The flow field induced by a vertical plate accelerating into a stationary fluid of finite depth with a free surface and a gravitational restoring force is investigated. This is a model problem for some technologically important design issues such as the bow splash of a ship moving at forward speed. Experimentally it is found that a thin jet forms on the plate and rises rapidly upwards. We investigate this jet in the small-time approximation and find an analytical solution for the flow field in which the jet emerges out of a thin region where the horizontal momentum of the main flow is converted by inertial effects into a rising jet.

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Flow down an open inclined channel is considered. Dressler ( Communs pure appl. Math. 2, 149-194 (1949),) using the equations of the shallow water theory augmented by the Chezy formula for drag, has shown that the uniform flow becomes unstable when the Froude number F exceeds 4, and in this case he was able to construct a one-parameter family of discontinuous periodic solutions by piecing together continuous sections of wave profile and a series of hydraulic jumps. Here the work of Dressler is extended by the inclusion of a further term that accounts for energy dissipation by shearing normal to the flow. It is shown that the inclusion of such a term does not alter the condition for stability of the uniform flow, and that when the uniform flow is unstable, a one-parameter family of quasi-steady periodic solutions exists (parametrized by the propagation speed U ), appearing as a Hopf bifurcation out of the uniform flow at the critical value U c = 1 + F -½ . After the existence of these periodic solutions has been shown, uniformly valid expansions for the periodic solutions are obtained by using the Krylov─Bogoliubov─Mitropolski averaging method, and the results are also extended to larger amplitudes by numerical integration.

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The propagating reaction-diffusion waves that develop in the isothermal autocatalytic system A + B -* 2B from a local initial input of reactant B are considered. A solution valid for a small initial input of B is obtained first, and this is augmented by numerical solutions of the general problem. These show that, asymptotically, the reaction-diffusion wave propagates with the minimum, physically acceptable, wave speed. The large-time solution for the general case is then discussed and it is shown that ahead of the reaction-diffusion front is a weak diffusion-controlled region. It is the matching between these two regions that fixes the wave speed, so that the speed of propagation of the reaction-diffusion front is controlled by the rate at which B can spread forward by diffusion.