Energy exchange analysis in droplet dynamics via the Navier–Stokes–Cahn–Hilliard model

We develop the energy budget equation of the coupled Navier–Stokes–Cahn–Hilliard (NSCH) system. We use the NSCH equations to model the dynamics of liquid droplets in a liquid continuum. Buoyancy effects are accounted for through the Boussinesq assumption. We physically interpret each quantity involved in the energy exchange to gain further insight into the model. Highly resolved simulations involving density-driven flows and the merging of droplets allow us to analyse these energy budgets. In particular, we focus on the energy exchanges when droplets merge, and describe flow features relevant to this phenomenon. By comparing our numerical simulations to analytical predictions and experimental results available in the literature, we conclude that modelling droplet dynamics within the framework of NSCH equations is a sensible approach worthy of further research.

[1]  R. Clift,et al.  Bubbles, Drops, and Particles , 1978 .

[2]  A. Peirce.  Computer Methods in Applied Mechanics and Engineering , 2010 .

[3]  Giancarlo Sangalli,et al.  Isogeometric Discrete Differential Forms in Three Dimensions , 2011, SIAM J. Numer. Anal..

[4]  G. Sangalli,et al.  Isogeometric analysis in electromagnetics: B-splines approximation , 2010 .

[5]  Giancarlo Sangalli,et al.  IsoGeometric Analysis: Stable elements for the 2D Stokes equation , 2011 .

[6]  John A. Evans,et al.  ISOGEOMETRIC DIVERGENCE-CONFORMING B-SPLINES FOR THE STEADY NAVIER–STOKES EQUATIONS , 2013 .

[7]  Xesús Nogueira,et al.  A new space–time discretization for the Swift–Hohenberg equation that strictly respects the Lyapunov functional , 2012 .

[8]  Jintai Chung,et al.  A Time Integration Algorithm for Structural Dynamics With Improved Numerical Dissipation: The Generalized-α Method , 1993 .

[9]  Victor M. Calo,et al.  An energy-stable convex splitting for the phase-field crystal equation , 2014, 1405.3488.

[10]  Patrick Patrick Anderson,et al.  On scaling of diffuse-interface models , 2006 .

[11]  John W. Cahn,et al.  Free Energy of a Nonuniform System. II. Thermodynamic Basis , 1959 .

[12]  Alex M. Andrew,et al.  Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science (2nd edition) , 2000 .

[13]  R. Spatschek,et al.  Phase field modeling of fracture and stress-induced phase transitions. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[14]  W. Rider,et al.  Reconstructing Volume Tracking , 1998 .

[15]  M. Gurtin.  Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance , 1996 .

[16]  Armando Miguel Awruch,et al.  A NURBS‐based finite element model applied to geometrically nonlinear elastodynamics using a corotational approach , 2015 .

[17]  J. Lowengrub,et al.  Quasi–incompressible Cahn–Hilliard fluids and topological transitions , 1998, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[18]  John R. Lister,et al.  Coalescence of liquid drops , 1999, Journal of Fluid Mechanics.

[19]  Ju Liu.  Thermodynamically consistent modeling and simulation of multiphase flows , 2014 .

[20]  D. M. Anderson,et al.  DIFFUSE-INTERFACE METHODS IN FLUID MECHANICS , 1997 .

[21]  H. Kavehpour,et al.  Coalescence of Drops , 2015 .

[22]  M. Gurtin,et al.  The Mechanics and Thermodynamics of Continua , 2010 .

[23]  P. Laplace,et al.  Traité de mécanique céleste , 1967 .

[24]  Victor M. Calo,et al.  On the computational efficiency of isogeometric methods for smooth elliptic problems using direct solvers , 2014 .

[25]  Heike Emmerich,et al.  The Diffuse Interface Approach in Materials Science: Thermodynamic Concepts and Applications of Phase-Field Models , 2003 .

[26]  Victor M. Calo,et al.  Performance evaluation of block-diagonal preconditioners for the divergence-conforming B-spline discretization of the Stokes system , 2015, J. Comput. Sci..

[27]  S. Laizet,et al.  High-fidelity simulations of the lobe-and-cleft structures and the deposition map in particle-driven gravity currents , 2015 .

[28]  Victor M. Calo,et al.  PetIGA-MF: A multi-field high-performance toolbox for structure-preserving B-splines spaces , 2016, J. Comput. Sci..

[29]  Drew Myers,et al.  Surfaces, interfaces, and colloids , 1991 .

[30]  P. Lin,et al.  A thermodynamically consistent phase-field model for two-phase flows with thermocapillary effects , 2014, Journal of Fluid Mechanics.

[31]  Victor M. Calo,et al.  The Cost of Continuity: Performance of Iterative Solvers on Isogeometric Finite Elements , 2012, SIAM J. Sci. Comput..

[32]  Victor M. Calo,et al.  The cost of continuity: A study of the performance of isogeometric finite elements using direct solvers , 2012 .

[33]  Victor M. Calo,et al.  Phase Field Modeling Using PetIGA , 2013, ICCS.

[34]  O. Lebaigue,et al.  The second gradient method for the direct numerical simulation of liquid—vapor flows with phase change , 2001 .

[35]  Xesús Nogueira,et al.  An unconditionally energy-stable method for the phase field crystal equation , 2012 .

[36]  Daniel Bonn,et al.  Hydrodynamics of droplet coalescence. , 2005, Physical review letters.

[37]  Victor M. Calo,et al.  Coupling Navier-stokes and Cahn-hilliard Equations in a Two-dimensional Annular flow Configuration , 2015, ICCS.

[38]  J. E. Hilliard,et al.  Free Energy of a Nonuniform System. I. Interfacial Free Energy , 1958 .

[39]  John A. Evans,et al.  Isogeometric divergence-conforming b-splines for the darcy-stokes-brinkman equations , 2013 .

[40]  D. Jacqmin.  Regular Article: Calculation of Two-Phase Navier–Stokes Flows Using Phase-Field Modeling , 1999 .

[41]  M. Gurtin,et al.  TWO-PHASE BINARY FLUIDS AND IMMISCIBLE FLUIDS DESCRIBED BY AN ORDER PARAMETER , 1995, patt-sol/9506001.

[42]  Victor M. Calo,et al.  Solving Nonlinear, High-Order Partial Differential Equations Using a High-Performance Isogeometric Analysis Framework , 2014, CARLA.

[43]  Ping Lin,et al.  A numerical method for the quasi-incompressible Cahn-Hilliard-Navier-Stokes equations for variable density flows with a discrete energy law , 2014, J. Comput. Phys..

[44]  John E. Hilliard,et al.  Free Energy of a Nonuniform System. III. Nucleation in a Two‐Component Incompressible Fluid , 1959 .

[45]  Thomas Young,et al.  An Essay on the Cohesion of Fluids , 1800 .

[46]  John A. Evans,et al.  Isogeometric Analysis , 2010 .

[47]  T. Etoh,et al.  The coalescence speed of a pendent and a sessile drop , 2005, Journal of Fluid Mechanics.

[48]  W. Kühnel.  Differential Geometry: Curves - Surfaces - Manifolds , 2002 .

[49]  Patrick Patrick Anderson,et al.  Capillary spreading of a droplet in the partially wetting regime using a diffuse-interface model , 2007, Journal of Fluid Mechanics.

[50]  Chun Liu,et al.  Approximation of Liquid Crystal Flows , 2000, SIAM J. Numer. Anal..

[51]  Lisandro Dalcin,et al.  PetIGA: High-Performance Isogeometric Analysis , 2013, ArXiv.

[52]  Gustav Amberg,et al.  Phase-field simulations of non-isothermal binary alloy solidification , 2001 .

[53]  Thomas J. R. Hughes,et al.  Isogeometric divergence-conforming B-splines for the unsteady Navier-Stokes equations , 2013, J. Comput. Phys..

[54]  James J. Feng,et al.  A diffuse-interface method for simulating two-phase flows of complex fluids , 2004, Journal of Fluid Mechanics.

[55]  T. Hughes,et al.  Isogeometric analysis of the isothermal Navier-Stokes-Korteweg equations , 2010 .

[56]  T. N. Stevenson,et al.  Fluid Mechanics , 2021, Nature.

[57]  G. Hulbert,et al.  A generalized-α method for integrating the filtered Navier–Stokes equations with a stabilized finite element method , 2000 .

[58]  T. Hughes,et al.  Isogeometric analysis of the Cahn–Hilliard phase-field model , 2008 .

[59]  Eckart Meiburg,et al.  Turbidity Currents and Their Deposits , 2010 .

[60]  Victor M. Calo,et al.  PetIGA: A Framework for High-Performance Isogeometric Analysis , 2013 .

[61]  Ruben Juanes,et al.  Three-dimensional simulation of unstable gravity-driven infiltration of water into a porous medium , 2013, J. Comput. Phys..