A continuum description of the buckling of a line of spheres in a transverse harmonic confining potential

A line of contacting hard spheres, placed in a transverse confining potential, buckles under compression or when tilted away from the horizontal, once a critical tilt angle is exceeded. This interesting nonlinear problem is enriched by the combined application of both compression and tilt. In a continuous formulation, the profile of transverse sphere displacement is well described by numerical solutions of a second-order differential equation (provided that buckling is not of large amplitude). Here we provide a detailed discussion of these solutions, which are approximated by analytic expressions in terms of Jacobi, Whittaker and Airy functions. The analysis in terms of Whittaker functions yields an exact result for the critical tilt for buckling without compression.

Automated summary

A line of contacting hard spheres placed in a transverse confining potential buckles under compression or tilt beyond a critical angle. The displacement profile of transverse spheres can be well described by numerical solutions of a second-order differential equation, approximated by analytic expressions involving Jacobi, Whittaker, and Airy functions. Exact results for the critical tilt without compression are obtained through the analysis in terms of Whittaker functions.