Socolowsky * On a viscous two-fluid channel flow including evaporation

In this contribution a particular plane steady-state channel ow including evaporation e ects is investigated from analytical point of view. The channel is assumed to be horizontal. The motion of two heavy viscous immiscible uids is governed by a free boundary value problem for a coupled system of Navier-Stokes and Stephan equations. The ow domain is unbounded in two directions and the free interface separating partially both liquids is semi-in nite, i.e. in nite in one direction. The free interface begins in some point Q where the half-line Σ1 separating the two parts of the channel in front of Q ends. Existence and uniqueness of a suitable solution inweightedHÖLDER spaces can be proved for small data (i.e. small uxes) of the problem.


Introduction
In this paper we are concerned with the investigation of a particular free boundary value problem (= BVP) for a two-uid non-isothermal channel ow. The in nite channel is assumed to be horizontal and it contains a partial inner wall (cf. the thin red line Σ in Figure 1) which is semi-in nite. The ow problem is assumed to be stationary and 2D. In Figure 1 the blue line denotes the lower channel wall which moves with constant speed R in x -direction. The red line Σ denotes the upper channel wall that is at rest. Finally, by the cyan curve Γ we understand the a priori unknown free interface between the two uid layers. It has the representation x = φ(x ) where the function φ has to be found as well as the ow elds for velocity v(x), for the pressure p(x) and for the temperature θ(x).
Models of the described kind are quite important in many technological and scienti c applications. Corresponding examples may be found in the eld of materials science, particularly in coating and solidi cation processes with evaporation or in crystal-growth processes (cf. [1][2][3][4][5][6][7][8][9][10][11][12]). The investigations of such problems are performed from technical point of view as well as from analytical and/or numerical point of view. It was our main objective to obtain statements about the existence and/or uniqueness of free BVP for evaporation problems.
The ow describes a coupled heat-and mass transfer (Stephan equations). The (positive) uxes Fm are prescribed in each uid layer Ωm(m = , ) (cf. Fig. 1). The lower liquid layer is characterized by red color whereas the upper one is marked by green color. Both liquids are heavy, viscous, heat-conducting, incompressible and immiscible. Therefore, the mathematical model can incorporate evaporation e ects. The surface tension σ(θ) is temperature-depending in a known manner. By νm, ηm and λm (m = , ) we understand the domain-wise (i.e. regional) constant values of the kinematic viscosity, of the density and of the thermal conductivity, respectively, of the m-th uid. By h∞ we denote the (asymptotic) position of the free interface Γ when x goes to +∞. By n, τ, respectively, the unit normal and the unit tangential vectors with respect to Γ are denoted. Their orientation (direction) is the same as for x , x . By g and eg we understand the acceleration and the direction of gravity, respectively. Concerning the interface tension σ we suppose the following linear function of temperature θ which is frequently used in the literature. This leads to an e ect which is called Benard-Marangoni-e ect or thermo-capillary convection. Finally, the following symbols and abbreviations have been used throughout this paper: δ j (t) := {x = t} ∩ Ω j , j = , } is some cross section of Ω j . The frictional stress tensor has the subsequent elements: . The symbol w(x ) | Γ represents the jump of the eld w crossing the interface Γ from below to above:

Mathematical model
The governing equations (Navier -Stokes & Stephan) of the problem which yield in Ω := Ω ∪ Ω read as follows (1) They are supplied by the boundary conditions at the lower moving wall Σ : Let us emphasize that the value 0 in Eq. (2) does not represent the absolute temperature but some dimensionless value which is in fact the di erence to some reference temperature related to a characteristic temperature di erence.
The boundary conditions at the walls at rest Σ k (k = , ) look like: Let us explain that the boundary conditions (3) for k = mean both sides Σ ± of the partial inner wall Σ . Finally, the conditions at the free interface Γ are: As a consequence one gets the relation: In order to prove the unique solvability of the BVP in appropriate functional spaces the following twocycle iteration scheme was applied.
This scheme was introduced by V.V. Pukhnachev and V.A. Solonnikov about 45 years ago (cf. e.g. [13,14] or [15]). The two-cycle iteration scheme was also applied in the papers [14,15] and by the author in [9,16]. In the references [11,17,18] other methods are used to handle di erent free BVP.
The scheme (5) is very senseful in cases where the free boundary is semi-in nite. In a rst cycle the three ow elds v, p, θ are computed in a ow domain with xed boundaries neglecting one of the boundary conditions -mostly the normal stress condition (4) , i.e. the 5th equation in (4). This rst cycle is then divided into several steps: The linear problem with xed boundary containing the corresponding estimates for the solution, a model problem at the separation point Q for the determination of the weight functions, the regularity of the solutions at in nity and then the nonlinear problem with xed boundary.
In a second stage the neglected boundary condition is used in order to compute a new shape of the free boundary (and simultaneously a new shape of the entire ow domain). This equation is usually where K(x ) denotes the curvature of Γ in x and it is equal to the left-hand side of Eq. (4) . In both cycles a related linear problem is solved and the continuous dependence of the solutions on the boundary data is also proved. Then BANACH's xed point argument related to some contraction operator B shows the remaining parts for small data.

Function spaces
First of all we de ne some weighted HÖLDER spaces. Let B be an arbitrary domain in R and N ⊂B a manifold of dimensionn < . De ne further ϱ N (x) := dist(x, N). By β = (β , β ) we understand a multiindex, and r is the integer part of r. Then by C r (B)(r > , non-integer) we mean the well-known HÖLDER space with a nite norm |u| (r) B . Now we obtain the subsequent weighted HÖLDER space Let us remark that the weight functions in (6) The spaces C r s (B ∪ B , N) represent the natural generalization of the last ones to the case of two separate subdomains B k as we have.
Finally, the weighted HÖLDER spaces containing the generalized solutions, are They are essentially used throughout this paper and their norms are given by || u || r,z Ωm ,s := |u| C r s (Ω m ,Q) + | exp(zx )u| (r) The weight functions here in formula (8) are exponential functions and they decay at the in nities. As above, we obtain for our double channel C r s,z (Ω) := C r s,z (Ω ∪ Ω , Q). At the end, for functions of one real variable we deal with the space C r s,z (R + ) supplied with the norm || f || r,z R + ,s = |f | C r s (I , ) + |f (x ) exp(zx )| (r) I + .

On the Basic Flow for Large x
In this section we are interested in getting an approriate starting (or initial) solution for the iteration scheme (5). For this purpose, and under the assumptions we calculate for given values F , F , R, θ = , θ , θ and associated rheological parameters the ow elds and values v(x), p(x), θ(x), p , h∞, θ∞ . The value θ∞ which has not been de ned before describes the (asymptotic) value of the temperature θ at the free interface when x goes to +∞. Let us emphasize that the assumptions guarantee solution elds that are uniform and unidirectional (not depending on main-stream direction x ). Under the assumptions (9) the governing Eqs. (1) take the subsequent reduced form: where the second equation replaces the continuity Eq. (1) Now it is possible to divide the original problem into three independent problems for the ow elds. Let us start with the problem for velocities v: For the pressure p one obtains the following equations Finally, the problem for temperature θ reads In Eqs. (10), (11), (12) the superscripts (k), (k = , ) or (+) denote the corresponding uid layer and the subregion x . The solutions of these three (independent) problems are of NUSSELT type (cf. also [19]) and allow the representation The coe cients in (13) are given by Note, that the values h∞ and p are already known for these expressions (see Eq. (16) below). That is why it follows θ∞ = (λ θ h∞)/[λ ( − h∞) + λ h∞] and for the complete temperature and pressure elds one obtains Since the associated linear problem is completely decomposed, we got the same polynomial equation for the determination of the value h∞ as in the former paper [17].
In [17] the subsequent two lemmas were proved.  In Ω − one obtains In Ω − one gets, respectively, It is well-known that the pressure p can be determined only up to an additive constant in channel ows (cf. k , k in formulae (17), (18)).
which can be obtained from the 5-th condition (4) of (4) by setting v = , p = const., θ = as the initial solution for F = F = R = θ = θ = . Let ξ = ξ (x ) be a smooth cut-o function vanishing for |x | and being equal to 1 for |x | . Finally, assume that η > η is satis ed. This makes physically sense. Now, the di erence function ω(x ) := (φ(x ) − φ (+) (x )) is equivalent to exp(− g(η − η )/ a x ) as x → +∞. For the unknown function ω(x ) we get a two-point BVP like BVP (8.8) from [20] subtracting Eq. (19) from Eq. (4) . A di erence to BVP (8.8) consists in the following. We have to replace β by g(η − η )/ a everywhere and, furthermore, we have to introduce the operator T ( ) by The remaining part of the proof of the main theorem is a slightly modi ed repetition of the proof of Theorem 8.1 in [20]. First of all, one has to study the dependence of the solution to the nonlinear auxiliary problem with xed boundary on small variations of the boundary. After getting the corresponding estimates one applies BANACH's xed point principle to the subsequent operator equation. Instead of the operator Eq. (8.10) from [20] we have to study the following one: with T ( ) given in (20) and the other parts taken from [20]. Since T ( ) is a contraction operator for small θ, we can conclude as in [20] that B is a contraction operator in the ball || ω || +s,z R + , +s < ε. Consequently, we have proved the main result of this paper.

Results
Let us formulate the main result of this contribution. A sketch of the proof has been given before. A very detailed application of this method can be found in the thesis [16] as well as in the article [20]. where ξ is the cut-o function described above, (v (−) , p (−) , θ (−) ) is the basic exact solution given by (17), (18) in both channels on left-hand side. The function φ (+) is the solution to the free BVP (19). Moreover, ϑ , w ∈ C s+ s,z (Ω), q ∈ C s+ s− ,z (Ω ∪ Ω + ), ∇q ∈ C s s− ,z (Ω) and ω ∈ C +s +s,z (R + ) hold.
Remark 6.2. If Eq. (16) has more than one real root h∞ between 0 and 1 then the statements of Theorem 6.1 remain true in the neighbourhood of each value.