Wall-mode instability in plane shear flow of viscoelastic fluid over a deformable solid

The linear stability analysis of a plane Couette flow of an Oldroyd-B viscoelastic fluid past a flexible solid medium is carried out to investigate the role of polymer addition in the stability behavior. The system consists of a viscoelastic fluid layer of thickness $R$, density $\rho$, viscosity $\eta$, relaxation time $\lambda$, and retardation time $\beta \lambda$ flowing past a linear elastic solid medium of thickness $HR$, density $\rho$, and shear modulus $G$. The emphasis is on the high-Reynolds-number wall-mode instability, which has recently been shown in experiments to destabilize the laminar flow of Newtonian fluids in soft-walled tubes and channels at a significantly lower Reynolds number than that for flows in rigid conduits. For Newtonian fluids, the linear stability studies have shown that the wall modes become unstable when flow Reynolds number exceeds a certain critical value ${\mathrm{Re}}_c$ which scales as ${\Sigma }^{3/4}$, where Reynolds number $\mathrm{Re}=\rho VR/\eta ,V$ is the top-plate velocity, and dimensionless parameter $\Sigma =\rho G{R}^2/{\eta }^2$ characterizes the fluid-solid system. For high-Reynolds-number flow, the addition of polymer tends to decrease the critical Reynolds number in comparison to that for the Newtonian fluid, indicating a destabilizing role for fluid viscoelasticity. Numerical calculations show that the critical Reynolds number could be decreased by up to a factor of 10 by the addition of small amount of polymer. The critical Reynolds number follows the same scaling ${\mathrm{Re}}_c\sim {\Sigma }^{3/4}$ as the wall modes for a Newtonian fluid for very high Reynolds number. However, for moderate Reynolds number, there exists a narrow region in $\beta \text{−}H$ parametric space, corresponding to very dilute polymer solution $(0.9\lesssim \beta <1)$ and thin solids $(H\lesssim 1.1)$, in which the addition of polymer tends to increase the critical Reynolds number in comparison to the Newtonian fluid. Thus, Reynolds number and polymer properties can be tailored to either increase or decrease the critical Reynolds number for unstable modes, thus providing an additional degree of control over the laminar-turbulent transition.

Figure 1

Schematic of plane Couette flow over a flexible surface showing a dimensional coordinate system.

Figure 2

Effect of fluid elasticity on Newtonian wall mode: critical Reynolds number ${\mathrm{Re}}_c$ as a function of Weissenberg number $\mathrm{We}$ for $\Sigma =5000,\beta =0.5,k=1$ and different values of solid thickness $H$. S denotes a stable region and U denotes an unstable region.

Figure 3

Variation of critical Reynolds number ${\mathrm{Re}}_c$ with disturbance wave number for $\Sigma =5000,H=2.0,\mathrm{We}=1.0$, and different values of $\beta$. The point of minimum ${\Gamma }_t$ on this curve represents the critical point $(k_c,{\Gamma }_c)$. The curve for $\beta =1$ represents the Newtonian wall mode.

Figure 4

Neutral stability curve showing the critical Reynolds number as function of dimensionless parameter $\Sigma$ for $H=5$ and for varying values of $\beta$. Polymer addition (decreasing $\beta$ below unity) tends to reduce ${\mathrm{Re}}_c$ for a given value of $\Sigma$. Viscoelastic wall-mode instability follows the scaling law $\mathrm{Re}\sim {\Sigma }^{3/4}$.

Figure 5

Stability diagram in the ${\mathrm{Re}}_c\text{−}\Sigma$ plane for $H=5$ and varying values of $\beta$ and $\mathrm{We}$. The destabilizing role of fluid elasticity is evident. For different values of viscoelasticity parameters ${\mathrm{Re}}_c$ scales as ${\Sigma }^{3/4}$.

Figure 6

Effect of wall thickness $H$ on critical Reynolds number for $\beta =0.9$ and $\mathrm{We}=10$. Thicker the solid the lower the critical Reynolds number.

Figure 7

Neutral stability curve in the ${\mathrm{Re}}_c\text{−}\Sigma$ plane for $H=0.5$. The broken lines are for $\beta =0.95$, representing a very dilute solution. The dual role of polymer addition is visible. The upper branch which follows ${\mathrm{Re}}_c\sim \Sigma$ exists for all Reynolds number. At $\mathrm{Re}\approx 7760$, the lower branch appears and it is the most critical mode higher Reynolds number. For $\mathrm{Re}\lesssim 7760$, the role of the polymer is strongly stabilizing.

Figure 8

Schematic diagram showing qualitatively the region of stabilizing and destabilizing effects of polymer addition on the least stable Newtonian wall mode in $\beta \text{−}H$ space valid for $\mathrm{Re}\approx 1100$ and any Weissenberg number under 100. Here S denotes stabilizing influence and $\alpha$ is the exponent in the power-law scaling $\mathrm{Re}\sim {\Sigma }^{\alpha }$.

Figure 9

Neutral stability curve in the ${\overline{\Gamma }}_t\text{−}\overline{\mathrm{We}}$ plane for $k=1,\beta =0.5$ and different values of $H$. (a) For wall mode at $\mathrm{Re}=1000$. (b) For viscous mode at $\mathrm{Re}\to 0$. The upper branch for the wall mode has the same value as the viscous mode for $\overline{\mathrm{We}}>1$.

Figure 10

Continuation of wall mode in Fig. 9 to $\mathrm{Re}\ll 1$. (a) For $\overline{\mathrm{We}}=1$; (b) for $\overline{\mathrm{We}}=100$. Two lower branches merge and disappear while upper branch continues to viscous mode.

Figure 11

Stability diagram for the unstable inviscid mode for $H=5$. The inviscid instability mode is seen to be more unstable than the most unstable wall mode and follows the scaling law $\mathrm{Re}\sim {\Sigma }^{1/2}$. The inviscid instability is unaffected by fluid elasticity.

Figure 12

Variation of growth rate with shear rate $\Gamma$ for Newtonian fluid flow at high Reynolds number for $H=1$ and $k=1$. (a) Numerical results for $\mathrm{Re}={10}^4$ and $\mathrm{Re}={10}^6$. Both inviscid-mode and wall-mode instability are shown. (b) Asymptotic results are compared with the numerical results for growth rate for $\mathrm{Re}={10}^6$. While the numerical growth rate is positive for a range of $\Gamma$, the asymptotically calculated growth rate remains negative, indicating stable inviscid mode.