Details for deriving constitutive laws
Differentials
The term \(\mathcal{I}\) is the differential of free energy density \(\mathcal{F}\):
(499)\[d\mathcal{F}(\phi,\boldsymbol{\nabla}\phi)=\frac{\partial\mathcal{F}}{\partial\phi}d\phi+\underbrace{\frac{\partial\mathcal{F}}{\partial(\boldsymbol{\nabla}\phi)}}_{\hat{=}\boldsymbol{\mathcal{F}}}\cdot d(\boldsymbol{\nabla}\phi)\]
where we note \(\boldsymbol{\mathcal{F}}\hat{=}\partial\mathcal{F}/\partial(\boldsymbol{\nabla}\phi)\). For \(\mathcal{F}\) defined by Eq. (476), \(\boldsymbol{\mathcal{F}}=\zeta\boldsymbol{\nabla}\phi\). For more generality we keep using the notation \(\boldsymbol{\mathcal{F}}\) in the rest of this section. After division of Eq. (499) by dt, we obtain:
(500)\[\frac{d\mathcal{F}(\phi,\boldsymbol{\nabla}\phi)}{dt}=\frac{\partial\mathcal{F}}{\partial\phi}{\frac{d\phi}{dt}}+\boldsymbol{\mathcal{F}}\cdot\underbrace{\frac{d(\boldsymbol{\nabla}\phi)}{dt}}_{\text{re-write}}\]
Replace \(d\phi/dt\) by Eq. (474) and re-write \(d(\boldsymbol{\nabla}\rho)/dt\)
Term \(\mathcal{K}\) kinetic energy: use impulsion balance Eq. (473)
(501)\[\frac{1}{2}\rho_{0}\frac{d\bigl|\boldsymbol{u}\bigr|^{2}}{dt}=\rho_{0}\boldsymbol{u}\cdot\frac{d\boldsymbol{u}}{dt}=\boldsymbol{u}\cdot\boldsymbol{\nabla}\cdot{\color{red}\overline{\overline{\boldsymbol{T}}}}\]
Algebraic calculation: usual tips and tricks
Tips and tricks
Trick 1
\[f\boldsymbol{\nabla}g=\boldsymbol{\nabla}(fg)-g\boldsymbol{\nabla}f\]Trick 2
\[f\boldsymbol{\nabla}f=\frac{1}{2}\boldsymbol{\nabla}f^{2}\]
Use chain rule
\[\boldsymbol{\nabla}f(\phi) =\frac{\partial f}{\partial\phi}\boldsymbol{\nabla}\phi=f^{\prime}(\phi)\boldsymbol{\nabla}\phi\]Use of index notations
Examples
Example 1
\[\mu_{\phi}\boldsymbol{\nabla}\phi =\boldsymbol{\nabla}f_{dw}-\zeta\boldsymbol{\nabla}^{2}\phi\boldsymbol{\nabla}\phi\]
Demo
\[\begin{split}\mu_{\phi}\boldsymbol{\nabla}\phi&=\left[\partial_{\phi}f_{dw}-\zeta\boldsymbol{\nabla}^{2}\phi\right]\boldsymbol{\nabla}\phi\\
&=\underbrace{\partial_{\phi}f_{dw}\boldsymbol{\nabla}\phi}_{\text{chain rule}}-\zeta\boldsymbol{\nabla}^{2}\phi\boldsymbol{\nabla}\phi\\
&=\boldsymbol{\nabla}f_{dw}-\zeta\boldsymbol{\nabla}^{2}\phi\boldsymbol{\nabla}\phi\end{split}\]
Example 2
\[\boldsymbol{\nabla}\cdot(\boldsymbol{\nabla}\phi\otimes\boldsymbol{\nabla}\phi)=\frac{1}{2}\boldsymbol{\nabla}(\bigl|\boldsymbol{\nabla}\phi\bigr|^{2})+(\boldsymbol{\nabla}^{2}\phi)\boldsymbol{\nabla}\phi\]
Demo
\[\begin{split}\partial_{\beta}(\partial_{\alpha}\phi\partial_{\beta}\phi)&=(\underbrace{\partial_{\beta}\partial_{\alpha}}_{\text{intervert}}\phi)(\partial_{\beta}\phi)+(\partial_{\alpha}\phi)(\partial_{\beta\beta}^{2}\phi)\\
&=\underbrace{[\partial_{\alpha}(\partial_{\beta}\phi)](\partial_{\beta}\phi)}_{\text{form }f\boldsymbol{\nabla}f=\boldsymbol{\nabla}f^{2}/2}+(\partial_{\alpha}\phi)(\partial_{\beta\beta}^{2}\phi)\\
&=\partial_{\alpha}(\partial_{\beta}\phi)^{2}/2+(\partial_{\alpha}\phi)(\partial_{\beta\beta}^{2}\phi)\end{split}\]
Other useful relations
Relation 1
\[\boxed{\frac{d}{dt}(\boldsymbol{\nabla}\phi)=\boldsymbol{\nabla}\left(\frac{d\phi}{dt}\right)-(\boldsymbol{\nabla}\boldsymbol{u})(\boldsymbol{\nabla}\phi)}\]
Demo
\[\begin{split}\frac{d}{dt}(\boldsymbol{\nabla}\phi)&=\frac{d}{dt}(\partial_{\alpha}\phi)\\
&=(\partial_{t}+u_{\beta}\partial_{\beta})(\partial_{\alpha}\phi)\\
&=\underbrace{\partial_{t}(\partial_{\alpha}}_{\text{intervert}}\phi)+u_{\beta}\underbrace{\partial_{\beta}(\partial_{\alpha}}_{\text{intervert}}\phi)\\
&=\partial_{\alpha}(\partial_{t}\phi)+\underbrace{u_{\beta}\partial_{\alpha}(\partial_{\beta}\phi)}_{\text{trick}}\\
&=\partial_{\alpha}(\partial_{t}\phi)+\partial_{\alpha}[u_{\beta}(\partial_{\beta}\phi)]-(\partial_{\alpha}u_{\beta})(\partial_{\beta}\phi)\\
&=\partial_{\alpha}[(\partial_{t}\phi)+u_{\beta}(\partial_{\beta}\phi)]-(\partial_{\alpha}u_{\beta})(\partial_{\beta}\phi)\\
&=\boldsymbol{\nabla}\left(\frac{d\phi}{dt}\right)-(\boldsymbol{\nabla}\boldsymbol{u})(\boldsymbol{\nabla}\phi)\end{split}\]
Relation 2
\[\boxed{\boldsymbol{\mathcal{F}}\cdot\frac{d}{dt}(\boldsymbol{\nabla}\phi)=\boldsymbol{\mathcal{F}}\cdot\boldsymbol{\nabla}\left(\frac{d\phi}{dt}\right)-\boldsymbol{\mathcal{F}}\otimes\boldsymbol{\nabla}\phi:\boldsymbol{\nabla}\boldsymbol{u}}\]
Demo (use Eq. ([eq:Trick_Material-Derivative-1]))
\[\begin{split}\boldsymbol{\mathcal{F}}\cdot\frac{d}{dt}(\boldsymbol{\nabla}\phi)&={\color{gray}\boldsymbol{\mathcal{F}}\cdot\boldsymbol{\nabla}\left(\frac{d\phi}{dt}\right)}-\mathcal{F}_{\alpha}(\partial_{\alpha}u_{\beta})(\partial_{\beta}\phi)\\
&={\color{gray}\boldsymbol{\mathcal{F}}\cdot\boldsymbol{\nabla}\left(\frac{d\phi}{dt}\right)}-\underbrace{\mathcal{F}_{\alpha}(\partial_{\beta}\phi)}_{\equiv\boldsymbol{\mathcal{F}}\otimes\boldsymbol{\nabla}\phi}\underbrace{(\partial_{\alpha}u_{\beta})}_{\equiv\boldsymbol{\nabla}\boldsymbol{u}}\end{split}\]
Relation 3
\[\boxed{\boldsymbol{\nabla}\cdot\boldsymbol{u}=\overline{\overline{\boldsymbol{I}}}:\boldsymbol{\nabla}\boldsymbol{u}}\]
Demo
\[\begin{split}\overline{\overline{\boldsymbol{I}}}:\boldsymbol{\nabla}\boldsymbol{u}&=\delta_{\alpha\beta}\partial_{\alpha}u_{\beta}\\
&=\partial_{\alpha}(u_{\beta}\delta_{\alpha\beta})-u_{\beta}\cancel{\partial_{\alpha}\delta_{\alpha\beta}}\\
&=\partial_{\alpha}u_{\alpha}\end{split}\]
Derivation
Terms \(\mathcal{I}\) and \(\mathcal{K}\) of Eq. ([eq:Reynolds-Transport_Appli])
\[\begin{split}\int_{V}\mathcal{I}dV&=\int_{V}\biggl[\frac{\partial\mathcal{F}}{\partial\phi}\frac{d\phi}{dt}+\underbrace{\boldsymbol{\mathcal{F}}\cdot\boldsymbol{\nabla}\left(\frac{d\phi}{dt}\right)}_{\text{integration by parts}}-\boldsymbol{\mathcal{F}}\otimes\boldsymbol{\nabla}\phi:\boldsymbol{\nabla}\boldsymbol{u}+\mathcal{F}\boldsymbol{\nabla}\cdot\boldsymbol{u}\biggr]dV\\ \int_{V}(\mathcal{K}+\mathscr{L})dV&=\int_{V}[\underbrace{\boldsymbol{u}\cdot\boldsymbol{\nabla}\cdot{\color{red}\overline{\overline{\boldsymbol{T}}}}}_{\text{ibp}}]-[\lambda\boldsymbol{\nabla}\cdot\boldsymbol{u}]dV\end{split}\]Results of integration by parts (ibp):
\[\begin{split}\int_{V}\left[\boldsymbol{\mathcal{F}}\cdot\boldsymbol{\nabla}\left(\frac{d\phi}{dt}\right)\right]dV&=\int_{\partial V}\frac{d\phi}{dt}\boldsymbol{\mathcal{F}}\cdot\boldsymbol{n}d(\partial V)-\int_{V}\boldsymbol{\nabla}\cdot\boldsymbol{\mathcal{F}}\frac{d\phi}{dt}dV\\ -\int_{V}\frac{\partial\mathcal{F}}{\partial\phi}\boldsymbol{\nabla}\cdot\boldsymbol{j}_{\phi}dV&=-\int_{\partial V}\frac{\partial\mathcal{F}}{\partial\phi}\boldsymbol{j}_{\phi}\cdot\boldsymbol{n}d(\partial V)+\int_{V}\boldsymbol{j}_{\phi}\cdot\boldsymbol{\nabla}\left(\frac{\partial\mathcal{F}}{\partial\phi}\right)dV\\ \int_{V}\boldsymbol{u}\cdot\boldsymbol{\nabla}\cdot\overline{\overline{\boldsymbol{T}}}dV&=\int_{\partial V}\boldsymbol{u}\cdot\overline{\overline{\boldsymbol{T}}}\boldsymbol{n}d(\partial V)-\int_{V}\overline{\overline{\boldsymbol{T}}}:\boldsymbol{\nabla}\boldsymbol{u}dV\end{split}\]Hypothesis: all \(\int_{\partial V}d(\partial V)\) terms are neglected
Group terms \(\cdot\boldsymbol{\nabla}\mu_{\phi}\) and \(:\boldsymbol{\nabla}\boldsymbol{u}\)
Sum Eq. ([eq:Integral_I])+Eq. ([eq:Integrals_K+L])
\[\begin{split}\int_{V}(\mathcal{I}+\mathcal{K}+\mathcal{L})dV&=\int_{V}\biggl\{\biggl[\underbrace{\frac{\partial\mathcal{F}}{\partial\phi}-\boldsymbol{\nabla}\cdot\boldsymbol{\mathcal{F}}}_{\equiv\mu_{\phi}}\biggr]\underbrace{\frac{d\phi}{dt}}_{\text{Eq. }(\ref{eq:Balance_phi_Incompr})}-\boldsymbol{\mathcal{F}}\otimes\boldsymbol{\nabla}\phi:\boldsymbol{\nabla}\boldsymbol{u}+\mathcal{F}\boldsymbol{\nabla}\cdot\boldsymbol{u}-{\color{red}\overline{\overline{\boldsymbol{T}}}}:\boldsymbol{\nabla}\boldsymbol{u}-\lambda\boldsymbol{\nabla}\cdot\boldsymbol{u}\biggr\} dV\\ &=\int_{V}\biggl\{\mu_{\phi}\biggl[-\phi\boldsymbol{\nabla}\cdot\boldsymbol{u}-\boldsymbol{\nabla}\cdot{\color{red}\boldsymbol{j}_{\phi}}\biggr]{\color{gray}-\boldsymbol{\mathcal{F}}\otimes\boldsymbol{\nabla}\phi:\boldsymbol{\nabla}\boldsymbol{u}+\mathcal{F}\boldsymbol{\nabla}\cdot\boldsymbol{u}-{\color{red}\overline{\overline{\boldsymbol{T}}}}:\boldsymbol{\nabla}\boldsymbol{u}-\lambda\boldsymbol{\nabla}\cdot\boldsymbol{u}}\biggr\} dV\\ &=\int_{V}\biggl\{(-\mu_{\phi}\phi+\mathcal{F}-\lambda)\underbrace{\boldsymbol{\nabla}\cdot\boldsymbol{u}}_{\text{Eq. }(\ref{eq:Equiv-divu})}-\biggl[\boldsymbol{\mathcal{F}}\otimes\boldsymbol{\nabla}\phi+{\color{red}\overline{\overline{\boldsymbol{T}}}}\biggr]:\boldsymbol{\nabla}\boldsymbol{u}-\mu_{\phi}\boldsymbol{\nabla}\cdot{\color{red}\boldsymbol{j}_{\phi}}\biggr\} dV\\ &=\int_{V}\biggl\{\biggl[(-\mu_{\phi}\phi+\mathcal{F}-\lambda)\overline{\overline{\boldsymbol{I}}}-\boldsymbol{\mathcal{F}}\otimes\boldsymbol{\nabla}\phi-{\color{red}\overline{\overline{\boldsymbol{T}}}}\biggr]:\boldsymbol{\nabla}\boldsymbol{u}-\underbrace{\mu_{\phi}\boldsymbol{\nabla}\cdot{\color{red}\boldsymbol{j}_{\phi}}}_{\text{ibp}}\biggr\} dV \\ &=-\int_{V}\biggl\{\biggl[\underbrace{(\mu_{\phi}\phi-\mathcal{F}+\lambda)\overline{\overline{\boldsymbol{I}}}+\boldsymbol{\mathcal{F}}\otimes\boldsymbol{\nabla}\phi+{\color{red}\overline{\overline{\boldsymbol{T}}}}}_{\overline{\overline{\boldsymbol{T}}}\text{ def such as }=-\overline{\overline{\boldsymbol{P}}}+\overline{\overline{\boldsymbol{\tau}}}}\biggr]:\boldsymbol{\nabla}\boldsymbol{u}dV+\int_{V}\underbrace{{\color{red}\boldsymbol{j}_{\phi}}\cdot\boldsymbol{\nabla}\mu_{\phi}}_{\boldsymbol{j}_{\phi}\text{ def such as }\propto-\boldsymbol{\nabla}\mu_{\phi}}dV\end{split}\]Appropriate choice of \(\boldsymbol{j}_{\phi}\) and \(\overline{\overline{\boldsymbol{T}}}\)
Appropriate choice of \(\boldsymbol{j}_{\phi}\) and \(\overline{\overline{\boldsymbol{T}}}\)
\[\begin{split}\boldsymbol{j}_{\phi}&=-\mathcal{M}_{\phi}\boldsymbol{\nabla}\mu_{\phi}\\ \overline{\overline{\boldsymbol{T}}}&=-\overline{\overline{\boldsymbol{P}}}+\overline{\overline{\boldsymbol{\tau}}}\\ \overline{\overline{\boldsymbol{\tau}}}&=\eta(\boldsymbol{\nabla}\boldsymbol{u}+\boldsymbol{\nabla}\boldsymbol{u}^{T})\end{split}\]where pressure tensor \(\overline{\overline{\boldsymbol{P}}}\) and hydrodynamic pressure \(p_{h}\)
\[\begin{split}\overline{\overline{\boldsymbol{P}}}&=(p_{h}-\mathcal{F})\overline{\overline{\boldsymbol{I}}}-\boldsymbol{\mathcal{F}}\otimes\boldsymbol{\nabla}\phi\\ p_{h}&=\lambda+\phi\mu_{\phi}\end{split}\]are appropriate constitutive laws such as the dissipation is positive
\[\mathcal{D}=\int_{V}\overline{\overline{\boldsymbol{T}}}:\boldsymbol{\nabla}\boldsymbol{u}dV-\int_{V}\boldsymbol{j}_{\phi}\cdot\boldsymbol{\nabla}\mu_{\phi}dV\geqslant0\]Remarks:
\(\overline{\overline{\boldsymbol{\tau}}}\): viscous stress tensor is such as \(\overline{\overline{\boldsymbol{\tau}}}:\boldsymbol{\nabla}\boldsymbol{u}\geqslant0\)
External force \(\rho\boldsymbol{g}\) could have been considered and set in \(\mathcal{W}(V)\)
If the integrals of surface are not neglected they must be considered in \(\Phi(\partial V)\)
Section author: Alain Cartalade