Numerical Simulation and Linearized Theory of Vortex Waves in a Viscoelastic, Polymeric Fluid

Robert A Handler,Michael J Buckingham

doi:10.3390/fluids6090325

Robert A Handler, Michael J Buckingham

Open Access

https://doi.org/10.3390/fluids6090325

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Abstract

In a high viscosity, polymeric fluid initially at rest, the release of elastic energy produces vorticity in the form of coherent motions (vortex rings). Such behavior may enhance mixing in the low Reynolds number flows encountered in microfluidic applications. In this work, we develop a theory for such flows by linearizing the governing equations of motion. The linear theory predicts that when elastic energy is released in a symmetric manner, a wave of vorticity is produced with two distinct periods of wave motion: (1) a period of wave expansion and growth extending over a transition time scale, followed by (2) a period of wave translation and viscous decay. The vortex wave speeds are predicted to be proportional to the square root of the initial fluid tension, and the fluid tension itself scales as the viscosity. Besides verifying the predictions of the linearized theory, numerical solutions of the equations of motion for the velocity field, obtained using a pseudo-spectral method, show that the flow is composed of right- and left-traveling columnar vortex pairs, called vortex waves for short. Wave speeds obtained from the numerical simulations are within 1.5% of those from the linear theory when the assumption of linearity holds. Vortex waves are found to decay on a time scale of the order of the vortex size divided by the solution viscosity, in reasonable agreement with the analytical solution of the linearized model for damped vortex waves. When the viscoelastic fluid is governed by a nonlinear spring model, as represented by the Peterlin function, wave speeds are found to be larger than the predictions of the linear theory for small polymer extension lengths.

Highlights

[1], it was shown theoretically and through numerical simulations that coherent vortex motions can be generated by the release of elastic energy in a viscoelastic, polymeric fluid initially at rest
We show that the release of elastic energy generates vorticity that propagates as a wave, whose speed is governed by the initial stress field in the fluid
0, which is the case in the situations we investigate through in which f/λ ν0 β α f C11 numerical simulations, the vortex wave speed can be accurately approximated by

Summary

Introduction

[1], it was shown theoretically and through numerical simulations that coherent vortex motions can be generated by the release of elastic energy in a viscoelastic, polymeric fluid initially at rest. Elastic stress gradients in viscoelastic flows of dilute polymer solutions were shown to generate torques on fluid elements, which in turn generate coherent vortex motions (e.g., vortex rings). In the situation where the initial elastic stress field was spatially symmetric, two vortex rings were shown to form and move in opposite directions before being dissipated by viscous effects. Important aspects of the kinematics and dynamics associated with these coherent vortical structures were not explored in [1]. We show that the release of elastic energy generates vorticity that propagates as a wave, whose speed is governed by the initial stress field in the fluid

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