I. The fingering problem in flow through porous media. II. The kinetic equation for Hamiltonian systems

Part I: The interface between two fluids in a porous media is stable or unstable depending on the densities and viscosities of the fluids. Unstable flows tend to develop into long "fingers" of fluids. Saffman and Taylor (1958) analyzed the two-dimensional steady state shape of a finger neglecting interfacial tension. They found that the solutions to the equations of motion are not unique: the width of the finger is arbitrary. In this paper, the problem is formulated including the effects of surface tension at the interface. The equations of motion are reduced to a pair of nonlinear integro-differential equations for the shape of the finger. The equations are solved numerically and analyzed by perturbation methods. The numerical results indicate that the system of equations has a unique solution for nonzero surface tension. Solutions are computed for a wide range of physical parameters. The computed profiles agree well with experimental observations. The perturbation analysis yields contradictory results. A formal expansion in the surface tension parameter can be obtained for arbitrary finger widths, suggesting that the equations do not have a unique solution. The conflict between the numerics and the perturbations is discussed but not resolved. The stability of the steady fingers to small disturbances is discussed. Linearized stability analysis indicates the two-dimensional fingers are unstable, a result which is at variance with experiment. The stability analysis of the plane interface reveals some new steady profiles. These profiles are computed for finite amplitude. Part II: The kinetic equation describing weak nonlinear interactions between wave components in a Hamiltonian system is obtained by perturbation methods. The analysis is facilitated by the use of Wyld diagrams. The results include some new terms not included in previous work. The inconsistency of some previous investigations is pointed out, and the significance of the new terms is discussed