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.

Introduction configuration of an experimental set-up-error-calibration – uncertainty analysis, er- ror propagation formula, analysis of scatter, design of experiments based on uncertainty – re- view of probes and transducers – integral measurements of volume, velocity and temperature – introduction to wind tunnels, open and closed circuit tunnels – optical instrumentation, lasers and coherent optics, refractive index variation in transparent media, interferometry, schlieren and shadowgraph methods, analysis of interferograms, Rayleigh scattering technique – tran- sient response and instrume

Mathematical models for fluid dynamics – classification of partial differential equations – discretization methods – finite difference formulation – numerical solution of elliptic equations – linear system of algebraic equations – numerical solution of parabolic equations – stability analysis – numerical solution of hyperbolic equations – finite volume method – time integration schemes – isentropic flow through CD nozzle – simulation of shockwave formation – incompressible Navier–Stokes equations and their solution algorithms – basics of grid generation.

1. Flame speed measurement

2. Void fraction measurement

3. Cryogenic flow regimes identification

4. Experiments on film cooling

5. Flow over blade cascades

6. Nozzle flow characterization