Waves and instabilities of viscoelastic fluid film flowing down an inclined wavy bottom

Evolution of waves and hydrodynamic instabilities of a thin viscoelastic fluid film flowing down an inclined wavy bottom of moderate steepness have been analyzed analytically and numerically. The classical long-wave expansion method has been used to formulate a nonlinear evolution equation for the development of the free surface. A normal-mode approach has been adopted to discuss the linear stability analysis from the viewpoint of the spatial and temporal study. The method of multiple scales is used to derive a Ginzburg-Landau-type nonlinear equation for studying the weakly nonlinear stability solutions. Two significant wave families, viz., ${\gamma }_1$ and ${\gamma }_2$, are found and discussed in detail along with the traveling wave solution of the evolution system. A time-dependent numerical study is performed with Scikit-FDif. The entire investigation is conducted primarily for a general periodic bottom, and the detailed results of a particular case study of sinusoidal topography are then discussed. The case study reveals that the bottom steepness $\zeta$ plays a dual role in the linear regime. Increasing $\zeta$ has a stabilizing effect in the uphill region, and the opposite occurs in the downhill region. While the viscoelastic parameter $\mathrm{\Gamma }$ has a destabilizing effect throughout the domain in both the linear and the nonlinear regime. Both supercritical and subcritical solutions are possible through a weakly nonlinear analysis. It is interesting to note that the unconditional zone decreases and the explosive zone increases in the downhill region rather than the uphill region for a fixed $\mathrm{\Gamma }$ and $\zeta$. The same phenomena occur in a particular region if we increase $\mathrm{\Gamma }$ and keep $\zeta$ fixed. The traveling wave solution reveals the fact that to get the ${\gamma }_1$ family of waves we need to increase the Reynolds number a bit more than the value at which the ${\gamma }_2$ family of waves is found. The spatiotemporal evolution of the nonlinear surface equation indicates that different kinds of finite-amplitude permanent waves exist.

Figure 1

Diagrammatic picture.

Figure 2

Spatial growth rate curve for different viscoelastic parameters at a point in the uphill (U; $x=4.71$) and at a point in the downhill (D; $x=1.57$) region for $\zeta =0.1\pi$, $\mathrm{Re}=1.5$.

Figure 3

Phase speed along the growth rate curve at a point in the uphill (U; $x=4.71$) and at a point in the downhill (D; $x=1.57$) region for $\mathrm{\Gamma }=0.4$, $\zeta =0.1\pi$, $\mathrm{Re}=1.5$.

Figure 4

Marginal stability curve (a) for different viscoelastic parameters keeping $\zeta$ the same and (b) for different bottom steepnesses keeping $\mathrm{\Gamma }$ the same, at a point in the uphill (U; $x=4.71$) and at a point in the downhill (D; $x=1.57$) region.

Figure 5

Temporal growth rate curve (a) for different viscoelastic parameters keeping $\zeta$ the same and (b) for different bottom steepnesses keeping $\mathrm{\Gamma }$ the same, at a point in the uphill (U; $x=4.71$) and at a point in the downhill (D; $x=1.57$) region for $\mathrm{Re}=2$. (a) Supercritical, $\mathrm{Re}=1:3$, $\mathrm{Bo}=0:04$ and (b) Subcritical, $\mathrm{Re}=5$, $\mathrm{Bo}=0:04$.

Figure 6

Critical Reynolds number (a) as a function of the bottom steepness for different $\mathrm{\Gamma }$ values and (b) as a function of the viscoelastic parameter for different $\zeta$ values, at a point in the uphill (U; $x=4.71$) and at a point in the downhill (D; $x=1.57$) region.

Figure 7

Stability curve at a point in the uphill (U; $x=4.71$) and at a point in the downhill (D; $x=1.57$) region for different $\mathrm{\Gamma }$ values. Regions indicated in the plot: I, unconditional; II, supercritical; III, subcritical; and IV, explosive. (a) $\mathrm{\Gamma }=0.06$, U; (b) $\mathrm{\Gamma }=0.2$, U; (c) $\mathrm{\Gamma }=0.4$, U; (d) $\mathrm{\Gamma }=0.06$, D; (e) $\mathrm{\Gamma }=0.2$, D; (f) $\mathrm{\Gamma }=0.4$, D.

Figure 8

Threshold amplitude of disturbances in the supercritical and subcritical regions at a point in the uphill (U; $x=4.71$) and at a point in the downhill (D; $x=1.57$) region for different $\mathrm{\Gamma }$ values. (a) Supercritical, $\mathrm{Re}=1.3,\mathrm{Bo}=0.04$ and (b) Subcritical, $\mathrm{Re}=5,\mathrm{Bo}=0.04$.

Figure 9

Nonlinear wave speed in the supercritical and subcritical regions at a point in the uphill (U; $x=4.71$) and at a point in the downhill (D; $x=1.57$) region for different $\mathrm{\Gamma }$ values. (a) Supercritical Uphill, $\mathrm{Re}=1.3$, $\mathrm{Bo}=0.04$, (b) Supercritical Downhill, $\mathrm{Re}=1.3$, $\mathrm{Bo}=0.04$, (c) Subcritical Uphill, $\mathrm{Re}=5$, $\mathrm{Bo}=0.04$, and (d) Subcritical Downhill, $\mathrm{Re}=5$, $\mathrm{Bo}=0.04$.

Figure 10

Bifurcation diagram for different $\mathrm{\Gamma }$ and $\mathrm{Re}$ for $\zeta =0.1\pi$. PD, period doubling; HP, Hopf bifurcation. Left column: $\mathrm{\Gamma }=0.4$, $\mathrm{Re}=12$. Right column: $\mathrm{\Gamma }=0.06$, $\mathrm{Re}=15$.

Figure 11

Bifurcation diagram for different $\mathrm{\Gamma }$ and $\mathrm{Re}$ for $\zeta =0.02\pi$. PD, period doubling; HP, Hopf bifurcation. Left column: $\mathrm{\Gamma }=0.4$, $\mathrm{Re}=15$. Right column: $\mathrm{\Gamma }=0.06$, $\mathrm{Re}=25$.

Figure 12

Periodic stationary solutions of the ${\gamma }_1$ and ${\gamma }_2$ families for different $\mathrm{\Gamma }$, $\mathrm{Re}$, $k$ and $\zeta$. (a) $\mathrm{\Gamma }=0.4$, $k=0.3$, $\mathrm{Re}=15$, ${\gamma }_1$ family; (b) $\mathrm{\Gamma }=0.4$, $k=0.3$, $\mathrm{Re}=15$, ${\gamma }_2$ family; (c) $\mathrm{\Gamma }=0.06$, $k=0.25$, $\mathrm{Re}=30$, ${\gamma }_1$ family; (d) $\mathrm{\Gamma }=0.06$, $k=0.25$, $\mathrm{Re}=30$, ${\gamma }_2$ family.

Figure 13

Phase plane portraits for the traveling wave solution in Fig. 12. (a) $\mathrm{\Gamma }=0.4$, $k=0.3$, $\mathrm{Re}=15$, ${\gamma }_1$ family; (b) $\mathrm{\Gamma }=0.4$, $k=0.3$, $\mathrm{Re}=15$, ${\gamma }_2$ family; (c) $\mathrm{\Gamma }=0.06$, $k=0.25$, $\mathrm{Re}=30$, ${\gamma }_1$ family; (d) $\mathrm{\Gamma }=0.06$, $k=0.25$, $\mathrm{Re}=30$, ${\gamma }_2$ family.

Figure 14

Periodic stationary solutions of the ${\gamma }_2$ family for different $\mathrm{\Gamma }$, $\mathrm{Re}$, and $k$ for $\zeta =0.1\pi$. (a) $\mathrm{\Gamma }=0.4$, $k=0.1$; (b) $\mathrm{\Gamma }=0.4$, $k=0.5$; (c) $\mathrm{\Gamma }=0.06$, $k=0.1$; (d) $\mathrm{\Gamma }=0.06$, $k=0.4$.

Figure 15

Periodic stationary solutions of the ${\gamma }_2$ family for different $\mathrm{\Gamma }$, $\mathrm{Re}$, $\zeta$, and $k$. Upper graph: $\mathrm{\Gamma }=0.4$, $\mathrm{Re}=12$, $k=0.5$. Lower graph: $\mathrm{\Gamma }=0.06$, $\mathrm{Re}=15$, $k=0.4$.

Figure 16

Periodic stationary solutions of the ${\gamma }_2$ family for different $\mathrm{\Gamma }$, $\mathrm{Re}$, and $\zeta$. Upper graph: $\zeta =0.1\pi$, $\mathrm{Re}=15$, $k=0.4$. Lower graph: $\zeta =0.02\pi$, $\mathrm{Re}=15$, $k=0.4$.

Figure 17

Free surface configuration for various $\mathrm{\Gamma }$, $\zeta$, and $\mathrm{Re}$ from time-dependent simulations.