Adopted diffuse free energy, mobility and capillary stress

Adopt the following isothermal phase-field model in a Cartesian inertial chart with absolute time $t$ in seconds and spatial coordinates $x$ in metres. Let $L>0$ metres, $U>0$ metres per second, ${\rho }_{\ast }>0$ kilograms per cubic metre, and ${\mathrm{\Phi }}_{\ast }>0$ in a chosen phase unit $[\mathrm{\Phi }]$ be fixed scales. Let $S>0$, $T_{\ast }=L\mathrm{/}U$, ${\mathrm{\Omega }}_L={\mathbb{R}}^3\mathrm{/}(2\pi L\mathbb{Z}{)}^3$, and $t\in [0,T_{\ast }S]$. A physical field is represented by a $(2\pi L\mathbb{Z}{)}^3$-periodic function on ${\mathbb{R}}^3$, and its spatial integral is the Riemann integral over $Q_L=[0,2\pi L{]}^3$. Set $s=t\mathrm{/}{T}_{\ast }$, $y=x\mathrm{/}L$, $v=u\mathrm{/}U$, $\varphi =\phi \mathrm{/}{\mathrm{\Phi }}_{\ast }$, and $\pi =p\mathrm{/}({\rho }_{\ast }{U}^2)$. The velocity is $u:[0,T_{\ast }S]\times {\mathrm{\Omega }}_L\to {\mathbb{R}}^3$ in metres per second, the scalar order parameter is $\phi :[0,T_{\ast }S]\times {\mathrm{\Omega }}_L\to \mathbb{R}$ in phase units $[\mathrm{\Phi }]$, and scalar pressure $p$ is in pascals. The order parameter is an internal phase variable, not total mass density; no positivity or pointwise interval constraint is imposed on it.

On ${\mathbb{T}}^3={\mathbb{R}}^3\mathrm{/}(2\pi \mathbb{Z}{)}^3$, require $v\in C^1([0,S];C_{\mathrm{p}\mathrm{e}\mathrm{r}}^2({\mathbb{T}}^3;{\mathbb{R}}^3))$, $\varphi \in C^1([0,S];C_{\mathrm{p}\mathrm{e}\mathrm{r}}^1({\mathbb{T}}^3;\mathbb{R}))\cap C([0,S];C_{\mathrm{p}\mathrm{e}\mathrm{r}}^2({\mathbb{T}}^3;\mathbb{R}))$, and $\pi \in C([0,S];C_{\mathrm{p}\mathrm{e}\mathrm{r}}^1({\mathbb{T}}^3;\mathbb{R}))$, with ${\mu }_c={\epsilon }^{-1}{W}^{\mathrm{\prime }}(\varphi )-\epsilon {\mathrm{\Delta }}_y\varphi \in C([0,S];C_{\mathrm{p}\mathrm{e}\mathrm{r}}^2)$. Here $C_{\mathrm{p}\mathrm{e}\mathrm{r}}^k$ means periodic lifts whose coordinate derivatives through order $k$ are continuous, bounded, and uniformly continuous on $[0,2\pi {]}^3$; time differentiability in $C_{\mathrm{p}\mathrm{e}\mathrm{r}}^k$ is in the corresponding supremum norms.

For a scalar field $g$, ${\mathrm{\nabla }}_{x}g=({\mathrm{\partial }}_{x_1}g,{\mathrm{\partial }}_{x_2}g,{\mathrm{\partial }}_{x_3}g)$ and ${\mathrm{\Delta }}_{x}g={\sum }_{j=1}^3{\mathrm{\partial }}_{x_j}^{2}g$; for a vector field $u$, $({\mathrm{\nabla }}_{x}u{)}_{ij}={\mathrm{\partial }}_{x_j}{u}_i$. On the dimensionless torus ${\mathbb{T}}^3={\mathbb{R}}^3\mathrm{/}(2\pi \mathbb{Z}{)}^3$, use periodic lifts and ${\mathrm{\Delta }}_y={\sum }_{j=1}^3{\mathrm{\partial }}_{y_j}^2$. For the smooth branch, require $v\in C^1([0,S];C_{\mathrm{p}\mathrm{e}\mathrm{r}}^2({\mathbb{T}}^3;{\mathbb{R}}^3))$, $\varphi \in C^1([0,S];C_{\mathrm{p}\mathrm{e}\mathrm{r}}^1({\mathbb{T}}^3;\mathbb{R}))\cap C([0,S];C_{\mathrm{p}\mathrm{e}\mathrm{r}}^2({\mathbb{T}}^3;\mathbb{R}))$, and $\pi \in C([0,S];C_{\mathrm{p}\mathrm{e}\mathrm{r}}^1({\mathbb{T}}^3;\mathbb{R}))$, with ${\mu }_c={\epsilon }^{-1}{W}^{\mathrm{\prime }}(\varphi )-\epsilon {\mathrm{\Delta }}_y\varphi \in C([0,S];C_{\mathrm{p}\mathrm{e}\mathrm{r}}^2)$. Here $C_{\mathrm{p}\mathrm{e}\mathrm{r}}^k$ means periodic lifts whose coordinate derivatives through order $k$ are continuous, bounded, and uniformly continuous on the fundamental cube; time differentiability in $C_{\mathrm{p}\mathrm{e}\mathrm{r}}^k$ is in the corresponding supremum norms.

Use dimensionless parameters $\epsilon ,M,\nu >0$, a potential $W\in C^2(\mathbb{R})$ bounded below, and the dimensional coefficients

$\beta =\frac{{\rho }_{\ast }{U}^2}{\epsilon },{\lambda }_{\varphi }=\frac{{\rho }_{\ast }{U}^2{L}^2\epsilon }{{\mathrm{\Phi }}_{\ast }^2},\eta ={\rho }_{\ast }UL\nu ,\mathcal{M}=\frac{M{\mathrm{\Phi }}_{\ast }^{2}L}{{\rho }_{\ast }U}\mathrm{.}$

The bulk phase free-energy density is $f_b(\phi )=\beta W(\phi \mathrm{/}{\mathrm{\Phi }}_{\ast })$ in joules per cubic metre. Define the chemical-potential field and the symmetric rate tensor by

$\mu =f_b^{\mathrm{\prime }}(\phi )-{\lambda }_{\varphi }{\mathrm{\Delta }}_x\phi ,D=\frac{1}{2}({\mathrm{\nabla }}_{x}u+({\mathrm{\nabla }}_{x}u{)}^{\mathsf{T}})\mathrm{.}$

For vectors $a,b\in {\mathbb{R}}^3$, $(a\otimes b{)}_{ij}=a_i{b}_j$; $I_3$ is the $3\times 3$ identity; and $({\mathrm{div}}_x\sigma {)}_i={\sum }_j{\mathrm{\partial }}_{x_j}{\sigma }_{ij}$. Adopt the phase-field stress and total Cauchy stress

${\sigma }_c=(f_b(\phi )-\phi \mu +\frac{{\lambda }_{\varphi }}{2}\mathrm{\mid }{\mathrm{\nabla }}_x\phi {\mathrm{\mid }}^2)I_3-{\lambda }_{\varphi }{\mathrm{\nabla }}_x\phi \otimes {\mathrm{\nabla }}_x\phi ,\sigma =-p{I}_3+2\eta D+{\sigma }_c,$

where each stress has units pascals and traction is $\sigma n$ on an outward unit normal $n$. The adopted pointwise equations are

${\mathrm{\partial }}_t\phi +u\cdot {\mathrm{\nabla }}_x\phi =\mathcal{M}{\mathrm{\Delta }}_x\mu ,{\mathrm{\nabla }}_x\cdot u=0,{\rho }_{\ast }({\mathrm{\partial }}_{t}u+(u\cdot {\mathrm{\nabla }}_x)u)={\mathrm{div}}_x\sigma \mathrm{.}$

The model uses these smooth periodic fields, so its spatial integrals have no boundary terms. It adopts constant density, constant viscosity, constant mobility, and no external body force. The phase equation describes evolution of the internal order parameter and is not a total-mass diffusion equation. This is a reduced isothermal model: $\mathcal{G}={\int }_{{\mathrm{\Omega }}_L}({\rho }_{\ast }\mathrm{\mid }u{\mathrm{\mid }}^2\mathrm{/}2+f_b(\phi )+{\lambda }_{\varphi }\mathrm{\mid }{\mathrm{\nabla }}_x\phi {\mathrm{\mid }}^2\mathrm{/}2)dx$ is kinetic plus phase-field free energy, not the complete thermodynamic energy. No heat or entropy balance, thermostat coupling, species-mass interpretation, sharp-interface limit, contact-line law, empirical calibration, or claim of universal physical validity is adopted.

SI scales, physical coefficients, phase free energy, and chemical-potential units

Adopted smooth continuum, Newtonian chart, mass and momentum balance framework, and its total-energy and entropy statements

Dimensionless periodic phase-flow formulation

J. Tinsley Oden, Lecture Notes, Fall 2016, Chapter 11 §§11.2 and 11.4 (printed pp. 209–212)

Stefanos Georgiadis, “Energy identity for the incompressible Cahn–Hilliard/Navier–Stokes system with non-degenerate mobility,” §1 (2024)

V.E. Badalassi, H.D. Ceniceros, and S. Banerjee, “Computation of multiphase systems with phase field models,” §§2.1–2.4