This study is concerned with the mathematical modelling of the motion of arthropod filiform hairs in general, and of spider trichobothria specifically, in oscillating air flows. Analysis of the behaviour of hair motion is based on numerical calculations of the equation for conservation of hair angular momentum. In this equation the air-induced drag and virtual mass forces driving the hair about the point of attachment to the substrate are both significant and require a correct prescription of the air velocity. Two biologically significant cases are considered. In one the air oscillates parallel to the axis of the cylindrical substrate supporting the hair. In the other the air oscillates normal to that axis. It is shown that the relative orientation between the respective directions of the air motion and the substrate axis has a marked effect on the magnitudes of hair displacement, velocity and acceleration but not on the resonance frequency of the hair. It is also shown that the variation of velocity with distance from the substrate depends on the value of the parameter Re s St s , the product of the Reynolds number and the Strouhal number characterizing the motion of air past the substrate. In the case of air motion parallel to the substrate axis the analytical result derived by Stokes (1851), for a fluid oscillating along a flat surface of infinite extent, applies if Re s St s >10 or, equivalently, if fD 2 / v >20/n where f is the air oscillation frequency, D the substrate diameter and v the kinematic viscosity of the air. In contrast, in the case of air motion perpendicular to the substrate axis Stokes’ (1851) analysis never applies due to a substrate curvature dependence of the velocity profile for all biologically significant values of Re s St s . Present theoretical considerations point to a new method for simultaneously determining R , the damping constant, and S, the torsional restoring constant of a filiform hair from measurements of the phase difference between hair displacement and air velocity as a function of the air oscillation frequency. For the filiform hairs of crickets we find from the data available that S = 0 (10 -11 ) N m rad -1 and R = 0 (10 -13 ) N m rad -1 . All major qualitative aspects of known hair motion in response to air motion are correctly predicted by the numerical model.
The starting jet produced by the impulsively started discharge of a submerged gas stream of constant velocity through a circular orifice in a plane wall is investigated by integrating numerically the axisymmetric Navier-Stokes equations for moderately large values of the jet Reynolds number. The analysis is restricted to low-Mach-number jets, for which the jet-to-ambient temperature ratio ␥ = T j / T o emerges as the most relevant parameter. It is seen that the leading vortex approaches a quasisteady structure propagating at an almost constant velocity, which is larger for smaller values of ␥. The action of the baroclinic torque in regions of nonuniform temperature leads to significant vorticity production, with a constant overall rate equal to that of an inviscid starting jet.
The slowly reacting mode of combustion of gaseous mixtures in spherical vessels. Part 1: Transient analysis and explosion limits.
Unsteady numerical simulations of axisymmetric reactive jets with one-step Arrhenius kinetics are used to investigate the problem of deflagration initiation in a premixed fuel-air mixture by the sudden discharge of a hot jet of its adiabatic reaction products. For the moderately large values of the jet Reynolds number considered in the computations, chemical reaction is seen to occur initially in the thin mixing layer that separates the hot products from the cold reactants. This mixing layer is wrapped around by the starting vortex, thereby enhancing mixing at the jet leading head, which is followed by an annular mixing layer that trails behind, connecting the leading vortex with the orifice rim. A successful deflagration is seen to develop for values of the orifice radius larger than a critical value, ac, of the order of the flame thickness of the planar deflagration, δL. Introduction of appropriate scales for the different flow variables provides the dimensionless formulation of the problem, with flame initiation characterized in terms of a critical Damköhler number ∆c = (ac/δL) 2 , whose parametric dependence is investigated. The numerical computations reveal that, while the jet Reynolds number exerts a limited influence on the criticality conditions, the effect of the reactant diffusivity on ignition is much more pronounced, with the value of ∆c increasing significantly with increasing Lewis numbers Le. The reactant diffusivity affects also the way ignition takes place, so that for reactants with Le > ∼ 1 the flame develops as a result of ignition in the annular mixing layer surrounding the developing jet stem, whereas for highly diffusive reactants with Lewis numbers sufficiently smaller than unity combustion is initiated in the mixed core formed around the starting vortex. Steady computations of weakly reactive subcritical jets are also employed to determine ∆c, giving results in close agreement with those of unsteady computations for Le > ∼ 1, when the role of the leading vortex is secondary. The boundary-layer problem that emerges in the limit of high jet Reynolds numbers is used to ascertain the effect of density variations and variable transport properties on the predicted critical Damköhler numbers, and to confirm the small influence of other dimensionless parameters such as the Zeldovich number, provided that its value is sufficiently large. The analysis provides increased understanding of deflagration initiation processes, including effects of differential diffusion, and points the need for further investigations incorporating detailed chemistry models for specific fuel-air mixtures.
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