Conditional phase-field free-energy dissipation

Let $S,L,U,{\rho }_{\ast },{\mathrm{\Phi }}_{\ast }>0$ with units respectively dimensionless, metres, metres per second, kilograms per cubic metre, and a chosen phase unit $[\mathrm{\Phi }]$. Put $T_{\ast }=L\mathrm{/}U$, ${\mathbb{T}}^3={\mathbb{R}}^3\mathrm{/}(2\pi \mathbb{Z}{)}^3$, ${\mathrm{\Omega }}_L={\mathbb{R}}^3\mathrm{/}(2\pi L\mathbb{Z}{)}^3$, and $t\in [0,T_{\ast }S]$. Fields on each torus are represented by periodic lifts and physical integrals are over $Q_L=[0,2\pi L{]}^3$. Let $\epsilon ,M,\nu >0$ be dimensionless and let $W\in C^2(\mathbb{R})$ be bounded below. Define

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

For periodic physical fields $u:[0,T_{\ast }S]\times {\mathrm{\Omega }}_L\to {\mathbb{R}}^3$, $\phi :[0,T_{\ast }S]\times {\mathrm{\Omega }}_L\to \mathbb{R}$, and scalar pressure $p$, define

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

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

Assume the dimensionless fields $v(s,y)=u(T_{\ast }s,Ly)\mathrm{/}U$, $\varphi (s,y)=\phi (T_{\ast }s,Ly)\mathrm{/}{\mathrm{\Phi }}_{\ast }$, and $\pi (s,y)=p(T_{\ast }s,Ly)\mathrm{/}({\rho }_{\ast }{U}^2)$ satisfy

$v\in C^1([0,S];C_{\mathrm{p}\mathrm{e}\mathrm{r}}^2({\mathbb{T}}^3;{\mathbb{R}}^3)),\varphi \in C^1([0,S];C_{\mathrm{p}\mathrm{e}\mathrm{r}}^1({\mathbb{T}}^3;\mathbb{R}))\cap C([0,S];C_{\mathrm{p}\mathrm{e}\mathrm{r}}^2({\mathbb{T}}^3;\mathbb{R})),\pi \in C([0,S];C_{\mathrm{p}\mathrm{e}\mathrm{r}}^1({\mathbb{T}}^3;\mathbb{R})),$

with ${\mu }_c={\epsilon }^{-1}{W}^{\mathrm{\prime }}(\varphi )-\epsilon {\mathrm{\Delta }}_y\varphi \in C([0,S];C_{\mathrm{p}\mathrm{e}\mathrm{r}}^2)$, and satisfy the adopted dimensional equations

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

Here $C_{\mathrm{p}\mathrm{e}\mathrm{r}}^k$ consists of periodic lifts whose coordinate derivatives through order $k$ are continuous, bounded, and uniformly continuous on $[0,2\pi {]}^3$; $C^1([0,S];C_{\mathrm{p}\mathrm{e}\mathrm{r}}^k)$ means time differentiability 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 $u$, $({\mathrm{\nabla }}_{x}u{)}_{ij}={\mathrm{\partial }}_{x_j}{u}_i$.

For the kinetic-plus-phase free-energy functional

$\mathcal{G}(t)={\int }_{{\mathrm{\Omega }}_L}(\frac{{\rho }_{\ast }}{2}\mathrm{\mid }u(t,x){\mathrm{\mid }}^2+\frac{{\lambda }_{\varphi }}{2}\mathrm{\mid }{\mathrm{\nabla }}_x\phi (t,x){\mathrm{\mid }}^2+\beta W(\phi (t,x)\mathrm{/}{\mathrm{\Phi }}_{\ast }))dx,$

where $\mathrm{\mid }{\mathrm{\nabla }}_{x}u{\mathrm{\mid }}^2={\sum }_{i,j=1}^3\mathrm{\mid }{\mathrm{\partial }}_{x_j}{u}_i{\mathrm{\mid }}^2$, the conditional model identity is

$\frac{d\mathcal{G}}{dt}(t)=-\eta {\int }_{{\mathrm{\Omega }}_L}\mathrm{\mid }{\mathrm{\nabla }}_{x}u(t,x){\mathrm{\mid }}^{2}dx-\mathcal{M}{\int }_{{\mathrm{\Omega }}_L}\mathrm{\mid }{\mathrm{\nabla }}_x\mu (t,x){\mathrm{\mid }}^{2}dx(0<t<T_{\ast }S)\mathrm{.}$

The functional $\mathcal{G}$ is not the complete thermodynamic energy. This deduction neither asserts existence of such a solution nor derives the adopted isothermal closure from a total-energy or entropy balance.

Adopted dimensional periodic phase-field equations and capillary stress

Scale map and dimensional coefficient definitions

Dimensionless smooth periodic energy identity