Steady streaming of a viscoelastic fluid within a periodically rotating sphere

Abstract

Motivated by understanding mass transport processes occurring in the vitreous chamber of the eye, we consider the steady streaming component of the flow generated in a viscoelastic fluid contained within a hollow, rigid sphere performing small-amplitude, periodic, torsional oscillations about an axis passing through its centre. The problem is solved semi-analytically, assuming that the amplitude of the oscillations is small. The paper extends the work by Repetto et al. (J. Fluid Mech., vol. 608, 2008, pp. 71-80), in which the case of a purely viscous fluid was analysed. However, in reality, in young and healthy subjects, the vitreous humour has complex rheological properties, and so here we model it as a viscoelastic fluid. A similar problem was studied by Nikolakis (Eine Theorie für stationäre Drifterscheinungen viskoelastischer Flüssigkeiten, 1992, VDI). In the present model, the steady streaming flow is governed by four dimensionless parameters. We show that, when we account for the viscoelasticity of the fluid, there is a considerably more complex set of possible flow regimes than was found in the purely viscous case, and the flows can be classified into five qualitatively different types. Whereas there was only one circulation cell in each hemisphere in the viscous case, accounting for viscoelasticity it is possible have either one, two or three circulation cells, with different senses of rotation, depending on the parameter values.

An experimental model of vitreous motion induced by eye rotations

“…The latter changes in space and time and the complex modulus does so accordingly. This effect is not accounted for in theory and may play a more important role than nonlinearity as discussed by Repetto et al [ 17 ], who showed that nonlinear corrections to the azimuthal velocity profiles would appear at the third order (in the small strain) and it is thus likely to contribute little to the measured velocity profiles.

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“…This is because considering for instance the case of linearly viscoelastic fluids, one should preserve between the prototype and the scaled model the values of two dimensionless parameters, e.g. the Womersley number and the ratio between the elastic and viscous components of the fluid (see for instance [ 17 ]). Once a certain fluid has been chosen for the experiments this should be accomplished by changing the model rotation frequency.…”

Fluid Dynamic Assessment of Hypersonic and Guillotine Vitrectomy Probes in Viscoelastic Vitreous Substitutes

“…The occurrence of steady flows produced by oscillating bodies in a fluid is a well-known phenomenon in fluid mechanics and is referred to as steady streaming, which occurs in both viscous 25,26 and viscoelastic fluids. 27 Cavitation has never been observed.…”

Numerical investigation of the solute dispersion in finite‐length microchannels with the interphase transport

“…Extensive studies of solute dispersion in microchannels with the interphase transport were carried out from the perspectives of the solute separation efficiency [19,23,24] or the solute dispersion theory [25][26][27][28][29]. Also, the dispersion of reactive solutes [29][30][31][32], non-Newton fluids [33,34], and stratified liquids [35] has been widely investigated. These studies obtain numerous expressions of the dispersion coefficient for varying flow fields and wall interactions, which are summarized in Ref.…”