A two species micro–macro model of wormlike micellar solutions and its maximum entropy closure approximations: An energetic variational approach

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 of polymeric fluids in a unified variational framework. We also study different maximum entropy closure approximations to the micro-macro model by “variation-then-closure” and “closure-then-variation” approaches. By imposing a proper dissipation in the coarse-grained level, the closure model, obtained by “closure-then-variation”, 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.