Table 2 – Table of dimensionless parameters used in model derivation.
Here, for application towards the syringe experiments, the reference velocity \(\mathbf{v}_{\mathbf{0}}\) is equal to\(\left|v_{H}\right|\), where \(v_{H}\) is the plunger velocity. Therefore \(v_{H}=\mathbf{\ -}\mathbf{v}_{\mathbf{0}}\) to ensure that the reference timescale isn’t negative. For application to the centrifuge experiments, \(\mathbf{v}_{\mathbf{0}}\) is chosen to be\(\left|\frac{\rho g}{\beta}\right|\). Solving for the derivative of\(\phi\) with respect to and both in terms of \(y\) and plugging into
eq. (1) alongside relevant non-dimensional quantities, we obtain:
\begin{equation} \left(10\right)\ \frac{\partial\phi}{\partial t^{{}^{\prime}}}\ =\ -\frac{H_{0}}{H}\frac{\partial}{\partial y}\left(\phi\left[y+\ \ v_{f}^{\prime}\right]\right)+\frac{H_{0}}{H}\text{ϕ.}\nonumber \\ \end{equation}
Inserting eq. (4) into the gradient of eq. (6), inserting the gradient of eq. (6) and eq (7) into eq. (5), using the condition\(v_{m}+S=\ v_{H}\), and making use of non-dimensional variables, we arrive at:
\begin{equation} \left(11\right)\ \beta^{{}^{\prime}}S^{\prime}=-\frac{\left(1-\phi\right)\rho g}{\mathbf{v}_{\mathbf{0}}\mathbf{\beta}_{\mathbf{0}}}+\frac{\mathbf{\xi}_{\mathbf{0}}}{{\mathbf{\beta}_{\mathbf{0}}H}^{2}}\left(\frac{\partial}{\partial y}\left[\left(1-\phi\right)\frac{\pi^{2}}{{\phi\mathbf{\phi}}_{\mathbf{0}}}\xi^{{}^{\prime}}\frac{\partial}{\partial y}S^{\prime}\right]+\frac{\partial}{\partial y}\left[\left(1-\phi\right)\xi^{{}^{\prime}}\frac{\partial}{\partial y}S^{\prime}\right]\right).\nonumber \\ \end{equation}
Here, with dimensions, \(\beta=\ \frac{c}{\phi^{2}}\) and \(\xi\) is the effective matrix viscosity assuming a 1D geometry. The equations we model directly are eqs. (11) and (10), solving for \(S^{{}^{\prime}}\) and\(\phi\) using a finite volumes approach. The full discretization can be found in the supplements.
Equating the drag and compaction stresses, eq. (11) shows they are commensurate when\(\delta_{c}\ =\ \sqrt{\frac{\mathbf{\xi}_{\mathbf{0}}}{\mathbf{\beta}_{\mathbf{0}}}}\), which is often referred to as the reference compaction length.