Tue April 17th 2018
16:00 – 17:00
Seminar Modeling and simulation of elasto-capillary fluid-solid interaction
Harald van Brummelen


Binary fluids are fluids that comprise two constituents, viz. two phases of the same fluid (gas or liquid) or two distinct species (e.g. water and air). A distinctive feature of binary-fluids is the presence of a fluid–fluid interface that separates the two components. This interface generally carries surface energy and accordingly it introduces capillary forces. The interaction of a binary-fluid with a deformable solid engenders a variety of intricate physical phenomena, collectively referred to as elasto-capillarity. The solid–fluid interface also carries surface energy and, generally, this surface energy is distinct for the two components of the binary fluid. Consequently, the binary-fluid–solid problem will exhibit wetting behavior [1,2]. Moreover, the
fluid-solid surface tension plays an essential role in the elastic deformations in the vicinity of the contact line [3,4]. Elasto-capillarity underlies miscellaneous complex physical phenomena such as durotaxis [5], i.e. seemingly spontaneous migration of liquid droplets on solid substrates with an elasticity gradient; capillary origami [6], i.e. large-scale solid deformations by capillary forces. Binary-fluid–solid interaction is moreover of fundamental technological relevance in a wide variety of high-tech industrial applications, such as inkjet printing and additive manufacturing. As opposed to the significant progress that has been made in the past two decades in modeling and simulation of conventional FSI problems, modeling and simulation of elasto-capillary FSI is still in its infancy, and many aspects are still open, even including fundamental modeling aspects such as the Shuttleworth effect [7].

In this presentation, I will present a computational model for elasto-capillary fluid-solid interaction based on a diffuse-interface model for the binary fluid and a hyperelastic-material model for the solid. The diffuse-interface binary-fluid model is described by the incompressible Navier–Stokes–Cahn–Hilliard equations [8] with preferential-wetting boundary conditions at the fluid-solid interface. To resolve the fluid-fluid interface and the localized displacements in the solid, we apply adaptive hierarchical spline approximations. A monolithic solution scheme is applied to enable robust solution of the coupled FSI problem. I will consider several aspects of the formulation and of the simulation techniques. Notably, I will address aspects
of the thermodynamic substructure of elasto-capillary FSI, and of appropriate (weak) formulations.

[1 ] P.-G. de Gennes, F. Brochard-Wyart, and D. Quer ́e. Capillarity and Wetting Phe- nomena: Drops, Bubbles, Pearls, Waves. Springer, 2004.

[2] P.G. de Gennes. Wetting: statics and dynamics. Rev. Mod. Phys., 57:827–863, 1985.

[3] E.H. van Brummelen, H. Shokrpour Roudbari, and G.J. van Zwieten. Elasto- capillarity simulations based on the Navier-Stokes-Cahn-Hilliard equations.
In Advances in Computational Fluid-Structure Interaction and Flow Simulation, Modeling and Simulation in Science, Engineering and Technology, pages 451–462. Birkhauser, 2016.

[4] E.H. van Brummelen, M. Shokrpour Roudbari, G. Simsek, and K.G. van der Zee. Fluid Structure Interaction, volume 20 of Radon Series on Computational and Applied Mathematics,
chapter Binary-fluid–solid interaction based on the Navier–Stokes– Cahn–Hilliard Equations. De Gruyter, 2017.

[5] R.W. Style et al. Patterning droplets with durotaxis. PNAS, 110(31):12541–12544, 2013.

[6] C. Py, P. Reverdy, L. Doppler, J. Bico, B. Roman, and C.N. Baroud. Capillary origami: Spontaneous wrapping of a droplet with an elastic sheet. Phys. Rev. Lett., 98:156103, Apr 2007.

[7] J.H. Snoeijer, E. Rolley, and B. Andreotti, The paradox of contact angle selection on stretched soft solids, arXiv:1803.04428 [cond-mat.soft] (2018).

[8] M. Shokrpour Roudbari, G. Simsek, E.H. van Brummelen, and K.G. van der Zee, Diffuse-interface two-phase flow models with different densities: a new quasi-incompressible form
and a linear energy-stable method, Math. Mod. Meth. Appl. Sci. (accepted) (2018).
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The 10th Complex Motion in Fluids 2021
Max Planck Gesellschaft
Centre for Scientific Computing