**7. Conclusion**

90 Hydrodynamics – Advanced Topics

 

(in the zero approximation with respect to the small

; this conforms to the rigorous result we

. Although capillary forces generally

(67)

, respectively. If we equate

<sup>2</sup> <sup>2</sup> <sup>1</sup> ln . *H z R R Hz R R R HR*

tend to flatten the boundary perturbation, in this case they produce the opposite effect. Acting to reduce the length of the cut, the capillary forces generate a flow of scale *H* in the

the work of surface-tension force with the rate of energy dissipation by viscous forces, we

 

 

In spite of the logarithm, (67) is a single-valued analytical function in *G* , because the boundary perturbation constitutes a branch cut. If we insert (67) in (62), we find that the

Fig. 5. Cavity perturbation. whence, using (35), we have

> 

normal velocity of the cut edges 2

parameter *<sup>H</sup>*

find that

obtained before.

*V*

). The edges close up after a time

region. The velocities along 1 *<sup>x</sup>* and 2 *<sup>x</sup>* have the scales *H* and

2

*<sup>H</sup>* <sup>2</sup> *H H H*

 or *<sup>H</sup>*

  We presented a method to calculate two-dimensional Stokes flow with free boundary, based on the expansion of pressure in a complete system of harmonic functions. The theory forms the basis for strict analytical results and numerical approximations. Using this approach we analyse the collapse of bubbles and relaxation of boundary perturbation. The results obtained by this method are correlating well with numerical calculations performed using commercial FEM software.
