Connecting the dots between Mathematics and Physics of the hydrodynamic instability known as Viscous Fingering (VF)

DST, Government of India, sanctioned a total outlay of Rs.14.63 lakhs to Dr Tapan Kumar Hota, Assistant Professor, Department of Mathematics, SRM University-AP, Andhra Pradesh, for pursuing research on “Mathematical analysis and Adjoint Based Stability for a Coupled Convection-Diffusion equation in Miscible Displacement”. Hydrodynamic Stability (Instability) is a study of an equilibrium point subject to small and/or medium disturbances. This assists in understanding several complicated natural phenomena such as fluid transition from laminar to turbulent, chemical mixing, formation of cloud, and others. The available stability analysis is unable to address the early-time evolution of the VF process. The early-time behaviour of the system helps us to choose the parameters that are responsible for the long-time behaviour of the coupled PDEs. To achieve the target, we need to address the linear stability of the system from the study of the singular-value-decomposition instead of traditional eigenvalue analysis. As the system is non-autonomous, the eigenvectors/eigenvalues may not be physically relevant. Further, it is required to confirm our linear stability analysis by mathematical analysis of the coupled non-linear PDEs.

‘The nonmonotonic viscosity profile when a high mobile fluid is displacing a less mobile fluid. There is a potentially unstable region, where the viscosity increases in the flow direction, followed in the downstream direction by a potentially stable region, where the viscosity decreases in the flow direction. The disturbance structure is computed using the singular-value decomposition of the propagator matrix. Reference: Hota & Mishra, Journal of Fluid Mech, Vol. 856, pp:552-579, 2018.’

Dr Tapan is keen on studying other branches of science to provide the correct mathematical approach to address the issues. His project will address one of the hydrodynamic instabilities known as viscous fingering (VF). It forms when a high mobile fluid displaces a low mobile fluid. It has enormous implications in the field of Chemical Engineering Science, Petroleum Engineering, Chromatography, and Oil Extraction process. He explains, “I will study this instability from a mathematician’s point of view, more precisely, to address the mathematical analysis of coupled partial differential equations (PDEs). The system that governs the VF is non-autonomous and there is no readily available stability analysis that can represent Physics, based on robust mathematical analysis. My aim is to connect the dots between mathematics and physics of the VF.”

Infrastructure and laboratory facilities are crucial to conduct research. Upon receiving approval from a prestigious organization like SERB (Science and Engineering Research Board), Govt. of India, Dr. Tapan is entitled to this unique opportunity to serve the scientific community. He divulges, “The award is not only a recognition to me but also to the SRM University, AP. The University will be more visible in the world of research and can attract more eminent scientists and researchers to make the University their second home.” Quoting Master Oogway, ‘Yesterday is history, Tomorrow is a mystery… Today is a gift, that’s why it is called present.’, Dr Tapan is eager to commence working on his project at the earliest, and looks forward to enjoying the process of accomplishing the task.