  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.
