Analytical solution for two-dimensional displacement flow in a curved down-sloping duct

Exact solution of the Navier-Stokes equations is used in this study to get a closed form equation for the transient and the steady state displacement flow of two Newtonian iso-viscous fluids in a two-dimensional curved plane channel. First, the displacement flow in a straight plane channel is considered. An available analysis is adapted in order to enhance its capability in explaining more details of the flow as well as prepare an appropriate frame to study the flow in curved channels. The behavior of the interface is predicted qualitatively both immediately and a while after the gate valve separating the fluids opens. It is shown that with the aid of the derivative of the flux of the heavier fluid, the viscous dominated flow regime may be analyzed with less difficulty compared to the available explanations in the literature and can also describe the behavior of the interface over a wider range of variations of the imposed flow. For the case of curved channels, the evolution of the interface at different rates of the imposed flow is analyzed again by considering the rate of change of the flux of the heavier displacing fluid in the viscous dominated regime. The results of the current study suggest that introducing the curvature in a displacement channel flow can facilitate the removal of the lighter fluid.