“Phase diagram” for viscoelastic Poiseuille flow over a wavy surface

Simon J Haward,Tamer A Zaki,Amy Q Shen,Jacob Page

doi:10.1063/1.5057392

Abstract

We experimentally examine the Poiseuille flow of viscoelastic fluids over wavy surfaces. Five precision microfabricated flow channels are utilized, each of depth 2d = 400 μm, spanwise width w = 10d and with a sinusoidal undulation of amplitude A = d/20 on one of the spanwise walls. The undulation wavelength λ is varied between each of the channels, providing dimensionless channel depths α in the range 0.2π ≤ α = 2πd/λ ≤ 3.2π. Nine viscoelastic polymer solutions are formulated, spanning more than four orders in elasticity number El and are tested in the wavy channels over a wide range of Reynolds and Weissenberg numbers. Flow velocimetry is used to observe and measure the resulting flow patterns. Perturbations to the Poiseuille base flow caused by the wavy surfaces are quantified by the depth of their penetration P into the flow domain. Consistent with theoretical predictions made for wavy plane-Couette flow [J. Page and T. A. Zaki, “Viscoelastic shear flow over a wavy surface,” J. Fluid Mech. 801, 392–429 (2016)], we observe three distinct flow regimes (“shallow elastic,” “deep elastic” and “transcritical”) that can be assembled into a “phase diagram” spanned by two dimensionless parameters: α and the depth of the theoretically predicted critical layer Σ∼El. Our results provide the first experimental verification of this phase diagram and thus constitute strong evidence for the existence of the predicted critical layer. In the inertio-elastic transcritical regime, a surprising amplification of the perturbation occurs at the critical layer, strongly influencing P. These effects are of likely importance in widespread inertio-elastic flows in pipes and channels, such as in polymer turbulent drag reduction.

Highlights

For a laminar Newtonian flow in a plane-Couette geometry with a sinusoidal wavy perturbation on the stationary wall [see schematic diagram in Fig. 1(a)], linear analysis has shown that vorticity perturbations induced by the wavy surface can be classified using two non-dimensionalized parameters: (1) the depth of the flow domain α = kd and (2) the viscous length θ =1/3
Fluid Mech. 801, 392–429 (2016)], we observe three distinct flow regimes (“shallow elastic,” “deep elastic” and “transcritical”) that can be assembled into a “phase diagram” spanned by√two dimensionless parameters: α and the depth of the theoretically predicted critical layer Σ ∼ El
In this work we used a series of five wavy-walled channels and nine polymer solutions of distinct rheology in order to perform an experimental investigation of viscoelastic shear flows over wavy surfaces

Summary

Introduction

A “shallow viscous” regime occurs for α 1, θ > α; in this regime, the channel depth is small compared to the roughness wavelength so the vorticity perturbation fills the flow domain (Pω ≈ α). The viscous layer is deeper than the flow domain, and so the flow is unaffected by inertia. A “deep viscous” regime occurs for α 1, θ 1; in this regime, the channel depth is large compared to the roughness wavelength, and the vorticity perturbation decays within the flow domain over approximately one wavelength from the surface (Pω ≈ 1). An “inviscid” regime occurs for the conditions α > θ, θ 1 In this case, the viscous layer is confined within the flow domain and inertial effects are appreciable. The perturbation is confined within the viscous layer (Pω ∼ θ) and is tilted forward due to the inertia.

Methods

Results

Conclusion