Isogeometric analysis of the advective Cahn–Hilliard equation: Spinodal decomposition under shear flow

Ju, Liu; Dedè, Luca; Evans, John A.; Borden, Michael J.; Hughes, Thomas J.R.

doi:10.1016/j.jcp.2013.02.008

Cited by 100 publications

(79 citation statements)

References 53 publications

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“…Complex fluids such as polymeric liquids, which exhibit elastic instabilities [9,31], or phase-separating fluid mixtures [32][33][34] show are more complex intrinsic dynamics than the simple vortex diffusion. Certainly, the oscillatory motion of the circular domain, induced by the feedback strategies, will couple to the characteristic modes and generate new flow patterns.…”

Section: Discussionmentioning

confidence: 99%

Feedback control of flow vorticity at low Reynolds numbers

2015

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Abstract. Our aim is to explore strategies of feedback control to design and stabilize novel dynamic flow patterns in model systems of complex fluids. To introduce the control strategies, we investigate the simple Newtonian fluid at low Reynolds number in a circular geometry. Then, the fluid vorticity satisfies a diffusion equation. We determine the mean vorticity in the sensing area and use two control strategies to feed it back into the system by controlling the angular velocity of the circular boundary. Hysteretic feedback control generates self-regulated stable oscillations in time, the frequency of which can be adjusted over several orders of magnitude by tuning the relevant feedback parameters. Time-delayed feedback control initiates unstable vorticity modes for sufficiently large feedback strength. For increasing delay time, we first observe oscillations with beats and then regular trains of narrow pulses. Close to the transition line between the resting fluid and the unstable modes, these patterns are relatively stable over long times.

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Section: Discussionmentioning

confidence: 99%

Feedback control of flow vorticity at low Reynolds numbers

2015

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“…The difference between the two solutions is used as an error estimator to modify the time step size. Even though shown to be robust and used since in other works [23,35,39], this strategy is inefficient given that the solution must be computed twice at each time step. In [27,63], no recovery strategies are proposed when the numerical solver fails.…”

Section: Time Adaptivitymentioning

confidence: 99%

“…The method has successfully been applied in its Galerkin version to solve the Cahn-Hilliard equation [21,23], the advective [39] and Navier-Stokes-CahnHilliard equations [19,56], the Swift-Hohenberg equation [24] and the phasefield crystal equation [25,54,57]. IGA possesses some advantages over standard finite element methods, which include being able to easily generate high-order, globally continuous basis functions as well as exact geometrical representations as the finite element space is refined.…”

Section: Introductionmentioning

confidence: 99%

An energy-stable time-integrator for phase-field models

Vignal

Collier

Dalcín

et al. 2017

Computer Methods in Applied Mechanics and Engineering

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We introduce a provably energy-stable time-integration method for general classes of phase-field models with polynomial potentials. We demonstrate how Taylor series expansions of the nonlinear terms present in the partial differential equations of these models can lead to expressions that guarantee energy-stability implicitly, which are second-order accurate in time. The spatial discretization relies on a mixed finite element formulation and isogeometric analysis. We also propose an adaptive time-stepping discretization that relies on a first-order backward approximation to give an error-estimator. This error estimator is accurate, robust, and does not require the computation of extra solutions to estimate the error. This methodology can be applied to any second-order accurate time-integration scheme. We present numerical examples in two and three spatial dimensions, which confirm the stability and robustness of the method. The implementation of the numerical schemes is done in PetIGA, a high-performance isogeometric analysis framework.

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“…The first article [27] uses the advective Cahn-Hilliard equation and presents an Isogeometric Analysis-based numerical study of spinodal decomposition of a binary fluid undergoing shear flow. In contrast to this work, however, they use a passive and externally provided velocity field and in particular do not solve a coupled NSCH system.…”

Section: Introductionmentioning

confidence: 99%

Isogeometric Analysis of the Navier–Stokes–Cahn–Hilliard equations with application to incompressible two-phase flows

Hosseini

Turek

Möller

et al. 2017

Journal of Computational Physics

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In this work, we present our numerical results of the application of Galerkin-based Isogeometric Analysis (IGA) to incompressible Navier-Stokes-Cahn-Hilliard (NSCH) equations in velocity-pressurephase field-chemical potential formulation. For the approximation of the velocity and pressure fields, LBB compatible non-uniform rational B-spline spaces are used which can be regarded as smooth generalizations of Taylor-Hood pairs of finite element spaces. The one-step θ-scheme is used for the discretization in time. The static and rising bubble, in addition to the nonlinear Rayleigh-Taylor instability flow problems, are considered in two dimensions as model problems in order to investigate the numerical properties of the scheme.

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Isogeometric analysis of the advective Cahn–Hilliard equation: Spinodal decomposition under shear flow

Cited by 100 publications

References 53 publications

Feedback control of flow vorticity at low Reynolds numbers

Feedback control of flow vorticity at low Reynolds numbers

An energy-stable time-integrator for phase-field models

Isogeometric Analysis of the Navier–Stokes–Cahn–Hilliard equations with application to incompressible two-phase flows

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