=Paper=
{{Paper
|id=Vol-2928/paper5
|storemode=property
|title=Modeling of the Two-Layer Flows With Inhomogeneous Type of Evaporation at the Interface
|pdfUrl=https://ceur-ws.org/Vol-2928/paper5.pdf
|volume=Vol-2928
|authors=Olga N. Goncharova,,Victoria B. Bekezhanova
}}
==Modeling of the Two-Layer Flows With Inhomogeneous Type of Evaporation at the Interface==
https://ceur-ws.org/Vol-2928/paper5.pdf
Modeling of the Two-Layer Flows With Inhomogeneous
Type of Evaporation at the Interface
Olga N. Goncharova Victoria B. Bekezhanova
Altai State University Institute of Computational
Barnaul, Russia 656049 Modelling SB RAS
gon@math.asu.ru Krasnoyarsk, Russia 660036
vbek@icm.krasn.ru
Abstract
The theoretical approach for modeling of the heat and mass exchange
in a two-phase system on the basis of an exact solution of equations
of thermal concentration convection is discussed. Characteristics of bi-
layer liquid-gas system are determined with the help of explicit expres-
sions for all the required functions which are derived in the framework
of a two-dimensional evaporative convection problem. The exact solu-
tion allows one to correctly take into account the non-uniform character
of phase changes through the sharp interface, thermocapillary proper-
ties of the liquid – gas boundary and the impact of direct and inverse
thermodiffusion effects in the vapor-gas layer. The influence of gas
flow rate on the parameters of convective regimes appearing in a plane
horizontal channel is investigated in the case when the first kind con-
ditions are imposed for the temperature function on the substrate and
for the vapor concentration function on the outer boundary which con-
fines the gas layer from above, whereas the second type condition is set
for the temperature on the upper channel wall. Feasibility of obtained
theoretical results is examined.
1 Introduction
The problems of convection under phase transitions is of interest due to wide application of fluid media in
different technological equipments. Bright examples of fluidic technologies are liquid cooling in various systems
of thermal control and thermal coating. The experimental testing of any set-up or technique is preceded by
theoretical study of regularities of heat and mass transfer processes, behavior of working fluids under different
conditions. The main problem under theoretical investigation of evaporative convection is the choice of a way of
description of considered phenomena [Bek18a]. The most widely used mathematical model to research convection
accompanied by phase changes is built upon the use of the Navier – Stokes equations or their approximations. To
consider the temperature effects in whole two-phase system and concentration effects in a gas phase appearing
due to the liquid vaporization, the equations of heat and mass transfer are invoked additionally to the motion
equations. The resulting system of governing equation possesses natural properties of symmetry with respect to
Copyright © 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 2021 , Omsk, Russia, 24-04-2021, published at http://ceur-ws.org
1
space and time, and as a consequence of this, it provides the implementation of the fundamental conservation
laws. Except that the system of thermal concentration convection equations is nonlinear one and has high order,
it is irrelevant to certain type. All of these heightens the value of exact solutions which take the basic symmetry
properties peculiar to the original system of equations. It implies physical plausibility of results obtained with
the help of the solutions.
There are a few available exact solution to describe convection in the two-phase systems [Bek18a]. Ex-
amples of meaningful solutions for studying steady flows with phase transitions are the Ostroumov – Birikh
type solutions, which can be effectively applied to define basic characteristics of the convective regimes
for various conditions. Two-dimensional solutions of such a class allowed one to specify the features
of the flow regimes with evaporation/condensation depending on the different parameters of the problem
[Bek16, Bek18b, Bek18c, She18, She19, Bek20a, Bek20b]. In all the cited works the theoretical results were
obtained on the assumption on uniform character of evaporation. In the present work, the generalization of the
Ostroumov – Birikh type solution is derived for the case of inhomogeneous evaporation when the evaporation
rate is not constant and changes along the channel.
2 Governing Equations And Functional Form Of Partially Invariant Solution
The joint flow of evaporating liquid and co-current gas-vapor flux in a plane infinite horizontal channel is
considered. The system is oriented so that the mass force vector g has the coordinates g = (0, −g) in the
Cartesian coordinate system (x, y). The lower and upper boundaries of the channel y = −l and y = h are
the solid impermeable walls. The fluids contact along the common thermocapillary surface Γ admitting phase
transition due to evaporation/condensation. The interface remains to be flat, it is given by the equation y = 0.
The Navier – Stokes equations in the Oberbeck – Boussinesq approximation is utilized for description of the
flow in both phases. In plane case the governing equations have the following form:
2
∂ u ∂2u
∂u ∂u 1 ∂p
u +v =− +ν + , (1)
∂x ∂y ρ ∂x ∂x2 ∂y 2
2
∂2v
∂v ∂v 1 ∂p ∂ v
u +v =− +ν + + g (βT + γC) , (2)
∂x ∂y ρ ∂y ∂x2 ∂y 2
∂u ∂v
+ = 0, (3)
∂x ∂y
∂2T ∂2T ∂2C ∂2C
∂T ∂T
u +v =χ + +δ + . (4)
∂x ∂y ∂x2 ∂y 2 ∂x2 ∂y 2
The terms containing function C are taken into account in modelling the flows in the gas phase. The vapor
transfer in the carrier gas is described by the diffusion equation [Lan1987]:
2
∂2C
2
∂2T
∂C ∂C ∂ C ∂ T
u +v =D + +α + . (5)
∂x ∂y ∂x2 ∂y 2 ∂x2 ∂y 2
In equations (1) – (5) the following notations are used: u, v are the longitudinal and transversal components
of the velocity vector, correspondingly, p is the modified pressure function which defines the deviation of real
pressure from the hydrostatic one, T is the fluid temperature, C is the vapor concentration in background gas,
ρ is the fluid density (for each phase it is some reference value of the density), ν, χ, β are the coefficients of
kinematic viscosity, heat diffusivity and thermal expansion, respectively, D is the coefficient of vapor diffusion in
the gas, γ is the concentration coefficient of density, the parameters δ and α are the Dufour and Soret coefficients
respectively, characterizing the effects of diffusive thermal conductivity and thermodiffusion.
System (1) – (5) admits an exact solution of a special type [Bek16]:
ui = ui (y), vi = 0, Tj = Tj (x, y) = ai1 + ai2 y x + ϑj (y),
(6)
C = C(x, y) = (b1 + b2 y) x + φ(y), pj = pj (x, y).
Here, index i denotes characteristics of working fluid: i = 1 and i = 2 regard to the liquid in the lower layer and
the gas-vapor mixture in the upper one, respectively. Explicit expressions for pj functions and additive terms
ϑj (y), φ(y) as well as exact values of the parameters ai2 , bj (i = 1, 2; j = 1, 2) which are the constants, are
determined with the help of the boundary conditions.
2
3 Boundary Conditions And Closing Relations
The boundary conditions are formulated taking into account the character of dependence of the unknown func-
tions on spatial variables defined by (6). The no-slip conditions on the rigid walls are valid for the velocity
functions:
u1 (−l) = 0, u2 (h) = 0, (7)
It is supposed that distributed thermal load is applied on the substrate
T1 (x, −l) = A1 x + ϑ− , (8)
whereas the upper wall is thermally-insulated:
T2y + δCy y=h = 0. (9)
The condition of zero concentration of vapor is prescribed on the upper wall:
C (x, h) = 0. (10)
The last condition defines property of the boundary to instantaneously absorb the vapor. In the experimental
setup it is provided by a freezing out of the vapor on the wall. The applicability of the first kind boundary
condition for the vapor concentration function is discussed in [Bek18d] in the framework of three-dimensional
statement of the evaporative convection problem.
Setting the boundary conditions on the interface Γ, one should take into account that it is not deformed and
given by the equation y = 0. The kinematic condition is fulfilled identically in view of the exact solution form.
The continuity conditions for the longitudinal component of velocity vector and temperature at the interface are
valid:
u1 = u2 , T1 = T2 . (11)
The continuity of the transversal component of the velocity vector follows from the kinematic condition. Pro-
jecting the dynamic condition on the unit tangential and normal vectors to the interface, there result two scalar
relations:
ρ1 ν1 u1y = ρ2 ν2 u2y − σT Tx y=0 , p1 = p2 . (12)
Here, the impact of the surface forces is characterized by σT > 0 which gives the temperature coefficient of the
surface tension σ, σ = σ0 − σT (T − T0 ); σ0 > 0, T0 are the reference values of the surface tension and the liquid
temperature. The heat balance condition is written considering the thermodiffusion effects and mass balance
equation:
κ1 T1y − κ2 T2y − δκ2 Cy y=0 = −λM, M = −Dρ2 (C y + αT2y |y=0 ) . (13)
Here, λ is the latent heat of evaporation, M defines the evaporation mass flow rate of the liquid at the interface.
In the present paper, the exact solution is derived under assumption, that the evaporation rate is not constant
and changes along the channel according to the linear law:
M = M (x) = M0 + Mx x. (14)
The evaporation mass flow rate M is one of the important qualitative characteristics of the convective regimes
with phase change. The positive (negative) values of M correspond to evaporation of the liquid into the gas flow
(vapor condensation from the gas phase).
The condition for C function on the phase boundary sets the saturated vapor concentration and presents the
linearized form of the equation which is the consequence of the Clapeyron – Clausius and Mendeleev – Clapeyron
equations:
C y=0 = C∗ [1 + ε(T2 y=0 − T0 )]. (15)
Here, ε = λµ0 /(RT02 ), µ0 is the molar mass of the evaporating liquid, R is the universal gas constant, C∗ is the
saturated vapor concentration at T2 = T0 .
To close the problem statement, the gas flow rate is preset:
Zh
Q= ρ2 u2 (y) dy. (16)
0
3
Thus, we have two sets of equations, each of them governs the heat and mass transfer in a certain fluid, and
interface conditions which relate these equations. The conditions imposed on the liquid – gas boundary allow
one to correctly describe the phase transition of diffusive type. The problem statement used is substantiated in
[Bek16, Bek20b]. Solving the problem, one can define the basic characteristics of the bilayer system: the velocity
ui , pressure pi , temperature Ti in ith phase and vapor concentration C in gas layer.
4 Explicit Expressions For Required Functions
The fulfilment of condition of temperature continuity at the interface (the second equality in (11)) results
in the following equality: ai1 = A (i = 1, 2). The value determines the interface longitudinal temperature
gradient characterizing the intensity thermal effects on the phase boundary, and as consequence, the intensity of
evaporation and thermocapillary convection.
Substitution of (6) in equations allows one to derive the explicit expressions for all the required functions:
y2 y3 y4
ui (y) = ci3 + ci2 y + ci1 + Li3 + Li4 ,
2 6 24
2
y2 y3 y4
y
pi (x, y) = di1 + di2 y + di3 x + ci8 + K1i y + K2i + K3i + K4i +
2 2 3 4
5 6 7 8
y y y y
+K5i + K6i + K7i + K8i , (17)
5 6 7 8
2
y y3 y4 y5 y6 y7
Ti (x, y) = A + ai2 y x + ci5 + ci4 y + N2i + N3i + N4i + N5i + N6i + N7i
,
2 6 24 120 720 1008
y2 y3 y4 y5 y6 y7
C (x, y) = (b1 + b2 y) x + c7 + c6 y + S2 + S3 + S4 + S5 + S6 + S7 .
2 6 24 120 720 1008
Coefficients cij (i = 1, 2; j = 1, ..., 5; 8), c6 , c7 are the integration constants; the common scheme for finding
of cij will be outlined below. Coefficients Li4 , Li3 , Sj , Km i
(i = 1, 2; j = 2, ..., 7; m = 1, ..., 8) are calculated via
physical parameters of the problem g, βi , νi , χi , ρi , D, γ, coefficients defining the longitudinal gradients for the
temperature and vapor concentration A, ai2 , bi (i = 1, 2), and integration constants cij .
5 Algorithm For Calculating The Integration Constants
Let the gas flow rate (16) and values of the longitudinal temperature gradients A1 and ϑ− (see (9)) be preset.
The fulfilment of boundary conditions (7) – (15) and relationship (16) results to a system of equations for finding
the integration constants cij (i = 1, 2; j = 1, ..., 5; 8), c6 , c7 . Determining these constants, the velocity and
temperature profiles, the pressure distributions for both phases and the vapor concentration in the gas layer are
calculated based on formulas (17). Below, the algorithm for finding all the unknown parameters and constants
is proposed.
(i) Condition (10) leads to expression connecting the parameters b1 and b2 which determine the vapor concen-
tration function C in (6), b2 = −b1 /h, and to the first linear algebraic equation: φ(h) = 0.
(ii) Thermal insulation condition (9) gives the connection a22 = −δb2 and the second linear algebraic equation:
ϑ0 2 (h) + δφ0 (h) = 0. The prime symbol denotes a derivative with respect to y.
(iii) Consequence of the mass balance condition (the second equality in (13)) allows one to write the relations for
Mx and M0 : Mx = −Dρ2 b2 (1 − αδ)), M0 = −Dρ2 (c6 + αc24 ). Condition for saturated vapor concentration
(15) leads to expressions for b1 and b2 in terms of interfacial gradient A, b1 = C∗ εA, b2 = −C∗ εA/h. As
consequence we can get: Mx = Dρ2 (1 − αδ)C∗ εA/h.
(iv) Condition (15) at the interface entails one more linear algebraic equations: c7 = C∗ + C∗ ε(c25 − T0 ).
(v) Temperature distribution (8) determines the parameter a12 = (A − A1 )/l, and heat transfer condition (13) at
the interface with the help of the
expressions from (iii) leads toa relationship for the longitudinal temperature
gradients A and A1 : A1 = A 1 + (l/h)Dρ2 (1 − αδ)λC∗ ε/κ1 .
4
(vi) Two linear algebraic equations for the integration constants follow from the conditions defining the thermal
boundary regime (8) and (13): ϑ1 (−l) = ϑ− , κ1 c14 − κ2 c24 − δκ2 c6 = −λM0 .
(vii) Values of the unknown integration constants {ci1 , ci2 , ci3 } (i = 1, 2) are calculated by solving the linear
algebraic equation system that is a consequence of no-slip conditions (7), balance relations (12), condition
of velocity continuity in (11) and equality (16) giving the gas flow rate.
(viii) Formulas determining the thermal and vapor concentration boundary regimes (9), (10), conditions at in-
terface setting the temperature continuity (11) and saturated vapor concentration (15), and heat balance
equation (13) lead to the system of the linear algebraic equations for calculation {ci4 , ci5 , c6 , c7 }, i = 1, 2.
(ix) The value of M0 will be computed with the help of obtained values c24 and c6 .
Realizing this algorithm, all the integration constants {ci1 , ci2 , ci3 } and {ci4 , ci5 , c6 , c7 } are found. Hence, the
mass flow rate of evaporation M at the interface can be determined in the form (14). It should be clear that
a part of parameters contained in the exact solution is associated with the some compatibility conditions dictated
by the boundary regimes.
6 Influence Of Gas Flow Rate On Characteristics Of Two-Phase Flows
The liquid – gas two-layer system is considered. The HFE-7100 fluid (HydroFluoroEther) and nitrogen are chosen
as working liquid and gas respectively. The HFE-7100 liquid volatilizes from the lower layer, so that a mixture
of nitrogen and HFE-7100–liquid vapor is in the upper layer. The physical parameters of the working media are
given below in the order {HFE-7100, nitrogen} (or only HFE-7100): ρ = {1.5 · 103 , 1.2} kg/m3 ; ν = {0.38 · 10−6 ,
0.15 · 10−4 } m2 /s; β = {1.8 · 10−3 , 3.67 · 10−3 } K−1 ; χ = {0.4 · 10−7 , 0.3 · 10−4 } m2 /s; κ = {0.07, 0.02717}
W/(m·K); σT = 1.14 · 10−4 N/(m·K); D = 0.7 · 10−5 m2 /s; L = 1.11 · 105 W·s/kg; C∗ = 0.45 (correspond to
equilibrium temperature T0 = 293.15 K); µ0 = 0.25 kg/mol; γ = −0.5; ε∗ = 0.04 K−1 ; δ = 10−5 K; α = 5 · 10−3
K−1 .
The influence of the gas flow rate Q on the flow patterns which arise in the bilayer system is investigated.
The typical distributions of the basic characteristics of the convective regimes with evaporation obtained on the
basis of the exact solutions under study are presented in Figs. 1, 2. The parameters defining the external thermal
action applied on the substrate are the longitudinal temperature gradient A1 and value ϑ− (see condition (8)).
In the calculation, the values of A1 equal to −10 or 10 K/m and ϑ− = 293.15 K are set. The negative (positive)
values of A1 correspond to cooling (heating) of the lower boundary of the flow domain in the direction of the
longitudinal axes. Values of the gas flow rate Q are assumed to be 0.1R0 , R0 and 10R0 , where R0 = 9.6 · 10−6
kg/(m2 ·s). The following parameters are fixed for all the cases under consideration: thicknesses of the liquid and
gas layers l = 2 mm and h = 6 mm, respectively, and gravity acceleration g = 9.81 m2 /s.
Changes in gas flow rate can lead to significant alteration of the flow topology. If the lower wall cools along
the longitudinal axes x with intensity A1 = −10 K/m, then negative thermal gradient A is formed on the phase
boundary (see corresponding value of A in Table 1). It induces the thermocapillary effect action causing the
liquid motion from hot domain in zone with lower temperature along the interface (see velocity profiles shown
in Figs. 1(a,d,g)). In this case, action of the Marangoni effect is co-directional to shear stresses generated by gas
flux. At small and moderate Q, the peculiar layering of the velocity profile occurs, which is appeared by the
formation of a wedge near the interface (Fig. 1(a,d)). One can tell about dominant impact of the thermocapillary
effect in these cases, since the maximum velocity is achieved in the liquid layer. Further increase in gas flow rate
leads to change of dominant mechanism, the velocity field is governed by large shear stresses. Upon that, the
maximum velocity is observed in the gas layer (Fig. 1(g)).
If the substrate heats with temperature gradient A1 = 10 K/m, then positive interface gradient A is set at the
interface (see Table 2). Due to the Marangoni effect the reverse flow appears in the liquid layer (Fig. 2), and this
effect is dominant mechanism defining the flow pattern under small gas flow rate Q = 0.1R0 . In this case the
maximum values of velocity is achieved in the liquid phase (Fig. 2(a)). At moderate values of Q, there occurs the
competition of the influence of the dynamical effect arising due to the shear stresses and thermocapillary one.
The intensity of motion in the gas and liquid layers is almost the same (Fig. 2(d)). At large Q the competition
is maintained. The global counterflow in the liquid persists, although significant increase of the velocity in gas
phase is observed (Fig. 2(g)).
For both types of external thermal load, the variations of the gas flow rate do not give rise to alteration
of the fields of temperature and vapor concentration as well as to significant quantitative changes in thermal
5
and concentration characteristics. Some quantitative parameters of the convective regimes are presented in
Tables 1, 2. According to point (v) of the aforesaid algorithm (see Section 5), the interface temperature gradient
A is determined by the values of A1 for certain pair of the working media and system geometry. One can see, that
the variations of Q does not result to the change in the interface thermal gradient. In the tables, the following
characteristics are given: 4T defines the temperature drop in the bilayer system, Tmax and |u|max are the
maximum values of the temperature and velocity modulus in whole system respectively, Cmax is the maximum
value of the vapor concentration in the gas phase, 4M specifies the percentage change of the evaporative mass
flow rate on the working segment.
Table 1: Parameters of the HFE-7100 – nitrogen system in the channel with the thermally-insulated upper wall
with changes in Q at fixed A1 = −10 K/m
Q, A, 4T , Tmax , Cmax |u|max ×, M0 · 104 , Mx · 104 , 4M ,
kg/(m2 ·s) K/m K K ×10−3 , m/s kg/(m2 ·s) kg/(m·s) %
0.1R0 −9.281 1.6 293.2 0.432 2.985 6.0414 −2.2685 1.913
R0 −9.281 1.6 293.2 0.433 3.081 6.0447 −2.2685 1.912
10R0 −9.281 1.2 293.2 0.439 19.045 6.0773 −2.2685 1.901
Table 2: Parameters of the HFE-7100 – nitrogen system in the channel with the thermally-insulated upper wall
with changes in Q at fixed A1 = 10 K/m
Q, A, 4T , Tmax , Cmax |u|max ×, M0 · 104 , Mx · 104 , 4M ,
kg/(m2 ·s) K/m K K ×10−3 , m/s kg/(m2 ·s) kg/(m·s) %
0.1R0 9.281 1.7 293.7 0.44 2.964 6.0407 2.2685 1.878
R0 9.281 1.7 293.7 0.44 2.868 6.0374 2.2685 1.879
10R0 9.281 2.1 293.7 0.434 20.494 6.0047 2.2685 1.890
One can see from the data given in the tables, the impact of the gas flow rate on the evaporation rate as well
as on the vapor content in the gas layer is rather weak. Two order of magnitude increase in Q does practically
not change of evaporation rate and its variations along the working segment being characterized by 4M . It is
provided by thermal regime on the upper boundary.
7 Summary
In the present work an exact solution of the problem of evaporative convection problem is constructed. The
physical interpretation of the solution is the following: it allows one to describe a flow of the evaporating liquid
driven by the gas flux on the working segment subjected by distributed thermal load from the substrate in
a quite long minichannel with thermally-insulated upper boundary. The new solution takes into account the
inhomogeneous character of phase transition. The explicit expressions for all the unknown functions are derived.
The impact of the flow rate in the gas layer is studied at an example of the bilayer system of certain working
fluids. It is established that changes in gas flow rate slightly influence the thermal characteristics of the system
and the parameters of vapor content in the considered conditions (i. e. for chosen the geometrical configuration,
intensity of external thermal load and gravity action) despite the significant changes in the flow topology. One
can assume, that due to thermal isolation of the upper boundary one can control the evaporation rate and vapor
concentration in the carrier gas in quite large range of variations of the gas flow rate.
8 Acknowledgements
The work of the first author was carried out in accordance with the State Assignment of the Russian Ministry
of Science and Higher Education entitled “Modern methods of hydrodynamics for environmental management,
6
industrial systems and polar mechanics” (Govt. contract code: FZMW-2020-0008) (the problem statement;
construction of the presented type of the Ostroumov – Birikh solution; idea to apply this solution for description of
the evaporative convection; obtaining the system of the linear algebraic equations for the integration constants).
The second author is grateful for financial supports the Krasnoyarsk Mathematical Center 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- 2021-1384)
(verification of the constructed exact solution; explanation of the group-analytical properties; visualization of
the Ostroumov – Birikh solution; explanation of the physical mechanisms of the convective flows described by
presented problem statement).
References
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[Bek16] V.B. Bekezhanova, O.N. Goncharova. Stability of the exact solutions describing the two-layer ows with
evaporation at interface. Fluid Dyn. Res., 48(6):061408, 2016.
[Bek18b] V.B. Bekezhanova, O.N. Goncharova. Analysis of the exact solution for the evaporative convection
problem and properties of the characteristic perturbations. Int. J. Therm. Sci., 130:323–332, 2018.
[Bek18c] V.B. Bekezhanova, I.A. Shefer. Influence of gravity on the stability of evaporative convection regimes.
Microgravity Sci. Technol., 30(4):543–560, 2018.
[She18] I.A. Shefer. Effect of the system geometry on the flow stability of an evaporating liquid. Fluid Dyn.,
53(Suppl.1):S59–S68, 2018.
[She19] I.A. Shefer. Influence of the transverse temperature drop on the stability of two-layer fluid flows with
evaporation. Fluid Dyn., 54(5):603–613, 2019.
[Bek20a] V.B. Bekezhanova, O.N. Goncharova. Analisys of characteristics of two-layer convective flows with
diffusive type evaporation based on exact solutions. Microgravity Sci. Technol., 32(2):139–154, 2020.
[Bek20b] V.B. Bekezhanova, O.N. Goncharova. Influence of the Dufour and Soret effects on the characteristics
of evaporating liquid flows. Int. J. Heat Mass Transfer, 154:119696, 2020.
[Lan1987] L.D. Landau, E.M. Lifshitz. Course of Theoretical Physics, Vol. 6, Fluid Mechanics (2nd ed.). But-
terworth – Heinemahh, Oxford, 1987.
[Bek18d] V.B. Bekezhanova, O.N. Goncharova. Modeling of three dimensional thermocapillary flows with evap-
oration at the interface based on the solutions of a special type of the convection equations. Appl.
Math. Model., 62:145–162, 2018.
7
6 293.2 0.432
y, mm T, K C
292.4 0.216
0
-2 291.6 0
-0.8271 1.0789 2.9848 0 0.02 0.04 0 0.02 0.04
u, mm/s x, m x, m
(a) (b) (c)
6 293.2 0.433
y, mm T, K C
292.4 0.216
0
-2 291.6 0
0 1.5404 3.0808 0 0.02 0.04 0 0.02 0.04
u, mm/s x, m x, m
(d) (e) (f)
6 293.2 0.439
y, mm T, K C
292.6 0.219
0
-2 292 0
0 9.5226 19.0452 0 0.02 0.04 0 0.02 0.04
u, mm/s x, m x, m
(g) (h) (i)
Figure 1: Velocity profiles (a,d,e), structures of the thermal field (b,e,h) and distributions of the vapor concentra-
tion in the background gas (c,f,i) in the two-layer HFE-7100 – nitrogen system with evaporation at A1 = −10 K/m:
(a,b,c) Q = 0.1R0 ; (d,e,f) Q = R0 ; (g,h,i) Q = 10R0
8
6 293.7 0.44
y, mm T, K C
292.85 0.219
0
-2 292 0
-2.9635 -0.8940 1.1756 0 0.02 0.04 0 0.02 0.04
u, mm/s x, m x, m
(a) (b) (c)
6 293.7 0.44
y, mm T, K C
292.85 0.219
0
-2 292 0
-2.8675 -0.0093 2.8488 0 0.02 0.04 0 0.02 0.04
u, mm/s x, m x, m
(d) (e) (f)
6 293.7 0.434
y, mm T, K C
292.65 0.217
0
-2 291.6 0
-1.9075 9.2932 20.4938 0 0.02 0.04 0 0.02 0.04
u, mm/s x, m x, m
(g) (h) (i)
Figure 2: Velocity profiles (a,d,e), structures of the thermal field (b,e,h) and distributions of the vapor concentra-
tion in the background gas (c,f,i) in the two-layer HFE-7100 – nitrogen system with evaporation at A1 = 10 K/m:
(a,b,c) Q = 0.1R0 ; (d,e,f) Q = R0 ; (g,h,i) Q = 10R0
9