DYNAMICS OF ARTHROPOD FILIFORM HAIRS .1. MATHEMATICAL-MODELING OF THEHAIR AND AIR MOTIONS

This study is concerned with the mathematical modelling of the motion of arthropod filiform hairs in general, and of spider trichobothria sp ecifically, in oscillating air flows. Analysis of the behaviour of hai r motion is based on numerical calculations of the equation for conser vation of hair angular momentum. In this equation the air-induced drag and virtual mass forces driving the hair about the point of attachmen t to the substrate are both significant and require a correct prescrip tion of the air velocity. Two biologically significant cases are consi dered. In one the air oscillates parallel to the axis of the cylindric al substrate supporting the hair. In the other the air oscillates norm al 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 acc eleration but not on the resonance frequency of the hair. It is also s hown that the variation of velocity with distance from the substrate d epends on the value of the parameter Re(S) St(S), the product of the R eynolds number and the Strouhal number characterizing the motion of ai r past the substrate. In the case of air motion parallel to the substr ate axis the analytical result derived by Stokes (1851), for a fluid o scillating along a flat surface of infinite extent, applies if Re(S) S t(S) > 10 or, equivalently, if fD2/nu>20/pi where f is the air oscilla tion frequency, D the substrate diameter and v the kinematic viscosity of the air. In contrast, in the case of air motion perpendicular to t he substrate axis Stokes' (1851) analysis never applies due to a subst rate 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 c onstant, and S, the torsional restoring constant of a filiform hair fr om measurements of the phase difference between hair displacement and air velocity as a function of the air oscillation frequency. For the f iliform hairs of crickets we find from the data available that S=O(10( -11)) N m rad-1 and R=O(10(-13)) N m s rad-1. All major qualitative as pects of known hair motion in response to air motion are correctly pre dicted by the numerical model.