Developing flow in inclined laminar-laminar stratified systems: Investigation of the multiple holdup problem

In this study, the multiple holdup solutions problem for stratified laminar-laminar flow in a channel is investigated. A stationary but developing monodimensional flow model is adopted here to follow the evolution of the holdup value from a non-equilibrium inlet condition to the final downstream and fully developed solution. A first order Ordinary Differential Equation (ODE) solver is used to perform this analysis under the assumption that the flow remains supercritical all along the pipe. The possibility of having a hydraulic shock during the longitudinal evolution of the system is investigated too. A second order ODE model is then proposed to handle situations with shocks, by including the effects of the longitudinal viscous stress diffusion when large interface level gradients occur. The results are also discussed regarding the approach of minimization of a potential function, showing a good consistency between the two methods.