In this thesis, I present one- and three-dimensional numerical solutions to a two-phase fluid flow problem. The context of these investigations is the evolution of a viscous permeable matrix with a small fraction of melt that is representative of partial melt in the Earth's mantle. The matrix compacts under gravity as melt moves upward. In
addition to the simple compaction solution, a range of solutions representing stably propagating waves are possible.
I first present a coherent mathematical development of the governing equations for the three-dimensional problem. I then describe a one-dimensional numerical algorithm
(1D2PF) that solves the second-order inhomogeneous P.D.E. for the velocity of the viscous matrix, V, for arbitrary melt fraction distribution, φ (the volume fraction
occupied by melt). Combined with a time-stepping algorithm which advances the melt fraction in time, fully time-dependent 1D solutions are obtained. With an initial constant base melt fraction φ0 with a superposed localised concentration of melt, I explore the evolution and formation of solitary compaction waves.
Using (1D2PF) I investigate the width, amplitude and phase velocity of stable solitary waves, and examine how these parameters depend on the initial conditions, permeability
coefficient (k0) and melt and matrix viscosities (ηf and ηm). I demonstrate the existence of a threshold initial width above which secondary solitary waves form, with larger
widths producing longer wave trains and smaller widths producing a small-amplitude oscillatory disturbance to the background melt fraction. Experiments with k0, ηf and ηm
reveal that the width of the stable solitary wave is simply proportional to the compaction length parameter δ=√k0ηm/ ηf and its velocity varies as δ16/ 9/ηm . I also show that the
width of the solitary waves varies as λS=4.6δ and the amplitude follows the relation AS≃89/δ . For initial melt fractions whose distribution is wider than the threshold
width, secondary waves are produced with progressively smaller amplitude, and hence slower propagation velocity. I demonstrate that smaller values of δ result in the same
volume of melt being partitioned over increasing numbers of relatively thinner solitary waves. The amplitude of the initial perturbation to the background melt fraction
however is shown to have no effect on the number of solitary waves produced. A train of such waves arriving at the surface could provide an explanation of intermittentvolcanic activity above a region of partial melt.
In a preliminary study of two-phase flow in three-dimensions I have also made significant progress toward the development of a three-dimensional two-phase flow
simulation program. To do so, I have adapted the three-dimensional viscous fluid convection program (TDCON) by Houseman (1990). The new program TD2PF depends
on a potential-function formulation similar to that of Spiegelman (1993a), in which the divergence of the matrix velocity field, D=∇·V, and the vector potential, A, are the
primary variables. I have introduced new functionality to a significantly expanded threedimensional
Poisson solver (program TDPOTS) but lack of time prevented a successful conclusion to the development of a general 3D solver for the divergence field D.
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