Uniform Flow past a Sphere

A second straightforward example, solved previously by other means, is Stokes original problem of uniform flow past a stationary sphere. To apply the methods of the preceding subsection to this problem, it is convenient to transform to the disturbance flow problem, 

To construct a solution for (u p ), we again begin with the pressure p. In this case, p is a decaying harmonic function, linear in U and a true scalar, that must be constructed solely from U and the vector harmonic functions. Examination of the decaying harmonics, (8-5), shows that there is a single combination of U and the vector harmonics that satisfy those conditions, namely, 

c is an arbitrary constant, and in this case the quantity on the right-hand side is a true scalar and thus has the same parity as p.  

The function u(H is harmonic, decaying, linear in U and a true vector. Again, examining the decaying vector harmonic functions (8-5), we find that there are only two products of U and the decaying harmonics that are true vectors. Thus the most general form for u(//) is 

Thus the general form for u that satisfies continuity is 

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To this point, the solution is essentially identical to that obtained for uniform flow past a sphere in Section A. Beyond this, however, the two differ. To satisfy boundary conditions at the surface of a sphere, a nonzero value of a is necessary. However, for a point force, a = 0. The simplest way to see this is to note that c is dimensionless, whereas a must have dimensions of (length)2. But a point force has no characteristic length scale, so a = 0. Thus,... 

We now seek a solution of (9 7) and (9-8) for small values of the Peclet number, Pe , by using the matched asymptotic expansion procedure that was detailed for uniform flow past a sphere in Section C. Although the reader may not immediately see that the derivation of an asymptotic solution for this new problem necessitates use of the matched asymptotic expansion technique, an attempt to develop a regular expansion for 9 for Pe 1 leads to a Whitehead-type paradox similar to that encountered for the uniform-flow problem. 

An especially powerful result is obtained if we apply the formula (8-215) to a stationary solid sphere of radius a in the undisturbed flow u00(x). In this case, the solution for uniform flow past a stationary solid sphere yields the general result... 

D. UNIFORM FLOW PAST A SOLID SPHERE AT SMALL, BUT NONZERO, REYNOLDS NUMBER... 

D. Uniform Flow Past a Solid Sphere at Small, but Nonzero, Reynolds Number... 

Figure 9-11. A schematic representation of the solution domain for forced convection heat transfer that is due to uniform flow past a solid sphere with a uniform surface temperature for 1 but Pe 3> 1.

Here, d is diameter, U is velocity, is particle number concentration (m ) in a control volume AV, ij is coUision efficiency, and At is a time step for a droplet to move from cme positirai to another. The subscripts p and d denote the physical quantities relative to particle and droplet, respectively. The particle-droplet relative velocity and the particle number concentration along the droplet moving path can be obtained based on a Lagrangian tracking. The collision efficiency i/ is defined as the ratio of the number of particles which collide with the spheroid to the number of particles which could collide wifli the spheroid if their trajectories were straight lines. In the case of laminar flow past a sphere, where the particles are uniformly distributed in the incident flow, the collision efficiency can be determined as t] = 2ycildd), where is the distance from the central symmetry axis of the flow, at which the particles only touch the sphere while flowing past it and is the diameter of the sphere (as shown in Fig. 18.30). The particles, whose coordinates in the incident flow are y > jcn will not collide with the sphere. In Schuch and Loffler [33], the collision efficiency rj is correlated with Stokes number (St) as... 

Figure 7-11. A schematic representation of the domain for a uniform flow past an arbitrary, axisymmetric body. For the case of a solid sphere, this is Stokes problem.

F. UNIFORM STREAMING FLOW PAST A SOLID SPHERE - STOKES LAW... 

F. Uniform Streaming Flow past a Solid Sphere - Stokes Law... 

Figure 7-12. The streamlines and contours of constant vorticity for uniform streaming flow past a solid sphere (Stokes problem). The streamfunction and vorticity values are calculated from Eqs. (7-158) and (7-162). Contour values plotted for the streamfunction are in increments of 1/16, starting from zero at the sphere surface, whereas the vorticity is plotted at equal increments equal to 0.04125.

As an example of the application of (7-131), we consider creeping flow past an arbitrary axisymmetric body with a uniform streaming motion at infinity. For the case of a solid sphere, this is known as Stokes problem. In the present case, we begin by allowing the geometry of the body to be arbitrary (and unspecified) except for the requirement that the symmetry axis be parallel to the direction of the uniform flow at infinity so that the velocity field will be axisymmetric. A sketch of the flow configuration is shown in Fig. 7 11. We measure the polar angle 9 from the axis of symmetry on the downstream side of the body. Thus ij = I on this axis, and ij = — 1 on the axis of symmetry upstream of the body. 

Here, we consider Stokes problem of uniform, streaming motion in the positive z direction, past a stationary solid sphere. The problem corresponds to the schematic representation shown in Fig. 7-11 when the body is spherical. This problem may also be viewed as that of a solid spherical particle that is translating in the negative z direction through an unbounded stationary fluid under the action of some external force. From a frame of reference whose origin is fixed at the center of the sphere, the latter problem is clearly identical with the problem pictured in Fig. 7-11. Because we have already derived the form for the stream-function under the assumption of a uniform flow at infinity, we adopt the latter frame of reference. The problem then reduces to applying boundary conditions at the surface of the sphere to determine the constants C and Dn in the general equation (7-149). The boundary conditions on the surface of a solid sphere are the kinematic condition and the no-slip condition,... 

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