Research Projects

In view of its wide range of applications in analysing the dynamics of aircraft wings, stability of bridges, blood flow through arteries etc., the mathematical analysis of fluid-structure interaction (FSI) problems has become an active area of research. We propose the study of well-posedness issues and numerical analysis of FSI models describing the physics behind devices known as `heat-exchangers' which allow transfer of heat between fluids (resulting in cooling or heating of the fluid stream) separated by a structure possessing high thermal conductivity. We propose the analysis of two systems of PDEs describing the dynamics of heat-exchangers: (a) System I: We consider two Newtonian incompressible heat-conducting fluids (modelled by Navier-Stokes equations) confined in 3D domains and separated by a Koiter shell. The shell allows heat transfer between the fluids. The fluids involved have different viscosities, which depend on the respective temperatures in a non-linear manner (the viscosity coefficient is roughly of the form exp(1/T-c), where T is the fluid temperature and c is a constant). The motion of the shell is driven by the resultant forces exerted by the fluids. The system is coupled with an internal energy balance equation with a singular source term. (b) System II: Unlike `System I', here one of the fluids involved in the bi-fluid FSI model is compressible in nature. The pressure of the compressible fluid is given as a function of density and temperature (involving a quartic polynomial of the temperature), where the density solves the continuity equation. In contrast with incompressible fluids, the viscosity of compressible gases is proportional to some power of the temperature and is generally modelled by `Hard-sphere kinetic theory'. To the best of our knowledge, the mathematical analysis of `heat exchanger' devices has never been performed before in the literature. Most, if not all, articles dealing with FSI models assume the temperature of the system to be constant. Whereas in reality the interaction between a fluid and an elastic medium is strongly influenced by temperature variations. A handful of articles that take the variation of temperature into account in FSI models deal with a single viscous fluid. The goal of the present proposal is to mathematically model `heat exchangers', construct solutions (weak as well as strong in suitable Sobolev spaces) to the models (both for System I and II), analyse uniqueness (weak-strong type/conditional) and to develop stable numerical schemes. The schemes we develop can be implemented to perform simulations, which will play a key role in observing the dynamics, studying the long-time behaviour and designing suitable controllers for optimal and efficient performance of `heat exchanger' devices widely used now a days in power plant broilers, HVAC systems, refrigeration and many other condensers and chillers.