# MAE 259A Winter 2018 Advanced Topics in Fluid Dynamics

## Interfacial dynamics for viscous fluids - thin films and surface tension

Tues/Thur 4-6pm

This course assumes good command of all the material taught in MAE 182B and
some of the material taught in MAE182A (most notably Fourier transform methods).Some material in 182C may also be used. Those students from Mathematics should
have taken 266 or 269 B/C. The course will not assume extensive training in fluid dynamics - we will review the Navier Stokes equations as part of the course.

This is a seminar style course. Lecture attendance is required. Students will work on projects and give presentations. Some of the work may be original research potentially leading to conference or journal publications.
Some homework assignments will be given to give students better command of the basic mathematical methods and modeling ideas. Most of the course will be
from reading journal papers rather than material in a textbook.
All papers will be made available through the CCLE.
Material is taken from journal publications spanning the early 1970s to the present day. Some of this material requires more mathematical background and the math will be taught alongside the science. This papers are mainly from the fluid dynamics and physics literature rather than the math literature and the instructor will make every effort to make the mathematical ideas accessible to students from engineering.

Course grade will be based on 10% attendance; 20% homeworks; 20% project prospectus; 50% final project

# Approximate Syllabus

## Mathematical background ideas: The heat equation, higher order diffusion, Navier-Stokes equations, Scalar conservation laws and shocks, similarity solutions

D. J. Acheson, Fluid Dynamics, Oxford U Press, 1990;
Michael Shearer and Rachel Levy, Partial Differential Equations: An Introduction to Theory and Applications, Princeton Univ. Press, 2015;
G. I. Barenblatt, Scaling, Self-similarity, and intermediate asymptotics, Cambr. Univ. Press 1996.
## The lubrication approximation for thin films with and without surface tension, Moving contact lines and slip

H. Huppert, "Flow and instability of a viscous current down a slope,
Nature 300, 427 (1982)
L. M. Hocking, "Rival contact-angle models and the spreading of drops", J. Fluid. Mech. 239, 671 (1992);
E. B. Dussan V, "The moving contact line: the slip boundary condition", J. Fluid Mech.77, 665 (1976).;
Oron, A., S. H. Davis, and S. George Bankoff, Long-scale evolution of thin liquid films, Rev. Mod. Phys. 69(3), 1997, p. 931
## Geometric motions: mean curvature flow and surface diffusion

T. P. Witelski and A. J. Bernoff, Self-similar asympotics for linear and non-linear diffusion equations, Stud. Appl. Math. 100(2): 153-193, 1998;
Andrew Bernoff, Andrea L Bertozzi, Thomas Witelski,Axisymmetric Surface Diffusion: Dynamics and Stability of Self-Similar Pinch-OffÂ, J. Stat. Phys., 93(3/4), November 1998, pp. 725-776.
## Spreading drops

L. Tanner, "The spreading of silicone oil drops on horizontal surfaces, "J. Phys. D 12, 1473 (1979).
V.M. Starov, Spreading of droplets of nonvolatile liquids over a flat solid surface, Coll. J. USSR, 45 (1983), pp. 1009 - 1015.
A.M. Cazabat and M. A. Cohen Stuart, "Dynamics of wetting: Effects of surface roughness,J. Phys. Chem. 90, 5845 (1986)
Gary F. Teletzke, H. Ted Davis, L.E. Scriven. Wetting hydrodynamics. Revue de Physique Appliquee, 1988, 23 (6), pp.989-1007
Michael Brenner and Andrea Bertozzi, "Spreading of droplets on a solid surface, Phys. Rev. Lett.71, 593 (1993).
## Driven Fronts with surface tension, linear instability and transient growth

S. M. Troian, E. Herbolzheimer, S. A. Safran, and J. F. Joanny, "Fingering instabilities of driven spreading films", Europhys. Lett.10, 25 (1989);
John M. Jerrett and John R. de Bruyn, "Finger instability of a gravitationally driven contact line", Phys. Fluids A 4, 234 (1992);
Nathalie Fraysse and George M. Homsy, "An experimental study of rivulet instabilities in centrifugal spin coating of viscous Newtonian and non-Newtonian fluids",Phys. Fluids6, 6 (1994);
A.L. Bertozzi and M. P. Brenner Linear Stability and Transient Growth in Driven Contact Lines, Physics of Fluids, 9(3) 530 doi: 10.1063/1.869217;
Lloyd N. Trefethen, Anne E. Trefethen, Satish C. Reddy, and Tobin A. Driscoll, Hydrodynamic stability without eigenvalues, Science261, 578 (1993).
## Marangoni stresses and Undercompressive shocks in driven films

V. Ludviksson and E. N. Lightfoot, The Dynamics of Thin Liquid Films in Presence of Surface Tension Gradients, AI Ch E Journal, 17(5), p. 1176, 1971
Teletzke, G. F., Davis, H. T., and Scriven, L. E., Chem. Eng. Commun. 55, 41 (1987)
A. M. Cazabat, F. Heslot, S. M. Troian, and P. Carles, Fingering instability of thin spreading films driven by temperature gradients, Letters to Nature, 346, p. 824, 1990
D. E. Kataoka and S. M. Troian, Stabilizing the advancing front of thermally driven climbing films, J. Coll. Interf. Aci. 203, 1998, pp. 334-344;
A. L. Bertozzi, A. MÃ¼nch, X. Fanton, and A. M. Cazabat, Contact Line Stability and Undercompressive Shocks in Driven Thin Film Flow Phys. Rev. Lett.81, 5169
;
A. Munch and A. L. Bertozzi, Rarefaction-undercompressive fronts in driven films, Physics of Fluids, 11, 2812, 1999 http://dx.doi.org/10.1063/1.870177
;
G. A. El, M. A. Hoefer, and M. Shearer, Dispersive and diffusive-dispersive shock waves for nonconvex conservation laws, SIAM Review, 59(1), p. 3-61.
;
A. L. Bertozzi, A. Munch, and M. Shearer, Undercompressive shocks in driven thin film flows, Physica D, 134, p. 431-464, 1999.
;
Munch, A. L. Bertozzi, Rarefaction-undercompressive fronts in driven films, Physics of Fluids, 11, 2812, 1999.
;
Jeanman Sur, Andrea L. Bertozzi, Robert P. Behringer,Â Reverse undercompressive shock structures in driven thin film flow, Phys. Rev. Lett. 90 (12) 126105, 2003.
;
Mark Bowen, Jeanman Sur, Andrea L. Bertozzi, Robert P. Behringer,Â Nonlinear dynamics of two-dimensional undercompressive shocks, Physica D, Volume 209, Issues 1-4 , 15 September 2005, Pages 36-48.
;
T. M. Segin, B. S. Tilley, L. Kondic, On undercompressive shocks in a constrained two-layer flows, Phys. D., 209(1-4), pp. 135-144, 2005.
;
P. G. LeFloch, M. Mohammadian, Why many theories of shock waves are necessary: kinetic functions, equivalent equations, and fourth order models, J. Comp. Phys. 227(8), (2008), pp. 4162-4189.
## Numerical methods for thin film equations

L. Zhornitskaya and A. L. Bertozzi, Positivity preserving numerical schemes for lubrication-type equations, SIAM J. Numer. Anal. 37(2), pp. 523-555, 2000
G. GrÂ¨un and M. Rumpf, Nonnegativity preserving convergent schemes for the thin film equation, Num. Math., 87 (2000), 113-152.
B. P. Vollmayr-Lee and A. D. Rutenberg, Fast and accurate coarsening simulation with an unconditionally stable time step, Physical Review E, 68 (2003), 1-13.
;
T. P. Witelski and M. Bowen, ADI methods for high order parabolic equations, Appl. Num. Anal., 45 (2003), 331-35.
;
J. Greer, A. Bertozzi and G. Sapiro, Fourth order partial differential equations on general geometries, J. Computational Physics, 216 (2006), 216-246.
;
Andrea L. Bertozzi, Ning Ju, and Hsiang-Wei Lu, A biharmonic modified forward time stepping method for fourth order nonlinear diffusion equations, DCDS, Vol. 29, No. 4, special issue on Trends and Developments in DE/Dynamics, April 2011, pages 1367-1391;
Matthew R. Mata and Andrea L. Bertozzi, A Numerical Scheme for Particle-Laden Thin Film Flow in two dimensions J. Comp. Phys., Vol. 230, No. 16, 10 July 2011, Pages 6334-6353
## Fiber coating

Abolfazl Sadeghpour, Zezhi Zeng, and Y. Sungtaek Ju, Effects of Nozzle Geometry on the Fluid Dynamics of Thin Liquid Films Flowing down Vertical Strings in the Rayleigh-Plateau Regime, Langmuir 2017, 33, 6292-6299
Zeng, Z.; Sadeghpour, A.; Warrier, G.; Ju, Y. S. Experimental
Study of Heat Transfer between Thin Liquid Films Flowing down a Vertical String in the Rayleigh-Plateau Instability Regime and a Counterflowing Gas Stream. Int. J. Heat Mass Transfer 2017, 108 (Part A), 830-840
Quere,D. Fluid Coating on a Fiber Annu Rev. Fluid Mech. 1999, 31 (1), 347-384.
Kalliadasis, S.; Chang, H.-C. Drop Formation during Coating of Vertical Fibres. J. Fluid Mech. 1994, 261, 135-168.
R. V. Craster and O. K. Matar, On viscous beads flowing down a vertical fibre, J. Fluid Mech., Volume 553 25 April 2006 , pp. 85-105
## Particle Laden Flow

Background on systems of conservation laws and the Riemann Problem
Junjie Zhou, B. Dupuy, A.L. Bertozzi and A. E. Hosoi, Theory for shock dynamics in particle-laden thin films, Physical Review Letters, March 25, 2005, 117803.
Benjamin Cook, Oleg Alexandrov, and Andrea Bertozzi, Linear stability of particle laden thin films, European Physical Journal - Special Topics, 166(1), pp. 77-81, 2009.
Cook B. P. 2008 Theory for particle settling and shear-induced migration in thin-film liquid flow. Phys. Rev. E 78, 045303.
N. Murisic, J. Ho, V. Hu, P. Latterman, T. Koch, K. Lin, M. Mata, and A. Bertozzi, Particle-laden viscous thin-film flows on an incline: experiments compared with an equilibrium theory based on shear-induced migration and particle settling, Physica D, 240(20), pp. 1661-1673, 2011
N. Murisic, B. Pausader, D. Peschka, and A. L. Bertozzi, Dynamics of particle settling and resuspension in viscous liquid films , J. Fl. Mech., vol. 717, pp. 203-231, 2013.
Sungyon Lee, Yvonne Stokes, and Andrea L. Bertozzi, Behavior of a particle-laden flow in a spiral channel, Physics of Fluids 26, 043302, 2014
S. Lee, J. Wong, and A. L. Bertozzi, Equilibrium theory of bidensity particle-laden flow on an incline, Mathematical Modelling and Numerical Simulation of Oil Pollution Problems, The Reacting Atmosphere Volume 2, pp. 85-97, Mattias Ehrhardt, Editor, 2015.
Jeffrey T. Wong and Andrea L. Bertozzi, A conservation law model for bidensity suspensions on an incline Physica D, Volume 330, 1 September 2016, Pages 47-57
L. Wang, A. Mavromoustaki, A. L. Bertozzi, G. Urdaneta, and K. Huang, Rarefaction-singular shock dynamics for conserved volume gravity driven particle-laden thin film, Physics of Fluids, 27, 033301, 2015.
A. Mavromoustaki and A. L. Bertozzi, Hyperbolic systems of conservation laws in gravity-driven, particle-laden thin-film flows, J. Eng. Math., 88(1), pp. 29-48, 2014