Marangoni instability in a heated viscoelastic liquid film: Long-wave versus short-wave perturbations

We investigate the Marangoni instability in a thin layer of viscoelastic fluid, confined between its deformable free surface and a substrate of low thermal conductivity. Following a theoretical analysis, we study the stability of the present system for the case when the fluid layer is subjected to heating from below. Here, we use the Maxwell model to depict the rheology of the viscoelastic fluid. Linear stability analysis of the quiescent base state reveals that, in addition to the conventional short-wave mode, a long-wave instability can also emerge in this system. We demonstrate the appearance of both the long-wave monotonic and oscillatory instabilities in such a system. We study this long-wave mode analytically using the scaling $k\sim \sqrt{\mathrm{Bi}}$ ($k$ is the wave number and Bi is the Biot number), whereas the short-wave mode is examined numerically. The influential role of elasticity of the fluid and the other involved parameters on the stability of the system is aptly discussed, and their ranges are identified within which a particular instability mode gets critical.

Figure 1

Sketch defining the physical system under study. A thin viscoelastic liquid film, resting on a solid substrate of very low thermal conductivity and having a deformable free surface, is subjected to heating from below. The dashed-dotted line corresponds to the undisturbed interface in the equilibrium state.

Figure 2

Neutral stability curves for a Newtonian liquid layer confined between a horizontal substrate of low thermal conductivity and a nondeformable free surface. The results obtained from the present study (shown by solid lines) are compared with the results of Pearson [7] (shown by marker “o”) for two different Bi (0 and 5, respectively). To mimic the characteristics of a nondeformable free surface, here we consider $\mathrm{Ga}=100$ and $\mathrm{\Sigma }={10}^6$.

Figure 3

Variation of the monotonic (Mon.: dashed-dotted line) and oscillatory (Osc.: solid and dotted lines) instability threshold for a viscoelastic liquid layer for $\mathrm{Ga}=0.1, \mathrm{\Sigma }=1000, \mathrm{Bi}=0.1, Pr=10$, and various values of Deborah number, De. The domain of stability is situated below the lines. Note the variation in the critical instability modes for $\mathrm{De}=0.1$ and 1. For the long-wave oscillatory mode $\mathrm{M}{\mathrm{a}}_c{|}_{\mathrm{De}=0.1}=39.15, \mathrm{M}{\mathrm{a}}_c{|}_{\mathrm{De}=1}=39.6$, while for the short-wave oscillatory mode $\mathrm{M}{\mathrm{a}}_c{|}_{\mathrm{De}=0.1}=42, \mathrm{M}{\mathrm{a}}_c{|}_{\mathrm{De}=1}=2$.

Figure 4

(a) Neutral stability curves for the long-wave monotonic (Mon.: solid lines) and oscillatory (Osc.: dashed lines) modes of instability at $\mathrm{Ga}=0.1, \mathrm{\Sigma }=1000$, and $\mathrm{De}=0.1$. The solid and dashed lines represent the result of the asymptotic analysis, while the dotted lines represent the numerical result. (b) Variation of the stability threshold for the long-wave oscillatory mode with De at $\mathrm{Bi}=0.1, \mathrm{Ga}=0.1$, and $\mathrm{\Sigma }=1000$ (inset shows the zoomed-in view of the neutral curves for $\mathrm{De}=0$ and 0.1, respectively). (c) Variation of the oscillation frequency of neutral perturbations for $\mathrm{De}=0$ (dashed and dotted lines) and $\mathrm{De}=1$ (dashed-dotted and solid lines) at $\mathrm{Ga}=0.1$ and $\mathrm{\Sigma }=1000$. In panels (a), (c) lines marked by 1 and 2 correspond to $\mathrm{Bi}=0.05$ and $\mathrm{Bi}=0.1$, respectively.

Figure 5

Variation of the stability threshold with Biot number for $\mathrm{Ga}=0.1, \mathrm{Pr}=10$, and $\mathrm{\Sigma }=1000$. Panels (a,b) display the neutral stability curves for monotonic (Mon.: dashed and dashed-dotted lines) and oscillatory (Osc.: solid and dotted lines) modes for $\mathrm{De}=0.1$ and 1, respectively. Panel (c) displays the variation of the frequency of neutral perturbations for $\mathrm{Bi}=0.05$ (solid lines) and $\mathrm{Bi}=0.1$ (dotted and dashed-dotted lines). Lines marked by 1 and 2 in panel (c) correspond to $\mathrm{De}=1$ and 0.1, respectively. Domains of stability are situated below the lines.

Figure 6

Variation of the critical Marangoni number (a), (c), and critical wave number (b) with Bi at $\mathrm{Ga}=1, Pr=10$, and $\mathrm{\Sigma }=1000$. Panels (a)–(c) represent the variations for $\mathrm{De}=0.1$ and 1, respectively. The solid line corresponds to the monotonic (Mon.) mode, the dashed one to the long-wave oscillatory (LO) mode, and the dashed-dotted line represents the variation for the short-wave oscillatory (SO) mode. For each long-wave instability mode, the results of the asymptotic analysis (represented by dotted lines) are juxtaposed with the corresponding numerical results in each panel.

Figure 7

Variation of the stability margin for the monotonic (Mon.: dashed-dotted and dashed lines) and oscillatory (Osc.: dotted and solid lines) instability modes for different values of De, Ga, and $\mathrm{\Sigma }$ at $\mathrm{Bi}=0.05,Pr=10$. Panel (a) corresponds to $\mathrm{De}=0.1$, panel (b) corresponds to $\mathrm{De}=1$, and panel (c) corresponds to $\mathrm{Ga}=0.1$. In panels (a,b), the lines marked by 1 and 2 correspond to $\mathrm{Ga}=0.1,\mathrm{\Sigma }={10}^3$ and $\mathrm{Ga}={10}^2,\mathrm{\Sigma }={10}^5$, respectively. For panel (c), the dashed and dotted lines correspond to $\mathrm{\Sigma }={10}^3$, while the dashed-dotted and solid lines correspond to $\mathrm{\Sigma }=10$. Here, the lines marked by 1 and 2 correspond to $\mathrm{De}=0.1$ and $\mathrm{De}=1$, respectively.

Figure 8

Variation of the critical Marangoni number (a), (d), the corresponding critical wave number (b), (e), and the frequency of critical perturbations (c), (f) with Deborah number at $\mathrm{Bi}=0.05$ and $Pr=10$. Panels (a)–(c) and (d)–(f) represent the variations for $\mathrm{Ga}=0.1,\mathrm{\Sigma }={10}^3$ and $\mathrm{Ga}={10}^2,\mathrm{\Sigma }={10}^5$, respectively. In each panel, the solid line corresponds to the monotonic (Mon.) mode, the dashed one to the long-wave oscillatory (LO) mode, and the dashed-dotted line represents the short-wave oscillatory (SO) mode, respectively.

Figure 9

The domains of monotonic and oscillatory instability modes. Panel (a) corresponds to $\mathrm{Ga}=0.1, \mathrm{\Sigma }={10}^3$; panel (b) corresponds to $\mathrm{Ga}={10}^2, \mathrm{\Sigma }={10}^5$. $Pr=10$. LM, LO, SM, and SO represent the domains of long-wave monotonic, long-wave oscillatory, short-wave monotonic, and short-wave oscillatory mode, respectively.