Inertioelastic Poiseuille flow over a wavy surface

Streamwise boundary undulations induce spanwise vorticity in flows. In Newtonian flow, vorticity decays exponentially with distance from the undulation. However, recent theory for inertioelastic polymeric flow predicts vorticity amplification in a “critical layer” far from the site of disturbance. Here we present the first experimental evidence for the existence of such critical layers, demonstrating their measurable role in real polymeric flows with inertia. These vorticity amplification effects should be considered in all evaluations of vorticity dynamics in inertioelastic flows with streamline curvature.

Figure 1

(a) Schematic representation of Couette flow over a wavy surface, indicating the nature of vorticity perturbations predicted in the three regimes of viscoelastic flow [16]. (b) Schematic representation of Poiseuille flow over a wavy surface, as employed in the present study.

Figure 2

Normalized $y$-velocity component perturbations $v^{\prime }/U$ in the wavy walled channel with $\alpha =3.2\pi$ at the dimensionless viscous length scale $\theta$ indicated at the top. (a) Newtonian fluid $\left(\mathrm{\Sigma }=0\right)$ at the indicated Re. (b) Polymeric fluid A $\left(\mathrm{\Sigma }=1.0\right)$ at the indicated Re and Wi. (c) Difference between the polymeric and the Newtonian response at each $\theta$ value.

Figure 3

Penetration depth $\mathcal{P}$ as a function of the viscous length $\theta$ for Newtonian and polymeric fluids in the deep and shallow channels. Inset shows the difference between the penetration of fluid A and that of the Newtonian fluid in the deep channel as a function of the Weissenberg number for the polymeric flow. The sigmoidal dashed line is to guide the eye.

Figure 4

(a) Normalized transverse velocity and (b) spanwise vorticity perturbations predicted by the linear equations in the elastic-Rayleigh limit $\left(\text{Wi}\gg 1,\theta \ll 1\right)$ in a deep channel $\left(\alpha =3.2\pi \right)$. In all three cases $\beta =0.96$. A finite Reynolds number regularizes the solution at the critical layer. The critical layer is centered about a dimensionless distance $ky=\mathrm{\Sigma }$, in agreement with Ref. [16].