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.
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.
|||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.|
|||M. Griebel, T. Dornseifer, T. Neunhoeffer, Numerical Simulation in Fluid Dynamics, a Practical Introduction, SIAM, Philadelphia, 1998.|
|||Y.D. Shikhmurzaev, Moving Contact Lines in Liquid/Liquid/Solid Systems, J. Fluid Mech., 334 (1997), pp. 211-249.|
The SFB-project C3 is related to the SFB-projects B2, C1, C2, and C4. The connection to B2 is due to the considered applications. The focus of B2, however, is more on the analytical side of the DCAP. Furthermore, there are connections to C4 where interfaces between two fluids are considered on the micro- and macro-scale. Next, the computational bottleneck of pressure correction schemes is the solution of the Poisson equation for the pressure. In multi-phase flows this is further complicated due to the discontinuity of the density, i.e., the discontinuity of the diffusion coefficient. Hence, we need to be concerned with robust multilevel solvers to overcome this challenge. This is just one aspect where we will interact with C2. Other important facets of our cooperation with C2 are aimed at the self-adaptive refinement control and parallel load balancing techniques.
Iterative solution techniques for PDE-constrained optimization problems with nonlinear partial differential equations as constraints are investigated in C1. With respect to the question how to handle the nonlinearities within the iterative solution of the optimization problem we expect new insights regarding the optimization of wetting processes.