Weakly nonlinear interfacial waves in a duct of arbitrary cross section

We study two kinds of weakly nonlinear waves which can occur in ducts of any cross section containing two layered fluids. In part 1 we consider internal waves of permanent form and small amplitude which can travel in horizontal or slightly inclined cylindrical ducts containing two layered co-flowing or counter-flowing fluids. In the frame of quasi-one dimensional theory with conventional shear stresses on the wall and at the interface, a linear temporal stability analysis predicting stratified/non-stratified transitional boundaries is performed. Weakly nonlinear roll-waves in which periodic waves are constructed by joining together piecewise continuous solutions through hydraulic jumps are then presented. The investigation requires the analysis of internal jumps. This problem is considered in the case of any cross section and, from energy loss consideration in every stream, criteria for internal shocks are derived. In part 2, starting from exact, frictionless equations, a perturbation method is used in order to show that weakly nonlinear, long gravity waves of cnoidal type can exist at the interface of two layered fluids flowing in a horizontal duct.