Ideal Profile Analysis illustration

It is necessary to caution that the foregoing analysis is a highly idealized one, which has been used primarily to illustrate the effects of magnetic fields on heat transfer. A more realistic analysis would consider the variation of electrical conductivity of the fluid and take into account the exact velocity profile rather than the slug-flow model. A survey of more exact relations for heat transfer in MFD systems is given in Refs. 1 and 2. 

The primary ion beam etches a well defined crater in a sample by a beam deflector as illustrated in Figure 8.23. Ideally, only the secondary ions emitted from the crater bottom should be collected for depth profiling. However, secondary ions will be collected at the same time as when the primary ions etch the crater. Thus, the secondary ions from the crater walls may also contribute to the secondary ions in analysis. Such secondary ions from crater walls do not represent the true concentrations of elements at the crater bottom. To avoid this problem,... 

One should realize that these calculations are based on an expression for Vr which corresponds to potential flow past a stationary nonde-formable bubble, as seen by an observer in a stationary reference frame. However, this analysis rigorously requires the radial velocity profile for potential flow in the Uquid phase as a nondeformable bubble rises through an incompressible liquid that is stationary far from the bubble. When submerged objects are in motion, it is important to use liquid-phase velocity components that are referenced to the motion of the interface for boundary layer mass transfer analysis. This is accomplished best by solving the flow problem in a body-fixed reference frame which translates and, if necessary, rotates with the bubble such that the center of the bubble and the origin of the coordinate system are coincident. Now the problem is equivalent to one where an ideal fluid impinges on a stationary nondeformable gas bubble of radius R. As illustrated above, results for the latter problem have been employed to estimate the maximum error associated with the neglect of curvature in the radial term of the equation of continuity. 

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