11 research outputs found
Numerical approximation of a phase-field surfactant model with fluid flow
Modelling interfacial dynamics with soluble surfactants in a multiphase
system is a challenging task. Here, we consider the numerical approximation of
a phase-field surfactant model with fluid flow. The nonlinearly coupled model
consists of two Cahn-Hilliard-type equations and incompressible Navier-Stokes
equation. With the introduction of two auxiliary variables, the governing
system is transformed into an equivalent form, which allows the nonlinear
potentials to be treated efficiently and semi-explicitly. By certain subtle
explicit-implicit treatments to stress and convective terms, we construct first
and second-order time marching schemes, which are extremely efficient and
easy-to-implement, for the transformed governing system. At each time step, the
schemes involve solving only a sequence of linear elliptic equations, and
computations of phase-field variables, velocity and pressure are fully
decoupled. We further establish a rigorous proof of unconditional energy
stability for the first-order scheme. Numerical results in both two and three
dimensions are obtained, which demonstrate that the proposed schemes are
accurate, efficient and unconditionally energy stable. Using our schemes, we
investigate the effect of surfactants on droplet deformation and collision
under a shear flow, where the increase of surfactant concentration can enhance
droplet deformation and inhibit droplet coalescence
Transient electrohydrodynamic flow with concentration dependent fluid properties: modelling and energy-stable numerical schemes
Transport of electrolytic solutions under influence of electric fields occurs
in phenomena ranging from biology to geophysics. Here, we present a continuum
model for single-phase electrohydrodynamic flow, which can be derived from
fundamental thermodynamic principles. This results in a generalized
Navier-Stokes-Poisson-Nernst-Planck system, where fluid properties such as
density and permittivity depend on the ion concentration fields. We propose
strategies for constructing numerical schemes for this set of equations, where
solving the electrochemical and the hydrodynamic subproblems are decoupled at
each time step. We provide time discretizations of the model that suffice to
satisfy the same energy dissipation law as the continuous model. In particular,
we propose both linear and non-linear discretizations of the electrochemical
subproblem, along with a projection scheme for the fluid flow. The efficiency
of the approach is demonstrated by numerical simulations using several of the
proposed schemes
Asymptotically and exactly energy balanced augmented flux-ADER schemes with application to hyperbolic conservation laws with geometric source terms
In this work, an arbitrary order HLL-type numerical scheme is constructed using the flux-ADER methodology. The proposed scheme is based on an augmented Derivative Riemann solver that was used for the first time in Navas-Montilla and Murillo (2015) 1]. Such solver, hereafter referred to as Flux-Source (FS) solver, was conceived as a high order extension of the augmented Roe solver and led to the generation of a novel numerical scheme called AR-ADER scheme. Here, we provide a general definition of the FS solver independently of the Riemann solver used in it. Moreover, a simplified version of the solver, referred to as Linearized-Flux-Source (LFS) solver, is presented. This novel version of the FS solver allows to compute the solution without requiring reconstruction of derivatives of the fluxes, nevertheless some drawbacks are evidenced. In contrast to other previously defined Derivative Riemann solvers, the proposed FS and LFS solvers take into account the presence of the source term in the resolution of the Derivative Riemann Problem (DRP), which is of particular interest when dealing with geometric source terms. When applied to the shallow water equations, the proposed HLLS-ADER and AR-ADER schemes can be constructed to fulfill the exactly well-balanced property, showing that an arbitrary quadrature of the integral of the source inside the cell does not ensure energy balanced solutions. As a result of this work, energy balanced flux-ADER schemes that provide the exact solution for steady cases and that converge to the exact solution with arbitrary order for transient cases are constructed
Ideal GLM-MHD - a new mathematical model for simulating astrophysical plasmas
Magnetic fields are ubiquitous in space. As there is strong evidence that magnetic fields play an important role in a variety of astrophysical processes, they should not be neglected recklessly. However, analytic models in astrophysical either do often not take magnetic fields into account or can do this after limiting simplifications reducing their overall predictive power. Therefore, computational astrophysics has evolved as a modern field of research using sophisticated computer simulations to gain insight into physical processes.
The ideal MHD equations, which are the most often used basis for simulating magnetized plasmas, have two critical drawbacks: Firstly, they do not limit the growth of numerically caused magnetic monopoles, and, secondly, most numerical schemes built from the ideal MHD equations are not conformable with thermodynamics.
In my work, at the interplay of math and physics, I developed and presented the first thermodynamically consistent model with effective inbuilt divergence cleaning. My new Galilean-invariant model is suitable for simulating magnetized plasmas under extreme conditions as those typically encountered in astrophysical scenarios. The new model is called the "ideal GLM-MHD" equations and supports nine wave solutions.
The accuracy and robustness of my numerical implementation are demonstrated with a number of tests, including comparisons to other schemes available within in the multi-physics, multi-scale adaptive mesh refinement (AMR) simulation code FLASH. A possible astrophysical application scenario is discussed in detail