Linearized instability of Couette flow in stress-power law fluids

TL;DR

The paper develops a thermodynamically consistent, non-convex stress–power law constitutive model for stress–strain rate behavior and analyzes the linear stability of planar Couette flow. By nondimensionalizing the governing equations and deriving an Orr–Sommerfeld–type eigenvalue problem, it evaluates stability under velocity and mixed boundary conditions using a pseudospectral method. Key results show that, with velocity boundary conditions, multiple steady states can exist: the two on ascending branches are unconditionally stable, while the descending-branch state is unconditionally unstable; with mixed boundary conditions, the base state is unique and stability depends on whether the applied boundary stress lies on an ascending (stable) or descending (unstable) branch. The findings highlight that boundary conditions and constitutive non-monotonicity fundamentally govern flow stability in complex fluids, and point to future work on curved geometries like Taylor–Couette to explore curvature–nonlinearity interactions.

Abstract

This paper examines the linearized stability of plane Couette flow for stress-power law fluids, which exhibit non-monotonic stress-strain rate behavior. The constitutive model is derived from a thermodynamic framework using a non-convex rate of dissipation potential. Under velocity boundary conditions, the system may admit three steady-state solutions. Linearized stability analysis reveals that the two solutions on ascending constitutive branches are unconditionally stable, while the solution on the descending branch is unconditionally unstable. For mixed traction-velocity boundary conditions, the base state is unique. Stability depends solely on whether the prescribed traction lies on an ascending (stable) or descending (unstable) branch of the constitutive curve. The results demonstrate that flow stability in these complex fluids is fundamentally governed by both boundary conditions and constitutive non-monotonicity.

Figures (10)

• Figure 1: Constitutive response of the generalized stress power--law model. Left:$\|\mathbf{D}\|$ as a function of $\|\mathbf{S}\|$ for representative values of $n$ with $a=0$, $\alpha=1$, $b=1$, $\beta=1$. Right: Variation with $a$ for $n=-2$, $\alpha=1$, $b=1$, $\beta=0.1$, illustrating the emergence of non-monotonicity and the regime in which multiple solutions may arise.
• Figure 2: Plot of the function $F(c_1)$ defined in \ref{['4.12b']} with $a=0.032$, $b=1$, $\Gamma=10^{-3}$, $n=-1.2$ and for various $Re$. The range in which 3 solutions occur is $(Re_m,Re_M)$, with $Re_m=23.2$ and $Re_M=42.2$.
• Figure 3: Regions of existence of the three solutions.
• Figure 4: $m_i(Re_l,Re_u)$ corresponding to the three base states $v_x^{(b,i)}(y)$, $i=1,2,3$. States (i) and (iii), lying on the ascending branches of the constitutive curve, are unconditionally stable, whereas state (ii), associated with the descending branch, is unconditionally unstable.
• Figure 5: Plot of the functions $m_1$, $m_3$ versus (i) $Re$ with $k=1$ and (ii) $k$ with $Re=32$.