Spherical drop, Stokes flow

In summary then, the leading-order problem is just the translation of a spherical drop through a quiescent fluid. The solution of this problem is straightforward and can again be approached by means of the eigenfunction expansion for the Stokes equations in spherical coordinates that was used in section F to solve Stokes problem. Because the flow both inside and outside the drop will be axisymmetric, we can employ the equations of motion and continuity, (7-198) and (7-199), in terms of the streamfunctions f<(>> and < l)), that is,... 

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

Figure 2.2. Translational Stokes flow past a spherical drop...

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. 

The spherical form of a drop or a bubble in Stokes flow follows from the fact that the flow is inertia-free. However, even for the case in which the inertia forces dominate viscous forces and the Reynolds number cannot be considered small, the drop remains undeformed if the inertia forces are small compared with the capillary forces. The ratio of inertial to capillary forces is measured by the Weber number We = p U a/a, where cr is the surface tension at the drop boundary. For small We, a deformable drop will conserve the spherical form. 

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. 

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. 

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

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... 

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. 

Drops and bubbles. For two spherical drops (bubbles) of equal radius arranged one behind the other on the axis of a translational Stokes flow, the following limit equation holds [169] ... 

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] ... 

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. 

The three Navier-Stokes equations can be put in very compact form by using the shorthand notation of vector calculus [6, p. 66 7 8, p. 80]. Furthermore, it is often convenient to use these equations in polar or spherical coordinates their transformations to those coordinate systems are shown in many texts [6, p. 66 8, p. 80]. The corresponding equations for fluids with variable density are also shown in numerous texts [6, p. 66 7 8, p. 80]. If we set /A = 0 in the Navier-Stokes equations, thus dropping the rightmost term, we find the Euler equation which is often used for three-dimensional flow where viscous effects are negligible. 

The Stokes experiment consists in dropping a spherical particle of diameter D where Ps>Pf The ball rapidly reaches a constant fall velocity Wc parallel to the gravitational acceleration g. When steady-state velocity is reached, the equilibrium of the forces applied on the ball reduces to the equilibrium between the reduced weight Fg (difference between the weight and Archimedes force) and the hydrodynamic drag force Fr exerted by the fluid flow on the particle. 

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