Droplets on the liquid substrate: Thermocapillary oscillatory instability

The influence of heating on the shape and the dynamics of a droplet on a liquid substrate has been investigated. The problem is studied numerically in the framework of longwave amplitude equations. The effect of gravity is taken into account. We show that the thermocapillary stresses produce a significant deformation of droplet's interfaces. Heating the droplet from above can create an oscillatory instability generating periodic and quasiperiodic oscillations, change of the droplet shape, and the droplet's decomposition. Heating from below can lead to the substrate layer's rupture due to its monotonic instability.

Figure 1

Geometric configuration of the region and coordinate axes: (a) the case of partial wetting; (b) the case of pseudopartial wetting.

Figure 2

The shapes of (a) interface $h_2(X,Y,\tau )$ and (b) interface $h_1(X,Y,\tau )$. $M=0, \text{Bo}=0, \text{Bi}=10, \tau ={10}^6$.

Figure 3

The cross section of the droplet and the precursor at $Y=L/2$ ($M=0, \text{Bo}=0, \text{Bi}=10, \tau =7\times {10}^6$). The solid line corresponds to the upper surface [$z=h_2(X,L/2)$], the dashed line corresponds to the lower surface [$z=h_1(X,L/2)$].

Figure 4

The shapes of (a) interface $h_2(X,Y,\tau )$ and (b) interface $h_1(X,Y,\tau )$. $M=0, \text{Bo}=0, \text{Bi}=10, \tau =7\times {10}^6$.

Figure 5

The cross section of the droplet and the precursor at $Y=L/2$ ($M=-0.1, \text{Bo}=0, \text{Bi}=10$). The solid line corresponds to the upper interface [$z=h_2(X,L/2)$], the dashed line corresponds to the lower surface [$z=h_1(X,L/2)$].

Figure 6

The stationary shapes of (a) interface $h_2(X,Y)$ and (b) interface $h_1(X,Y)$ for $M=-0.1, \text{Bo}=0, \text{Bi}=10$.

Figure 7

The stationary fields of (a) $h_2(X,Y)$ and (b) $h_1(X,Y)$ for $M=-0.1, \text{Bo}=0, \text{Bi}=10$.

Figure 8

The stationary shapes of (a) interface $h_2(X,Y)$ and (b) interface $h_1(X,Y)$ for $M=-0.5, \text{Bo}=0, \text{Bi}=10$.

Figure 9

The cross section of the droplet and the precursor at $Y=L/2$ ($M=-0.1, \text{Bo}=0.1, \text{Bi}=10$). The solid line corresponds to the upper interface [$z=h_2(X,L/2)$], the dashed line corresponds to the lower surface [$z=h_1(X,L/2)$].

Figure 10

The cross section of the droplet and the precursor at $Y=L/2$ ($M=-1, \text{Bo}=0.1, \text{Bi}=10$). The solid line corresponds to the upper interface [$z=h_2(X,L/2)$], the dashed line corresponds to the lower surface [$z=h_1(X,L/2)$].

Figure 11

The stationary shapes of (a) interface $h_2(X,Y)$ and (b) interface $h_1(X,Y)$ for $M=-0.5, \text{Bo}=0.1, \text{Bi}=10$.

Figure 12

The cross section of the droplet and the precursor at $Y=L/2$ ($M=0.01, \text{Bo}=0.1, \text{Bi}=10$). The solid line corresponds to the upper interface [$z=h_2(X,L/2)$], the dashed line corresponds to the lower surface [$z=h_1(X,L/2)$].

Figure 13

The stationary shapes of (a) interface $h_2(X,Y)$ and (b) interface $h_1(X,Y)$ for $M=-2, \text{Bo}=0, \text{Bi}=20$.

Figure 14

The fields of (a) $h_2(X,Y,\tau )$ and (b) $h_1(X,Y,\tau )$ for $M=-2.5, \text{Bo}=0, \text{Bi}=20, \tau =7\times {10}^5$.

Figure 15

The fields of (a) $h_2(X,Y,\tau )$ and (b) $h_1(X,Y,\tau )$ for $M=-2.5, \text{Bo}=0, \text{Bi}=20, \tau =7.15\times {10}^5$.

Figure 16

The oscillations of $h_{\max ,2}\left(\tau \right)$ (solid line) and $h_{\max ,1}\left(\tau \right)$ (dashed line) for $M=-2.5, \text{Bo}=0, \text{Bi}=20$.

Figure 17

The fields of (a) $h_2(X,Y,\tau )$ and (b) $h_1(X,Y,\tau )$ for $M=-4, \text{Bo}=0, \text{Bi}=20, \tau =6.9\times {10}^5$.

Figure 18

The fields of (a) $h_2(X,Y,\tau )$ and (b) $h_1(X,Y,\tau )$ for $M=-4, \text{Bo}=0, \text{Bi}=20, \tau =7\times {10}^5$.

Figure 19

The shapes of (a) interface $h_2(X,Y,\tau )$ and (b) interface $h_1(X,Y,\tau )$ for $M=-4, \text{Bo}=0, \text{Bi}=20, \tau =6.9\times {10}^5$.

Figure 20

The shapes of (a) interface $h_2(X,Y,\tau )$ and (b) interface $h_1(X,Y,\tau )$ for $M=-4, \text{Bo}=0, \text{Bi}=20, \tau =7\times {10}^5$.

Figure 21

The oscillations of $h_{\max ,2}\left(\tau \right)$ (solid line) and $h_{\max ,1}\left(\tau \right)$ (dashed line) for $M=-4, \text{Bo}=0, \text{Bi}=20$.

Figure 22

A snapshot of the fields of (a) $h_2(X,Y,\tau )$ and (b) $h_1(X,Y,\tau )$ for $M=-2.75, \text{Bo}=0, \text{Bi}=20, \tau =1\times {10}^6$.

Figure 23

A snapshot of the fields of (a) $h_2(X,Y,\tau )$ and (b) $h_1(X,Y,\tau )$ for $M=-5.5, \text{Bo}=0, \text{Bi}=20, \tau =1\times {10}^6$.

Figure 24

The shapes of (a) interface $h_2(X,Y,\tau )$ and (b) interface $h_1(X,Y,\tau )$ for $M=-5.5, \text{Bo}=0, \text{Bi}=20, \tau =1\times {10}^6$.

Figure 25

The oscillations of $h_{\max ,2}\left(\tau \right)$ (solid line), $h_{\max ,1}\left(\tau \right)$ (dashed line) and $h_{\min ,1}\left(\tau \right)$ (dotted line) for $M=-5.5, \text{Bo}=0, \text{Bi}=20$.

Figure 26

The shapes of (a) interface $h_2(X,Y,\tau )$ and (b) interface $h_1(X,Y,\tau )$ for $M=-8, \text{Bo}=0, \text{Bi}=20, \tau =1\times {10}^6$.

Figure 27

The oscillations of $h_{\max ,2}\left(\tau \right)$ (solid line)and $h_{\max ,1}\left(\tau \right)$ (dashed line) for $M=-8, \text{Bo}=0, \text{Bi}=20$.

Figure 28

The shapes of (a) interface $h_2(X,Y,\tau )$ and (b) interface $h_1(X,Y,\tau )$ for $M=-2.5, \text{Bo}=0, \text{Bi}=20, \tau =2\times {10}^6$.

Figure 29

Snapshots of the isolines of $h_2(X,Y,\tau )$; (a) $\tau =1000000$; (b) $\tau =1000250$; (c) $\tau =1000500$; (d) $\tau =1000750$; $M=-12, \text{Bo}=0, \text{Bi}=20$.

Figure 30

The shapes of (a) interface $h_2(X,Y,\tau )$ and (b) interface $h_1(X,Y,\tau )$ for $M=-12, \text{Bo}=0, \text{Bi}=20, \tau =2\times {10}^6$.

Figure 31

The fields of $h_2(X,Y,\tau )$ for $M=-4, \text{Bo}=0.1, \text{Bi}=20$; (a) $\tau =400$; (b) $\tau =1,200$; (c) $\tau =3,500$; (d) $\tau ={10}^5$.

Figure 32

The shapes of (a) interface $h_2(X,Y,\tau )$ and (b) interface $h_1(X,Y,\tau )$ for $M=-4, \text{Bo}=0.1, \text{Bi}=20, \tau ={10}^5$.

Figure 33

The shapes of (a) $h_2(X,Y,\tau )$ and (b) $h_1(X,Y,\tau )$ for $M=-2.5, \text{Bo}=0, \text{Bi}=20, \tau =7\times {10}^5$.

Figure 34

The evolution of $h_{\max ,2}\left(\tau \right)$ (solid line), $h_{\max ,1}\left(\tau \right)$ (dashed line) and $h_{\min ,1}\left(\tau \right)$ (dotted line) for $M=0.01, \text{Bo}=0, \text{Bi}=10$.

Figure 35

The cross section of the droplet and the precursor at $Y=L/2$ ($M=-0.01, \text{Bo}=0, \text{Bi}=10, \tau ={10}^6$). The solid line corresponds to the upper surface [$z=h_2(X,L/2)$], the dashed line corresponds to the lower surface [$z=h_1(X,L/2)$].

Figure 36

The cross section of the droplet and the precursor at $Y=L/2$ ($M=0.01, \text{Bo}=0, \text{Bi}=10, \tau ={10}^6$). The solid line corresponds to the upper surface [$z=h_2(X,L/2)$], the dashed line corresponds to the lower surface [$z=h_1(X,L/2)$].