this case of a linear shear flow far from the sphere. The appropriate choice for the characteristic velocity is... 

Before we leave the present problem, the reader s attention is called to several generalizations of the predicted relationship (9-161) between Nu and Pe for Pe <asymptotic method to provide insight into the form of correlations between dimensionless parameters, with a minimum of detailed analysis. The first is due to Batchelor14 and Acrivos,15 who showed that the correlation (9-191), first derived for a sphere in linear shear flow, could be generalized easily and extended to the much more general case of a rigid, heated sphere in an arbitrary linear flow... 

Figure 9-18. Streamlines for a freely rotating circular cylinder in simple, linear shear flow (9-306). Contours 0 to 0.75 in increments of 1/16.

Among references that discuss closed-streamline patterns and eddies in low-Reynolds-number flows, the reader may wish to refer to Ref. 13, Chap. 7, and D. J. Jeffrey and J. D. Sherwood, Streamline patterns and eddies in low-Reynolds-number flow, J. Fluid Mech. 96, 315-34 (1980) A. M. J. Davis and M. B. O Neill, The development of viscous wakes in a Stokes flow when a particle is near a large obstacle, Chem. Eng. Sci. 32, 899-906 (1977) A. M. J. Davis and M. B. O Neill, Separation in a slow linear shear flow past a cylinder and a plane, J. Fluid Mech., 81, 551-64 (1977). 

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. 

Examples of shear flows. Now let us consider the most frequently encountered types of linear shear flows [518]. 

Statement of the problem. Let us consider a linear shear flow at low Reynolds numbers past a solid spherical particle of radius a. In the general case, the Stokes equation (2.1.1) must be completed by the no-slip condition (2.2.1) on the particle boundary and the following boundary conditions remote from the particle (see Section 1.1) ... 

In the problem of linear shear flow past a spherical drop (bubble), the Stokes equations (2.1.1) and the boundary conditions at infinity (2.5.1) must be completed by the boundary conditions on the interface and the condition that the solution is bounded inside the drop. In particular, in the axisymmetric case, the boundary conditions (2.2.6)-(2.2.10) are used. 

Fixed cylinder. Let us consider a fixed circular cylinder in an arbitrary steady-state linear shear flow of viscous incompressible fluid in the plane normal to the cylinder axis. The velocity field of such a flow remote from the cylinder in the Cartesian coordinates X, X% can be represented in the general case as follows ... 

Figure 2.10. Linear shear flow past a fixed circular cylinder (a) straining flow (Ei = 0 and Cl = 0) (b) simple shear flow (Ei = 0 and Ei = -Cl)...

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