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**Workshop Description**

The general topic of the event is computational methods to study multiphase flows [1,2]. Such methods are applied in very different disciplines, such as statistical physics, materials science, applied mathematics, and engineering, with applications ranging from geophysical to micro scales. Examples include volcano eruptions, oil recovery, and the dynamics of droplets on structured surfaces (“lotus effect”). The computational approaches to tackle these problems are as disparate as the phenomena themselves and the corresponding scientific communities, which rarely communicate amongst each other. The purpose of this school and workshop is to bring these various practitioners together for a fruitful exchange with the aim of improving the methodological toolbox which is still facing significant problems.

From the computational point of view, three major approaches (which shall all be covered) are commonly used: (i) sharp interface methods that keep track of the interface position [3]; (ii) smeared interface methods, which again may be subdivided into level set approaches [4-6] and methods based upon a Cahn-Hilliard free energy, or similar (to be discussed in the next paragraph) and finally (iii) methods which average over several phases being present in one volume element [7-9].

Concerning Cahn-Hilliard based approaches and similar, a whole plethora of methods has been developed. In metallurgy and other branches of materials science, phase-field models are fairly popular and have been particularly successful in the prediction of solid structures and their dynamic formation [10-15]. For fluid systems, the usual approach has been standard Computational Fluid Dynamics, based upon Finite Elements / Finite Differences / Finite Volume discretizations. These have recently been generalized to also include thermal fluctuations [16], which are typically needed for modeling phenomenena in the soft-matter domain, i.e. the micro- and nanoscale. Instead of using an Eulerian grid, an alternative discretization of the Navier-Stokes equations is also possible in terms of Lagrangian particles; this is the so-called Smoothed Particle Hydrodynamics (SPH) method, which has been used for macroscale multiphase flows for quite a while [17,18]. An exciting recent development has generalized SPH to also include thermal fluctuations [20,21], which was subsequently combined with the multiphase methodology [22,23].

A substantial body of work is based on the Lattice Boltzmann method [24]. While the original version was for an ideal gas on the macroscale, it has been generalized to include thermal fluctuations [25] and also multiphase flows, where typically the Shan-Chen model [26], the Swift-Yeomans model [27,28], or variants thereof [29,30] are being used. Thermal fluctuations have been included as well [31]. Quite successful applications include spinodal decomposition [32], Pickering emulsions [33-35], and flow of droplets past structured surfaces [36]. The Lattice Boltzmann method is particularly well-suited for modern parallel computer architectures and hence considerations of computational efficiency have played an important role in the literature [37,38].

A problem that has so far not been solved fully satisfactorily is the appearance of so-called “spurious currents” at an interface, which are a mere discretization artifact. Though also present in standard grid-based CFD calculations [39], they seem to have mainly been discussed in the Lattice Boltzmann literature [40-42]. An important goal of the event will be to critically discuss such artifacts, as well as issues of thermodynamic consistency. This will be targeted at (i) avenues toward systematic understanding, reduction and ultimate elimination of such undesired effects, but also at (ii) the more pragmatic question of how far these issues matter in practical applications.

### References

[1] Prosperetti, A. & Tryggvason, G., ed. (2009), Computational Methods for Multiphase Flow, Cambridge University Press, Cambridge; New York.

[2] Tryggvason, G.; Scardovelli, R. & Zaleski, S. (2011), Direct Numerical Simulations of Gas-Liquid Multiphase Flows, Cambridge University Press, Cambridge; New York.

[3] Tryggvason, G.; Bunner, B.; Esmaeeli, A.; Juric, D.; Al-Rawahi, N.; Tauber, W.; Han, J.; Nas, S. & Jan, Y. J. (2001), A Front-Tracking Method for the Computations of Multiphase Flow, Journal of Computational Physics 169(2), 708–759.

[4] Olsson, E. & Kreiss, G. (2005), A conservative level set method for two phase flow, Journal of Computational Physics 210(1), 225–246.

[5] Olsson, E.; Kreiss, G. & Zahedi, S. (2007), A conservative level set method for two phase flow II, Journal of Computational Physics 225(1), 785–807.

[6] Zahedi, S.; Gustavsson, K. & Kreiss, G. (2009), A conservative level set method for contact line dynamics, Journal of Computational Physics 228(17), 6361–6375.

[7] Hassanizadeh, M. & Gray, W. G. (1979), General conservation equations for multi-phase systems: 1. Averaging procedure, Advances in Water Resources 2, 131–144.

[8] Hassanizadeh, M. & Gray, W. G. (1979), General conservation equations for multi-phase systems: 2. Mass, momenta, energy, and entropy equations, Advances in Water Resources 2, 191–203.

[9] Hassanizadeh, M. & Gray, W. G. (1980), General conservation equations for multi-phase systems: 3. Constitutive theory for porous media flow, Advances in Water Resources 3(1), 25–40.

[10] Echebarria, B.; Folch, R.; Karma, A. & Plapp, M. (2004), Quantitative phase-field model of alloy solidification, Physical Review E 70(6), 061604.

[11] Folch, R. & Plapp, M. (2005), Quantitative phase-field modeling of two-phase growth, Physical Review E 72(1), 011602.

[12] Plapp, M. (2011), Unified derivation of phase-field models for alloy solidification from a grand-potential functional, Physical Review E 84(3), 031601.

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[14] Nestler, B.; Garcke, H. & Stinner, B. (2005), Multicomponent alloy solidification: Phase-field modeling and simulations, Physical Review E 71(4), 041609.

[15] Janssens, K. G. F. (2007), Computational Materials Engineering: An Introduction to Microstructure Evolution, Academic Press, Amsterdam; Boston.

[16] Chaudhri, A.; Bell, J. B.; Garcia, A. L. & Donev, A. (2014), Modeling multiphase flow using fluctuating hydrodynamics, Physical Review E 90(3), 033014.

[17] Monaghan, J. J. & Kocharyan, A. (1995), SPH simulation of multi-phase flow, Computer Physics Communications 87(1), 225–235.

[18] Monaghan, J. J. & Rafiee, A. (2012), A simple SPH algorithm for multi-fluid flow with high density ratios, International Journal for Numerical Methods in Fluids 71(5), 537–561.

[19] Morris, J. P. (2000), Simulating surface tension with smoothed particle hydrodynamics, International Journal for Numerical Methods in Fluids 33(3), 333–353.

[20] Espanol, P. & Revenga, M. (2003), Smoothed dissipative particle dynamics, Physical Review E 67(2), 026705.

[21] Vazquez-Quesada, A.; Ellero, M. & Espanol, P. (2009), Consistent scaling of thermal fluctuations in smoothed dissipative particle dynamics, The Journal of Chemical Physics 130(3), 034901.

[22] Hu, X. Y. & Adams, N. A. (2006), A multi-phase SPH method for macroscopic and mesoscopic flows, Journal of Computational Physics 213(2), 844–861.

[23] Hu, X. Y. & Adams, N. A. (2007), An incompressible multi-phase SPH method, Journal of Computational Physics 227(1), 264–278.

[24] Krueger, T.; Kusumaatmaja, H.; Kuzmin, A.; Shardt, O.; Silva, G. & Viggen, E. M. (2017), The Lattice Boltzmann Method: Principles and Practice, Springer International Publishing.

[25] Duenweg, B. & Ladd, A. J. C. (2009), Lattice Boltzmann Simulations of Soft Matter Systems, in Advanced Computer Simulation Approaches for Soft Matter Sciences III, Springer, Berlin, Heidelberg, , pp. 89–166.

[26] Shan, X. & Chen, H. (1994), Simulation of nonideal gases and liquid-gas phase transitions by the lattice Boltzmann equation, Physical Review E 49(4), 2941–2948.

[27] Swift, M. R.; Osborn, W. R. & Yeomans, J. M. (1995), Lattice Boltzmann Simulation of Nonideal Fluids, Physical Review Letters 75(5), 830–833.

[28] Swift, M. R.; Orlandini, E.; Osborn, W. R. & Yeomans, J. M. (1996), Lattice Boltzmann simulations of liquid-gas and binary fluid systems, Physical Review E 54(5), 5041–5052.

[29] Sbragaglia, M.; Benzi, R.; Biferale, L.; Succi, S.; Sugiyama, K. & Toschi, F. (2007), Generalized lattice Boltzmann method with multirange pseudopotential, Physical Review E 75(2), 026702.

[30] Krueger, T.; Frijters, S.; Guenther, F.; Kaoui, B. & Harting, J. (2013), Numerical simulations of complex fluid-fluid interface dynamics, The European Physical Journal Special Topics 222(1), 177–198.

[31] Thampi, S. P.; Pagonabarraga, I. & Adhikari, R. (2011), Lattice-Boltzmann-Langevin simulations of binary mixtures, Physical Review E 84(4), 046709.

[32] Kendon, V. M.; Cates, M. E.; Pagonabarraga, I.; Desplat, J.-C. & Bladon, P. (2001), Inertial effects in three-dimensional spinodal decomposition of a symmetric binary fluid mixture: a lattice Boltzmann study, Journal of Fluid Mechanics 440, 147–203.

[33] Stratford, K.; Adhikari, R.; Pagonabarraga, I.; Desplat, J.-C. & Cates, M. E. (2005), Colloidal Jamming at Interfaces: A Route to Fluid-Bicontinuous Gels, Science 309(5744), 2198–2201.

[34] Jansen, F. & Harting, J. (2011), From bijels to Pickering emulsions: A lattice Boltzmann study, Physical Review E 83(4), 046707.

[35] Michele, L. D.; Fiocco, D.; Varrato, F.; Sastry, S.; Eiser, E. & Foffi, G. (2014), Aggregation dynamics, structure, and mechanical properties of bigels, Soft Matter 10(20), 3633–3648.

[36] Asmolov, E. S.; Schmieschek, S.; Harting, J. & Vinogradova, O. I. (2013), Flow past superhydrophobic surfaces with cosine variation in local slip length, Physical Review E 87(2), 023005.

[37] Cates, M. E.; Desplat, J.-C.; Stansell, P.; Wagner, A. J.; Stratford, K.; Adhikari, R. & Pagonabarraga, I. (2005), Physical and computational scaling issues in lattice Boltzmann simulations of binary fluid mixtures, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 363(1833), 1917–1935.

[38] Schmieschek, S.; Shamardin, L.; Frijters, S.; Krueger, T.; Schiller, U. D.; Harting, J. & Coveney, P. V. (2017), LB3D: A parallel implementation of the Lattice-Boltzmann method for simulation of interacting amphiphilic fluids, Computer Physics Communications 217, 149–161.

[39] Zahedi, S.; Kronbichler, M. & Kreiss, G. (2011), Spurious currents in finite element based level set methods for two-phase flow, International Journal for Numerical Methods in Fluids 69(9), 1433–1456.

[40] Shan, X. (2006), Analysis and reduction of the spurious current in a class of multiphase lattice Boltzmann models, Physical Review E 73(4), 047701.

[41] Lee, T. & Fischer, P. F. (2006), Eliminating parasitic currents in the lattice Boltzmann equation method for nonideal gases, Physical Review E 74(4), 046709.

[42] Pooley, C. M. & Furtado, K. (2008), Eliminating spurious velocities in the free-energy lattice Boltzmann method, Physical Review E 77(4), 046702.