Nonisothermal viscous two-fluid flows occur in numerous kinds of coating devices. The corresponding mathematical models often represent two-dimensional free boundary value problems for the Navier-Stokes equations or their modifications. In the present paper we are concerned with a particular problem of coupled heat and mass transfer. Marangoni convection is incorporated, too. The solvability of a corresponding stationary problem is discussed. The obtained results generalize previous results for a similar isothermal problem.

1. Introduction

Thermocapillary convection describes a fluid motion driven by surface-tension gradients on a liquid-liquid interface, where these gradients arise from the temperature dependence of interface tension. This type of convection is quite important in several technological and scientific applications; interesting examples may be found in the field of materials science, particularly in coating and solidification processes or in crystal-growth processes (cf. [15]).

In this paper we study a problem for a 2D stationary flow with two viscous incompressible heat-conducting fluids (having kinematic viscosities 𝜈𝑖>0, densities 𝜚𝑖>0, and thermal conductivities 𝜆𝑖, 𝑖=1,2) down an inclined bottom 𝑆0 having a slope 𝛼 (cf. Figure 1). In fact, the bottom 𝑆0 represents a perturbed plane. Assume that the bottom is given by the formula 𝑆0={𝐱=(𝑥1,𝑥2)2𝑥2=𝜀𝜑0(𝑥1),<𝑥1<+} with 𝜑0 having a compact support, that is, 𝜑0(𝑥1)0 for |𝑥1|𝑅0>0, and suppose that the direction 𝐞𝑔 of the gravitational force is the vector 𝐞𝑔=(sin𝛼,cos𝛼)𝑇 which makes (with respect to the chosen coordinate system) an angle 𝛼=𝜋/2𝛼(0<𝛼𝜋/2) with the 𝑥1-axis. Note that the corresponding problem will be formulated in dimensionless form. The concrete transition to that formulation can be found in [6].

Let us formulate the problem. We consider the two-fluid flow down the inclined bottom 𝑆0 caused by gravity 𝑔𝐞𝑔, only. This means mathematically that the positive (final) layer thickness in each liquid layer Ω𝑖(𝑖=1,2) is a priori prescribed. In slide coaters, such flows occur on some parts of the coater. The corresponding flow fields and layer profiles are essential there.

Suppose that the free interface Γ1 separating the two fluid layers and the upper free surface Γ2 admit the parametrizations Γ𝑖={𝐱2𝑥2=𝜓𝑖(𝑥1),<𝑥1<+}(𝑖=1,2), where the functions 𝜓𝑖(𝑖=1,2) are a priori unknown and have to be found. Let 𝑖>0(0<1<2) be the prescribed constant limits of 𝜓𝑖(𝑥1)(𝑖=1,2), at infinity. The problem under consideration has the following form: to find a vector of velocity 𝐯=(𝑣1(𝑥1,𝑥2),𝑣2(𝑥1,𝑥2))𝑇, a pressure 𝑝(𝑥1,𝑥2), a temperature 𝜃(𝑥1,𝑥2), and functions 𝜓𝑖(𝑥1)(𝑖=1,2) satisfying in the domain Ω=Ω1Ω2 with Ω1={𝐱2𝜀𝜑0(𝑥1)<𝑥2<𝜓1(𝑥1),<𝑥1<+} and Ω2={𝐱2𝜓1(𝑥1)<𝑥2<𝜓2(𝑥1),<𝑥1<+}, then the following equations of a coupled heat and mass transfer are

(𝐯)𝐯𝜈21𝐯+𝜚𝑝=𝑔𝐞𝑔,𝐯=0,(𝐯)𝜃𝜆2𝜃=0,(1.1) and the boundary and integral conditions are


𝜃𝜃𝑎+𝜆2𝜕𝜃|||𝜕𝑛Γ2=0,𝐯𝐧|Γ2||=0,𝝉𝐒(𝐯)𝐧Γ2=𝑏2𝜕𝜃|||𝜕𝜏Γ2,dd𝑥1𝜓2𝑥11+𝜓2𝑥12=1𝜎2𝑝(𝜃)𝑎||𝑝+𝐧𝐒(𝐯)𝐧Γ2,lim||𝑥1||+𝜓2𝑥1=2.(1.4) In [1] it was shown that for a large number of liquids the surface tensions 𝜎𝑖 can be regarded as linear functions of the temperature 𝜃 along the free interface Γ𝑖(𝑖=1,2) (cf. also [3, 5]) as follows:

𝜎𝑖(𝜃)=𝑎𝑖𝑏𝑖𝜃𝑎𝑖,𝑏𝑖>0,𝑖=1,2.(1.5) By 𝜆𝑚 we denote the thermal conductivity of the 𝑚th fluid (𝑚=1,2) in Problem (1.1)–(1.4). The symbol 𝑔 means the acceleration of gravity. The value 𝜃0 is the (constant) given temperature of the wall 𝑆0. Without loss of generality one can suppose that 𝜃0=0 and that 𝜃 is in fact the difference between the physical temperature and 𝜃0. By 𝑝𝑎 and 𝜃𝑎 we denote the given (constant) pressure and temperature of the ambient air, respectively.

Furthermore, the subsequent notations have been used: 𝐧 and 𝝉 are unit vectors normal and tangential to Γ1 and oriented as 𝑥2,𝑥1, respectively. By 𝐚𝐛 we mean the inner product of 𝐚,𝐛2,=(𝜕/𝜕𝑥1,𝜕/𝜕𝑥2)𝑇 is the gradient operator, 𝑝=grad𝑝,𝐯=div𝐯, 𝜚|Ω𝑚=𝜚𝑚(𝑚=1,2) is the restriction of 𝜚 to Ω𝑚 (analogously for 𝜈 and 𝜆) and 2 denotes the Laplace operator. By 𝐒(𝐯) we denote the deviatoric stress tensor, that is, a matrix with elements 𝑆𝑖𝑗(𝐯)=𝜚𝜈(𝜕𝑣𝑖/𝜕𝑥𝑗+𝜕𝑣𝑗/𝜕𝑥𝑖)(𝑖,𝑗=1,2). The symbol [𝑤]|Γ1 denotes the jump of 𝑤 crossing the free interface Γ1, that is,

𝑤𝐱0||Γ1=lim𝐲𝐱0𝑤(𝐲)lim𝐱𝐱0𝑤𝐱(𝐱),0Γ1,𝐲Ω1,𝐱Ω2,(1.6) and the symbol 𝑤|Γ1 denotes the limit from below at the interface Γ1; more precisely

𝑤𝐱0||Γ1=lim𝐲𝐱0𝑤𝐱(𝐲),0Γ1,𝐲Ω1.(1.7) Note that the left-hand side of (1.3)6 (i.e., of the sixth equation in (1.3)) is equal to the curvature 𝐾1(𝑥1) of Γ1. The same is true for 𝐾2(𝑥1) in case of Γ2. Furthermore, (1.3)5 represents the mathematical expression of the so-called Marangoni convection.

2. General Solution Technique

Mathematical problems for the stationary flows of a viscous incompressible fluid with a free boundary were studied by many authors. Numerous references on this topic can be found, for example, in the bibliographies of [710]. In the analytical investigations [13, 5, 11], the temperature dependence was additionally taken into account. Numerical studies of nonisothermal free boundary problems can be found in the papers [1, 6].

For free boundary problems in which the unknown flow domain is unbounded in two directions as in Problem (1.1)–(1.4), a special linearization scheme is necessary (cf. [7, 8] and others).

In order to solve such kind of problems—in [7], and independently in [12], an appropriate scheme was proposed based on a linearization of the original problem on a corresponding exact solution in the unperturbed “uniform” flow domain, say: Π={𝐱20<𝑥2<11<𝑥2<2}. The main difference of this scheme from previous applied ones is that on each step of iterations the determination of 𝐯,𝑝,𝜃 is not separated from the determination of the free boundaries Γ𝑖(𝑖=1,2) (i.e., from the determination of the functions 𝜓𝑖 describing Γ𝑖). For Problem (1.1)–(1.4) this scheme can be illustrated by the diagram

𝐯0,𝑝0,𝜃0,𝜓01,𝜓02𝐯1,𝑝1,𝜃1,𝜓11,𝜓12𝐯𝑚,𝑝𝑚,𝜃𝑚,𝜓𝑚1,𝜓𝑚2,(2.1) where on each step of iterations the linearized problem is solved in the same “strip-like” domain and the functions 𝐯,𝑝,𝜃, and 𝜓𝑖(𝑖=1,2) are determined simultaneously.

A significant part in deriving the correct linearization takes the determination of exact solutions of the nonlinear problems in a “uniform” (not distorted) flow domain. These exact (basic) solutions in the uniform domain Π will be calculated in the appendix. They are also important for the numerical flow simulation: they can be used as inlet boundary data in more complicated problems. In [8] the analogous isothermal problem (without any inclusion of temperature) to Problem (1.1)–(1.4) was solved by numerical methods.

3. Function Spaces

When studying Problem (1.1)–(1.4), it is useful to work with weighted Sobolev spaces. Let Π𝑚(𝑚=1,2) be the strip-like domains

Π1=𝐱20<𝑥2<1,<𝑥1,Π<+2=𝐱21<𝑥2<2,<𝑥1,<+(3.1) and Π=Π1Π2 their union. We introduce the space 𝑊𝛽𝑙,2(Π) of functions 𝐮 on Π with restrictions 𝐮(𝑚)=𝐮|Π𝑚 belonging to 𝑊𝛽𝑙,2(Π𝑚)(𝑚=1,2) having the finite norms

𝐮(𝑚);𝑊𝛽𝑙,2Π𝑚=𝐮(𝑚)𝛽exp1+𝑥21;𝑊𝑙,2Π𝑚,(𝑚=1,2),(3.2) where 𝑊𝑙,2(Π𝑚) is the usual Sobolev space. The norm in 𝑊𝛽𝑙,2(Π) is given by

𝐮;𝑊𝛽𝑙,2=(Π)2𝑚=1𝐮(𝑚)𝛽exp1+𝑥21;𝑊𝑙,2Π𝑚.(3.3) If 𝛽>0, then elements of 𝑊𝛽𝑙,2(Π) vanish exponentially as |𝑥1|, and if 𝛽<0, then elements 𝐮𝑊𝛽𝑙,2(Π) might exponentially increase as |𝑥1|.

The spaces 𝑊𝛽𝑙1/2,2() of functions defined on can be introduced analogously. Let 𝑆={𝑥Π𝑥1,𝑥2={0,1,2}} be a line. Denote by 𝑊𝛽𝑙1/2,2(𝑆) the spaces of traces on 𝑆 of functions from 𝑊𝛽𝑙,2(Π). Then 𝑊𝛽𝑙1/2,2() coincides with 𝑊𝛽𝑙1/2,2(𝑆), that is, if 𝐮𝑊𝛽𝑙1/2,2(Π), then 𝐮(,)𝑊𝛽𝑙1/2,2().

In the paper the spaces of scalar and vector-valued functions are not distinguished in notations. The norm for vector-valued functions is then the sum of the norms of the corresponding coordinate functions.

4. Solvability Results

Problem (1.1)–(1.4) can be handled by the same methods as in [8, 13]. Let us start with the main result about this problem.

Theorem 4.1. Let 𝑆0={𝐱2𝑥2=𝜀𝜑0(𝑥1),<𝑥1<+},𝜑0𝑊𝛽𝑙+5/2,2() with 𝑙0, and 𝛽=|𝛽0|sin𝛼>0, where 𝛼 denotes the slope of the inclined bottom 𝑆0. Assume that 𝛼 is sufficiently small. Then there exist positive numbers ̂𝜀,̂𝑟 such that for every 𝜀(0,̂𝜀). Problem (1.1)–(1.4) has a unique solution (𝐯,𝑝,𝜃,𝜓1,𝜓2)𝑇. The solution admits the representation 𝐯(𝐱)=𝐯0(𝐱)+𝜀𝐮(𝐱),𝑝(𝐱)=𝑝0(𝐱)+𝜀𝑞(𝐱),𝜃(𝐱)=𝜃0𝜓(𝐱)+𝜀𝜗(𝐱),(4.1)1𝑥1=1+𝜀Ψ1𝑥1,𝜓2𝑥1=2+𝜀Ψ2𝑥1,(4.2) where {𝐯0,𝑝0,𝜃0} are the functions of the basic solution from (A.3)–(A.6), while 𝐔=𝐮,𝑞,𝜗,Ψ1,Ψ2𝑇𝑊𝛽𝑙+2,2(Π)2×𝑊𝛽𝑙+1,2(Π)×𝑊𝛽𝑙+2,2𝑊(Π)×𝛽𝑙+5/2,2()2𝒟𝛽𝑙,2𝑊(Π),(4.3) and the following inequalities hold: 𝐔;𝒟𝛽𝑙,2𝑊(Π)̂𝑟,̂𝜀constsin2𝛼.(4.4)

We would like to present a short sketch of the proof of this theorem by successive approximations. First, the original (perturbed) and unknown flow domain Ω (cf. Figure 1) is transformed onto the uniform (strip-like) domain Π. Then, using this transformation mapping, the original flow Problem (1.1)–(1.4) is linearized over the basic solution (A.3)–(A.6) (see the appendix) in domain Π. By 𝔑 we denote the operator of the left-hand side of that linearized auxiliary problem. In virtue of a corresponding theorem for the linear auxiliary problem (see, e.g., [13]), there exists a bounded inverse operator 𝔑1 such that

𝔑1𝛽𝑙,2𝑊(Π)𝒟𝛽𝑙,2𝑊(Π),(4.5) with 𝛽=|𝛽0|sin𝛼. Herein 𝛽0 is independent of 𝛼 and depends on eigenvalues of the operator pencils associated with the corresponding linear problem (cf. [13]). Also, the multidimensional space 𝛽𝑙,2𝑊(Π) to which the right-hand side of the linearized problem belongs is introduced in a similar way as that of the space 𝒟𝛽𝑙,2𝑊(Π) (see above). Moreover, one can show that there holds the estimate

𝔑1𝐶,sin𝛼(4.6) with a constant 𝐶 being independent of 𝛼. Therefore, Problem (1.1)–(1.4) is equivalent to the following operator equation in the space 𝒟𝛽𝑙,2𝑊(Π):

𝐔=𝔑1𝔉(𝐔)𝔎(𝐔),(4.7) where 𝐔=(𝐮,𝑞,𝜗,Ψ1,Ψ2)𝑇, and 𝔉(𝐔)=(𝑓1(𝐮,𝑞,𝜗,Ψ1,Ψ2),𝑓2(𝐮,𝑞,𝜗,Ψ1,Ψ2),0,𝑓3(𝐮,𝑞,𝜗, Ψ1,Ψ2),)𝑇 denotes the long vector of right-hand side after the linearization. The elements of the right-hand side vector depend on 𝜀 via the transformation mapping of the original flow domain. In order to show the convergence of the successive approximations (𝐮(𝑛),𝑞(𝑛),𝜗(𝑛),Ψ1(𝑛),Ψ2(𝑛))𝑇, it is sufficient to show that the operator 𝔎 is a contraction mapping in a ball of the space 𝒟𝛽𝑙,2𝑊(Π) for small values of 𝜀 and 𝛼.

Note that the corresponding isothermal problem to Problem (1.1)–(1.4) (i.e., without any inclusion of temperature) was analytically examined in details in the papers [8, 13]. In order to prove Theorem 4.1 in details, one has to repeat and to modify all the investigations from those papers. Since the temperature equation is also nonlinear elliptic there, are not essential changes in the proof. Thus we omit the detailed proof here.


On the Nonisothermal Base Flow

Let us now investigate the two-fluid flow in the “uniform” two-fluid domain Π down the really inclined plane, that is, in the case where 𝑆0 is replaced by the inclined straight line 𝑆={𝐱=(𝑥1,𝑥2)2𝑥20,<𝑥1<+}. We suppose that the (unknown) free interface and the free surface Γ𝑖 permit the representations Γ𝑖={𝐱2𝑥2=𝜓𝑖(𝑥1)=𝑖=const.,<𝑥1<+}, and we are looking for stationary unidirectional flows. Such flows fulfil the assumptions

𝑣20,𝜕𝑣1𝜕𝑥10,𝜕𝜃𝜕𝑥10.(A.1) As a consequence we obtain that the pressure gradient downstream is a constant 𝜕𝑝/𝜕𝑥1=𝑝0=const. Since the motion is generated by gravity, only the pressure gradient downstream 𝑝0 must be equal to 0. Under these assumptions, the nonisothermal Navier-Stokes equations reduce to

𝜈𝜚2𝑣1=𝜚𝑔sin𝛼,𝜕𝑝𝜕𝑥2=𝜚𝑔cos𝛼,𝜆2𝜃=0,(A.2) and the equation of continuity (1.1)2 is automatically fulfilled. Problem (1.1)–(1.4) can now be transformed to three systems of equations containing the unknowns 𝑣1(𝑥2),𝑝(𝑥2), and 𝜃(𝑥2), independently. These three systems have the following solutions:

𝑣01𝑥2=1𝑔sin𝛼2𝜈1𝑥22+21𝑟𝜈2+1𝜈1𝑥2,0𝑥21,1𝑔sin𝛼2𝜈22𝑥22+12𝜈2212+12𝜈121+1𝑟𝜈2211,1𝑥22,𝑣(A.3)02𝑥2𝜈0,with𝑟=1𝜚1𝜈2𝜚2,(A.4) while

𝑝0𝑥2=1𝑥2𝜚1𝑔cos𝛼+21𝜚2𝑔cos𝛼+𝑝𝑎,0𝑥21,2𝑥2𝜚2𝑔cos𝛼+𝑝𝑎,1𝑥22,𝜃(A.5)0𝑥2=𝜃𝑎𝜆2𝜆21+𝜆121+𝜆1𝜆2𝑥2,0𝑥21,𝜃𝑎𝜆1𝑥2+1𝜆2𝜆1𝜆21+𝜆121+𝜆1𝜆2,1𝑥22.(A.6) The so-called fluxes 𝐹𝑖(𝑖=1,2) for both liquid layers that are the integrals

𝐹1=10𝑣1(1)𝑥2d𝑥2,𝐹2=21𝑣1(2)𝑥2d𝑥2(A.7) play an important role in the theoretical proof of the solvability to Problem (1.1)–(1.4). Their values can be calculated as

𝐹11=𝑔sin𝛼3𝜈131+12𝑟𝜈22121,𝐹21=𝑔sin𝛼3𝜈2213+2112𝜈121+1𝑟𝜈2121.(A.8) Recall that if sin𝛼 is small, then for the fluxes 𝐹𝑖 the same is true. Furthermore, let us emphasize that solution (A.3)–(A.6) to problem (A.2) is frequently called Nusselt  solution of the inclined film flow in the fluid mechanical literature.


The author is grateful for the comments and suggestions of the anonymous referee.