Three-dimensional viscous steady streaming in a rectangular channel past a cylinder

TL;DR

This paper models three-dimensional viscous steady streaming caused by oscillatory flow past a cylinder in a rectangular channel, analyzing how confinement and geometry influence flow patterns using asymptotic and numerical methods.

Contribution

It introduces a combined Fourier and asymptotic approach to study confinement effects on steady streaming in a rectangular channel with a cylinder, including symmetry breaking and higher-order corrections.

Findings

Steady streaming depends on domain width and cylinder position.

Symmetry breaking occurs due to domain shape.

Higher-order solutions improve accuracy of flow predictions.

Abstract

We consider viscous steady streaming induced by oscillatory flow past a cylinder between two plates, where the cylinder's axis is normal to the plates. While this phenomenon was first studied in the 1930s, it has received renewed interest recently for possible applications in particle manipulations and non-Newtonian flows. The flow is driven at the ends of the channel by the boundary condition which is a series solution of the oscillating flow problem in a rectangular channel in the absence of a cylinder. We use a combination of Fourier series and an asymptotic expansion to study the confinement effects for steady-streaming. The Fourier series in time naturally simplifies to a finite series. In contrast, it is necessary to truncate the Fourier series in z, which is in the direction of the axis of the cylinder, to solve numerically. The successive equations for the Fourier coefficients resulting from the asymptotic expansion are then solved numerically using finite element methods. We use our model to evaluate how steady streaming depends on the domain width and distance from the cylinder to the outer walls, including the possible breaking of the four-fold symmetry due to the domain shape. We utilize the tangential steady-streaming velocity along the radial chord at an angle of pi/4 to analyze our solutions over an extensive range of oscillating frequencies and multiple levels in the z-direction. Finally, higher-order solutions are computed and an asymptotic correction to steady streaming is included.