Resonance in laminar pipe flow of non-linear viscoelastic fluids

Abstract

Resonance phenomenon is investigated in pulsating flows in straight circular tubes for the class of constitutively non-linear affine viscoelastic fluids represented by the Johnson–Segalman constitutive model. The relaxation response of non-linear viscoelastic fluids associated with the natural frequencies of oscillation that arise as a consequence of the structure of the constitutive equation is explored. An analytical solution, which circumvents the high-Weissenberg number problem, which plagues numerical solutions of highly viscoelastic fluids, based on an asymptotic expansion in terms of a small material parameter is developed. At the lowest order of the asymptotic expansion the velocity field of the Maxwell fluid in round tubes is recovered, therefore providing validation of the computations. The analytical solution allows insights into the high Weissenberg number behavior of highly elastic fluids and thus avoids the difficulties the numerical solutions of high Weissenberg number problems are fraught with such as instability and lack of convergence. The analysis reveals that the forcing frequency associated with the pressure gradient generates a sequence of resonances of rapidly decaying intensity with increasing forcing frequency for a fixed value of the Weissenberg number. The intensity of the resonance is a rapidly increasing function of the increasing Weissenberg numbers, in particular at the first natural frequency. The latter is a decreasing function of the increasing elasticity of the fluid that is of the increasing Weissenberg numbers. The intensity of the resonance at the first natural frequency is a rapidly increasing function of the increasing elasticity. We show the existence of a limit point for the enhancement in energy savings with increasing elasticity. The effect of resonance on the flow rate, maximum amplitude of the average velocity, the mean flow rate and the oscillating velocity field is explored.