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  • 1.
    Pålsson, Sara
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Sorgentone, Chiara
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Adaptive time-stepping for surfactant-laden drops2017In: Proceedings of the Eleventh UK Conference on Boundary Integral Methods (UKBIM 11) / [ed] Chappell, D.J., 2017Conference paper (Refereed)
    Abstract [en]

    An adaptive time-stepping scheme is presented aimed at computing the dynamics of surfactant-covered deforming droplets. This involves solving a coupled system, where one equation corresponds to the evolution of the drop interfaces and one to the surfactant concentration. The first is discretised in space using a boundary integral formulation which can be treated explicitly in time. The latter is a convection-diffusion equation solved with a spectral method and is advantageously solved with a semi-implicit method in time. The scheme is adaptive with respect to drop deformation as well as surfactant concentration and the adjustment of time-steps takes both errors into account. It is applied and demonstrated for simulation of the deformation of surfactant-covered droplets, but can easily be applied to any system of equations with similar structure. Tests are performed for both 2D and 3D formulations and the scheme is shown to meet set error tolerances in an efficient way.

  • 2.
    Sorgentone, Chiara
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Favini, B.
    A systematic method to construct mimetic Finite-Difference schemes for incompressible flows2017In: International Journal of Numerical Analysis & Modeling, ISSN 1705-5105, Vol. 14, no 3, p. 419-436Article in journal (Refereed)
    Abstract [en]

    We present a general procedure to construct a non-linear mimetic finite-difference operator. The method is very simple and general: it can be applied for any order scheme, for any number of grid points and for any operator constraints. In order to validate the procedure, we apply it to a specific example, the Jacobian operator for the vorticity equation. In particular we consider a finite difference approximation of a second order Jacobian which uses a 9x9 uniform stencil, verifies the skew-symmetric property and satisfies physical constraints such as conservation of energy and enstrophy. This particular choice has been made in order to compare the present scheme with Arakawa’s renowned Jacobian, which turns out to be a specific case of the general solution. Other possible generalizations of Arakawa’s Jacobian are available in literature but only the present approach ensures that the class of solutions found is the widest possible. A simplified analysis of the general scheme is proposed in terms of truncation error and study of the linearised operator together with some numerical experiments. We also propose a class of analytical solutions for the vorticity equation to compare an exact solution with our numerical results.

  • 3.
    Sorgentone, Chiara
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    La Cognata, Cristina
    Nordstrom, Jan
    A new high order energy and enstrophy conserving Arakawa-like Jacobian differential operator2015In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 301, p. 167-177Article in journal (Refereed)
    Abstract [en]

    A new high order Arakawa-like method for the incompressible vorticity equation in two-dimensions has been developed. Mimetic properties such as skew-symmetry, energy and enstrophy conservations for the semi-discretization are proved for periodic problems using arbitrary high order summation-by-parts operators. Numerical simulations corroborate the theoretical findings.

  • 4.
    Sorgentone, Chiara
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    A highly accurate boundary integral equation method for surfactant-laden drops in 3D2018In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 360, p. 167-191Article in journal (Refereed)
    Abstract [en]

    The presence of surfactants alters the dynamics of viscous drops immersed in an ambient viscous fluid. This is specifically true at small scales, such as in applications of droplet based microfluidics, where the interface dynamics become of increased importance. At such small scales, viscous forces dominate and inertial effects are often negligible. Considering Stokes flow, a numerical method based on a boundary integral formulation is presented for simulating 3D drops covered by an insoluble surfactant. The method is able to simulate drops with different viscosities and close interactions, automatically controlling the time step size and maintaining high accuracy also when substantial drop deformation appears. To achieve this, the drop surfaces as well as the surfactant concentration on each surface are represented by spherical harmonics expansions. A novel reparameterization method is introduced to ensure a high-quality representation of the drops also under deformation, specialized quadrature methods for singular and nearly singular integrals that appear in the formulation are evoked and the adaptive time stepping scheme for the coupled drop and surfactant evolution is designed with a preconditioned implicit treatment of the surfactant diffusion.

  • 5.
    Sorgentone, Chiara
    et al.
    KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Vlahovska, Petia M.
    Northwestern Univ, Engn Sci & Appl Math, Evanston, IL 60208 USA..
    A 3D boundary integral method for the electrohydrodynamics of surfactant-covered drops2019In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 389, p. 111-127Article in journal (Refereed)
    Abstract [en]

    We present a highly accurate numerical method based on a boundary integral formulation and the leaky dielectric model to study the dynamics of surfactant-covered drops in the presence of an applied electric field. The method can simulate interacting 3D drops (no axisymmetric simplification) in close proximity, can consider different viscosities, is adaptive in time and able to handle substantial drop deformation. For each drop global representations of the variables based on spherical harmonics expansions are used and the spectral accuracy is achieved by designing specific numerical tools: a specialized quadrature method for the singular and nearly singular integrals that appear in the formulation, a general preconditioner for the implicit treatment of the surfactant diffusion and a reparametrization procedure able to ensure a high-quality representation of the drops also under deformation. Our numerical method is validated against theoretical, numerical and experimental results available in the literature, as well as a new second-order theory developed for a surfactant-laden drop placed in a quadrupole electric field.

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