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 t$^\alpha$ or to exp ($\alpha t$) 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 $\alpha$ 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.