title: Mathematical modeling and Volume-of-Fluid based simulation of dynamic wetting
Creators: name: Fricke, Mathis
Publisher: name: UNSPECIFIED
Year: 2021

description: Dynamic wetting phenomena are omnipresent in nature and technology. The legs of the water-strider make use of a sophisticated hierarchical surface structure to achieve superhydrophobicity and to allow the insect to stand and run easily on a water surface. The ability to understand and control dynamic wetting processes is crucial for a variety of industrial and technical processes such as inkjet- or bioprinting or mass transport in microfluidic devices. On the other hand, the moving contact line problem, even in a largely simplified setting, still poses considerable challenges regarding fundamental mathematical modeling as well as numerical methods. The present work addresses both the fundamental modeling and the development of numerical methods based on the geometrical Volume-of-Fluid (VOF) method. The spreading dynamics of droplets on a bare silicon wafer and on a silicon wafer coated with a polymer brush is studied in cooperation with experimentalists within the Collaborative Research Center (CRC) 1194. An ordinary differential equation describing the spreading dynamics of spherical drops is derived and compared with experimental results. The model is a generalization of a classical model for perfectly wetting drops to the case of partial wetting. Besides these simplified modeling approaches, the main focus of the present work lies on the continuum mechanical description of dynamic wetting. The moving contact line singularity in the classical hydrodynamic description based on the no-slip boundary condition motivated a lot of research in the past 50 years, aiming at a physically sound model. It has been shown that the Navier slip condition combined with a fixed contact angle leads to a so-called "weak singularity" and it was suspected by Ren and E that the solution may become completely regular for Navier slip combined with a dynamic contact angle (Phys. Fluids, 2007). The central mathematical tool developed in the present work allows to prove that the latter conjecture is false (as long as the slip length is finite). The basic idea is to study the kinematics of wetting in the sharp-interface sharp-contact-line setting independently from the specific continuum mechanical model. An evolution equation for the dynamic contact angle is derived and proven rigorously, assuming that a sufficiently regular velocity field is given on the moving hypersurface with boundary. Thanks to this very general setting, the result is applicable to a large class of continuum mechanical models including the mechanisms of mass transfer between the phases or mass transfer to a surface phase like in the Interface Formation Model. The kinematic result is applied to regular solutions of the "standard model" of dynamic wetting based on the Navier slip condition. It is shown that the system cannot relax to the equilibrium state with a regular solution. Hence, it is concluded that physically sound solutions in the standard model cannot be regular. Moreover, regular solutions to generalizations of the standard model are studied. In particular, it is shown that surface tension gradients at the contact line may give rise to regular solutions. Furthermore, the compatibility of the boundary conditions at the contact line is studied for the standard model and an adaptation of the Interface Formation Model proposed recently by Lukyanov and Pryer (Langmuir, 2017). It is shown that, depending on the model parameters, the boundary conditions in the model by Lukyanov and Pryer may be compatible at the contact line. In this case, one can even compute explicit expressions for the curvature and the pressure at the moving contact line. The second part of the present thesis is devoted to numerical methods for dynamic wetting. In order to make the kinematic results easily accessible, an open-source demonstrator code based on a level set representation of the interface is developed and published in an open repository. The current state-of-the-art methods for dynamic wetting based on the geometrical Volume-of-Fluid approach are briefly reviewed. In particular, it is shown that the method to enforce the dynamic contact angle proposed by Afkhami and Bussmann (Int. J. Numer. Methods Fluids, 2008) delivers inconsistent values for the curvature at the contact line. Motivated by the fundamental results on the kinematics of moving contact lines, novel interface reconstruction methods are developed and implemented that allow to reconstruct the free surface close to the domain boundary. In particular, the Boundary ELVIRA method delivers an accurate numerical transport of the contact angle that is consistent with the fundamental kinematics. The latter method greatly improves the accuracy of the VOF method in the presence of contact lines. Moreover, the numerical approximation of the mean curvature based on the height function technique is studied thoroughly. A rigorous error analysis for the two-dimensional height function method in the presence of data errors is given. In particular, the discrete error amplification is estimated and studied in detail. The latter type of error is rarely discussed in the scientific literature on the topic. But in fact, the impact of the discrete error amplification on the total error can be significant, in particular when disturbances due to transport errors are present in the volume fraction data. The kinematic evolution equation for the mean curvature, which is derived in the first part of this work, serves as a reference solution to validate the numerical transport of the curvature at the contact line. As can be expected from the height function error analysis, the transport error for the curvature is found to be first-order divergent for the Boundary Youngs method and constant for the Boundary ELVIRA method. The latter results clearly show the need for higher-order interface advection methods. The third part of this work closely investigates two particular wetting flow configurations, namely the capillary rise and the breakup of a liquid bridge on a chemically structured surface. A novel numerical benchmark for wetting flows based on the capillary rise is established with four numerical methods developed within the CRC 1194 at TU Darmstadt. Moreover, a novel adaptation of the Navier slip condition called "staggered slip" is introduced. The goal of the staggered slip condition is to reduce the "numerical slip" inherent to the method. This is achieved by defining the slip length with respect to a virtual boundary that is located in between the physical boundary and location of the face-centered velocity used to transport the volume fraction field. As a result, the discrete viscous dissipation is increased compared to the standard Navier slip implementation. It is shown that the convergence for the capillary rise can be significantly improved if the slip length is not yet resolved. On the other hand, the order of convergence is reduced compared to the standard implementation for a single-phase channel flow example. The wetting of structured surfaces is studied in joint work with experimentalists in the CRC 1194. The goal is to quantitatively describe the breakup dynamics of a wetting capillary bridge on a structured surface. A major problem for the interpretation of both the experimental and the numerical data arises from the uncertainty in the precise time of pinch-off of the capillary bridge. In order to solve this problem, we introduce a systematic way to analyze the data without the need to determine the pinch-off time. The basic idea, which has been applied before by Li Sprittles (J. Fluid Mech., 2016), is to plot the speed of the breakup process (i.e. the time derivative of the minimum diameter) as a function of the minimum diameter itself. This procedure is well-defined since the minimum diameter is strictly decreasing with time. Indeed, we show that the transformation that maps from the standard representation to the phase space representation is invertible up to a shift in absolute time. With this technique, we are able to study the breakup process in great detail in both the three-dimensional VOF simulation and the experiment. In general, a good agreement is found between experiment and simulation both qualitatively and quantitatively in terms of the time evolution of the minimum diameter. The numerical simulations allowed to identify different regimes in the breakup dynamics that were also found in the experimental data. Remarkably, dynamic surface tension may play a significant role in the breakup dynamics. The agreement between simulation and experiment close to the breakup can be improved by increasing the value of the (constant) surface tension to 90 mN/m. The latter value has been proposed by Hauner et al. (J. Phys. Chem. Lett., 2017) for a freshly created water surface. Moreover, the local rate of interface generation is found to be quite high. However, a fully quantitative assessment of this phenomenon can only be achieved in future work. Finally, we revisit the capillary rise problem in the case of structured surfaces. Interestingly, the surface pattern can be used to construct an energy functional with two stable configurations. The resulting dynamic rise behavior is quite complex with a bifurcation between the two stable configurations at a critical initial rise height.