2. Formulation of the problem

We consider a droplet of liquid 2 that floats on a layer of liquid 1, and it is in contact with gas phase 3 (see figure 1). Later on, we do not consider any processes in fluid 3: at the liquid–gas interface, the viscous stresses are neglected and the heat transfer is described using the heat exchange coefficient $q$ . The $m$ th fluid has density $\rho _m$ , dynamic viscosity $\eta _m$ and thermal conductivity $\kappa _m$ , $m=1,2$ .

Figure 1.

Geometric configuration of the region and coordinate axes.

The contact angles on the triple line surrounding the droplet are determined by the balance of interfacial tensions $\sigma _{12}$ , $\sigma _{23}$ and $\sigma _{13}$ between the corresponding fluids according to the Neumann triangle construction (Neumann Reference Neumann1894). The droplet exists in two cases: (i) when the spreading coefficient $S=\sigma _{13}-\sigma _{12}-\sigma _{23}\lt 0$ (partial wetting) and (ii) when $S\gt 0$ , but only a small part of fluid 2 is spread between fluids 1 and 3 forming an ultrathin film (pseudo-partial wetting) due to the attractive interaction of those fluids through the film of fluid 2 (positive Hamaker constant A).

The temperature of the solid substrate is $T_s+{\tilde A}x$ and the temperature of the gas near the interface is $T_g+{\tilde A}x$ . Thus, it is assumed that the same constant temperature gradient is imposed in the direction of the axis $x$ in the solid substrate and in the gas phase. Assuming that the temperature differences in the system are not too large, we disregard the dependence of liquid parameters on the temperature, with one exception: because we are interested in the investigation of the thermocapillary convection, we take into account the thermocapillary stresses proportional to derivatives of the interfacial tensions with respect to the temperature. The interfacial tension coefficients on the lower and upper surfaces of the droplet, $\sigma _1\equiv \sigma _{12}$ and $\sigma _2\equiv \sigma _{23}$ , are assumed to be linear functions of temperature $T$ : $\sigma _1=\sigma _1^0-\alpha _1 T$ , $\sigma _2=\sigma _2^0-\alpha _2 T$ , where $\alpha _1$ and $\alpha _2$ are constants. It is assumed that $|\alpha _1T|\ll \sigma _1$ and $|\alpha _2T|\ll \sigma _2$ ; therefore we disregard that dependence in the relations that contain the interfacial tensions as a whole, i.e. in the stress balances on the triple line and in the expressions for Laplace pressures.

The description of the temporal evolution of the triple line surrounding the droplet is technically difficult (see Kriegsmann & Miksis Reference Kriegsmann and Miksis2003). In Craster & Matar (Reference Craster and Matar2006), the precursor model was suggested for the description of a floating droplet: the interface between fluids 1 and 3 outside the droplet is replaced by an ultrathin precursor layer of fluid 2 (see figure 1). The latter model describes the droplet on the liquid substrate as a two-layer film. The same equations are used in the whole region, but outside the droplet, where the top layer is ultrathin, the corresponding disjoining pressure is taken into account. Let us emphasize that the latter approach can be applied in the case of both pseudo-partial wetting and partial wetting, because macroscopically both cases are identical. In the present paper, we apply that approach for the description of the dynamics of a non-isothermal floating droplet.

The deformable interfaces are described by equations $z=H_1(x,y,t)$ and $z=H_2(x,y,t)$ . Far from the droplet, the equilibrium thickness of layer 1 is $H_1^0$ , and the thickness of the precursor film is $H_{\infty }$ (see figure 1). The gravitational acceleration is $g$ .

Later on, we consider a slender droplet, i.e. the slopes of both the droplet’s interfaces are small. Also, we assume that the characteristic horizontal scale of the interface deformations is large as compared with the characteristic vertical size of the droplet and the substrate. Those assumptions allow applying the mathematical model governing the longwave dynamics of non-isothermal liquid layers that has been derived using the lubrication approximation by Pototsky et al. (Reference Pototsky, Oron and Bestehorn2005) (see also Nepomnyashchy & Simanovskii Reference Nepomnyashchy and Simanovskii2007). In the framework of the longwave approach, the shapes of the interfaces $z=H_1$ and $z=H_2$ depend on the scaled horizontal coordinates $X=\epsilon x$ and $Y=\epsilon y$ , $\epsilon \ll 1$ , rather than on $x$ and $y$ . Also, it is assumed that they depend on the scaled time variable $\tau =\epsilon ^2 t$ . A comprehensive description of the longwave approach can be found in the review paper by Oron et al. (Reference Oron, Davis and Bankoff1997).

The problem is presented in non-dimensional form using the equilibrium thickness of the lower layer, $H_1^0$ , as the vertical length scale. The choice of the horizontal scale $L^*$ is arbitrary (Nepomnyashchy & Simanovskii Reference Nepomnyashchy and Simanovskii2012). We choose

(2.1) \begin{equation} \tau ^*=\frac {\eta _1(L^*)^4}{\sigma _1^0(H_1^0)^3} \end{equation}

as a time scale and

(2.2) \begin{equation} p^*=\frac {\sigma _1^0H_1^0}{(L^*)^2} \end{equation}

as a pressure scale.

The non-dimensional system of equations looks as follows (lower-case letters correspond to non-dimensional quantities):

(2.3) \begin{equation} h_{1\tau }+\nabla \cdot \textbf { q}_1=0,\;h_{2\tau }+\nabla \cdot \textbf { q}_2=0, \end{equation}

(2.4) \begin{equation} \textbf { q}_1=f_{11}\nabla \bar {p}_1+f_{12}\nabla \bar {p}_2+\textbf { q}_1^T,\;\textbf { q}_2=f_{21}\nabla \bar {p}_1+f_{22}\nabla \bar {p}_2+\textbf { q}_2^T, \end{equation}

(2.5) \begin{equation} f_{11}=-\frac {1}{3}h_1^3,\;f_{12}=-\frac {1}{2}h_1^2(h_2-h_1), \end{equation}

(2.6) \begin{equation} f_{21}=\frac {1}{6}h_1^3-\frac {1}{2}h_1^2h_2,\;f_{22}= (h_2-h_1)\left [h_1^2\left (\frac {1}{2}-\frac {\eta }{3}\right ) +h_1h_2\left (-1+\frac {2\eta }{3}\right )-\frac {\eta }{3}h_2^2\right ]. \end{equation}

The expressions for pressures,

(2.7) \begin{equation} \bar {p}_1=-\nabla ^2h_1-\sigma \nabla ^2h_2+Boh_1+Bo\rho (h_2-h_1)+\pi _1(h_1,h_2), \end{equation}

(2.8) \begin{equation} \bar {p}_2=-\sigma \nabla ^2h_2+Bo\rho h_2+\pi _2(h_1,h_2), \end{equation}

include the contributions of the Laplacian pressures, hydrostatic pressures and disjoining pressures:

(2.9) \begin{equation} \textbf { q}_1^T=\frac {M_{\bot }}{2}h_1^2\nabla [d(Bi h_1-\alpha \kappa )]-\frac {M_{\|}}{2}(1+\alpha )h_1^2\textbf { e}_x, \end{equation}

(2.10) \begin{equation} \textbf { q}_2^T=\frac {M_{\bot }}{2} \{-h_2^2\nabla (d\eta \alpha \kappa )+ (2h_2-h_1)h_1\nabla \{d[Bi h_1-\alpha \kappa (1-\eta )]\} \} \end{equation}

(2.11) \begin{equation} -\frac {M_{\|}}{2}[\eta \alpha (h_2-h_1)^2+(1+\alpha )h_1(2h_2-h_1)]\textbf {e}_x, \end{equation}

(2.12) \begin{equation} d=[\kappa +Bi(1-\kappa )h_1+Bi\kappa h_2]^{-1}. \end{equation}

Note that the choice of scale is different from that used in Nepomnyashchy & Simanovskii (Reference Nepomnyashchy and Simanovskii2012), because here we consider the Marangoni instability rather than the vanderWaals instability. The derivation of the longwave evolution equations is given in Appendix A.

The non-dimensional parameters of the problem are as follows:

(2.13) \begin{equation} M_{\bot }=\frac {\alpha _1(T_s-T_g)}{\sigma _1^0}\left (\frac {L^*}{H_1^0}\right )^2, \end{equation}

(2.14) \begin{equation} M_{\|}=\frac {\alpha _1AL^*}{\sigma _1^0}\left (\frac {L^*}{H_1^0}\right )^2 \end{equation}

are the modified transverse and longitudinal Marangoni numbers, correspondingly,

(2.15) \begin{equation} Bi=\frac {qH_1^0}{\kappa _2} \end{equation}

is the Biot number, $\eta =\eta _1/\eta _2$ , $\kappa =\kappa _1/\kappa _2$ , $\sigma =\sigma _2^0/\sigma _1^0$ , $\alpha =\alpha _2/\alpha _1$ , $\rho =\rho _2/\rho _1$ and

(2.16) \begin{align} Bo=\frac {g\rho _1(L^*)^2}{\sigma _1^0} \end{align}

is the Bond number.

One can see that each flow rate, $\textbf { q}_1^T$ and $\textbf {q}_2^T$ , contains two contributions, one proportional to $M_{\|}$ and another to $M_{\bot }$ . The former contribution, which is caused by the imposed horizontal temperature gradient, is always directed opposite to vector $\textbf { e}_x$ , i.e. towards the region with lower temperature and hence higher interfacial tensions. Note that the contribution of the horizontal temperature gradient to $\nabla \cdot \textbf { q}_1$ can be written as $u_1h_{1x}$ , where

(2.17) \begin{equation} u_1=-M_{\|}(1+\alpha ) h_1. \end{equation}

(The subscript means the derivative, and the way of non-dimensionalization is described above.) Thus, each point of the droplet’s bottom surface moves to the left with velocity (2.17). The term with $M_{\|}$ in the expression for $\nabla \cdot \textbf { q}_2$ is $u_2h_{2x}+w_2$ , where

(2.18) \begin{equation} u_2=-M_{\|}[\eta \alpha (h_2-h_1)+(1+\alpha )h_1], \end{equation}

(2.19) \begin{equation} w_2=M_{\|}[\eta \alpha -(1+\alpha )](h_2-h_1)h_{1x}. \end{equation}

Thus, in addition to the motion to the left with velocity (2.18), a point on the top surface of the droplet moves upward or downward with velocity (2.19).

Figure 2.

A snapshot of the fields of (a) $h_2(X,Y,\tau )$ and (b) $h_1(X,Y,\tau )$ for $M_{\bot }=-0.25$ , $M_{\|}=0.0001$ , $Bo=0$ , $Bi=20$ , $\tau =1\times 10^6$ . The horizontal axis is $X$ and the vertical axis is $Y$ .

Figure 3.

The shapes of interfaces (a) $h_2(X,Y,\tau )$ and (b) $h_1(X,Y,\tau )$ for $M_{\bot }=-0.25$ , $M_{\|}=0.0001$ , $Bo=0$ , $Bi=20$ , $\tau =1\times 10^6$ .

The second contribution proportional to $M_{\bot }$ is generated by non-uniformities of the interfacial temperature field, which are caused by the deformations of interfaces.

Because the thickness of liquid layer 1 is always macroscopic, we can neglect the contribution of $\pi _1$ . In layer 2, we apply the following expression for the disjoining pressure:

(2.20) \begin{equation} \pi _2=\frac {a}{(h_2-h_1)^3} \left [1-\left (\frac {h_{\infty }}{h_2-h_1}\right )^3\right ], \end{equation}

where $a$ is the non-dimensional Hamaker constant, which is related to the dimensional Hamaker constant $A$ as

(2.21) \begin{equation} a=\frac {A(L^*)^2}{6\pi \sigma _1^0(H_1^0)^4}, \end{equation}

and $h_{\infty }=H_{\infty }/H_1^0$ (for details, see Craster & Matar Reference Craster and Matar2006; Pototsky et al. Reference Pototsky, Oron and Bestehorn2019).

Note that in the absence of gravity, a liquid layer with a deformable interface is subject to a monotonic Marangoni instability for arbitrary $M_{\bot }\gt 0$ , i.e. by any heating from below (Sternling & Scriven Reference Scriven and Sternling1964). That instability is not saturable, and it leads to the rupture of the substrate layer. The temperature disturbance caused by the droplet acts as a ‘seed’ of instability. Therefore, in the presence of the temperature gradient, one can expect the existence of a stable configuration containing a droplet on a layer flat on the infinity only if $M_{\bot }\lt 0$ , i.e. by cooling from below.

Figure 4.

The shapes of interfaces (a) $h_2(X,Y,\tau )$ and (b) $h_1(X,Y,\tau )$ for $M_{\bot }=-0.5$ , $M_{\|}=0.0001$ , $Bo=0$ , $Bi=20$ , $\tau =4.12\times 10^5$ .

3. Nonlinear simulations

3.1. Methodology

The problem governed by equations (2.3)–(2.20) was solved numerically with some initial conditions. The evolution equations were discretized by central differences for spatial derivatives and solved using an explicit scheme. Periodic boundary conditions have been applied on the boundaries of the computational region $L \times L$ .

Recall that we assume that the characteristic horizontal scale of the surface deformations is significantly larger than the thickness of the droplet and that of the substrate layer typically chosen as the length scale. The distance between the grid nodes can be chosen rather large. The computations have been performed in the region $L \times L= 240\times 240$ using an $80\times 80$ grid. Some trial simulations on $100\times 100$ and $120\times 120$ grids did not reveal any qualitative changes. The time step typically was changed between 0.00125 and 0.005, which is significantly below the linear scheme stability boundary. No quantitative changes have been observed for simulations made with different time steps.

We used two kinds of initial conditions. First, we applied the initial conditions

(3.1) \begin{equation} h_1(X,Y,0)=1;\;h_2(X,Y,0)=h_-+h_+\exp \{-[(X-L/2)^2+(Y-L/2)^2]/R^2\}, \end{equation}

i.e. a droplet of liquid 2 with a Gaussian shape was imposed on a flat layer of liquid 1, where $h_{-}$ , $h_{+}$ , corresponding to the initial shape of the droplet, are constants and $R$ is the radius of the droplet. In some other simulations, we used a solution formerly obtained for another set of parameters as the initial condition.

In the present paper, we perform computations for the system of fluorinert FC70 (liquid 1) and silicone oil 10 (liquid 2). This system of liquids was used in microgravity experiments (see e.g. Géoris et al. Reference Géoris, Hennenberg, Lebon and Legros1999). We applied the following set of liquid parameters that was formerly used in simulations of Marangoni instability in two-layer systems (Nepomnyashchy & Simanovskii Reference Nepomnyashchy and Simanovskii2012): $\eta =3.04$ , $\kappa =0.522$ , $\alpha =2$ , $\rho =0.482$ , $\sigma =2.6$ , $h_-=1.02$ , $h_+=1$ , $R=60$ , $h_{\infty }=0.01$ and $Bi=20$ . The value of $a$ is chosen equal to $0.31\times 10^{-6}$ (see Nepomnyashchy & Simanovskii Reference Nepomnyashchy and Simanovskii2021).

The primary analysis of the obtained nonlinear regimes has been performed using snapshots of the fields of $h_j(X,Y,\tau ),$ $j=1,2.$ The analysis has been supplemented by the investigation of the time dependence for the maximum values of the variables $h_j$ :

(3.2) \begin{equation} h_{max,j}(\tau )=\max h_{j}(X,Y,\tau ). \end{equation}

Figure 5.

The oscillations of $h_{max,2}(\tau )$ (solid line) and $h_{max,1}(\tau )$ (dashed line) for $M_{\bot }=-1.07$ , $M_{\|}=0.0001$ , $Bo=0$ , $Bi=20$ .

Figure 6.

A snapshot of the fields of $h_2(X,Y,\tau )$ for $\bar {M}=-1.07$ , $M_{\|}=0.0001$ , $Bo=0$ , $Bi=20$ : (a) $\tau =3\times 10^5$ ; (b) $\tau =3.6\times 10^5$ .

Figure 7.

The oscillations of $h_{max,2}(\tau )$ (solid line) and $h_{max,1}(\tau )$ (dashed line) for $M_{\bot }=-2$ , $M_{\|}=0.0001$ , $Bo=0$ , $Bi=20$ .

Figure 8.

The fields of $h_2(X,Y,\tau )$ for $M_{\bot }=-2$ , $M_{\|}=0.0001$ , $Bo=0$ , $Bi=20$ : (a) $\tau =912\,500$ ;(b) $\tau =925\,000$ ;(c) $\tau =937\,000$ ; (d) $\tau =942\,500$ ; (e) $\tau =948\,000$ ;(f) $\tau =962\,000$ .

Figure 9.

The shapes of interfaces (a) $h_2(X,Y,\tau )$ and (b) $h_1(X,Y,\tau )$ for $M_{\bot }=-2$ , $M_{\|}=0.0001$ , $Bo=0$ , $Bi=20$ , $\tau =912\,500$ .

3.2. Nonlinear regimes

3.2.1. Travelling droplets

Let us describe now the results of nonlinear simulations. The simplest structure observed for sufficiently small values of $M_{\|}$ and $|M_{\bot }|$ is the droplet, moving from the right-hand side to the left-hand side as a whole:

(3.3) \begin{align} h_j=h_j(X+c\tau ),\;j=1,2,\;c\gt 0. \end{align}

The snapshot of the fields of $h_2(X,Y,\tau )$ and $h_1(X,Y,\tau )$ is shown in figure 2. For all the figures showing the droplet, the horizontal axis is $X$ and the vertical axis is $Y$ . The shapes of interfaces are presented in figure 3. This type of motion has also been found for larger values of $M_{\bot }$ (at $M_{\bot }=-0.5$ ). As in the previous case, the droplet moves from the right-hand side to the left-hand side. The shapes of interfaces are shown in figure 4. Let us emphasize that while in the case of an isothermal droplet the lower interface of the droplet is below the substrate level, in the case of the non-isothermal droplet and sufficiently large value of the Biot number, the lower interface of the droplet is mainly above the substrate. The change of the droplet’s shape is caused by the Marangoni effect.

3.2.2. Excitation of oscillations

Let us consider the evolution of the regimes at $M_{\|}=0.0001$ for sufficiently large values of $|M_{\bot }|$ . At $M_{\bot }=-1.07$ , we observed periodic oscillations. The temporal evolution of $h_{max,2}$ and $h_{max,1}$ is shown in figure 5. The period of oscillations $T=146\,700$ .The two-dimensional snapshots of the fields $h_2(X,Y,\tau )$ at different instants of time are shown in figure 6. The droplet moves from right to left, changing its shape.

At $M_{\bot }=-2$ ( $M_{\|}=0.0001$ ), we get periodic oscillations of the droplet of a rather complex form. The temporal evolution of $h_{max,2}$ and $h_{max,1}$ is shown in figure 7. The period of oscillations $T=75\,500$ . One can see that with an increase of $M_{\bot }$ , the period of oscillations decreases (cf. Figures 5 and 7). The two-dimensional snapshots of the fields $h_2(X,Y,\tau )$ at different instants of time are shown in figure 8 . The droplet moves from the right-hand side to the left-hand side, changing its shape during an oscillatory process. The shapes of interfaces at $\tau =912\,500$ are shown in figure 9 (see also figure 8 a).

Let us emphasize that for the same values of $M_{\|}$ and $M_{\bot }$ , different spatial structures can be obtained, depending on the initial conditions.

With a further increase of $|M_{\bot }|$ ( $M_{\bot }=-4$ , $M_{\|}=0.0001$ ), the system makes a transition to another stable mode through quasi-periodic oscillations. The temporal evolution of $h_{max,1}$ and $h_{max,2}$ is shown in figure 10. The two-dimensional snapshots of the fields $h_2(X,Y,\tau )$ at different instants of time are shown in figure 11. One can see that the system of droplets moves from right to left, changing shape during an oscillatory process. The shapes of interfaces at $\tau =310\,000$ are shown in figure 12 (see also figure 11 e).

Figure 10.

The oscillations of $h_{max,2}(\tau )$ (solid line) and $h_{max,1}(\tau )$ (dashed line) for $M_{\bot }=-4$ , $M_{\|}=0.0001$ , $Bo=0$ , $Bi=20$ .

Figure 11.

The fields of $h_2(X,Y,\tau )$ for $M_{\bot }=-4$ , $M_{\|}=0.0001$ , $Bo=0$ , $Bi=20$ : (a) $\tau =300\,000$ ; (b) $\tau =302\,500$ ; (c) $\tau =305\,000$ ; (d) $\tau =307\,500$ ; (e) $\tau =310\,000$ ; (f) $\tau =312\,700$ .

Figure 12.

The shapes of interfaces (a) $h_2(X,Y,\tau )$ and (b) $h_1(X,Y,\tau )$ for $M_{\bot }=-4$ , $M_{\|}=0.0001$ , $Bo=0$ , $Bi=20$ , $\tau =310\,000$ .

The thermocapillary flow influences the interaction between droplets. Let us take as the initial conditions the structure consisting of several fragments (see figure 13). This structure was created at $M_{\bot }=-4$ , $M_{\|}=0$ as the result of droplet destruction by an intensive Marangoni flow. The motion of droplets under the action of an inclined temperature gradient ( $M_{\bot }=-4$ , $M_{\|}=0.0001)$ can simplify their coalescence leading to the creation of a film covering the whole computational region. The film instability creates a more or less ordered system of in-phase deformations of interfaces (see figure 14). One can compare figures 14(a) and 11(e) obtained for the same values of parameters from different initial conditions.

Figure 13.

A snapshot of the fields of (a) $h_2(X,Y,\tau )$ and (b) $h_1(X,Y,\tau )$ for $M_{\bot }=-4$ , $M_{\|}=0$ , $Bo=0$ , $Bi=20$ , $\tau =1\times 10^6$ .

Figure 14.

A snapshot of the fields of (a) $h_2(X,Y,\tau )$ and (b) $h_1(X,Y,\tau )$ for $M_{\bot }=-4$ , $M_{\|}=0.0001$ , $Bo=0$ , $Bi=20$ , $\tau =1\times 10^6$ ; figure 13 has been taken as initial conditions.

For sufficiently large values of $|M_{\bot }|$ and $M_{\|}$ (we take $M_{\bot } =-4$ and $M_{\|}=0.0005$ ), the nonlinear growth of disturbances with non-commensurable frequencies leads to the appearance of quasi-periodic motion. The temporal evolution of $h_{max,2}$ and $h_{max,1}$ is shown in figure 15. The two-dimensional snapshots of the fields $h_2(X,Y,\tau )$ at different instants of time are shown in figure 16. The pattern moves from the right-hand side to the left-hand side, changing shape during an oscillatory process. The shapes of interfaces at $\tau =402\,500$ are shown in figure 17 (see also figure 16 b).

Figure 15.

The oscillations of $h_{max,2}(\tau )$ (solid line) and $h_{max,1}(\tau )$ (dashed line) for $M_{\bot }=-4$ , $M_{\|}=0.0005$ , $Bo=0$ , $Bi=20$ .

Figure 16.

The fields of $h_2(X,Y,\tau )$ for $M_{\bot }=-4$ , $M_{\|}=0.0005$ , $Bo=0$ , $Bi=20$ : (a) $\tau =400\,000$ ; (b) $\tau =402\,500$ ; (c) $\tau =405\,000$ ; (d) $\tau =407\,500$ .

Figure 17.

The shapes of interfaces (a) $h_2(X,Y,\tau )$ and (b) $h_1(X,Y,\tau )$ for $M_{\bot }=-4$ , $M_{\|}=0.0005$ , $Bo=0$ , $Bi=20$ , $\tau =402\,500$ .

Figure 18.

A snapshot of the fields of $h_2(X,Y,\tau )$ for $M_{\bot }=-0.25$ , $M_{\|}=0.004$ , $Bo=0$ , $Bi=20$ : (a) $\tau =430\,000$ ; (b) $\tau =442\,000$ . The structure has been obtained from asymmetricinitial conditions with respect to axis $Y=L/2$ . The horizontal axis is $X$ and the vertical axis is $Y$ .

Let us take as initial conditions the steady droplet moving from the right-hand side to the left-handside at $M_{\bot }=-0.25$ and $M_{\|}=0.0001$ . With an increase of $M_{\|}$ (at $M_{\|}=0.004$ ), the droplet is split, and we get a pair of pulsating droplets. The height and the shape of the droplets change with a small amplitude and large period ( $\tau =110\,000$ ) during an oscillatory process. The snapshots of the fields $h_2(X,Y,\tau )$ at different instants of time are shown in figure 18.

3.2.3. Suppression of oscillations

Now, let us fix $M_{\bot }$ ( $M_{\bot }=-2$ ), changing $M_{\|}$ . For sufficiently large values of $M_{\|}$ , spreading of the droplet leads to the creation of a film covering the substrate. Periodic oscillations are suppressed. In a certain region of parameters, film instability leads to the formation of ordered structures that consist of equally sized deformations of interfaces moving from the right to the left with equal velocities and located at equal distances from each other. An example of the formation of the ordered structure is shown in figure 19 ( $M_{\bot }=-2$ , $M_{\|}=0.0002$ , $\tau =1\times 10^6$ ). Note that the locations of the deformations on both interfaces are correlated. Let us emphasize that the system of ordered patterns is not a result of a casual monodispersity of droplets due to a specific choice of initial conditions but a result of a natural evolution of the system to its stable configuration. The diagram of flow regimes in the plane ( $M_{\|}$ , $M_{\bot }$ ) is constructed in figure 20. Let us note that we observed two stable regimes that have been obtained from different initial conditions: (i) oscillations of a droplet, obtained from the oscillatory state, corresponding to $M_{\|}=0.0001$ and $M_{\bot }=-2$ , and (ii) stationary droplet, obtained from the state corresponding to $M_{\|}=0.0003$ and $M_{\bot }=-2$ . Thus, bistability at several points takes place.

Figure 19.

A snapshot of the fields of (a) $h_2(X,Y,\tau )$ and (b) $h_1(X,Y,\tau )$ for $M_{\bot }=-2$ , $M_{\|}=0.0002$ , $Bo=0$ , $Bi=20$ , $\tau =1\times 10^6$ . The horizontal axis is $X$ and the vertical axis is $Y$ .

Figure 20.

The diagram of flow regimes in the plane ( $M_{\|}$ , $M_{\bot }$ ). Squares, two-dimensional waves; asterisks, time-periodic three-dimensional waves; circles, non-periodic three-dimensional waves.

4. Droplets with an inclined temperature gradient in the presence of gravity

Let us consider the influence of the gravity force on droplet dynamics. It is known that gravity stabilizes the monotonic deformational instability (Smith Reference Smith1966) and the oscillatory Marangoni instability in two-layer systems (Simanovskii & Nepomnyashchy Reference Simanovskii and Nepomnyashchy2006), because the gravity force tends to suppress the surface deformation and flatten the surface. Similarly, gravity flattens the droplet and suppresses the oscillatory instability of the droplet. Let us take as initial conditions periodic oscillations presented in figures 7 and 8(a) at $M_{\bot }=-2$ , $M_{\|}=0.0001$ , $Bo=0$ . Under the action of a sufficiently small value of the gravity force (we take $Bo=0.02$ ), periodic oscillations become of a rather simple form. The temporal evolution of $h_{max,2}$ and $h_{max,1}$ is shown in figure 21. One can see that the amplitude of oscillations decreases in comparison with the case $Bo=0$ (cf. figures 7 and 21). The two-dimensional snapshots of the fields $h_2(X,Y,\tau )$ at different instants of time are presented in figure 22 and three-dimensional shapes of interfaces corresponding to figure 22(f) are shown in figure 23. Let us note that for droplets of the same volume, in the presence of gravity, both the top and the bottom interfaces are significantly lower and flatter than in the absence of gravity (cf., for example, figures 9 and 23).

Figure 21.

The oscillations of $h_{max,2}(\tau )$ (solid line) and $h_{max,1}(\tau )$ (dashed line) for $M_{\bot }=-2$ , $M_{\|}=0.0001$ , $Bo=0.02$ , $Bi=20$ .

Figure 22.

The fields of $h_2(X,Y,\tau )$ for $M_{\bot }=-2$ , $M_{\|}=0.0001$ , $Bo=0.02$ , $Bi=20$ : (a) $\tau =872\,700$ ; (b) $\tau =893\,000$ ;(c) $\tau =898\,500$ ;(d) $\tau =908\,500$ ;(e) $\tau =928\,750$ ;(f) $\tau =942\,500$ .

Figure 23.

The shapes of (a) $h_2(X,Y,\tau )$ and (b) $h_1(X,Y,\tau )$ for $\bar {M}=-2$ , $M_{\|}=0.0001$ , $Bo=0.02$ , $Bi=20$ , $\tau =942\,500$ .

With an increase of the gravity force (we take $Bo=0.05$ ), the oscillations are suppressed. The temporal evolution of $h_{max,2}$ , $h_{max,1}$ and $h_{min,1}$ is shown in figure 24. The droplets presented in figure 8 at $Bo=0$ are destroyed and finally transform to a new stable configuration–one large wave, moving from the right-hand side to the left-hand side. The two-dimensional snapshots of the fields $h_2(X,Y,\tau )$ and $h_1(X,Y,\tau )$ corresponding to a ‘vertical stripe’ are shown in figure 25. The shapes of interfaces are presented in figure 26.

Figure 24.

The oscillations of $h_{max,2}(\tau )$ (solid line), $h_{max,1}(\tau )$ (dashed line) and $h_{min,1}(\tau )$ (dotted line)for $M_{\bot }=-2$ , $M_{\|}=0.0001$ , $Bo=0.05$ , $Bi=20$ .

Figure 25.

The fields of (a) $h_2(X,Y,\tau )$ and (b) $h_1(X,Y,\tau )$ for $M_{\bot }=-2$ , $M_{\|}=0.0001$ , $Bo=0.05$ , $Bi=20$ , $\tau =1\times 10^6$ . The horizontal axis is $X$ and the vertical axis is $Y$ .

Let us take as the initial conditions the oscillatory flow presented in figures 15 and 16(a) for the larger value of $M_{\bot }$ ( $M_{\bot }=-4$ , $M_{\|}=0.0001$ , $Bo=0$ ). Under the action of the gravity force (we take $Bo=0.04$ ), the amplitude of oscillations decreases. The two-dimensional snapshots of the fields $h_2(X,Y,\tau )$ at different instants of time are presented in figure 27 and three-dimensional shapes of interfaces corresponding to figure 27(a) are shown in figure 28. With an increase of $Bo$ (at $Bo\gt 0.045$ ), oscillations are suppressed and we get the ‘vertical stripe’ similar to that presented in figures 25 and 26.