Sphere, flow past

The first of these problems involves relative motion between a rigid sphere and a liquid as analyzed by Stokes in 1850. The results apply equally to liquid flowing past a stationary sphere with a steady-state (subscript s) velocity v or to a sphere moving through a stationary liquid with a velocity -v the relative motion is the same in both cases. If the relative motion is in the vertical direction, we may visualize the slices of liquid described above as consisting of... 

A particular fluid flow problem must have an associated characteristic length L and characteristic velocity V. These values may be more or less arbitrarily specified, with the only constraint being that they represent some typical scales. For example, if the problem involves a flow past a sphere, L could be the diameter of the sphere and V could be the velocity of the fluid at infinity. The characteristic length and characteristic velocity also fix a characteristic time scale T = L/V. 

For flow past a circular cylinder with L/d > normal to the cylinder axis, the flow is similar to over for a sphere. An equation that adequately represents the cylinder drag coefficient over the entire range of NRc (up to... 

Beetstra, R., van der Hoef, M. A., and Kuipers, J. A. M. Drag force from lattice Boltzmann simulations of intermediate Reynolds number flow past mono- and bidisperse arrays of spheres, Manuscript submitted to AIChE J. (2006). 

Inviscid Flow and Potential Flow Past a Sphere... 

These results are useful reference conditions for real flows past spherical particles. For example, comparisons are made in Chapter 5 between potential flow and results for flow past a sphere at finite Re. Other potential flow solutions exist for closed bodies, but none has the same importance as that outlined here for the motion of solid and fluid particles. 

This result may be contrasted with potential flow past a sphere, where the streamlines again have fore-and-aft symmetry but p is an even function of 9 so that there is no net pressure force (see Chapter 1). Additional drag components arise from the deviatoric normal stress ... 

Stokes s solution (S9) for steady creeping flow past a rigid sphere may be obtained directly from the results of the previous section with co. The same results are obtained by solving Eq. (3-1) with Eqs. (3-4) to (3-6) replaced by the single condition that Uq O a.tr = a. The corresponding streamlines are shown in Figs. 3.3a and 3.4a. As for fluid spheres, the particle causes significant... 

Equation (3-39) has been solved for steady Stokes flow past a rigid sphere (B6, M2). The resulting values of Sh, obtained numerically for a wide range of Pe, are shown as the k = oo curve in Fig. 3.10. For small Pe, Sh approaches Sho, while for large Pe, Sh becomes proportional to Pe. The numerical solution... 

Fig. 5.1 Streamlines for flow past a sphere. Numerical results of Masliyah (M2). Flow from right to left. Values of T indicated, (a) Re = 1.0 (b) Re = 10 (c) Re = 50 (d) Re = 100.

Since the flow is only slightly perturbed from irrotational, a first approximation for the drag on a spherical bubble may be obtained by calculating the viscous energy dissipation for potential flow past a sphere. This gives (Lll) ... 

The mechanism of mass transfer to the external flow is essentially the same as for spheres in Chapter 5. Figure 6.8 shows numerically computed streamlines and concentration contours with Sc = 0.7 for axisymmetric flow past an oblate spheroid (E = 0.2) and a prolate spheroid (E = 5) at Re = 100. Local Sherwood numbers are shown for these conditions in Figs. 6.9 and 6.10. Figure 6.9 shows that the minimum transfer rate occurs aft of separation as for a sphere. Transfer rates are highest at the edge of the oblate ellipsoid and at the front stagnation point of the prolate ellipsoid. 

For CO 0, Eq. (11-7) reduces to the stream function for steady creeping flow past a rigid sphere, i.e., Eq. (3-7) with k = co. The parameter 3 may be regarded as a characteristic length scale for diffusion of vorticity generated at the particle surface into the surrounding fluid. When co is very large, 3 is small, and the flow can be considered irrotational except in the immediate vicinity of the particle. In the limit co go, Eq. (11-7) reduces to Eq. (1-29), the result for potential flow past a stationary sphere. 

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