Study of Saffman-Taylor Problem and its Relation with Statistical Properties of Dla

In this thesis, we have studied the classic Saffman -Taylor problem with asymmetric forcing, in different geometry and higher dimension through various analytic and numerical methods systematically. Previously unknown solutions in these cases have been found analytically (in the case of asymmetric forcing and sector geometry) and numerically (for axisymmetric 3D case). Different methods have been applied to these steady state solutions to determine the selected pattern by the presence of the surface tension effect. A more general selection picture has merged from the study of the Saffman-Taylor problem in the sector geometry and in the axisymmetric 3D geometry. The relation between this selection picture and the stability of the selected pattern has been discussed. We have also constructed and studied a meanfield theory describing the statistical properties of the Diffusion Limited Aggregation and its relation to the Saffman-Taylor pattern in the different geometries that we have done the ST problem.