Motion of a sphere oscillating at low Reynolds numbers in a viscoelastic-fluid-filled cylindrical tube

Abstract

The motion at low Reynolds numbers of a sphere oscillating with a small amplitude inside a cylindrical tube containing a viscous fluid (Newtonian or viscoelastic) is investigated. The linear Jeffreys model is used to describe the constitutive relation for the viscoelastic fluid. The governing equations are transformed to, and solved in, the frequency domain where the Fourier components for the flow and drag are obtained using a finite difference method. The numerical results for the Newtonian fluid velocity agree well with those given by a high frequency asymptotic solution. The effects of the Stokes number, ϵ, on the unsteady drag, D R + iD I, are discussed. At small ϵ, the real component of the unsteady drag, D R, is dominated by the quasi-steady term for both Newtonian and viscoelastic fluids. For a Newtonian fluid, the acceleration-dependent terms of D R and D I are proportional to ϵ 4 and ϵ 2 respectively. Whereas D R increases monotonically with ϵ for a Newtonian fluid, it reaches a minimum for a Jeffreys fluid when ϵ ≈ O(1). The difference observed is caused by the shear-thinning property of the viscoelastic fluid. At large ϵ, D R becomes proportional to ϵ, and the imaginary component of the unsteady drag, D I, for both Newtonian and viscoelastic fluids, is dominated by the added-mass force. However, as opposed to a Newtonian fluid, D I changes sign near ϵ ≈ O(1) for a viscoelastic fluid due to shear thinning. Finally, a method for determining the rheological properties of a general viscoelastic fluid from measurements of the unsteady drag in a ball rheometer is presented.