The stability of the displacement front between 2 immiscible fluids of radial flow between 2 parallel plates (Hele-Shaw model) is studied mathematically by superposing onto the circular displacement front a sinusoidal perturbation. The equations are reduced to dimensionless variables, and it is shown that the stable and unstable domains in a plot: dimensionless viscosity vs. dimensionless time are separated by a polygonal contour, each side of the contour being characterized by the (integer) number of perturbations along the circumference. There is a critical reduced time below which the perturbations are amortized but beyond which they are amplified. Experimental results have been in fair general agreement with theoretical results, the divergence between them being attributable to neglecting capillary phenomena, which may become very important at large radial distances. One test with miscible fluids has shown that even in this case, there is a critical time or an equivalent critical radius.