PaperlayerView exact revision on arXiv

Source-linked AI summary

Linear and entropy-stable scheme for the dynamic van der Waals model of non-isothermal liquid-vapor flows

Yunzhuo Guo, Zhonghua Qiao, Xianmin Xu

arXiv:2610.13029v1math.NAcs.NA

TL;DR

The dynamic van der Waals model describes non-isothermal phase transitions in single-component liquid-vapor systems, but discrete chain-rule failure and decoupling complicate thermodynamic structure preservation. This paper constructs a linear, decoupled, fully discrete entropy-stable scheme, which captures droplet evaporation and Leidenfrost effects in simulations. The scheme’s positivity analysis cannot be established because the logarithmic term is treated essentially explicitly.

  • Problem

    Fully discrete discretization must address chain-rule failure, while decoupled schemes make energy conservation difficult even at the semi-discrete level.

    Source: §3 · p. 7, §3.3 · p. 14

  • Method

    The scheme reformulates the equations using stress divergence identities and uses component-wise convex splitting with explicit velocity treatment to solve the conservation equations linearly and sequentially.

    Source: §3 · p. 7, §3.3 · p. 12

  • Results

    The scheme captures droplet evaporation and two- and three-dimensional Leidenfrost effects, with simulations indicating robust, efficient, and physically consistent behavior.

    Source: §5 · p. 20

  • Takeaways & Limitations

    The simulations demonstrate that the scheme reproduces complex non-isothermal phase-transition dynamics while numerically exhibiting discrete entropy production.

    Source: §3.3 · p. 13, §5 · p. 20

    Because the logarithmic term is treated essentially explicitly, positivity analysis for density and temperature cannot be established.

    Source: §3.3 · p. 13

Abstract

from arXiv · show

In this paper, we propose a linear and decoupled fully discrete scheme that preserves the entropy production property for the dynamic van der Waals model. This model provides a thermodynamically consistent diffuse-interface framework for describing non-isothermal phase transitions in single-component liquid-vapor systems, containing the coupled conservation laws of mass, momentum and energy. We first introduce an equivalent reformulation of the governing equations that avoids the use of chain-rule identities, which generally fail under discrete differentiation. Further, a component-wise convex-splitting method is employed to calculate the discrete chemical potential and temperature. This treatment provides a discrete entropy-stable inequality without a time step restriction. Based on the explicitly treated velocity, the scheme allows the three conservation equations to be solved linearly and sequentially in a decoupled manner. Since the momentum equation does not directly affect the entropy-stability analysis, we use a classical linear semi-implicit scheme to solve it. Numerical experiments verify the expected numerical accuracy. Simulations of droplet evaporation and two- and three-dimensional Leidenfrost effects further demonstrate the proposed method can capture complex non-isothermal phase-transition dynamics while numerically reproducing the discrete entropy production behavior.

1. Introduction.

The dynamic van der Waals model describes non-isothermal phase transitions in single-component liquid-vapor systems. The paper addresses the challenge of designing a fully discrete scheme that preserves the model’s thermodynamic structures while remaining linear and decoupled.

1. Introduction.

  • The dynamic van der Waals model is a thermodynamically consistent continuum framework for non-isothermal phase transitions in single-component liquid-vapor systems.The paper connects this setting to phenomena including the Leidenfrost and thermocapillary effects.Source: §1 · p. 1
  • Mass, momentum, and energy conservation are coupled, with the energy balance incorporating stress work, heat conduction, and gravity.The reversible stress includes pressure and density-gradient contributions.Source: §1 · p. 2
  • Preserving energy conservation and entropy production is a major scheme-design challenge; Kou et al. developed an entropy-stable semi-discrete component-wise convex-splitting scheme, but artificial viscosity for energy conservation prevented fully discrete generalization.Source: §1 · p. 3
  • The paper reformulates the equations using stress identities because chain-rule identities generally fail under discrete temporal and spatial differentiation.The authors also introduce a non-dimensionalization to reduce physical-scale differences and the number of independent parameters.Source: §1 · p. 3
  • Explicit velocity treatment and component-wise convex splitting yield a linear, decoupled scheme for the mass, energy, and momentum equations.The chemical potential and temperature are computed by component-wise convex splitting; the momentum equation uses a classical linear semi-implicit scheme.Source: §1 · p. 4

2. The dimensionless equations.

The governing equations are non-dimensionalized under stated constitutive assumptions and expressed in dimensionless variables. The resulting system satisfies energy conservation and entropy production, with boundary heat flux and gravity represented in the energy balance.

2. The dimensionless equations.

  • The dimensionless setup assumes density-proportional shear and bulk viscosities and thermal conductivity, with the energy gradient coefficient set to zero and C positive.If viscosity or conductivity coefficients instead depend nonlinearly on density and temperature, the non-dimensionalization changes.Source: §2 · p. 4
  • Number density is scaled as ñ = v0n, placing dimensionless density between 0 and 1; the length reference scales gradient contributions consistently.The time scale is the viscous relaxation time associated with the length scale, and the Reynolds number is consequently set to unity.Source: §2 · pp. 4–5
  • The dimensionless equations satisfy total energy conservation and entropy production.The energy balance includes a surface heat-flux term and gravity; the surface heat flux vanishes under homogeneous Neumann temperature conditions.Source: §2 · pp. 5–6
  • The entropy-production derivation uses identities based on the chemical potential, temperature, and mass and energy equations.The paper derives the entropy production law from these identities and the chain-rule relations.Source: §2 · pp. 6–7

3. The fully discrete numerical scheme.

The scheme is reformulated to avoid discrete chain-rule identities, then discretized and solved linearly in a decoupled sequence. Its component-wise convex splitting yields a discrete entropy-production inequality without a time-step restriction, subject to the theorem’s assumptions.

3.1. The equivalent governing equations.

  • The equivalent governing equations put the stress in divergence form in both momentum and energy, avoiding reliance on chain-rule identities that fail under discrete differentiation.This reformulation makes it possible to demonstrate entropy increase without recourse to the chain rule.Source: §3 · p. 7, §3.1 · p. 8

3.2. The spatial discretization.

  • Spatial discretization uses centered finite differences on a uniform square mesh, with a staggered marker-and-cell grid for cell- and face-centered functions.The notation extends naturally to three dimensions.Source: §3.2 · pp. 8–9

3.3. The decoupled component-wise convex-splitting scheme.

  • Component-wise convex splitting constructs the discrete chemical potential and temperature, while explicit velocity treatment enables linear sequential solves for density, energy and temperature, then velocity.Once density is fixed, temperature depends linearly on internal energy; the momentum equation does not directly affect the entropy analysis.Source: §3.2 · p. 11, §3.3 · pp. 11–12
  • The linear construction leaves positivity unproved because the logarithmic term is treated essentially explicitly, while fully discrete energy conservation remains unresolved for decoupled schemes.The authors identify this positivity issue as a trade-off inherent in constructing a linear scheme.Source: §3.3 · pp. 13–14
  • Under the stated boundary conditions, zero gravity, and pointwise-positive density and internal energy, the scheme satisfies a discrete entropy-production law without a time-step restriction.The proof uses convex–concave splitting of the entropy function.Source: §3.3 · p. 13

4. Numerical simulations.

The simulations test the scheme’s convergence and its behavior in droplet evaporation and two- and three-dimensional Leidenfrost flows. The reported results match the expected convergence rates, reproduce classical droplet-evaporation scaling, and show vapor-driven droplet lift and evaporation in the Leidenfrost cases.

4.1. Convergence test.

  • The scheme’s spatial convergence results match the expected second-order accuracy, while its temporal results verify first-order convergence.The spatial test uses fixed ∆t = 0.01 and Tfinal = 10; the temporal test fixes h = L/256 and Tfinal = 10.Source: §4.1 · p. 15, Table 1 · p. 15, Table 2 · p. 16

4.2. Droplet evaporation.

  • In droplet evaporation, the interface thickens and merges into the surroundings, while entropy production, mass, and energy are preserved.The temperature exhibits similar diffusive behavior.Source: §4.2 · p. 16, Figure 1 · p. 16, Figure 2 · p. 17
  • The droplet’s squared diameter decreases approximately linearly with time, consistent with the classical D2-law and showing excellent agreement in Figure 3.The interface is located using nmid ≈ 0.1170.Source: §4.2 · pp. 16–17, Figure 3 · p. 17

4.3. Leidenfrost effect in 2D and 3D.

  • In the 2D Leidenfrost simulation, vapor rapidly vaporizes and lifts the droplet, forming an indentation as it evaporates and merges with the background gas.The initially resting droplet does not show the pronounced vertical bouncing reported in.Source: §4.3 · p. 18, Figure 4 · p. 18
  • The 3D droplet is likewise lifted by vapor, develops an upward indentation, and gradually evaporates; its energy and entropy increase oscillatory, with smoother overall evolution than in 2D.The simulation uses a semi-ellipsoidal droplet and reduces the spatial step to h = L/128.Source: §4.3 · pp. 18–19

5. Conclusion.

The proposed linear, fully discrete scheme solves the density, internal-energy/temperature pair, and velocity through sequential linear subproblems while preserving fully discrete entropy production. Numerical experiments report expected accuracy and capture droplet evaporation and two- and three-dimensional Leidenfrost effects.

5. Conclusion.

  • The component-wise convex-splitting strategy defines discrete chemical potential and temperature, and explicit velocity treatment enables sequential linear solves for density, internal energy-temperature, and velocity.The reformulated reversible stress tensor avoids reliance on continuous chain-rule identities at the discrete level.Source: §5 · p. 19
  • The numerical experiments verify first-order temporal and second-order spatial accuracy.Source: §5 · p. 19
  • The 3D Leidenfrost evolution is visualized using the nmid = 0.1170 half-isosurface and central-cross-section density contours at times t = 20 through 200.Source: Figure 7 · p. 20
  • The semi-elliptical-initial-density 3D Leidenfrost case tracks variations in energy, entropy, and mass.Source: Figure 8 · p. 20
  • The scheme captures droplet evaporation and two- and three-dimensional Leidenfrost effects, while preserving entropy production at the fully discrete level.The authors identify simultaneous energy conservation and entropy production, and positivity preservation for density and temperature, as future development directions.Source: §5 · pp. 19–20
Loading 2610.13029v1…