Viscous Effects on Capillary Instability

We now turn to the case of a cylinder in which viscous effects play a dominant role. In this case, the initial analysis is the same as that of the preceding section, up to the point of nondimensionalizing. Now the appropriate choice for a characteristic pressure is pc = pud a, and hence, instead of (12-12), we have 

In this case, it is convenient to express / in the more general form 

as before, the pressure perturbation is a harmonic function and thus, corresponding to (12 43), must be expressible in the form 

The first thing to note is that the proposed solution (12-46) satisfies the continuity equation (12-39) for any choice of U(r). Of course, U(r) must also satisfy the equation of motion (12-38). To determine the appropriate form for U(r), let us first consider the r component of this equation  

If we substitute for ur and dp/dr from (12-45) and (12-46) and factor out the common exponential factor, we obtain 

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