The Rising Bubble Problem Reprise

In Example 15, we saw that v, the concentration at the center, appears naturally as a parameter in terms of which 17 and p can be calculated. It is, then, a very natural thing to plot the two against one another, regarding v as a marker along the curve. We shall see later in the chapter that this aids the asymptotic analysis. 

An awareness of the possible parameterization of solution curves is helpful in setting up the numerical procedure on the computer, as may be illustrated 

In the first place, we see that the right-hand sides are infinite along the axis = 0 and the curve ( = im. However, the solution paths are the same as those for the system 

In the ascending bubble problem (Example 8), the requirement that the pressure at the surface should be positive rules out the dramatic behavior near the singular curve ( = but, by suitably defining the value of the derivative at the discontinuity, solution paths of perilous stability can be found4 (Cf. [312]). In Fig. 13, the broken line A = 0 is manifestly unstable. However, the broken line B = 0 can be reached in a finite time by a solution starting above A = 0 in the lower part of the phase plane and below A - 0 in the upper part. If the value at the singularity, B = 0, which is otherwise undefined, is correctly chosen, the solution turns and follows the path B = 0. In practice, it falls victim to the peril, even when stable. 

3 Isodic = means having the same paths. I am not sure if this term, obvious though it is, is in current usage. 

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