Lundgrens Infinite Hierarchy of Probability Density Functions for Linear Viscoelastic Materials

The statistical theory of inertial turbulence at high Reynolds numbers has been well-established and thoroughly studied for the better part of a century. The treatment in terms of multi-point probability density functions, governed by Lundgren's infinite hierarchy of transport equations, is particularly tractable.

The governing equations have recently been extended to the treatment of compressible flows, and the treatment of active- and magnetohydrodynamic turbulence. However, no such description of viscoelastic turbulence yet exists.

To obtain a formally closed description of viscoelastic turbulence, the introduction of multi-time probability density functions is proposed, capturing the temporally non-local nature of the Cauchy stress tensor. These density functions convey the probability of observing the velocities v₁ through v_N at N separate spatiotemporal points, within some infinitesimal phase-space volumes dv_i. Finally, a Lungren-type hierarchy is obtained by utilizing the deterministic one-point momentum equations. Although the current framework only allows for the treatment of linear viscoelastic materials, its extension to the more general Walter’s fluids is discussed.