Sphere lift, rotating

Drew DA, Lahey RT Jr (1987) The virtual mass and lift force on a sphere in rotating and straining inviscid flow. Int J Multiphase Flow 13 (1) 113-121... 

Drew DA, Lahey RT Jr (1979) Application of general constitutive principles to the derivation of multidimensional two-phase flow equations. Int J Multiph How 5 243-264 Drew DA (1983) Mathematical modeling of two-phase flow. Ann Rev Huid Mech 15 261-291 Drew DA, Lahey RT Jr (1987) The virtual mass and lift force on a sphere in rotating and straining inviscid flow. Int J Multiph Flow 13(1) 113-121... 

The angular velocity of the sphere, H, is taken as positive for rotation in the same sense as that of the fluid. The resulting lift on the particle is taken as positive in the direction Q x Ur. ... 

Theoretical attempts to explain lift have concentrated on flow at small but nonzero Re, using matched asymptotic expansions in the manner of Proudman and Pearson for a nonrotating sphere (see Chapter 3). In the absence of shear, Rubinow and Keller (R6) showed that the drag is unchanged by rotation. With... 

Drag and lift coefficients for rotating spheres. All data plotted are for smooth spheres. 

At high Reynolds numbers, the rotation of the sphere yields an asymmetric wake, as shown in Fig. 3.4. In this case, the theoretical analysis of the Magnus force and the drag force becomes rather complex because of the difficulties in obtaining the expressions for the pressure and velocity distributions around the surface of the sphere. Thus, the determination of the lift force as well as the drag force relies mainly on the empirical approach. 

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). 

A rotating sphere in uniform flow will experience a lift which causes the particle to drift across the flow direction. This is called the Magnus effect (or force). The physics of this phenomenon are complex. 

Rubinow and Keller [123] calculated the flow around a rotating sphere moving in a viscous fluid for small Reynolds numbers. They determined the drag, torque, and lift force (Magnus) on the sphere to O(Rep). The results were ... 

Drew and Wallis [37] (p 61) examined the forces on spheres in two-phase suspensions based on theoretical analyzes. Their result included lift forces that give rise to a net transverse force on particle swarms if the group of spheres are translating and rotating as a unit. Note that this force is different from... 

Clift et al [22] summarized the measurements of drag and lift on rotating spheres, and concluded that the phenomena involved are so complex that drag and lift forces on rotating spheres should be determined experimentally. 

Fig. 5.5. The Saffman force on a particle in a shear flow. The sketch illustrates that this lift force is caused by the pressure distribution developed around the sphere due to particle rotation induced by the shear flow velocity gradient.

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... 

The Lift on a Sphere That is Rotating in a Simple Shear Flow... 

It is a simple matter to see that the lift is zero for each of these component problems. For a sphere rotating in a stationary fluid, it is evident that there is no difference between any of the possible directions in the plane that is orthogonal to fl. Hence, because there is no direction in the orthogonal direction that is distinguishable from any other, the force produced by rotation alone must be zero. 

The reader may find the result (7-16) surprising. As already noted, it is well known that a rotating and translating sphere in a stationary fluid will often experience a sideways force (that is, lift) that will cause it to travel in a curved path-think, for example, of a curve ball in baseball or an errant slice or hook in golf. The difference between these familiar examples and the problem previously analyzed is that the Reynolds numbers are not small and the governing equations are the full, nonlinear Navier-Stokes equations rather than the linear creeping-flow approximation. Thus the decomposition to a set of simpler component problems cannot be used, and it is not possible to deduce anything about the forces on the... 

A few additional experiments were conducted in which the sphere was allowed to rotate. Though not wholly conclusive, no statistically significant effect of particle rotation on lift was found. 

The data of Theodore (T3a) for the lift force on a nonrotating sphere in a Poiseuille flow, and, to a lesser extent, the comparable data of Oliver (02), show that lateral forces arise at small Reynolds numbers even in the absence of particle rotation. Thus, inertial lift forces can arise from slip-shear as well as from slip-spin. That the character of these two forces is very different is shown clearly by the theoretical analysis of Saffman (SIa). 

Saffman had interests in turbulence, viscous flows, vortex motion and water waves. He made valuable theoretical contributions to different areas of low-Reynolds-number hydrodynamics. These included the lifting force on a sphere in a shear flow at small but finite Reynolds numbers, the Brownian motion in thin liquid films, and particle motion in rapidly rotating flows. Saffinan s other contributions include dispersion in porous media, average velocity of sedimenting suspensions, and compressible low-Reynolds-number flows. 

Maxey and Riley [47] derived an equation of motion for a small rigid sphere of radius R in a nonuniform flow. If one considers small bubbles moving in a polar liquid, this equation might be appropriate because surfactants would tend to immobilize the surface of a bubble and make it behave like a rigid sphere. Maxey and Riley assumed that the Reynolds number based on the difference between the sphere velocity and the undisturbed fluid velocity was small compared to unity. In addition, they assumed that the spatial nonuni-formity of the undisturbed flow was sufficiently small that the modified drag due to particle rotation and the Saffman [48] lift force could be neglected. Finally, they ignored interactions between particles. 

Drew DA, Lahey RT Jr (1990) Some supplemental analysis concerning the virtual mass and lift force on a sphere in a rotating and straining flow. Int J Multiph How 16 1127-1130 Drew DA (1992) Analytical modeling of multiphase flows modern developments and advances. In Lahey RT jr (ed) Boiling heat transfer. Elsevier Science Publishers BV, Amsterdam, pp 31-83... 

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