Flow of Viscoelastic Fluid in a Curved Pipe

Abstract

The flow rate of a steady flow of viscoelastic fluid under a constant axial pressure gradient in a toroidal pipe of circular cross-section is analytically solved with the White-Metzner constitutive equation by a perturbation method. It is examined how the shear thinning viscosity and elasticity affect the flow rate of the fluid at high shear-rate to which former analyses could not be applied.The analysis shows that the characteristics of the flow in a curved pipe are determined not only by the Dean number but also by the non-dimensional value We; /√ where We; is the Weissenberg number and R is the ratio of the toroidal radius to the pipe cross-sectional one. The results calculated for the ratio fr; of the flow rate in a toroidal pipe to that in a straight pipe under the same axial pressure gradient at low Reynolds number are as follows:(1) In the case of the Power law fluid in which We; is zero, fr; decreases with increment of the Reynolds number Re; and with decrement of the viscosity index n representing the shear thinning viscosity. Thus, the pipe resistance in a curved pipe is higher than that in a straight one.(2) The larger We; gives the larger fr; and thus the smaller pipe resistance.(3) Large Re; and small n give large increment of fr; with increasing We; .(4) The effect of the elasticity index s, representing the shear thinning elasticity, on fr; is insignificant unless n is small.