Detailed Description (scroll down for links to publications, gallery,etc.) as of 05/28/2017
On Hele-Shaw Flows: We are interested in linear and nonlinear stability studies of multi-layer Hele-Shaw (HS) flows of simple and complex fluids in rectilinear and radial geometries. One of the goals of such studies has been to find novel ways to slow down and/or completely arrest development of fingering instabilities. Most studies to-date has been for a single interface HS flows. Multi-layer counterpart brings in many interesting challenges some of which we have been addressing for quite sometime. For a single-interface Hele-Shaw flows, the dispersion relation results explicitly or implicitly primarily from the dynamic interfacial condition after it has been reformulated using the normal mode ansatz, the field equation, and the kinematic condition. In contrast, in multi-layer flows stability analysis gives rise to a non-standard eigenvalue problem for each layer specific to the fluid in that layer and all such eigenvalue problems are coupled through interfacial conditions. Eigenvalue problems can be distinctly different in different layers depending on the profiles of fluid properties in the layers and the rheological properties of the fluids in the layers. Many of these eigenvalue problems do not fall into classical standard types such as Sturm-Liouville problems. Characterization of eigenvalues and eigenfunctions of these problems and their faithful computation may pose challenges. Simplest of these problems is the one when each layer contains Newtonian fluid of constant viscosity with adjacent layers having different constant viscosity. This set-up is relevant from practical standpoint. We have obtained upper bounds on the growth rate using variational principles, computed eigenvalues, and also developed the non-autonomous system of ODEs (evolution equation) for the linearized motion of interfaces. We have obtained more interesting and useful results in the radial case than in the rectilinear case, specially when the injection rate is time dependent. Current work is in progress on various aspects of these problems. These problems even with the Newtonian fluid layers become more interesting, challenging and useful when fluid layer have variable viscous profiles which is the subject of an upcoming paper in preparation. Nonlinear stability of these flows are relatively much harder. But the starting point of this must be nonlinear stability study of a potentially unstable single-layer HS flow in which the viscous profile is monotonically increasing in the direction of basic flow. We have published results on this and looking into extending this study for single- and multi-layer HS flows. There are several challenging mathematical problems on this topic which need to be solved analytically. We are also investigating possibility of development of singularity formation one one or more interfaces as they evolve according to the full nonlinear initial value problem. We are also interested in developing fast and efficient algorithms for computing nonlinear evolution of these interfaces by solving the associated initial value problems.
On Porous-Media Flows: Porous media flows are usually modeled using Buckley-Leverett equations. The Buckley-Leverett model allows mixing macroscopically between immiscible phases whereas the Hele-Shaw model discussed above does not. Hele-Shaw flows thus model porous media flows to a very good approximation when such mixing effects driven by nonlinear waves remain minimal and/or negligible compared to the advective and capillary fluxes. We are interested in developing and analyzing fast and accurate methods for Buckley-Leverett model and then using these for direct numerical simulation of multi-phase heterogeneous porous media fluid flows. When capillary and dispersive fluxes can be neglected, the Buckley-Leverett equations form a hyperbolic system of conservation laws in most situations. One can take advantage of huge knowledge base available for solving such systems for a variety of initial data using direct numerical simulations. One of the methods that we have developed in conjunction with collaborators is Wave Front Tracking method. We are interested in further development and refinement of these methods in various directions for applications to problems to two- and three-dimensions. Some of these efforts require, among other things, development of Riemann solvers in higher dimensions. We have been also working for a while on the development of efficient MMOC and discontinuous finite element methods for solving equations arising here. We are also interested in estimating the relative effects, analytically and numerically, of advective, dispersive and capillary fluxes on the development of instabilities in such flows. More on this topic can be found here.
Publications on this topic