 Coordinators IIT Madras

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This course concentrates on obtaining analytical and semi-analytical solutions to multiphase flow problems. The emphasis is on using a very fundamental approach : equations of conservation of mass, momentum and energy without any empiricism. The focus is on getting approximate  solutions using perturbation theory, and analysing stability of systems using linear stability analysis. This will help prepare students for doing research in these areas.

 Lec No Topic 1 Introduction and overview of the course: Multiphase flows 2 Stratified flow in a micro channel: Velocity profiles. 3A Stratified flow in a micro channel: Effects of physical parameters 3B Flow regimes in microchannels: Modeling and Experiments 4 Scaling Analysis: Introduction 5 Scaling Analysis: Worked Examples 6 Interfacial tension and its role in Multiphase flows 7 Eulerian and Lagrangian approaches 8 Reynolds Transport Theorem and the Equation of Continuity 9 Derivation of Navier-Stokes equation 10 Vector operations in general orthogonal coordinates: Grad., Div., Lapacian 11 Normal and shear stresses on arbitrary surfaces: Force balance 12 Normal and shear stresses on arbitrary surfaces: Stress Tensor formulation 13 Stresses on deforming surfaces: Introduction to Perturbation Theory 14 Pulsatile flow: Analytical solution 15 Pulsatile flow: Analytical solution and perturbation solution for Rw<<1 16 Pulsatile flow: Perturbation solution for Rw >> 1 17 Viscous heating: Apparent viscosity in a viscometer 18 Domain perturbation methods: Flow between wavy walls 19 Flow between wavy walls: Velocity profile 20 Introduction to stability of dynamical systems: ODEs 21 Stability of distributed systems (PDEs): reaction diffusion example 22 Stability of a reaction-diffusion system contd. 23 Rayleigh-Benard convection: Physics and governing equations 24A Rayleigh-Benard convection: Linear stability analysis part 1 24B Rayleigh-Benard convection: Linear stability analysis part 2 24C Rayleigh-Benard convection: Linear stability analysis part 3 25 Rayleigh Benard convection: Discussion of results 26 Rayleigh-Taylor ‘heavy over light’ instability 27 Rayleigh-Taylor instability contd. 28 Capillary jet instability: Problem formulation 29 Capillary jet instability: Linear stability analysis 30 Capillary jet instability: Rayleigh’s Work Principle 31 Tutorial Session: Solution of Assignment 4 on linear stability 32 Turing patterns: Instability in reaction-diffusion systems 33 Turing patterns: Results 34 Marangoni convection: Generalised tangential and normal stress boundary conditions 35 Marangoni convection: Stability analysis 36 Flow in a circular curved channel: Governing equations and scaling 37 Flow in a circular curved channel: Solution by regular perturbation 38 Stability of flow through curved channels: Problem formulation 39 Stability of flow through curved channels: Numerical calculation 40 Viscous Fingering: Darcy’s law 41 Viscous Fingering: Stability analysis 42 Shallow Cavity flows

It is desirable to have done a first course on Fluid Mechanics, and have Exposure to Partial Differential Equations, Fourier series and Linear Algebra. Proficiency in programming in Matlab will be useful.

Textbooks

1. Leal, L.G (2008). Advanced transport phenomena: Fluid mechanics and convective transport processes. Cambridge: Cambridge University Press.

Reference Books

1. Krantz, W. B (2007). Scaling analysis in modeling transport and reaction processes. A Systematic Approach to Model Building and the Art of Approximation. New Jersey: John Wiley and Sons Inc.
2. Pierre-Gilles de Gennes, Francoise Brochard-Wyart, David Quere (2003). Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves. New York: Springer Science Business Media Inc.
3. White, F.M (1991). Viscous Fluid Flow (third edition). Tata Mcgraw Hill
4. Rutherford, A. (1990). Vectors, Tensors and the Basic Equations of Fluid Mechanics. Dover Publications Inc.
5. Gupta, V., Gupta, S.K (1984). Fluid Mechanics and Its Applications. Wiley Eastern
6. Newell, H. E (2008). Vector Analysis. Dover publications.
7. Van Dyke, M. (1975). Perturbation Methods In Fluid Mechanics. Stanford, California: The Parabolic Press
8. Johns, L. E., & Narayanan, R. (2002). Interfacial Instability. New York: Springer-Verlag.
9. Pushpavanam, S. (2012). Mathematical Methods for Chemical Engineers (Reprint ed.). PHI Learning Pvt.
10. Strogatz, S. (2000). Nonlinear dynamics and chaos with applications to physics, biology, chemistry, and engineering. Cambridge, MA: Westview Press.
11. Cross, M., & Greenside, H. (2009). Pattern formation and dynamics in nonequilibrium systems. Ch.2.Cambridge, UK: Cambridge University Press
12. Chandrasekhar, S. (1961). Hydrodynamic and hydromagnetic stability. Oxford University Press.
13. Drazin, P. G., Reid, W. H. (2004). Hydrodynamic Stability (2nd Ed. p.108). New York: Cambridge University Press.

Journal publications

1. B. Malengier, S. Pushpavanam, Comparison of Co-Current and Counter-Current Flow Fields on Extraction Performance in Micro-Channels, Adv. Chem. Eng. Sci. 02 (2012) 309â320.
2. A.B. Vir, S.R. Kulkarni, J.R. Picardo, A. Sahu, S. Pushpavanam, Holdup characteristics of two-phase parallel microflows, Microfluid. Nanofluidics. (2013).
3. Joseph, D. D. Domain perturbations: the higher order theory of infinitesimal water waves. Arch. Rational Mech. Anal. 51 (1975) 295â303.
4. A.M. Turing. The Chemical Basis of Morphogenesis, Philos. Trans. R. Soc. Lond. B. Biol. Sci. 237 (2007) 37â72
5. Dean, W. R. Note on the motion of fluid in a curved pipe. Phil. Mag., 4(1927), 208â223
6. Sparrow, E. M. On the onset of flow instability in a curved channel of arbitrary height. ZAMP. 15(1964) 638â642