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## Multiscale Modeling of Simple and Complex Liquid Flow Using Particle-Continuum Hybrids

#### ZCAM, Zaragoza, Spain

#### Organisers

Particle and continuum hydrodynamics are intimately connected for several reasons. First, rheological properties of liquids are ultimately determined by the structure of their constituent molecules, and second, many liquids contain dispersed particles (e.g. pollutants) or large molecules (polymers, colloids) in suspension, which can substantially alter their dynamical behaviour. Another important aspect of this general question is how interactions at molecular scale may affect the fluid boundary conditions, at walls, soft interfaces etc... The range of fundamental and applied problems requiring some description of the interaction between particle and continuum fluid dynamics is huge and covers many time and length scales: from molecular dynamics (MD) research (e.g. the effect of flow on polymeric chains at low Reynolds number), to computational fluid dynamics (CFD) macroscopic problems (e.g. dispersion of pollutants in turbulent flows), including Non-Newtonian hydrodynamics (polymeric liquids), fluid-structure interaction (relevant for many biological applications), and so forth. Due to the large scope in the applied field, recent advances in computational methods for particle-continuum coupling are being developed by quite different communities whose work and ideas remain, quite often, isolated from each other. Albeit, an eagle-eye inspection on the literature reveals that most of these techniques share fundamental aspects and face similar challenges. On the other hand, due to the fast and independent growth of these different communities, there is a growing need to connect researchers working in theoretical modeling, applied science and software/hardware development.

Averaged particle effects or molecular information of the liquid bulk and of its physical boundaries have been traditionally included into the (not-closed) set of hydrodynamic equations via constitutive relations and phenomenological boundary conditions. In a sequential multiscale research program these relations are obtained from independent molecular simulations. However, the sequential approach fails whenever the particles and fluid structure are dynamically coupled due to their mutual interactions. This workshop focuses on multiscale computational methods which concurrently combine different techniques to solve particle and fluid dynamics. In this sense these type of methods are usually called hybrids.

Particle-continuum hybrids can be divided in three classes.

A: Domain decomposition. Some problems require to resolve a certain domain of interest at molecular detail and connect its dynamics with a hydrodynamic description of the surrounding flow field. These techniques are particularly suited at non-trivial or complex boundary conditions, which might contain singularities, interfaces or even macromolecules. Among the different techniques, one can mention , state coupling, flux coupling, Schwartz method or density control [1,2,3,4].

B: Constitutive modeling based on molecular simulations. A way to overcome the need of constitutive relations is to obtain the time evolution of the pressure tensor at each Eulerian node from a set of independent microscopic (molecular or stochastic) non-equilibrium simulations with the local velocity gradient thereby imposed. Since the pioneer work of Laso and Ottinger [5], there have been several important contributions, setting the general mathematical framework [6,7] and applications in scale-bridging molecular and macroscopic scales of polymeric liquid flow [8].

C: Particles in flow. There are several computational methods to solve the hydrodynamic interactions between solute (particles) and solvent (fluid) flow. Typical problems are the dynamics of colloidal or polymeric suspensions or larger particles dispersed in a liquid. Eulerian-Lagrangian hybrids solve the fluid component in an Eulerian mesh, while the particle dynamics are described in a Lagrangian fashion (continuum space) [9]. The same idea is applied in hybrids based on lattice Boltzmann (LB) and molecular dynamics [10]; the flexibility of the LB method makes feasible generalizations to allow for simulations of charged fluid flow [11], or colloids in binary fluids [11b]. Interpolation techniques are required to distribute the force between the particle and the fluid. Details of this force are highly dependent on the flow parameters (Reynolds number, Stokes number, particle size). In small devices it is usually approximated by the Stokes friction force [9-11] but it may demand more involved schemes inspired on the Immersed Boundary Method [12] for fluid-structure interaction. For instance, the direct forcing approach derives the force imposing the no-slip boundary condition at the particle surface [13,14]. Finally, fully Lagrangian (partlcle-particle) methods like Smooth Particle Hydrodynamics [16] or Multiparticle Collision Dynamics treat similar problems, with the benefit of simplifying the treatment of complex boundary conditions [15]. Albeit, comparisons among different schemes (benchmarks) and ranges of applicability are lacking in the literature.

## References

**Spain**

Rafael Delgado-Buscalioni ( Universidad Autonoma de Madrid ) - Organiser

**Switzerland**

Ignacio Pagonabarraga ( CECAM ) - Organiser