  Wormlike micelles are self-assemblies of polymer chains that can break and
recombine reversibly. In this paper, we derive a thermodynamically consistent
two species micro-macro model of wormlike micellar solutions by employing an
energetic variational approach. The model incorporates a breakage and
combination process of polymer chains into the classical micro-macro dumbbell
model for polymeric fluids in a unified variational framework. We also study
different maximum entropy closure approximations to the new model by
"variation-then-closure" and "closure-then-variation" approaches. By imposing
proper dissipation in the coarse-grained level, the closure model, obtained by
"closure-then-approximation", preserves the thermodynamical structure of both
mechanical and chemical parts of the original system. Several numerical
examples show that the closure model can capture the key rheological features
of wormlike micellar solutions in shear flows.
