The dynamics of a bilayer system filling a rectangular cuvette subjected to external heating is studied. The influence of two types of thermal exposure on the flow pattern and on the dynamic contact angle is analyzed. In particular, the cases of local heating from below and distributed thermal load from the lateral walls are considered. The simulation is carried out within the frame of a two-sided evaporative convection model based on the Boussinesq approximation. A benzine–air system is considered as reference system. The variation in time of the contact angle is described for both heating modes. Under lateral heating, near-wall boundary layers emerge together with strong convection, whereas the local thermal load from the lower wall results in the formation of multicellular motion in the entire volume of the fluids and the appearance of transition regimes followed by a steady-state mode. The results of the present study can aid the design of equipment for thermal coating or drying and the development of methods for the formation of patterns with required structure and morphology.

Thermocapillary convection is a subject of extensive research due to the great importance of the results which can be applied in developing various technologies in the fields of materials science and microfluidic systems. The mechanism of occurrence of convective motion is related to changes in the surface tension of a liquid. The changes can result from an external thermal action [

Fluidic systems with working media being in different aggregate states and filling domains with outer solid boundaries as well as limited volumes of the liquid contacting with a rigid substrate have a three-phase contact line near which the convection characteristics can also significantly vary [

In the present paper, we study the structure and characteristics of convective regimes in a liquid–gas system under diffusive-type evaporation based on numerical modeling. We investigate the influence of different types of thermal action on the flow structure, behavior of the phase boundary and dynamic contact angle. We consider the cases of local heating from the substrate and distributed lateral thermal impact. The simulation is carried out in the framework of a two-dimensional mathematical model whose implementation admits the correct agreement between the no-slip boundary condition and the moving contact points [

We investigate joint non-stationary flows in a two-layer system filling a plane rectangular cuvette (_{1} and Ω_{2} are occupied by gas and liquid media, respectively, and divided by a common interface which is a thermocapillary surface admitting the mass transfer of the diffusive type due to evaporation/condensation. The phase boundary is considered as a smooth curve embedded in two-dimensional space. The curve is defined by the equation

Initially, both fluids are at rest, and the interface is straight so that the left and right contact angles

We assume that the system is in the terrestrial gravity field ^{2}, and both fluids are incompressible viscous heat-conducting media. Then, the basic mechanisms governing the convective flows with evaporation in the bilayer system are buoyancy and thermocapillary forces caused by external heating of the cell walls. For describing the dynamics of the system we use a two-sided model of evaporative convection on the basis of the Oberbeck–Boussinesq approximation proposed in [

To model the convective heat and mass transfer in the domains Ω_{j}, we use the governing equations written in

Assuming that the vapor is a passive admixture which does not change any properties of the background gas, we add the convection-diffusion equation_{1} and lower Ω_{2} domains, respectively (see _{0}. The problem contains the following similarity criteria: Reynolds numbers

The following conditions define the initial state of the system at the time moment

The initial location of the interface is set by the equation

On the outer boundary ∂Ω the conditions for the stream functions

Here,

The interface conditions are derived on the basis of the conservation laws for mass, momentum and energy, while other ones are the result of using additional suppositions with regard to the character of the processes under study [

(i) the thermocapillary surface

(ii) we consider that

Then, complementing the kinematic, dynamic and energy conditions with the standard continuity conditions for the velocity and temperature and the relation for the vapor concentration on the phase boundary, we can write the interface conditions at

Here,

The sign of

A detailed substantiation of the problem statement presented is given in [

We use the numerical algorithm proposed in [

The dynamics of convective heat and mass transfer in a closed cuvette filled with the liquid and gas media divided by an interface under various heating conditions is studied on the example of a benzene–air system. The physicochemical parameters of the working fluids are given in

Parameter | Benzine |
Air |
---|---|---|

Density |
1.35 | |

Kinematic viscosity |
||

Thermal expansion |
||

Heat diffusivity |
||

Heat conductivity |
0.186 | 0.026 |

Molar mass |
0.112 | 0.029 |

Reference value of the surface tension |
0.0237 | |

Temperature coefficient of the surface tension |
||

Latent heat of vaporization |
||

Concentration expansion |
–0.62 | |

Coefficient of vapor diffusion |
||

Dufour coefficient |
||

Soret coefficient |

Parameter | Value | Parameter | Value |
---|---|---|---|

18.4211 | |||

2.1588 | |||

10 | Ca | ||

10 | |||

0.215 | |||

0.048 | Ga | 400.408 | |

0.1398 | Ma | 268.53 | |

19.2857 |
Pe | 1.657 |

We consider the cases of non-stationary heating from below (Heating mode I) and from the side of the lateral walls (Heating mode II). In Heating mode I, the temperature of the lateral walls does not change, ^{∘}C so that the maximum temperature drop by 4.5 degrees is provided). In Heating mode II, the substrate temperature remains to be constant,

We note that the convective mechanism completely defines the flow structure and character of changes in the basic characteristics. The heat transfer from the thermal elements triggers cascade responses affecting the system dynamics. The onset of motion is caused by the buoyancy force; here, the topological pattern of the flow is determined by the number and size of the heaters. The quantity of vortices forming typical convective cells at the first stage is equal to 2

The initial convective regime is replaced by a multicellular mode whose characteristics are shown in

The jump-like change in the temperature of Heater 2 leads to the development of oscillatory regimes with the behavior of the system characterized by the oscillations of the basic parameters near certain values. The wave behavior of the interface distinguishes these regimes from the first two modes (

After switching off Heater 1, the double-vortex motion in each fluid is settled. The flow occupies almost the entire cuvette. There is one zone with an ascendant flow above Heater 2 (

In contrast to the case of local heating from below, the dynamical characteristics as well as the behavior of the liquid–gas interface do not qualitatively change with time upon lateral heating.

The specific temperature distribution entails other behavior of the interface. The phase boundary has a convex upward form in response to lateral heating. It results from the combined action of the surface tension forces and arising pressure gradients. The lateral heating attenuates the interface surface tension. At the same time, the pressure from the gas side near the cuvette edges is higher than that from the liquid, and therefore, the phase boundary hangs down therein. Out of the thermal “spots”, the pressure drop vanishes and the interface is forced out upwards. It leads to the formation of a thermocapillary bump. The higher the intensity of the external thermal load, the larger the pressure drop and the larger the height of the interface rise (compare images (b) in

As to the velocity field pattern, we note that the wall swirls near both side boundaries evolve into the vortex structures with a complex symmetry. The eventual flow regime is characterized by the coexistence of pronounced boundary layers with a near-wall core and a slow-convecting zone. Here, the velocity of motion induced by the distributed lateral heating is well above those in the case of the local thermal load from the substrate (compare the values of

The behavior of the dynamic contact angle is illustrated by an example of the angle ^{∘}. Thus, a concave meniscus is formed in the ultimate mode under local heating from below. Small variations of

Another character of the contact angle alterations is observed under lateral heating. The intensive supply of heat from the side walls results in a rapid significant decrease in the ^{∘} persists in the eventual regime under Heating mode II.

The dynamics of the two-phase system with a deformable interface admitting the mass transfer due to evaporation in the box-shaped cuvette subjected to various heating conditions is investigated at an example of the system of working media like benzine–air. The study is carried out in the framework of a full two-sided model based on the Boussinesq approximation. The original approach used to state the problem in terms of new auxiliary functions allows one to derive the corresponding numerical model that enables one not only to calculate all the system characteristics, including the mass evaporation rate, but also to scrutinize the behavior of the dynamic contact angles, depending on the heating mode.

Based on the numerical simulation, considerable differences in the behavior of the bilayer fluidic system are elucidated. An eventual regime under the local heating from below is established through the transient modes, whereas the inertia of the system behavior is typical for the case of the distributed lateral heating, giving rise to monotonous changes in the hydrodynamic, temperature and concentration characteristics. Here, the amplitudes of the interface deformations under the thermal load applied on the side walls is lower than those under the other type of heating. Two various scenarios of the interface evolution are realized, depending on the type of the external thermal action. A thermocapillary flexure and a thermocapillary bump appear upon heating from the substrate and side walls, respectively. In the first case, convective heat and mass exchange occurs in the whole volume of the fluids, and equilibrium concave menisci are formed on the edges in the ultimate steady regime. The second heating regime leads to the appearance of a boundary layer and near-wall temperature and concentration patterns; here, convex menisci are observed on the heated lateral walls. In both cases, the contact angle variation in time is described by the numerical model.

The results obtained contribute to a better understanding of the interaction of the basic mechanisms governing the convective regimes in systems with internal interfaces and points of three-phase contact. The results of the present study can aid in designing equipment for thermal coating or drying and working sections employed in control tools for various applications as well as in developing methods for the formation of patterns of the required structure and morphology on solid surfaces.

Concentration (mass fraction of volatile component)

Capillary number

Vapor diffusion coefficient

Interface parametrization function

Auxiliary function in the boundary conditions

Evaporation number

Galilei number

Grashof number

Gravity acceleration vector

Fluid layer thickness

Thermal conductivity coefficient

Latent heat of vaporization

Unit normal vector at the interface

Evaporation mass flow rate

Marangoni number

Prandtl number

Diffusive Peclet number

Heater temperature

Domain of the substrate with the embedded thermal element

Interface curvature radius

Universal gas constant

Reynolds number

Unit tangent vector on interface

Time variable

Temperature

Auxiliary function

Velocity vector

Cartesian coordinates

Cuvette length

Cuvette height

Dufour coefficient

Soret coefficient

Thermal expansion coefficient

Concentration expansion coefficient

Kronecker delta

Dimensional coefficient in the boundary condition for the vapor concentration at the interface

Molar mass of the evaporating liquid

Kinematic viscosity coefficient

Density

Surface tension

Temperature coefficient of the surface tension

Thermal diffusivity coefficient

Stream function

Vorticity function

Flow domain

Analogues for the spatial coordinates in the numerical model

Reference flow parameters (specified at

Denotes belonging to the upper (

Integer-valued index, denotes the order number

Related to the normal vector

Related to the substrate or to the tangential vector

Denotes the characteristic values or dimensional quantities

The authors express gratitude to Dr. O.M. Lavrenteva for useful discussions.

The work of O.N. Goncharova was carried out in accordance with the State Assignment of the Russian Ministry of Science and Higher Education entitled “Modern Models of Hydrodynamics for Environmental Management, Industrial Systems and Polar Mechanics”, Govt. Contract Code: FZMW-2024-0003,

The authors confirm equal contribution to the paper: study conception and design, data collection, analysis and interpretation of results, draft manuscript preparation: V.B. Bekezhanova. O.N. Goncharova. All authors reviewed the results and approved the final version of the manuscript.

All the data are based on publicly available documents, whose data sources were cited in the manuscript.

The authors declare that they have no conflicts of interest to report regarding the present study.