Laminar oscillatory flow of Maxwell and Oldroyd-B fluids: Theoretical analysis

Abstract

We study the laminar oscillatory flow of Maxwell and Oldroyd-B viscoelastic fluids. We consider two basic geometries (rectangular and cylindrical), under two different modes of driving. Our results show that in the inertialess regime ( Re ≪ 1) the flow properties depend only on three characteristic lengths: the wavelength λ 0 and damping length x 0 of viscoelastic shear waves, and the characteristic transverse size of the system a. The three length scales are generic functions – that we compute – of three independent dimensionless groups: t v / λ (viscous to relaxation time), De (relaxation time to oscillation period) and X (viscosity ratio). The oscillatory behavior can be classified in two broad classes, corresponding to ‘wide’ ( a/ x 0 > 1) and ‘narrow’ ( a/ x 0 < 1) systems. In wide systems the oscillation is confined to the sidewalls and the flow in the central core is inviscid. In narrow systems shear waves cross the whole system and interfere, eventually leading to constructive resonances that result in a dramatic increase of the amplitude of the velocity profiles. Our analysis shows that resonant peaks are located at universal values of the ratio a/ λ 0. For the UCM fluid, the amplitude of the velocity profiles depends only on t v / λ. A study of the oscillatory flow of the Oldroyd-B fluid as a function of X under resonant conditions ( t v / λ < 1) reveals that a very small additive Newtonian solvent contribution is sufficient to largely suppress the resonant behavior.