The relation between CDM instability and Deborah number in differential type rheological equations

Abstract

Previous studies have argued that rheological equations of the differential type, such as second-order fluid models, are inadequate because they result in unstable solution after cessation of steady shear. If the sign of the viscoelastic coefficient is selected so that the storage modulus is positive, the fluid velocity increases indefinitely and the flow does not decay by viscous dissipation, in contradiction to thermodynamic laws. This study mitigates this problem by demonstrating that the solution of such equations is actually stable at low values of Deborah number De, where these equations are only valid for other reasons. In fact, second order and higher order differential type equations are applicable only if the relaxation time of the fluid is low relative to a characteristic time of the flow. The study shows how to determine the characteristic time and thus clarifies, in practical terms, the limits of the region where differential type equations can be applied.