Sphere in a uniform flow

The first term on the right-hand side of Eq. (3.80) is the total drag force in the opposite direction of Up, including both Stokes drag and Oseen drag. The second term represents a lift force in the direction perpendicular to Up. Thus, the lift force or Magnus force for a spinning sphere in a uniform flow at low Reynolds numbers is obtained as... 

Heat Transfer of a Single Sphere in a Uniform Flow... 

For forced convective heat transfer over a sphere in a uniform flow, a frequently used empirical relation was proposed by Ranz and Marshall (1952) as... 

Formost gases, Pris nearly constant ( 0.7). Therefore, for convective heat transfer of an individual solid sphere in a gaseous medium, Nup is only a function of Rep. An alternative estimation of Nup for thermal convection of a sphere in a uniform flow was recommended by McAdams (1954) as... 

Rigid spheres sometimes experience a lift force perpendicular to the direction of the flow or motion. For many years it was believed that only two mechanisms could cause such a lift. The first one described is the so-called Magnus force which is caused by forced rotation of a sphere in a uniform flow field. This force may also be caused by forced rotation of a sphere in a quiescent fluid. The second mechanism is the Saffman lift. This causes a particle in a shear flow to move across the flow field. This force is not caused by forced rotation of the particle, as particles that are not forced to rotate also experience this lift (i.e., these particles may also rotate, but then by an angular velocity induced by the flow field itself). 

The limit Pe 0 yields the pure conduction heat transfer case. However, for a fluid in motion, we find that the pure conduction limit is not a uniformly valid first approximation to the heat transfer process for Pe 1, but breaks down far from a heated or cooled body in a flow. We discuss this in the context of the Whitehead paradox for heat transfer from a sphere in a uniform flow and then show how the problem of forced convection heat transfer from a body in a flow can be understood in the context of a singular-perturbation analysis. This leads to an estimate for the first correction to the Nusselt number for small but finite Pe - this is the first small effect of convection on the correlation between Nu and Pe for a heated (or cooled) sphere in a uniform flow. 

The lift on a nonrotating sphere in a uniform flow is also zero because of the symmetry of the problem. Again, because no direction perpendicular to U can be distinguished from any other, the only possibility is that there can be no force on the sphere in any of these directions. The only force acting on the sphere in this case will be a drag force that is collinear with U. 

Figure 9-3. A schematic representation of the solution domain for heat transfer from a sphere in a uniform flow in the limit of small Peclet numhers, that is, Pe —> 0.

Figure 9-4. Contours of constant temperature (isotherms) for heat transfer from a sphere in a uniform flow at low Peclet numbers according to Eq. (9-51). Note that the sphere appears in this representation, in which Ic=k/Uoo, as a point source (sink) of heat (the radius of the sphere, a, is vanishingly small compared with k/Uoo in the limit as Pe - 0). In the inner region, near to the sphere, the isotherms at leading order of approximation are still spherical as illustrated in Fig. 9-2. The three contours plotted are o = 0.25, 2.75, and 5.25.

The cases of a sphere and slightly deformed sphere in a uniform flow field are considered first in Sections 4 and 5. The mathematical method used conventionally in these problems is the regular asymptotic expansion. The reader is introduced to this method. In Section 6, the dip coating problem under the lubrication theory approximation is examined. (The closely related slender body approximation is outlined in Problem 7.5.) A more sophisticated method of matched asymptotic expansions is used to solve this problem and its main features... 

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... 

The long-range, purely hydrodynamic interaction between two suspended spheres in a shear flow was first calculated by Guth and Simha (1936), yielding a value of kx = 14.1 via a reflection method. Saito (1950,1952) proposed two alternative modifications, obtaining kt = 12.6 and 2.5, respectively the latter value is obtained upon supposing a spatially uniform distribution of particles. 

The rotation of a rigid sphere will cause the surrounding fluid to be entrained. When the sphere is placed in a uniform flow, this results in higher fluid velocity on one side of the particle, and lower velocity on the other side. 

For steady flow of an aerosol around a collector such as a cylinder or sphere, a force balance can be written on a particle by considering it to be fixed in a uniform flow of elocity u — uy ... 

Figure 7-2. Illustration of the decomposition of the problem of a freely rotating sphere in a simple shear flow as the sum of three simpler problems (a) a sphere rotating in a fluid that is stationary at infinity, (b) a sphere held stationary in a uniform flow, and (c) a nonrotating sphere in a simple shear flow that is zero at the center of the sphere. The angular velocity Cl in (a) is the same as the angular velocity of the sphere in the original problem. The translation velocity in (b) is equal to the undisturbed fluid velocity evaluated at the position of the center of the sphere. The shear rate in (c) is equal to the shear rate in the original problem.

C. HEAT TRANSFER FROM A SOLID SPHERE IN A UNIFORM STREAMING FLOW AT SMALL, BUT NONZERO, PECLET NUMBERS... 

To complete the specification of the problem for 9, we must specify a particular velocity field u. In the case of Re <creeping-flow solutions of Chaps. 7 and 8, and it is again convenient to focus our attention on the case of a sphere in a uniform streaming flow, in which a first approximation to the velocity field is given by the Stokes solution, Eq. (7-158), from which we can calculate the velocity components by means of (7-102). 

C. Heat Transfer From a Solid Sphere in a Uniform Streaming Flow... 

To obtain a valid approximate solution for heat transfer from a sphere in a uniform streaming flow at small, but nonzero, Peclet numbers, we must resort to the method of matched (or singular) asymptotic expansions.4 In this method, as we have already seen in Chap. 4, two (or more) asymptotic approximations are proposed for the temperature field at Pe 1, each valid in different portions of the domain but linked in a so-called overlap or matching region where it is required that the two approximations reduce to the same functional form. The approximate forms of (9-1), from which these matched expansions are derived, can be obtained by nondimensionalization by use of characteristic length scales that are appropriate to each subdomain. 

We have invested considerable effort to analyze the rate of heat transfer from a heated sphere in a uniform streaming flow at low Peclet number. Now that we understand the asymptotic structure of the low-Peclet-number limit, however, we can very easily extend our result for the first two terms in the expression for Nu, (9-60), for the same flow to bodies of arbitrary shape, which may be either solid or fluid, and to arbitrary values of the Reynolds number provided only that Pe <. This extension was first demonstrated by Brenner,12 and our discussion largely follows his original analysis. 

But before the virtues of the results and the approach are extolled, the method must be described in detail. Let us therefore return to a systematic development of the ideas necessary to solve transport (heat or mass transfer) problems (and ultimately also fluid flow problems) in the strong-convection limit. To do this, we begin again with the already-familiar problem of heat transfer from a solid sphere in a uniform streaming flow at sufficiently low Reynolds number that the velocity field in the domain of interest can be approximated adequately by Stokes solution of the creeping-flow problem. In the present case we consider the limit Pe I. The resulting analysis will introduce us to the main ideas of thermal (or mass transfer) boundary-layer theory. 

In the 1970s, Shilov and Estrella-Lopis first recognized that electrohydrodynamics (what we now call ICEO) can contribute to the motion of particles in low-frequency, nonuniform electric fields [17], in addition to DEP, although the effect has not been studied much in theory or experiment. Shilov and Simonova analyzed the problem of an ideally polarizable sphere in a uniform field gradient and made the remarkable prediction that the particle does not move. Due to equal and opposite motions by DEP and ICEP, the sphere levitates in the field while driving a steady ICEO flow, but this is a unique case. 

Example 4.2 Show that the convective heat transfer from a sphere in a flowing medium at low Reynolds number can be expressed by Eq. (4.42). It is assumed that the sphere is kept at a constant temperature Tp and the temperature inside the sphere is uniform. The flow is uniform with velocity of Uco and temperature of Too at infinity. All the thermal properties are constant. 

Maxey, M.R. and Riley, J.J. (1983), Equation of motion for a small rigid sphere in a non uniform flow, Phys. Fluids, 26(4), 883. 

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