We consider the Kakinuma model for the motion of interfacial gravity waves. The Kakinuma model is a system of Euler–Lagrange equations for an approximate Lagrangian, which is obtained by approximating the velocity potentials in the Lagrangian of the full model. Structures of the Kakinuma model and the well-posedness of its initial value problem were analysed in the companion paper [14]. In this present paper, we show that the Kakinuma model is a higher order shallow water approximation to the full model for interfacial gravity waves with an error of order $O(\delta _1^{4N+2}+\delta _2^{4N+2})$ in the sense of consistency, where $\delta _1$ and $\delta _2$ are shallowness parameters, which are the ratios of the mean depths of the upper and the lower layers to the typical horizontal wavelength, respectively, and $N$ is, roughly speaking, the size of the Kakinuma model and can be taken an arbitrarily large number. Moreover, under a hypothesis of the existence of the solution to the full model with a uniform bound, a rigorous justification of the Kakinuma model is proved by giving an error estimate between the solution to the Kakinuma model and that of the full model. An error estimate between the Hamiltonian of the Kakinuma model and that of the full model is also provided.

Most of two-phase heat transfer analysis involve estimation of dry-out phenomenon in order to find whether it takes place and if so, at what location in the tube it will happen. Dry-out phenomenon considerably reduces heat transfer rate and it should be avoided as far as possible. Entrainment highly affects dry-out occurrence chance and location. Therefore a great attention should be paid on simulation of entrainment in order to have a good estimation of heat transfer and to safely design thermal systems.Although there are many models available for simulation of entrainment phenomenon, few are instrumentally employed in heat and fluid flow analyses, since most of those are empirical and limitedly can be applied due to the restricted conditions of experiments. There also are few theoretical approaches in the literature for simulating entrainment phenomenon by applying the physical potential of entrainment but some of those have a considerable deviation from the available empirical models. In the present study a conceptual approach is employed to simulate entrainment. In this regard the most important potential of this phenomenon, known as interfacial waves, are studied and modeled to find how they will lead to entrainment and how much the amount of entrainment is in vertical annular two-phase flows. An entrainment correlation is proposed at the end and will be compared with the available empirical ones for the verification objectives.

Composition gradients in miscible liquids can create volume forces resulting in variousinterfacial phenomena. Experimental observations of these phenomena are related to some difficultiesbecause they are transient, sufficiently weak and can be hidden by gravity driven flows.As a consequence, the question about their existence and about adequate mathematical models isnot yet completely elucidated. In this work we present some experimental evidences of interfacialphenomena in miscible liquids and numerical simulations of miscible drops and diffuse interfaces.