Simulation of viscous fingering due to saffman-taylor instability in hele-shaw cell

Abstract

We simulated the growth of the viscous fingers in a Hele-Shaw cell, a Laplacian growth, by a numerical solution. We solved a Laplace equation numerically with boundary conditions that indicated a pressure jumping due to the surface tension in the interface of the two fluids. By using Darcy’s law, we gained the time evolution of the interface and then visualized it in MATLAB. Subsequently, we examined the effects of several parameters in forming the fingers in rectangular cells and compared the results with the theoretical predictions which had a desirable agreement with our experimental findings. Our findings indicated that wave-lengths scale well with the control parameter in all conditions. Furthermore, we estimated the time evolution of the interface for Newtonian and non-Newtonian fluids in a circular cell, for Newtonian fluid, in agreement with experimental finding and theoretical prediction for dominant pattern was tip splitting. For non-Newtonian fluids from Shear-Thinning kind, we used two generalized Darcy’s equation, we found that in both cases the tip dose not split but it will be sharped, finally we found that the two different model suggested for generalized Darcy’s low (Bonn’s model and Kondic’s model) are in good agreement with each other and also with the experimental findin