Nonlinearly stable multilayer viscoplastic flows

When two purely viscous fluids flow in a parallel multi-layer shear flow down a pipe or other duct, the flow is frequently unstable. The primary instability arises at the interface, which is linearly unstable to certain wavelengths. Physically, for any perturbation of a planar interface, the less viscous fluid accelerates the more viscous fluid; see [5]. Although it is possible to suppress short wavelength instabilities through surface tension and also to achieve a more stable flow through arranging the viscosities of the fluid layers, (see e.g. [9]), none of these methods actually eliminate the underlying interfacial instability. In this thesis we demonstrate that by using a visco-plastic fluid, (i.e. a fluid with a yield stress), as the lubricating fluid in a multi-layer pipe flow, we do actually eliminate interfacial instabilities. This fact has been established recently in [2] using a linear stability analysis. In this thesis the analysis is extended to nonlinear stability, applying the methods developed in [10]. Our basic flow is an axisymmetric flow in which a Bingham fluid surrounds a Newtonian fluid. The flow rates and rheologies are such that the Bingham fluid is unyielded at the interface. We consider the nonlinear stability of this flow. This nonlinear stability problem is interesting for a number of reasons. Firstly, for a nonlinear perturbation it is possible for the unyielded plug region, (and the Newtonian region that it surrounds), to be perturbed and move. For the linear stability problem in [2], such motions do not occur since the stress perturbations on the interface and yield surface are linear and periodic, (via a normal mode approach). Thus, they integrate to zero over the volume of the plug and there is no motion. To allow for motion of the plug and Newtonian fluid region, whilst preserving a stable interface, is the main challenge of this thesis. The stable interface is preserved by bounding the perturbation in the deviatoric stress, which ensures that an unyielded region will still surrounds the Newtonian region. To carry out the analysis for a moving region we then consider a perturbation about a basic flow that is asymmetric. We find stability bounds, (conditional on the stress perturbation), below which the velocity perturbation from the asymmetric basic flow decays. We then show that the displacement of the plug region is also bounded for all time. Thus, in the above way we prove that the axisymmetric basic flow is stable but not asymptotically stable, i.e. conditional nonlinear perturbations will decay to an asymmetric solution that is close to the axisymmetric basic flow. Secondly, the asymmetric basic flows are themselves interesting. These flows fall into a class of flows that have been studied in [3], from the point of view of proving existence and uniqueness, in the context of examining various problems in cementing. Here we compute these flows using the augmented Lagrangian approach. This method is special in that it manages to compute the unyielded behaviour exactly in regions where the stress is below the yield stress. This is the first time, to our knowledge, that this method has been used to compute multi-fluid flows of visco-plastic fluids. Thirdly, the flows we study are special in that the structure is preserved, (i.e. due to one of the fluids having a yield stress). For any multi-phase fluid problem this is unusual. It is really this preservation of structure that has allowed the application of nonlinear stability methods. As such, this study is one of very few multi-phase flow problems that we know of, where an energy method has been effectively applied. Finally, in demonstrating weak nonlinear stability of these flows, we open the way for a range of applications to industrial processes where multi-layer processing might be of interest, e.g. coated wires and tapes, co-extruded laminates, etc.

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