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The gap function $g$ , representing the distance between the contact surfaces, is defined by $$\begin{equation} \mathbf{x}^{\left(2\right)}=\mathbf{x}^{\left(1\right)}+g\mathbf{n}^{\left(1\right)},\quad g=\left(\mathbf{x}^{\left(2\right)}-\mathbf{x}^{\left(1\right)}\right)\cdot\mathbf{n}^{\left(1\right)}\,.\label{eq630} \end{equation}$$ The linearization of variables associated with motion, pressure, and concentration, is given by $$\begin{equation} \begin{aligned}D\mathbf{x}^{\left(1\right)} & =\Delta\mathbf{u}^{\left(1\right)} & D\mathbf{x}^{\left(2\right)} & =\Delta\mathbf{u}^{\left(2\right)}+\mathbf{g}_{\alpha}^{\left(2\right)}\Delta\eta_{\left(2\right)}^{\alpha}\\ D\tilde{p}^{\left(1\right)} & =\Delta\tilde{p}^{\left(1\right)} & D\tilde{p}^{\left(2\right)} & =\Delta\tilde{p}^{\left(2\right)}+\frac{\partial\tilde{p}^{\left(2\right)}}{\partial\eta_{\left(2\right)}^{\alpha}}\Delta\eta_{\left(2\right)}^{\alpha}\\ D\tilde{c}^{\left(1\right)} & =\Delta\tilde{c}^{\left(1\right)} & D\tilde{c}^{\left(2\right)} & =\Delta\tilde{c}^{\left(2\right)}+\frac{\partial\tilde{c}^{\left(2\right)}}{\partial\eta_{\left(2\right)}^{\alpha}}\Delta\eta_{\left(2\right)}^{\alpha}\\ D\delta\mathbf{v}^{\left(1\right)} & =\mathbf{0} & D\mathbf{v}^{\left(2\right)} & =\frac{\partial\delta\mathbf{v}^{\left(2\right)}}{\partial\eta_{\left(2\right)}^{\alpha}}\Delta\eta_{\left(2\right)}^{\alpha}\\ D\delta\tilde{p}^{\left(1\right)} & =0 & D\delta\tilde{p}^{\left(2\right)} & =\frac{\partial\delta\tilde{p}^{\left(2\right)}}{\partial\eta_{\left(2\right)}^{\alpha}}\Delta\eta_{\left(2\right)}^{\alpha}\\ D\delta\tilde{c}^{\left(1\right)} & =0 & D\delta\tilde{c}^{\left(2\right)} & =\frac{\partial\delta\tilde{c}^{\left(2\right)}}{\partial\eta_{\left(2\right)}^{\alpha}}\Delta\eta_{\left(2\right)}^{\alpha} \end{aligned} \,,\label{eq631} \end{equation}$$ where $$\begin{equation} \Delta\eta_{\left(2\right)}^{\alpha}=\left(\Delta\mathbf{u}^{\left(1\right)}-\Delta\mathbf{u}^{\left(2\right)}\right)\cdot a^{\alpha\beta}\mathbf{g}_{\beta}^{\left(1\right)}-a^{\alpha\beta}g\mathbf{n}^{\left(1\right)}\cdot\frac{\partial\Delta\mathbf{u}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{\beta}}\,,\label{eq632} \end{equation}$$ with $a^{\alpha\beta}=\left(A_{\alpha\beta}\right)^{-1}$ and $A_{\alpha\beta}=\mathbf{g}_{\alpha}^{\left(1\right)}\cdot\mathbf{g}_{\beta}^{\left(2\right)}$ .