Numerical methods for free surface flows

As in many areas of science and technology, the use of computational mathematics and numerical algorithms has become an important part of modern coating technology development and other applications involving micro-scale free surface flows such as complex liquid two-phase flow in porous media.

With the more complex product structure and increasing competitive pressures, refinements in coating technology rely more and more on fully understanding the physical mechanisms involved in the process and that will determine its success or failure.  The most critical step of a coating process is the flow in the region where the liquid is deposited on the substrate.  This region is the coating bead.  Its steady states and their stability to disturbances and upsets are crucial.  In the same way, the complete understanding of the two-phase oil-water flow in porous media, emulsion formation during oil production and other multiphase flows are crucial for better design of different processes in the oil industries.

Research on these free surface flows is challenging.  The dimensions of the coating bead and porous space is in the order of a few micrometers and it is usually bounded by liquid-air interfaces (and liquid-liquid interfaces) with large radius of curvature, that make capillary effects very important.

Theoretical analysis demands computer-aided solutions of the governing equations, which are strongly nonlinear due to the capillarity in free surfaces, inertia if it is appreciable, and non-Newtonian behavior, which is often present.  Analysis can be further complicated by elastohydrodynamic effects of compliant confining surfaces.  Stability of steady-state solutions, elastohydrodynamic effects, and the consequences of non-Newtonian liquid behavior are three areas least well examined, much less least well understood in scientific terms.

The goal of our research is to develop computational methods to analyze the steady state and stability of microscale free surface flows bounded by rigid and resilient boundaries.  The equation systems for steady flows, for their stability, and for their transient evolution are solved by Galerkin’s method and extensions of it together with finite element basis functions, Newton’s method, and continuation strategies.  Because of the free surface, the flow domain is unknown a priori. Capillary forces are comparable to viscous forces in coating and pore-scale two-phase flows; fully coupled solution methods – the position of the interface and the flow solution are computed simultaneously – are superior to fixed-point iterative method where the domain shape and the flow are computed alternatively. Fully coupled methods rely on mapping the unknown physical domain into a known reference domain by means of a bijective mapping that connects the physical and reference domains.  The linear stability analysis of the flows leads to a large non-symmetric, generalized eigenvalue problem.  The solution of the large eigenspectrum is also an important challenge being addressed in our research.

The research also extends to the development of asymptotic methods and solvers to study different free surface flows with applications in coating and oil industries.  In some cases, the developments lead to simulation software that is then used in the design of industrial processes.  Recent examples include drying of coated substrates and displacement flow through annular space during cementation process.