We study the nonlinear flow which results when two immiscible inviscid incompressible fluids of different densities and separated by an interface which is free to move and which supports surface tension, are caused to flow in a straight infinite channel. Gravity is taken into consideration and the velocities of each phase can be different, thus giving rise to the Kelvin-Helmholtz instability. Our objective is to study the competing effects of the Kelvin-Helmholtz instability coupled with a stably or unstably stratified fluid system (Rayleigh-Taylor instability) when surface tension is present to regularize the dynamics. Our approach involves the derivation of two-and three-dimensional model evolution equations using long-wave asymptotics and the ensuing analysis and computation of these models. In addition, we derive the appropriate Birkhoff-Rott integro-differential equation for two-phase inviscid flows in channels of arbitrary aspect ratios.
A long wave asymptotic analysis is undertaken to develop a theory for fully nonlinear interfacial waves allowing amplitudes as large as the channel thickness. The result is a set of evolution equations for the interfacial shape and the velocity jump across the interface. Linear stability analysis reveals that capillary forces stabilize short-wave disturbances in a dispersive manner and we study their effect on the fully nonlinear dynamics described by our models. In the case of two-dimensional interfacial deflections, traveling waves of permanent form are constructed and it is shown that solitary waves are possible for a range of physical parameters. All solitary waves are expressed implicitly in terms of incomplete elliptic integrals of the third kind. When the upper layer has zero density, two explicit solitary-wave solutions have been found whose amplitudes are equal to h/4 or h/9 where 2h is the channel thickness. In the absence of gravity, solitary waves are not possible but periodic ones are. Numerically constructed traveling and solitary waves are given for representative physical parameters. The initial value problem for the partial differential equations is also addressed numerically in periodic domains, and the regularizing effect of surface tension is investigated. In particular, when surface tension is absent it is shown that the system of governing evolution equations terminates in a singularity after a finite time. This is achieved by studying a 2 x 2 system of nonlinear conservation laws in the complex plane and by numerical solution of the evolution equations. The analysis shows that a sinusoidal perturbation of the flat interface and a cosine perturbation to the unit velocity jump across the interface, develop a singularity at time tc = ln 1/ε+0 (ln(ln 1/ε)) where ε is the initial amplitude of the disturbances. This result is asymptotic for small ε and is derived by studying the asymptotic form of the flow characteristics in the complex plane.
We also derive the analogous three-dimensional evolution equations by assuming that the wavelengths in the principal horizontal directions are large compared to the channel thickness. Surface tension is again incorporated to regularize short-wave Kelvin-Helmholtz instabilities and the equations are solved numerically subject to periodic boundary conditions. Evidence of singularity formation is found. In particular, we observe that singularities occur at isolated points starting from general initial conditions. This finding is consistent with numerical studies of unbounded three-dimensional vortex sheets (see Introduction for a discussion and references).
In the final part of this work we consider the vortex-sheet formulation of the exact nonlinear two-dimensional flow of a vortex sheet which is bounded in a channel. We derive a Birkhoff-Rott type integro-differential evolution equation for the velocity of the interface in terms of the vorticity as well as the evolution equation for the unnormalized vortex sheet strength. For the case of a spatially periodic vortex sheet, this Birkhoff-Rott type equation is written in terms of Jacobi's functions. The equation is shown to recover the limits of unbounded and non-periodic flows which are known in the literature.