Stability of plane Couette flow of Carreau fluids past a deformable solid at arbitrary Reynolds numbers

Abstract

The linear stability of the plane Couette flow of both power-law and Carreau fluids past a deformable, neo-Hookean solid is analyzed at arbitrary Reynolds numbers. An algebraic error in the mathematical formulation of the earlier studies (for the power-law fluid) is corrected and is shown to result in quantitative differences in the predictions for the stability of the flow. Due to the lack of a proper (zero-shear) viscosity scale and a time scale for the onset of shear thinning in the power-law model, we show that the stability analysis of the flow yields vastly different scalings for the unstable mode depending on the way the problem is scaled to render it dimensionless. When the deformable solid properties are used to non-dimensionalize, we show that for the unstable modes (the so-called “wall modes” at high Re) Γc∝Re−1(2n+1), while when flow properties are used to non-dimensionalize, Γc∝Re−13 much akin to a Newtonian fluid, where Γ=Vm*η*/G*R* is the dimensionless shear rate in the flow, and Γc denotes the minimum value required for instability. Here, Vm* is the velocity of the top plate, G* is the shear modulus of the solid, R* is the fluid thickness, and η* is the (arbitrary) viscosity scale in the power-law model. Within the framework of the power-law model, it is not possible to discriminate between the two predicted scalings. To resolve this in an unambiguous manner, we used the Carreau model to account for shear thinning and to study its role on the stability of flow past deformable solid surfaces. The Carreau model has a well-defined zero-shear viscosity η0* as well as a time scale λ* that characterizes the onset of shear thinning. For fixed λ*η0*/(ρ*R*2), we show that the unstable wall modes scale as Γc∼Re(1−2n)3 at high Re, thus providing a resolution to the ambiguity in the results obtained using the power-law model. The present work thus shows that, at moderate to high Re, shear thinning has a strongly stabilizing effect on the wall mode instability in flow past deformable solid surfaces.

Purely elastic instabilities in the airways and oral area

“…The viscous mode of instability predicted by Kumaran et al (1994), Kumaran & Muralikrishnan (2000) and Chokshi & Kumaran (2007) for the plane Couette flow past a deformable solid in the creeping-flow limit can be destabilised by increasing inertia. For Re > 10, the viscous mode of instability exhibits a characteristic scaling Γ c ∼ Re −1 (Chokshi & Kumaran 2008a;Tanmay et al 2018). However, from figure 10, the inertia has a stabilising effect on the solid elastic mode, which further sets apart the viscous mode of instability from the solid elastic mode predicted here.…”

“…Their analysis predicted a non-monotonic effect of the variation in the power-law index on the viscous and short-wave modes of instability. Their work was extended to arbitrary Reynolds number by and Tanmay, Patne & Shankar (2018).…”

Soft hydraulics: from Newtonian to complex fluid flows through compliant conduits

“…As recently as 2019, inconsistencies have been found in previous calculations [196]. A theoretical re-analysis of a planar Couette flow over a compliant surface of a non-Newtonian fluid under the Carreau model showed that shear-thinning has a strongly stabilizing effect [197], with Re c ∼ Γ 3/(2n−1) , where Γ = Gh 0 /(V p η 0 ) and V p is the velocity of the rigid plate driving the Couette flow. Meanwhile, another experimental study [157] motivated the scaling Re c ∼ El −3/2 , where El = λ r η 0 /(a 2 ρ) is an elasticity number that characterizes the viscoelasticity of the polyacrylamide into water solution used.…”

Soft hydraulics: from Newtonian to complex fluid flows through compliant conduits

“…As recently as 2019, inconsistencies have been found in previous calculations [193]. A theoretical re-analysis of a planar Couette flow over a compliant surface of a non-Newtonian fluid under the Carreau model showed that shear-thinning has a strongly stabilizing effect [194], with Re c ∼ Γ 3/(2n−1) , where Γ = Gh 0 /(V p η 0 ) and V p is the velocity of the rigid plate driving the Couette flow. Meanwhile, another experimental study [154] motivated the scaling Re c ∼ El −3/2 , where El = λ r η 0 /(a 2 ρ) is an elasticity number that characterizes the viscoelasticity of the polyacrylamide into water solution used.…”