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.

Interaction between arthropod filiform hairs in a fluid environment

“…The height of the boundary layer is on order of the lengths of the filiform hairs and thus has a non-negligible effect on hair motion. We compute the boundary layer, which we denote u b , for axial flow along an infinite cylinder using the work of Humphrey et al [12].…”

“…The Navier-Stokes equations can be solved explicitly both for a periodically driven flow over an infinite plane [35] and axial flow over a bi-infinite cylinder [12]. Since each cercus is approximately a long finite cone, Shimozawa and collaborators [31,32,33] argued that both infinite planar and cylindrical approximations can be used successfully.…”

“…In this section, following Humphrey et al [12] and Osborne [27], we solve the Navier-Stokes equations in cylindrical coordinates for axial flow over a bi-infinite cylinder, and assume that this is a reasonable approximation to the boundary layer over a finite cylinder. First allow to be the representation of the boundary layer in cylindrical coordinates, where the z-axis extends length-wise through the center of the cylinder.…”

“…Now we are able to convert (2) into a dimensionless equation using the following factors: D is the diameter of the cylinder, U 0 is the peak far-field air velocity, ω is the angular frequency of the air motion, Re is the Reynolds number, and St is analogous to the Strouhal number. We set Then, through a substitution and a short computation (see [12,27]) equation (2) becomes…”

“…This work has been extended by Humphrey et al [12], Shimozawa et al [33] and Osborne [27]. More recently Bathellier et al [3] studied viscosity-mediated coupling between pairs of hairs aligned with the direction of air motion on the leg of the spider Cupiennius salei.…”