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 bilayer liquid-gas system are determined with the help of explicit expressions for all the required functions which are derived in the framework of a two-dimensional evaporative convection problem. The exact solution allows one to correctly take into account the non-uniform character of phase changes through the sharp interface, thermocapillary properties of the liquid { gas boundary and the impact of direct and inverse thermodi usion e ects in the vapor-gas layer. The in uence of gas ow rate on the parameters of convective regimes appearing in a plane horizontal channel is investigated in the case when the rst kind conditions are imposed for the temperature function on the substrate and for the vapor concentration function on the outer boundary which connes 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.
Copyright © by the paper's authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). 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]. Examples of meaningful solutions for studying steady ows with phase transitions are the Ostroumov { Birikh type solutions, which can be e ectively applied to de ne basic characteristics of the convective regimes for various conditions. Two-dimensional solutions of such a class allowed one to specify the features of the ow regimes with evaporation/condensation depending on the di erent 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
The joint ow of evaporating liquid and co-current gas-vapor ux in a plane in nite 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 uids contact along the common thermocapillary surface admitting phase transition due to evaporation/condensation. The interface remains to be at, it is given by the equation y = 0.
The Navier { Stokes equations in the Oberbeck { Boussinesq approximation is utilized for description of the ow in both phases. In plane case the governing equations have the following form: The terms containing function C are taken into account in modelling the ows in the gas phase. The vapor transfer in the carrier gas is described by the di usion equation [Lan1987]: 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 modi ed pressure function which de nes the deviation of real pressure from the hydrostatic one, T is the uid temperature, C is the vapor concentration in background gas, is the uid density (for each phase it is some reference value of the density), ; ; are the coe cients of kinematic viscosity, heat di usivity and thermal expansion, respectively, D is the coe cient of vapor di usion in the gas, is the concentration coe cient of density, the parameters and are the Dufour and Soret coe cients respectively, characterizing the e ects of di usive thermal conductivity and thermodi usion.
System (1) { (5) admits an exact solution of a special type [Bek16]: ui = ui(y); vi = 0;
Tj = Tj (x; y) = ai1 + ai2y x + #j (y); C = C(x; y) = (b1 + b2y) x + (y); pj = pj (x; y): Here, index i denotes characteristics of working uid: 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 #j (y); (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. It is supposed that distributed thermal load is applied on the substrate whereas the upper wall is thermally-insulated:
The boundary conditions are formulated taking into account the character of dependence of the unknown functions on spatial variables de ned by (6). The no-slip conditions on the rigid walls are valid for the velocity functions: Here, " = 0=(RT02), 0 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 ow rate is preset:
M = M (x) = M0 + Mxx: h Z Q = 2u2(y) dy:
Thus, we have two sets of equations, each of them governs the heat and mass transfer in a certain uid, and interface conditions which relate these equations. The conditions imposed on the liquid { gas boundary allow one to correctly describe the phase transition of di usive type. The problem statement used is substantiated in [Bek16, Bek20b]. Solving the problem, one can de ne 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
The ful lment 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 e ects 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: ui (y) = pi (x; y) = Ti (x; y) = C (x; y) =
2 ci3 + ci2y + ci1 y2 + Li3 y63
+ Li4 2y44 ; 5 +K5i y5
Coe cients cij (i = 1; 2; j = 1; :::; 5; 8), c6; c7 are the integration constants; the common scheme for nding of cij will be outlined below. Coe cients L4; Li3; Sj ; Kmi (i = 1; 2; j = 2; :::; 7; m = 1; :::; 8) are calculated via i physical parameters of the problem g; i; i; i; i; D; , coe cients de ning the longitudinal gradients for the temperature and vapor concentration A; ai2, bi (i = 1; 2), and integration constants cij . 5
Let the gas ow rate (
Realizing this algorithm, all the integration constants fci1; ci2; ci3g and fci4; ci5; c6; c7g are found. Hence, the mass ow 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
The liquid { gas two-layer system is considered. The HFE-7100 uid (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 fHFE-7100, nitrogeng (or only HFE-7100): = f1:5 103, 1:2g kg/m3; = f0:38 10 6, 0:15 10 4g m2/s; = f1:8 10 3, 3:67 10 3g K 1; = f0:4 10 7, 0:3 10 4g m2/s; = f0:07, 0:02717g W/(m K); T = 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); 0 = 0:25 kg/mol; = 0:5; " = 0:04 K 1; = 10 5 K; = 5 10 3 K 1.
The in uence of the gas ow rate Q on the ow 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 de ning the external thermal action applied on the substrate are the longitudinal temperature gradient A1 and value # (see condition (8)). In the calculation, the values of A1 equal to 10 or 10 K/m and # = 293:15 K are set. The negative (positive) values of A1 correspond to cooling (heating) of the lower boundary of the ow domain in the direction of the longitudinal axes. Values of the gas ow rate Q are assumed to be 0:1R0, R0 and 10R0, where R0 = 9:6 10 6 kg/(m2 s). The following parameters are xed 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 ow rate can lead to signi cant alteration of the ow 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 e ect action causing the liquid motion from hot domain in zone with lower temperature along the interface (see velocity pro les shown in Figs. 1(a,d,g)). In this case, action of the Marangoni e ect is co-directional to shear stresses generated by gas ux. At small and moderate Q, the peculiar layering of the velocity pro le 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 e ect in these cases, since the maximum velocity is achieved in the liquid layer. Further increase in gas ow rate leads to change of dominant mechanism, the velocity eld 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 e ect the reverse ow appears in the liquid layer (Fig. 2), and this e ect is dominant mechanism de ning the ow pattern under small gas ow 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 in uence of the dynamical e ect 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 counter ow in the liquid persists, although signi cant increase of the velocity in gas phase is observed (Fig. 2(g)).
For both types of external thermal load, the variations of the gas ow rate do not give rise to alteration of the elds of temperature and vapor concentration as well as to signi cant quantitative changes in thermal 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 de nes the temperature drop in the bilayer system, Tmax and jujmax 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 speci es the percentage change of the evaporative mass ow rate on the working segment.
One can see from the data given in the tables, the impact of the gas ow 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
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 ow of the evaporating liquid driven by the gas ux 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 ow rate in the gas layer is studied at an example of the bilayer system of certain working uids. It is established that changes in gas ow rate slightly in uence the thermal characteristics of the system and the parameters of vapor content in the considered conditions (i. e. for chosen the geometrical con guration, intensity of external thermal load and gravity action) despite the signi cant changes in the ow 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 ow rate. 8
The work of the rst 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, 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 nancial supports the Krasnoyarsk Mathematical Center nanced 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) (veri cation 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 ows described by presented problem statement). [Bek18d] V.B. Bekezhanova, O.N. Goncharova. Modeling of three dimensional thermocapillary ows with evaporation at the interface based on the solutions of a special type of the convection equations. Appl. Math. Model., 62:145{162, 2018. 6 y, mm 0 -2 -0.8271
6 y, mm 0 -2 -2 -2.9635 -0.8940 1.1756 0
u, mm/s 0.02 0.04
x, m -2.8675 -0.0093 2.8488 0
u, mm/s 0.02 0.04
x, m (g)
20.4938 0 u, mm/s 0.02 (h) 291.6 0 0.02 0.02 C 0.219