Sphere rotating

The behavior of a rotating sphere or hemisphere in an otherwise undisturbed fluid is like a centrifugal fan. It causes an inflow of the fluid along the axis of rotation toward the spherical surface as shown in Fig. 1(a). Near the surface, the fluid flows in a spirallike motion towards the equator as shown in Fig. 1(b) and (c). On a rotating sphere, two identical flow streams develop on the opposite hemispheres. The two streams interact with each other at the equator, where they form a thin swirling jet toward the bulk fluid. The Reynolds number for the rotating sphere or hemisphere is defined as ... 

Table 1. Numerical results for the first four terms in the power series expansions of the velocity profile near a rotating sphere [2].

The transition from laminar to turbulent flow on a rotating sphere occurs approximately at Re = 1.5 4.0 x 104. Experimental work by Kohama and Kobayashi [39] revealed that at a suitable rotational speed, the laminar, transitional, and turbulent flow conditions can simultaneously exist on the spherical surface. The regime near the pole of rotation is laminar whereas that near the equator is turbulent. Between the laminar and turbulent flow regimes is a transition regime, where spiral vortices stationary relative to the surface have been observed. The direction of these spiral vortices is about 4 14° from the negative direction of the azimuthal angle,. The phenomenon is similar to the flow transition on a rotating disk [19]. 

For turbulent flow on a rotating sphere or hemisphere, Sawatzki [53] and Chin [22] have analyzed the governing equations using the Karman-Pohlhausen momentum integral method. The turbulent boundary layer was assumed to originate at the pole of rotation, and the meridional and azimuthal velocity profiles were approximated with the one-seventh power law. Their results can be summarized by the... 

Eqs. (33)- 35) have been solved analytically by Chin [13]. For large Schmidt numbers, Sc, the concentration distribution in the viscinity of the rotating sphere is given by ... 

ShaJSc113 as indicated by the thin solid line. This 0.67 power of Re agrees with the result of a turbulent heat transfer measurement on a rotating sphere [40], Since the flow induced by a rotating sphere is also characterized by an outflowing radial jet at the equator caused by the collosion of two opposing flow boundary layers on the sphere, the 0.67 power dependence on Re is clearly related to the radial flow stream away from the equator. 

In electrochemistry, spherical and hemispherical electrodes have been commonly used in the laboratory investigations. The spherical geometry has the advantage that in the absence of mass transfer effect, its primary and secondary current distributions are uniform. However, the limiting current distribution on a rotating sphere is not uniform. The limiting current density is highest at the pole, and decreases with... 

The diffusion coefficient is estimated using Stokes law, D = kTjQmrja, where rj is the viscosity and a the radius of the rotating sphere. This rough model allows a calculation of the correlation function. It turns out that a Boltzmann distribution remains a Boltzmann distribution so that the system of equations (12)... 

Fig. 10,8 Schematic diagram showing Magnus effect on rotating sphere.

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

A turbulent eddy can be visualized as a large number of different-sized rotating spheres or ellipsoids. Each sphere has subspheres and so on until the smallest eddy size is reached. The smallest eddies are dissipated by viscosity, which explains why turbulence does not occur in narrow passages there is simply no room for eddies that will not be dissipated by viscosity. 

A nucleus with an odd atomic number or an odd mass number has a nuclear spin that can be observed by the NMR spectrometer. A proton is the simplest nucleus, and its odd atomic number of 1 implies it has a spin. We can visualize a spinning proton as a rotating sphere of positive charge (Figure 13-1). This movement of charge is like an... 

Here, v(r) is the velocity field at position r, p(r) the pressure field, and o(r) the rate-of-strain tensor defined as the symmetric part of the velocity gradient tensor. In the calculation below, n(r) is assumed to be spherically symmetric around a solute. v(r) around a rotating sphere can be expressed in the form... 

We numerically solve Eq. (4) by a method" similar to the cases of and B with a condition that the long-distance form of/ r) at large r is given by Eq. (3). Stokes theorem tells us that the friction exerted on the rotating sphere can be calculated from the long-distance behavior of/(r) With S in Eq (3) obtained by the numerical solution of Eq (4), is given by... 

Lower values were obtained when the presistence length was estimated by the beginning of a sharp kink on theqsp/c vs. c plot (c is the concentration). This kink indicates the overlapping of the rotation spheres of asymmetric particles. The persistence length for PBA was found to be equal to 180-240 A, and for PPTA, 150-180 A %... 

K. Nisancioglu and J. Newman, "Current Distribution on a Rotating Sphere below the Limiting Current," Journal of The Electrochemical Society, 121 (1974) 241-246. 

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. 

Maccoll [95] studied the aerodynamics of a spinning sphere, and observed a negative Magnus effect when the ratio of the equatorial speed of the rotating sphere to the flow speed, Ve ua/v, was less than 0.5. 

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 ... 

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. 

The quantity of interest to us is the average behavior of the component of the dipole moment in the field direction, namely () (we postulate an assembly of such rotating spheres). Since... 

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.

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