Anisotropic Viscous Fingering.
Abstract
We have qualitatively explained the experiments of McCloud and Maher (McCloud and Maher (95)) for the viscous fingering problem in which an anisotropy in the surface tension parameter was imposed by engraving a grid in one of the plates of the HeleShaw cell. We saw the need to approach the problem in an analytical form. Therefore we decided to extend solvability theory to incorporate the effect of anisotropy. We have introduced the anisotropy through a moving boundary condition by considering an effective anisotropic surface tension with an anisotropy entering as the simplest cosine term having the right symmetry for a square lattice. We carried out the singular perturbation appropriate for the surface tension parameter assuming the length scale introduced by the anisotropy is small in comparison with the length scale introduced by the surface tension. In this sense, the perturbation can be said to be microscopic. For the case in which the surface tension has a maximum at the finger tip, our theory provides two possible solutions: one corresponding to the solution of the isotropic case and a new solution which, below a threshold of the surface tension parameter, predicts a wider finger than the isotropic solution. Intuitively, we expect the "old" solution, namely the one that does not differ from the isotropic case, to be the selected solution for large values of the surface tension parameter and we expect the new solution to be selected for small values of the surface tension parameter. This was confirmed by dynamical simulations of the interface done by David Jasnow. His simulation predicts that for the case in which the surface tension has a maximum at the finger tip, anisotropy is irrelevant for large values of the surface tension parameter. Furthermore below a threshold in this surface tension parameter, the selected finger width is systematically wider than the corresponding isotropic case. We conclude that our solvability theory together with the dynamic simulation of the flow satisfactorily explains all the experimental observations for which the strength of the anisotropy can be considered microscopic.
 Publication:

Ph.D. Thesis
 Pub Date:
 1995
 Bibcode:
 1995PhDT.......236C
 Keywords:

 Physics: Condensed Matter