3.8 Weak Formulation for Fluid-Solutes Analyses

Virtual Work and Weak form for Fluid-Solutes

The virtual work statement is used to enforce the governing equations needed to solve for the nodal \(\mathbf{v}^{f}\), \(e^{f}\), and all \(\tilde{c}^{\iota}\), namely the mixture mass balance (2.16-2), the fluid momentum balance (2.16-10), and the solute mass balances (2.16-3), in addition to the current density constraint (2.16-14). The virtual work statement for a Galerkin finite element formulation ¹ is \(\delta W=0\), where

\[ \begin{equation} \begin{aligned}\delta W= & \int_{\Omega^{f}}\delta\mathbf{v}^{f}\cdot\left(-\grad\tilde{p}+\divg\boldsymbol{\tau}-\sum_{\iota}R\theta\tilde{\kappa}^{\iota}\grad\tilde{c}^{\iota}+\rho^{f}\left(\mathbf{b}^{f}-\mathbf{a}^{f}\right)+M^{\iota}c^{\iota}\mathbf{b}^{\iota}\right)dv\\ & +\int_{\Omega^{f}}\delta J^{f}\left(-\frac{1}{J^{f}}\frac{D^{f}J^{f}}{Dt}+\divg\mathbf{v}^{f}\right)dv\\ & +\sum_{\iota}\int_{\Omega^{f}}\delta\tilde{c}^{\iota}\left(\frac{1}{J^{f}}\frac{D^{f}\left(J^{f}c^{\iota}\right)}{Dt}+\divg\mathbf{j}^{\iota}-\hat{c}^{\iota}+\sum_{\beta}z^{\beta}\divg\mathbf{j}^{\iota}\right)dv\,. \end{aligned} \label{eq:CFDSol-Virtual-Work} \end{equation} \]

These integrals are evaluated in the current (time-invariant) configuration of \(\Omega^{f}\). Here, \(\delta\mathbf{v}^{f}\) is the virtual fluid velocity, \(\delta J^{f}\) is the virtual fluid energy density, and \(\delta\tilde{c}^{\iota}\) is the virtual molar energy of solute \(\iota\). Integrating by parts and using the divergence theorem, the weak form of this statement may be written as \(0=\delta W=\delta W_{ext}-\delta W_{int}\) where the internal virtual work is

\[ \begin{equation} \begin{aligned}\delta W_{int}= & \int_{\Omega^{f}}\delta\mathbf{v}^{f}\cdot\left(\grad\tilde{p}+\rho^{f}\mathbf{a}^{f}+\sum_{\iota}R\theta\tilde{\kappa}^{\iota}\grad\tilde{c}^{\iota}\right)+\boldsymbol{\tau}:\grad\delta\mathbf{v}^{f}dv\\ & +\int_{\Omega^{f}}\delta J^{f}\frac{1}{J^{f}}\frac{D^{f}J^{f}}{Dt}+\grad\delta J^{f}\cdot\mathbf{v}^{f}dv\\ & +\sum_{\iota}\int_{\Omega^{f}}\left(\grad\delta\tilde{c}^{\iota}\cdot\left(\mathbf{j}_{d}^{\iota}+\sum_{\gamma}z^{\gamma}\mathbf{j}_{d}^{\gamma}\right)-\delta\tilde{c}^{\iota}\left(\frac{1}{J^{f}}\frac{D^{f}\left(J^{f}c^{\iota}\right)}{Dt}-\hat{c}^{\iota}\right)\right)dv\,. \end{aligned} \label{eq:Int-Virtual-Work-CFDSol} \end{equation} \]

The external virtual work is

\[ \begin{equation} \begin{aligned}\delta W_{ext}= & \int_{\Gamma^{f}}\delta\mathbf{v}^{f}\cdot\mathbf{t}^{\tau}da+\int_{\Gamma^{f}}\delta J^{f}v_{n}^{f}da+\int_{\Gamma^{f}}\delta\tilde{c}^{\iota}\tilde{j}_{n}^{\iota}da\\ & +\int_{\Omega^{f}}\delta\mathbf{v}^{f}\cdot\left(\rho^{f}\mathbf{b}^{f}+\sum_{\iota}M^{\iota}c^{\iota}\mathbf{b}^{\iota}\right)dv\\ & -\sum_{\iota}\int_{\Omega^{f}}\grad\delta\tilde{c}^{\iota}\cdot\tilde{\mathbf{j}}_{b}^{\iota}dv\,, \end{aligned} \label{eq:Ext-Virt-Work-CFDSol} \end{equation} \]

where

\[ \begin{equation} \tilde{j}_{n}^{\iota}=\left(\mathbf{j}^{\iota}+\sum_{\gamma}z^{\gamma}\mathbf{j}^{\gamma}\right)\cdot\mathbf{n}\,.\label{eq:Normal-Sol-Flux-CFDSol} \end{equation} \]

Here, \(\mathbf{t}^{\tau}=\boldsymbol{\tau}\cdot\mathbf{n}\) is the fluid viscous traction, \(v_{n}^{f}=\mathbf{v}^{f}\cdot\mathbf{n}\) is the normal component of the fluid velocity on the boundary \(\Gamma^{f}\), and \(\tilde{j}_{n}^{\iota}\) is the normal effective diffusive flux of solute \(\iota\) on the boundary \(\Gamma^{f}\), where \(\mathbf{n}\) is the outward normal to \(\Gamma^{f}\). The integrands of the surface integrals represent the natural boundary conditions for this formulation. If boundary conditions are not set explicitly on \(\Gamma^{f}\), the natural boundary conditions are \(\mathbf{t}^{\tau}=\mathbf{0}\), \(v_{n}^{f}=0\), and \(\tilde{j}_{n}^{\iota}=0\). These natural boundary conditions are consistent with the jump conditions presented above in (2.16-18), (2.16-16), and (2.16-17).

Essential boundary conditions are prescribed on the fluid velocity \(\mathbf{v}^{f}\), fluid dilatation \(e^{f}\), and effective solute concentration \(\tilde{c}^{\iota}\), which are also consistent with the above jump conditions in (2.16-15), (2.16-16), and (2.16-19). For example, an essential no-slip boundary condition may be prescribed on \(\Gamma^{f}\) by setting \(\mathbf{v}^{f}=\mathbf{0}\). A symmetry plane may be prescribed from the natural boundary conditions \(\mathbf{t}^{\tau}=\mathbf{0}\), \(v_{n}^{f}=0\), and \(\tilde{j}_{n}^{\iota}=0\). In this formulation, the mixture traction is defined as \(\mathbf{t}=-p\mathbf{n}+\mathbf{t}^{\tau}\). Because of the way we chose to split the internal and external virtual work in \eqref{eq:Int-Virtual-Work-CFDSol}-\eqref{eq:Ext-Virt-Work-CFDSol}, \(\mathbf{t}\) is not a natural boundary condition in this formulation. Thus, when \(\mathbf{t}=\mathbf{0}\) is desired, it has to be enforced by satisfying the natural boundary condition \(\mathbf{t}^{\tau}=\mathbf{0}\), and selecting suitable values for the essential boundary conditions \(J^{f}\) and \(\tilde{c}^{\iota}\) using (2.16-11)-(2.16-12) to produce \(p=0\), as explained next.

The actual fluid gauge pressure in this fluid-solutes mixture is \(p\), as given in eq.(2.16-11). To prescribe a desired value \(p=p^{*}\) on a boundary, we implemented a “fluid pressure” boundary condition that prescribed the correct value of \(J^{f}\) according to the current values of all \(\tilde{c}^{\iota}\), using the user-selected constitutive model for \(\tilde{p}\left(J^{f}\right)\). For example, using eq.(2.16-12), this “fluid pressure” boundary condition actually prescribes \(J^{f}=1/\left(1+\left(p^{*}-R\theta\Phi\sum_{\iota}\tilde{\kappa}^{\iota}\tilde{c}^{\iota}\right)/K\right)\) as an essential boundary condition at nodes of \(\Gamma^{f}\). The values of \(\Phi\) and \(\tilde{\kappa}^{\iota}\) are obtained from the finite element of the domain \(\Omega^{f}\) right underneath the face \(\Gamma^{f}\).

Linearization for Fluid-Solutes

We can rewrite the internal work of eq.\eqref{eq:CFDSol-Virtual-Work} as

\[ \begin{aligned}\delta W_{int} & =\delta W_{int}^{v}+\delta W_{int}^{J}+\delta W_{int}^{c}\end{aligned} \]

where

\[ \begin{aligned}\delta W_{int}^{v} & =\int_{\Omega^{f}}\delta\mathbf{v}^{f}\cdot\left(\grad\tilde{p}+\rho^{f}\mathbf{a}^{f}+\sum_{\iota}R\theta\tilde{\kappa}^{\iota}\grad\tilde{c}^{\iota}\right)+\boldsymbol{\tau}:\grad\delta\mathbf{v}^{f}dv\\ \delta W_{int}^{J} & =\int_{\Omega^{f}}\delta J^{f}\frac{1}{J^{f}}\frac{D^{f}J^{f}}{Dt}+\grad\delta J^{f}\cdot\mathbf{v}^{f}dv\\ \delta W_{int}^{c} & =\sum_{\iota}\int_{\Omega^{f}}\left(\grad\delta\tilde{c}^{\iota}\cdot\left(\mathbf{j}_{d}^{\iota}+\sum_{\gamma}z^{\gamma}\mathbf{j}_{d}^{\gamma}\right)-\delta\tilde{c}^{\iota}\left(\frac{1}{J^{f}}\frac{D^{f}\left(J^{f}c^{\iota}\right)}{Dt}-\hat{c}^{\iota}\right)\right)dv\,. \end{aligned} \]

Recall that \(\hat{c}^{\iota}=\nu^{\iota}\hat{\zeta}\). Also note that

\[ \begin{aligned}\mathbf{a}^{f} & =\frac{\partial\mathbf{v}^{f}}{\partial t}+\grad\mathbf{v}^{f}\cdot\mathbf{v}^{f}\\ \frac{D^{f}J^{f}}{Dt} & =\frac{\partial J^{f}}{\partial t}+\grad J^{f}\cdot\mathbf{v}^{f}\\ \frac{D^{f}\left(J^{f}c^{\iota}\right)}{Dt} & =\tilde{c}^{\iota}\left(\tilde{\kappa}^{\iota}+J^{f}\frac{\partial\tilde{\kappa}^{\iota}}{\partial J^{f}}\right)\frac{D^{f}J^{f}}{Dt}+J^{f}\left(\tilde{\kappa}^{\iota}\frac{D^{f}\tilde{c}^{\iota}}{Dt}+\tilde{c}^{\iota}\sum_{\gamma}\frac{\partial\tilde{\kappa}^{\iota}}{\partial\tilde{c}^{\gamma}}\frac{D^{f}\tilde{c}^{\gamma}}{Dt}\right)\\ \frac{D^{f}c^{\iota}}{Dt} & =\frac{\partial c^{\iota}}{\partial t}+\grad c^{\iota}\cdot\mathbf{v}^{f}\\ & =\tilde{\kappa}^{\iota}\frac{\partial\tilde{c}^{\iota}}{\partial t}+\tilde{c}^{\iota}\frac{\partial\tilde{\kappa}^{\iota}}{\partial t}+\left(\tilde{\kappa}^{\iota}\grad\tilde{c}^{\iota}+\tilde{c}^{\iota}\grad\tilde{\kappa}^{\iota}\right)\cdot\mathbf{v}^{f}\\ \frac{\partial\tilde{\kappa}^{\iota}}{\partial t} & =\frac{\partial\tilde{\kappa}^{\iota}}{\partial J^{f}}\frac{\partial J^{f}}{\partial t}+\sum_{\gamma}\frac{\partial\tilde{\kappa}^{\iota}}{\partial\tilde{c}^{\gamma}}\frac{\partial\tilde{c}^{\gamma}}{\partial t}\\ \grad\tilde{\kappa}^{\iota} & =\frac{\partial\tilde{\kappa}^{\iota}}{\partial J^{f}}\grad J+\sum_{\gamma}\frac{\partial\tilde{\kappa}^{\iota}}{\partial\tilde{c}^{\gamma}}\grad\tilde{c}^{\gamma} \end{aligned} \]

Using the temporal discretization approach outlined in Section Temporal Discretization and Linearization, the linearizations of each of these terms along increments in the degrees of freedom are obtained using

\[ \begin{aligned}D\mathbf{a}^{f}\left[\Delta\mathbf{v}^{f}\right] & =\frac{\alpha_{m}}{\gamma}\frac{\Delta\mathbf{v}^{f}}{\Delta t}+\alpha_{f}\left(\grad\Delta\mathbf{v}^{f}\cdot\mathbf{v}^{f}+\mathbf{L}^{f}\cdot\Delta\mathbf{v}^{f}\right)\\ D\left(\frac{D^{f}J^{f}}{Dt}\right)\left[\Delta\mathbf{v}_{f}\right] & =\alpha_{f}\grad J^{f}\cdot\Delta\mathbf{v}_{f}\\ D\boldsymbol{\tau}\left[\Delta\mathbf{v}^{f}\right] & =\alpha_{f}\boldsymbol{\mathcal{C}}^{\tau}:\frac{1}{2}\left(\grad\Delta\mathbf{v}^{f}+\grad^{T}\Delta\mathbf{v}^{f}\right)\\ D\left(\frac{D^{f}\left(J^{f}c^{\iota}\right)}{Dt}\right)\left[\Delta\mathbf{v}^{f}\right] & =\alpha_{f}\grad\left(J^{f}c^{\iota}\right)\cdot\Delta\mathbf{v}^{f} \end{aligned} \]

\[ \begin{aligned}D\left(\grad\tilde{p}\right)\left[\Delta J^{f}\right] & =\alpha_{f}\left(\tilde{p}_{J}^{\prime}\grad\Delta J^{f}+\Delta J^{f}\tilde{p}_{J}^{\prime\prime}\grad J^{f}\right)\\ D\rho^{f}\left[\Delta J^{f}\right] & =-\alpha_{f}\frac{\rho^{f}}{J^{f}}\Delta J^{f}\\ D\boldsymbol{\tau}\left[\Delta J^{f}\right] & =\alpha_{f}\boldsymbol{\tau}_{J}^{\prime}\Delta J^{f}\\ D\left(\frac{D^{f}J^{f}}{Dt}\right)\left[\Delta J^{f}\right] & =\alpha_{f}\left(\frac{\alpha_{m}}{\alpha_{f}\gamma}\frac{\Delta J^{f}}{\Delta t}+\grad\Delta J^{f}\cdot\mathbf{v}^{f}\right)\\ D\tilde{\kappa}^{\iota}\left[\Delta J^{f}\right] & =\alpha_{f}\frac{\partial\tilde{\kappa}^{\iota}}{\partial J^{f}}\Delta J^{f}\\ D\left(\frac{D^{f}\left(J^{f}c^{\iota}\right)}{Dt}\right)\left[\Delta J^{f}\right] & =\alpha_{f}\tilde{c}^{\iota}\left(\tilde{\kappa}^{\iota}+J^{f}\frac{\partial\tilde{\kappa}^{\iota}}{\partial J^{f}}\right)\left(\frac{\alpha_{m}}{\alpha_{f}\gamma}\frac{\Delta J^{f}}{\Delta t}+\grad\Delta J^{f}\cdot\mathbf{v}^{f}\right)\\ & +\alpha_{f}\tilde{c}^{\iota}\frac{D^{f}J^{f}}{Dt}\left(2\frac{\partial\tilde{\kappa}^{\iota}}{\partial J^{f}}+J^{f}\frac{\partial^{2}\tilde{\kappa}^{\iota}}{\partial\left(J^{f}\right)^{2}}\right)\Delta J^{f}\\ & +\alpha_{f}\left(\tilde{\kappa}^{\iota}\frac{D^{f}\tilde{c}^{\iota}}{Dt}+\tilde{c}^{\iota}\sum_{\gamma}\frac{\partial\tilde{\kappa}^{\iota}}{\partial\tilde{c}^{\gamma}}\frac{D^{f}\tilde{c}^{\gamma}}{Dt}\right)\Delta J^{f}\\ & +\alpha_{f}J^{f}\left(\frac{\partial\tilde{\kappa}^{\iota}}{\partial J^{f}}\frac{D^{f}\tilde{c}^{\iota}}{Dt}+\tilde{c}^{\iota}\sum_{\gamma}\left(\frac{\partial^{2}\tilde{\kappa}^{\iota}}{\partial J^{f}\partial\tilde{c}^{\gamma}}\right)\frac{D^{f}\tilde{c}^{\gamma}}{Dt}\right)\Delta J^{f}\\ D\hat{c}^{\iota}\left[\Delta J^{f}\right] & =\alpha_{f}\nu^{\iota}\frac{\partial\hat{\zeta}}{\partial J^{f}}\Delta J^{f}\\ D\mathbf{j}_{d}^{\iota}\left[\Delta J^{f}\right] & =-\alpha_{f}\frac{\partial\tilde{\kappa}^{\iota}}{\partial J^{f}}\Delta J^{f}d_{0}^{\iota}\grad\tilde{c}^{\iota} \end{aligned} \]

\[ \begin{aligned}D\left(\tilde{\kappa}^{\iota}\tilde{c}^{\iota}\right)\left[\Delta\tilde{c}^{\beta}\right] & =\alpha_{f}\left(\delta_{\iota\beta}\tilde{\kappa}^{\beta}+\frac{\partial\tilde{\kappa}^{\iota}}{\partial\tilde{c}^{\beta}}\tilde{c}^{\iota}\right)\Delta\tilde{c}^{\beta}\\ D\left(\tilde{\kappa}^{\iota}\grad\tilde{c}^{\iota}\right)\left[\Delta\tilde{c}^{\beta}\right] & =\alpha_{f}\left(\delta_{\iota\beta}\tilde{\kappa}^{\beta}\grad\Delta\tilde{c}^{\beta}+\frac{\partial\tilde{\kappa}^{\iota}}{\partial\tilde{c}^{\beta}}\Delta\tilde{c}^{\beta}\grad\tilde{c}^{\iota}\right)\\ D\left(\frac{D^{f}\left(J^{f}c^{\iota}\right)}{Dt}\right)\left[\Delta\tilde{c}^{\beta}\right] & =\alpha_{f}J^{f}\left(\delta_{\iota\beta}\tilde{\kappa}^{\beta}+\tilde{c}^{\iota}\frac{\partial\tilde{\kappa}^{\iota}}{\partial\tilde{c}^{\beta}}\right)\left(\left(\frac{\alpha_{m}}{\alpha_{f}\gamma}\frac{1}{\Delta t}+\frac{1}{J^{f}}\frac{D^{f}J^{f}}{Dt}\right)\Delta\tilde{c}^{\beta}+\grad\Delta\tilde{c}^{\beta}\cdot\mathbf{v}^{f}\right)\\ D\hat{c}^{\iota}\left[\Delta\tilde{c}^{\beta}\right] & =\alpha_{f}\nu^{\iota}\frac{\partial\hat{\zeta}}{\partial\tilde{c}^{\beta}}\Delta\tilde{c}^{\beta}\\ D\mathbf{j}_{d}^{\iota}\left[\Delta\tilde{c}^{\beta}\right] & =-\alpha_{f}\left(\delta_{\iota\beta}\tilde{\kappa}^{\beta}d_{0}^{\beta}\grad\Delta\tilde{c}^{\beta}+\Delta\tilde{c}^{\beta}\left(\frac{\partial\tilde{\kappa}^{\iota}}{\partial\tilde{c}^{\beta}}d_{0}^{\iota}+\tilde{\kappa}^{\iota}\frac{\partial d_{0}^{\iota}}{\partial\tilde{c}^{\beta}}\right)\grad\tilde{c}^{\iota}\right) \end{aligned} \]

Then we can evaluate the linearizations as follows,

\[ \begin{aligned}D\left(\delta W_{int}^{v}\right)\left[\Delta\mathbf{v}^{f}\right] & =\alpha_{f}\int_{\Omega^{f}}\delta\mathbf{v}^{f}\cdot\rho^{f}\left(\frac{\alpha_{m}}{\alpha_{f}\gamma}\frac{\Delta\mathbf{v}^{f}}{\Delta t}+\left(\grad\Delta\mathbf{v}^{f}\cdot\mathbf{v}^{f}+\mathbf{L}^{f}\cdot\Delta\mathbf{v}^{f}\right)\right)dv\\ & +\alpha_{f}\int_{\Omega^{f}}\grad\delta\mathbf{v}^{f}:\boldsymbol{\mathcal{C}}^{\tau}:\frac{1}{2}\left(\grad\Delta\mathbf{v}^{f}+\grad^{T}\Delta\mathbf{v}^{f}\right)dv\\ D\left(\delta W_{int}^{v}\right)\left[\Delta J^{f}\right] & =\alpha_{f}\int_{\Omega^{f}}\delta\mathbf{v}^{f}\cdot\left(\tilde{p}_{J}^{\prime}\grad\Delta J^{f}+\Delta J^{f}\tilde{p}_{J}^{\prime\prime}\grad J^{f}+\Delta J^{f}\sum_{\iota}R\theta\frac{\partial\tilde{\kappa}^{\iota}}{\partial J^{f}}\grad\tilde{c}^{\iota}\right)dv\\ & -\alpha_{f}\int_{\Omega^{f}}\delta\mathbf{v}^{f}\cdot\frac{\rho^{f}}{J^{f}}\Delta J^{f}\mathbf{a}^{f}dv+\alpha_{f}\int_{\Omega^{f}}\boldsymbol{\tau}_{J}^{\prime}\Delta J^{f}:\grad\delta\mathbf{v}^{f}dv\\ D\left(\delta W_{int}^{v}\right)\left[\Delta\tilde{c}^{\beta}\right] & =\alpha_{f}\int_{\Omega^{f}}\delta\mathbf{v}^{f}\cdot R\theta\sum_{\iota}\left(\delta_{\iota\beta}\tilde{\kappa}^{\beta}\grad\Delta\tilde{c}^{\beta}+\frac{\partial\tilde{\kappa}^{\iota}}{\partial\tilde{c}^{\beta}}\Delta\tilde{c}^{\beta}\grad\tilde{c}^{\iota}\right)dv \end{aligned} \]

\[ \begin{aligned}D\left(\delta W_{int}^{J}\right)\left[\Delta\mathbf{v}^{f}\right] & =\alpha_{f}\int_{\Omega^{f}}\left(\delta J^{f}\frac{1}{J^{f}}\grad J^{f}+\grad\delta J^{f}\right)\cdot\Delta\mathbf{v}^{f}dv\\ D\left(\delta W_{int}^{J}\right)\left[\Delta J^{f}\right] & =\alpha_{f}\int_{\Omega^{f}}\delta J^{f}\frac{1}{J^{f}}\left(\left(\frac{\alpha_{m}}{\alpha_{f}\gamma}\frac{1}{\Delta t}-\frac{1}{J^{f}}\frac{D^{f}J^{f}}{Dt}\right)\Delta J^{f}+\grad\Delta J^{f}\cdot\mathbf{v}^{f}\right)dv\\ D\left(\delta W_{int}^{J}\right)\left[\Delta\tilde{c}^{\beta}\right] & =0 \end{aligned} \]

\[ \begin{aligned}D\left(\delta W_{int}^{c}\right)\left[\Delta\mathbf{v}^{f}\right] & =\sum_{\iota}\int_{\Omega^{f}}-\delta\tilde{c}^{\iota}\frac{1}{J^{f}}\grad\left(J^{f}c^{\iota}\right)\cdot\alpha_{f}\Delta\mathbf{v}_{f}\\ D\left(\delta W_{int}^{c}\right)\left[\Delta J^{f}\right] & =\sum_{\iota}\int_{\Omega^{f}}\delta\tilde{c}^{\iota}\frac{\alpha_{f}\Delta J^{f}}{\left(J^{f}\right)^{2}}\frac{D^{f}\left(J^{f}c^{\iota}\right)}{Dt}\\ & -\delta\tilde{c}^{\iota}\alpha_{f}\frac{\tilde{c}^{\iota}}{J^{f}}\left(\tilde{\kappa}^{\iota}+J^{f}\frac{\partial\tilde{\kappa}^{\iota}}{\partial J^{f}}\right)\left(\frac{\alpha_{m}}{\alpha_{f}\gamma}\frac{\Delta J^{f}}{\Delta t}+\grad\Delta J^{f}\cdot\mathbf{v}^{f}\right)\\ & -\delta\tilde{c}^{\iota}\alpha_{f}\frac{\tilde{c}^{\iota}}{J^{f}}\frac{D^{f}J^{f}}{Dt}\left(2\frac{\partial\tilde{\kappa}^{\iota}}{\partial J^{f}}+J^{f}\frac{\partial^{2}\tilde{\kappa}^{\iota}}{\partial\left(J^{f}\right)^{2}}\right)\Delta J^{f}\\ & -\delta\tilde{c}^{\iota}\alpha_{f}\frac{1}{J^{f}}\left(\tilde{\kappa}^{\iota}\frac{D^{f}\tilde{c}^{\iota}}{Dt}+\tilde{c}^{\iota}\sum_{\gamma}\frac{\partial\tilde{\kappa}^{\iota}}{\partial\tilde{c}^{\gamma}}\frac{D^{f}\tilde{c}^{\gamma}}{Dt}\right)\Delta J^{f}\\ & -\delta\tilde{c}^{\iota}\alpha_{f}\left(\frac{\partial\tilde{\kappa}^{\iota}}{\partial J^{f}}\frac{D^{f}\tilde{c}^{\iota}}{Dt}+\tilde{c}^{\iota}\sum_{\gamma}\left(\frac{\partial^{2}\tilde{\kappa}^{\iota}}{\partial J^{f}\partial\tilde{c}^{\gamma}}\right)\frac{D^{f}\tilde{c}^{\gamma}}{Dt}\right)\Delta J^{f}\\ & +\alpha_{f}\delta\tilde{c}^{\iota}\nu^{\iota}\frac{\partial\hat{\zeta}}{\partial J^{f}}\Delta J^{f}-\alpha_{f}\grad\delta\tilde{c}^{\iota}\cdot\left(\frac{\partial\tilde{\kappa}^{\iota}}{\partial J^{f}}d_{0}^{\iota}\grad\tilde{c}^{\iota}+\sum_{\gamma}z^{\gamma}\frac{\partial\tilde{\kappa}^{\gamma}}{\partial J^{f}}d_{0}^{\gamma}\grad\tilde{c}^{\gamma}\right)\Delta J^{f}dv\\ D\left(\delta W_{int}^{c}\right)\left[\Delta\tilde{c}^{\beta}\right] & =\sum_{\iota}\int_{\Omega^{f}}-\delta\tilde{c}^{\iota}\alpha_{f}\left(\delta_{\iota\beta}\tilde{\kappa}^{\beta}+\tilde{c}^{\iota}\frac{\partial\tilde{\kappa}^{\iota}}{\partial\tilde{c}^{\beta}}\right)\left(\left(\frac{\alpha_{m}}{\alpha_{f}\gamma}\frac{1}{\Delta t}+\frac{1}{J^{f}}\frac{D^{f}J^{f}}{Dt}\right)\Delta\tilde{c}^{\beta}+\grad\Delta\tilde{c}^{\beta}\cdot\mathbf{v}^{f}\right)\\ & +\delta\tilde{c}^{\iota}\alpha_{f}\nu^{\iota}\frac{\partial\hat{\zeta}}{\partial\tilde{c}^{\beta}}\Delta\tilde{c}^{\beta}\\ & -\alpha_{f}\grad\delta\tilde{c}^{\iota}\cdot\left(\delta_{\iota\beta}\tilde{\kappa}^{\beta}d_{0}^{\beta}\grad\Delta\tilde{c}^{\beta}+\Delta\tilde{c}^{\beta}\left(\frac{\partial\tilde{\kappa}^{\iota}}{\partial\tilde{c}^{\beta}}d_{0}^{\iota}+\tilde{\kappa}^{\iota}\frac{\partial d_{0}^{\iota}}{\partial\tilde{c}^{\beta}}\right)\grad\tilde{c}^{\iota}\right)\\ & -\alpha_{f}\grad\delta\tilde{c}^{\iota}\cdot\left(z^{\beta}\tilde{\kappa}^{\beta}d_{0}^{\beta}\grad\Delta\tilde{c}^{\beta}+\Delta\tilde{c}^{\beta}\sum_{\gamma}z^{\gamma}\left(\frac{\partial\tilde{\kappa}^{\gamma}}{\partial\tilde{c}^{\beta}}d_{0}^{\gamma}+\tilde{\kappa}^{\gamma}\frac{\partial d_{0}^{\gamma}}{\partial\tilde{c}^{\beta}}\right)\grad\tilde{c}^{\gamma}\right)dv \end{aligned} \]

Similarly, we can rewrite eq.\eqref{eq:Ext-Virt-Work-CFDSol} as

\[ \begin{aligned}\delta W_{ext}= & \int_{\Gamma^{f}}\delta\mathbf{v}^{f}\cdot\mathbf{t}^{\tau}da+\int_{\Gamma^{f}}\delta J^{f}v_{n}^{f}da+\int_{\Gamma^{f}}\delta\tilde{c}^{\iota}\tilde{j}_{n}^{\iota}da\\ & +\delta G_{ext}^{v}+\delta G_{ext}^{c} \end{aligned} \]

where

\[ \begin{aligned}\delta G_{ext}^{v} & =\int_{\Omega^{f}}\delta\mathbf{v}^{f}\cdot\left(\rho^{f}\mathbf{b}^{f}+\sum_{\iota}M^{\iota}c^{\iota}\mathbf{b}^{\iota}\right)dv\\ \delta G_{ext}^{c} & =-\sum_{\iota}\int_{\Omega^{f}}\grad\delta\tilde{c}^{\iota}\cdot\left(\mathbf{j}_{b}^{\iota}+\sum_{\gamma}z^{\gamma}\mathbf{j}_{b}^{\gamma}\right)dv \end{aligned} \]

and we can linearize \(\delta G_{ext}^{v}\) and \(\delta G_{ext}^{c}\) as

\[ \begin{aligned}D\left(\delta G_{ext}^{v}\right)\left[\Delta\mathbf{v}^{f}\right] & =\mathbf{0}\\ D\left(\delta G_{ext}^{v}\right)\left[\Delta J^{f}\right] & =\alpha_{f}\int_{\Omega^{f}}\delta\mathbf{v}^{f}\cdot\left(-\frac{\rho^{f}}{J^{f}}\mathbf{b}^{f}+\sum_{\iota}M^{\iota}\frac{\partial\tilde{\kappa}^{\iota}}{\partial J^{f}}\tilde{c}^{\iota}\mathbf{b}^{\iota}\right)\Delta J^{f}dv\\ D\left(\delta G_{ext}^{v}\right)\left[\Delta\tilde{c}^{\beta}\right] & =\alpha_{f}\int_{\Omega^{f}}\delta\mathbf{v}^{f}\cdot\left(M^{\beta}\tilde{\kappa}^{\beta}\mathbf{b}^{\beta}+\sum_{\gamma}M^{\gamma}\frac{\partial\tilde{\kappa}^{\gamma}}{\partial\tilde{c}^{\beta}}\tilde{c}^{\gamma}\mathbf{b}^{\gamma}\right)\Delta\tilde{c}^{\beta}dv \end{aligned} \]

and

\[ \begin{aligned}D\left(\delta G_{ext}^{c}\right)\left[\Delta\mathbf{v}^{f}\right] & =\mathbf{0}\\ D\left(\delta G_{ext}^{c}\right)\left[\Delta J^{f}\right] & =-\sum_{\iota}\alpha_{f}\int_{\Omega^{f}}\grad\delta\tilde{c}^{\iota}\cdot\left(s^{\iota}\frac{\partial\tilde{\kappa}^{\iota}}{\partial J^{f}}\tilde{c}^{\iota}\mathbf{b}^{\iota}+\sum_{\gamma}z^{\gamma}s^{\gamma}\frac{\partial\tilde{\kappa}^{\gamma}}{\partial J^{f}}\tilde{c}^{\gamma}\mathbf{b}^{\gamma}\right)\Delta J^{f}dv\\ D\left(\delta G_{ext}^{c}\right)\left[\Delta\tilde{c}^{\beta}\right] & =-\sum_{\iota}\alpha_{f}\int_{\Omega^{f}}\grad\delta\tilde{c}^{\iota}\cdot\left(\left(\delta_{\iota\beta}+z^{\beta}\right)s^{\beta}\tilde{\kappa}^{\beta}\mathbf{b}^{\beta}+s^{\iota}\frac{\partial\tilde{\kappa}^{\iota}}{\partial\tilde{c}^{\beta}}\tilde{c}^{\iota}\mathbf{b}^{\iota}+\sum_{\gamma}z^{\gamma}s^{\gamma}\frac{\partial\tilde{\kappa}^{\gamma}}{\partial\tilde{c}^{\beta}}\tilde{c}^{\gamma}\mathbf{b}^{\gamma}\right)\Delta\tilde{c}^{\beta}\,dv \end{aligned} \]

under the assumption that \(D\mathbf{b}^{\alpha}\left[\Delta\mathbf{v}^{f}\right]=\mathbf{0}\)

Discretization for Fluid-Solutes

To evaluate the discretized form of the virtual work and its linearizations we use

\[ \begin{aligned}\delta\mathbf{v}^{f} & =\sum_{a}N_{a}\delta\mathbf{v}_{a}^{f} & \Delta\mathbf{v}^{f} & =\sum_{b}N_{b}\Delta\mathbf{v}_{b}^{f}\\ \delta J^{f} & =\sum_{a}N_{a}\delta J_{a}^{f} & \Delta J^{f} & =\sum_{b}N_{b}\Delta J_{b}^{f}\\ \delta\tilde{c}^{\iota} & =\sum_{a}N_{a}\delta\tilde{c}_{a}^{\iota} & \Delta\tilde{c}^{\beta} & =\sum_{b}N_{b}\Delta\tilde{c}_{b}^{\beta} \end{aligned} \]

where \(N_{a}\) are shape functions. Then, the discretized form of the virtual work takes the form

\[ \begin{aligned}\delta W_{int}^{v} & =\sum_{a}\delta\mathbf{v}_{a}^{f}\cdot\mathbf{f}_{a}^{v}\\ \delta W_{int}^{J} & =\sum_{a}\delta J_{a}^{f}f_{a}^{J}\\ \delta W_{int}^{c} & =\sum_{\iota}\sum_{a}\delta\tilde{c}_{a}^{\iota}f_{a}^{\iota} \end{aligned} \]

where

\[ \begin{aligned}\mathbf{f}_{a}^{v} & =\int_{\Omega^{f}}N_{a}\left(\grad\tilde{p}+\rho^{f}\mathbf{a}^{f}+\sum_{\iota}R\theta\tilde{\kappa}^{\iota}\grad\tilde{c}^{\iota}\right)+\boldsymbol{\tau}\cdot\grad N_{a}dv\\ f_{a}^{J} & =\int_{\Omega^{f}}N_{a}\frac{1}{J^{f}}\frac{D^{f}J^{f}}{Dt}+\grad N_{a}\cdot\mathbf{v}^{f}dv\\ f_{a}^{\iota} & =\int_{\Omega^{f}}-N_{a}\left(\frac{1}{J^{f}}\frac{D^{f}\left(J^{f}c^{\iota}\right)}{Dt}-\hat{c}^{\iota}\right)+\grad N_{a}\cdot\left(\mathbf{j}_{d}^{\iota}+\sum_{\gamma}z^{\gamma}\mathbf{j}_{d}^{\gamma}\right)dv \end{aligned} \]

\[ \begin{aligned}\frac{1}{J^{f}}\frac{D^{f}\left(J^{f}c^{\iota}\right)}{Dt} & =\frac{c^{\iota}}{J^{f}}\frac{D^{f}J^{f}}{Dt}+\tilde{\kappa}^{\iota}\frac{D^{f}\tilde{c}^{\iota}}{Dt}+\tilde{c}^{\iota}\frac{D^{f}\tilde{\kappa}^{\iota}}{Dt}\end{aligned} \]

and

\[ \begin{aligned}\delta G_{ext}^{v} & =\sum_{a}\delta\mathbf{v}_{a}^{f}\cdot\mathbf{b}_{a}^{v}\\ \delta G_{ext}^{c} & =\sum_{\iota}\sum_{a}\delta\tilde{c}_{a}^{\iota}b_{a}^{\iota} \end{aligned} \]

where

\[ \begin{aligned}\mathbf{b}_{a}^{v} & =\int_{\Omega^{f}}N_{a}\left(\rho^{f}\mathbf{b}^{f}+\sum_{\iota}M^{\iota}c^{\iota}\mathbf{b}^{\iota}\right)dv\\ b_{a}^{\iota} & =\int_{\Omega^{f}}N_{a}\divg\left(\mathbf{j}_{b}^{\iota}+\sum_{\gamma}z^{\gamma}\mathbf{j}_{b}^{\gamma}\right)\,dv \end{aligned} \]

Note that

\[ \begin{aligned}\frac{D^{f}J^{f}}{Dt} & =\frac{\partial J^{f}}{\partial t}+\grad J^{f}\cdot\mathbf{v}^{f}\\ \frac{D^{f}c^{\iota}}{Dt} & =\frac{\partial c^{\iota}}{\partial t}+\grad c^{\iota}\cdot\mathbf{v}^{f}\\ & =\tilde{\kappa}^{\iota}\frac{\partial\tilde{c}^{\iota}}{\partial t}+\tilde{c}^{\iota}\frac{\partial\tilde{\kappa}^{\iota}}{\partial t}+\left(\tilde{\kappa}^{\iota}\grad\tilde{c}^{\iota}+\tilde{c}^{\iota}\grad\tilde{\kappa}^{\iota}\right)\cdot\mathbf{v}^{f}\\ \frac{D^{f}\tilde{\kappa}^{\iota}}{Dt} & =\frac{\partial\tilde{\kappa}^{\iota}}{\partial J^{f}}\frac{D^{f}J^{f}}{Dt}+\sum_{\gamma}\frac{\partial\tilde{\kappa}^{\iota}}{\partial\tilde{c}^{\gamma}}\frac{D\tilde{c}^{\gamma}}{Dt}\\ \grad\tilde{\kappa}^{\iota} & =\frac{\partial\tilde{\kappa}^{\iota}}{\partial J^{f}}\grad J+\sum_{\gamma}\frac{\partial\tilde{\kappa}^{\iota}}{\partial\tilde{c}^{\gamma}}\grad\tilde{c}^{\gamma}\\ \frac{D^{f}\left(J^{f}c^{\iota}\right)}{Dt} & =c^{\iota}\frac{D^{f}J^{f}}{Dt}+J^{f}\frac{D^{f}c^{\iota}}{Dt} \end{aligned} \]

The linearization of \(\delta W_{int}^{v}\) gets discretized into

\[ \begin{aligned}D\left(\delta W_{int}^{v}\right)\left[\Delta\mathbf{v}^{f}\right] & =\sum_{a}\delta\mathbf{v}_{a}^{f}\cdot\sum_{b}\alpha_{f}\mathbf{K}_{ab}^{vv}\cdot\Delta\mathbf{v}_{b}^{f}\\ D\left(\delta W_{int}^{v}\right)\left[\Delta J^{f}\right] & =\sum_{a}\delta\mathbf{v}_{a}^{f}\cdot\sum_{b}\alpha_{f}\mathbf{k}_{ab}^{vJ}\Delta J_{b}^{f}\\ D\left(\delta W_{int}^{v}\right)\left[\Delta\tilde{c}^{\beta}\right] & =\sum_{a}\delta\mathbf{v}_{a}^{f}\cdot\sum_{b}\alpha_{f}\mathbf{k}_{ab}^{v\beta}\Delta\tilde{c}_{b}^{\beta} \end{aligned} \]

where

\[ \begin{aligned}\mathbf{K}_{ab}^{vv} & =\int_{\Omega^{f}}\grad N_{a}\cdot\boldsymbol{\mathcal{C}}^{\tau}\cdot\grad N_{b}dv\\ & +\int_{\Omega^{f}}N_{a}\rho^{f}\left(\left(\frac{\alpha_{m}}{\alpha_{f}\gamma}\frac{N_{b}}{\Delta t}+\grad N_{b}\cdot\mathbf{v}^{f}\right)\mathbf{I}+N_{b}\mathbf{L}^{f}\right)dv\\ \mathbf{k}_{ab}^{vJ} & =\int_{\Omega^{f}}N_{a}\left(\tilde{p}_{J}^{\prime}\grad N_{b}+N_{b}\left(\tilde{p}_{J}^{\prime\prime}\grad J^{f}+R\theta\sum_{\gamma}\frac{\partial\tilde{\kappa}^{\gamma}}{\partial J^{f}}\grad\tilde{c}^{\gamma}\right)\right)dv\\ & +\int_{\Omega^{f}}N_{b}\boldsymbol{\tau}_{J}^{\prime}\cdot\grad N_{a}dv-\alpha_{f}\int_{\Omega^{f}}N_{a}N_{b}\frac{\rho^{f}}{J^{f}}\mathbf{a}^{f}dv\\ \mathbf{k}_{ab}^{v\beta} & =\int_{\Omega^{f}}N_{a}R\theta\left(\tilde{\kappa}^{\beta}\grad N_{b}+N_{b}\sum_{\gamma}\frac{\partial\tilde{\kappa}^{\gamma}}{\partial\tilde{c}^{\beta}}\grad\tilde{c}^{\gamma}\right)dv \end{aligned} \]

The linearization of \(\delta W_{int}^{J}\) gets discretized into

\[ \begin{aligned}D\left(\delta W_{int}^{J}\right)\left[\Delta\mathbf{v}^{f}\right] & =\sum_{a}\delta J_{a}^{f}\sum_{b}\alpha_{f}\mathbf{k}_{ab}^{Jv}\cdot\Delta\mathbf{v}_{b}^{f}\\ D\left(\delta W_{int}^{J}\right)\left[\Delta J^{f}\right] & =\sum_{a}\delta J_{a}^{f}\sum_{b}\alpha_{f}k_{ab}^{JJ}\Delta J_{b}^{f}\\ D\left(\delta W_{int}^{J}\right)\left[\Delta\tilde{c}^{\beta}\right] & =0 \end{aligned} \]

where

\[ \begin{aligned}\mathbf{k}_{ab}^{Jv} & =\int_{\Omega^{f}}N_{b}\left(N_{a}\frac{1}{J^{f}}\grad J^{f}+\grad N_{a}\right)dv\\ k_{ab}^{JJ} & =\int_{\Omega^{f}}\frac{N_{a}}{J^{f}}\left(\left(\frac{\alpha_{m}}{\alpha_{f}\gamma}\frac{1}{\Delta t}-\frac{1}{J^{f}}\frac{D^{f}J^{f}}{Dt}\right)N_{b}+\grad N_{b}\cdot\mathbf{v}^{f}\right)dv \end{aligned} \]

Finally, the linearization of \(\delta W_{int}^{c}\) gets discretized into

\[ \begin{aligned}D\left(\delta W_{int}^{c}\right)\left[\Delta\mathbf{v}^{f}\right] & =\sum_{\iota}\sum_{a}\delta\tilde{c}_{a}^{\iota}\sum_{b}\alpha_{f}\mathbf{k}_{ab}^{\iota v}\cdot\Delta\mathbf{v}_{b}^{f}\\ D\left(\delta W_{int}^{c}\right)\left[\Delta J^{f}\right] & =\sum_{\iota}\sum_{a}\delta\tilde{c}_{a}^{\iota}\sum_{b}\alpha_{f}k_{ab}^{\iota J}\Delta J_{b}^{f}\\ D\left(\delta W_{int}^{c}\right)\left[\Delta\tilde{c}^{\beta}\right] & =\sum_{\iota}\sum_{a}\delta\tilde{c}_{a}^{\iota}\sum_{b}\alpha_{f}k^{\iota\beta}\,\Delta\tilde{c}_{b}^{\beta} \end{aligned} \]

where

\[ \begin{aligned}\mathbf{k}_{ab}^{\iota v} & =-\int_{\Omega^{f}}N_{a}N_{b}\frac{1}{J^{f}}\grad\left(J^{f}c^{\iota}\right)\,dv\\ k_{ab}^{\iota J} & =\int_{\Omega^{f}}N_{a}N_{b}\frac{1}{\left(J^{f}\right)^{2}}\frac{D^{f}\left(J^{f}c^{\iota}\right)}{Dt}\\ & -N_{a}\tilde{c}^{\iota}\left(\frac{\tilde{\kappa}^{\iota}}{J^{f}}+\frac{\partial\tilde{\kappa}^{\iota}}{\partial J^{f}}\right)\left(\frac{\alpha_{m}}{\alpha_{f}\gamma}\frac{N_{b}}{\Delta t}+\grad N_{b}\cdot\mathbf{v}^{f}\right)\\ & -N_{a}N_{b}\frac{\tilde{c}^{\iota}}{J^{f}}\frac{D^{f}J^{f}}{Dt}\left(2\frac{\partial\tilde{\kappa}^{\iota}}{\partial J^{f}}+J^{f}\frac{\partial^{2}\tilde{\kappa}^{\iota}}{\partial\left(J^{f}\right)^{2}}\right)\\ & -N_{a}N_{b}\left(\left(\frac{\tilde{\kappa}^{\iota}}{J^{f}}+\frac{\partial\tilde{\kappa}^{\iota}}{\partial J^{f}}\right)\frac{D^{f}\tilde{c}^{\iota}}{Dt}+\tilde{c}^{\iota}\sum_{\gamma}\left(\frac{1}{J^{f}}\frac{\partial\tilde{\kappa}^{\iota}}{\partial\tilde{c}^{\gamma}}+\frac{\partial^{2}\tilde{\kappa}^{\iota}}{\partial J^{f}\partial\tilde{c}^{\gamma}}\right)\frac{D^{f}\tilde{c}^{\gamma}}{Dt}\right)\\ & +N_{a}N_{b}\nu^{\iota}\underbrace{\frac{\partial\hat{\zeta}}{\partial J^{f}}}_{=0\text{ currently}}-N_{b}\grad N_{a}\cdot\left(\frac{\partial\tilde{\kappa}^{\iota}}{\partial J^{f}}d_{0}^{\iota}\grad\tilde{c}^{\iota}+\sum_{\gamma}z^{\gamma}\frac{\partial\tilde{\kappa}^{\gamma}}{\partial J^{f}}d_{0}^{\gamma}\grad\tilde{c}^{\gamma}\right)dv\\ k^{\iota\beta} & =\int_{\Omega^{f}}-N_{a}\left(\delta_{\iota\beta}\tilde{\kappa}^{\beta}+\tilde{c}^{\iota}\frac{\partial\tilde{\kappa}^{\iota}}{\partial\tilde{c}^{\beta}}\right)\left(\left(\frac{\alpha_{m}}{\alpha_{f}\gamma}\frac{1}{\Delta t}+\frac{1}{J^{f}}\frac{D^{f}J^{f}}{Dt}\right)N_{b}+\grad N_{b}\cdot\mathbf{v}^{f}\right)\\ & +N_{a}N_{b}\nu^{\iota}\frac{\partial\hat{\zeta}}{\partial\tilde{c}^{\beta}}\\ & -\grad N_{a}\cdot\left(\delta_{\iota\beta}\tilde{\kappa}^{\beta}d_{0}^{\beta}\grad N_{b}+N_{b}\left(\frac{\partial\tilde{\kappa}^{\iota}}{\partial\tilde{c}^{\beta}}d_{0}^{\iota}+\tilde{\kappa}^{\iota}\frac{\partial d_{0}^{\iota}}{\partial\tilde{c}^{\beta}}\right)\grad\tilde{c}^{\iota}\right)\\ & -\grad N_{a}\cdot\left(z^{\beta}\tilde{\kappa}^{\beta}d_{0}^{\beta}\grad N_{b}+N_{b}\sum_{\gamma}z^{\gamma}\left(\frac{\partial\tilde{\kappa}^{\gamma}}{\partial\tilde{c}^{\beta}}d_{0}^{\gamma}+\tilde{\kappa}^{\gamma}\frac{\partial d_{0}^{\gamma}}{\partial\tilde{c}^{\beta}}\right)\grad\tilde{c}^{\gamma}\right)dv \end{aligned} \]

\[ \begin{aligned}\frac{1}{J^{f}}\grad\left(J^{f}c^{\iota}\right) & =\frac{1}{J^{f}}c^{\iota}\grad J^{f}+\tilde{\kappa}^{\iota}\grad\tilde{c}^{\iota}+\tilde{c}^{\iota}\grad\tilde{\kappa}^{\iota}\\ \frac{1}{J^{f}}\frac{D^{f}\left(J^{f}c^{\iota}\right)}{Dt} & =\frac{c^{\iota}}{J^{f}}\frac{D^{f}J^{f}}{Dt}+\tilde{\kappa}^{\iota}\frac{D^{f}\tilde{c}^{\iota}}{Dt}+\tilde{c}^{\iota}\frac{D^{f}\tilde{\kappa}^{\iota}}{Dt} \end{aligned} \]

For the external work contributions from body forces, the discretized forms are

\[ \begin{aligned}\delta G_{ext}^{v} & =\sum_{a}\delta\mathbf{v}_{a}^{f}\cdot\mathbf{b}_{a}^{v}\\ \delta G_{ext}^{c} & =\sum_{\iota}\sum_{a}\delta\tilde{c}_{a}^{\iota}b_{a}^{\iota} \end{aligned} \]

where

\[ \begin{aligned}\mathbf{b}_{a}^{v} & =\int_{\Omega^{f}}N_{a}\left(\rho^{f}\mathbf{b}^{f}+\sum_{\iota}M^{\iota}c^{\iota}\mathbf{b}^{\iota}\right)dv\\ b_{a}^{\iota} & =-\int_{\Omega^{f}}\grad N_{a}\cdot\left(\mathbf{j}_{b}^{\iota}+\sum_{\gamma}z^{\gamma}\mathbf{j}_{b}^{\gamma}\right)dv \end{aligned} \]

and

\[ \begin{aligned}D\left(\delta G_{ext}^{v}\right)\left[\Delta\mathbf{v}^{f}\right] & =\mathbf{0}\\ D\left(\delta G_{ext}^{v}\right)\left[\Delta J^{f}\right] & =\sum_{a}\delta\mathbf{v}_{a}^{f}\cdot\sum_{b}\alpha_{f}\mathbf{b}_{ab}^{vJ}\Delta J_{b}^{f}\\ D\left(\delta G_{ext}^{v}\right)\left[\Delta\tilde{c}^{\beta}\right] & =\sum_{a}\delta\mathbf{v}_{a}^{f}\cdot\sum_{b}\alpha_{f}\mathbf{b}_{ab}^{v\beta}\Delta\tilde{c}_{b}^{\beta} \end{aligned} \]

where

\[ \begin{aligned}\mathbf{b}_{ab}^{vJ} & =\int_{\Omega^{f}}N_{a}N_{b}\left(-\frac{\rho^{f}}{J^{f}}\mathbf{b}^{f}+\sum_{\gamma}M^{\gamma}\frac{\partial\tilde{\kappa}^{\gamma}}{\partial J^{f}}\tilde{c}^{\iota}\mathbf{b}^{\gamma}\right)dv\\ \mathbf{b}_{ab}^{v\beta} & =\int_{\Omega^{f}}N_{a}N_{b}\left(M^{\beta}\tilde{\kappa}^{\beta}\mathbf{b}^{\beta}+\sum_{\gamma}M^{\gamma}\frac{\partial\tilde{\kappa}^{\gamma}}{\partial\tilde{c}^{\beta}}\tilde{c}^{\gamma}\mathbf{b}^{\gamma}\right)dv \end{aligned} \]

Similarly,

\[ \begin{aligned}D\left(\delta G_{ext}^{c}\right)\left[\Delta\mathbf{v}^{f}\right] & =\mathbf{0}\\ D\left(\delta G_{ext}^{c}\right)\left[\Delta J^{f}\right] & =\sum_{\iota}\sum_{a}\delta\tilde{c}_{a}^{\iota}\sum_{b}\alpha_{f}b_{ab}^{\iota J}\Delta J_{b}^{f}\\ D\left(\delta G_{ext}^{c}\right)\left[\Delta\tilde{c}^{\beta}\right] & =\sum_{\iota}\sum_{a}\delta\tilde{c}_{a}^{\iota}\sum_{b}\alpha_{f}b_{ab}^{\iota\beta}\Delta\tilde{c}_{b}^{\beta} \end{aligned} \]

where

\[ \begin{aligned}b_{ab}^{\iota J} & =-\int_{\Omega^{f}}N_{b}\grad N_{a}\cdot\left(s^{\iota}\frac{\partial\tilde{\kappa}^{\iota}}{\partial J^{f}}\tilde{c}^{\iota}\mathbf{b}^{\iota}+\sum_{\gamma}z^{\gamma}s^{\gamma}\frac{\partial\tilde{\kappa}^{\gamma}}{\partial J^{f}}\tilde{c}^{\gamma}\mathbf{b}^{\gamma}\right)dv\\ b_{ab}^{\iota\beta} & =-\int_{\Omega^{f}}N_{b}\grad N_{a}\cdot\left(\left(\delta_{\iota\beta}+z^{\beta}\right)s^{\beta}\tilde{\kappa}^{\beta}\mathbf{b}^{\beta}+s^{\iota}\frac{\partial\tilde{\kappa}^{\iota}}{\partial\tilde{c}^{\beta}}\tilde{c}^{\iota}\mathbf{b}^{\iota}+\sum_{\gamma}z^{\gamma}s^{\gamma}\frac{\partial\tilde{\kappa}^{\gamma}}{\partial\tilde{c}^{\beta}}\tilde{c}^{\gamma}\mathbf{b}^{\gamma}\right)\,dv \end{aligned} \]

In the special case when the body force acting on each constituent is the same, \(\mathbf{b}^{f}=\mathbf{b}^{\iota}\equiv\mathbf{b}\), these expressions simplify further,

\[ \begin{aligned}b_{ab}^{\iota J} & =-\int_{\Omega^{f}}N_{b}\left(w^{\iota J}+w_{e}^{cJ}\right)\grad N_{a}\cdot\mathbf{b}dv\\ b_{ab}^{\iota\beta} & =-\int_{\Omega^{f}}N_{b}\left(\left(\delta_{\iota\beta}+z^{\beta}\right)s^{\beta}\tilde{\kappa}^{\beta}+w^{\iota\beta}+w_{e}^{c\beta}\right)\grad N_{a}\cdot\mathbf{b}\,dv \end{aligned} \]

where

\[ \begin{aligned}w^{\iota J} & \equiv s^{\iota}\frac{\partial\tilde{\kappa}^{\iota}}{\partial J^{f}}\tilde{c}^{\iota} & w_{e}^{cJ} & \equiv\sum_{\gamma}z^{\gamma}s^{\gamma}\frac{\partial\tilde{\kappa}^{\gamma}}{\partial J^{f}}\tilde{c}^{\gamma}\\ w^{\iota\beta} & \equiv s^{\iota}\frac{\partial\tilde{\kappa}^{\iota}}{\partial\tilde{c}^{\beta}}\tilde{c}^{\iota} & w_{e}^{c\beta} & \equiv\sum_{\gamma}z^{\gamma}s^{\gamma}\frac{\partial\tilde{\kappa}^{\gamma}}{\partial\tilde{c}^{\beta}}\tilde{c}^{\gamma} \end{aligned} \]

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1. Bonet, Javier; Wood, Richard D.. "Nonlinear continuum mechanics for finite element analysis." Cambridge University Press (1997). ↩