3.3 Conservation of momentum in matrix
Similar to above, the stresses per unit length that act across
matrix-melt interface are expressed from the perspective of the matrix
as \(-\nabla\bullet H_{m}\). Momentum conservation in the matrix may
then be simplified to the following:
\begin{equation}
\left(5\right)\ 0=\ -\left(1-\phi\right)\nabla P_{m}+\ \nabla\phi\left(P_{m}-\ P_{f}\right)\ +\ \nabla\bullet\left[\left(1-\phi\right)\tau_{m}\right]+\ \rho_{m}\left(1-\phi\right)g-c\left(v_{m}-v_{f}\right),\nonumber \\
\end{equation}where \(P_{m}\) is the pressure, \(\rho_{m}\) is the density, and\(\tau_{m}\) is the viscous stress tensor in the matrix phase. Bercovici
et al. (2001) derived an expression for the interphase pressure
difference using a materially invariant relation and a micromechanical
model of pore collapse:
\begin{equation}
{\left(6\right)\text{\ P}}_{m}-\ P_{f}=-\zeta\nabla\bullet v_{m},\nonumber \\
\end{equation}where \(\ \zeta=\frac{\text{πξ}}{\phi},\ \)and \(\pi\) is a geometric
constant, \(\xi\) is the effective matrix viscosity, equal to\(\frac{4}{3}\eta+K,\) and \(\eta\) and \(K\) are the shear and bulk
viscosity of the matrix, respectively. Eq. (6) is consistent with the
derivation of Bercovici et al. (2001) (see full derivation in the
supplements), however, it deviates from the derivation of McKenzie
(1984) (his eq. (C11)) and expressions often used in rock mechanics by a
factor of 1/\(\phi\) (Connolly & Schmidt, 2022).
3.4.1 Boundary conditions – syringe experiments
To best approximate the phase separation experiments of Hoyos et
al. (2022), we consider a compacting medium in a 1-D domain with a
permeable upper boundary (at z = 0) and whose height, \(H\), decreases
over time due to the upward migration of the lower boundary at a fixed
boundary velocity, \(v_{H}\) (Fig. 2b ).
To simplify the boundary conditions for the momentum conservation
equations, we introduce the segregation flux, \(S\), which is defined
as:
\(\left(7\right)\ S=\ \phi\left(v_{f}-v_{m}\right)\).
At the bottom boundary, z = H(t), a no slip condition is assumed and\(v_{f}=v_{m}=\ v_{H}\) as the movement of matrix and fluid is
coupled to the bottom boundary, which leads to \(S(z\ =\ H(t))=0.\)The combined mass conservation of fluid and matrix, upon substitution of\(S\), yields:
\(\left(8\right)\ \frac{\ \partial}{\partial z}\left[v_{m}+S\right]\)= 0,
which indicates that the sum \(v_{m}+S\) is constant and equals to\(v_{H}\) for 0 < z < H because of the bottom
boundary condition. The upper boundary condition is impermeable to the
solid material but not the interstitial liquid, so that\(S(z\ =\ 0)=\ v_{H}\).
3.4.2 Boundary conditions – centrifuge experiments
For application of the model to the centrifuge experiments, the boundary
conditions are similar except that at the top of the column\(\frac{\ \partial}{\partial z}S=0\). This is because in this case,
the matrix pressure, \(P_{m}\), and fluid pressure, \(P_{f}\), are equal
at the top of the crystal column. Furthermore, \(v_{H}\) is no longer
imposed and instead we need to track the location of the top of the
sedimented (mushy) layer with time using the condition\(\frac{\ \partial}{\partial z}S=0\) at the top of the domain. This
treatment allows for the development of a pure melt layer above the
crystal column, which is an accurate description of the centrifuge
experiments summarized in Connolly and Schmidt (2022). We assume that
the particle settling is rapid with respect to the duration of each
experiment. This is likely the case for all experiments other than ZOB9
(Connolly & Schmidt, 2022).