## 3 Numerical Simulation
### Variational Formulation  
Note that the fluid flow influences the species transport; however, the species concentration does not influence the fluid flow. Due to this, the flow can be computed only once in advance. The FEM approximation of the steady state flow problem [41] is based on the variational formulation of the considered boundary value problem (1)-(6). The following functional space \(\boldsymbol{V}\) is defined for the velocity \(\boldsymbol{u}\) (\(\boldsymbol{u}\in\boldsymbol{V}\)):  
\[\boldsymbol{V}=\{\boldsymbol{u}\in\boldsymbol{H}^{1}(\Omega): \boldsymbol{u}\cdot\boldsymbol{n}=1,\;\boldsymbol{u}\times\boldsymbol{n}=0 \text{ on }\Gamma_{in},\] \[\boldsymbol{u}=0\text{ on }\Gamma_{s},\;\boldsymbol{u}\cdot \boldsymbol{n}=0\text{ on }\Gamma_{sim}\}.\]  
Test function \(\boldsymbol{v}\in\hat{\boldsymbol{V}}\), where  
\[\hat{\boldsymbol{V}}=\{\boldsymbol{v}\in\boldsymbol{H}^{1}(\Omega_{f}):\; \boldsymbol{v}=0\text{ on }\Gamma_{in},\;\boldsymbol{v}=0\text{ on }\Gamma_{s},\; \boldsymbol{v}\cdot\boldsymbol{n}=0\text{ on }\Gamma_{sim}\}.\]  
For the pressure \(p\) and the related test functions \(q\), it is required that \(p,q\in Q\), where  
\[Q=\{q\in L_{2}(\Omega_{f}):\;q=0\text{ on }\Gamma_{out}\}.\]  
Multiplying (1) by \(\boldsymbol{v}\), (2) by \(q\), integrating over the computational domain and taking into account the boundary conditions (3)-(6), the following system of equations is obtained with respect to \(\boldsymbol{v}\in\boldsymbol{V}\), \(p\in Q\):  
\[a(\boldsymbol{u},\boldsymbol{v})-b(\boldsymbol{v},p)=0\;\forall\boldsymbol{v} \in\hat{\boldsymbol{V}}, \tag{16}\]  
\[b(\boldsymbol{u},q)=0\;\forall q\in Q. \tag{17}\]  
where  
\[a(\boldsymbol{u},\boldsymbol{v}):=\int_{\Omega_{f}}\
abla\boldsymbol{u}\cdot \
abla\boldsymbol{v}\,d\boldsymbol{x},\]  
\[b(\boldsymbol{v},q):=\int_{\Omega_{f}}(\
abla\cdot\boldsymbol{v})q\,d \boldsymbol{x}.\]  
The following finite dimensional subspaces are used for the basis and test functions in the FEM approximation of the velocity and the pressure: \(\boldsymbol{V}_{h}\subset\boldsymbol{V}\), \(\hat{\boldsymbol{V}}_{h}\subset\hat{\boldsymbol{V}}\) and \(Q_{h}\subset Q\). The unsteady species transport problem (7)-(10) is solved numerically using finite elements. Here we define  
\[S=\{c\in H^{1}(\Omega_{f}):\;c=1\text{ on }\Gamma_{in}\},\]  
\[\hat{S}=\{s\in H^{1}(\Omega_{f}):\;s=0\text{ on }\Gamma_{in}\}.\]The approximate solution \(c\in S\) is sought from  
\[\boldsymbol{u}=0\text{ on }\Gamma_{s},\;\boldsymbol{u}\cdot \boldsymbol{n}=0\text{ on }\Gamma_{sim}\}.\]0  
where the following notations are used:  
\[\boldsymbol{u}=0\text{ on }\Gamma_{s},\;\boldsymbol{u}\cdot \boldsymbol{n}=0\text{ on }\Gamma_{sim}\}.\]1  
\[\boldsymbol{u}=0\text{ on }\Gamma_{s},\;\boldsymbol{u}\cdot \boldsymbol{n}=0\text{ on }\Gamma_{sim}\}.\]2  
For determining \(m\in G=L_{2}(\Gamma_{s})\) (see (11)) we use  
\[\boldsymbol{u}=0\text{ on }\Gamma_{s},\;\boldsymbol{u}\cdot \boldsymbol{n}=0\text{ on }\Gamma_{sim}\}.\]3  
The symmetric Crank-Nicolson discretization method is used for time discretization. Let \(\tau\) be a step-size of a uniform grid in time such that \(c^{n}=c(t^{n}),t^{n}=n\tau,\;n=0,1,...\). Equation (18) is approximated in time as follows  
\[\boldsymbol{u}=0\text{ on }\Gamma_{s},\;\boldsymbol{u}\cdot \boldsymbol{n}=0\text{ on }\Gamma_{sim}\}.\]4  
In a similar way, for (19), we get  
\[\boldsymbol{u}=0\text{ on }\Gamma_{s},\;\boldsymbol{u}\cdot \boldsymbol{n}=0\text{ on }\Gamma_{sim}\}.\]5  
We will not show pictures of the solution of the direct problem in this paper, since it differs from the previous work only by a length of time. This length was \(T=40\), but is now \(T=6000\), because this time, we are considering a case of a dominant reaction; therefore, the breakthrough curve looms over a long period of time. We have performed a direct solution to the problem on exactly the same geometry of the computational domain. Therefore, interested readers are encouraged to reference the computational fields in our previous work [37].