View from the Pennines: Sticky Waves

There is a lot of mud on the moors. Thankfully this does not mean that my boots need cleaning after every walk. If the weather has been dry or there is a light drizzle, then my boots are covered by a thin film of peaty dust which I can live with. If it is very wet then the water cascades down the paths and my boots return sodden but with only the odd stem of grass or heather stuck to them. All they need is time to dry. It is the intermediate case that is troublesome. Steady, undramatic rain penetrates the surface of the peat paths which rapidly revert to boggy quagmires. Mud adheres to mud, and with each step my boots get heavier until a critical mass is reached and gravitational effects remove a large chunk of mud. I stride on and the process begins anew. Mathematicians have recently been thinking about another sticky problem, but being mathematicians their description is cleaner and drier. Like the problem with my boots, though, the behaviour of their sticky liquid depends on the relative strengths of gravity, viscosity and surface tension. Kelmanson and colleagues at Leeds and Cambridge have been working on a classic problem in fluid mechanics. They consider the motion of a sticky liquid coating the outside of a rotating cylinder when the axis of the cylinder is horizontal and gravity acts vertically downwards. This should be familiar to anyone who has used a honey spoon – to keep the honey on the spoon it is necessary to rotate the spoon, creating complicated waves on the surface of the honey as a drip begins to form and is then swept round the spoon by the rotation. The analysis of [2, 3] describes these waves, and in particular the different instabilities that form on different time scales. The early advances in this problem are due to Moffatt [4] (also at Cambridge) and Pukhnachev [6]. Numerical simulations found surface waves, but it was hard to tell whether these were damped or undamped on the time scales that could reasonably be investigate [1, 5]. When computers cannot give clear answers a little more thought is called for, and this is what Hinch and Kelmanson have provided [2, 3].