Research Group of Prof. Dr. M. Griebel
Institute for Numerical Simulation

Numerical Simulation of Dynamic Wetting Processes


Prof. Dr. Jens Frehse
Prof. Dr. Michael Griebel
Prof. Dr. Angela Kunoth
Dr. Marcel Arndt
Dr. Roberto Croce
Dr. Marc Alexander Schweitzer


In this project we are concerned with the efficient numerical simulation of wetting processes in three space dimensions on a parallel computer. To this end, we need to consider two immiscible flow phases separated by a moving interface which in our implementation is described by a level-set approach. Furthermore, surface tension plays an important role. The simulation of wetting processes such as curtain or dip coating is further complicated by the dynamic contact angle problem (DCAP), where the free surface as well as the fluid and the solid interact at a triple point/line (contact point/line). The main reason why the DCAP is a challenging problem is the fact that it involves a non-integrable singularity if the classical no-slip boundary conditions are employed at the solid boundaries together with the classical condition of zero tangential stresses at the free liquid surface. The shear stress as well as the pressure at the triple-point become infinite. Therefore, many models replace the no-slip boundary conditions by various slip boundary conditions based on physical considerations to avoid this effect. Since there are arbitrarily many slip conditions which eliminate the singularity, it is not clear which one is appropriate and gives reliable results. Besides, this approach oftentimes does not provide a direct relation between the dynamic contact angle and the bulk velocity. Therefore, this interplay still has to be pre-defined in most of these models. Usually the contact angle is taken as a function of the bulk velocity and several material parameters of the interacting media and is then calibrated with empirical data from measurements. Special cases are for example Tanner's law and Hoffman's Master-curve.

Recently, Yulii D. Shikhmurzaev developed a model which avoids the aforementioned problems. This model considers the contact line problem from a thermodynamic perspective. Roughly speaking, the idea is that fluid particles on the free surface traverse the contact line in finite time due to the true kinematics of the flow. Therefore the properties of the surface have to relax to new equilibrium values giving rise to surface tension gradients in the immediate neighborhood of the advancing contact line. Further, the deviation between the dynamic contact angle and the static contact angle together with the tangential momentum balance law (the classical Young equation) also imply that the surface tension along the free surface and the solid boundaries are not in equilibrium in the vicinity of the moving contact line. Hence, a local surface tension relaxation scheme is provided by the Shikhmurzaev model along the liquid/liquid and liquid/solid interfaces in the vicinity of the contact point/line. An important property of this model is that the values of the dynamic contact angles are not input parameters for the problem, but are part of the solution.

Our major research task is the development of appropiate discretization techniques for the complete Shikhmurzaev model.


We consider the moving contact line problem for a liquid-liquid-solid system. In other words, two immiscible viscous liquids are in contact with a solid surface and one of the liquids displaces the other so that the contact line is constrained to move across the solid surface.

These liquid-liquid-solid contact lines have an essential affect on processes as wetting and dewetting. In particular, the influence of surface tension on the contact line behaviour is shown in the following Figures which respect a viscous drop - with and without surface tension - on an inclined surface.

1) First example of drop sliding due to gravity. It shows the dynamic contact angle behaviour of a viscous drop with surface tension on an inclined surface. Depicted are snapshots of the free surface at different time steps.

2) Second example of drop sliding due to gravity. It shows the dynamic contact angle behaviour of the same viscous drop problem as in the first example, but now without surface tension. Depicted are snapshots of the free surface at the same time steps as in the first example.

3) Simulation of a dynamic curtain coating process. The coating viscous fluid contains surface tension and the substrate (bottom) is moving with constant velocity in x-direction. Top: global view in chronological order from left to right. Bottom: back view of the curtain coating showing 3D effects near the contact-line. Depicted are snapshots of the free surface at different time steps.


[1] R. Croce, Ein paralleler, dreidimensionaler Navier-Stokes-Löser für inkompressible Zweiphasenströmungen mit Oberflächenspannung, Hindernissen und dynamischen Kontaktflächen. Diplomarbeit, Institut für Angewandte Mathematik, Universität Bonn, 2002.
[2] M. Griebel, T. Dornseifer, T. Neunhoeffer, Numerical Simulation in Fluid Dynamics, a Practical Introduction, SIAM, Philadelphia, 1998.
[3] Y.D. Shikhmurzaev, Moving Contact Lines in Liquid/Liquid/Solid Systems, J. Fluid Mech., 334 (1997), pp. 211-249.

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