### Reactive fingering instability:

In the porous media when a more mobile fluid is injected into a less mobile one, the interface between the two fluids deforms into fingers growing in time, known as fingering instability. It can be driven by gravity (without injection of one fluid into another) if the flow geometry is horizontal containing two fluids of different densities, being the heavier one above the other. Such a fingering instability occurs when less viscous, for example, water displaces more viscous oil in underground reservoirs leading to viscous fingering which produces flow heterogeneity and reduces oil recovery. Fingering instability can affect in contaminant transport in aquifers, in packed bed reactors, and is particularly detrimental to chromatographic separation techniques. The situation is often more tedious if the chemical reactions affect (i) the flow properties themselves such as its density or viscosity and/or, (ii) the porous medium such as permeability and porosity. In that case, the chemistry can be the fundamental reason behind the origin of hydrodynamic instabilities, as these instabilities are not seen in non-reactive systems. Thus the flow induced by hydrodynamic instability can in turn affect the chemical reaction and then feedback between reaction and hydrodynamics starts in, which enhances fingering and hence mixing. In this context, my aim is to study the coupling between chemical reactions and hydrodynamic instabilities from a fundamental theoretical perspective. My objective is to study simple reaction-diffusion-convection model systems in which only one mechanism of hydrodynamic instability is at play and for which a one or two-variable model can describe the chemical reactions, for instance chemical reaction of type: A+B --> C, A+B <==>C, A+B --> C+D, where A and B are two reactants, and C and D are the products.### Granular flows and pattern formation:

In the same spirit of hydrodynamic stability, the rapid granular flow shows various instability-induced patterns. Particularly granular plane Couette flow is a prototypical example that shows stationary and travelling wave instabilities, and shear-banding instability. When a granular material is sheared between two rotating cylinders or parallel plates, shearing remains localized within zones near walls leaving rest of material untouched. Such shearbanding phenomenon, where the non-uniformity of the shear rate appears along the flow gradient direction, is very generic in complex fluids for example colloidal suspensions, worm-like micelles, lyotropic liquid crystals, etc. My objective is to propose a possible theoretical description of shear-banding phenomena, travelling and stationary wave instabilities, in terms of phenomenological order parameter model based on flow equations. Such description has been successfully used to understand patterns in simple fluids (Newtonian fluid) from decades. However in dissipative system like complex fluid such studies are lacking. One could of course ask why order parameter models should be applied to complex fluids? The goal is to derive order parameter equations starting from the flow equations and to study the transition from one pattern forming state to another. The critical parameters for the onset of hydrodynamic instability could provide better understanding of the dynamics of complex fluids which remains a major challenge for researchers in fluid dynamics.#### Other research problems are listed below:

- Linear and nonlinear stability of active matter
- Two layer flow of immiscible fluids down an incline
- Poiseuille flow of two immiscible fluids in a channel