On the growth rate for three-layer Hele-Shaw flows: Variable and constant viscosity cases

The linear stability of three-layer Hele-Shaw flows with middle-layer having variable viscosity is considered. Based on application of the Gerschgorin's theorem on finite-difference approximation of the linearized disturbance equations, an upper bound of the growth rate is given and its limiting case for the case of constant viscosity middle-layer is considered. A weak formulation of this equation, we obtained after some analysis. The upper bound in this case has also been derived here by analyzing an weak formulation of the problem.