A phenomenological model for shear-thickening in wormlike micelle solutions

We present a simple phenomenological model to describe shear-thickening in a wormlike micelle solution. The solution is assumed to undergo a reaction above a critical shear stress, to form a gel-like phase. There is no flow within this "gel" phase, and it is insoluble in the solution. The amount, of gel in the system is determined by a competition between a stress-dependent reaction rate, which creates gel, and a constant breakage rate, which destroys gel. We recover a re-entrant region in the stress-shear rate curve, which is only accessible under controlled stress conditions. When the apparent shear rate is fixed, the corresponding stress-shear rate relation is discontinuous. Our results are in qualitative agreement with the experimental observations of Pine et al.