Unsteady and quasi-steady shear stress under oscillating flow of a viscoelastic fluid in a cylindrical pipe

Abstract

This paper examines the oscillatory laminar flow of a viscoelastic fluid in a pipe with circular cross-section. The linear Oldroyd equation is adopted as the rheological equation. Expressions for the transfer and frequency functions in the Laplace transforms are derived. The ratio of unsteady and quasi-steady wall shear stress, in contrast to the quasi-steady approach, depends on the parameter expressing the acceleration of fluid flow. Computational experiments were carried out to determine the unsteady wall shear stress. It is established that with increasing acceleration of the fluid flow, the unsteady shear stress increases for a viscous fluid, while for a viscoelastic fluid it also depends on the relaxation parameters. For certain frequencies, the transient shear stress depends almost linearly on frequency. For a Maxwell fluid, the dependence of unsteady shear stress on frequency has an oscillatory character. The differences between these results and the results obtained based on the quasi-steady approach are shown. Influence of viscoelastic properties of the fluid on dynamics of unsteady shear stress is also studied. The results obtained show that the viscoelastic properties of the fluid act as a limiting factor when using a quasi-steady approach to determine the unsteady wall shear stress. It was shown that the acceleration of fluid flow acts in the same manner.