Stability to Nonspherical Disturbances

In the present case, we consider a bubble whose interface is described in terms of a spherical coordinate system in the asymptotic form 

e is a small parameter that will form the basis of an asymptotic approximation for the dynamics of the bubble surface. The question here is whether a bubble with a nonspherical initial shape of small amplitude 0(e) will return to a sphere - that is, fn(9, p,t) 0 as t —r oc - or whether the initial disturbance in shape will grow with increase of /. 

To analyze this problem, we need to go back to a statement of the problem in general fluid dynamical terms. Thus we begin by restating the governing equations and boundary conditions for an oscillating bubble in a quiescent, incompressible fluid. These are the Navier-Stokes and continuity equations the three boundary conditions 

ps is the pressure inside the bubble, y is the interfacial tension, and t, represents one of the pair of orthogonal unit vectors that are tangent at any point to the bubble surface. The problem is to calculate velocity and pressure fields in the fluid, as well as to determine the functions R(t) and fn(0, p, t) that describe the geometry of the bubble surface. In the present context, it is the latter part of the problem that is the focus of our interest. 

One complication is that the boundary conditions (4-264)-(4-266) must be applied at the bubble surface, which is both unknown [that is, specified in terms of functions R(t) and fn(9, p,t) that must be calculated as part of the solution] and nonspherical. Further, the normal and tangent unit vectors n and t, that appear in the boundary conditions are also functions of the bubble shape. In this analysis, we use the small-deformation limit s 1 to simplify the problem by using the method of domain perturbations that was introduced earlier in this chapter. First, we note that the unit normal and tangent vectors can be approximated for small e in the forms 

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