Constitutive equations for extensional flow of wormlike micelles: Stability analysis of the Bautista-Manero model

We carry out a stability analysis of the Bautista-Manero (B-M) constitutive equations for extensional flow of wormlike micelles. We show that all solutions for the steady-state extensional viscosity η_E are unstable when the elongational rates ε exceed some critical value. In some cases the only real solution for the extensional viscosity is negative at intermediate values of the elongational rate. This critical elongational rate is not unfeasibly large, e.g., 250 s^-1 for a typical EHAC solution. We note that the extension rates at which enhanced pressure drop is observed experimentally in porous media flow is generally well below the onset of the instability in the B-M model. However, the extension rates presented here are only average values for the porous media in question and at the pore scale a wide range of values will be encountered. For this reason, we require an improved rheological equation of state. We first separate the contributions to the viscosity of the solution and the wormlike micelles. Then, we remove the term containing the retardation time λ_J. We now find that the extensional flow behaviour is well defined for both the transient and steady-state cases.