Displacement flows in horizontal, narrow, eccentric annuli with a moving inner cylinder

We analyze the effects of rotation and axial motion of the inner cylinder of an eccentric annular duct during the displacement flow between two Newtonian fluids of differing density and viscosity. The annulus is assumed narrow and is oriented near the horizontal. The main application is the primary cementing of horizontal oil and gas wells, in which casing rotation and reciprocation is becoming common. In this application it is usual for the displacing fluid to have a larger viscosity than the displaced fluid. We show that steady traveling wave displacements may occur, as for the situation with stationary walls. For small buoyancy numbers and when the annulus is near to concentric, the interface is nearly flat and a perturbation solution can be found analytically. This solution shows that rotation reduces the extension of the interface in the axial direction and also results in an azimuthal phase shift of the steady shape away from a symmetrical profile. Numerical solution is used for larger buoyancy numbers. We see that the phase shift results in the positioning of heavy fluid over light fluid along segments of the interface. When the axial extension of the interface is sufficiently large, this leads to a local buoyancy-driven fingering instability, for which a simple predictive theory is advanced. Over longer times, the local fingering is replaced by steady propagation of a diffuse interfacial region that spreads slowly due to dispersion. Slow axial motion of the annulus walls on its own is apparently less interesting. There is no breaking of the symmetry of the interface and hence no instability. However, axial wall motion does generate secondary flows which may combine with those from cylinder rotation resulting in enhanced dispersive effects.