Instabilities in strongly shear-thinning viscoelastic flows through channels and tubes

TL;DR

This paper analyzes the linear stability of shear-thinning viscoelastic flows in channels and tubes, revealing universal short-wavelength instabilities near walls and finite-wavelength instabilities influenced by geometry, consistent with experimental data.

Contribution

It demonstrates the existence of universal short-wavelength wall-localized instabilities in shear-thinning viscoelastic flows and characterizes finite-wavelength instabilities depending on geometry and boundary conditions.

Findings

Short wavelength instabilities occur at shear-thinning exponents less than 0.3.

Finite wavelength instabilities depend on geometry and boundary conditions.

Predictions align with experimental observations of polymer solution flows.

Abstract

The linear stability of a shear-thinning, viscoelastic fluid undergoing any of the canonical rectilinear shear flows, viz., plane Couette flow and pressure-driven flow through a channel or a tube is analyzed in the creeping-flow limit using the White--Metzner model with a power-law variation of the viscosity with shear rate. While two-dimensional disturbances are considered for plane Couette and channel flows, axisymmetric disturbances are considered for pressure-driven flow in a tube. For all these flows, when the shear-thinning exponent is less than $0.3$, there exists an identical instability at wavelengths much smaller than the relevant geometric length scale (gap between the plates or tube radius). There is also a finite-wavelength instability in these configurations governed by the details of the geometry and boundary conditions at the centerline of the channel or tube. The most unstable mode could be either of the short-wave or finite-wavelength instabilities depending on model parameters. For pressure-driven channel flow, it is possible to have sinuous or varicose unstable modes depending on the symmetry of the normal velocity eigenfunction about the channel centerline. This difference in symmetry is relevant only for the finite wavelength instability, in which case sinuous modes turn out to be more unstable, in accordance with experimental observations. In all the three configurations, the short wavelength unstable modes are localized near the walls, and are insensitive to symmetry conditions at the centerline. It is argued that this instability should be a generic feature in any wall-bounded shear flow of strongly shear-thinning viscoelastic fluids. Our predictions for the finite-wavelength instability in pressure-driven channel and pipe flows are in good agreement with experimental observations for the flow of concentrated polymer solutions in these geometries.