Stability of a thin viscoelastic film falling down an inclined plane

The stability of a thin viscoelastic film of Oldroyd-B fluid falling down an incline is investigated. In the weak viscoelasticity limit, a reduced model is derived using the weighted residual techniques, which consists of two coupled equations for the film thickness and local flow rate. Through the normal mode analysis, temporal growth rates and neutral stability curves are calculated to explore linear stability of the film. Results show that the viscoelasticity acts to destabilize the film and decrease the phase speed of linear waves. Good agreement is found between the reduced model and full linearized equations solved by the Chebyshev spectral collocation method when viscoelastic effect is relatively weak. Nonlinear traveling waves are further determined. The speed of fast/slow-wave families is promoted/reduced in the presence of the viscoelasticity, resulting in a dispersion effect on the system; while the wave amplitudes are augmented for both fast and slow waves. Besides, the temporal evolution of surface waves is numerically resolved, which validates the linear prediction of the instability threshold. Steady permanent waves are observed in the final stage; the surface deformation and perturbation energy are enhanced by the viscoelasticity as expected.

Figure 1

Schematic of a thin film flowing down an incline.

Figure 2

Influence of the Deborah number on (a) temporal growth rate ${\omega }_r$ and wave speed $c_l=-{\omega }_i/k$ at $\mathrm{Re}=2$; and (b) neutral stability curves on the $(\mathrm{Re},k)$ plane when $We=20, \theta =\pi /4$, and $r=0.5$. Thick and thin lines correspond to the full equations and WRIBL, respectively (same notation is adopted in Figs. 3 and 4).

Figure 3

Influence of the time constant ratio on (a) temporal growth rate ${\omega }_r$ and wave speed $c_l=-{\omega }_i/k$ at $\mathrm{Re}=2$; and (b) neutral stability curves on the $(\mathrm{Re},k)$ plane when $We=20, \theta =\pi /4$, and $De=0.1$.

Figure 4

Results of elastic instability when $\mathrm{Re}=0, We=20, \theta =18\pi /37$. (a) and (b) temporal growth rate ${\omega }_r$ and wave speed $c_l=-{\omega }_i/k$ at different $De$ and $r$; (c) neutral stability curves on the $(De,k)$ plane.

Figure 5

Effect of the reduced viscoelastic parameter $M_0$ on the dynamic wave speeds $c_{d\pm }$.

Figure 6

(a) The speed of the traveling waves and (b) the gap between the maximum and minimum amplitudes $\mathrm{\Delta }{h}_m=h_{max}-h_{min}$ at different viscoelastic parameters when $\theta =\pi /4, \mathrm{Re}=5$, and $We=20$.

Figure 7

Wave profiles of the film thickness $h$: (a) fast ${\gamma }_2$ family and (b) slow ${\gamma }_1$ family at different viscoelastic parameters when $k=0.03$. Other parameters are the same with those in Fig. 6.

Figure 8

(a) The speed and (b) the amplitude gap at different viscoelastic parameters when $\theta =\pi /4, We=20$, and $k=0.03$.

Figure 9

Wave evolution (a) $\mathrm{Re}=0.5$, (b) $\mathrm{Re}=2$, and (c) $\mathrm{Re}=5$. (d) Evolution of perturbation energy $E$. The other parameters are $\theta =\pi /4, We=20, M=0.1$, and $k=0.03$.

Figure 10

Comparison of the critical Reynolds number when $\theta =\pi /4, We=20, r=0.5$.

Figure 11

Wave evolution at different viscoelastic numbers: (a) wave profiles at $t=1000$ and (b) evolution of perturbation energy $E$ when $\theta =\pi /4, \mathrm{Re}=5, We=20$, and $k=0.03$.