We develop a theoretical formulation for the large-scale dynamics of galaxy clusters involving two spherical ‘isothermal fluids’ coupled by their mutual gravity and derive asymptotic similarity solutions analytically. One of the fluids roughly approximates the massive dark matter halo, while the other describes the hot gas, the relatively small mass contribution from the galaxies being subsumed in the gas. By properly choosing the self-similar variables, it is possible to consistently transform the set of time-dependent two-fluid equations of spherical symmetry with self-gravity into a set of coupled nonlinear ordinary differential equations (ODEs). We focus on the analytical analysis and discuss applications of the solutions to galaxy clusters.