Interfacial convection in multilayer systems

513 Pages·2012·17.507 MB·English

by  A. Nepomnyashchy, I. Simanovskii, J.C. Legros (auth.)

Applied Mathematical Sciences
Volume 179
Editors
S.S.Antman
DepartmentofMathematics
and
InstituteforPhysicalScienceandTechnology
UniversityofMaryland
CollegePark,MD20742-4015
USA
[email protected]
P.Holmes
DepartmentofMechanicalandAerospaceEngineering
PrincetonUniversity
215FineHall
Princeton,NJ08544
USA
[email protected]
L.Sirovich
LaboratoryofAppliedMathematics
DepartmentofBiomathematicalSciences
MountSinaiSchoolofMedicine
NewYork,NY10029-6574
USA
[email protected]
K.Sreenivasan
DepartmentofPhysics
NewYorkUniversity
70WashingtonSquareSouth
NewYorkCity,NY10012
USA
[email protected]
Advisors
L.Greengard J.Keener
J.Keller R.Laubenbacher B.J.Matkowsky
A.Mielke C.S.Peskin A.Stevens A.Stuart
Forfurthervolumes:
http://www.springer.com/series/34

Alexander Nepomnyashchy • Ilya Simanovskii
Jean Claude Legros
Interfacial Convection
in Multilayer Systems
Second Edition
Alexander Nepomnyashchy Ilya Simanovskii
Department of Mathematics Department of Mathematics
Technion - Israel Technion - Israel
Institute of Technology Institute of Technology
Haifa, Israel Haifa, Israel
[email protected] [email protected]
Jean Claude Legros
Microgravity Research Center (MRC)
Universite Libre de Bruxelles
Bruxelles, Belgium
[email protected]
ISSN 0 066-5452
ISBN 978-0-387-87713-6 e-ISBN 978-0-387-87714-3
DOI 10 .1007/978-0-387-87714-3
Springer New York Dordrecht He idelberg London

Library of Congress Control Number: 2011944277
© Springer Science+Business Media, LLC 2012
All rights reserved. This work may not be translated or copied in whole or in part without the written
permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York,
NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in
connection with any form of information storage and retrieval, electronic adaptation, computer
software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.
The use in this publication of trade names, trademarks, service marks, and similar terms, even if they
are not identified as such, is not to be taken as an expression of opinion as to whether or not they are
subject to proprietary rights.

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)
Contents
Preface ........................................................ ix
1 Introduction. Models of Interfacial Convection............. 1
1.1 Motivation of the Problem................................ 1
1.2 Models of Heat Convection ............................... 3
1.2.1 Sharp-Interface Approach .......................... 3
1.2.2 One-Layer Model.................................. 6
1.2.3 Two-Layer Model ................................. 11
1.2.4 Three-Layer Model ................................ 15
1.3 Models of Interfacial Convection in Multicomponent Fluids ... 17
1.3.1 Solutocapillary Convection in Isothermal Fluids ....... 17
1.3.2 Interfacial Convection in Nonisothermal Solutions ..... 20
1.3.3 Convection in Nanofluids ........................... 22
1.4 Models of Convection with Phase Transitions ............... 24
1.4.1 Convection with Evaporation ....................... 24
1.4.2 Convection with Solidification....................... 29
1.5 Models of Ultrathin Film Flows ........................... 31
1.5.1 Lubrication Approximation ......................... 31
1.5.2 Intermolecular Forces .............................. 33
1.5.3 Nonisothermal Films............................... 35
1.6 Mesoscopic Models ...................................... 36
1.6.1 Diffuse–Interface Models ........................... 36
1.6.2 Lattice Boltzmann Approach ....................... 38
1.6.3 Dissipative Particle Dynamics....................... 38
1.7 Microscopic Models...................................... 39
2 Types of Convective Instability in Systems with
an Interface ................................................ 41
2.1 The Problem of Stability ................................. 41
2.2 Rayleigh–B´enard Convection.............................. 43
2.2.1 Linear Stability ................................... 43
2.2.2 Nonlinear Flow Regimes ........................... 49
v
vi Contents
2.3 Anticonvection .......................................... 57
2.4 Stationary Marangoni Patterns............................ 64
2.4.1 Exact Formulas ................................... 64
2.4.2 Shortwave Marangoni Patterns...................... 66
2.4.3 Longwave Marangoni Patterns; the Case of Poorly
Conducting Boundaries ............................ 69
2.4.4 Longwave Deformational Instabilities ................ 70
2.5 Marangoni Waves in Systems with a Nondeformable
Interface ............................................... 75
2.5.1 Oscillatory Marangoni Instability.................... 75
2.5.2 Competition Between Marangoni and Rayleigh
Instability Mechanisms............................. 82
2.5.3 Mode Mixing of Interfacial and Internal Waves........116
2.5.4 Oscillatory Instability in the Presence of a Thermal
Gradient and a Surfactant..........................120
2.6 Marangoni Waves in Systems with a Deformable Interface ....123
2.6.1 Transverse Marangoni Instability in One-Layer
Systems..........................................123
2.6.2 The Limit of Large Ga and M ......................125
2.6.3 Linear Theory of Transverse Instability: Numerical
Results...........................................127
2.6.4 Nonlinear Theory of Transverse Instability............131
2.6.5 Oscillations Generated by a Surfactant...............135
2.6.6 Transverse and Longitudinal Marangoni Instabilities
in the Case of Mass Transfer........................140
2.7 Instabilities in the Presence of Evaporation .................145
2.7.1 Mechanisms of Instability in Evaporating Fluid
Systems..........................................145
2.7.2 Evaporation of Pure Liquids ........................145
2.7.3 Evaporation of Binary Liquids ......................148
3 B´enard Problem in Multilayer Systems with
Undeformable Interfaces...................................151
3.1 General Equations and Boundary Conditions................151
3.2 Linear Stability Theory ..................................153
3.2.1 Marangoni Convection. The Case of a Symmetric
System and Equal Layer Thicknesses.................154
3.2.2 Onset of Marangoni Convection in Asymmetric
Three-Layer Systems...............................160
3.2.3 Combined Action of Marangoni and Rayleigh
Instability Mechanisms.............................163
3.3 Nonlinear Simulations....................................169
3.3.1 Marangoni Convection. The Case of a Symmetric
System...........................................172
Contents vii
3.3.2 Marangoni Convection. The Case of an Asymmetric
System...........................................182
3.3.3 Rayleigh Convection ...............................194
3.3.4 Mixed Rayleigh–Marangoni Convection ..............201
3.3.5 Anticonvection....................................218
3.4 Space Experiments ......................................225
3.4.1 Experiment Description ............................226
3.4.2 Experiment I .....................................228
3.4.3 Experiment II.....................................233
4 B´enard Problem in Multilayer Systems with Deformable
Interfaces..................................................237
4.1 Formulation of the Problem...............................237
4.2 Linear Stability Analysis .................................238
4.2.1 Longwave Asymptotics.............................240
4.2.2 Neutral Stability Curves ...........................244
4.3 Nonlinear Theory........................................251
4.3.1 Derivation of the Amplitude Equations...............251
4.3.2 Traveling Wave Solutions...........................257
4.3.3 Results of Numerical Simulations....................262
5 Stability of Flows..........................................267
5.1 Mechanisms of Instabilities for Flows Generated by a
Surface-Tension Gradient Applied Along the Surface .........267
5.1.1 Purely Thermocapillary Flows ......................267
5.1.2 Flows Under Combined Action of Thermocapillarity
and Buoyancy ....................................269
5.2 Thermocapillary Flows in Two-Layer Systems...............270
5.2.1 Basic Equations and Boundary Conditions............270
5.2.2 Stationary Flow Profiles............................272
5.2.3 Linear Stability Theory ............................274
5.2.4 Nonlinear Patterns ................................281
5.3 Buoyancy–Thermocapillary Convection in Two-Layer
Systems ................................................287
5.4 Buoyancy–Thermocapillary Convection in Three-Layer
Systems ................................................294
5.4.1 Formulation of the Problem ........................294
5.4.2 Results of Numerical Simulations. Periodic Boundary
Conditions .......................................299
5.4.3 Results of Numerical Simulations. Closed Cavities .....306
5.5 Deformational Instabilities of Thermocapillary Flows in
Three-Layer Systems.....................................317
5.5.1 Formulation of the Model ..........................320
5.5.2 Derivation of the Interface Evolution Equations .......323
viii Contents
5.5.3 Linear Stability Analysis ...........................324
5.5.4 Weakly Nonlinear Model ...........................327
6 Flows in Ultrathin Two-Layer Films .......................331
6.1 Isothermic Films ........................................331
6.1.1 Longwave Evolution Equations......................331
6.1.2 Stability of Isothermic Films........................336
6.2 Films with a Vertical Temperature Gradient ................340
6.2.1 Formulation of the Problem ........................340
6.2.2 Derivation of Longwave Amplitude Equations.........344
6.2.3 Linear Stability Theory: General Dispersion Relation ..347
6.2.4 Marangoni Instability in the Absence of Gravity.......349
6.2.5 Marangoni Instability in the Presence of Gravity ......353
6.2.6 Van der Waals Instability ..........................361
6.3 Stability of Convective Flows .............................385
6.3.1 Formulation of the Problem ........................385
6.3.2 Flows with a Horizontal Temperature Gradient .......393
6.3.3 Flows with an Inclined Temperature Gradient without
Gravity ..........................................396
6.3.4 Flows with an Inclined Temperature Gradient in the
Presence of Gravity................................422
7 Outlook ...................................................429
7.1 Extension of the Linear Stability Theory. Influence of Lateral
Boundaries .............................................429
7.1.1 Oscillatory Instability in a Confined Container .......429
7.1.2 Monotonic Instability in a Confined Container ........430
7.1.3 Nonmodal Approach...............................431
7.2 Three-Dimensional Convective Flows.......................432
7.3 Deformation of the Interface ..............................433
7.4 Transition to Chaos and Interfacial Turbulence..............435
7.5 Multicomponent Fluids...................................435
7.6 Chemical Reactions......................................438
7.7 Porous Layers...........................................439
7.8 Contact Line Dynamics ..................................441
7.9 Feedback Control of Interfacial Instabilities .................444
7.10 Biological Surface-Tension-Driven Flows....................449
Bibliography...................................................453
Index..........................................................493
Preface
Interfacialconvectionisawidespreadphenomenonthatisofgreatimportance
in numerous branches of technology, including chemical engineering, material
processing, space technologies, microfluidics, coating etc. The role of the in-
terfacial convection is especially large under microgravity conditions and on
small scales. During the last decades, an essential progress was achieved in
understandingbasicphysicalprocessesunderlyingthephenomenonofinterfa-
cialconvectionandinthedevelopmentofadequateandreliablemathematical
models of those processes. A detailed description of the state of art in that
field can be found in the books by Simanovskii and Nepomnyashchy (1993),
Colinet et al. (2001), Nepomnyashchy et al. (2002) (see also a collection of
review papers edited by Narayanan and Schwabe (2003)).
The subject of the present book is the interfacial convection in multilayer
systems. Studying this kind of interfacial convection has its own engineering
rootsandscientificgoals.Thebest-knownmodernengineeringtechniquethat
requires an investigation of interfacial convection in systems with many in-
terfaces is the liquid-encapsulation crystal-growth technique (Johnson, 1975;
G´eoris and Legros, 1996; Simanovskii et al., 2003) used in space lab missions.
It is known that time-dependent thermocapillary convection leads to solute
segregation and hence to striation in crystals (Eyer et al., 1985). The liquid-
encapsulation technique allows one to reduce the convection significantly and
to grow high-quality, striation-free crystals by putting the melt between the
fluid layers (Eyer and Leiste, 1985).
Another important problem that needs a multilayer approach for its
self-consistent description is the coalescence between droplets or bubbles
(Leshansky, 2001; Yeo et al., 2001; Yeo and Matar, 2003; Narsimhan, 2009)
or between droplets and a bulk liquid (Savino et al., 2003) under conditions
of heat/mass transfer. The influence of the interfacial convection on the co-
alescence rate, as well as droplet migration under the action of an applied
temperature/concentration gradient (Subramanian, 1981; Balasubramaniam
and Subramanian, 2000) are significant for many engineering processes, in-
cluding various extraction processes (Groothuis and Zuiderweg, 1960), steel
ix