Philip W. Sharp

Self-similar solutions for elastohydrodynamic cavity flow

The enlargement of a lens-shaped cavity lying in a plane of cleavage between two elastic half spaces and filling with viscous fluid from a source on the axis of symmetry is considered. The internal flow is modelled by lubrication theory, which gives a nonlinear partial differential equation connecting the pressure to the cavity shape, and the same two quantities are also related by the singular integral equation of linear elasticity. If the total volume of fluid Q(t) in the cavity at time t is proportional either to tα or to exp (αt) the resulting boundary value problem can be reduced to a self-similar form in which time does not appear explicitly. The solution in non-dimensional terms depends on a single parameter, which may be interpreted as the stress-intensity factor K at the tip. Calculations have been made for the two-dimensional version of the problem for a range of values of α and for a range of stress intensities. The numerical method is to expand the cavity height in a Chebyshev series, the coefficients being found by a nonlinear optimization technique to yield a least squares fit to the Reynolds equation. These lead to expressions for the rate of cavity growth and other quantities of physical interest.

Buoyancy-driven crack propagation: a mechanism for magma migration

A solution has been obtained for steady propagation of a two-dimensional fluid fracture driven by buoyancy in an elastic medium. The problem is formulated in terms of an integro-differential equation governing the elastic deformation, coupled with the differential equation of lubrication theory for viscous flow in the crack. The numerical treatment of this system is carried out in terms of an eigenfunction expansion of the cavity shape, in which the coefficients are found by use of a nonlinear constrained optimization technique. When suitably non-dimensionalized, the solution appears to be unique. It exhibits a semi-infinite crack of constant width following the propagating fracture. For each value of the stress intensity factor of the medium, the width and propagation speed are determined. The results are applied to the problem of the vertical ascent of magma through the earths mantle and crust. Values obtained for the crack width and ascent velocity are in accord with observations. This mechanism can explain the high ascent velocities required to quench diamonds during a Kimberlite eruption. The mechanism can also explain how basaltic eruptions can carry large mantle rocks (xenoliths) to the surface.

Completely imbedded Runge-Kutta pairs

Recently, pairs of explicit Runge–Kutta methods of orders 5 and 6 based on a new design have been derived independently by several authors. These pairs may be implemented so that the approximation of order 6 may be propagated using eight stages, and the extra derivative evaluation required for the error-estimating approximation of order 5 may be used again in the next step.An improved derivation of these pairs leads to a convenient generalization and thus to the derivation of families of higher-order pairs of the same design. For arbitrary p, pairs of orders

N-body simulations: The performance of some integrators

p - 1

On order 5 symplectic explicit Runge-Kutta Nyström methods

and p require

A Multirate Störmer Algorithm for Close Encounters

s \leq {{(p^2 - 7p + 22)} / 2}

Generation of high-order interpolants for explicit Runge-Kutta pairs

internal stages (the inequality holds for

Distortion and Necking of a Viscous Inclusion in Stokes Flow

p \geq 8

High order explicit Runge---Kutta Nyström pairs

) and the derivative evaluation of the propagating stage. Furthermore, for each p, imposing an additional constraint on the nodes removes the necessity for the extra derivative evaluation. For each pair of this restricted subfamily, the method of order p utilizes all s stages so that the method of order

The performance of phase-lag enhanced explicit Runge-Kutta Nyström pairs on N-body problems

p - 1