3.7 Load at bottom boundary for application to syringe experiments
To fit the model to the experimental data of Hoyos et al. (2022), we solve for the load, \(\Sigma\), required to keep the bottom boundary migrating upwards at a constant velocity of \(v_{H}\), where:
\begin{equation} \left(12\right)\ \sum\ =-P_{f}\phi+\ \left(-P_{m}+\tau_{m_{\text{zz}}}\right)\left(1-\phi\right),\nonumber \\ \end{equation}
assuming \(\tau_{f_{\text{zz}}}\) is negligible (Ricard et al. , 2001). Using eq. (4) to solve for \(P_{f}\), substituting eq. (6) for\(P_{m}\) and eq. (8) for \(\tau_{m_{\text{zz}}}\), and introducing dimensionless variables, we obtain:
\begin{equation} \left(13\right)\ \sum\ =\ {-\rho}_{f}gH+H\mathbf{v}_{\mathbf{0}}\int_{0}^{y=1}\ \frac{\eta_{f}}{\kappa}S^{\prime}dy+{\mathbf{\xi}_{\mathbf{0}}\mathbf{v}}_{\mathbf{0}}\frac{\left(1-\phi\right)}{H}\xi^{{}^{\prime}}\frac{d}{\text{dy}}v_{m}^{\prime}\left(\frac{\pi}{\phi}+1\right),\nonumber \\ \end{equation}
where \(v_{m}=\ v_{H}-\ S\).