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

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

Reference Books

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.

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.

Rutherford, A. (1990). Vectors, Tensors and the Basic Equations of Fluid Mechanics. Dover Publications Inc.

Gupta, V., Gupta, S.K (1984). Fluid Mechanics and Its Applications. Wiley Eastern

Newell, H. E (2008). Vector Analysis. Dover publications.

Van Dyke, M. (1975). Perturbation Methods In Fluid Mechanics. Stanford, California: The Parabolic Press

Johns, L. E., & Narayanan, R. (2002). Interfacial Instability. New York: Springer-Verlag.

Pushpavanam, S. (2012). Mathematical Methods for Chemical Engineers (Reprint ed.). PHI Learning Pvt.

Strogatz, S. (2000). Nonlinear dynamics and chaos with applications to physics, biology, chemistry, and engineering. Cambridge, MA: Westview Press.

Cross, M., & Greenside, H. (2009). Pattern formation and dynamics in nonequilibrium systems. Ch.2.Cambridge, UK: Cambridge University Press

Chandrasekhar, S. (1961). Hydrodynamic and hydromagnetic stability. Oxford University Press.

Drazin, P. G., Reid, W. H. (2004). Hydrodynamic Stability (2nd Ed. p.108). New York: Cambridge University Press.

Journal publications

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.

A.B. Vir, S.R. Kulkarni, J.R. Picardo, A. Sahu, S. Pushpavanam, Holdup characteristics of two-phase parallel microflows, Microfluid. Nanofluidics. (2013).

Joseph, D. D. Domain perturbations: the higher order theory of infinitesimal water waves. Arch. Rational Mech. Anal. 51 (1975) 295â€“303.

A.M. Turing. The Chemical Basis of Morphogenesis, Philos. Trans. R. Soc. Lond. B. Biol. Sci. 237 (2007) 37â€“72

Dean, W. R. Note on the motion of fluid in a curved pipe. Phil. Mag., 4(1927), 208â€“223

Sparrow, E. M. On the onset of flow instability in a curved channel of arbitrary height. ZAMP. 15(1964) 638â€“642

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