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arxiv: 2604.21200 · v1 · submitted 2026-04-23 · 🧮 math.AP · cs.NA· math.NA

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A Temperature-Coupled Cahn-Hilliard-Stokes-Heat Model for Thermally Driven Phase Separation

Maria Deliyianni , Boris Muha , Andrej Novak
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Pith reviewed 2026-05-09 21:47 UTC · model grok-4.3

classification 🧮 math.AP cs.NAmath.NA
keywords Cahn-HilliardStokes equationsheat equationphase separationweak solutionsfinite element methodconvex splittingthermal coupling
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The pith

Local weak solutions exist for a regularized temperature-coupled Cahn-Hilliard-Stokes-Heat model.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper analyzes a diffuse interface model coupling the Cahn-Hilliard equation to Stokes flow and heat transport to describe phase separation driven by temperature in viscous incompressible fluids. Temperature modifies the Landau free energy while the phase field influences density and viscosity. The central result is a proof of local-in-time existence of weak solutions for a suitably regularized version of the system, obtained via a priori estimates on the subproblems. A corresponding finite element scheme using convex splitting for the Cahn-Hilliard component is shown to conserve mass and to be energy stable in the isothermal limit. Simulations demonstrate the effects of thermal gradients on pattern formation in two-dimensional confined domains.

Core claim

We study a diffuse-interface model for thermally driven phase separation in viscous incompressible mixtures by coupling a convective Cahn-Hilliard equation for the order parameter with a Stokes subsystem for the velocity-pressure field and a heat equation for the temperature. Temperature enters the bulk free energy through a Landau-type coefficient, while the phase field feeds back on the flow through concentration-dependent density and viscosity. We prove local-in-time existence of weak solutions for a regularized coupled system and propose a fully discrete finite element scheme with convex-splitting time discretization that conserves mass under impermeable boundary conditions and isuncondi

What carries the argument

Regularized temperature-coupled Cahn-Hilliard-Stokes-Heat system with temperature-dependent Landau free energy and concentration-dependent density and viscosity.

Load-bearing premise

The assumptions on the temperature-dependent Landau free energy, concentration-dependent density and viscosity, impermeable boundary conditions, and the regularization together allow derivation of the a priori estimates needed for local existence of weak solutions.

What would settle it

A concrete counterexample consisting of initial data and parameter values for which no weak solution to the regularized system exists on a positive time interval.

Figures

Figures reproduced from arXiv: 2604.21200 by Andrej Novak, Boris Muha, Maria Deliyianni.

Figure 1
Figure 1. Figure 1: Comparison of the logarithmic potential and the polynomial double-well potential. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Transition of the potential f(c, Θ) from a double–well form (Θ < ΘS, phase separation) to a single–well form (Θ > ΘS, homogeneous phase). denote the local volume fraction of phase 1 in a representative volume element. The partial mass densities in that element are ρe1 = ρ1ϕ, ρe2 = ρ2(1 − ϕ), and the total density is ρ = ρ1ϕ + ρ2(1 − ϕ). (2.7) The mass balance for phase 1 reads ∂ρe1 ∂t + ∇ · m1 = 0, (2.8) w… view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of the simulated velocity profiles (a) with the reference profiles reported by Khatavkar et al. [PITH_FULL_IMAGE:figures/full_fig_p032_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of the interface profiles c = 0 at t = 1 for λ = 1 (left) and λ = 10 (right). The bottom row reproduces the corresponding results of Khatavkar et al. from [23]. 4.5. Experiment 2: Isothermal phase separation above and below the transition temperature In this experiment we suppress advection by setting u ≡ 0 and do not solve the heat equation. The evolution is therefore governed solely by the Cah… view at source ↗
Figure 5
Figure 5. Figure 5: Snapshots of the phase field for Θ = 1.5 ΘS (single-well regime). The solution remains close to a homogeneous state. 4.6. Experiment 3: Gravity-driven phase separation with variable density and viscosity The third experiment activates the coupling between the Cahn–Hilliard equation and the quasi-static Stokes system in the presence of gravity. The goal is to illustrate the effect of buoyancy when the two p… view at source ↗
Figure 6
Figure 6. Figure 6: Snapshots of the phase field for Θ = 0.2 ΘS (double-well regime). Phase separation sets in rapidly and is followed by coarsening. so that β(Θ) is fixed and the heat equation is not solved in this test. Consistently with the affine coefficient laws ρ(c) and η(c) used in the numerical scheme, the density and viscosity contrasts are kept fixed throughout the run. We take λρ = 0.0009, λη = 0.08, together with … view at source ↗
Figure 7
Figure 7. Figure 7: Phase distribution at t = 2 for Θ = 0.2 ΘS, visualized with an HSV colormap to emphasize the diffuse interfaces. Parameter Symbol Value Domain Ω [0, 1] × [0, 1] Mesh size 100 × 100 Boundary conditions u = 0 on Γ1; [−pI + 2ηE(u)] · n = 0 on Γ2 ∪ Γ3 ∪ Γ4 Initial phase field c0(x) random perturbation of a 70:30 mean state Initial velocity u0 0 Prescribed temperature Θ(x, t) Θ0 = 0.3 ΘS Péclet number P e 1000 … view at source ↗
Figure 8
Figure 8. Figure 8: Phase-field snapshots for Experiment 3. Dense domains migrate toward the lower wall under the action of gravity [PITH_FULL_IMAGE:figures/full_fig_p037_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: Heat diffuses from the left boundary Γ4 into the interior, increasing β(Θ) near that boundary and locally suppressing the double-well structure of the bulk potential. Consequently, phase-separated structures adjacent to Γ4 shrink and the solution relaxes locally toward a more homogeneous state. Case 2 (cooled boundary). In the second run the initial temperature is above the transition value, Θ(x, 0) = Θ0 … view at source ↗
Figure 9
Figure 9. Figure 9: Phase-field snapshots for the heated-boundary configuration: [PITH_FULL_IMAGE:figures/full_fig_p038_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Temperature distribution at t = 1 for the heated-wall configuration. Parameter Symbol Value Domain Ω [0, 2] × [0, 1] Mesh size 128 × 64 Boundary conditions see text Initial phase field c0(x) random perturbation of a 70:30 mean state Initial velocity u0 0 Initial temperature (case 1) Θ0 0.3 ΘS Boundary temperature on Γ4 (case 1) Θ|Γ4 1.5 ΘS Initial temperature (case 2) Θ0 1.5 ΘS Boundary temperature on Γ4 … view at source ↗
Figure 11
Figure 11. Figure 11: Phase-field snapshots for the cooled-boundary configuration: [PITH_FULL_IMAGE:figures/full_fig_p040_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Temperature distribution at t = 2 for the cooled-wall configuration. 40 [PITH_FULL_IMAGE:figures/full_fig_p040_12.png] view at source ↗
read the original abstract

We study a diffuse-interface model for thermally driven phase separation in viscous incompressible mixtures. The system couples a convective Cahn-Hilliard equation for the order parameter with a Stokes subsystem for the velocity-pressure field and a heat equation for the temperature. Temperature enters the bulk free energy through a Landau-type coefficient, while the phase field feeds back on the flow through concentration-dependent density and viscosity, yielding a phenomenological temperature-coupled Cahn-Hilliard-Stokes-Heat system. We motivate the chemical potential by a temperature-dependent Landau free energy and derive a priori estimates for the regularized subproblems. On the analytical side, we prove local-in-time existence of weak solutions for a regularized coupled system. On the numerical side, we propose a fully discrete finite element scheme combining a convex-splitting time discretization for the Cahn-Hilliard equation with an implicit treatment of viscous and thermal diffusion terms and a an implicit Stokes solve. Under impermeable velocity boundary conditions, the Cahn-Hilliard substep conserves mass, in the purely diffusive isothermal case, the convex-splitting discretization is unconditionally energy-stable for the Cahn-Hilliard free energy. Numerical experiments in two dimensions illustrate thermally driven spinodal decomposition, wall-induced phase separation near cooled walls, and phase separation in narrow channels under imposed thermal gradients. The simulations show the qualitative influence of key nondimensional parameters (such as the mass and thermal P\'eclet numbers, the Cahn number, the density and viscosity ratios, and the gravitational parameter $G$) on pattern formation, interface motion, and flow structure, and confirm that the proposed framework is a robust tool for studying thermally driven phase separation in confined geometries.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces a diffuse-interface model coupling a convective Cahn-Hilliard equation, Stokes flow, and a heat equation to describe thermally driven phase separation in incompressible viscous mixtures, with temperature entering via a Landau-type free energy and phase field affecting density and viscosity. It proves local-in-time existence of weak solutions to a regularized version of the coupled system via a priori estimates, and develops a fully discrete finite-element scheme using convex splitting for the Cahn-Hilliard subproblem together with implicit treatment of diffusion and Stokes terms. The scheme is shown to conserve mass under impermeable boundary conditions and to be unconditionally energy stable for the Cahn-Hilliard free energy in the purely diffusive isothermal case; 2D numerical experiments illustrate the effects of mass and thermal Péclet numbers, Cahn number, density/viscosity ratios, and gravitational parameter on spinodal decomposition and interface motion.

Significance. If the local existence result and the stability properties hold, the work supplies a mathematically grounded framework for analyzing and simulating temperature-driven phase separation in confined flows, with direct relevance to materials processing and microfluidics. The explicit unconditional energy stability of the convex-splitting discretization in the isothermal limit is a concrete strength that supports robust long-time computations without artificial time-step restrictions. The numerical illustrations of parameter dependence on pattern formation provide useful qualitative insight even if the full non-isothermal scheme stability remains open.

minor comments (3)
  1. [Abstract] Abstract: the phrase 'with a an implicit Stokes solve' contains a repeated article and should be corrected to 'with an implicit Stokes solve'.
  2. [Section on existence proof] The existence theorem is stated only for the regularized system; a brief remark on whether the regularization parameter can be sent to zero while retaining the a priori estimates would clarify the scope of the result.
  3. [Numerical scheme section] The unconditional stability is proved only in the isothermal diffusive limit; the manuscript should explicitly note that the full temperature-coupled scheme lacks this guarantee and state any conditional stability results that may hold.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful and positive assessment of our manuscript on the temperature-coupled Cahn-Hilliard-Stokes-Heat model. The referee's summary accurately captures the main analytical result (local-in-time existence of weak solutions for the regularized system) and the numerical contribution (convex-splitting finite-element scheme with mass conservation and unconditional energy stability in the isothermal case). We appreciate the recognition of the framework's relevance to materials processing and microfluidics, as well as the value of the 2D numerical illustrations. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper constructs a phenomenological model from standard Landau free-energy considerations with temperature dependence, then proves local existence of weak solutions for the regularized system via standard Galerkin/fixed-point arguments and a priori energy estimates that close under the stated assumptions on the free energy, density, viscosity, and impermeable boundary conditions. The numerical scheme's mass conservation and energy stability (in the isothermal case) follow from direct, explicit calculations on the convex-splitting discretization without any fitted parameters, self-referential definitions, or load-bearing self-citations. No step in the claimed derivation chain reduces by construction to its own inputs; the regularization and energy methods supply independent compactness and bounds.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard mathematical assumptions for nonlinear PDE existence rather than new physical postulates or fitted entities.

axioms (2)
  • domain assumption Regularity and growth conditions on the temperature-dependent Landau free energy and on the concentration-dependent density and viscosity functions
    Required to close the a priori estimates for the regularized system in the existence proof.
  • domain assumption Impermeable velocity boundary conditions and suitable initial data
    Used to obtain mass conservation and energy stability in the numerical scheme.

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Reference graph

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