Sphere Stokes flow

In a series of papers, Felderhof has devised various methods to solve anew one- and two-sphere Stokes flow problems. First, the classical method of reflections (Happel and Brenner, 1965) was modified and employed to examine two-sphere interactions with mixed slip-stick boundary conditions (Felderhof, 1977 Renland et al, 1978). A novel feature of the latter approach is the use of superposition of forces rather than of velocities as such, the mobility matrix (rather than its inverse, the grand resistance matrix) was derived. Calculations based thereon proved easier, and convergence was more rapid explicit results through terms of 0(/T7) were derived, where p is the nondimensional center-to-center distance between spheres. In a related work, Schmitz and Felderhof (1978) solved Stokes equations around a sphere by the so-called Cartesian ansatz method, avoiding the use of spherical coordinates. They also devised a second method (Schmitz and Felderhof, 1982a), in which... 

This is known as Stokes flow, and Eq. (11-3) has been found be accurate for flow over a sphere for NRe < 0.1 and to within about 5% for NRe < 1. Note the similarity between Eq. (11-3) and the dimensionless Hagen-Poiseuille equation for laminar tube flow, i.e.,/ = 16/tVRe. 

The viscosity of a Newtonian fluid can be determined by measuring the terminal velocity of a sphere of known diameter and density if the fluid density is known. If the Reynolds number is low enough for Stokes flow to apply (fVRe < 0.1), then the viscosity can be determined directly by rearrangement of Eq. (11-10) ... 

The Stokes flow criterion is rather stringent. (For example, a 1 mm diameter sphere would have to fall at a rate of 1 mm/s or slower in a fluid with a viscosity of 10 cP and SG = 1 to be in the Stokes range, which means that the density of the solid would have to be within 2% of the density of the... 

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. 3.10 External Sh for spheres in Stokes flow (1) Exact numerical solution for rigid and circulating spheres (2) Brenner (B6) rigid sphere, Pe 0, Eq. (3-45) (3) Levich (L3) rigid sphere, Pe 00, Eq. (3-47) (4) Acrivos and Goddard (A 1) rigid sphere, Pe oo, Eq. (3-48) (5) Approximate values fluid spheres.

Fig. 3.11 Local Sherwood number for rigid sphere in Stokes flow (1) Exact numerical solution Pe = 10 (2) High Pe asymptotic solution (L3) Pe = 10 (3) Low Pe asymptotic solution (A2) Pe = 0.1.

Harper and Chang (H4) generalized the analysis for any three-dimensional body and defined a lift tensor related to the translational resistances in Stokes flow. Lin et al (L3) extended Saffman s treatment to give the velocity and pressure fields around a neutrally buoyant sphere, and also calculated the first correction term for the angular velocity, obtaining... 

The analytical solution for convective heat transfer from an isolated particle in a Stokes flow can be obtained by using some unique perturbation methods, noting that the standard perturbation technique of expanding the temperature field into a power series of the Peclet number (Pe = RepPr) fails to solve the problem [Kronig and Bruijsten, 1951 Brenner, 1963]. The Nup for the thermal convection of a sphere in a uniform Stokes flow is given by... 

Available results pertinent to the hydrodynamics of fractal suspensions are sparse thus far, encompassing only three physical situations. Gilbert and Adler (1986) determined the Stokes rotation-resistance dyadic for spheres arranged in a Leibniz packing [Fig. 7(a)], With the gap between any two spheres assumed small compared with their radii, lubrication-type approximations suffice. In this analysis, the inner spheres are assumed to rotate freely, whereas external torques T( (i = 1, 2, 3) are applied to the three other spheres. For Stokes flow, these torques are linearly related to the sphere angular velocities by the expression... 

Small Reynolds Number Flow, Re < 1. The slow viscous motion without interfacial mass transfer is described by the Hadamard (66)-Rybcynski (67) solution. For infinite liquid viscosity the result specializes to that of the Stokes flow over a rigid sphere. An approximate transient analysis to establish the internal motion has been performed (68), Some simplified heat and mass transfer analyses (69, 70) using the Hadamard-Rybcynski solution to describe the flow field also exist. These results are usually obtained through numerical integration since analytical solutions are usually difficult to obtain. 

The bubbles occurring in the present electrochemical system are small enough to fulfil the criterion of Stokes flow. Thus, turbulent wakes arising behind bubbles can be neglected [26], Furthermore, we consider all bubbles in the present simulations as small, non-deformable and rigid spheres. This hypothesis holds for bubbles of low Eotvos numbers Eo ... 

Friedlander S.K., A note on transport to spheres in Stokes flow, AIChE ). 

The Basset s expression is found in the limit of unsteady Stokes flow for a rigid sphere and is given by ... 

Solve for the drag on a sphere in a flowing stream with a uniform velocity profile upstream. Solve for zero Reynolds number (Stokes flow). Compare the solution with an analytical solution in your textbook. (Hint Set the density to zero to simulate Stokes flow. The drag is obtained by integrating certain stresses over the boundary of the sphere.)... 

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.

Problem 7-26. An Alternative Derivation of the Solution for Stokes Flow. In Subsection B.4, we showed that the force acting on a sphere that translates through a fluid at low Reynolds number can be expressed in terms of a resistance tensor A in the form F = A U. A generalization of this idea is that the pressure and velocity fields around the sphere must also be a linear function of the vector U, and thus expressible in terms of a vector pressure and a tensor velocity in the form... 

Problem 8-5. The annular region between two concentric rigid spheres of radii a and A.a (with k > 1) is filled with Newtonian fluid of viscosity /i and density p. The outer sphere is held stationary whereas the inner sphere is made to rotate with angular velocity fl. Assume that inertia is negligible so that the fluid is in the Stokes flow regime. 

Problem 8-9. Torque on a Sphere in a General Stokes Flow. Use the reciprocal theorem for Stokes equations to derive the following expression for the torque exerted on a sphere of radius a that is held fixed in the Stokes flow u°°(x) ... 

According to this solution, which is plotted in Fig. 9-2, the temperature falls off inversely with distance from the sphere in all directions. This very simple solution is the equivalent of Stokes solution for creeping motion past a sphere. Indeed, it may be recalled from Chap. 7 that the disturbance to the velocity field that is due to a sphere in Stokes flow also decreased as r 1 for large r. 

The original analysis of this problem was pubhshed by A. Acrivos and T. E. Taylor, Heat and mass transfer from single spheres in Stokes flow, Phys. Fluids 5, 387-94 (1962). 

A. A. Zick and G. M. Homsy, Stokes flow through periodic arrays of spheres, J. Fluid Mech. 115, 13-26 (1982) A. S. Sangani and A. Acrivos, Slow flow through a periodic array of spheres, Int. J. Multiphase Flow 8, 343-60 (1982) G. Liu and K. E. Thompson, A domain decomposition method for modelling Stokes flow in porous materials, Int. J. Numer. Meth. Fluids 38, 1009-25 (2002). 

Numerous available numerical solutions the Navier-Stokes equations, as well as experimental data (see a review in [94]), provide a detailed analysis of the flow pattern for increasing Reynolds numbers. For 0.5 < Re < 10, there is no flow separation, although the fore-and-aft symmetry typical of inertia-free Stokes flow past a sphere is more and more distorted. Finally, at Re = 10, flow separation occurs at the rear of the particle. 

Let us investigate convective mass transfer to the surface of a solid sphere freely suspended in an arbitrary plane shear Stokes flow. In this case, the fluid velocity distribution remote from the particle is given by formulas (4.5.1) with... 

By virtue of the no-slip condition on the surface, a sphere freely suspended in a plane shear flow will rotate at a constant angular velocity fl equal to the flow rotation velocity at infinity. The solution of the corresponding three-dimensional hydrodynamic problem on a particle in a Stokes flow is given in [343]. 

---