The threshold of the instability in miscible displacements in a Hele-Shaw cell at high rates

For sufficiently large viscosity ratios and injection rates, miscible displ acements in a vertical Hele-Shaw cell at high rates become unstable, leadin g to three-dimensional (3D) fingering patterns. Below the instability thres hold, the base state is 2D in the form of a "tongue" of constant thickness. We apply the long wave Saffman-Taylor stability analysis to find an expres sion for the threshold of instability as a function of the viscosity ratio and the injection rate. The results are in agreement with the experimental data. (C) 2001 American Institute of Physics.