1 Introduction
During recent decades, a great deal of attention has been paid to the investigation of nonlinear patterns created by instabilities (Cross & Hohenberg Reference Cross and Hohenberg1993). The nonlinear dynamics of systems subjected to oscillatory instabilities is especially rich (Aranson & Kramer Reference Aranson and Kramer2002). For nonlinear patterns, a multistability is characteristic; specifically, in spatially extended systems different pattern planforms are possible.
An important problem is to control the development of instability, i.e. the suppression of undesired kinds of patterns or generation of desired kinds of patterns (Mikhailov & Showalter Reference Mikhailov and Showalter2006; Schöll & Schuster Reference Schöll and Schuster2008).
A possible way of controlling the pattern selection is a spatial modulation of the control parameter. It is necessary to distinguish between patterns created by short-wave instabilities, characterized by a non-zero critical wavenumber
$k_{c}\neq 0$
, and patterns generated by long-wave instabilities, where the critical wavenumber
$k_{c}=0$
, so that the wavenumber interval of instability is
$0<k<k_{m}$
. The influence of the spatial modulation of the control parameter on short-wave patterns with moduli of wavevectors close to
$k_{c}$
is rather well understood. The action of a resonant spatially periodic forcing on the onset of stationary patterns created by a short-wave instability has been explored extensively (Vozovoi and Nepomnyashchy Reference Vozovoi, Nepomnyashchy and Zhukhovitskii1974; Vozovoi & Nepomnyashchy Reference Vozovoi and Nepomnyashchy1979; Coullet, Elphick & Repaux Reference Coullet, Elphick and Repaux1987; Pismen Reference Pismen1987; Manor, Hagberg & Meron Reference Manor, Hagberg and Meron2008; Freund, Pesch & Zimmermann Reference Freund, Pesch and Zimmermann2011; Mau et al.
Reference Mau, Haim, Hagberg and Meron2013; Haim, Mau & Meron Reference Haim, Mau and Meron2014; Weiss, Seiden & Bodenschatz Reference Weiss, Seiden and Bodenschatz2014). The influence of resonant and non-resonant spatial modulations on short-wave oscillatory instabilities was studied on the background of a complex Ginzburg–Landau equation by Nepomnyashchy (Reference Nepomnyashchy1988), Malomed (Reference Malomed1993), Utzny, Zimmermann & Bär (Reference Utzny, Zimmermann and Bär2002), Hammele & Zimmermann (Reference Hammele and Zimmermann2006), Abarzhi et al. (Reference Abarzhi, Desjardins, Nepomnyashchy and Pitsch2007) and Nepomnyashchy & Abarzhi (Reference Nepomnyashchy and Abarzhi2010).
In the case of a long-wave instability, where disturbances with significantly different wavenumbers are relevant, the prediction of the pattern dynamics is much more difficult. In the absence of a control, the pattern evolution is governed by the nonlinear interaction of disturbances with different wavevectors, which can lead to complex spatio-temporal dynamics. It is tempting to impose a spatial modulation of the control parameter with a prescribed wavelength, which can simplify the dynamics and make it more predictable. However, the effect of the spatial modulation on long-wave patterns has still hardly been explored.
This paper is devoted to the investigation of an important class of long-wave oscillatory instabilities which cannot be described by the Ginzburg–Landau equation (Nepomnyashchy & Shklyaev Reference Nepomnyashchy and Shklyaev2016). This kind of instability is typical for liquid systems with deformable interfaces. To the best of our knowledge, the influence of spatial modulations on long-wave oscillatory instabilities has never been studied before.
We consider the nonlinear dynamics of waves generated by the deformational Marangoni instability in a two-layer film under the action of a spatial temperature modulation on the solid substrate. In the absence of a temperature modulation, a two-layer film is subject to a long-wave oscillatory instability by heating from above (Nepomnyashchy & Simanovskii Reference Nepomnyashchy and Simanovskii2007). In contradistinction to a one-layer film, where the deformational instability is monotonic and leads to film rupture, an oscillatory instability in a two-layer film creates various wavy patterns (Nepomnyashchy & Simanovskii Reference Nepomnyashchy and Simanovskii2007, Reference Nepomnyashchy and Simanovskii2012). The formulation of the problem is given in § 2. In § 3, we derive a system of long-wave equations that govern the evolution of the surface deformations. The results of nonlinear simulations are presented in § 4. Section 5 contains concluding remarks.
2 Formulation of the problem
Consider a system of two superposed layers of immiscible liquids with different physical properties (see figure 1). The bottom layer rests on a solid substrate and the top layer is in contact with the adjacent gas phase. The temperature of the solid substrate is
$T_{s}(x,y)$
(we assume that
$T_{s}$
is a slow function of
$x$
and
$y$
), and the temperature of the gas is
$T_{g}$
. All of the variables referring to the bottom layer are marked by a subscript 1 and all of the variables referring to the top layer are marked by a subscript 2. The coordinates of the interfaces in a quiescent state are
$z=H_{m}^{0}$
,
$m=1,2$
. The deformable interfaces are described by the equations
$z=H_{1}(x,y,t)$
(liquid–liquid interface) and
$z=H_{2}(x,y,t)$
(liquid–gas interface). The
$m$
th fluid has density
$\unicode[STIX]{x1D70C}_{m}$
, kinematic viscosity
$\unicode[STIX]{x1D708}_{m}$
, dynamic viscosity
$\unicode[STIX]{x1D702}_{m}=\unicode[STIX]{x1D70C}_{m}\unicode[STIX]{x1D708}_{m}$
, thermal diffusivity
$\unicode[STIX]{x1D712}_{m}$
and heat conductivity
$\unicode[STIX]{x1D705}_{m}$
. The surface tension coefficients on the lower and upper interfaces,
$\unicode[STIX]{x1D70E}_{1}$
and
$\unicode[STIX]{x1D70E}_{2}$
, are linear functions of temperature
$T$
:
$\unicode[STIX]{x1D70E}_{1}=\unicode[STIX]{x1D70E}_{1}^{0}-\unicode[STIX]{x1D6FC}_{1}T$
,
$\unicode[STIX]{x1D70E}_{2}=\unicode[STIX]{x1D70E}_{2}^{0}-\unicode[STIX]{x1D6FC}_{2}T$
. We do not consider the effect of gravity, which is negligible for sufficiently thin layers.
Figure 1. Geometric configuration of the region and coordinate axes.
The complete system of nonlinear equations governing Marangoni convection is written in the following form (Simanovskii & Nepomnyashchy Reference Simanovskii and Nepomnyashchy1993):
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{v}_{m}}{\unicode[STIX]{x2202}t}+(\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{v}_{m}=-\frac{1}{\unicode[STIX]{x1D70C}_{m}}\unicode[STIX]{x1D735}p_{m}+\unicode[STIX]{x1D708}_{m}\unicode[STIX]{x0394}\boldsymbol{v}_{m}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}T_{m}}{\unicode[STIX]{x2202}t}+\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}T_{m}=\unicode[STIX]{x1D712}_{m}\unicode[STIX]{x0394}T_{m}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{v}_{m}=0,\quad m=1,2. & \displaystyle\end{eqnarray}$$
Here,
$\boldsymbol{v}_{m}$
and
$p_{m}$
are the velocity and the difference between the overall pressure and the atmospheric pressure in the
$m$
th liquid respectively. The boundary conditions on the rigid boundary are
$$\begin{eqnarray}\displaystyle \boldsymbol{v}_{1}=0,\quad T_{1}=T_{s}(x,y);\quad \text{at }z=0. & & \displaystyle\end{eqnarray}$$
On the deformable interface
$z=H_{1}$
, the following boundary conditions hold: the balance of normal stresses,
$$\begin{eqnarray}\displaystyle p_{2}-p_{1}+2\unicode[STIX]{x1D70E}_{1}K_{1}=\left[-\unicode[STIX]{x1D702}_{1}\left(\frac{\unicode[STIX]{x2202}v_{1i}}{\unicode[STIX]{x2202}x_{k}}+\frac{\unicode[STIX]{x2202}v_{1k}}{\unicode[STIX]{x2202}x_{i}}\right)+\unicode[STIX]{x1D702}_{2}\left(\frac{\unicode[STIX]{x2202}v_{2i}}{\unicode[STIX]{x2202}x_{k}}+\frac{\unicode[STIX]{x2202}v_{2k}}{\unicode[STIX]{x2202}x_{i}}\right)\!\right]n_{1i}n_{1k};\quad \!i,k=1,2,3; & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
the balance of tangential stresses,
$$\begin{eqnarray}\displaystyle & & \displaystyle \left[-\unicode[STIX]{x1D702}_{1}\left(\frac{\unicode[STIX]{x2202}v_{1i}}{\unicode[STIX]{x2202}x_{k}}+\frac{\unicode[STIX]{x2202}v_{1k}}{\unicode[STIX]{x2202}x_{i}}\right)+\unicode[STIX]{x1D702}_{2}\left(\frac{\unicode[STIX]{x2202}v_{2i}}{\unicode[STIX]{x2202}x_{k}}+\frac{\unicode[STIX]{x2202}v_{2k}}{\unicode[STIX]{x2202}x_{i}}\right)\right]\unicode[STIX]{x1D70F}_{1i}^{(l)}n_{1k}\nonumber\\ \displaystyle & & \displaystyle \quad -\,\unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D70F}_{1i}^{(l)}\frac{\unicode[STIX]{x2202}T_{1}}{\unicode[STIX]{x2202}x_{i}}=0;\quad l=1,2;i,k=1,2,3;\end{eqnarray}$$
the continuity of the velocity field,
$$\begin{eqnarray}\displaystyle \boldsymbol{v}_{1}=\boldsymbol{v}_{2}; & & \displaystyle\end{eqnarray}$$
the kinematic equation for the interface motion,
$$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}H_{1}}{\unicode[STIX]{x2202}t}+v_{1x}\frac{\unicode[STIX]{x2202}H_{1}}{\unicode[STIX]{x2202}x}+v_{1y}\frac{\unicode[STIX]{x2202}H_{1}}{\unicode[STIX]{x2202}y}=v_{1z}; & & \displaystyle\end{eqnarray}$$
the continuity of the temperature field,
$$\begin{eqnarray}\displaystyle T_{1}=T_{2}; & & \displaystyle\end{eqnarray}$$
and the balance of normal heat fluxes,
$$\begin{eqnarray}\displaystyle \left(\unicode[STIX]{x1D705}_{1}\frac{\unicode[STIX]{x2202}T_{1}}{\unicode[STIX]{x2202}x_{i}}-\unicode[STIX]{x1D705}_{2}\frac{\unicode[STIX]{x2202}T_{2}}{\unicode[STIX]{x2202}x_{i}}\right)n_{1i}=0. & & \displaystyle\end{eqnarray}$$
Similar boundary conditions are imposed on the deformable interface
$z=H_{2}$
:
$$\begin{eqnarray}\displaystyle & \displaystyle -p_{2}+2\unicode[STIX]{x1D70E}_{2}K_{2}=-\unicode[STIX]{x1D702}_{2}\left(\frac{\unicode[STIX]{x2202}v_{2i}}{\unicode[STIX]{x2202}x_{k}}+\frac{\unicode[STIX]{x2202}v_{2k}}{\unicode[STIX]{x2202}x_{i}}\right)n_{2i}n_{2k}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle -\unicode[STIX]{x1D702}_{2}\left(\frac{\unicode[STIX]{x2202}v_{2i}}{\unicode[STIX]{x2202}x_{k}}+\frac{\unicode[STIX]{x2202}v_{2k}}{\unicode[STIX]{x2202}x_{i}}\right)\unicode[STIX]{x1D70F}_{2i}^{(l)}n_{2k}-\unicode[STIX]{x1D6FC}_{2}\unicode[STIX]{x1D70F}_{2i}^{(l)}\frac{\unicode[STIX]{x2202}T_{3}}{\unicode[STIX]{x2202}x_{i}}=0,\quad l=1,2,i,k=1,2,3, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}H_{2}}{\unicode[STIX]{x2202}t}+v_{2x}\frac{\unicode[STIX]{x2202}H_{2}}{\unicode[STIX]{x2202}x}+v_{2y}\frac{\unicode[STIX]{x2202}H_{2}}{\unicode[STIX]{x2202}y}=v_{2z}. & \displaystyle\end{eqnarray}$$
In the formulae presented above,
$K_{1}$
and
$K_{2}$
are the mean curvatures,
$\boldsymbol{n}_{1}$
and
$\boldsymbol{n}_{2}$
are the normal vectors and
$\unicode[STIX]{x1D749}_{1}^{(l)}$
and
$\unicode[STIX]{x1D749}_{2}^{(l)}$
are the tangential vectors of the lower and upper interfaces. In the quantities with two subscripts, the first subscript corresponds to the number of the liquid (
$m=1,2$
) and the second subscript determines the number of the Cartesian coordinate (
$i,k=1,2,3$
;
$x_{1}=x$
,
$x_{2}=y$
,
$x_{3}=z$
). The usual summation convention is applied. For a heat flux on the liquid–gas interface we use an empirical condition,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}_{2}\frac{\unicode[STIX]{x2202}T_{2}}{\unicode[STIX]{x2202}x_{i}}n_{2i}=-q(T_{2}-T_{g}), & & \displaystyle\end{eqnarray}$$
where
$q$
is the heat exchange coefficient, which is assumed to be constant.
3 Derivation of the long-wave amplitude equations
The system of equations and boundary conditions (2.1)–(2.14) is rather complicated. However, in this paper we will consider the case where the characteristic spatial scale of the temperature modulation of the substrate is much larger than the thickness of the layer, i.e. the temperature modulation depends on the scaled coordinates
$\tilde{X}=\unicode[STIX]{x1D716}x$
and
${\tilde{Y}}=\unicode[STIX]{x1D716}y$
,
$\unicode[STIX]{x1D716}\ll 1$
, rather than on
$x$
and
$y$
. Later on, we assume that the solution of the equations and boundary conditions (2.1)–(2.14) itself depends only on the slow variables,
$\tilde{\boldsymbol{X}}=(\tilde{X},{\tilde{Y}})$
(the reason for this assumption will be discussed below). Moreover, it is assumed that the solution depends on the scaled time variable
$\tilde{\unicode[STIX]{x1D70F}}=\unicode[STIX]{x1D716}^{2}t$
. In this case, the nonlinear model governing three-dimensional flows with a deformable interface can be drastically simplified by means of a long-wavelength expansion. The details of the long-wave approach applied to thermocapillary flows can be found in the review papers of Davis (Reference Davis1987) and Oron, Davis & Bankoff (Reference Oron, Davis and Bankoff1997).
Actually, the long-wave approach applied above is justified only in the case of a strong surface tension. A strong surface tension suppresses short-wave deformations of the surfaces; therefore the instability takes place only in the region of the long waves. Later on, we assume that
$\unicode[STIX]{x1D70E}_{m}=\unicode[STIX]{x1D70E}_{m}^{0}\unicode[STIX]{x1D716}^{-2}$
,
$\unicode[STIX]{x1D70E}_{m}^{0}=O(1)$
,
$m=1,2$
. Moreover, we assume that the dependence of the interfacial tensions on the temperature is relatively weak and can be neglected in the boundary conditions for normal stresses (but not in the boundary conditions for tangential stresses where it is the source of a thermocapillary motion).
The appropriate scaling of variables for a long-scale flow governed by the system (2.1)–(2.14) is as follows:
$$\begin{eqnarray}\displaystyle (v_{mx},v_{my})=\unicode[STIX]{x1D716}\boldsymbol{V}_{m}+o(\unicode[STIX]{x1D716}),\quad v_{mz}=\unicode[STIX]{x1D716}^{2}W_{m}+o(\unicode[STIX]{x1D716}^{2}),\quad p_{m}=P_{m}+o(\unicode[STIX]{x1D716}^{2});\quad m=1,2. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
At the leading order, the evolution of the system is governed by the following equations and boundary conditions:
$$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle P_{1zz}=0;\quad -\tilde{\unicode[STIX]{x1D735}}_{\bot }P_{1}+\unicode[STIX]{x1D702}_{1}\boldsymbol{V}_{1zz}=0;\quad \tilde{\unicode[STIX]{x1D735}}_{\bot }\boldsymbol{\cdot }\boldsymbol{V}_{1}+W_{1z}=0;\\ \displaystyle T_{1zz}=0;\quad 0<z<H_{1}(\tilde{\boldsymbol{X}},\tilde{\unicode[STIX]{x1D70F}});\end{array}\right\} & & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle P_{2zz}=0;\quad -\tilde{\unicode[STIX]{x1D735}}_{\bot }P_{2}+\unicode[STIX]{x1D702}_{2}\boldsymbol{V}_{2zz}=0;\quad \tilde{\unicode[STIX]{x1D735}}_{\bot }\boldsymbol{\cdot }\boldsymbol{V}_{2}+W_{2z}=0;\\ \displaystyle T_{2zz}=0;\quad H_{1}(\tilde{\boldsymbol{X}},\tilde{\unicode[STIX]{x1D70F}})<z<H_{2}(\tilde{\boldsymbol{X}},\tilde{\unicode[STIX]{x1D70F}});\end{array}\right\} & & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle z=0:\boldsymbol{V}_{1}=0;\quad W_{1}=0;\quad T_{1}=T_{s}; & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle z=H_{1}:\boldsymbol{V}_{1}=\boldsymbol{V}_{2};\quad W_{1}=W_{2}; & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle P_{2}-P_{1}-\unicode[STIX]{x1D70E}_{1}^{0}\tilde{\unicode[STIX]{x1D735}}_{\bot }^{2}H_{1}=0; & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D702}_{2}\boldsymbol{V}_{2z}-\unicode[STIX]{x1D702}_{1}\boldsymbol{V}_{1z}-\unicode[STIX]{x1D6FC}_{1}(\tilde{\unicode[STIX]{x1D735}}_{\bot }T_{1}+T_{1z}\tilde{\unicode[STIX]{x1D735}}_{\bot }H_{1})=0; & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle H_{1\tilde{\unicode[STIX]{x1D70F}}}+\boldsymbol{V}_{1}\boldsymbol{\cdot }\tilde{\unicode[STIX]{x1D735}}_{\bot }H_{1}=W_{1}; & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle T_{1}=T_{2};\quad \unicode[STIX]{x1D705}_{1}T_{1z}=\unicode[STIX]{x1D705}_{2}T_{2z}+Q_{\ast }; & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle z=H_{2}:-\unicode[STIX]{x1D702}_{2}\boldsymbol{V}_{2z}-\unicode[STIX]{x1D6FC}_{2}(\tilde{\unicode[STIX]{x1D735}}_{\bot }T_{2}+T_{2z}\tilde{\unicode[STIX]{x1D735}}_{\bot }H_{2})=0; & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle -P_{2}-\unicode[STIX]{x1D70E}_{2}^{0}\tilde{\unicode[STIX]{x1D735}}_{\bot }^{2}H_{2}=0; & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle H_{2\tilde{\unicode[STIX]{x1D70F}}}+\boldsymbol{V}_{2}\boldsymbol{\cdot }\tilde{\unicode[STIX]{x1D735}}_{\bot }H_{2}=W_{2}; & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D705}_{2}T_{2z}=-q(T_{2}-T_{g}). & \displaystyle\end{eqnarray}$$
Here,
$\tilde{\unicode[STIX]{x1D735}}_{\bot }=(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\tilde{X},\unicode[STIX]{x2202}/\unicode[STIX]{x2202}{\tilde{Y}})$
, and the subscripts
$z$
and
$\tilde{\unicode[STIX]{x1D70F}}$
denote partial derivatives with respect to the corresponding variables.
Relations (3.6) and (3.11) determine expressions for the Laplace pressures,
$$\begin{eqnarray}\displaystyle & \displaystyle P_{1}=-\unicode[STIX]{x1D70E}_{1}\tilde{\unicode[STIX]{x1D735}}_{\bot }^{2}H_{1}-\unicode[STIX]{x1D70E}_{2}\tilde{\unicode[STIX]{x1D735}}_{\bot }^{2}H_{2}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle P_{2}=-\unicode[STIX]{x1D70E}_{2}\tilde{\unicode[STIX]{x1D735}}_{\bot }^{2}H_{2}. & \displaystyle\end{eqnarray}$$
Solving the problem for the temperature fields, we find
$$\begin{eqnarray}\displaystyle & \displaystyle T_{1}(\tilde{\boldsymbol{X}},\tilde{\unicode[STIX]{x1D70F}})=T_{s}(\tilde{\boldsymbol{X}})-(T_{s}(\tilde{\boldsymbol{X}})-T_{g})D(\tilde{\boldsymbol{X}},\tilde{\unicode[STIX]{x1D70F}})q\unicode[STIX]{x1D705}_{2}z, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle T_{2}(\tilde{\boldsymbol{X}},\tilde{\unicode[STIX]{x1D70F}})=T_{s}(\tilde{\boldsymbol{X}})-(T_{s}(\tilde{\boldsymbol{X}})-T_{g})D(\tilde{\boldsymbol{X}},\tilde{\unicode[STIX]{x1D70F}})q[(\unicode[STIX]{x1D705}_{2}-\unicode[STIX]{x1D705}_{1})H_{1}(\tilde{\boldsymbol{X}},\tilde{\unicode[STIX]{x1D70F}})+\unicode[STIX]{x1D705}_{1}z], & \displaystyle\end{eqnarray}$$
where
$$\begin{eqnarray}\displaystyle D(\tilde{\boldsymbol{X}},\tilde{\unicode[STIX]{x1D70F}})=[\unicode[STIX]{x1D705}_{1}\unicode[STIX]{x1D705}_{2}+q(\unicode[STIX]{x1D705}_{2}-\unicode[STIX]{x1D705}_{1})H_{1}(\tilde{\boldsymbol{X}},\tilde{\unicode[STIX]{x1D70F}})+q\unicode[STIX]{x1D705}_{1}H_{2}(\tilde{\boldsymbol{X}},\tilde{\unicode[STIX]{x1D70F}})]^{-1}. & & \displaystyle\end{eqnarray}$$
The horizontal components of the flow velocities
$\boldsymbol{V}_{m}$
(
$m=1,2$
), which are generated by the thermocapillary stresses and the gradients of the Laplace pressures, are determined by linear equations and boundary conditions. Therefore, we can present
$\boldsymbol{V}_{m}$
as a sum of two terms
$\boldsymbol{V}_{m}^{T}$
and
$\boldsymbol{V}_{m}^{\unicode[STIX]{x1D70E}}$
, which are related by those two factors respectively. Similarly, the vertical component of the flow velocity
$W_{m}$
can be presented as
$W_{m}=W_{m}^{T}+W_{m}^{\unicode[STIX]{x1D70E}}$
. Let us consider
$(\boldsymbol{V}_{m}^{T},W_{m}^{T})$
and
$(\boldsymbol{V}_{m}^{\unicode[STIX]{x1D70E}},W_{m}^{\unicode[STIX]{x1D70E}})$
separately.
Disregarding Laplace pressures, we obtain the following expressions for the horizontal components of the thermocapillary flows:
$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{V}_{1}^{T}=-\left(\frac{\unicode[STIX]{x1D6FC}_{1}}{\unicode[STIX]{x1D702}_{1}}\tilde{\unicode[STIX]{x1D735}}_{\bot }A+\frac{\unicode[STIX]{x1D6FC}_{2}}{\unicode[STIX]{x1D702}_{1}}\tilde{\unicode[STIX]{x1D735}}_{\bot }B\right)z, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{V}_{2}^{T}=-\frac{\unicode[STIX]{x1D6FC}_{1}}{\unicode[STIX]{x1D702}_{1}}H_{1}\tilde{\unicode[STIX]{x1D735}}_{\bot }A-\unicode[STIX]{x1D6FC}_{2}\left(\frac{z-H_{1}}{\unicode[STIX]{x1D702}_{2}}+\frac{H_{1}}{\unicode[STIX]{x1D702}_{1}}\right)\tilde{\unicode[STIX]{x1D735}}_{\bot }B, & \displaystyle\end{eqnarray}$$
where
$$\begin{eqnarray}\displaystyle & \displaystyle A(\tilde{\boldsymbol{X}},\tilde{\unicode[STIX]{x1D70F}})=T_{s}(\tilde{\boldsymbol{X}})-(T_{s}(\tilde{\boldsymbol{X}})-T_{g})D(\tilde{\boldsymbol{X}},\tilde{\unicode[STIX]{x1D70F}})q\unicode[STIX]{x1D705}_{2}H_{1}(\tilde{\boldsymbol{X}},\tilde{\unicode[STIX]{x1D70F}}), & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle B(\tilde{\boldsymbol{X}},\tilde{\unicode[STIX]{x1D70F}})=T_{s}(\tilde{\boldsymbol{X}})-(T_{s}(\tilde{\boldsymbol{X}})-T_{g})D(\tilde{\boldsymbol{X}})q[(\unicode[STIX]{x1D705}_{2}-\unicode[STIX]{x1D705}_{1})H_{1}(\tilde{\boldsymbol{X}})+\unicode[STIX]{x1D705}_{1}H_{2}(\tilde{\boldsymbol{X}})]. & \displaystyle\end{eqnarray}$$
Solving the continuity equations with respect to
$W_{1}$
and
$W_{2}$
with corresponding boundary conditions, we find that
$$\begin{eqnarray}\displaystyle & \displaystyle W_{1}^{T}(\tilde{\boldsymbol{X}},\tilde{\unicode[STIX]{x1D70F}})=-\int _{0}^{H_{1}}\tilde{\unicode[STIX]{x1D735}}_{\bot }\boldsymbol{\cdot }\boldsymbol{V}_{1}\,\text{d}z, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle W_{2}^{T}(\tilde{\boldsymbol{X}},\tilde{\unicode[STIX]{x1D70F}})=-\int _{0}^{H_{1}}\tilde{\unicode[STIX]{x1D735}}_{\bot }\boldsymbol{\cdot }\boldsymbol{V}_{1}\,\text{d}z-\int _{H_{1}}^{H_{2}}\tilde{\unicode[STIX]{x1D735}}_{\bot }\boldsymbol{\cdot }\boldsymbol{V}_{2}\,\text{d}z. & \displaystyle\end{eqnarray}$$
Substituting the expressions for the flow velocities obtained above into equations (3.8) and (3.12), we arrive at a closed system of equations that governs the evolution of a heated two-layer film under the action of the thermocapillary effect:
$$\begin{eqnarray}\displaystyle H_{1\tilde{\unicode[STIX]{x1D70F}}}+\tilde{\unicode[STIX]{x1D735}}_{\bot }\boldsymbol{\cdot }\boldsymbol{Q}_{1}^{T}=0,\quad H_{2\tilde{\unicode[STIX]{x1D70F}}}+\tilde{\unicode[STIX]{x1D735}}_{\bot }\boldsymbol{\cdot }\boldsymbol{Q}_{2}^{T}=0. & & \displaystyle\end{eqnarray}$$
The expressions for the fluxes produced by the thermocapillary effect are as follows:
$$\begin{eqnarray}\displaystyle \boldsymbol{Q}_{1}^{T} & = & \displaystyle -\frac{\unicode[STIX]{x1D6FC}_{1}H_{1}^{2}}{2\unicode[STIX]{x1D702}_{1}}\tilde{\unicode[STIX]{x1D735}}_{\bot }A-\frac{\unicode[STIX]{x1D6FC}_{2}H_{1}^{2}}{2\unicode[STIX]{x1D702}_{1}}\unicode[STIX]{x1D705}_{1}\unicode[STIX]{x1D705}_{2}\tilde{\unicode[STIX]{x1D735}}_{\bot }C,\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \boldsymbol{Q}_{2}^{T} & = & \displaystyle -\frac{\unicode[STIX]{x1D6FC}_{1}H_{1}(2H_{2}-H_{1})}{2\unicode[STIX]{x1D702}_{1}}\tilde{\unicode[STIX]{x1D735}}_{\bot }A\nonumber\\ \displaystyle & & \displaystyle -\,\frac{\unicode[STIX]{x1D6FC}_{2}\unicode[STIX]{x1D705}_{1}\unicode[STIX]{x1D705}_{2}}{2\unicode[STIX]{x1D702}_{1}\unicode[STIX]{x1D702}_{2}}[H_{2}^{2}\unicode[STIX]{x1D702}_{1}+(2H_{2}-H_{1})H_{1}(\unicode[STIX]{x1D702}_{2}-\unicode[STIX]{x1D702}_{1})]\tilde{\unicode[STIX]{x1D735}}_{\bot }C,\end{eqnarray}$$
where
$$\begin{eqnarray}\displaystyle C=(T_{s}-T_{g})D. & & \displaystyle\end{eqnarray}$$
Equations (3.25) were formerly derived by Nepomnyashchy & Simanovskii (Reference Nepomnyashchy and Simanovskii2007) in the case
$T_{s}=\text{const}$
.
Similarly, disregarding the thermocapillary stresses, one can calculate the velocities
$(\boldsymbol{V}_{m}^{\unicode[STIX]{x1D70E}},W_{m}^{\unicode[STIX]{x1D70E}})$
,
$m=1,2$
, and the fluxes
$\boldsymbol{Q}_{1}^{\unicode[STIX]{x1D70E}}$
,
$\boldsymbol{Q}_{2}^{\unicode[STIX]{x1D70E}}$
. Because these fluxes were computed formerly by Pototsky et al. (Reference Pototsky, Bestehorn, Merkt and Thiele2004), Bandyopadhyay, Gulabani & Sharma (Reference Bandyopadhyay, Gulabani and Sharma2005) and Fisher & Golovin (Reference Fisher and Golovin2005), we present here only the final result:
$$\begin{eqnarray}\displaystyle \boldsymbol{Q}_{1}^{\unicode[STIX]{x1D70E}}=F_{11}\tilde{\unicode[STIX]{x1D735}}_{\bot }P_{1}+F_{12}\tilde{\unicode[STIX]{x1D735}}_{\bot }P_{2},\quad \boldsymbol{Q}_{2}^{\unicode[STIX]{x1D70E}}=F_{21}\tilde{\unicode[STIX]{x1D735}}_{\bot }P_{1}+F_{22}\tilde{\unicode[STIX]{x1D735}}_{\bot }P_{2}, & & \displaystyle\end{eqnarray}$$
where the pressures
$P_{1}$
and
$P_{2}$
are determined by the expressions (3.14) and (3.15), and the mobility functions are
$$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle F_{11}=-\frac{1}{3\unicode[STIX]{x1D702}_{1}}H_{1}^{3};\quad F_{12}=-\frac{1}{2\unicode[STIX]{x1D702}_{1}}H_{1}^{2}(H_{2}-H_{1});\quad F_{21}=\frac{1}{6\unicode[STIX]{x1D702}_{1}}H_{1}^{3}-\frac{1}{2\unicode[STIX]{x1D702}_{1}}H_{1}^{2}H_{2};\\ \displaystyle F_{22}=(H_{2}-H_{1})\left[H_{1}^{2}\left(\frac{1}{2\unicode[STIX]{x1D702}_{1}}-\frac{1}{3\unicode[STIX]{x1D702}_{2}}\right)+H_{1}H_{2}\left(-\frac{1}{\unicode[STIX]{x1D702}_{1}}+\frac{2}{3\unicode[STIX]{x1D702}_{2}}\right)-\frac{1}{3\unicode[STIX]{x1D702}_{2}}H_{2}^{2}\right].\end{array}\right\}\qquad & & \displaystyle\end{eqnarray}$$
Finally, we obtain the following evolution equations:
$$\begin{eqnarray}\displaystyle H_{1\tilde{\unicode[STIX]{x1D70F}}}+\tilde{\unicode[STIX]{x1D735}}_{\bot }\boldsymbol{\cdot }(\boldsymbol{Q}_{1}^{T}+\boldsymbol{Q}_{1}^{\unicode[STIX]{x1D70E}})=0,\quad H_{2\tilde{\unicode[STIX]{x1D70F}}}+\tilde{\unicode[STIX]{x1D735}}_{\bot }\boldsymbol{\cdot }(\boldsymbol{Q}_{2}^{T}+\boldsymbol{Q}_{2}^{\unicode[STIX]{x1D70E}})=0. & & \displaystyle\end{eqnarray}$$
Let us transform equations (3.31) to a non-dimensional form. The natural vertical length scale is the equilibrium thickness of the lower layer,
$H_{1}^{0}$
. The choice of the horizontal length scale,
$L^{\ast }$
, is arbitrary (see Nepomnyashchy & Simanovskii Reference Nepomnyashchy and Simanovskii2012). We choose
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70F}^{\ast }=\frac{\unicode[STIX]{x1D702}_{1}(L^{\ast })^{4}}{\unicode[STIX]{x1D70E}_{1}^{0}(H_{1}^{0})^{3}} & & \displaystyle\end{eqnarray}$$
as a time scale and
$$\begin{eqnarray}\displaystyle p^{\ast }=\frac{\unicode[STIX]{x1D70E}_{1}^{0}H_{1}^{0}}{(L^{\ast })^{2}} & & \displaystyle\end{eqnarray}$$
as a pressure scale.
The non-dimensional parameters of the problem are the Biot number,
$$\begin{eqnarray}\displaystyle Bi=\frac{qH_{1}^{0}}{\unicode[STIX]{x1D705}_{2}}, & & \displaystyle\end{eqnarray}$$
$\unicode[STIX]{x1D702}=\unicode[STIX]{x1D702}_{1}/\unicode[STIX]{x1D702}_{2}$
,
$\unicode[STIX]{x1D705}=\unicode[STIX]{x1D705}_{1}/\unicode[STIX]{x1D705}_{2}$
,
$\unicode[STIX]{x1D70E}=\unicode[STIX]{x1D70E}_{2}^{0}/\unicode[STIX]{x1D70E}_{1}^{0}$
and
$\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}_{2}/\unicode[STIX]{x1D6FC}_{1}$
. We define non-dimensional variables
$$\begin{eqnarray}\displaystyle \boldsymbol{X}=\tilde{\boldsymbol{X}}/L^{\ast },\quad \unicode[STIX]{x1D70F}=\tilde{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D70F}^{\ast },\quad h_{j}=H_{j}/H_{1}^{0},\quad \unicode[STIX]{x03C0}_{j}=P_{j}/p^{\ast },\quad j=1,2. & & \displaystyle\end{eqnarray}$$
We introduce also the local modified Marangoni number,
$$\begin{eqnarray}\displaystyle M(\boldsymbol{X})=\frac{\unicode[STIX]{x1D6FC}_{1}(T_{s}(\boldsymbol{X})-T_{g})}{\unicode[STIX]{x1D70E}_{1}^{0}}\left(\frac{L^{\ast }}{H_{1}^{0}}\right)^{2} & & \displaystyle\end{eqnarray}$$
and
$$\begin{eqnarray}\displaystyle d(\boldsymbol{X},\unicode[STIX]{x1D70F})=[\unicode[STIX]{x1D705}+Bi(1-\unicode[STIX]{x1D705})h_{1}(\boldsymbol{X},\unicode[STIX]{x1D70F})+Bi\unicode[STIX]{x1D705}h_{2}(\boldsymbol{X},\unicode[STIX]{x1D70F})]^{-1}. & & \displaystyle\end{eqnarray}$$
Equations (3.31) written in the non-dimensional form look as follows:
$$\begin{eqnarray}\displaystyle h_{1\unicode[STIX]{x1D70F}}+\unicode[STIX]{x1D735}_{\bot }\boldsymbol{\cdot }\boldsymbol{q}_{1}=0,\quad h_{2\unicode[STIX]{x1D70F}}+\unicode[STIX]{x1D735}_{\bot }\boldsymbol{\cdot }\boldsymbol{q}_{2}=0, & & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \boldsymbol{q}_{1}=f_{11}\unicode[STIX]{x1D735}_{\bot }p_{1}+f_{12}\unicode[STIX]{x1D735}_{\bot }p_{2}+\boldsymbol{q}_{1}^{T},\quad \boldsymbol{q}_{2}=f_{21}\unicode[STIX]{x1D735}_{\bot }p_{1}+f_{22}\unicode[STIX]{x1D735}_{\bot }p_{2}+\boldsymbol{q}_{2}^{T}, & & \displaystyle\end{eqnarray}$$
where
$$\begin{eqnarray}\displaystyle f_{11}=-{\textstyle \frac{1}{3}}h_{1}^{3},\quad f_{12}=-{\textstyle \frac{1}{2}}h_{1}^{2}(h_{2}-h_{1}), & & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle f_{21}=\frac{1}{6}h_{1}^{3}-\frac{1}{2}h_{1}^{2}h_{2},\quad f_{22}=(h_{2}-h_{1})\left[h_{1}^{2}\left(\frac{1}{2}-\frac{\unicode[STIX]{x1D702}}{3}\right)+h_{1}h_{2}\left(-1+\frac{2\unicode[STIX]{x1D702}}{3}\right)-\frac{\unicode[STIX]{x1D702}}{3}h_{2}^{2}\right]. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
The capillary pressures are
$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x03C0}_{1}=-\unicode[STIX]{x1D6FB}_{\bot }^{2}h_{1}-\unicode[STIX]{x1D70E}\unicode[STIX]{x1D6FB}_{\bot }^{2}h_{2}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x03C0}_{2}=-\unicode[STIX]{x1D70E}\unicode[STIX]{x1D6FB}_{\bot }^{2}h_{2}. & \displaystyle\end{eqnarray}$$
The non-dimensional expressions for the fluxes generated by the thermocapillary effect are
$$\begin{eqnarray}\displaystyle \boldsymbol{q}_{1}^{T} & = & \displaystyle -\frac{h_{1}^{2}}{2}\unicode[STIX]{x1D735}_{\bot }\{M[1+d(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D705}-Bih_{1})]\},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \boldsymbol{q}_{2}^{T} & = & \displaystyle -\frac{\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D705}}{2}h_{2}^{2}\unicode[STIX]{x1D735}_{\bot }(Md)\nonumber\\ \displaystyle & & \displaystyle +\,\frac{(2h_{2}-h_{1})h_{1}}{2}\unicode[STIX]{x1D735}_{\bot }\{M[-1+Bih_{1}d-\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D705}(1-\unicode[STIX]{x1D702})d]\}.\end{eqnarray}$$
4 Nonlinear simulations
4.1 Methodology
We have performed nonlinear simulations of equations (3.38) with a spatially periodic modulation of the Marangoni number,
$$\begin{eqnarray}\displaystyle M(X+L,Y)=M(X,Y+L)=M(X,Y). & & \displaystyle\end{eqnarray}$$
Let us emphasize that
$L$
is a non-dimensional parameter; the dimensional modulation period is
$\tilde{L}=LL^{\ast }$
. Equations (3.38)–(3.45) have been discretized by central differences for spatial derivatives and solved using an explicit scheme. Initial conditions for
$h_{j}$
,
$j=1,2$
, have been chosen in such a way that the mean value of
$h_{1}(X,Y,0)$
is equal to 1 and the mean value of
$h_{2}(X,Y,0)$
is equal to
$h$
, where
$h>1$
. It should be noted that the solutions of the problem depend on the additional geometric parameter,
$h=H_{2}^{0}/H_{1}^{0}$
. Small random deviations of
$h_{j}(X,Y,0)$
from their mean values were imposed using a code creating pseudo-random numbers. The computations were performed in the region
$L\times L=240\times 240$
with periodic boundary conditions using a grid of
$80\times 80$
. In this paper, we consider the system of fluorinert FC70 (liquid 1) and silicone oil 10 (liquid 2) formerly used in microgravity experiments (see, e.g., Géoris et al.
Reference Géoris, Hennenberg, Lebon and Legros1999). It is characterized by the following set of parameters:
$\unicode[STIX]{x1D702}=3.04$
,
$\unicode[STIX]{x1D705}=0.522$
,
$\unicode[STIX]{x1D6FC}=2$
,
$\unicode[STIX]{x1D70C}=0.482$
,
$\unicode[STIX]{x1D70E}=2.6$
. The computations are carried out for
$h=2.5$
and
$Bi=10$
.
The primary analysis of the obtained nonlinear regimes has been performed using snapshots of the fields of
$h_{j}(X,Y,\unicode[STIX]{x1D70F}),$
$j=1,2.$
This analysis has been supplemented by the investigation of the Fourier components
$$\begin{eqnarray}\displaystyle & \displaystyle c_{mn}(\unicode[STIX]{x1D70F})=\frac{2}{L^{2}}\int _{0}^{L}\int _{0}^{L}h_{1}(X,Y,\unicode[STIX]{x1D70F})\cos \left[\frac{2\unicode[STIX]{x03C0}}{L}(mX+nY)\right]\,\text{d}X\,\text{d}Y, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle s_{mn}(\unicode[STIX]{x1D70F})=\frac{2}{L^{2}}\int _{0}^{L}\int _{0}^{L}h_{1}(X,Y,\unicode[STIX]{x1D70F})\sin \left[\frac{2\unicode[STIX]{x03C0}}{L}(mX+nY)\right]\,\text{d}X\,\text{d}Y, & \displaystyle\end{eqnarray}$$
where
$m$
,
$n$
are integer numbers.
We have used also variables
$$\begin{eqnarray}\displaystyle r_{mn}(\unicode[STIX]{x1D70F})=\sqrt{c_{mn}^{2}(\unicode[STIX]{x1D70F})+s_{mn}^{2}(\unicode[STIX]{x1D70F})}, & & \displaystyle\end{eqnarray}$$
characterizing the amplitudes of the corresponding complex Fourier harmonics, and quantities
$$\begin{eqnarray}\displaystyle h_{max,j}(\unicode[STIX]{x1D70F})=\max _{X,Y}h_{j}(X,Y,\unicode[STIX]{x1D70F}),\quad j=1,2, & & \displaystyle\end{eqnarray}$$
which describe the deformations of the surfaces.
4.2 One-dimensional spatial modulation with period
$L$
In this subsection, we discuss the nonlinear regimes observed in the case of a one-dimensional modulation of the local Marangoni number,
$$\begin{eqnarray}\displaystyle M(\boldsymbol{X})=M\left(1+\unicode[STIX]{x1D6FF}_{X}\sin \frac{X}{L}\right)=M-\unicode[STIX]{x1D6E5}_{X}\sin \frac{X}{L}, & & \displaystyle\end{eqnarray}$$
where
$M<0$
,
$\unicode[STIX]{x1D6FF}_{X}>0$
and
$\unicode[STIX]{x1D6E5}_{X}=|M|\unicode[STIX]{x1D6FF}_{X}$
. The diagram of regimes is shown in figure 2.
Figure 2. General diagram of regimes on the plane
$(\unicode[STIX]{x1D6E5}_{X},M)$
: black triangle, mechanical equilibrium; white circle, time-periodic three-dimensional standing wave; white triangle, quasiperiodic three-dimensional standing wave; black square, two-dimensional standing wave; white square, two-dimensional stationary pattern; black circle, three-dimensional travelling wave; white diamond, time-periodic standing wave with spatial period
$L/2$
; big star, travelling wave with spatial period
$L/2$
; black diamond, quasiperiodic wave with spatial period
$L/2$
; asterisk, standing wave with spatial period
$L/3$
.
Figure 3. Oscillations of
$s_{10}(\unicode[STIX]{x1D70F})$
(solid line) and
$s_{01}$
(dashed line) for
$M=-2$
,
$\unicode[STIX]{x1D6E5}_{X}=0$
.
4.2.1 Nonlinear regime in the absence of modulation
The stability of the mechanical equilibrium, and the nonlinear flow regimes in the absence of the substrate temperature modulation (
$\unicode[STIX]{x1D6E5}_{X}=0$
) have been studied by Nepomnyashchy & Simanovskii (Reference Nepomnyashchy and Simanovskii2012). For
$Bi=10$
, the mechanical equilibrium state is unstable with respect to oscillatory disturbances by heating from above. For the region
$L\times L=240\times 240$
with periodic boundary conditions, the linear stability theory predicts the instability threshold
$M_{\ast }\approx 1.02$
. Below the threshold, the system tends to the mechanical equilibrium state. Just above the threshold, the nonlinear development of the instability creates a two-dimensional time-periodic standing wave with the wavevectors directed either along the
$X$
axis or along the
$Y$
axis (see the black square in figure 2). With the growth of
$|M|$
, we observe a transition to the wavy pattern known as alternating rolls. This motion is a superposition of two periodic standing waves with the wavevectors directed along the axes
$X$
,
$Y$
. The oscillations of waves with orthogonal wavevectors have a mutual phase shift corresponding to a quarter of the period,
$T/4$
(see figure 3). The mean values of the Fourier components
$s_{10}(\unicode[STIX]{x1D70F})$
,
$c_{10}(\unicode[STIX]{x1D70F})$
,
$s_{01}(\unicode[STIX]{x1D70F})$
,
$c_{01}(\unicode[STIX]{x1D70F})$
are equal to zero; the amplitudes of both standing waves are equal,
$$\begin{eqnarray}\displaystyle \max _{\unicode[STIX]{x1D70F}}r_{10}(\unicode[STIX]{x1D70F})=\max _{\unicode[STIX]{x1D70F}}r_{01}(\unicode[STIX]{x1D70F}). & & \displaystyle\end{eqnarray}$$
The quantities
$h_{max,j}(\unicode[STIX]{x1D70F})$
,
$j=1,2$
, oscillate with the period
$T/4$
(see figure 4).
4.2.2 Two-dimensional stationary flow
As mentioned above, in the absence of modulation no flow is observed in the subcritical region
$|M|<M_{\ast }$
. The inhomogeneity of the substrate temperature generates a two-dimensional stationary flow (3.19), (3.20), and that flow creates a two-dimensional stationary deformation of the surfaces
$h_{j}=h_{j}(x)$
,
$j=1,2$
. For this solution,
$c_{10}=0$
and
$s_{10}=\text{const}$
; all
$c_{mn}$
,
$s_{mn}$
with
$n\neq 0$
are equal to zero. In figure 2, this regime occupies the region indicated by white squares. For moderate values of
$\unicode[STIX]{x1D6E5}_{X}$
, the stationary flow generated by temperature modulation is more stable than the mechanical equilibrium in the absence of modulation: it is observed up to
$|M|\approx 3$
. Isolines of
$h_{2}(X,Y)-h$
for this flow are shown in figure 5.
Figure 4. Oscillations of
$h_{max,1}(\unicode[STIX]{x1D70F})$
(dashed line) and
$h_{max,2}(\unicode[STIX]{x1D70F})$
(solid line) for
$M=-2$
,
$\unicode[STIX]{x1D6E5}_{X}=0$
.
Figure 5. Isolines of
$h_{2}(X,Y)-h$
;
$M=-2$
,
$\unicode[STIX]{x1D6E5}_{X}=0.5$
.
4.2.3 Two-dimensional standing waves
The instability of a two-dimensional stationary flow can lead to the development of two-dimensional standing waves. In this regime,
$s_{10}(\unicode[STIX]{x1D70F})$
oscillates with a non-zero mean value, while
$c_{mn}$
,
$s_{mn}$
with
$n\neq 0$
are equal to zero (see figure 6).
The quantities
$h_{max,j}$
,
$j=1,2$
, oscillate with the period
$T$
, and they have two maxima during the period (see figure 7).
It should be noted that the selection of the phase of the standing wave (
$c_{10}(\unicode[STIX]{x1D70F})\rightarrow 0$
) is an extremely slow process. This process is shown in figure 8 for
$\unicode[STIX]{x1D6E5}_{X}=0.2$
. For smaller values of
$\unicode[STIX]{x1D6E5}_{X}$
, it is even slower.
Figure 6. The temporal dependence of
$s_{10}(\unicode[STIX]{x1D70F})$
(solid line) and
$s_{01}(\unicode[STIX]{x1D70F})$
(dashed line) for
$M=-2$
,
$\unicode[STIX]{x1D6E5}_{X}=0.2$
.
Figure 7. Oscillations of
$h_{max,2}(\unicode[STIX]{x1D70F})$
for
$M=-2$
,
$\unicode[STIX]{x1D6E5}_{X}=0.2$
.
Figure 8. Decay of
$c_{10}(\unicode[STIX]{x1D70F})$
for
$M=-2$
,
$\unicode[STIX]{x1D6E5}_{X}=0.2$
.
Figure 9. Oscillations of
$s_{10}(\unicode[STIX]{x1D70F})$
(solid line) and
$s_{01}(\unicode[STIX]{x1D70F})$
(dashed line) for
$M=-3$
,
$\unicode[STIX]{x1D6E5}_{X}=0.15$
.
The region of two-dimensional standing waves is indicated in figure 2 by black squares.
4.2.4 Three-dimensional standing waves
The most typical nonlinear regime at moderate values of
$\unicode[STIX]{x1D6E5}_{X}$
is the regime of three-dimensional standing waves, which are developed due to the influence of modulation on the alternating rolls.
The non-uniformity of heating determined by (4.1) breaks (i) the translational symmetry along the
$X$
axis and (ii) the symmetry between standing waves with wavevectors directed along the
$X$
and
$Y$
axes. While for
$\unicode[STIX]{x1D6E5}_{X}=0$
the phase of the standing wave
$(s_{10}(\unicode[STIX]{x1D70F}),c_{10}(\unicode[STIX]{x1D70F}))$
,
$\unicode[STIX]{x1D719}=\tan ^{-1}(c_{10}/s_{10})$
, is arbitrary, in the case
$\unicode[STIX]{x1D6E5}_{X}\neq 0$
a definite value of
$\unicode[STIX]{x1D719}$
is selected. Typically,
$c_{10}(\unicode[STIX]{x1D70F})\rightarrow 0$
, i.e.
$\unicode[STIX]{x1D719}=0$
is selected, but the transition to the equilibrium value of the phase is a very slow process. The imposed temperature non-uniformity generates a thermocapillary flow, which creates a certain mean deformation of surfaces. Therefore, the mean value of
$s_{10}(\unicode[STIX]{x1D70F})$
is non-zero (see figure 9), and it increases with the growth of
$\unicode[STIX]{x1D6E5}_{X}$
. The phase of the standing wave
$(s_{01}(\unicode[STIX]{x1D70F}),c_{01}(\unicode[STIX]{x1D70F}))$
is arbitrary, the mean values of
$s_{01}(\unicode[STIX]{x1D70F})$
and
$c_{01}(\unicode[STIX]{x1D70F})$
are equal to zero (figure 9) and the amplitude of this standing wave decreases with the growth of
$\unicode[STIX]{x1D6E5}_{X}$
.
The phase trajectory in the plane
$(r_{10},r_{01})$
is shown in figure 10. One can see that each of the functions
$r_{10}(\unicode[STIX]{x1D70F})$
,
$r_{01}(\unicode[STIX]{x1D70F})$
has two different local maxima during the period. The quantities
$h_{max,j}$
,
$j=1,2$
, oscillate now with the period
$T$
, but they have four non-equal maxima during the period (see figure 11).
The regime of time-periodic three-dimensional standing waves described above is observed in a large region in the plane
$(\unicode[STIX]{x1D6E5}_{X},M)$
(see figure 2, white circles).
With the growth of
$\unicode[STIX]{x1D6E5}_{X}$
, the frequencies of standing waves with the wavevectors directed along the
$X$
and
$Y$
axes become different from each other (see figure 12). The region of these quasiperiodic oscillations is shown in figure 2 by white triangles.
Figure 10. Phase trajectory in the plane
$(r_{10},r_{01})$
for
$M=-3$
,
$\unicode[STIX]{x1D6E5}_{X}=0.15$
.
Figure 11. Oscillations of
$h_{max,2}(\unicode[STIX]{x1D70F})$
for
$M=-3$
,
$\unicode[STIX]{x1D6E5}_{X}=0.15$
.
Figure 12. Oscillations of
$s_{10}(\unicode[STIX]{x1D70F})$
(solid line) and
$s_{01}$
(dashed line) for
$M=-4$
,
$\unicode[STIX]{x1D6E5}_{X}=0.7$
.
Figure 13. Snapshots of the isolines of (a)
$h_{2}(X,Y,\unicode[STIX]{x1D70F})-h$
and (b)
$h_{1}(X,Y,\unicode[STIX]{x1D70F})-1$
;
$M=-2$
,
$\unicode[STIX]{x1D6E5}_{X}=0.7$
.
4.2.5 Three-dimensional travelling waves
When
$\unicode[STIX]{x1D6E5}_{X}$
increases, the stationary flow described in § 4.2.2 becomes unstable with respect to three-dimensional oscillatory disturbances with the period
$L$
in the
$Y$
-direction. The instability creates travelling waves propagating in the
$Y$
-direction:
$$\begin{eqnarray}\displaystyle h_{j}(X,Y,\unicode[STIX]{x1D70F})=h_{j}^{\pm }(X,Y\pm c\unicode[STIX]{x1D70F}), & & \displaystyle\end{eqnarray}$$
where
$c$
is the phase velocity of the wave (see figure 13).
For the travelling wave solutions,
$c_{10}=0$
,
$s_{10}=const$
,
$r_{01}^{2}=c_{01}^{2}(\unicode[STIX]{x1D70F})+s_{01}^{2}(\unicode[STIX]{x1D70F})=const$
(see figure 14).
4.2.6 Re-entrant three-dimensional standing waves
At higher values of
$\unicode[STIX]{x1D6E5}_{X}$
, we have observed the transition to the regime of standing waves with almost proportional
$c_{01}(\unicode[STIX]{x1D70F})$
and
$s_{01}(\unicode[STIX]{x1D70F})$
. Through the nonlinear terms in the equations, the
$T$
-periodic oscillations of
$c_{01}(\unicode[STIX]{x1D70F})$
and
$s_{01}(\unicode[STIX]{x1D70F})$
induce oscillations of
$s_{10}$
with the period
$T/2$
(see figure 15).
Figure 14. The phase trajectory in the plane
$s_{01}$
,
$c_{01}$
;
$M=-2$
,
$\unicode[STIX]{x1D6E5}_{X}=0.75$
.
Figure 15. Oscillations of
$s_{10}(\unicode[STIX]{x1D70F})$
(solid line) and
$s_{01}(\unicode[STIX]{x1D70F})$
(dashed line) for
$M=-2$
,
$\unicode[STIX]{x1D6E5}_{X}=1$
. The indicated values of time correspond to
$\unicode[STIX]{x1D70F}-2\times 10^{6}$
.
Figure 16. Snapshots of the isolines of
$h_{2}(X,Y,\unicode[STIX]{x1D70F})-h$
: (a)
$\unicode[STIX]{x1D70F}=2\,300\,640$
, (b)
$\unicode[STIX]{x1D70F}=2\,302\,400$
, (c)
$\unicode[STIX]{x1D70F}=2\,308\,000$
, (d)
$\unicode[STIX]{x1D70F}=2\,308\,800$
, (e)
$\unicode[STIX]{x1D70F}=2\,310\,200$
, (f)
$\unicode[STIX]{x1D70F}=2\,312\,400$
;
$M=-2$
,
$\unicode[STIX]{x1D6E5}_{X}=1$
.
Figure 17. The phase trajectory in the plane
$s_{02}$
,
$c_{02}$
;
$M=-4$
,
$\unicode[STIX]{x1D6E5}_{X}=1$
.
The snapshots of the isolines of
$h_{2}(X,Y,\unicode[STIX]{x1D70F})-h$
are shown in figure 16.
4.2.7 Short-scale three-dimensional structures
For larger values of
$\unicode[STIX]{x1D6E5}_{X}$
, the stationary two-dimensional flow becomes unstable with respect to three-dimensional oscillatory disturbances with the spatial period
$L/2$
in the
$Y$
-direction. Its nonlinear development creates a time-periodic regime of standing waves with
$c_{02}$
proportional to
$s_{02}$
(see figure 17). Due to the cubic nonlinearity in the equation, oscillations of
$c_{02}$
and
$s_{02}$
generate oscillations of
$s_{10}$
with a doubled frequency, i.e. with a temporal period that is two times smaller than that of
$(c_{02},s_{02})$
(see figure 18). The snapshots of the isolines for this regime are shown in figures 19 and 20. It should be noted that the fields of
$h_{j}(X,Y)$
are symmetric with respect to some horizontal axes. The shapes of the upper surface and the interface between the liquids are presented in figure 21.
At larger values of
$\unicode[STIX]{x1D6FF}_{X}=\unicode[STIX]{x1D6E5}_{X}/M$
, we observe travelling waves similar to those described in § 4.2.5 (see formula (4.8)), but with the spatial period
$L/2$
in the
$Y$
-direction. The symmetry of
$h_{j}(X,Y)$
with respect to horizontal axes is broken. The snapshots of the waves moving in the direction opposite to the direction of the
$Y$
axis are shown in figure 22.
Figure 18. The phase trajectory in the plane
$s_{10}$
,
$s_{02}$
;
$M=-4$
,
$\unicode[STIX]{x1D6E5}_{X}=1$
.
Figure 19. Snapshots of the isolines of
$h_{2}(X,Y,\unicode[STIX]{x1D70F})-h$
: (a)
$\unicode[STIX]{x1D70F}=2\,199\,805$
, (b)
$\unicode[STIX]{x1D70F}=2\,200\,100$
, (c)
$\unicode[STIX]{x1D70F}=2\,200\,250$
, (d)
$\unicode[STIX]{x1D70F}=2\,201\,050$
;
$M=-4$
,
$\unicode[STIX]{x1D6E5}_{X}=1$
.
Figure 20. Snapshots of the isolines of
$h_{1}(X,Y,\unicode[STIX]{x1D70F})-1$
: (a)
$\unicode[STIX]{x1D70F}=2\,199\,805$
, (b)
$\unicode[STIX]{x1D70F}=2\,200\,100$
, (c)
$\unicode[STIX]{x1D70F}=2\,200\,250$
, (d)
$\unicode[STIX]{x1D70F}=2\,201\,050$
;
$M=-4$
,
$\unicode[STIX]{x1D6E5}_{X}=1$
.
Figure 21. Shapes of (a) the upper surface
$z=h_{2}(X,Y,\unicode[STIX]{x1D70F})$
and (b) the interface
$z=h_{1}(X,Y,\unicode[STIX]{x1D70F})$
;
$M=-4$
,
$\unicode[STIX]{x1D6E5}_{X}=1$
.
Figure 22. Snapshots of the isolines of (a)
$h_{2}(X,Y,\unicode[STIX]{x1D70F})-h$
and (b)
$h_{1}(X,Y,\unicode[STIX]{x1D70F})-1$
;
$M=-2$
,
$\unicode[STIX]{x1D6E5}_{X}=0.7$
.
Figure 23. The phase trajectory in the plane
$r_{10}$
,
$r_{02}$
;
$M=-8$
,
$\unicode[STIX]{x1D6E5}_{X}=2$
.
In a certain region of parameters, we observe a wave that can be considered as an intermediate structure between a standing wave and a travelling wave. Similarly to the case of a standing wave,
$r_{02}=\sqrt{c_{02}^{2}+s_{02}^{2}}$
and
$r_{10}=|s_{10}|$
(note that
$c_{10}=0$
) oscillate periodically with the same temporal period
$T$
(see figure 23), but now
$r_{02}$
is never equal to 0. The fields
$h_{j}(X,Y,\unicode[STIX]{x1D70F})$
, which have a spatial period
$L/2$
in the
$Y$
-direction (see figures 24 and 25), satisfy the relation
$$\begin{eqnarray}\displaystyle h_{j}(X,Y,\unicode[STIX]{x1D70F}+T)=h_{j}(X,Y+l,\unicode[STIX]{x1D70F}), & & \displaystyle\end{eqnarray}$$
where
$l$
and
$L$
are generally non-commensurate. Therefore,
$h_{max,j}(\unicode[STIX]{x1D70F})$
are periodic functions (see figure 26), while
$c_{02}(\unicode[STIX]{x1D70F})$
and
$s_{02}(\unicode[STIX]{x1D70F})$
are quasiperiodic functions of
$\unicode[STIX]{x1D70F}$
(see figure 26).
Figure 24. Snapshots of the isolines of
$h_{2}(X,Y,\unicode[STIX]{x1D70F})-h$
;
$M=-8$
,
$\unicode[STIX]{x1D6E5}_{X}=2$
. (a)
$\unicode[STIX]{x1D70F}=2\,050\,000$
; (b)
$\unicode[STIX]{x1D70F}=2\,050\,420$
; (c)
$\unicode[STIX]{x1D70F}=2\,050\,620$
; (d)
$\unicode[STIX]{x1D70F}=2\,051\,340$
; (e)
$\unicode[STIX]{x1D70F}=2\,051\,600$
; (f)
$\unicode[STIX]{x1D70F}=2\,051\,980$
.
For
$M=-6$
,
$\unicode[STIX]{x1D6E5}_{X}=3$
we have observed a structure with a spatial period
$L/3$
in the
$Y$
-direction (see figures 28–30). The dynamics of the regime is fully similar to that found for
$M=-4$
,
$\unicode[STIX]{x1D6E5}_{X}=1$
:
$c_{03}$
and
$s_{03}$
are proportional, and
$s_{10}$
oscillates with a doubled frequency with respect to that of
$(c_{03},s_{03})$
.
The existence regions of the regimes described above are shown in figure 2.
4.3 Two-dimensional spatial modulation with period
$L$
Let us discuss now the modification of the basic dynamical regimes described above in the presence of a two-dimensional spatial temperature modulation,
$$\begin{eqnarray}\displaystyle M(\boldsymbol{X})=M\left(1+\unicode[STIX]{x1D6FF}_{X}\sin \frac{X}{L}+\unicode[STIX]{x1D6FF}_{Y}\sin \frac{Y}{L}\right)=M-\unicode[STIX]{x1D6E5}_{X}\sin \frac{X}{L}-\unicode[STIX]{x1D6E5}_{Y}\sin \frac{Y}{L}, & & \displaystyle\end{eqnarray}$$
where
$M<0$
,
$\unicode[STIX]{x1D6FF}_{X}>0$
,
$\unicode[STIX]{x1D6FF}_{Y}>0$
;
$\unicode[STIX]{x1D6E5}_{X}=|M|\unicode[STIX]{x1D6FF}_{X}$
,
$\unicode[STIX]{x1D6E5}_{Y}=|M|\unicode[STIX]{x1D6FF}_{Y}$
. Because of the symmetry of the region, it is sufficient to consider the case
$\unicode[STIX]{x1D6E5}_{Y}\geqslant \unicode[STIX]{x1D6E5}_{X}$
.
4.3.1 Stationary flow
First, consider the stationary flow generated by the non-uniform distribution of the Marangoni number (4.1). The two-dimensional stationary deformation of surfaces
$h_{j}=h_{j}(X,Y)$
,
$j=1,2$
, reflects the spatial symmetry of the perturbation (4.1). In the case
$\unicode[STIX]{x1D6E5}_{X}=\unicode[STIX]{x1D6E5}_{Y}$
, a square pattern is imposed (see figure 30). It should be recalled that in the case
$\unicode[STIX]{x1D6E5}_{X}\neq 0$
,
$\unicode[STIX]{x1D6E5}_{Y}=0$
, the pattern is one-dimensional (see figure 5).
4.3.2 Standing waves
With the growth of
$\unicode[STIX]{x1D6E5}_{Y}$
, the stationary regime becomes unstable with respect to a periodic standing wave characterized by vanishing
$c_{10}(\unicode[STIX]{x1D70F})=c_{01}(\unicode[STIX]{x1D70F})=0$
. The basic harmonics
$s_{10}(\unicode[STIX]{x1D70F})$
and
$s_{01}(\unicode[STIX]{x1D70F})$
oscillate with the same period (see figure 31). The snapshots of the oscillations are shown in figure 32.
4.4 One-dimensional spatial modulation with period
$L/2$
An interesting resonant phenomenon has been observed in the case where the wavelength of the temperature modulation is half of the length of the region, i.e.
$$\begin{eqnarray}\displaystyle M(\boldsymbol{X})=M-\unicode[STIX]{x1D6E5}_{X}\sin \frac{2X}{L}. & & \displaystyle\end{eqnarray}$$
The imposed temperature modulation creates a thermocapillary flow that generates the deformations of surfaces described by the Fourier component
$s_{20}$
. For instance, at
$M=-1$
,
$\unicode[STIX]{x1D6E5}_{X}=0.75$
we observe stationary deformations with period
$L/2$
. However, because of the nonlinearity of the governing equations, that stationary flow can become unstable with respect to the excitation of the modes characterized by the Fourier components
$c_{10}(\unicode[STIX]{x1D70F})$
and
$s_{10}(\unicode[STIX]{x1D70F})$
(‘spatial parametric resonance’), which oscillate on the background of the
$s_{20}$
mode. Due to the nonlinear interaction, the oscillations of
$c_{10}(\unicode[STIX]{x1D70F})$
and
$s_{10}(\unicode[STIX]{x1D70F})$
are completely synchronized (see figure 34).
Figure 25. Snapshots of the isolines of
$h_{1}(X,Y,\unicode[STIX]{x1D70F})-1$
;
$M=-8$
,
$\unicode[STIX]{x1D6E5}_{X}=2$
. (a)
$\unicode[STIX]{x1D70F}=2\,050\,000$
; (b)
$\unicode[STIX]{x1D70F}=2\,050\,420$
; (c)
$\unicode[STIX]{x1D70F}=2\,050\,620$
; (d)
$\unicode[STIX]{x1D70F}=2\,051\,340$
; (e)
$\unicode[STIX]{x1D70F}=2\,051\,600$
; (f)
$\unicode[STIX]{x1D70F}=2\,051\,980$
.
Figure 26. Oscillations of
$h_{max,2}(\unicode[STIX]{x1D70F})$
for
$M=-8$
,
$\unicode[STIX]{x1D6E5}_{X}=2$
.
Figure 27. The phase trajectory in the plane
$s_{02}$
,
$c_{02}$
;
$M=-8$
,
$\unicode[STIX]{x1D6E5}_{X}=2$
.
The flow is two-dimensional, i.e. all of the Fourier components
$c_{mn}$
and
$s_{mn}$
vanish for
$n\neq 0$
. The snapshots of the two-dimensional standing wave are shown in figure 35. Typically, the wave has one maximum and one minimum, due to the Fourier components
$s_{10}(\unicode[STIX]{x1D70F})=c_{10}(\unicode[STIX]{x1D70F})$
(see figure 35
a,b,d). At the time instants when
$s_{10}(\unicode[STIX]{x1D70F})$
,
$c_{10}(\unicode[STIX]{x1D70F})$
are small, a wave with two maxima and two minima is observed, due to the Fourier component
$s_{20}$
(see figure 35
c).
The two-dimensional standing waves described above exist in a wide region of the parameters
$M$
and
$\unicode[STIX]{x1D6E5}_{X}$
.
5 Conclusions
The influence of a substrate temperature modulation on nonlinear Marangoni waves has been investigated. In the framework of a long-wave approach, the problem is reduced to a system of two coupled nonlinear equations for the surface deformations. Simulations have been carried out for several types of spatial temperature modulations. A number of physical phenomena have been revealed.
On one hand, the non-uniformity of heating breaks the translational and rotational symmetries of the original problem. It tends to create a stationary flow with a symmetry corresponding to that of the modulation. This tendency competes with the intrinsic oscillatory instability of the system which generates wave motions. As a result of this competition, a number of nonlinear regimes are produced.
Figure 28. Snapshots of the isolines of
$h_{2}(X,Y,\unicode[STIX]{x1D70F})-h$
: (a)
$\unicode[STIX]{x1D70F}=2\,050\,260$
, (b)
$\unicode[STIX]{x1D70F}=2\,050\,400$
, (c)
$\unicode[STIX]{x1D70F}=2\,050\,520$
, (d)
$\unicode[STIX]{x1D70F}=2\,050\,560$
, (e)
$\unicode[STIX]{x1D70F}=2\,050\,620$
, (f)
$\unicode[STIX]{x1D70F}=2\,050\,660$
;
$M=-6$
,
$\unicode[STIX]{x1D6E5}_{X}=3$
.
Figure 29. Snapshots of the isolines of
$h_{1}(X,Y,\unicode[STIX]{x1D70F})-1$
: (a)
$\unicode[STIX]{x1D70F}=2\,050\,260$
, (b)
$\unicode[STIX]{x1D70F}=2\,050\,400$
, (c)
$\unicode[STIX]{x1D70F}=2\,050\,520$
, (d)
$\unicode[STIX]{x1D70F}=2\,050\,560$
, (e)
$\unicode[STIX]{x1D70F}=2\,050\,620$
, (f)
$\unicode[STIX]{x1D70F}=2\,050\,660$
;
$M=-6$
,
$\unicode[STIX]{x1D6E5}_{X}=3$
.
Figure 30. Shapes of (a) the upper surface
$z=h_{2}(X,Y,\unicode[STIX]{x1D70F})$
and (b) the interface
$z=h_{1}(X,Y,\unicode[STIX]{x1D70F})$
;
$M=-6$
,
$\unicode[STIX]{x1D6E5}_{X}=3$
.
In the case of a small modulation amplitude, three-dimensional standing waves, similar to the alternating rolls of the original non-modulated problem, are developed. In the case of a one-dimensional modulation with a period equal to the length of the region, with the growth of the modulation parameter, this regime is replaced by two-dimensional standing waves, and then by a two-dimensional stationary flow. Wave motions, travelling and standing, reappear with a further growth of the modulation amplitude. For sufficiently strong modulation and large values of the Marangoni number, structures with a shorter wavelength in the direction perpendicular to the substrate temperature gradient are observed. The general diagram of regimes has been constructed.
Figure 31. Snapshot of the isolines of
$h_{2}(X,Y,\unicode[STIX]{x1D70F})-h$
;
$M=-2$
,
$\unicode[STIX]{x1D6E5}_{X}=\unicode[STIX]{x1D6E5}_{Y}=0.5$
.
Figure 32. Oscillations of
$s_{10}(\unicode[STIX]{x1D70F})$
(solid line) and
$s_{01}$
(dashed line) for
$M=-2$
,
$\unicode[STIX]{x1D6E5}_{X}=0.2$
,
$\unicode[STIX]{x1D6E5}_{Y}=1$
.
Figure 33. Snapshots of the isolines of
$h_{2}(X,Y,\unicode[STIX]{x1D70F})-h$
: (a)
$\unicode[STIX]{x1D70F}=2\,004\,000$
, (b)
$\unicode[STIX]{x1D70F}=2\,004\,057$
, (c)
$\unicode[STIX]{x1D70F}=2\,004\,100$
, (d)
$\unicode[STIX]{x1D70F}=2\,004\,126$
, (e)
$\unicode[STIX]{x1D70F}=2\,004\,190$
, (f)
$\unicode[STIX]{x1D70F}=2\,004\,299$
;
$M=-2$
,
$\unicode[STIX]{x1D6E5}_{X}=0.2$
,
$\unicode[STIX]{x1D6E5}_{Y}=1$
.
Figure 34. The temporal dependence of
$s_{10}(\unicode[STIX]{x1D70F})=c_{10}(\unicode[STIX]{x1D70F})$
for
$M=-2$
,
$\unicode[STIX]{x1D6E5}_{X}=0.75$
.
Figure 35. Snapshots of the isolines of
$h_{1}(X,Y,\unicode[STIX]{x1D70F})-1$
: (a)
$\unicode[STIX]{x1D70F}=492\,000$
, (b)
$\unicode[STIX]{x1D70F}=512\,330$
, (c)
$\unicode[STIX]{x1D70F}=522\,000$
, (d)
$\unicode[STIX]{x1D70F}=552\,000$
;
$M=-2$
,
$\unicode[STIX]{x1D6E5}_{X}=0.75$
.
In the case of a two-dimensional modulation, the stationary flow is replaced by three-dimensional standing waves.
On the other hand, the imposed periodic perturbation creates an interaction between waves with different wavevectors. This can lead to the generation of a standing wave with a period that is two times larger than the period of the substrate temperature modulation (‘spatial parametric resonance’).
