In 1967, Stuart [J. Fluid Mech. 29, 417 (1967)] found an exact nonlinear solution of the inviscid, incompressible two-dimensional Navier–Stokes equations, representing an infinite row of identical vortices which are now known as Stuart vortices. In this Brief Communication, the corresponding result for an infinite row of counter-rotating vortices, i.e., a row of vortices of alternating sign, is presented. While for Stuart’s solution, the streamfunction satisfied Liouville’s equation, the streamfunction presented here satisfies the sinh–Gordon equation [Solitons: An Introduction (Cambridge U.P., London, 1989)]. The connection with Stuart’s solution is discussed.

We address the pricing of American straddle options. We use partial Laplace transform techniques due to Evans et al. (1950) to derive a pair of integral equations giving the locations of the optimal exercise boundaries for an American straddle option with a constant dividend yield.

Abstract. American call options are financial derivatives that give the holder the right but not the obligation to buy an underlying security at a pre-determined price. They differ from European options in that they may be exercised at any time prior to their expiration, rather than only at expiration. Their value is described by the Black-Scholes PDE together with a constraint that arises from the possibility of early exercise. This leads to a free boundary problem for the optimal exercise boundary, which determines whether or not it is beneficial for the holder to exercise the option prior to expiration. However, an exact solution cannot be found, and therefore by using asymptotic techniques employed in the study of boundary layers in fluid mechanics, we find an asymptotic expression for the location of the optimal exercise boundary and the value of the option near to expiration.2000 Mathematics Subject Classification. 91B28, 41A58.

The motion of rigid spheroidal particles settling under gravity in a spatially periodic, cellular flow field has been studied. The particles are sufficiently small that their motion relative to the surrounding fluid satisfies the conditions for local Stokes flow, and the force and couple on the particle are linearly related to the local flow conditions. The motion of-each particle depends on the orientation of its symmetry axis, which turns in response to the local vorticity and rate of strain. For spherical shapes the cellular flow field generally can hold particles in permanent suspension, as they move in simple closed paths, over a significant portion of each cell. By comparison, for nonspherical shapes this suspension is greatly reduced, though not eliminated. The individual particles undergo a tumbling motion as they settle which, at large enough aspect ratios, is found to be chaotic.