Double bracket formulation for the distribution function approach to multibead-chain suspensions

Abstract: A suspension of elastic chains of small beads in a Newtonian fluid is a common model for a viscoelastic polymer solution. The configuration of these multibead chains can be described by a distribution function that evolves according to a Liouville or Fokker--Planck equation. For example, when we consider a suspension of Hookean bead-spring pairs, this approach leads to the well-known upper-convected Maxwell model. This is done by taking the second moment of the Fokker--Planck equation that governs the distribution function, which describes the stress of the bead-spring pairs on the fluid. The evolution of these multibead-chain suspensions can be described using a double bracket formulation with a Hamiltonian functional. The conservative part of the dynamics is described by a Poisson bracket, and the dissipative part by an additional symmetric bracket. We treat the configuration space of multibead chains as a higher order tangent bundle. Lifting the fluid velocity field to the bundle leads naturally to a semidirect product Lie--Poisson bracket for the conservative dynamics. The elastic stress exerted by the chains then follows directly from the Hamiltonian functional. For chains with three or more beads, the possible bending of the chain introduces an angular momentum flux that is absent for chains with two beads. This flux appear as an asymmetric elastic stress whose antisymmetric part is the divergence of a rank-$3$ tensor, as in the Cosserats' theory of couple stresses. We investigate the possibility of an exact closure, passing from a distribution function description to a closed internal state variable description of the fluid suspension, and obtain some sufficient conditions for their existence. The resulting exactly closable models are generalisations of the upper-convected Maxwell model to Hookean bead-spring chains.