Translational flow

To illustrate the relation between the different flows and the two reaction velocities, we remark that the flows 23, 34, and 45 are obviously the velocity of the main reaction, r, while the flows 12 and 50 equals the velocity s of the side reaction. This is shown in Fig. 5 by means of letters and arrows. The diagram also shows the symbolic analogy between our flows and real physical flows. Thus we may speak of sources and sinks, 1 being a source and 0 a sink, and of translational and rotational flows for example, we may say that the flow s62 is a superposition of a translational flow ( — s) and a rotational flow (r). s may be assumed to be always positive. The case s = 0 is in principle the same as the one treated above (p. 322), where we may speak of catalysis with X2 as a catalyst. As the chain (23452) is broken in this case only by the reaction 21, the chain length then has its maximum, but its numerical value cannot be defined unless we know the kinetics of the reactions (12) and (21), which may be unknown compare the discussion in the literature of the hydrogen-bromine reaction (see also p. 334). 

Translational flow. For uniform translational flow with velocity Uj around a finite body, the boundary condition remote from the body has the form... 

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. 

For a uniform translational flow, the velocity of the nonperturbed flow is independent of the coordinates therefore, all Gkm = 0. In this case we have the simplest flow around a body with the boundary condition (1.1.14) at infinity. 

In chemical technology one often meets the problem of a steady-state motion of a spherical particle, drop, or bubble with velocity U in a stagnant fluid. Since the Stokes equations are linear, the solution of this problem can be obtained from formulas (2.2.12) and (2.2.13) by adding the terms Vr = -U cos6 and V = U[ sin 6, which describe a translational flow with velocity U, in the direction opposite to the incoming flow. Although the dynamic characteristics of flow remain the same, the streamline pattern looks different in the reference frame fixed to the stagnant fluid. In particular, the streamlines inside the sphere are not closed. 

Spherical Particles in Translational Flow at Various Reynolds Numbers... 

For Re > 0.5, asymptotic solutions no longer give an adequate description of translational flow of a viscous fluid past a spherical particle. 

In the case of a spherical bubble in a translational flow at small Reynolds numbers, the solution of Oseen s equation (2.3.1) results is a two-term asymptotic expansion for the drag coefficient [476] ... 

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. 

Figure 2.6. Body of revolution in translational flow (arbitrary orientation)...

Low Reynolds numbers. In [216, 382] the problem on a circular cylinder of radius a in translational flow of viscous incompressible fluid with velocity Ul at low Reynolds numbers was solved by the method of matched asymptotic expansions. The study was carried out on the basis of the Navier-Stokes equations (1.1.4) in the polar coordinates 1Z, 6. Thus, the following expression for the stream function was obtained for IZ/a 1 ... 

Translational flow. At low Reynolds and Weber numbers, the axisymmetric problem on the slow translational motion of a drop with steady-state velocity U in a stagnant fluid was studied in [476] under the assumption that We = 0(Re2). Deformations of the drop surface were obtained from the condition that the jump of the normal stress across the drop surface is equal to the pressure increment associated with interfacial tension. It was shown that a drop has the shape of an oblate (in the flow direction) ellipsoid with the ratio of the major to the minor semiaxis equal to... 

Statement of the problem. Thermal boundary layer. Let us consider heat transfer to a flat plate in a longitudinal translational flow of a viscous incompressible fluid with velocity U at high Reynolds numbers. We assume that the temperature on the plate surface and remote from it is equal to the constants Ts and 7], respectively. The origin of the rectangular coordinates X, Y is at the front edge of the plate, the X-axis is tangent, and the Y-axis is normal to the plate. 

In this section, we study the one-dimensional translational flow with velocity JJ remote from the particle. 

Statement of the Problem. Let us consider mass exchange between a spherical particle of radius a and a translational flow. In the Stokes flow (Re —> 0), one can represent the dimensionless (divided by U ) fluid velocity components as... 

Peclet numbers, the problem of mass exchange between a particle of arbitrary shape and a uniform translational flow were studied by the method of matched asymptotic expansions in [62]. The following expression was obtained for the mean Sherwood number up to first-order infinitesimals with respect to Pe ... 

Formula (4.4.21) is quite general and holds for solid particles, drops, and bubbles of arbitrary shape in a uniform translational flow at any Re as Pe —F 0. 

For a particle of arbitrary shape in a translational flow, the first three terms of the asymptotic expansion of the dimensionless total diffusion flux as Pe — 0 have the form [62]... 

Circular cylinder. The mass exchange between a circular cylinder of radius a and a uniform translational flow whose direction is perpendicular to the generatrix of the cylinder was considered in [186,218] for low Peclet and Reynolds numbers Pe = Sc Re and Re = aU-Jv. For the mean Sherwood number (per unit length of the cylinder) determined with respect to the radius, the following two-term expansions were obtained ... 

Cylinder of arbitrary shape. Let us consider mass exchange for cylindrical bodies of arbitrary shape in a uniform translational flow of viscous fluid at small... 

Mass transfer to a particle in a translational flow, considered in Section 4.4, is a good model for many actual processes in disperse systems in which the velocity of the translational motion of particles relative to fluid plays the main role in convective transfer and the gradient of the nonperturbed velocity field can be neglected. 

In this section, some interpolation formulas are presented (see [367, 368]) for the calculation of the mean Sherwood number for spherical particles, drops, and bubbles of radius a in a translational flow with velocity U at various Peclet numbers Pe = aU /D and Reynolds numbers Re = aU-Jv. We denote the mean Sherwood number by Shb for a gas bubble and by Shp for a solid sphere. 

Spherical bubble as Re —> 0, 0 < Pe < oo. The problem of mass transfer to a spherical bubble in a translational flow as Re 0 was studied numerically in [321], The results for the mean Sherwood number can be approximated well by the expression... 

Spherical drop as Re —> 0, 0 < Pe < oo. In the range 0 < Pe < 200, the results of numerical calculations of mean Sherwood numbers for a spherical drop in a translational flow under a limiting resistance of the continuous phase is well described by the approximate formula [68]... 

Spherical particle at various Reynolds numbers. In the case of a spherical particle in a translational flow at 0.5 < Re < 200 and 0.125 < Sc < 50, numerical results concerning the mean Sherwood number (e.g., see [114, 281]) can be described [94] by the approximate formula... 

The analysis of available experimental data on heat and mass transfer to a solid sphere in a translational flow results in the following correlations [94]. Heat exchange with air at Pr = 0.7 ... 

Spherical bubble at any Peclet numbers for Re > 35. For a spherical bubble in a translational flow at moderate and high Reynolds numbers and high Peclet numbers, the mean Sherwood number can be calculated by the formula [94]... 

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... 

Thus the Sherwood number remains constant (is equal to the Sherwood number for a uniform translational flow) as u> varies in the range 0 < u> < 5/3, and grows with u for w > 5/3. 

Suppose that the axis of revolution makes an angle u> with the translation flow velocity at infinity. In [358] the following approximate formula for the mean Sherwood number was obtained ... 

For a noncirculatory translational flow of an ideal fluid past a sphere, the maximum error in (4.10.10) is about 3%. 

---