Linear stability of layered two-phase flows through parallel soft-gel-coated walls

The linear stability of layered two-phase Poiseuille flows through soft-gel-coated parallel walls is studied in this work. The focus is on determining the effect of the elastohydrodynamic coupling between the fluids and the soft-gel layers on the different instabilities observed in flows between parallel plates. The fluids are assumed Newtonian and incompressible, while the soft gels are modeled as linear viscoelastic solids. A long-wave asymptotic analysis is used to obtain an analytical expression for the growth rate of the disturbances. A Chebyshev collocation method is used to numerically solve the general linearized equations. Three distinct instability modes are identified in the flow: (a) a liquid-liquid long-wave mode; (b) a liquid-liquid short-wave mode; (c) a gel-liquid short-wave mode. The effect of deformability of the soft gels on these three modes is analyzed. From the long-wave analysis of the liquid-liquid mode a stability map is obtained, in which four different regions are clearly demarcated. It is shown that introducing a gel layer near the more viscous fluid has a predominantly stabilizing effect on this mode seen in flows between rigid plates. For parameters where this mode is stable for flow between rigid plates, introducing a gel layer near the less viscous and thinner fluid has a predominantly destabilizing effect. The liquid-liquid short-wave mode is destabilized by the introduction of soft-gel layers. Additional instability modes at the gel-liquid interfaces induced by the deformability of the soft-gel layers are identified. We show that these can be controlled by varying the thickness of the gel layers. Insights into the physical mechanism driving different instabilities are obtained using an energy budget analysis.

Figure 1

Schematic of the flow configuration under study. At the base state, the liquid-liquid interface is at $y=0$, top gel-liquid interface is at $y=1$, and the bottom gel-liquid interface is at $y=-n_{21}$. The perturbed liquid-liquid interface is given by $y=f\left(x,t\right)$, the perturbed top gel-liquid interface is at $y=1+g\left(x,t\right)$, and the perturbed bottom gel-liquid interface is given by $y=-n_{21}+h\left(x,t\right)$. The parabolic velocity profile in the base state for the two fluids is also shown.

Figure 2

Stability features in different regions in ${\mu }_{21}\text{−}{n}_{21}$ parameter space for two-phase layered flow between rigid flat plates.

Figure 3

Comparison of dispersion curves obtained from long-wave asymptotic analysis and numerical simulations for ${\mathrm{Re}}_1=6,8$. The arrow points in the direction of increasing Reynolds number. Other parameters: $n_{21}=1,n_{31}=1.5,n_{41}=1.5$, ${\mu }_{21}=0.1$, ${\mu }_{31}=1$, ${\mu }_{41}=1,{\rho }_{21}=1.16$, $S_{31}=S_{41}=0.0001$, $S_{21}=\frac{829}{{\mathrm{Re}}_1^2}$, $\mathrm{W}{\mathrm{i}}_3=\mathrm{W}{\mathrm{i}}_4=0.5$.

Figure 4

(a) Division of parameter space into different regions where each region has a specific sign of coefficients $b_3$ and $b_4$ in Eq. (55). (b) Dominant influence on stability in different regions. Parameters: $n_{31}=2$, $n_{41}=2$, ${\mu }_{31}=1$, ${\mu }_{41}=1$, ${\rho }_{21}=1.16$.

Figure 5

Dispersion curves for the LL long-wave mode for $\mathrm{W}{\mathrm{i}}_3=\mathrm{W}{\mathrm{i}}_4$ varying from 0.05 to 1 and ${\mu }_{21}=0.1$. The arrow points in the direction of decreasing Weissenberg number. For $\mathrm{W}{\mathrm{i}}_3=\mathrm{W}{\mathrm{i}}_4=1$ complete stabilization of the LL long-wave mode is observed. Other parameters: $n_{21}=1,n_{31}=1.5,n_{41}=1.5$, ${\mu }_{31}=1$, ${\mu }_{41}=1,{\rho }_{21}=1.16$, ${\mathrm{Re}}_1=8$, $S_{31}=S_{41}=0.0001$, $S_{21}=\frac{829}{{\mathrm{Re}}_1^2}$.

Figure 6

The perturbed axial velocity for the LL long-wave mode of instability for $n_{21}=1,n_{31}=1.5,n_{41}=1.5$, ${\mu }_{31}=1$, ${\mu }_{41}=1,{\rho }_{21}=1.16$, ${\mathrm{Re}}_1=8$, $S_{31}=S_{41}=0.0001$, $S_{21}=\frac{829}{{\mathrm{Re}}_1^2}$, ${\mu }_{21}=0.1$, $\mathrm{W}{\mathrm{i}}_3=\mathrm{W}{\mathrm{i}}_4=0.05$. The profiles are obtained for $x=2\pi /k$.

Figure 7

Contours of perturbed stream function for the LL long-wave instability for $0\le x\le 2l_{\mathrm{wave}}$ and $-1\le y\le 1$ where $l_{\mathrm{wave}}=2\pi /k$. Other parameters: $n_{21}=1,n_{31}=1.5,n_{41}=1.5$, ${\mu }_{31}=1$, ${\mu }_{41}=1,{\rho }_{21}=1.16$, ${\mathrm{Re}}_1=8$, $S_{31}=S_{41}=0.0001$, $S_{21}=\frac{829}{{\mathrm{Re}}_1^2}$, ${\mu }_{21}=0.1$, $\mathrm{W}{\mathrm{i}}_3=\mathrm{W}{\mathrm{i}}_4=0.05$.

Figure 8

Destabilization of LL long-wave mode. Parameters: $n_{21}=1.76,n_{31}=2,n_{41}=2$, ${\mu }_{21}=2$, ${\mu }_{31}=1$, ${\mu }_{41}=1,{\rho }_{21}=1.16$, ${\mathrm{Re}}_1=8$, $S_{31}=S_{41}=0.0001$, $S_{21}=\frac{829}{{\mathrm{Re}}_1^2}$. Arrow points in the direction of increasing $\mathrm{W}{\mathrm{i}}_3$.

Figure 9

Dispersion curves for the LL short-wave mode for $\mathrm{W}{\mathrm{i}}_3=\mathrm{W}{\mathrm{i}}_4$ varying from 0.05 to 0.25 and (a) ${\mu }_{21}=0.05$, ${\mathrm{Re}}_1=230$; (b)${\mu }_{21}=0.1$, ${\mathrm{Re}}_1=700$. The arrow points in the direction of increasing $\mathrm{W}{\mathrm{i}}_3$. Other parameters: $n_{21}=1,n_{31}=1.5,n_{41}=1.5$, ${\mu }_{31}=10,{\mu }_{41}=10$, ${\rho }_{21}=1.16$, $S_{21}=\frac{829}{{\mathrm{Re}}_1^2}$, $S_{31}=S_{41}=0.0001$.

Figure 10

The perturbed axial velocity for the LL short-wave mode instability for $n_{21}=1$, $n_{31}=1.5$, $n_{41}=1.5$, ${\mu }_{31}=10$, ${\mu }_{41}=10$, ${\rho }_{21}=1.16$, ${\mathrm{Re}}_1=700$, $S_{31}=S_{41}=0.0001$, $S_{21}=\frac{829}{{\mathrm{Re}}_1^2}$, ${\mu }_{21}=0.1$, $\mathrm{W}{\mathrm{i}}_3=\mathrm{W}{\mathrm{i}}_4=0.2$. The profiles are obtained for $x=2\pi /k$.

Figure 11

Contours of stream function perturbation for the LL short-wave instability for $0\le x\le 2l_{\mathrm{wave}}$ and $-1\le y\le 1$ where $l_{\mathrm{wave}}=2\pi /k$. Other parameters are $n_{21}=1,n_{31}=1.5,n_{41}=1.5$, ${\mu }_{31}=10$, ${\mu }_{41}=10,{\rho }_{21}=1.16$, ${\mathrm{Re}}_1=700$, $S_{31}=S_{41}=0.0001$, $S_{21}=\frac{829}{{\mathrm{Re}}_1^2}$, ${\mu }_{21}=0.1$, $\mathrm{W}{\mathrm{i}}_3=\mathrm{W}{\mathrm{i}}_4=0.2$.

Figure 12

Effect of ${\mu }_{21}$ on the neutral stability curve for the LL short-wave mode (arrow points in the direction of increasing ${\mu }_{21}$). Here $\mathrm{W}{\mathrm{i}}_3=\mathrm{W}{\mathrm{i}}_4$. Other parameters: $n_{21}=1,n_{31}=1.5,n_{41}=1.5$, ${\mu }_{31}=10,{\mu }_{41}=10$, ${\rho }_{21}=1.16$, ${\mathrm{Re}}_1=230$, $S_{21}=\frac{829}{{\mathrm{Re}}_1^2}$, $S_{31}=S_{41}=0.0001$.

Figure 13

Effect of ${\mu }_{21}$ on the ${\mathrm{Wi}}_3\text{−}{\mathrm{Re}}_1$ critical surfaces for the LL short-wave mode. Here $\mathrm{W}{\mathrm{i}}_3=\mathrm{W}{\mathrm{i}}_4$. Other parameters are $n_{21}=1,n_{31}=1.5,n_{41}=1.5$, ${\mu }_{31}=10,{\mu }_{41}=10$, ${\rho }_{21}=1.16$, $S_{21}=\frac{829}{{\mathrm{Re}}_1^2}$, $S_{31}=S_{41}=0.0001$.

Figure 14

Dispersion curves for the GL short-wave mode. The arrow points in the direction of the increasing $\mathrm{W}{\mathrm{i}}_3$. (a) Other parameters: $n_{21}=1$, $n_{31}=4$, $n_{41}=4$, ${\mu }_{21}=1$, ${\mu }_{31}={\mu }_{41}=1$, ${\rho }_{21}=1$, $\mathrm{W}{\mathrm{i}}_4=0.001$, ${\mathrm{Re}}_1=0.01$, $S_{21}=0$, $S_{31}=S_{41}=0.01$; (b) $n_{21}=1$, $n_{31}=4$, $n_{41}=4$, ${\mu }_{21}=1$, ${\mu }_{31}={\mu }_{41}=1$, ${\rho }_{21}=1$, $\mathrm{W}{\mathrm{i}}_4=0.001$, ${\mathrm{Re}}_1=1$, $S_{21}=0$, $S_{31}=S_{41}=10$.

Figure 15

Contours of stream function perturbation for the GL short-wave instability show in Fig. 14 for $0\le x\le 2l_{\mathrm{wave}}$ and $-1\le y\le 1$ where $l_{\mathrm{wave}}=2\pi /k$.

Figure 16

Perturbed velocity in the top liquid and top gel. Other parameters: $n_{21}=1$, $n_{31}=4$, $n_{41}=4$, ${\mu }_{21}=1$, ${\mu }_{31}={\mu }_{41}=1$, ${\rho }_{21}=1$, $\mathrm{W}{\mathrm{i}}_4=0.001$, ${\mathrm{Re}}_1=0.01$, $S_{21}=0$, $S_{31}=S_{41}=0.01$.

Figure 17

Effect of ${\mu }_{31}$ on the critical surface in the $\mathrm{W}{\mathrm{i}}_3\text{−}{n}_{31}$ plane for the GL short-wave mode instability. Other parameters: $n_{21}=1$, ${\mu }_{21}=1$, ${\rho }_{21}=1$, ${\mathrm{Re}}_1=0$, $S_{21}=0.0001$, $S_{31}=S_{41}=0$, $\mathrm{W}{\mathrm{i}}_4=0.001$. Arrow points in the direction of increase in the viscosity ratio ${\mu }_{31}$.

Figure 18

Dispersion curve when both the top and bottom GL short-wave modes of instabilities coexist in the flow (the arrow points in the direction of increasing $\mathrm{W}{\mathrm{i}}_4$). Other parameters: $n_{21}=1,n_{31}=6,n_{41}=6$, ${\mu }_{21}=1,{\mu }_{31}={\mu }_{41}=1$, ${\rho }_{21}=1$, $S_{21}={10}^{-5}$, $S_{31}=S_{41}=10,{\mathrm{Re}}_1={10}^{-3}$, $\mathrm{W}{\mathrm{i}}_3=10$, and $\mathrm{W}{\mathrm{i}}_4=5$.

Figure 19

Contours of stream function perturbation for the GL instability for $0\le x\le 2l_{\mathrm{wave}}$ and $-1\le y\le 1$ where $l_{\mathrm{wave}}=2\pi /k$. Other parameters: $n_{21}=1,n_{31}=6,n_{41}=6$, ${\mu }_{21}=1,{\mu }_{31}={\mu }_{41}=1$, ${\rho }_{21}=1$, $S_{21}={10}^{-5}$, ${\rho }_{31}={\rho }_{41}=1$, $S_{31}=S_{41}=10{\mathrm{Re}}_1={10}^{-3}$, $\mathrm{W}{\mathrm{i}}_3=10$, and $\mathrm{W}{\mathrm{i}}_4=5$.