The linear stability of a core–annular flow in an asymptotically corrugated tube

Abstract

This paper examines the core–annular flow of two immiscible fluids in a straight circular tube with a small corrugation, in the limit where the ratio ε of the mean undisturbed annulus thickness to the mean core radius and the corrugation (characterized by the parameter σ) are both asymptotically small and where the surface tension is small. It is motivated by the problems of liquid–liquid displacement in irregular rock pores such as occur in secondary oil recovery and in the evolution of the liquid film lining the bronchii in the lungs whose diameters vary over different generations of branching. We investigate the asymptotic base flow in this limit and consider the linear stability of its leading order (in the corrugation parameter) solution. For the chosen scalings of the non-dimensional parameters the core's base flow slaves that of the annulus. The equation governing the leading-order interfacial position for a given wall corrugation function shows a competition between shear and capillarity. The former tends to align the interface shape with that of the wall and the latter tends to introduce a phase shift, which can be of either sign depending on whether the circumferential or the longitudinal component of capillarity dominates. The asymptotic linear stability of this leading-order base flow reduces to a single partial differential equation with non-constant coefficients deriving from the non-uniform base flow for the time evolution of an interfacial disturbance. Examination of a single mode k wall function allows the use of Floquet theory to analyse this equation. Direct numerical solutions of the above partial differential equation agree with the predictions of the Floquet analysis. The resulting spectrum is periodic in α- space, α being the disturbance wavenumber space. The presence of a small corrugation not only modifies (at order σ2) the primary eigenvalue of the system. In addition, short-wave order-one disturbances that would be stabilized flowing to capillarity in the absence of corrugation can, in the presence of corrugation and over time scales of order ln(1/σ), excite higher wall harmonics (α±nk) leading to the growth of unstable long waves. Similar results obtain for more complicated wall shape functions. The main result is that a small corrugation makes a core–annular flow unstable to far more disturbances than would destabilize the same uncorrugated flow system. A companion paper examines that competition between this added destabilization due to pore corrugation with the wave steepening and stabilization in the weakly nonlinear regime.