Abstract
AbstractMotivated 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.
Highlights
It is well known that a purely oscillatory boundary condition typically induces a velocity field with a non-zero time average, due to nonlinear effects in the underlying equations of motion
We have studied the steady streaming flow of a viscoelastic fluid contained in a rigid sphere performing periodic torsional oscillations about an axis passing through its centre
A semi-analytical solution of the problem has been found by expanding all variables in terms of powers of the amplitude of oscillations, assumed small
Summary
It is well known that a purely oscillatory boundary condition typically induces a velocity field with a non-zero time average, due to nonlinear effects in the underlying equations of motion. The time-average flow is referred to as the steady streaming, and thorough reviews of this phenomenon for the case of Newtonian fluids are provided by Riley (1967, 2001). A general methodology for studying the steady streaming in a. Meskauskas viscoelastic fluid, which is of interest in the present paper, is described by Nikolakis (1992). Steady streaming can often be important, because it can play a major role in mass transport processes; this is the main reason why steady streaming flows have received so much attention in the literature
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