  We perform a linear stability analysis of three-layer radial porous media and
Hele-Shaw flows with variable viscosity in the middle layer. A nonlinear change
of variables results in an eigenvalue problem that has time-dependent
coefficients and eigenvalue-dependent boundary conditions. We study this
eigenvalue problem and find upper bounds on the spectrum. We also give a
characterization of the eigenvalues and prescribe a measure for which the
eigenfunctions are complete in the corresponding $L^2$ space. The limit as the
viscous gradient goes to zero is compared with previous results on multi-layer
radial flows. We then numerically compute the eigenvalues and obtain, among
other results, optimal profiles within certain classes of functions.
