content_image/398491.jpg content_image/398490.jpg
0 . 2 < a < 0 . 4 gives a reasonable fit to the upper bound). The shaded region between these bounds indicates the range of all allowable localisation lengths, as predicted by the Continuum Model. Filled and open symbols rep- resent the VF simulation results, which are the same as in Fig. 7, only with the minimum localisation length of l$_{min}$ = ¯ A 1 / 2 subtracted to coincide with the Continuum Model predictions which give l = 0 for V → ∞ . 

<img src='content_image/398491.jpg'>

The definition for the critical cross-over point, as given by eq. 22 may also be formulated in terms of λV . This is achieved by inserting eq. 25 into eq. 21 and find- ing the point at which this line intersects eq. 26. Alter- 

natively, one may insert eq. 24 and eq. 25 into eq. 22. 

This gives 

<img src='content_image/398490.jpg'>

which is illustrated by the dotted lines in Fig. 12. The calculated critical cross-over point ( λV )$_{c}$ = 0 . 056 γ/ ¯ A 1 / 2 fulfills its promise of offering a reasonable estimate of the point below which the system yields a wide range of localisation lengths. 

While the comparison between the VF Model and the Continuum Model presented in this section gives a fasci- nating theoretical explanation for the simulation results discussed, its details are far from precise. The location of the upper bound in Fig. 12 is simply an estimate, and further simulations may be needed to determine its exact location. In addition, many approximations were taken in relating the parameters of the two models. How- ever, it is remarkable that despite these approximations, a robust prediction can still me made. 

## 7 Outlook 

The apparent agreement of the simulation results in this paper with published experimental work suggests that the 2D VF Model may have further potential for describ- ing realistic foam dynamics. For more detailed studies to be conducted, however, the VF algorithm will need to be improved to decrease the required computation time for these types of simulations. Issues that we will address in- clude the effect of µ$_{2}$ ( A ) on localisation with the 2D VF 

