Flow of two immiscible uniformly rotating couple stress fluid layers

Abstract

In this article, the flow of two uniformly rotating immiscible couple stress fluid layers is examined. In the upper layer, the flow has different velocity [Formula: see text], density [Formula: see text], viscosity [Formula: see text], couple stress viscosity [Formula: see text], and pressure [Formula: see text], rotating with a constant angular velocity [Formula: see text] over another immiscible fluid layer with velocity [Formula: see text], density [Formula: see text], viscosity [Formula: see text], couple stress viscosity [Formula: see text], and pressure [Formula: see text], rotating with a constant angular velocity [Formula: see text]. The considered problem has a curious form, having characteristics of the famous von Karman and B[Formula: see text]dewadth flows of couple stress fluid below and above the interface, respectively. The flows are co-rotating at [Formula: see text] and counter-rotating at [Formula: see text], where [Formula: see text] is the ratio of angular velocities of the fluid layers. The lower layer would counter-rotate as compared to the upper layer. By utilizing similarity variables, the system of governing equations is transformed into an ordinary system. A finite-difference Keller–Box technique is applied to acquire the numerical results. For co-rotating flows ([Formula: see text]), the similarity solution exists for [Formula: see text], but for counter-rotating flows ([Formula: see text]), the solution exists up to some specific values of [Formula: see text] [i.e., [Formula: see text]]. In the limiting cases, there are some similarities between the B[Formula: see text]dewadt problem (an outflow of fluid) and the upper layer flow and similarities between the von Karman problem (an inflow of fluid) and the lower-layer flow. The lower fluid layer shows a recirculation region of the flow near the interface, where the fluid cannot transfer.