Model of fluid flow for three immiscible phases

That model is an extension of Model of Navier-Stokes/Conservative Allen-Cahn (CAC)/Composition for three immiscible fluids. The philosophy is identical as Model of Navier-Stokes/Allen-Cahn/Composition interacting with a solid phase. Two other phase-fields are introduced to model every fluid phase: \(\phi_{k}\) for \(k=0,1,2\). Two PDEs are solved, the first one for \(\phi_{1}\) and the second one for \(\phi_{2}\). The last phase-field is obtained by the conservation principle which writes:

(118)\[\phi_{0}=1-\phi_{1}-\phi_{2}\]

In what follows, the notation \(\boldsymbol{\phi}=(\phi_{0},\phi_{1},\phi_{2})\) will be used.

Mathematical model

Mass balance

The mass balance equation writes

(119)\[\boldsymbol{\nabla}\cdot\boldsymbol{u}=0\]

where \(\boldsymbol{u}=(u_{x},u_{y},u_{z})\) is the velocity vector.

Impulsion balance

The impulsion balance equation is

(120)\[\varrho(\boldsymbol{\phi})\left[\frac{\partial\boldsymbol{u}}{\partial t}+\boldsymbol{\nabla}\cdot(\boldsymbol{u}\boldsymbol{u})\right]=-\boldsymbol{\nabla}p_{h}+\boldsymbol{\nabla}\cdot\left[\varrho\vartheta(\boldsymbol{\phi})\left(\boldsymbol{\nabla}\boldsymbol{u}+\boldsymbol{\nabla}\boldsymbol{u}^{T}\right)\right]+\boldsymbol{F}_{tot}\]

where \(\varrho(\boldsymbol{\phi})\) is the total density and \(\vartheta(\boldsymbol{\phi})\) is the total viscosity. They are respectively defined by Eqs (130) and (131). The hydrodynamic pressure is noted \(p_{h}\) and \(\boldsymbol{F}_{tot}\) is the total force defined by Eq. (125).

First phase-field equation

(121)\[\frac{\partial\phi_{1}}{\partial t}+\boldsymbol{\nabla}\cdot(\boldsymbol{u}\phi_{1})=\boldsymbol{\nabla}\cdot\Bigl[M_{\phi}\bigl(\boldsymbol{\nabla}\phi_{1}-\bigl|\boldsymbol{\nabla}\phi_{1}\bigr|^{eq}\boldsymbol{n}_{1}+\mathscr{L}(\boldsymbol{\phi})\bigr)\Bigr]\]

where the Lagrange multiplier is defined by

\[\mathscr{L}(\boldsymbol{\phi})=\frac{1}{3}\sum\bigl|\boldsymbol{\nabla}\phi_{k}\bigr|^{eq}\boldsymbol{n}_{k}\]

and the unit normal vector for each phase \(k\) is defined by

(122)\[\boldsymbol{n}_{k}=\frac{\boldsymbol{\nabla}\phi}{\bigl|\boldsymbol{\nabla}\phi\bigr|}\]

Second phase-field equation

(123)\[\frac{\partial\phi_{2}}{\partial t}+\boldsymbol{\nabla}\cdot(\boldsymbol{u}\phi_{2})=\boldsymbol{\nabla}\cdot\Bigl[M_{\phi}\bigl(\boldsymbol{\nabla}\phi_{2}-\bigl|\boldsymbol{\nabla}\phi_{2}\bigr|^{eq}\boldsymbol{n}_{2}+\mathscr{L}(\boldsymbol{\phi})\bigr)\Bigr]\]

Composition equation

(124)\[\frac{\partial c}{\partial t}+\boldsymbol{\nabla}\cdot(\boldsymbol{u}c)=\boldsymbol{\nabla}\cdot\left\{ (D_{k}\phi_{k})\boldsymbol{\nabla}\bigl[\mu_{c}^{eq}+c(\phi,\mu_{c})-c_{k}^{eq}\phi_{k}\bigr]\right\}\]

Note

Compared to the Model of Navier-Stokes/Allen-Cahn/Composition interacting with a solid phase, the differences come from the form of PDE on \(\phi_2\) (resp. \(\psi\)) and the form of the Lagrange multiplier \(\mathscr{L}(\boldsymbol{\phi})\).

Force terms

Total force

The total force is the sum of the capillary force \(\boldsymbol{F}_{c}\) and the gravity force \(\boldsymbol{F}_{g}\):

(125)\[\boldsymbol{F}_{tot}=\boldsymbol{F}_{c}+\boldsymbol{F}_{g}\]

Capillary forces

The capillary force is the sum of contribution of every phase-field:

(126)\[\boldsymbol{F}_{c}=\mu_{\phi_{0}}\boldsymbol{\nabla}\phi_{0}+\mu_{\phi_{1}}\boldsymbol{\nabla}\phi_{1}+\mu_{\phi_{2}}\boldsymbol{\nabla}\phi_{2}\]

where the chemical potential \(\mu_{\phi_{k}}\) for each phase \(k\) is defined by

(127)\[\mu_{\phi_{k}}(\boldsymbol{x},t)=\frac{4\gamma_{T}}{W}\sum_{\ell\neq k}\left[\frac{1}{\gamma_{\ell}}\left(\frac{\partial f_{dw}}{\partial\phi_{k}}-\frac{\partial f_{dw}}{\partial\phi_{\ell}}\right)\right]-\frac{3}{4}W\gamma_{k}\boldsymbol{\nabla}^{2}\phi_{k}\]

and the spreading coefficient \(\gamma_{T}\) is the harmonic mean of each one:

(128)\[\frac{3}{\gamma_{T}}=\sum_{k}\frac{1}{\gamma_{k}}\]

and the double-well potential is:

\[f_{dw}(\phi_{0},\phi_{1},\phi_{2})=\sum_{k=0}^{2}\frac{12}{W}\left[\frac{\gamma_{k}}{2}\phi_{k}^{2}(1-\phi_{k})^{2}\right]\]

The spreading coefficient of each phase is defined by a combination of their surface tensions:

\[\begin{split}\begin{aligned} \gamma_{0} & =\sigma_{10}+\sigma_{20}-\sigma_{12}\\ \gamma_{1} & =\sigma_{10}+\sigma_{12}-\sigma_{20}\\ \gamma_{2} & =\sigma_{20}+\sigma_{12}-\sigma_{01}\end{aligned}\end{split}\]

Gravity force

The gravity force writes:

(129)\[\boldsymbol{F}_{g}=\varrho(\boldsymbol{\phi})\boldsymbol{g}\]

Closure terms

Total density

The total density is simply an interpolation of each bulk density:

(130)\[\varrho(\boldsymbol{\phi})=\sum_{k}\rho_{k}\phi_{k}(\boldsymbol{x},t)\]

Total kinematic viscosity

The total kinematic viscosity is obtained by a harmoni mean of bulk kinematic viscosities:

(131)\[\frac{1}{\vartheta(\boldsymbol{\phi})}=\sum_{k}\frac{\phi_{k}(\boldsymbol{x},t)}{\nu_{k}}\]

Examples of simulations with that model

Double-Serpentine

Spreading lenses

Spinodal decomposition of three immiscible fluids

Rayleigh-Taylor and splash of three immiscible fluids

Section author: Alain Cartalade