Finger Behavior of a Shear Thinning Fluid in a Hele-Shaw Cell

We make a theoretical study of the behavior of a simple fluid displacing a shear thinning fluid confined in a Hele-Shaw cell. To study the Saffman-Taylor instability when the displaced fluid is non-Newtonian we face the problem of having a field which is non-Laplacian. By means of a hodographic transformation we are able to solve the problem in the case of weak shear thinning while taking into account the non-Laplacian character of the equation. Our results predict that the finger width decreases towards zero for small values of the surface tension parameter which is inversely proportional to the finger velocity.