=Paper=
{{Paper
|id=Vol-2642/paper6
|storemode=property
|title=Thermocapillary Convection In A Locally Heated Liquid-Gas System
|pdfUrl=https://ceur-ws.org/Vol-2642/paper6.pdf
|volume=Vol-2642
|authors=Victoria B. Bekezhanova,,Artur I. Krom
}}
==Thermocapillary Convection In A Locally Heated Liquid-Gas System==
https://ceur-ws.org/Vol-2642/paper6.pdf
Thermocapillary Convection In A Locally Heated
Liquid – Gas System
Victoria B. Bekezhanova Artur I. Krom
Institute of Computational Modeling SB RAS Siberian Federal University
Krasnoyarsk, Russia 660036 Krasnoyarsk, Russia 660041
vbek@icm.krasn.ru dilirion111@yandex.ru
Abstract
Numerical simulation of the dynamics of a locally heated two-layer
system with the deformable interface is carried out. The system is
subjected to local thermal exposure due to action of the heaters of
finite size arranged on outer boundaries of a working section. Influence
of the lower heater size on a structure of arising convective regimes and
character of interface deformations is analyzed. Feasibility of the liquid
layer rupture in the heating zone are investigated for given liquid layer
thickness and intensity of thermal load.
1 Introduction
Processes of heat-and-mass transfer in fluidic micro- and mini-systems exposed to local thermal load are of both
fundamental and practical interests due to possible technical applications. The scientific concern is occasioned
by richness of liquid behavior modes under non-isothermal processes. The utilitarian interest relates to great
development of multifarious fluidic technologies in space branch, chemical and pharmaceutical industry, material
engineering, thermophysics, microelectronics. One of the examples of wide application of non-isothermal fluid
media in conditions of thermal impact is the thermostabilization systems with liquid cooling. Inter alia, they are
used for cooling of components of on-board equipment for space vehicles of different purposes. In developing and
testing the systems it becomes necessary to obtain preliminary characteristics, to determine influence of various
factors on the system dynamics, to work out the control ways of arising regimes of liquid coolant flows, as well as
to specify ways of suppression of undesirable effects caused by the external temperature action. In the present
work the numerical modeling is performed to investigate the influence of a heat element size on the structure
of the arising flows in working fluids and evolution of the liquid – gas phase boundary. Since a rupture of liquid
coolant layer leads to the critical drop in effective characteristics of heat removal, then the main question is: will
gap appear in the liquid layer with certain thickness under given size and temperature of heater?
2 Basic Assumptions And Problem Statement
A two-layer liquid – gas system filling the plane horizontal cuvette with solid walls is considered (Figure 1). The
media contact along the phase boundary Γt which is the deformable thermocapillary interface defined by the
equation y = f (t, x). The tangential forces act lengthwise of the interface. It is assumed that the surface tension
of liquid σ linearly depends on the temperature: σ = σ0 − σT (θl − θ0 ). Here, σ0 , θ0 are the reference values of
Copyright ⃝
c by the paper’s authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
In: Sergei S. Goncharov, Yuri G. Evtushenko (eds.): Proceedings of the Workshop on Applied Mathematics and Fundamental
Computer Science 2020 , Omsk, Russia, 30-04-2020, published at http://ceur-ws.org
1
the surface tension and temperature for the liquid, respectively, σT is the temperature coefficient, σ0 , σT > 0.
At the initial instant t = 0 both fluids are at rest, and surface Γt has zero curvature. The actuation of heaters of
finite size placed on the lower and upper walls of the cuvette results in onset of convective motion in the layers
and the interface deformations.
Figure 1: Configuration of the bilayered system
The flow regime pattern and character of the interface behavior depend on several factors; they are (i) thickness
of the liquid layer, (ii) thermophysical properties of working media, (iii) intensity and features of external actions
(gravitational, thermal etc). Here, the influence of the heater size on the system evolution is investigated. It is
supposed that the heaters have length 0.5, 1 or 2 cm and can functionate in the stationary or switchable mode.
In the first case the heater temperature remains constant, in another case it can be discontinuously increased or
decreased.
2.1 Governing Equations
It is assumed that the bilayered system is in the field of mass forces with the gravity acceleration vector g =
(0, −g). The Oberbeck – Boussinesq approximation of the Navier – Stokes equations is used for description of
motion induced by applied external thermal exposure in j-th medium. Upon that, j = 1 corresponds to the
upper (gas) layer, j = 2 refers to lower (liquid) layer (see Figure 1). In dimensionless form the governing equations
are the following:
( ) ( )
∂t ωj + ∂x ωj ∂y ψj − ∂y ωj ∂x ψj = Re−1 −2
j ∆ωj + Grj Rej ∂x Tj ,
( ) ( ) (1)
∆ψj + ωj = 0, ∂t Tj + ∂x Tj ∂y ψj − ∂y Tj ∂x ψj = Pr−1 −1
j Rej ∆Tj .
Functions ψj , ωj , Tj (stream function, vorticity and temperature in j-th layer, respectively) are the required
ones. Similarity criteria Grj = βj T∗ gh31 /νj2 (Grashof number), Rej = u∗ h1 /νj (Reynolds number), Prj = νj /χj
(Prandtl number) are introduced by usual way for each medium. They are assumed to be given and defined by the
thermophysical parameters (coefficients of thermal expansion βj , kinematic viscosity νj , thermal diffusivity χj ) as
well as the characteristic values (temperature drop T∗ , velocity of viscous stresses relaxation u∗ in the gas phase,
height of the upper layer h1 in the initial (unperturbed) state). Symbols ∂t , ∂x , ∂y denote the partial-differential
operators with respect to corresponding independent variables.
2.2 Boundary Conditions
Boundary conditions on the common interface Γt are the result of relations on a strong discontinuity, conservation
laws for mass, momentum and energy, and some additional assumptions [And12]. To state the conditions in
“stream function – vorticity” variables, the unit vectors of tangent and normal lines to Γt are introduced:
( ) ( )
1 ∂x f ∂x f 1
s= √ , √ , n= −√ , √ .
1 + ∂x2 f 1 + ∂x2 f 1 + ∂x2 f 1 + ∂x2 f
Here, it is taken into account that n is the vector of external normal to Γt for lower fluid (Figure 1). Further, the
normal and tangent components of the velocity are determined for points, which lie on the interface: vn = −∂s ψ,
vs = ∂n ψ; here, symbols ∂s , ∂n denote the derivatives in the tangential and normal direction. Thus, velocity of
any point on the interface is presented in the form v = vn n + vs s. Besides, the standard continuity conditions
2
for the total velocity vector, temperature and heat fluxes are assumed to be fulfilled on the phase boundary. The
continuity condition for the tangential velocities together with the volume conservation requirement for each of
the media ensure the fulfilment of the following relations for the stream functions on Γt :
ψ1 = ψ2 , ∂n ψ2 − ∂n ψ1 = 0.
The kinematic condition can be written in the form:
√
∂t f + 1 + ∂x2 f ∂s ψ2 = 0.
Solving the equation allows one to determine a profile of the phase boundary at every instant.
The analogues of the tangential and normal components of the dynamic condition are the matching conditions
for the vorticity functions on Γt :
ω2 − ρ̄ν̄ω1 = F1 (t, x), ∂n ω2 − ρ̄ν̄∂n ω1 = F2 (t, x). (2)
Here, function F1 has regard to the thermocapillary force action, and F2 takes into account a contribution
of pressure-jump and effects of the problem nonstationarity. Detailed derivation of relations (2) and exact
expressions for F1 and F2 functions are presented in [Bek19].
The outer boundaries of the cuvette (x = 0, x = X, y = 0, y = Y ) are the solid impermeable walls (Figure 1).
Conditions for the stream functions ψj on these boundaries correspond to the no-slip conditions for viscous fluid:
ψj = 0, ∂n ψj = 0. Relations for the temperature functions consider the presence of thermal elements on the
substrate (y = 0) and upper wall (y = Y ):
T1 | y=Y, x∈Q
/ up
p
= 0, T1 | y=Y, x∈Qup
p
= qpup (t), T2 | y=0, x∈Q
/ sl = 0, T2 | y=0, x∈Qsl = qls (t),
where Qsl is the area of the substrate occupied by l-th heater with the temperature qls , Qup
p is the part of the
upper boundary on that p-th thermal element with the temperature qpup is arranged.
The numerical algorithm based on the Ovcharova method [Ovch14] is used to solve the stated adjoint problem.
The outline of the algorithm and some details of the numerical method are given in [Bek20].
3 Results Of Numerical Simulation And Discussion
A series of calculations was performed for the ethanol – nitrogen system filling the vessel with length X = 0.2 m
and height Y = 0.01 m. The condition of terrestrial gravity with g = 9.81 m/s2 were considered. In the
unperturbed state the thickness of each layer was taken to be equal to 5 · 10−3 m. The dynamics of system were
numerically investigated under different operating modes of the heaters of various size. The case was considered
when the thermal elements were simultaneously arranged on the bottom and upper walls of the cuvette. Upon
that, one heater with the length of 0.01 or 5 · 10−3 m was placed on the lower boundary, and it was operated
in commutated mode. Two heaters with the length of 0.02 and 0.015 m were arranged on the upper wall; along
with this, one of them was located above the lower thermal element. The top heaters had the same constant
temperature. Considering that the primary influence on the amplitude of the interface deformations was governed
by action of the lower heater [Bek19], the parameters of this thermal element, namely, temperature and size were
changed.
The specific feature of the switchable (commutated) operating mode is the abrupt change in heater temperature
(both increase and drop) to some limit values. Similar regime simulates periodical heating (under actuation)
or cooling (after switching off) of a heater on the working area of real fluidic path. For practical systems of
thermal control the working range of the temperature is 0 − 25 o C. The modeling was carried out precisely for
such temperature drops of lower heater; the temperature q s was varied by 2.5 o C under each switching. The
character of changes for thermal and hydrodynamical fields and the behavior of the liquid – gas interface were
analyzed. The comparison of characteristics with those investigated in [Bek19] for case of heater with larger size
was performed.
When the lower heater of small size is switched on the solitary thermal plume (upper right picture in Figure 2)
evolves in zone of thermal impact due to the convective mechanism and dual-vortex flows occurs in each layer
(lower right picture in Figure 2). Along with this, the thermocapillary deflection caused by change in the surface
tension of Γt appears. The formation of similar flow pattern with the typical temperature plume in a locally
heated liquid with a free surface is experimentally confirmed [Kon16]. In contrast to the case when the lower
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thermal element has larger size (the thermal and hydrodynamical structures of flow for this case are presented
on the left pictures in Figure 2), in the system with the small heater the two-vortex regime in the liquid layer
is directly set in above zone of heating. In such a case, the transient process from the two-vortex flow to the
quadruple-vortex one and back is absent (the description of mechanisms generating such alteration is given in
[Bek19]). The differences are explained by the structure of the thermal field. In the case of large lower heater the
double-type thermal plume appears in the domain of thermal exposure (upper left picture in Figure 2), therefore,
four-vortex flow in the liquid layer is able to develop. Such a dependence of the plume form on the heater size
was established in experiments [Kon16].
Figure 2: Comparison of the thermal (from above) and topological (from below) pattern of the flows in the
heating zone in the systems with large (from the left) and small (from the right) heater
At each successive jump-like increasing of the heater temperature the interface undergoes oscillations accom-
panied by the changes in amplitude and shape of the flexure. It should be noted that the system responds on the
variations of thermal exposure intensity with certain time lag. The delay time corresponds to period in which
heat from the thermal element on the substrate comes to the liquid – gas surface. The convection speed depends
on both the rate of thermal attack and the liquid layer thickness, and also on the intensity of the gravity field
etc. [Bek20]. With time the boundary layer is formed near the interface. It deforms the convective cells and
leads to the formation of the regime with drifting from the heating zone vortices (Figure 3).
Figure 3: Field of velocity and temperature in the ethanol – nitrogen system at the lower heater temperature
q s = 22.5 o C (155 seconds after switching on the heater)
Action of the upper thermal elements has a little effect on the system dynamics. Heat from the sources
arranged on the top walls transfers to the interface due to the thermal conductivity and scarcely affects both the
interface deformation and the flow topology. In the upper layer the vortices are only slightly deformed (Figure 3).
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With increase of the heater temperature the amplitude of interface deformation in the zone of thermal attack
grows (Figure 4). If the temperature of lower heater is not increased with time, then steady thermocapillary
deflection with concave profile is formed above the thermal element.
Figure 4: Position of Γt and temperature distribution near the interface in the thermal exposure zone for the
ethanol – nitrogen system: at q s = 12.5 o C (50 seconds after switching on the heaters, from above), at q s = 17.5 o C
(137 seconds after switching on the heaters, from below)
4 Conclusions
It was established that for the considered range of working temperature of the heaters (from 0 to 25 o C), if the
thickness of the lower layer was 5 · 10−3 m then the rupture of liquid layer did not occur despite long duration of
heating and unfavourable (switchable) operation mode. In this range of working temperature the gap of liquid
layer can appear only if the initial thickness of the lower layer is significantly less than 5 · 10−3 m. Furthermore,
it was found that the size of the lower heater slightly influences the amplitude of the interface deformation. And
finally, it was shown that the arrangement of additional upper thermal sources (even directly above the lower
heater) does not result in significant alteration of the flow regimes and changes in the interface behavior. Thus,
one of the ways to avoid critical deformations of the phase boundary is to arrange the heater on the upper
wall of the working section. Also, upper thermal elements can be used to prevent the appearance of thermal
hysteresis where the parameters of an electronic device do not revert to the initial values after its heating and
subsequent cooling. Switching to these additional upper heaters allows one to ensure continuity of the operation
of onboard equipment without losing effective power. The lifetime of thermal elements can be prolonged by
placing a duplicate heater and periodic switching of the electric circuit from one heater to another while the
system relaxes.
5 Acknowledgements
This work was supported by the Krasnoyarsk Mathematical Center and financed by the Ministry of Science and
Higher Education of the Russian Federation in the framework of the establishment and development of regional
Centers for Mathematics Research and Education (Agreement No. 075-02-2020-1631).
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References
[And12] V. K. Andreev, Yu. A. Gaponenko, O. N. Goncharova, V. V. Pukhnachov. Mathematical models of
convection (de Gruyter Studies in Mathematical Physics). De Gruyter, Berlin, Boston, 2012.
[Bek19] V. B. Bekezhanova, A. S. Ovcharova. Convection regimes induced by local boundary heating in a
liquid – gas system. Journal of Fluid Mechanics, 873:441–458, 2019.
[Ovch14] A. S. Ovcharova. Rupture of liquid film, placed over deep fluid, under action of thermal load. Inter-
national Journal of Heat and Mass Transfer, 78:294–301, 2014.
[Bek20] V. B. Bekezhanova, O. N. Goncharova. Impact of gravity on the flow pattern in a locally heated
two-layer system. Microgravity Science and Technology, 32:229–243, 2020.
[Kon16] A. Kondrashov A, I. Sboev I, P. Dunaev. Evolution of convective plumes adjacent to localized heat
sources of various shapes. International Journal of Heat and Mass Transfer, 103:298–304, 2016.
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