Stability of elastoviscoplastic plane Couette flow

TL;DR

This work analyzes the linear stability of elastoviscoplastic plane Couette flow using the Saramito-2007 constitutive model. A pseudo-spectral Chebyshev approach solves the resulting eigenvalue problem, revealing that in the $B=0$ (UCM) limit the flow supports two stable Gorodtsov–Leonov modes, while increasing the Bingham number $B$ destabilizes these modes and generates centre modes traveling at the mean flow speed. The destabilization arises from an extra tangential stress term tied to yield stress, specifically $(t_3-t_1)\tilde{\tau}_{xx}$, which only appears when $B>0$. Additionally, the study identifies weak Hadamard instability at high wavenumbers, which is eliminated by including a stress-diffusion term with coefficient $\epsilon$, demonstrating a stabilizing mechanism within the EVP framework. Overall, the PCF of EVP fluids is shown to be linearly unstable, with implications for understanding transition pathways to turbulence in EVP flows.

Abstract

Several studies have investigated the turbulent flow of elastoviscoplastic (EVP) fluids, which exhibit yield stress in addition to viscoelasticity. The instabilities that could be responsible for the transition to turbulence in the EVP fluid flows remain unknown. Thus, the present explores the linear stability of EVP plane Couette flow (PCF) by employing the Saramito model. The eigenvalue problem is solved by using the pseudo-spectral method. In the limit of vanishing yield stress, EVP fluid behaves as Upper Convected Maxwell (UCM) fluid. The creeping flow of UCM fluid exhibits two stable Gorodotsov \& Leonov (GL) modes, thus a stable flow. As the Bingham number (i.e., yield stress) increases, the GL modes become unstable, implying an unstable flow. Additionally, there are new unstable modes with phase speed equalling the average velocity of the fluid. The analysis reveals an extra tangential stress term, arising due to yield stress, is responsible for the predicted instabilities. Also, the Saramito model exhibits weak Hadamard instability, i.e., unstable perturbations of arbitrarily small wavelengths. The present study demonstrates the removal of the Hadamard instability by adding a stress diffusion term in the Saramito constitutive equation. To conclude, the PCF of an EVP fluid is linearly unstable.

Figures (6)

• Figure 1: The variation of the discrete GL modes and continuous spectrum due to variation in the number of collocation points at $W=5, B=0$ and $k=0.7$. The GL modes exhibit negligible variation despite an increase in the number of collocation points.
• Figure 2: Eigenspectra for $W=1$ and $k=1$. Panel (a) shows the converged unstable GL modes for $B=12$. Panel (b) demonstrates GL mode destabilisation due to the increasing yield stress effect, i.e., Bingham number $B$. Note that $c_i>0$ implies an unstable mode.
• Figure 3: The manifestation of centre modes due to an increase in the Bingham number for $W=1$ and $k=0.5$. The GL modes merge and give rise to two modes with $c_r=0.5$, henceforth referred to as centre modes.
• Figure 4: Eigenspectra for $W=1$ and $k=0.5$ for center modes. Panel (a) shows the converged unstable centre mode for $B=12$. Panel (b) demonstrates centre mode destabilisation due to the increasing Bingham number $B$. Note that $c_i>0$ implies an unstable mode.
• Figure 5: The dispersion curves for $W=1$. Panel (a) shows the destabilisation of a range of wavenumber as $B$ increases. Panel (b) shows the switching of the most unstable mode from centre mode to GL mode characterised by a change in $c_r=0.5$ to $c_r>0.5$. There is one more GL mode (not shown here) with the same $c_i$ but $c_r<0.5$.