Flow Around a Sphere

Many authors (B4, G3, H2, L2, Sll) have considered the flow pattern and wakes involved in the flow of a fluid past a rigid sphere. Nearly every book on fluid mechanics contains a chapter on flow around submerged shapes. Flow around fluid shapes is only touched upon by a few advanced treatises such as that by Lamb (L2). 

Elzinga and Banchero (El) use Meksyn s boundary layer equation (M2) for flow around a rigid sphere, with the boundary condition that the interfacial velocity is not zero, to calculate a shift in the boundary-separation ring from an equivalent rigid-sphere location. Their calculated positions are slightly less than their observed shifts but confirm the thesis that these shifts are due to internal circulation. Similar quantitative results are reported by Garner and Tayeban (G7). 

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Figure 6. Flow around a sphere. The system size is 50 x 25 x 25 with y = 8 particles per cell. The gravitational field strength was g = 0.005 and the rotation angle for MPC dynamics was a — ti/2. Panel (a) is for a Reynolds number of Re — 24 corresponding to X = 1.8 while panel (b) is the flow for Re = 76 and X = 0.35. (From Ref. 30.)...

The flow toward the surface is caused by the pressure under the indenter. It is analogous to the upward flow around a sphere dropped into a liquid. It is also analogous to inverse extrusion. A model of the flow has been proposed by Brown (2007) in terms of rotational slip. This model reproduces some of the observed behavior, but it is a continuum model and does not define the mechanism of rotational slip. 

The value of the Reynolds number which approximately separates laminar from turbulent flow depends, as previously mentioned, on the particular configuration of the system. Thus the critical value is around 50 for a film of liquid or gas flowing down a flat plate, around 500 for flow around a sphere, and around 2500 for flow through a pipe. The characteristic length in the definition of the Reynolds number is, for example, the diameter of the sphere or of the pipe in two of these examples. 

Figure 10. Velocity field of creeping flow around a sphere [24], Each arrow represents the velocity at the origin of the depicted vector. The length of the arrow corresponding with the free velocity v would equal the radius of the sphere...

In contrast to single-phase flow in a pipe of constant cross section, flow around a sphere or other bluff object exhibits several different flow regimes at different values of the Reynolds number. 

This equation too is solved with the same boundary conditions as Eq. (148). A series of equations results when different combinations of fluids are used. There is no change for the first stage. All the terms of equation of motion remain the same except the force terms arising out of dispersed-phase and continuous-phase viscosities. The main information required for formulating the equations is the drag during the non-Newtonian flow around a sphere, which is available for a number of non-Newtonian models (A3, C6, FI, SI 3, SI 4, T2, W2). Drop formation in fluids of most of the non-Newtonian models still remains to be studied, so that whether the types of equations mentioned above can be applied to all the situations cannot now be determined. 

Now we need and h, the mass and heat transfer coefficients around a sphere. These come from Sherwood and Nusselt numbers, respectively, for flow around a sphere,... 

Johansson (Jl) reported numerical calculations of the flow around a sphere fixed on the axis of a Poiseuille flow (Fig. 9.1 with b = 0,U = 0). Only solutions for 2 = 0.1 were considered, and wake formation was predicted for Re = 20.4 based on the centerline velocity Uq. ... 

At the other extreme of Re, Achenbach (Al) investigated flow around a sphere fixed on the axis of a cylindrical wind tunnel in the critical range. Wall effects can increase the supercritical drag coefficient well above the value of 0.3 arbitrarily used to define Re in an unbounded fluid (see Chapter 5). If Re is based on the mean approach velocity and corresponds to midway between the sub- and super-critical values, the critical Reynolds number decreases from 3.65 x 10 in an unbounded fluid to 1.05 x 10 for k = 0.916. 

The values of n and m vary depending on the system and the geometry, but typical values are n = m = 0.5 for creeping flow around a sphere in a gas-liquid system, and n = m = 0.33 for creeping flow around a sphere in liquid-solid systems. 

When fluid flows around the outside of an object, an additional loss occurs separately from the frictional energy loss. This loss, called form drag, arises from Bernoulli s effect pressure changes across the finite body and would occur even in the absence of viscosity. In the simple case of very slow or creeping flow around a sphere, it is possible to compute this form drag force theoretically. In all other cases of practical interest, however, this is essentially impossible because of the difficulty of the differential equations involved. 

When droplets are not at rest relative to the oxidizing atmosphere, the quiescent results no longer hold so forced convection must again be considered. No one has solved this complex case. As discussed in Section 4.E, flow around a sphere can... 

The heat itransfer correlation relating the Nusselt nmtiber to the Prandtl and Reynolds numbers for flow around a sphere is ... 

The second boundary condition is related to the transformation of the flow profile in the surface layer into the flow profile for the Stokes problem on the flow around a sphere outside the thin surface layer near the particle surface [4] and can be written as... 

A geometric dimensionless number does not appear here, as a sphere is already geometrically characterised by its diameter d. The functions Ftube and Fsphere have different forms because flow fields and heat transfer conditions in flow through a tube differ from those in a fluid flowing around a sphere. 

As the resistance factor cR in flow around a sphere is dependent on the Reynolds number, according to (3.278), a relationship between the Reynolds number formed with the falling velocity urs and the Archimedes number can be derived, Res = f(Ar). 

Dandy and Dwyer [30] computed numerically the three-dimensional flow around a sphere in shear flow from the continuity and Navier-Stokes equations. The sphere was not allowed to move or rotate. The drag, lift, and heat flux of the sphere was determined. The drag and lift forces were computed over the surface of the sphere from (5.28) and (5.33), respectively. They examined the two contributions to the lift force, the pressure contribution and the viscous contribution. While the viscous contribution always was positive, the pressure contribution would change sign over the surface of the sphere. The pressure... 

Auton [7], Thomas et al [152] and Auton et al [8] determined a lift force due to inviscid flow around a sphere. In an Eulerian model formulation this lift force parameterization is usually approximated for dilute suspensions, giving ... 

The stream function and radial velocity distribution function for a low-Reynold.s-number flow around a sphere are given by the following expressions due to Stokes ... 

The first group of terms on the right-hand-side of Eq. 4 describes particle transport to a collector surface by Brownian diffusion. NPe is the Peclet number, a ratio of particle transport by fluid advection to transport by molecular or viscous processes. The term As is introduced to account for the effects of neighboring collectors or media grains on the fluid flow around a collector of interest. The results here assume Happel s model (Happel, 1958) for flow around a sphere in a packed bed 4S depends only on the porosity of the bed (Table 1). The derivation for diffusive transport is based on the early work of Levich (1962). 

A similar solution can be obtained for the flow in a cylindrical pipe where y is replaced by the radial distance from the axis of the cylinder. There are a few other simple analytic solutions of the Stokes equation, e.g. for the flow around a sphere, etc. (Lamb, 1932). 

Fig. 10.12. Liquid stream-lines of a potential flow around a sphere calculated for different values of the radius of collision (the point of inflection is shown which separates the near and the distant parts of stream-lines) 0, z are polar coordinates of points of inflection with a small (I) and a large (2) radius of collision.

Unlike the complete solution to the equation of motion (EOM) for two-dimensional flow around a sphere, it is only necessary to relate Vr and v via the... 

The virtual volume coefficient Cy for potential flow around a sphere is 0.5. For ellipsoidal bubbles with a ratio of semiaxes 1 2, Cy is 1.12. For ellipsoidal bubbles with random wobbling motions, Fopez de Bertodano [26] calculated to be about 2.0. In addition, Cy is a function of the specific gas holdup [27-29] ... 

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