3.2 Conservation of momentum in fluid
The velocity of the fluid and matrix and therefore how melt is
redistributed due to migration of fluid and matrix depends on how
buoyancy, compaction, drag, and sometimes external forces are balanced.
To solve for the evolution of the fluid velocity, we start with the
conservation of momentum in the fluid neglecting inertia (Bercovici &
Ricard, 2003, Bercovici et al. , 2001, McKenzie, 1984):
\begin{equation}
\left(3\right)\ 0=\int_{\text{δV}}{\ \nabla\bullet\ \sigma_{f}\ \phi\ dV\ +\int_{\text{δV}}{\rho_{f}\text{g\ ϕ\ dV}}+\int_{\text{δV}}{\nabla\bullet H_{f}\text{\ dV}}}.\nonumber \\
\end{equation}Here,\(\ \sigma_{f}\) is the total stress tensor in the fluid, \(g\) is
the gravitational constant, \(\rho_{f}\) is the density of the fluid
phase, and \(H_{f}\) is a term that accounts for the stresses that act
across the matrix-melt interface. This quantity includes the forces that
arise due to the differential velocity between the matrix and fluid, the
difference in pressure between phases, and the surface tension at the
interface between the two phases. Following the treatment of Bercoviciet al. (2001), \(\nabla\bullet H_{f}\) is expressed as\(\ \nabla\bullet H_{f}=\ c\left(v_{m}-v_{f}\right)+\ \left(P_{f}\right)\nabla\phi\),
ignoring surface tension. Here, \(P_{f}\) is the pressure in the fluid
phase and \(c\) = \(\frac{\eta_{f}\phi^{2}}{\kappa}\), where\(\eta_{f}\) is the fluid viscosity and \(\kappa\) is the permeability.
Rearranging and solving for \(\nabla P_{f},\) we arrive at:
\begin{equation}
\left(4\right)\ \nabla P_{f}=\rho_{f}g+\frac{c}{\phi}\left(v_{m}-v_{f}\right).\nonumber \\
\end{equation}