We study the linear stability of multi-layer Hele-Shaw flows. This topic has many useful applications including the design of efficient enhanced oil recovery techniques. We study four problems: two in a rectilinear flow geometry and two in a radial flow geometry. The first of these involves a characterization of the eigenvalues and eigenfunctions of the eigenvalue problem which results from the stability analysis of three-layer rectilinear flows in which the middle layer has variable viscosity. The resulting eigenvalue problem is a Sturm-Liouville problem in which the eigenvalues appear in the boundary conditions. For the case of an increasing viscous profile, we find that there is an infinite number of eigenvalues that increase without bound. By connecting the problem to a related regular Sturm-Liouville problem, we are able to prove the completeness of the eigenfunctions in a certain Sobolev space. We then provide an in-depth analysis of the case where the viscous profile of the middle layer is exponential. We find an explicit sequence of numbers which alternate with the eigenvalues. The second problem involves the stability of three-layer rectilinear Hele-Shaw flows in which there is diffusion of polymer within the middle layer of fluid. We first reformulate the eigenvalue problem using dimensionless quantities. We then revisit an old theorem about the stabilizing effect of diffusion and give a new proof. An efficient and accurate pseudo-spectral Chebyshev method is used to show that the stabilizing effect of diffusion is, in fact, drastic. We proceed to consider the stability of multi-layer Hele-Shaw flows in a radial flow geometry. We first study the case of an arbitrary number of fluid layers with constant viscosity. We provide upper bounds on the growth rate of disturbances and use them to provide conditions for stabilization of the flow. We also show that the equations for rectilinear flow can be obtained as a certain limit of radial flow. For the specific case of three-layer flows, we give exact expressions for the growth rate and explore the asymptotic limits of a thick and thin intermediate layer. We finish by using these exact expressions to study the effects of important parameters of the problem. We conclude that large values of interfacial tension can completely stabilize the flow and that decreasing the curvature of the interfaces by pumping in additional fluid has a non-monotonic effect on stability. We then consider three-layer radial flows in which the intermediate layer has variable viscosity. In order to use a similar analysis to that which is done in the previous problems, we define a change of variables that fixes the basic solution. In this new coordinate system, we are able to formulate the eigenvalue problem that governs the growth rate of disturbances. We define a measure based on the eigenvalue problem which leads to a Hilbert space in which the problem is self-adjoint. We also derive upper bounds on the growth rate, analogous to ones previously found for variable viscosity rectilinear flows. We then undertake a numerical study of the eigenvalue problem and find that variable viscosity flows, if chosen properly, can be less unstable than constant viscosity flows. Finally, we give details on our numerical method which is used throughout.
We study the linear stability of multi-layer Hele-Shaw flows. This topic has many useful applications including the design of efficient enhanced oil recovery techniques. We study four problems: two in a rectilinear flow geometry and two in a radial flow geometry. The first of these involves a characterization of the eigenvalues and eigenfunctions of the eigenvalue problem which results from the stability analysis of three-layer rectilinear flows in which the middle layer has variable viscosity. The resulting eigenvalue problem is a Sturm-Liouville problem in which the eigenvalues appear in the boundary conditions. For the case of an increasing viscous profile, we find that there is an infinite number of eigenvalues that increase without bound. By connecting the problem to a related regular Sturm-Liouville problem, we are able to prove the completeness of the eigenfunctions in a certain Sobolev space. We then provide an in-depth analysis of the case where the viscous profile of the middle layer is exponential. We find an explicit sequence of numbers which alternate with the eigenvalues.
The second problem involves the stability of three-layer rectilinear Hele-Shaw flows in which there is diffusion of polymer within the middle layer of fluid. We first reformulate the eigenvalue problem using dimensionless quantities. We then revisit an old theorem about the stabilizing effect of diffusion and give a new proof. An efficient and accurate pseudo-spectral Chebyshev method is used to show that the stabilizing effect of diffusion is, in fact, drastic.
We proceed to consider the stability of multi-layer Hele-Shaw flows in a radial flow geometry. We first study the case of an arbitrary number of fluid layers with constant viscosity. We provide upper bounds on the growth rate of disturbances and use them to provide conditions for stabilization of the flow. We also show that the equations for rectilinear flow can be obtained as a certain limit of radial flow. For the specific case of three-layer flows, we give exact expressions for the growth rate and explore the asymptotic limits of a thick and thin intermediate layer. We finish by using these exact expressions to study the effects of important parameters of the problem. We conclude that large values of interfacial tension can completely stabilize the flow and that decreasing the curvature of the interfaces by pumping in additional fluid has a non-monotonic effect on stability.
We then consider three-layer radial flows in which the intermediate layer has variable viscosity. In order to use a similar analysis to that which is done in the previous problems, we define a change of variables that fixes the basic solution. In this new coordinate system, we are able to formulate the eigenvalue problem that governs the growth rate of disturbances. We define a measure based on the eigenvalue problem which leads to a Hilbert space in which the problem is self-adjoint. We also derive upper bounds on the growth rate, analogous to ones previously found for variable viscosity rectilinear flows. We then undertake a numerical study of the eigenvalue problem and find that variable viscosity flows, if chosen properly, can be less unstable than constant viscosity flows.
Finally, we give details on our numerical method which is used throughout.