Translational Stokes flow

Spherical Particles, Drops, and Bubbles in Translational Stokes Flow... 

Let us consider a solid spherical particle of radius o in a translational Stokes flow with velocity U and dynamic viscosity /i (Figure 2.1). We assume that the fluid has a dynamic viscosity /z. We use the spherical coordinate system. R, 9, ip with origin at the center of the particle and with angle 0 measured from the direction of the incoming flow (that is, from the rear stagnation point on the particle surface). In view of the axial symmetry, only two components of the fluid velocity, namely, Vr and Vg, are nonzero, and all the unknowns are independent of the third coordinate [Pg.58]

Figure 2.1. Translational Stokes flow past a spherical particle...

Now let us consider a spherical drop of radius a in a translational Stokes flow of another fluid with velocity U (Figure 2.2). We assume that the dynamic viscosities of the outer and inner fluids are equal to /j, and fi2, respectively. The unknown variables outside and inside the drop are indicated by the superscripts (1) and (2), respectively. 

Let a be the outer radius of the compound drop, and let ae be the radius of the core (0 < e < 1). The exact solution of the problem on the flow past a compound drop in a translational Stokes flow with velocity U can be found in [416], where the stream functions in the phases are given. The drag force is also... 

Translational Stokes Flow Past Ellipsoidal Particles... 

The axisymmetric problem about a translational Stokes flow past an ellipsoidal particle admits an exact closed-form solution. Here we restrict our consideration to a brief summary of the corresponding results presented in [179],... 

Oblate ellipsoid of revolution. Let us consider an oblate ellipsoid of revolution (on the left in Figure 2.5) with semiaxes a and b a>b) in a translational Stokes flow with velocity U[. We assume that the fluid viscosity is equal to p. We pass from the Cartesian coordinates X, Y, Z to the reference frame... 

Prolate ellipsoid of revolution. To solve the corresponding problem about an ellipsoidal particle (on the right in Figure 2.5) in a translational Stokes flow, we use the reference frame cr, r, fixed to the prolate ellipsoid of revolution. The transformation to the coordinates (o, r, ip) is determined by the formulas... 

Translational Stokes Flow Past Bodies of Revolution... 

Translational Stokes Flow Past Particles of Arbitrary Shape... 

Following [270], we first consider steady-state diffusion to the surface of a solid spherical particle in a translational Stokes flow (Re - 0) at high Peclet numbers. In the dimensionless variables, the mathematical statement of the corresponding problem for the concentration distribution is given by Eq. (4.4.3) with the boundary conditions (4.4.4) and (4.4.5), where the stream function is determined by (4.4.2). 

Now let us consider the exterior problem about mass exchange between a spherical drop (bubble) of radius a and a translational Stokes flow with limiting diffusion resistance of the continuous phase. 

Spherical particle as Re —> 0, 0 < Pe < oo. The problem of mass transfer to a solid spherical particle in a translational Stokes flow (Re -f 0) was studied in the entire range of Peclet numbers by finite-difference methods in [1, 60, 281], To find the mean Sherwood number for a spherical particle, it is convenient to use the following approximate formula [94] ... 

For the special case of a translational Stokes flow past a spherical drop, Eq. (4.7.11) passes into (4.7.4). 

Let us consider diffusion to the surface of a solid ellipsoidal particle in a homogeneous translational Stokes flow (Re 0). The particle is an ellipsoid of revolution with semiaxes a and b oriented along and across the flow, respectively (b is the equatorial radius). We introduce the following notation ... 

For arbitrary Peclet numbers, the mean Sherwood number (corresponding to the characteristic length ae) for a translational Stokes flow past an ellipsoidal particle can be approximated by the formula [94]... 

The axisymmetric problem on mass exchange between an ellipsoidal particle and a translational Stokes flow was numerically studied in [281] by the finite-difference method. Two cases were considered, in which the length of the particle semiaxis oriented along the flow was, respectively, five times greater and five times smaller than the length of the semiaxis perpendicular to the flow. According to [94], it follows from the results of the numerical solution in [281] that the maximum error of formula (4.10.6) for an ellipsoidal particle does not exceed 10% in the cases under consideration. 

The case x -> oo (that is, a -> 0 and b = const) corresponds to diffusion to the surface of a thin circular disk of radius b perpendicular to a uniform translational Stokes flow. 

The following general statement was proved in [63] for the case of a uniform translational Stokes flow (Re -4 0) or a potential flow past a particle of an arbitrary shape the mean Sherwood number remains the same if the flow direction is changed to the opposite. 

At low Peclet numbers, for the translational Stokes flow past an arbitrarily shaped body of revolution, formula (4.10.8) coincides with the exact asymptotic expression in the first three terms of the expansion [358], Since (4.10.8) holds identically for a spherical particle at all Peclet numbers, one can expect that for particles whose shape is nearly spherical, the approximate formula (4.10.8) will give good results for low as well as moderate or high Peclet numbers. 

For a translational Stokes flow past a convex body of revolution of sufficiently smooth shape with symmetry axis parallel to the flow, the error (in percent) in formula (4.10.9) for the mean Sherwood number can be approximately estimated as follows ... 

In the case of nonstationary mass transfer in a steady-state translational Stokes flow past a spherical drop with limiting resistance of the continuous phase, the steady-state value Shst is presented in the first row of Table 4.7. By substituting this value into (4.12.3), we obtain... 

Drop, bubble Translational Stokes flow r 2Pe j1/2 U = U-, is the fluid velocity at infinity... 

Drop, bubble Translational Stokes flow Analytical, DBLA 0.7 [84,271,410]... 

Solid particle Translational Stokes flow Interpolation of numerical and analytical results 1.4 [94]... 

Solid particle Translational Stokes flow Finite-difference numerical method (at Pe = 500) 4 [68]... 

Statement of the problem. Preliminary remarks. Let us consider the transient convective mass and heat transfer between a spherical drop of radius a and a translational Stokes flow where the resistance to the transfer exists only in the disperse phase. We assume that at the initial time t = 0 the concentration inside the drop is constant and equal to Co, whereas for t > 0 the concentration on the interface is maintained constant and equal to Cs. 

Solid particles. Let us consider stationary diffusion to two axisymmetric solid particles arranged one after the other on the axis of a translational Stokes flow. We assume that the solid particles are symmetric with respect to some plane z = const (see Figure 4.8) and each of them has only two stagnation points on the surface, which lie on the flow axis (closed streamlines are absent). The surfaces of solid particles completely absorb the solute. 

Order of magnitude of dimensionless (related to the radius of drops or solid particles) characteristic sizes of regions of diffusion wake in translational Stokes flow at high Peclet number... 

The following limit relation for the total diffusion fluxes on the surface of two identical solid particles arranged on the axis of a translational Stokes flow (Figure 4.8) was derived in [169] ... 

Figure 4.8. Translational Stokes flow past two identical solid particles...

Relation (4.14.1), in particular, holds for spheres of equal radius arranged on the axis of a translational Stokes flow (the velocity distribution for this case is presented in [179,463], It also holds for a three-dimensional Stokes flow past two identical ellipsoids of rotation whose axes are parallel and perpendicular to the undisturbed flow. The direction of the line passing through their centers coincides with the direction of translational 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.)... 

At high Peclet numbers, for an nth-order surface reaction withn=l/2, 1,2, Eq. (5.1.5) was tested in the entire range of the parameter ks by comparing its root with the results of numerical solution of appropriate integral equations for the surface concentration (derived in the diffusion boundary layer approximation) in the case of a translational Stokes flow past a sphere, a circular cylinder, a drop, or a bubble [166, 171, 364], The comparison results for a second-order surface reaction (n = 2) are shown in Figure 5.1 (for n = 1/2 and n = 1, the accuracy of Eq. (5.1.5) is higher than for n = 2). Curve 1 (solid line) corresponds to a second-order reaction (n = 2). One can see that, the maximum inaccuracy is observed for 0.5 < fcs/Shoo < 5.0 and does not exceed 6% for a solid sphere (curve 2), 8% for a circular cylinder (curve 3), and 12% for a spherical bubble (curve 4). 

The dependence of the auxiliary Sherwood number Sho on the Peclet number Pe for a translational Stokes flow past a spherical particle or a drop is determined by the right-hand sides of (4.6.8) and (4.6.17). In the case of a linear shear Stokes flow, the values of Sho are shown in the fourth column in Table 4.4. 

For a first-order volume reaction and a translational Stokes flow past a spherical drop, the asymptotic solution of the inner problem (5.3.1), (5.3.2) as Pe -4 oo results in the following expression for the mean Sherwood number [104] ... 

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