SI dimensionalization of the periodic diffuse-interface model

Use the dimensionless periodic torus ${\mathbb{T}}^3={\mathbb{R}}^3\mathrm{/}(2\pi \mathbb{Z}{)}^3$ with dimensionless coordinates $y\in {\mathbb{T}}^3$ and $s\in [0,S]$, where $S>0$ is dimensionless. Its physical length-scale image is the periodic box ${\mathrm{\Omega }}_L={\mathbb{R}}^3\mathrm{/}(2\pi L\mathbb{Z}{)}^3$, with physical coordinates $x=Ly$ in metres and fundamental cube $Q_L=[0,2\pi L{]}^3$. Fields on ${\mathrm{\Omega }}_L$ are represented by $(2\pi L\mathbb{Z}{)}^3$-periodic lifts, spatial derivatives are coordinate derivatives of these lifts, and spatial integrals mean Riemann integrals over $Q_L$. For a scalar field $g$, set ${\mathrm{\nabla }}_{x}g=({\mathrm{\partial }}_{x_1}g,{\mathrm{\partial }}_{x_2}g,{\mathrm{\partial }}_{x_3}g)$ and ${\mathrm{\Delta }}_{x}g={\sum }_{j=1}^3{\mathrm{\partial }}_{x_j}^{2}g$; for a vector field $u$, set $({\mathrm{\nabla }}_{x}u{)}_{ij}={\mathrm{\partial }}_{x_j}{u}_i$. Choose fixed scales $L>0$ in metres, $U>0$ in metres per second, ${\rho }_{\ast }>0$ in kilograms per cubic metre, and ${\mathrm{\Phi }}_{\ast }>0$ in a specified phase unit $[\mathrm{\Phi }]$. Set $T_{\ast }=L\mathrm{/}U$ in seconds, so physical time is $t=T_{\ast }s$ and the interval is $[0,T_{\ast }S]$.

Let $\epsilon ,M,\nu >0$ be dimensionless coefficients and let $W\in C^2(\mathbb{R})$ be a dimensionless potential bounded below. Use the dimensionless classical field tuple from the earlier definition “Periodic Cahn–Hilliard incompressible flow system”: $v:[0,S]\times {\mathbb{T}}^3\to {\mathbb{R}}^3$ and $\varphi ,\pi :[0,S]\times {\mathbb{T}}^3\to \mathbb{R}$, with

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

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 $Q_3=[0,2\pi {]}^3$, and $C^1([0,S];C_{\mathrm{p}\mathrm{e}\mathrm{r}}^k)$ means time differentiability in the supremum norms of those spatial derivatives. Set ${\mathrm{\Delta }}_y={\sum }_{j=1}^3{\mathrm{\partial }}_{y_j}^2$. Define the dimensionless chemical-potential field

${\mu }_c={\epsilon }^{-1}{W}^{\mathrm{\prime }}(\varphi )-\epsilon {\mathrm{\Delta }}_y\varphi ,$

and require ${\mu }_c\in C([0,S];C_{\mathrm{p}\mathrm{e}\mathrm{r}}^2({\mathbb{T}}^3;\mathbb{R}))$. Define the corresponding 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 $p:[0,T_{\ast }S]\times {\mathrm{\Omega }}_L\to \mathbb{R}$ by

$u(t,x)=Uv(s,y),\phi (t,x)={\mathrm{\Phi }}_{\ast }\varphi (s,y),p(t,x)={\rho }_{\ast }{U}^2\pi (s,y),(s,y)=(t\mathrm{/}{T}_{\ast },x\mathrm{/}L)\mathrm{.}$

Thus $u$ has units metres per second, $\phi$ has unit $[\mathrm{\Phi }]$, and $p$ has units pascals.

Define the positive physical coefficients

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

Here $\beta$ is a bulk free-energy-density coefficient in joules per cubic metre, ${\lambda }_{\varphi }$ is a gradient free-energy coefficient in joules per metre per phase-unit squared, $\eta$ is dynamic viscosity in pascal seconds, and $\mathcal{M}$ has units $[\mathrm{\Phi }{]}^2{\mathrm{m}}^3\mathrm{s}\mathrm{k}{\mathrm{g}}^{-1}$. The physical chemical potential has units $\mathrm{J}{\mathrm{m}}^{-3}[\mathrm{\Phi }{]}^{-1}$ and the pressure has units pascals.

Define the bulk phase free-energy density, phase free-energy functional, and kinetic-plus-phase free-energy functional by

$f_b(\phi )=\beta W(\phi \mathrm{/}{\mathrm{\Phi }}_{\ast }),\mathcal{F}[\phi ]={\int }_{{\mathrm{\Omega }}_L}(f_b(\phi )+\frac{{\lambda }_{\varphi }}{2}\mathrm{\mid }{\mathrm{\nabla }}_x\phi {\mathrm{\mid }}^2)dx,$

$\mathcal{G}[u,\phi ]={\int }_{{\mathrm{\Omega }}_L}(\frac{{\rho }_{\ast }}{2}\mathrm{\mid }u{\mathrm{\mid }}^2+f_b(\phi )+\frac{{\lambda }_{\varphi }}{2}\mathrm{\mid }{\mathrm{\nabla }}_x\phi {\mathrm{\mid }}^2)dx\mathrm{.}$

Both functionals have units joules.

The dimensional chemical-potential field used by this model is

$\mu =f_b^{\mathrm{\prime }}(\phi )-{\lambda }_{\varphi }{\mathrm{\Delta }}_x\phi =\frac{\beta }{{\mathrm{\Phi }}_{\ast }}{W}^{\mathrm{\prime }}(\phi \mathrm{/}{\mathrm{\Phi }}_{\ast })-{\lambda }_{\varphi }{\mathrm{\Delta }}_x\phi \mathrm{.}$

The length, speed, density, and phase scales are model inputs; no measured calibration is assigned by this definition.

Dimensionless periodic phase-flow fields, regularity, chemical potential, and energy functional

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

V.E. Badalassi, H.D. Ceniceros, and S. Banerjee, §§2.1–2.4 (pp. 372–374)