Viscoelastic film flows over an inclined substrate with sinusoidal topography. II. Linear stability analysis

The linear hydrodynamic stability of a film of viscoelastic fluid flowing down an inclined wavy surface is studied. We investigate the stability of the flow with respect to infinitesimal two- and three-dimensional (2D and 3D) disturbances and employ the Floquet-Bloch theory to examine the effect of periodic disturbances of any wavelength. The study is based on the numerical solution of the momentum equations along with the Phan-Thien–Tanner (PTT) model to account for material viscoelasticity. The generalized eigenvalue problem is solved using Arnoldi's algorithm, in a Newton-like implementation in order to calculate faster the critical conditions for the onset of the instability. Our results are in excellent agreement with the previous experimental and theoretical results in the case of Newtonian liquids flowing over flat and undulating substrates and viscoelastic liquids over flat substrates. We present detailed stability maps for finite amplitude of the wall corrugations and a wide range of material parameters. Our calculations indicate that fluid elasticity is primarily stabilizing, while shear thinning of the fluid tends to destabilize the fluid flow. In order to investigate the mechanisms involved, we perform an energy analysis of the flow under long-wave disturbances indicating that the convection of stress-gradient disturbances provides an additional viscoelastic mechanism for the destabilization of the flow, in contrast to the base state stress gradients which contribute to stabilization of the flow. Besides the usual long-wave instability, conditions are identified which lead to unstable disturbances of wavelength equal or smaller than the wavelength of the substrate. Experimental observations for Newtonian liquids have indicated that these short-wave instabilities will dominate and a similar behavior is predicted for viscoelastic liquids. Sometimes, before the short-wave instabilities, a hysteresis loop in the steady flow can be identified, which leads to a sharp change in the critical frequency. Finally, we examine the stability of the flow when subjected to disturbances in the spanwise direction and show that for highly elastic liquids 3D instabilities may arise.

Figure 1

Eigenspectrum for $\mathrm{Re}=8$; the flowing liquid is Elbesil 65 (see Table 1), for $Q\in \left[0,1\right)$. The light blue dots indicate the eigenvalues which have been calculated for $Q=0$; the light red symbols indicate the eigenvalues for $Q$ in the range $\left(0,0.5\right]$ and gray symbols indicate the eigenvalues for $Q$ in the range $\left(0.5,1\right]$.

Figure 2

Stability map for the Elbesil 65 (see Table 1), in (a) the Floquet parameter space $\left(\mathrm{Re},Q\right)$ and in (b) the frequency space $\left(\mathrm{Re},f\right)$. The light blue area indicates the unstable regimes of the fluid flow.

Figure 3

Comparison of the predicted neutral curves for Newtonian liquids with previous studies. The liquids used are (a) Elbesil 145 and (b) Elbesil 100; see Table 1. Experimental data are shown with dots while the theoretical predictions are presented with lines; the continuous black line and orange dashed line refer to the current work and Schörner et al. [18], respectively.

Figure 4

Comparison of the predictions of this study with the experimental data of Cao et al. [17] as a function of inclination angle. Black dots depict the experiments; the continuous blue line indicates the critical Reynolds number with the minimum value for finite values of $Q$, and the dashed red line indicates the most unstable mode for $Q=0$. The Kapitza number is 110, whereas the geometric parameters are $L=4.9$, $A/L=0.167$.

Figure 5

(a) Critical Reynolds number for the Nusselt flow of a viscoelastic liquid subjected to 2D disturbances. The gray dashed line is the analytical expression derived by Lai [25] using the Oldroyd-B model with $\beta =0.1$. The rest of the lines correspond to our neutral curves for long-wave disturbances ($Q={10}^{-2}$) and different $\varepsilon$ values. (b) Critical Reynolds number for the Nusselt flow of a viscoelastic liquid subjected to 3D disturbances as a function of $k$. Comparison with the analytical expression by Kang and Chen [24] for $\mathrm{Ka}=10$, $L=100$.

Figure 6

The growth rate of the most unstable eigenvalue for an Oldroyd-B (continuous blue line) and a Newtonian (dashed black line) liquid over a sinusoidal topography. The remaining parameters are $\mathrm{Wi}=1$, $\varepsilon =0$.

Figure 7

Steady free surface (continuous blue line) and its corresponding long-wave disturbances for $Q=0.1$ (continuous green line), calculated at (a) $\mathrm{Re}=9.1$, (b) $\mathrm{Re}=16.25$. The red dash-dotted line in (a) and (b) indicates the superposition of the steady free surface and the disturbance, where the magnitude of the latter has been magnified 500 times. The black dotted lines indicate the long-wave disturbances. The remaining parameters are $\mathrm{Wi}=1$, $\varepsilon =0$.

Figure 8

Energy analysis diagrams for the leading mode for (a) Newtonian and (b) Oldroyd-B liquid, for $\varepsilon =0$ and $\mathrm{Wi}=1,$ under long-wave disturbances $Q={10}^{-2}$. (c) Steady free surface (continuous blue line) and its corresponding ${\tau }_{p,xx}$ disturbance for $Q={10}^{-2}$ (red dashed dotted line) calculated at $\mathrm{Re}=9.1$. The remaining parameters are $\mathrm{Wi}=1$, $\varepsilon =0$.

Figure 9

Effect of the $\mathrm{Wi}$ number in the stability diagrams using the Oldroyd-B model. (a) $\mathrm{Wi}=0.5$, (b) $\mathrm{Wi}=0.75$, (c) $\mathrm{Wi}=1.0$, (d) $\mathrm{Wi}=1.5$.

Figure 10

Effect of the $\mathrm{Wi}$ number in the stability diagrams using the ePTT model for $\varepsilon =0.05$. (a) $\mathrm{Wi}=0.5$, (b) $\mathrm{Wi}=1$.

Figure 11

(a) Stability diagram and (b) relative amplitude of the steady free surface, $A_{\mathrm{rel}}$, close to the hysteresis loop, for $\mathrm{Wi}=2$ and $\varepsilon =0.05$.

Figure 12

(a) Stability diagram for $\mathrm{Wi}=1$, $\varepsilon =0.25$ and (b) close-up of the relative amplitude of the free surface at steady solution, $A_{\mathrm{rel}}$, corresponding to the jump in frequency.

Figure 13

Eigenspectrum calculated at (a) $\mathrm{Re}=14.37$ and (b) $\mathrm{Re}=14.28$ for $\mathrm{Wi}=1$ and $\varepsilon =0.25$. The white circles indicate the eigenvalues for $Q=0$, while the blue lines indicate the continuous spectrum for $Q\ne 0$.

Figure 14

Steady free surface and its disturbances for $Q=0$ (black-orange dashed line) and $Q=0.2$ (dashed green line), for $\mathrm{Re}=14.28$, $\mathrm{Wi}=1$, $\varepsilon =0.25$. The red dash-dotted line indicates the superposition of the steady free surface (continuous blue line) and the corresponding disturbances, where the magnitude of the latter has been magnified 200 times. The black dotted line indicates the long-wave disturbances.

Figure 15

Spatial variation of the steady (a) shear stress, ${\tau }_{p,yx}$, and (b) normal stress component, ${\tau }_{p,xx}$, of the polymeric stress tensor for $\mathrm{Re}=30.22$, $\mathrm{Wi}=1$, $\varepsilon =0.25$.

Figure 16

Effect of $\mathrm{Ka}$ in the stability diagrams using the ePTT model for $\mathrm{Wi}=1$, $\varepsilon =0.05$. (a) $\mathrm{Ka}=0.5$, (b) $\mathrm{Ka}=1.28$, (c) $\mathrm{Ka}=3.75$, (d) $\mathrm{Ka}=10$.

Figure 17

Steady shear stress component of the polymeric stress tensor,${\tau }_{p,yx}$, for $\mathrm{Re}=3$, $\mathrm{Wi}=1$, $\varepsilon =0.05$, $\mathrm{K}\alpha =10$.

Figure 18

Effect of $\beta$ in the stability diagrams using the ePTT model for $\mathrm{Wi}=1$, $\varepsilon =0.05$, $\beta =0.4$.

Figure 19

Effect of $\mathrm{El}$ in the stability diagrams using the ePTT model for $\varepsilon =0.20$. (a) $\mathrm{El}=1$, (b) $\mathrm{El}=5$, and (c) $\mathrm{El}=10$.

Figure 20

(a) Dispersion curves for disturbances in the spanwise direction for $Q=0$ for $\mathrm{Wi}=1.0,1.5,2.0$ shown with dashed lines, while $\mathrm{Wi}=2.5$ is presented with the continuous line. (b) Disturbances of the spanwise velocity component, $u_{z,d}\left(x,y\right)$, for $k=0.75$, $\mathrm{Wi}=2.5$,$\varepsilon =0.15$, and $\mathrm{Re}=10$.

Figure 21

Stability map in the $\left(\mathrm{Wi},\varepsilon \right)$ plane for the onset of 3D instabilities for $\mathrm{Re}=10$, $\mathrm{Ka}=3.75$, and $\beta =0.10$.