Dynamics of arthropod filiform hairs. I. Mathematical modelling of the hair and air motions

Abstract

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.