Transport in Porous Media

, Volume 122, Issue 2, pp 253–277 | Cite as

A Two-Phase SPH Model for Dynamic Contact Angles Including Fluid–Solid Interactions at the Contact Line

  • P. KunzEmail author
  • S. M. Hassanizadeh
  • U. Nieken


The description of wetting phenomena on the continuum scale is a challenging problem, since intermolecular interactions, like van der Waals forces between liquid and solid, alter the flow field at the contact line. Recently, these effects were included in the smoothed particle hydrodynamics method by introducing a contact line force (CLF) on the continuum scale. This physically based contact line force model is employed here to simulate two-phase flow in a wide range of wetting dynamics parametrized by capillary number. In particular, dynamic contact angles at various capillary number values are calculated by CLF model and compared to measured values. We find that there is significant disagreement between simulated and measured results, specially at low wetting speeds. It is indeed expected that most of the driving force is dissipated to overcome strong liquid–solid interactions, which are not adequately accounted for in the existing CLF model. Therefore, we have extended that model to account for stick-slip (SSL) behavior of the contact line caused by solid–fluid interactions. The new SSL model results in dynamic contact angle values that are in good agreement with experimental data for the full range of wetting dynamics.


Dynamic contact angle Contact line force Two-phase flow Stick-slip behavior Smoothed particle hydrodynamics (SPH) 

List of symbols


Arbitrary field variable


Transition function for continuous surface force at solid boundary


Color function


Model parameter (capillary number at transition point)


Distance between walls/tape in simulation domain, L

\(\mathbf {d}\)

Distance to solid wall, L


Energy, L\(^2\)MT\(^{-2}\)

\(\mathbf {f}_\mathrm{wn}\)

Force per unit area acting on fluid–fluid interface, M L\(^{-1}\) T\(^{-2}\)

\(\mathbf {f}_\mathrm{wns}\)

Force per unit line acting at contact line, M T\(^{-2}\)

\(\mathbf {F}^\mathrm{VOL}\)

Force per unit volume, M L\(^{-2}\) T\(^{-2}\)

\(\mathbf {g}\)

Gravitational field strength, L T\(^{-2}\)


Smoothing length of kernel function, L


PI controller parameter


Domain length, L

\(\mathbf {L}\)

Kernel correction matrix


Mass, M

\(\hat{\mathbf {n}}\)

Unit normal vector


Pressure, M L\(^{-1}\) T\(^{-2}\)

\(\mathbf {r}\)

Distance, L


Shepard correction for kernel function


Time, T

\(\mathbf {v}\)

Velocity, L T\(^{-1}\)


Volume, L\(^3\)


Kernel function, L\(^{-3}\)

\(\mathbf {x}\)

Position, L

\(\alpha \)

Model parameter (fractional viscosity increase at the contact line)

\(\beta \)

Dimensionless relation of wall distances

\(\delta _\mathrm{wn}\)

Dirac delta distribution and volume reformulation at fluid–fluid interface, L\(^{-1}\)

\(\delta _\mathrm{wns}\)

Dirac delta distribution and volume reformulation at contact line, L\(^{-2}\)

\(\Delta x\)

Initial particle spacing, L

\(\zeta \)

Multiplicative factor for volume reformulation at contact line

\(\theta \)

Contact angle

\(\kappa \)

Curvature, L\(^{-1}\)

\(\mu \)

Dynamic viscosity, M L\(^{-1}\) T\(^{-1}\)

\(\hat{\mathbf {\nu }}\)

Unit vector

\(\rho \)

Density, M L\(^{-3}\)

\(\sigma \)

Surface tension, M T\(^{-2}\)



This work was funded by the German Research Foundation (DFG) within the framework of the International Research Training Group “NUPUS, Non-linearities and upscaling in porous media” (IRTG 1398) and the research unit “Multiskalen-Analyse komplexer Dreiphasensysteme” (FOR 2397). The second author would like to thank European Research Council (ERC) for the support received under the ERC Advanced Grant Agreement No. 341225. The constructive comments of three anonymous reviewers helped to improve the manuscript significantly.


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Authors and Affiliations

  1. 1.Institute of Chemical Process EngineeringUniversity of StuttgartStuttgartGermany
  2. 2.Department of Earth Sciences, Faculty of GeosciencesUtrecht UniversityUtrechtThe Netherlands

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