The Physics Society of Iran Iranian Journal of Physics Research 1682-6957 13 2 2019 11 26 Conformal mapping for the Saffman-Taylor fingering Conformal mapping for the Saffman-Taylor fingering 113 124 1021 FA M Goudarzi N Maleki-Jirsaraei Journal Article 2019 11 26  We studied the growth of viscous fingers as a Laplacian growth by conformal mapping. Viscous fingers grow due to Saffman-Taylor instability in the interface between two fluids, when a less viscous fluid pushes a more viscous fluid. As there was an interest in the rectangular Hele-Shaw cell, we solved the Laplacian equation with appropriate boundary conditions by means of conformal mapping techniques. The results were then visualized on a personal computer. Using these techniques, we studied singular effects of surface tension in the dynamics of the finger competition in the Saffman-Taylor problem with channel geometry. We also studied the motion of the interface between the two fluids in a pressure field. The more viscous fluid moves with a velocity proportional to the gradient of its pressure. In the two-dimensional case the interface can be described by a complex function which is analytic. Applying surface tension in the equations causes the tip-splitting at a longer finger (ahead). In zero order finger perturbation we had equal results for with and without surface tension. But for the first order perturbation, there was a difference. For limited surface tension in solutions for larger fingers (ahead), we observed tip splitting for larger fingers, which is completely in agreement with experimental observations.  We studied the growth of viscous fingers as a Laplacian growth by conformal mapping. Viscous fingers grow due to Saffman-Taylor instability in the interface between two fluids, when a less viscous fluid pushes a more viscous fluid. As there was an interest in the rectangular Hele-Shaw cell, we solved the Laplacian equation with appropriate boundary conditions by means of conformal mapping techniques. The results were then visualized on a personal computer. Using these techniques, we studied singular effects of surface tension in the dynamics of the finger competition in the Saffman-Taylor problem with channel geometry. We also studied the motion of the interface between the two fluids in a pressure field. The more viscous fluid moves with a velocity proportional to the gradient of its pressure. In the two-dimensional case the interface can be described by a complex function which is analytic. Applying surface tension in the equations causes the tip-splitting at a longer finger (ahead). In zero order finger perturbation we had equal results for with and without surface tension. But for the first order perturbation, there was a difference. For limited surface tension in solutions for larger fingers (ahead), we observed tip splitting for larger fingers, which is completely in agreement with experimental observations. https://ijpr.iut.ac.ir/article_1021_0768281a05da9f27df178b5c39a51263.pdf