Rotating hemisphere properties

To describe the velocity profile in laminar flow, let us consider a hemisphere of radius a, which is mounted on a cylindrical support as shown in Fig. 2 and is rotating in an otherwise undisturbed fluid about its symmetric axis. The fluid domain around the hemisphere may be specified by a set of spherical polar coordinates, r, 8, , where r is the radial distance from the center of the hemisphere, 0 is the meridional angle measured from the axis of rotation, and (j> is the azimuthal angle. The velocity components along the r, 8, and (j> directions, are designated by Vr, V9, and V. It is assumed that the fluid is incompressible with constant properties and the Reynolds number is sufficiently high to permit the application of boundary layer approximation [54], Under these conditions, the laminar boundary layer equations describing the steady-state axisymmetric fluid motion near the spherical surface may be written as ... 

Very true, Miss Muxdroozol. Also note that the art of navigation would have developed more slowly in the Southern Hemisphere since there is no equivalent to Polaris, the North Star, in the southern skies.6 Polaris appears to stand still, while other stars rotate around it, because it is almost exactly aligned along the celestial North Pole. This property is useful to navigators, because the star always lies in the direction of north. ... 

Fig. 1. A schematic diagram of the orbital positions of the moons of Jupiter discovered by Galileo. Their positions are scaled to Jupiter s radius (Rj) with properties given in Table 1. Note these satellites, like our moon, are all phase locked to the parent planet, hence the same side faces Jupiter throughout each moon s orbit. Since the magnetic field is attached to Jupiter, it rotates faster, therefore, in addition to ions and electrons moving up and down the field lines and impacting the satellites, there is a net preferential flow onto the hemisphere trailing the satellite s motion.

The Kepes balance rheometer is another device that determines dynamic properties in steady rotation. Material is confined between concentric hemispheres whose axes of rotation form an angle 6. Both hemispheres rotate with the same angular velocity cd. Ro — Rf = 8 Ri and 6 is small. 

Other devices have been developed that operate along similar lines. The so-called balance rheometer confines a test fluid between two concentric hemispheres that rotate at the same rate but whose axes of rotation are inclined at an angle to one another. Similarly, cone-and-plate geometry, in which the axis of the cone is not perpendicular to the plate, and Couette (cup-and-bob) geometry, with the bob not centered in the cup can also be used to obtain dynamic data. These devices are analyzed by Walters. Dynamic tensile properties can be obtained for relatively rigid materials by subjecting a rotating cylindrical rod to a cantilever deflection. ... 

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