Generalised flow field equation in two-phase flows – Eulerian averaging –modelling perspectives – one dimensional two-fluid model – modelling annular flow, slug flow and bubbly flow – fluid dispersions – flows with particles of one phase in turbulent field of another phase gas-particle flows in rocket nozzle, Compressible two-phase flows – 1d transient two-phase flow. Cavitation, Simple modelling of cavitating flows.

Static and dynamic instability, causes and mechanism of instability normal mode analysis, global stability, role of viscosity- dynamical systems, equilibrium points, linearization, concept of limit cycles and other fixed points of a non-linear dynamical system, asymptotic stability and Lyapunov stability, steady and dynamic bifurcations, bifurcations in 1-D and 2-D non-linear dynamical systems, attractor bifurcations for nonlinear evolutions.

Introduction to Physiological Fluid Mechanics; Review of Concepts in Fluid Mechanics, Kinematics, Hydrostatics, Conservation relations, Viscous Flow, Unsteady Flow; Clinical Fluid Dynamic Measurements; Analysis of Total Peripheral Flow; Circulatory Bio-fluid Mechanics, Blood Rheology, Blood Composition and Structure, Flow Properties of Blood, Blood Vessel Structure; Models of Biofluid Flow, Models of Blood Flow, Applications of Poiseuilli’s Law for the study of Blood Flow; Introduction to Non-Newtonian Fluids, Power Law Model, Herschel-Bulkley Model, Casson Model, Non-Newtonian Flow in Elas

Role of Optical Diagnostic Techniques in Combustion Studies- Planar Imaging Systems (Lasers, Camera, Optics, Signal and Noise) – Optical Diagnostics (Shadowgraphy, Schlieren, Luminosity, Chemiluminescence) – Scattering Processes (Elastic, Inelastic) – Laser Diagnostics (Background Physics, Absorption, LIF, Rayleigh, Raman, CARS, LII, PIV, LDV, PDPA) – High speed Diagnostics – Simultaneous Diagnostics- Safety Procedures

Special Functions, Bessel equation and related functions, Laplace transform methods: Inverse Laplace transform, Complex numbers, Bromwich Integral, bilateral laplace transforms, solution to ordinary and partial differential equation, Green function and boundary value problems, Fourier transform methods, Mellin transforms. Eigen Value problems and Eigen function expansions: Sturm-Liouville problems. Integral equations, Perturbation methods.

Vector Space, norm, and angle – linear independence and orthonormal sets – row reduction and echelon forms, matrix operations, including inverses – effect of round-off error, operation counts – block/banded matrices arising from discretization of differential equations – linear dependence and independence – subspaces and bases and dimensions – orthogonal bases and orthogonal projections – Gram-Schmidt process – linear models and least-squares problems – eigenvalues and eigenvectors – diagonalization of a matrix – symmetric matrices – positive definite matrices – similar matrices – linear tr

Introduction – fundamental conservation equations; Navier-Stokes equations and energy equation – dimensionless parameters and order of magnitude analysis – boundary layer flows, boundary layer separation– External laminar flow solutions, free shear flows: Similarity solution and Integral solution approaches – Internal laminar flows, typical solutions – Fundamentals of incompressible turbulent mean flows – Introduction to turbulence models – free convection flows : similarity solution, free convection in enclosure: Rayleigh Bernard Convection, Mixed convection.

Review of field equations in single phase flows and heat transfer – introduction to two-phase flows – basic averaging concepts – formulation and treatment of one-dimensional homogeneous flow model – separated flow model – drift flux model – predictive methodologies for flow pattern transition in adiabatic and diabatic flows – Liquid-Vapour Phase Change Phenomenon: pool boiling – wetting phenomenon – nucleation and bubble growth – bubble dynamics – convective boiling – heat transfer in partially and fully developed sub-cooled boiling – heat transfer in saturated boiling – Condensation – cond

Hugoniot relation – normal shocks – unsteady 1-D flows – finite amplitude waves – characteristics – Riemann invariants – unsteady shock waves – shock tubes – shock tunnels – weak and strong shocks – shock interactions – reflections – shock-boundary layer interaction – shock polar – diffraction – shock focussing – contact discontinuities – Richtmyer–Meshkov instability – spherical blast waves – internal, near-field and external ballistics – various shock structures.