Mark J. Cooker

The impact of liquid-coated solid bodies onto porous substrates, or equivalently the impact of porous bodies onto shallow water layers, is considered. This study investigates the role of air cushioning in the pre-impact dynamics of a wetted particle approaching a dry porous medium. By extending a shallow water air cushioning model, we develop an asymptotic theory that couples flow in a lubricating air layer, with an inviscid shallow water film and with Darcy air flow within the porous substrate. Two distinct regimes are identified, namely shallow- and intermediate-depth substrates. The formulated models are solved numerically to determine the influence of substrates on impact pressure, bubble formation, and air escape mechanisms. Numerical results, up to the instant of liquid–substrate impact, reveal that at high porosity the substrate significantly reduces the impact pressure and can suppress bubble entrapment. Permeability is shown generally to hasten the liquid film’s descent to the top of the substrate (so-called ‘touchdown’) as the air gap is closed but, perhaps most surprisingly, porosity can delay touchdown in the regime of shallow porous layers. These findings enhance our understanding of impact mechanics on porous media and offer insights into practical applications such as de-icing technologies and controlled particle adhesion in manufacturing processes.

An analytical model for droplet impact onto a porous substrate is presented, based on Wagner theory. An idealised substrate boundary condition is introduced, mimicking the effect of fluid entry into a genuinely porous substrate. The asymptotic analysis yields a solution for a small porous correction with free-surfaces and pressures compared with the impermeable case. On a global scale, it is found that the impact region on the substrate grows more slowly with porosity included due to loss of mass into the substrate. The spatial distribution of liquid volume flux into the substrate is also described. Locally near the turn-over regions, the expected jetting along the surface is calculated with the same volume flux but the jet is found to be slower and thicker than for an impermeable substrate.

Inviscid, incompressible liquid is released from rest by a sudden dam break, accelerating under gravity over a uniformly sloping impermeable plane bed. The liquid flows downhill or up a beach. A linearised model is derived from Euler's equations for the early stage of motion, of duration . Second, the triangle belongs to a class of finite-domain solutions with a curved front face. Third, an unbounded domain is treated, with a curved face resembling a steep-fronted breaking water wave flowing up a beach. The fluid goes uphill due to a nearshore pressure gradient. In all cases the free-surface-bed contact point is the most accelerated particle, exceeding the acceleration due to gravity. Physical consequences are discussed, and the pressure approximation of shallow water theory is found poor during this early stage, near the steep free surface exposed by a dam break.

This Article is about the many complicated tasks that one mathematician had to carry out, and the barriers he had to overcome in order to publish one very important book in the history of applied mathematics.

Sediment accumulation downstream of hydraulic jumps can occur in many settings but the architectures of such deposits are poorly documented. Here, three flume runs were used to examine the influence of sediment grain size and transport rate on the characteristics of hydraulic‐jump unit bars. In one of these runs six hydraulic‐jump unit bars formed a hydraulic‐jump bar complex. In another, the same sediment was supplied more quickly and only two unit bars formed. In the third run with the same sediment supply rate, but different grain size, only one large unit bar formed. All unit bars developed in a similar way but their size and internal architecture differed; they all resulted from a reduction in sediment transport capacity at the transition from supercritical flow to subcritical flow in the hydraulic jump. After initial onset of sedimentation and unit bar formation, generation of subsequent unit bars may be: (i) related to small changes in sediment flux; and (ii) independent of changes in the hydraulic jump. Continued sedimentation caused changes from oscillating to weak hydraulic jumps and hydraulic‐jump unit bars formed in both circumstances. The flow of water and suspended sediment becomes shallower over the lee of the bar complex. This leads to flow acceleration and a return to supercritical flow conditions. In turn, a chain of such features can form and generate a chute and pool bed morphology. There is an inherent upper size limit to a hydraulic‐jump bar complex due to the changing flow conditions over the growing deposit as the water above it becomes shallower. There is also an amplitude minimum for the development of foresets and subsequent unit bar growth. Hydraulic‐jump unit bars have architectures that should be recognizable in the rock record and because their size is constrained by the flow conditions, their identification should be useful for interpreting palaeoenvironment.

One day the Tortoise was discussing with mighty Achilles the behaviour of infinite series. Achilles boasted that his mental powers were so fast that he could exhaust any series summation problem by simply adding up enough terms. But Tortoise was unimpressed: ‘Is that so? What's the sum of this series, then?’ With his tail he drew a line in the sand, and wrote which is the same as ‘I'll give you until tomorrow to find K to four significant digits!’