**Analysis and Numerics of sharp and diffuse interface models for droplet dynamics**

## Team

### Prof. Dr. rer. nat. habil. Christian Rohde

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## Description

For the numerical simulation of droplet dynamic processes the liquid phase and the surrounding gaseous fluid have been previously modeled incompressible. But if we consider the interaction of droplets with high-speed flows, droplets near the critical point or even evaporation processes of highly volatile substances, compressible effects can no longer be neglected. In this subproject fully compressible models for the dynamics of free single droplets and droplet ensembles are developed and studied analytically and numerically. The sharp interface approach (SI) and the diffuse interface approach (DI) are considered comparatively.

The SI-approach describes the dynamics in time dependent regions by the hydrodynamic equations for the two fluids. These equations are coupled with jump conditions, so that an overall free boundary value problem is set. For the numerical approximation algorithms are developed, which, in particular, take into account the jumps in the equations and the curvature of the surface. These methods are incorporated in a ghost fluid approach in the code FS3D of the SFB-TRR 75 for solving the SI model numerically.

The SI-approach is invalid when the topology changes – e.g. in case of drop division – and questionable as modelling approach in the transition to supercritical states. Alternatively in phase transition problems phase field models (DI-models) are used very succesfully. Within the DI-models the location of the interface is not exactly given by a jump in the state variables, but smeared out over a small area, such that a system of evolution equations is valid in the whole area. The thickness of the boundary layer is controlled by a regularization parameter. In this subproject a fully compressible DI-model for any state equations will be developed and validated with tools of asymptotic analysis. The focus is on the numerical solution of the DI-model in several dimensions. Therefore an hp-adaptive discontinuous Galerkin method is developed on unstructured grids and implemented. The numerical method is used for the simulation of droplet experiments.

## Publications

#### 2021

M. Alkämper, J. Magiera, and C. Rhode:*An Interface Presenting Moving Mesh in Multiple Space Dimensions*

CoRR, Preprint

doi: 10.48550/arXiv.2112.11956

J. Magiera*A Molecular–Continuum Multiscale Solver for Liquid–Vapor Flow*

Small Collaboration: Advanced Numerical Methods for Nonlinear Hyperbolic Balance Laws and Their Applications (hybrid meeting).

Vol. 41. Oberwolfach Rep. Abstracts from the mini-workshop held 29 Aug – 4 Sep 2021,

Organized by Song Jiang, Jiequan Li, Maria Lukacova, Gerald Warnecke. 2021.

doi: 10.14760/OWR-2021-41. url: http://publications.mfo.de/handle/mfo/3891.

J. Magiera:*A Molecular–Continuum Multiscale Solver for Liquid–Vapor Flow: Modeling and Numerical Simulation*

Ph.D. Thesis. University of Stuttgart, 2021.

J. Magiera:*Data Sets for ’A Molecular–Continuum Multiscale Solver for Liquid–Vapor Flow: Modeling and Numerical Simulation’*

DaRUS. Version V1. DaRUS, 2021.

doi: 10.18419/darus-1258.

#### 2020

T. Hitz, J. Keim, C.-D. Munz, and C. Rohde:*A parabolic relaxation model for the Navier-Stokes-Korteweg equations*

J. Comput. Phys. 421 (2020), pp. 109714, 24.

doi: 10.1016/j.jcp.2020.109714.

J. Magiera, D. Ray, J. S. Hesthaven, and C. Rohde:*Constraint-aware neural networks for Riemann problems*

J. Comput. Phys. 409 (2020), pp. 109345, 27.

doi: 10.1016/j.jcp.2020.109345.

#### 2019

R. M. Colombo, P. G. LeFloch, C. Rohde, and K. Trivisa:*Nonlinear Hyperbolic Problems: Modeling, Analysis, and Numerics*

Oberwohlfach Rep. 16 (2019). Abstracts from the workshop held June 19–25, 2016,

Organized by Rinaldo M. Colombo, Philippe G. LeFloch and Christian Rohde, pp. 1419–1497.

V. Sharanya, G. R. Sekhar, and C. Rohde:*Surfactant-induced migration of a spherical droplet in non-isothermal Stokes flow*

Phys. Fluids 31, 012110 (2019).

doi: https://doi.org/10.1063/1.5064694

#### 2018

C. Chalons, J. Magiera, C. Rohde, and M. Wiebe:*A finite-volume tracking scheme for two-phase compressible flow*

Theory, numerics and applications of hyperbolic problems. I. Vol. 236. Springer Proc. Math. Stat. Springer, Cham, 2018, pp. 309–322.

doi: 10.1007/978-3-319-91545-6\_2.

S. Fechter, C.-D. Munz, C. Rohde, and C. Zeiler:*Approximate Riemann solver for compressible liquid vapor flow with phase transition and surface tension*

Comput. & Fluids 169 (2018), pp. 169–185.

doi: 10.1016/j.compfluid.2017.03.026.

J. Magiera and C. Rohde:*A particle-based multiscale solver for compressible liquid–vapor flow*

Theory, numerics and applications of hyperbolic problems. II. Vol. 237. Springer Proc. Math. Stat. Springer, Cham, 2018, pp. 291–304.

doi: 10.1007/978-3-319-91548-7\_23.

G. P. Raja Sekhar, V. Sharanya, and C. Rohde:*Effect of surfactant concentration and interfacial slip on the flow past a viscous drop at low surface Péclet number*

International Journal of Multiphase Flow (2018), pp. 82–103.

C. Rohde:*Fully resolved compressible two-phase flow: modelling, analytical and numerical issues*

New trends and results in mathematical description of fluid flows. Nečas Center Ser. Birkhäuser/Springer, Cham, 2018, pp. 115–181.

C. Rohde and C. Zeiler:*On Riemann solvers and kinetic relations for isothermal two-phase flows with surface tension*

Z. Angew. Math. Phys. 69.3 (2018), Paper No. 76, 40.

doi: 10.1007/s00033-018-0958-1.

#### 2017

A. Chertock, P. Degond, and J. Neusser.:*An asymptotic-preserving method for a relaxation of the Navier-Stokes-Korteweg equations*

J. Comput. Phys. 335 (2017), pp. 387–403. doi: 10.1016/j.jcp.2017.01.030.

S. Fechter, C.-D. Munz, C. Rohde, and C. Zeiler.:*A sharp interface method for compressible liquid-vapor flow with phase transition and surface tension*

J. Comput. Phys. 336 (2017), pp. 347–374.

doi: 10.1016/j.jcp.2017.02.001.

Magiera, J.; Rohde, C.: *A particle-based multiscale solver for compressible liquid-vapor flow*,

erscheint in: Klingenberg, C.; Westdickenberg, M. (eds.), Hyperbolic Problems. Springer Proc. Math. Stat., Springer, Cham, 2017.

Fechter, S.; Zeiler, C.; Munz, C.-D.; Rohde, C.: *Approximate Riemann solver for compressible liquid vapor flow with phase transition and surface tension*,

erscheint bei Comput. Fluids, http://doi.org/10.1016/j.compfluid.2017.03.026.

#### 2016

Rohde, C., Zeiler, C.: *On Riemann Solvers and Kinetic Relations for Isothermal Two-Phase Flows with Surface Tension*,

eingereicht bei ZAMP, 2016.

B. Kabil, C. Rohde.*Persistence of undercompressive phase boundaries for isothermal Euler equations including configurational forces and surface tension,*

Mathematical Methods in the Applied Sciences, 2016, 39, 5409-5426.

Schleper, V.: *A HLL-type Riemann solver for two-phase flow with surface forces and phase transitions,*

Appl. Numer. Math., 2016, 108, 256-270

Sharanya, V.; Raja Sekhar, G. P. & Rohde, C.: *Bed of polydisperse viscous spherical drops under thermocapillary effects*,

Zeitschrift für angewandte Mathematik und Physik, 2016, 67, 1-17

Dragomirescu, I.; Eisenschmidt, K.; Rohde, C. , Weigand, B.*Perturbation solutions for the finite radially symmetric Stefan problem,*

Inter. J. Thermal Sci., 104, 386-395, 2016.

#### 2015

Rohde, C.; Zeiler, C.: *A relaxation Riemann solver for compressible two-phase flow with phase transition and surface tension*,

Appl. Numer. Math., 95, 267-279, 2015

J. Neusser, C. Rohde, V. Schleper. *Relaxation of the Navier–Stokes–Korteweg equations for compressible two-phase flow with phase transition,*

Int. J. Numer. Meth. Fluids (2015), 79, 615-639.

#### 2014

Fechter, S.; Zeiler, C.; Munz, C.-D.; Rohde, C.: *Simulation of compressible multi-phase flows at extreme ambient conditions using a Discontinuous-Galerkin method*,

ILASS-Europe, 26th European Conference on Liquid Atomization and Spray Systems, 2014.

Engel, P.; Viorel, A.; Rohde, C.*A Low-Order Approximation for Viscous-Capillary Phase Transition Dynamics*,

Port. Math*., * 2014*, 70*, 319-344.

Corli, A.; Rohde, C. , Schleper, V. *Parabolic approximations of diff**usive-dispersive equations,*

J. Math. Anal. Appl., 2014, 414,773-798.

Eymard, R.,Schleper, V. *Study of a numerical scheme for miscible two-phase flow in porous media*,

Numer. Meth. Part. D. E., 2014, 30, 723-748.

Maria Wiebe.

*Ein Sharp-Interface-Ansatz für Phasenübergangsprobleme **(A Sharp-Interface Approach for Phase Transition Problems)*,

Master Thesis, Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, (2014).

Kabil, B.; Rohde, C.: *The influence of surface tension and configurational forces on the stability of liquid-vapor interfaces*,

Nonlinear Analysis-Theor., 107, 63-75, 2014.

**2013**

K. Eisenschmidt, P. Rauschenberger, C. Rohde, B. Weigand. *Modelling of freezing processes in super-cooled droplets on sub-grid scale*,

ILASS-Europe 2013, European Conference on Liquid Atomization and Spray Systems, Chania, Greece, 2013

Fechter, S., Jägle, F., Schleper, V.* Exact and approximate Riemann solvers at phase boundaries*

Computers & Fluids, 2013, 75, 112-126.

**2012**

Chalons, C., Coquel, F., Engel, P., Rohde, C.*Fast relaxation solvers for hyperbolic-elliptic transition problems*

SIAM J. Sci. Comput. 34 (2012), A1753-A1776

Colombo, R. M., Schleper, V.*Two-phase flow: Non-smooth well-posedness and the compressible to incompressible limit*

Nonlinear Anal. Real World Appl. 13 (2012), 2195-2213.

Corli, A., Rohde, C.*Singular limits for a parabolic regularization of scalar conservation laws*

J. Differential Equations 253 (2012), 1399-1421.

Dreyer, W., Giesselmann, J., Kraus, C., Rohde, C.*Asymptotic analysis for Korteweg models*

Interfaces Free Bound. 14 (2012), 105-143.

Schleper, V.*On the coupling of compressible and incompressible fluids*

Numerical Methods for Hyperbolic Equations (eds. E. Vazquez-Cendon, A. Hidalgo, P. Garcia-Navarro, L. Cea), (2012), 8 Seiten, Taylor & Francis.

Fechter, S., Jagle, F.; Schleper, V.,*A multiscale algorithm for compressible liquid-vapor flow with surface tension*

preprint 2012

Jaegle, F., Schleper, V., Rohde, C.*Exact and approximate Riemann solvers at phase boundaries*,

Preprint 2012.

Eymard, R., Schleper, V.*Study of a numerical scheme for miscible two-phase flow*

Numerical Methods for Partial Differential Equations (2012), 22 Seiten. http://hal.archives-ouvertes.fr/hal-00741425

Fechter, S.; Jaegle, F.; Boger, M.; Zeiler, C.; Munz, C.-D., Rohde, C.*A discontinuous Galerkin based multiscale method for compressible multiphase flow*,

Proceedings of the 7th ICCFD, Hawaii, 2012.

**2011**

Corli, A.; Rohde, C.

*Singular limits for a parabolic-elliptic regularization of scalar conservation laws*,

preprint, 2011.

Colombo, R. M.; Schleper, V. *Two-phase flows: Non-Smooth well posedness and the compressible to incompressible limit,*

preprint, 2011.

**2010**

Rohde, C*A local and low-order Navier-Stokes-Korteweg system*

Nonlinear partial differential equations and hyperbolic wave phenomena, 315-337, Contemp. Math., 526, Amer. Math. Soc., (2010)

Kissling, F., Rohde, C.*The computation of nonclassical shock waves with a heterogeneous multiscale method*

Netw. Heterog. Media 5 (2010), 661-674