Unsteady non-viscometric flows in tubes driven by rotational boundary waves—I: Longitudinal field

Abstract

The unsteady non-viscometric motion of a memory integral fluid of order three in a rigid, circular, straight tube has been investigated. The motion is driven by a constant longitudinal pressure gradient and a periodic forcing from the boundary made up of superposed sinusoidal, rotational boundary waves. A regular perturbation is used to study the time-periodic longitudinal field for a range of viscoelastic liquids and driving parameters. Qualitative results obtained are independent of the explicit forms of the constitutive functions, the kernels of the memory integral fluid of order three, which are needed only for quantitative predictions. It is shown that anomalous steady longitudinal viscometric flows may exist due to frequency cancellation. The elastic and shear rate dependent viscosity characteristics of the fluid play important parts in setting up the oscillatory velocity field. In particular, shear-thinning in periodic shear is the dominant mechanism in smoothing out the effects of elasticity, and in increasing the velocities. A parametric study of the longitudinal oscillatory velocity field is presented for a range of liquids and driving conditions. The integral fluid shows drastic longitudinal steady velocity increases due to the interaction of the oscillatory, zero mean time average, transversal shear field with the longitudinal simple steady shear. A parametric study of the change in mass transport is also presented for a range of liquids to show that contrary to what is sometimes assumed, both elasticity and shear-thinning in oscillatory shear play important roles in defining the enhancement. In particular, nonlinear-elastic properties of the integral fluid of order three act to decrease the enhancement driven by orthogonal waves.