Arbitrary 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. 

Note that in [523], a similar plane problem on an arbitrary linear shear flow past a porous cylinder was solved. The flow outside the cylinder was described by using the Stokes equations and the percolation of the outer fluid inside the porous cylinder was assumed to obey Darcy s law (2.2.24). The amount of the fluid penetrating into the cylinder per unit time was determined. 

For a spherical particle in an arbitrary linear shear flow (4.5.1), the first four terms of the asymptotic expansion as Pe - 0 of the mean Sherwood number have the form [5]... 

For an arbitrary linear shear flow, the expression (5.6.3) gives two terms of the expansion (up to terms of the order of %/Pe inclusive). In this special case, Sh(l, Pe) is determined by the ratio of the right-hand side of (4.5.4) to the dimensionless surface area of the particle with O(Pe) terms neglected. 

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). 

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... 

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 ... 

Freely rotating cylinder. Now let us consider a circular cylinder freely floating in an arbitrary linear shear Stokes flow (Re —i 0). The velocity distribution for such a flow remote from the cylinder is still given by relations (2.7.8). 

Particle of Arbitrary Shape in a Linear Shear Flow... 

The mass exchange for a particle of arbitrary shape freely suspended in a linear shear flow described by (4.5.1) was considered in [5], The three-term expansion... 

Particles in Linear Shear Flows. Arbitrary Peclet Numbers... 

The solution of hydrodynamic problems for an arbitrary straining linear shear flow (Gkm = Gmk) past a solid particle, drop, or bubble in the Stokes approximation (as Re -> 0) is given in Section 2.5. In the diffusion boundary layer approximation, the corresponding problems of convective mass transfer at high Peclet numbers were considered in [27, 164, 353]. In Table 4.4, the mean Sherwood numbers obtained in these papers are shown. 

For a spherical drop in an arbitrary straining linear shear flow under limiting resistance of the continuous phase, one can use the interpolation formula [353] ... 

Freely rotating cylinder. Now let us consider convective mass transfer to the surface of a circular cylinder freely suspended in an arbitrary linear shear Stokes flow (Re -> 0). In view of the no-slip condition, the cylinder rotates at a constant angular velocity equal to the angular velocity of the flow at infinity. The fluid velocity distribution is described by formulas (2.7.11). The streamline pattern qualitatively differs from that for the case of a fixed cylinder. For 0 0, there are no stagnation points on the surface of the cylinder and there exist two qualitatively different types of flow. For 0 < Ifigl < 1, there are both closed and open streamlines in the flow, the region filled with closed streamlines is adjacent to the surface of the cylinder, and streamlines far from the cylinder are open (Figure 2.11). For Ifl l > 1, all streamlines are open. 

Drop, bubble Arbitrary straining linear shear flow ( Gij - Gji) Hkf / 3 y/2 U = a[ GijGij, Gij are the shear matrix coefficients... 

We note that, in spite of the superficial similarity to a unidirectional flow, the solution for uo is actually quite different from the linear profile of a simple, planar shear flow. This is not at all surprising for an arbitrary ratio of the cylinder radii. However, we should expect that the approximation to a simple shear flow should improve as the gap width becomes... 

The expressions (8-44) and (8 45) represent a complete, exact solution of the creeping-flow equations for a completely arbitrary linear flow. Among the linear flows of special interest are axisymmetric pure strain, which was solved by means of the eigenfunction expansion for axisymmetric flows in the previous chapter, and simple shear flow, for which... 

For a solid spherical particle in an arbitrary linear straining shear flow, the following interpolation formula was suggested in [27] for the mean Sherwood number ... 

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. 

Formula (5.6.4) is valid for an arbitrary laminar flow without closed streamlines for particles and drops of an arbitrary shape. The quantity Sh(l,Pe) corresponds to the asymptotic solution of the linear problem (5.6.1) at Pe > 1. For spherical particles, drops, and bubbles in a translational or linear straining shear flow, the values of Sh(l, Pe) are shown in the fourth column in Table 4.7. 

Viscosity coefficients measured in these geometries when n is immobilised by boMiesowicz viscosities. (Note, that in the literature a variety of alternative notations are common in particular the definitions of r i and r 2 are frequently interchanged.) If the orientation of n is fixed in an arbitrary direction with respect to v and Vv, then the effective viscosity coefficient is given by a linear combination of the Miesowicz viscosities, and another viscosity constant Tju, which cannot be visualised in a pure shear-flow ... 

Steady parallel flow can be realized in ducts of essentially arbitrary cross section. A linear elliptic partial differential equation must be solved to determine the velocity field and the shear stresses on the walls. For an incompressible, constant-viscosity fluid, the axial momentum equation states that... 

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