  We study the Hele-Shaw immiscible displacements when all surfaces tensions on
the interfaces are zero. The Saffman-Taylor instability occurs when a less
viscous fluid is displacing a more viscous one, in a rectangular Hele-Shaw
cell. We prove that an intermediate liquid with a variable viscosity can almost
suppress this instability. On the contrary, a large number of constant
viscosity liquid-layers inserted between the initial fluids gives us boundless
growth rates with respect to the wave numbers of perturbations. The same amount
of intermediate liquid is used in both cases.
