Biomedical Engineering Reference
In-Depth Information
Boundary integrals allow for example a high order of accuracy in the description
of interfaces for potential flows (irrotational or Stokes flows) but are not easily
applicable to viscous flows governed by Navier-Stokes equations such as the one
describing droplet evaporation [48, 49]. On the other hand, Lagrangian approaches
allow an explicit description of the interface location and rely on the resolution of
the interface equations on a small number of grid nodes. However, they impose
non trivial handling of these latter for mass conservation purposes, when highly
distorted or complex geometries have to be treated. Approaches coupling finite vol-
ume methods with a structured grid together with an immersed set of connected
marker points for explicit interface tracking have also been developed [50]. Always
for front tracking, Volume of Fluid (VOF) methods are mass conservative and al-
low the description of systems with non trivial geometries without complex grid
nodes restructuring [51]. They belong to the family of Eulerian techniques and treat
the simulation domain as a whole (i.e., a single fluid), which makes them easier
to formulate for full 3D geometries. But the price of their simplicity is the loss of
interface sharpness and the necessity to introduce specific methods for its preser-
vation. In the case of multiphase systems like evaporating droplets, this approach
uses a function acting like a fluid tracer and taking a specific value for each fluid.
This function is advected as time evolves and used to compute the interface curva-
ture which is incorporated into the Navier-Stokes equation where it contributes as a
source term. Each of the previous numerical approaches is more or less adapted to
specific problems but, among all, VOF methods have the notable advantage of ro-
bustness even for basic structured discretization meshes when using implicit finite
volume schemes. When used with high order integration schemes, VOF methods
have the ability to handle most of the changes in the shape of the interfaces [52].
They hence appear to be adapted to catch important topological transitions in the
interface geometry such as the appearance of residual droplets in experiments that
sometimes show up in the ultimate evaporation regime. But, the use of implicit
integration schemes leads to constraints in time stepping that have to be treated
carefully. The requirement to fulfill the CFL condition for numerical convergence
can indeed lead to prohibitive CPU times due to too small time steps. Satisfactory
compromises between time and space discretization steps can usually be found to
overcome this difficulty.
In the context of free interfaces capturing, the generalization of VOF approaches
to systems with more than two phases and moving boundaries like in evaporating
droplets is still to be developed. Adapted expressions for the sources to be intro-
duced in the momentum equation are yet expected to provide stable simulations and
new insights not only for receding contact line regimes but also for the pinned one.
Indeed, besides liquid-gas interfaces, solid-liquid ones can also be time evolving
in this regime due to chemical reactions such as corrosion or electrolytic deposi-
tion. The actual contribution of these effects can be regarded as marginal in pinned
regimes, but are key in contact line de-pinning processes. They indeed modify the
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