June  2015, 35(6): 2325-2347. doi: 10.3934/dcds.2015.35.2325

Numerical simulation of two-phase flows with heat and mass transfer

1. 

AM III, Department Math, Cauerstr. 11 91058 Erlangen, Germany, Germany

2. 

Reinbeckstrasse 7, 12459 Berlin, Germany

Received  January 2014 Revised  May 2014 Published  December 2014

We present a finite element method for simulating complex free surface flow. The mathematical model and the numerical method take into account two-phase non-isothermal flow of an incompressible liquid and a gas phase, capillary forces at the interface of both fluids, Marangoni effects due to temperature variation of the interface and mass transport across the interface by evaporation/condensation. The method is applied to two examples from microgravity research, for which experimental data are available.
Citation: Eberhard Bänsch, Steffen Basting, Rolf Krahl. Numerical simulation of two-phase flows with heat and mass transfer. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2325-2347. doi: 10.3934/dcds.2015.35.2325
References:
[1]

E. Bänsch, Simulation of instationary, incompressible flows,, Acta Math. Univ. Com., 67 (1998), 101.   Google Scholar

[2]

E. Bänsch, Finite element discretization of the Navier-Stokes equations with a free capillary surface,, Numer. Math., 88 (2001), 203.  doi: 10.1007/PL00005443.  Google Scholar

[3]

J. Brackbill, D. Kothe and C. Zemach, A continuum method for modeling surface tension,, Journal of Computational Physics, 100 (1992), 335.  doi: 10.1016/0021-9991(92)90240-Y.  Google Scholar

[4]

M.-O. Bristeau, R. Glowinski and J. Pariaux, Numerical methods for the Navier-Stokes equations. applications to the simulation of compressible and incompressible viscous flow,, Computer Physics Report, 6 (1987), 73.  doi: 10.1007/978-3-322-87873-1.  Google Scholar

[5]

A. N. Brooks and T. J. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations,, Computer Methods in Applied Mechanics and Engineering, 32 (1982), 199.  doi: 10.1016/0045-7825(82)90071-8.  Google Scholar

[6]

S. Das and E. Hopfinger, Mass transfer enhancement by gravity waves at a liquid-vapour interface,, International Journal of Heat and Mass Transfer, 52 (2009), 1400.  doi: 10.1016/j.ijheatmasstransfer.2008.08.016.  Google Scholar

[7]

J. Donea, S. Giuliani and J. P. Halleux, An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interactions,, Comp. Meth. App. MechEng., 33 (1982), 689.  doi: 10.1016/0045-7825(82)90128-1.  Google Scholar

[8]

M. E. Dreyer, Free Surface Flows under Compensated Gravity Conditions,, no. 221 in Springer Tracts in Modern Physics, (2007).   Google Scholar

[9]

G. Dziuk, An algorithm for evolutionary surfaces,, Numerische Mathematik, 58 (1991), 603.  doi: 10.1007/BF01385643.  Google Scholar

[10]

C. Eck, M. Fontelos, G. Grün, F. Klingbeil and O. Vantzos, On a phase-field model for electrowetting,, Interfaces Free Bound., 11 (2009), 259.  doi: 10.4171/IFB/211.  Google Scholar

[11]

E. Fuhrmann and M. Dreyer, Description of the Sounding Rocket Experiment SOURCE,, Microgravity Science and Technology, 20 (2008), 205.  doi: 10.1007/s12217-008-9017-4.  Google Scholar

[12]

E. Fuhrmann and M. Dreyer, Heat transfer by thermo-capillary convection,, Microgravity Science and Technology, 21 (2009), 87.  doi: 10.1007/s12217-009-9125-9.  Google Scholar

[13]

E. Fuhrmann, M. Dreyer, S. Basting and E. Bänsch, Free surface deformation and heat transfer by thermocapillary convection, 2013,, Submitted for publication., ().   Google Scholar

[14]

J. Gerstmann, Numerische Untersuchung zur Schwingung freier Flüssigkeitsoberflächen,, no. 464 in Fortschritt-Berichte VDI, (2004).   Google Scholar

[15]

J. Gerstmann, M. Michaelis and M. E. Dreyer, Capillary driven oscillations of a free liquid interface under non-isothermal conditions,, PAMM, 4 (2004), 436.  doi: 10.1002/pamm.200410199.  Google Scholar

[16]

F. Gibou, L. Chen, D. Nguyen and S. Banerjee, A level set based sharp interface method for the multiphase incompressible Navier-Stokes equations with phase change,, J. Comp. Phys., 222 (2007), 536.  doi: 10.1016/j.jcp.2006.07.035.  Google Scholar

[17]

V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations,, Springer, (1986).  doi: 10.1007/978-3-642-61623-5.  Google Scholar

[18]

S. Gross and A. Reusken, Numerical Methods for Two-phase Incompressible Flows, vol. 40 of Springer Series in Computational Mathematics,, Springer-Verlag, (2011).  doi: 10.1007/978-3-642-19686-7.  Google Scholar

[19]

M. E. Gurtin, An Introduction to Continuum Mechanics,, Academic Press, (1981).   Google Scholar

[20]

C. W. Hirt and B. D. Nichols, Volume of fluid (VOF) method for the dynamics of free boundaries,, J. Comp. Phys., 39 (1981), 201.  doi: 10.1016/0021-9991(81)90145-5.  Google Scholar

[21]

C. Hirt, A. Amsden and J. Cook, An arbitrary Lagrangian-Eulerian computing method for all flow speeds,, Journal of Computational Physics, 135 (1997), 203.  doi: 10.1006/jcph.1997.5702.  Google Scholar

[22]

B. Höhn, Numerik für die Marangoni-Konvektion beim Floating-Zone Verfahren,, Dissertation, (1999).   Google Scholar

[23]

T. J. R. Hughes, W. Liu and T. K. Zimmermann, Lagrangian-Eulerian finite element formulation for incompressible viscous flows,, Computer Methods in Applied Mechanics and Engineering, 29 (1981), 329.  doi: 10.1016/0045-7825(81)90049-9.  Google Scholar

[24]

D. Jacqmin, Calculation of two-phase Navier-Stokes flows using phase-field modeling,, J. Comp. Phys., 155 (1999), 96.  doi: 10.1006/jcph.1999.6332.  Google Scholar

[25]

D. Jamet, O. Lebaigue, N. Coutris and J. M. Delhaye, The second gradient method for the direct numerical simulation of liquid-vapor flows with phase change,, J. Comp. Phys, 169 (2001), 624.  doi: 10.1006/jcph.2000.6692.  Google Scholar

[26]

E. Kennard, Kinetic theory of gases: with an introduction to statistical mechanics,, International series in pure and applied physics, (1938).   Google Scholar

[27]

R. Klein, N. Botta, T. Schneider, C. D. Munz, S. Roller, A. Meister, L. Hoffmann and T. Sonar, Asymptotic adaptive methods for multi-scale problems in fluid mechanics,, Eng. Math., 39 (2001), 261.  doi: 10.1023/A:1004844002437.  Google Scholar

[28]

R. Krahl, M. Adamov, M. Lozano Avilés and E. Bänsch, A model for two phase flow with evaporation,, in Two-Phase Flow Modelling and Experimentation 2004 (eds. G. P. Celata, (2004), 2381.   Google Scholar

[29]

R. Krahl, J. Gerstmann, P. Behruzi, E. Bänsch and M. E. Dreyer, Dependency of the apparent contact angle on nonisothermal conditions,, Physics of Fluids, 20 (2008).  doi: 10.1063/1.2899641.  Google Scholar

[30]

R. Krahl and E. Bänsch, Impact of marangoni effects on the apparent contact angle - a numerical investigation,, Microgravity Science and Technology, 17 (2005), 39.  doi: 10.1007/BF02872086.  Google Scholar

[31]

R. Krahl and E. Bänsch, On the stability of an evaporating liquid surface,, Fluid Dynamics Research, 44 (2012).  doi: 10.1088/0169-5983/44/3/031409.  Google Scholar

[32]

R. Krahl and J. Gerstmann, Non-isothermal reorientation of a liquid surface in an annular gap,, in $4^{th}$ International Berlin Workshop - IBW 4 on Transport Phenomena with Moving Boundaries (ed. F.-P. Schindler), (2007), 227.   Google Scholar

[33]

N. Kulev, S. Basting, E. Bänsch and M. Dreyer, Interface reorientation of cryogenic liquids under non-isothermal boundary conditions,, Cryogenics, 62 (2014), 48.  doi: 10.1016/j.cryogenics.2014.04.006.  Google Scholar

[34]

N. Kulev and M. Dreyer, Drop tower experiments on non-isothermal reorientation of cryogenic liquids,, Microgravity Science and Technology, 22 (2010), 463.  doi: 10.1007/s12217-010-9237-2.  Google Scholar

[35]

D. Meschede (ed.), Gerthsen Physik,, 22nd edition, (2004).   Google Scholar

[36]

M. Michaelis, Kapillarinduzierte Schwingungen Freier Flüssigkeitsoberflächen,, no. 454 in Fortschritt-Berichte VDI, (2003).   Google Scholar

[37]

S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations,, J. Comp. Phys., 79 (1988), 12.  doi: 10.1016/0021-9991(88)90002-2.  Google Scholar

[38]

S. Ostrach, Low-gravity fluid flows,, Ann. Rev. Fluid. Mech., 14 (1982), 313.  doi: 10.1146/annurev.fl.14.010182.001525.  Google Scholar

[39]

L. M. Pismen and Y. Pomeau, Disjoining potential and spreading of thin liquid layers in the diffuse-interface model coupled to hydrodynamics,, Phys. Rev. E, 62 (2000), 2480.  doi: 10.1103/PhysRevE.62.2480.  Google Scholar

[40]

M. Rumpf, A variational approach to optimal meshes,, Numerische Mathematik, 72 (1996), 523.  doi: 10.1007/s002110050180.  Google Scholar

[41]

Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems,, SIAM Sci. Comp., 7 (1986), 856.  doi: 10.1137/0907058.  Google Scholar

[42]

R. Scardovelli and S. Zaleski, Direct numerical simulation of free-surface and interfacial flow,, Ann. Rev. Fluid Mech., 31 (1999), 567.  doi: 10.1146/annurev.fluid.31.1.567.  Google Scholar

[43]

J. Schlottke and B. Weigand, Direct numerical simulation of evaporating droplets,, J. Comp. Phys., 227 (2008), 5215.  doi: 10.1016/j.jcp.2008.01.042.  Google Scholar

[44]

L. E. Scriven, Dynamics of a fluid interface equation of motion for Newtonian surface fluids,, Chem. Eng. Sci., 12 (1960), 98.  doi: 10.1016/0009-2509(60)87003-0.  Google Scholar

[45]

J. A. Sethian and P. Smereka, Level set methods for fluid interfaces,, Ann. Rev. Fluid Mech., 35 (2003), 341.  doi: 10.1146/annurev.fluid.35.101101.161105.  Google Scholar

[46]

G. Son and V. K. Dhir, Numerical simulation of film boiling near critical pressures with a level set method,, J. Heat Transfer, 120 (1998), 183.  doi: 10.1115/1.2830042.  Google Scholar

[47]

M. Sussman, P. Smereka and S. Osher, A level set approach for computing solutions to incompressible two-phase flow,, Journal of Computational Physics, 114 (1994), 146.  doi: 10.1006/jcph.1994.1155.  Google Scholar

[48]

S. Tanguy, T. Ménard and A. Berlemont, A level set method for vaporizing two-phase flows,, J. Comp. Phys., 221 (2007), 837.  doi: 10.1016/j.jcp.2006.07.003.  Google Scholar

[49]

M. Tenhaeff, Computation of Incompressible, Axisymmetric Flows in Electrically Conducting Fluids Under Influence of Rotating Magnetic Fields (in German),, Diploma thesis, (1997).   Google Scholar

[50]

T. Tezduyar and R. Benney, Mesh moving techniques for fluid-structure interactions with large displacements,, J. Applied Mechanics, 70 (2003), 58.   Google Scholar

[51]

S. W. J. Welch and J. Wilson, A volume of fluid based method for fluid flows with phase change,, J. Comp. Phys, 160 (2000), 662.  doi: 10.1006/jcph.2000.6481.  Google Scholar

[52]

T. Wick, Fluid-structure interactions using different mesh motion techniques,, Computers & Structures, 89 (2011), 1456.  doi: 10.1016/j.compstruc.2011.02.019.  Google Scholar

[53]

Y. F. Yap, J. C. Chai, K. C. Toh, T. N. Wong and Y. C. Lam, Numerical modeling of unidirectional stratified flow with and without phase change,, J. Int. Heat Mass Transfer, 48 (2005), 477.  doi: 10.1016/j.ijheatmasstransfer.2004.09.017.  Google Scholar

show all references

References:
[1]

E. Bänsch, Simulation of instationary, incompressible flows,, Acta Math. Univ. Com., 67 (1998), 101.   Google Scholar

[2]

E. Bänsch, Finite element discretization of the Navier-Stokes equations with a free capillary surface,, Numer. Math., 88 (2001), 203.  doi: 10.1007/PL00005443.  Google Scholar

[3]

J. Brackbill, D. Kothe and C. Zemach, A continuum method for modeling surface tension,, Journal of Computational Physics, 100 (1992), 335.  doi: 10.1016/0021-9991(92)90240-Y.  Google Scholar

[4]

M.-O. Bristeau, R. Glowinski and J. Pariaux, Numerical methods for the Navier-Stokes equations. applications to the simulation of compressible and incompressible viscous flow,, Computer Physics Report, 6 (1987), 73.  doi: 10.1007/978-3-322-87873-1.  Google Scholar

[5]

A. N. Brooks and T. J. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations,, Computer Methods in Applied Mechanics and Engineering, 32 (1982), 199.  doi: 10.1016/0045-7825(82)90071-8.  Google Scholar

[6]

S. Das and E. Hopfinger, Mass transfer enhancement by gravity waves at a liquid-vapour interface,, International Journal of Heat and Mass Transfer, 52 (2009), 1400.  doi: 10.1016/j.ijheatmasstransfer.2008.08.016.  Google Scholar

[7]

J. Donea, S. Giuliani and J. P. Halleux, An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interactions,, Comp. Meth. App. MechEng., 33 (1982), 689.  doi: 10.1016/0045-7825(82)90128-1.  Google Scholar

[8]

M. E. Dreyer, Free Surface Flows under Compensated Gravity Conditions,, no. 221 in Springer Tracts in Modern Physics, (2007).   Google Scholar

[9]

G. Dziuk, An algorithm for evolutionary surfaces,, Numerische Mathematik, 58 (1991), 603.  doi: 10.1007/BF01385643.  Google Scholar

[10]

C. Eck, M. Fontelos, G. Grün, F. Klingbeil and O. Vantzos, On a phase-field model for electrowetting,, Interfaces Free Bound., 11 (2009), 259.  doi: 10.4171/IFB/211.  Google Scholar

[11]

E. Fuhrmann and M. Dreyer, Description of the Sounding Rocket Experiment SOURCE,, Microgravity Science and Technology, 20 (2008), 205.  doi: 10.1007/s12217-008-9017-4.  Google Scholar

[12]

E. Fuhrmann and M. Dreyer, Heat transfer by thermo-capillary convection,, Microgravity Science and Technology, 21 (2009), 87.  doi: 10.1007/s12217-009-9125-9.  Google Scholar

[13]

E. Fuhrmann, M. Dreyer, S. Basting and E. Bänsch, Free surface deformation and heat transfer by thermocapillary convection, 2013,, Submitted for publication., ().   Google Scholar

[14]

J. Gerstmann, Numerische Untersuchung zur Schwingung freier Flüssigkeitsoberflächen,, no. 464 in Fortschritt-Berichte VDI, (2004).   Google Scholar

[15]

J. Gerstmann, M. Michaelis and M. E. Dreyer, Capillary driven oscillations of a free liquid interface under non-isothermal conditions,, PAMM, 4 (2004), 436.  doi: 10.1002/pamm.200410199.  Google Scholar

[16]

F. Gibou, L. Chen, D. Nguyen and S. Banerjee, A level set based sharp interface method for the multiphase incompressible Navier-Stokes equations with phase change,, J. Comp. Phys., 222 (2007), 536.  doi: 10.1016/j.jcp.2006.07.035.  Google Scholar

[17]

V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations,, Springer, (1986).  doi: 10.1007/978-3-642-61623-5.  Google Scholar

[18]

S. Gross and A. Reusken, Numerical Methods for Two-phase Incompressible Flows, vol. 40 of Springer Series in Computational Mathematics,, Springer-Verlag, (2011).  doi: 10.1007/978-3-642-19686-7.  Google Scholar

[19]

M. E. Gurtin, An Introduction to Continuum Mechanics,, Academic Press, (1981).   Google Scholar

[20]

C. W. Hirt and B. D. Nichols, Volume of fluid (VOF) method for the dynamics of free boundaries,, J. Comp. Phys., 39 (1981), 201.  doi: 10.1016/0021-9991(81)90145-5.  Google Scholar

[21]

C. Hirt, A. Amsden and J. Cook, An arbitrary Lagrangian-Eulerian computing method for all flow speeds,, Journal of Computational Physics, 135 (1997), 203.  doi: 10.1006/jcph.1997.5702.  Google Scholar

[22]

B. Höhn, Numerik für die Marangoni-Konvektion beim Floating-Zone Verfahren,, Dissertation, (1999).   Google Scholar

[23]

T. J. R. Hughes, W. Liu and T. K. Zimmermann, Lagrangian-Eulerian finite element formulation for incompressible viscous flows,, Computer Methods in Applied Mechanics and Engineering, 29 (1981), 329.  doi: 10.1016/0045-7825(81)90049-9.  Google Scholar

[24]

D. Jacqmin, Calculation of two-phase Navier-Stokes flows using phase-field modeling,, J. Comp. Phys., 155 (1999), 96.  doi: 10.1006/jcph.1999.6332.  Google Scholar

[25]

D. Jamet, O. Lebaigue, N. Coutris and J. M. Delhaye, The second gradient method for the direct numerical simulation of liquid-vapor flows with phase change,, J. Comp. Phys, 169 (2001), 624.  doi: 10.1006/jcph.2000.6692.  Google Scholar

[26]

E. Kennard, Kinetic theory of gases: with an introduction to statistical mechanics,, International series in pure and applied physics, (1938).   Google Scholar

[27]

R. Klein, N. Botta, T. Schneider, C. D. Munz, S. Roller, A. Meister, L. Hoffmann and T. Sonar, Asymptotic adaptive methods for multi-scale problems in fluid mechanics,, Eng. Math., 39 (2001), 261.  doi: 10.1023/A:1004844002437.  Google Scholar

[28]

R. Krahl, M. Adamov, M. Lozano Avilés and E. Bänsch, A model for two phase flow with evaporation,, in Two-Phase Flow Modelling and Experimentation 2004 (eds. G. P. Celata, (2004), 2381.   Google Scholar

[29]

R. Krahl, J. Gerstmann, P. Behruzi, E. Bänsch and M. E. Dreyer, Dependency of the apparent contact angle on nonisothermal conditions,, Physics of Fluids, 20 (2008).  doi: 10.1063/1.2899641.  Google Scholar

[30]

R. Krahl and E. Bänsch, Impact of marangoni effects on the apparent contact angle - a numerical investigation,, Microgravity Science and Technology, 17 (2005), 39.  doi: 10.1007/BF02872086.  Google Scholar

[31]

R. Krahl and E. Bänsch, On the stability of an evaporating liquid surface,, Fluid Dynamics Research, 44 (2012).  doi: 10.1088/0169-5983/44/3/031409.  Google Scholar

[32]

R. Krahl and J. Gerstmann, Non-isothermal reorientation of a liquid surface in an annular gap,, in $4^{th}$ International Berlin Workshop - IBW 4 on Transport Phenomena with Moving Boundaries (ed. F.-P. Schindler), (2007), 227.   Google Scholar

[33]

N. Kulev, S. Basting, E. Bänsch and M. Dreyer, Interface reorientation of cryogenic liquids under non-isothermal boundary conditions,, Cryogenics, 62 (2014), 48.  doi: 10.1016/j.cryogenics.2014.04.006.  Google Scholar

[34]

N. Kulev and M. Dreyer, Drop tower experiments on non-isothermal reorientation of cryogenic liquids,, Microgravity Science and Technology, 22 (2010), 463.  doi: 10.1007/s12217-010-9237-2.  Google Scholar

[35]

D. Meschede (ed.), Gerthsen Physik,, 22nd edition, (2004).   Google Scholar

[36]

M. Michaelis, Kapillarinduzierte Schwingungen Freier Flüssigkeitsoberflächen,, no. 454 in Fortschritt-Berichte VDI, (2003).   Google Scholar

[37]

S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations,, J. Comp. Phys., 79 (1988), 12.  doi: 10.1016/0021-9991(88)90002-2.  Google Scholar

[38]

S. Ostrach, Low-gravity fluid flows,, Ann. Rev. Fluid. Mech., 14 (1982), 313.  doi: 10.1146/annurev.fl.14.010182.001525.  Google Scholar

[39]

L. M. Pismen and Y. Pomeau, Disjoining potential and spreading of thin liquid layers in the diffuse-interface model coupled to hydrodynamics,, Phys. Rev. E, 62 (2000), 2480.  doi: 10.1103/PhysRevE.62.2480.  Google Scholar

[40]

M. Rumpf, A variational approach to optimal meshes,, Numerische Mathematik, 72 (1996), 523.  doi: 10.1007/s002110050180.  Google Scholar

[41]

Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems,, SIAM Sci. Comp., 7 (1986), 856.  doi: 10.1137/0907058.  Google Scholar

[42]

R. Scardovelli and S. Zaleski, Direct numerical simulation of free-surface and interfacial flow,, Ann. Rev. Fluid Mech., 31 (1999), 567.  doi: 10.1146/annurev.fluid.31.1.567.  Google Scholar

[43]

J. Schlottke and B. Weigand, Direct numerical simulation of evaporating droplets,, J. Comp. Phys., 227 (2008), 5215.  doi: 10.1016/j.jcp.2008.01.042.  Google Scholar

[44]

L. E. Scriven, Dynamics of a fluid interface equation of motion for Newtonian surface fluids,, Chem. Eng. Sci., 12 (1960), 98.  doi: 10.1016/0009-2509(60)87003-0.  Google Scholar

[45]

J. A. Sethian and P. Smereka, Level set methods for fluid interfaces,, Ann. Rev. Fluid Mech., 35 (2003), 341.  doi: 10.1146/annurev.fluid.35.101101.161105.  Google Scholar

[46]

G. Son and V. K. Dhir, Numerical simulation of film boiling near critical pressures with a level set method,, J. Heat Transfer, 120 (1998), 183.  doi: 10.1115/1.2830042.  Google Scholar

[47]

M. Sussman, P. Smereka and S. Osher, A level set approach for computing solutions to incompressible two-phase flow,, Journal of Computational Physics, 114 (1994), 146.  doi: 10.1006/jcph.1994.1155.  Google Scholar

[48]

S. Tanguy, T. Ménard and A. Berlemont, A level set method for vaporizing two-phase flows,, J. Comp. Phys., 221 (2007), 837.  doi: 10.1016/j.jcp.2006.07.003.  Google Scholar

[49]

M. Tenhaeff, Computation of Incompressible, Axisymmetric Flows in Electrically Conducting Fluids Under Influence of Rotating Magnetic Fields (in German),, Diploma thesis, (1997).   Google Scholar

[50]

T. Tezduyar and R. Benney, Mesh moving techniques for fluid-structure interactions with large displacements,, J. Applied Mechanics, 70 (2003), 58.   Google Scholar

[51]

S. W. J. Welch and J. Wilson, A volume of fluid based method for fluid flows with phase change,, J. Comp. Phys, 160 (2000), 662.  doi: 10.1006/jcph.2000.6481.  Google Scholar

[52]

T. Wick, Fluid-structure interactions using different mesh motion techniques,, Computers & Structures, 89 (2011), 1456.  doi: 10.1016/j.compstruc.2011.02.019.  Google Scholar

[53]

Y. F. Yap, J. C. Chai, K. C. Toh, T. N. Wong and Y. C. Lam, Numerical modeling of unidirectional stratified flow with and without phase change,, J. Int. Heat Mass Transfer, 48 (2005), 477.  doi: 10.1016/j.ijheatmasstransfer.2004.09.017.  Google Scholar

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