Hyperbolic relaxation models for thin films down an inclined plane

Dhaouadi, Firas; Gavrilyuk, Sergey; Vila, Jean‐Paul

doi:10.1016/j.amc.2022.127378

Cited by 8 publications

(5 citation statements)

References 21 publications

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“…This is not the case, as straightforward computations show that A is missing two right eigenvectors for the multiple eigenvalue u 1 , meaning that system (3.7) is only weakly hyperbolic. This does not come as a surprise since this shortcoming appears in several models where an independent field bound by a curl-free constraint is evolved [38][39][40][41][42][43]. Nonetheless, there are procedures allowing to recover strong hyperbolicity in this case.…”

Section: Hyperbolic Model Analysis (A) Eigenstructure Studymentioning

confidence: 95%

An Eulerian hyperbolic model for heat transfer derived via Hamilton’s principle: analytical and numerical study

Dhaouadi,

Gavrilyuk

2024

Proc. R. Soc. A.

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In this paper, we present a new model for heat transfer in compressible fluid flows. The model is derived from Hamilton’s principle of stationary action in Eulerian coordinates, in a setting where the entropy conservation is recovered as an Euler–Lagrange equation. A sufficient criterion for the hyperbolicity of the model is formulated. The governing equations are asymptotically consistent with the Euler equations for compressible heat conducting fluids, provided the addition of suitable relaxation terms. A study of the Rankine–Hugoniot conditions and Clausius–Duhem inequality is performed for a specific choice of the equation of state. In particular, this reveals that contact discontinuities cannot exist while expansion waves and compression fans are possible solutions to the governing equations. Evidence of these properties is provided on a set of numerical test cases.

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Section: Hyperbolic Model Analysis (A) Eigenstructure Studymentioning

confidence: 95%

An Eulerian hyperbolic model for heat transfer derived via Hamilton’s principle: analytical and numerical study

Dhaouadi,

Gavrilyuk

2024

Proc. R. Soc. A.

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“…Such a method of "extended" Lagrangian was efficiently used for dispersive models appearing both in classical and quantum fluids (e.g., Refs. [18][19][20][21][22][23][24]. map…”

Section: F I G U R Ementioning

confidence: 99%

Hyperbolicity study of the modulation equations for the Benjamin–Bona–Mahony equation

Gavrilyuk

Shyue

2023

Stud Appl Math

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A complete study of the modulation equations for the Benjamin-Bona-Mahony equation is performed. In particular, the boundary between the hyperbolic and elliptic regions of the modulation equations is found. When the wave amplitude is small, this boundary is approximately defined by 𝑘 = √ 3, where 𝑘 is the wave number.This particular value corresponds to the inflection point of the linear dispersion relation for the BBM equation. Numerical results are presented showing the appearance of the Benjamin-Feir instability when the periodic solutions are inside the ellipticity region.

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“…The system considered here for numerical simulation is the first-order hyperbolic relaxation model approximating the NSK equations presented in [32] and whose structure and properties are briefly recalled in this section. The model can be seen as the combination of the unified hyperbolic and thermodynamically compatible Godunov-Peshkov-Romenski (GPR) model of continuum mechanics [4][5][6][7], with an augmented Lagrangian approach [71,72], which allows to cast dispersive systems of the Euler-Korteweg type into first-order hyperbolic PDEs with stiff relaxation terms. The full system of equations, is given by…”

Section: First-order Hyperbolic Reformulation Of the Nsk Equationsmentioning

confidence: 99%

“…η can be seen as the proxy for ρ as the order parameter whose gradient is used in the Korteweg theory. The gradient and material derivative of η are promoted to new independent variables, respectively, p and w and whose evolution equations are established from their definitions [71,72]. Naturally, their evolution equations need also to be supplemented with the appropriate initial conditions w(x, t = 0) = η(x, t = 0) and p(x, t = 0) = ∇η(x, t = 0).…”

Section: First-order Hyperbolic Reformulation Of the Nsk Equationsmentioning

confidence: 99%

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A Structure-Preserving Finite Volume Scheme for a Hyperbolic Reformulation of the Navier–Stokes–Korteweg Equations

Dhaouadi

Dumbser

2023

Mathematics

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In this paper, we present a new explicit second-order accurate structure-preserving finite volume scheme for the first-order hyperbolic reformulation of the Navier–Stokes–Korteweg equations. The model combines the unified Godunov-Peshkov-Romenski model of continuum mechanics with a recently proposed hyperbolic reformulation of the Euler–Korteweg system. The considered PDE system includes an evolution equation for a gradient field that is by construction endowed with a curl-free constraint. The new numerical scheme presented here relies on the use of vertex-based staggered grids and is proven to preserve the curl constraint exactly at the discrete level, up to machine precision. Besides a theoretical proof, we also show evidence of this property via a set of numerical tests, including a stationary droplet, non-condensing bubbles as well as non-stationary Ostwald ripening test cases with several bubbles. We present quantitative and qualitative comparisons of the numerical solution, both, when the new structure-preserving discretization is applied and when it is not. In particular for under-resolved simulations on coarse grids we show that some numerical solutions tend to blow up when the curl-free constraint is not respected.

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Hyperbolic relaxation models for thin films down an inclined plane

Cited by 8 publications

References 21 publications

An Eulerian hyperbolic model for heat transfer derived via Hamilton’s principle: analytical and numerical study

An Eulerian hyperbolic model for heat transfer derived via Hamilton’s principle: analytical and numerical study

Hyperbolicity study of the modulation equations for the Benjamin–Bona–Mahony equation

A Structure-Preserving Finite Volume Scheme for a Hyperbolic Reformulation of the Navier–Stokes–Korteweg Equations

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