On the Nusselt Solution of a Nonisothermal Two-Fluid Inclined Film Flow

Abstract

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. [1–5]).

In this paper we study a problem for a 2D stationary flow with two viscous incompressible heat-conducting fluids (having kinematic viscosities 𝜈_𝑖>0, densities 𝜚_𝑖>0, and thermal conductivities 𝜆_𝑖, 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)∈ℝ²∶𝑥₂=𝜀𝜑₀(𝑥₁),−∞<𝑥₁<+∞} with 𝜑₀ having a compact support, that is, 𝜑₀(𝑥₁)≡0 for |𝑥₁|≥𝑅₀>0, and suppose that the direction 𝐞_𝑔 of the gravitational force is the vector 𝐞_𝑔=(sin𝛼,−cos𝛼)^𝑇 which makes (with respect to the chosen coordinate system) an angle 𝛼^∗∶=𝜋/2−𝛼(0<𝛼≤𝜋/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].

Figure 1 

Flow domain of a nonisothermal two-fluid inclined film flow.

Let us formulate the problem. We consider the two-fluid flow down the inclined bottom 𝑆₀ caused by gravity 𝑔𝐞_𝑔, only. This means mathematically that the positive (final) layer thickness in each liquid layer Ω_𝑖(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 Γ_𝑖={𝐱∈ℝ²∶𝑥₂=𝜓_𝑖(𝑥₁),−∞<𝑥₁<+∞}(𝑖=1,2), where the functions 𝜓_𝑖(𝑖=1,2) are a priori unknown and have to be found. Let ℎ_𝑖>0(0<ℎ₁<ℎ₂) be the prescribed constant limits of 𝜓_𝑖(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 𝜓_𝑖(𝑥₁)(𝑖=1,2) satisfying in the domain Ω=Ω₁∪Ω₂ with Ω₁={𝐱∈ℝ²∶𝜀𝜑₀(𝑥₁)<𝑥₂<𝜓₁(𝑥₁),−∞<𝑥₁<+∞} and Ω₂={𝐱∈ℝ²∶𝜓₁(𝑥₁)<𝑥₂<𝜓₂(𝑥₁),−∞<𝑥₁<+∞}, then the following equations of a coupled heat and mass transfer are

(𝐯⋅∇)𝐯−𝜈∇²𝐯+1𝜚∇𝑝=𝑔𝐞^𝑔(𝐯⋅∇)∇𝜃⋅−𝐯𝜆=∇0₂,,𝜃=0,(1.1) and the boundary and integral conditions are

[𝜃]||𝐯|_𝑆0=𝟎,𝜃||_𝑆0=𝜃₀,(1.2)_Γ1=0,𝜆𝜕𝜃𝜕𝑛|||_Γ1𝐯⋅𝐧|_Γ−1=0,[𝐯]||_Γ1=𝟎,=0,[𝝉⋅𝐒(𝐯)𝐧]||_Γ1=−𝑏₁𝜕𝜃𝜕𝜏|||_Γ⁻¹d,d𝑥₁𝜓^1𝑥₁1+𝜓^1𝑥₁²=1𝜎₁(𝜃)[−𝑝+𝐧⋅𝐒(𝐯)𝐧]||^Γ1lim_{||𝑥1||→+∞},𝜓₁𝑥₁=ℎ₁,(1.3)

𝜃−𝜃^𝑎+𝜆₂𝜕𝜃𝜕𝑛|||_Γ2𝐯⋅𝐧|_Γ2=0,=0,𝝉⋅𝐒(𝐯)𝐧||_Γ2=−𝑏₂𝜕𝜃𝜕𝜏|||_Γ2d,d𝑥₁𝜓^2𝑥₁1+𝜓^2𝑥₁²=1𝜎₂(𝜃)𝑝^𝑎lim_{||𝑥1||→+∞}−𝑝+𝐧⋅𝐒(𝐯)𝐧||_Γ2,𝜓₂𝑥₁=ℎ₂.(1.4) In [1] it was shown that for a large number of liquids the surface tensions 𝜎_𝑖 can be regarded as linear functions of the temperature 𝜃 along the free interface Γ_𝑖(𝑖=1,2) (cf. also [3, 5]) as follows:

𝜎_𝑖(𝜃)=𝑎_𝑖−𝑏_𝑖𝜃𝑎_𝑖,𝑏_𝑖>0,𝑖=1,2.(1.5) By 𝜆_𝑚 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 𝑝_𝑎 and 𝜃_𝑎 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 𝐚⋅𝐛 we mean the inner product of 𝐚,𝐛∈ℝ²,∇=(𝜕/𝜕𝑥₁,𝜕/𝜕𝑥₂)^𝑇 is the gradient operator, ∇𝑝=grad𝑝,∇⋅𝐯=div𝐯, 𝜚|_Ω𝑚=𝜚_𝑚(𝑚=1,2) is the restriction of 𝜚 to Ω_𝑚 (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 𝑆_𝑖𝑗(𝐯)=𝜚𝜈(𝜕𝑣_𝑖/𝜕𝑥_𝑗+𝜕𝑣_𝑗/𝜕𝑥_𝑖)(𝑖,𝑗=1,2). The symbol [𝑤]|_Γ1 denotes the jump of 𝑤 crossing the free interface Γ₁, that is,

𝑤𝐱₀||_Γ1∶=lim_𝐲→𝐱0𝑤(𝐲)−lim_𝐱→𝐱0𝑤(𝐱),𝐱₀∈Γ₁,𝐲∈Ω₁,𝐱∈Ω₂,(1.6) and the symbol 𝑤|_Γ⁻¹ denotes the limit from below at the interface Γ₁; more precisely

𝑤𝐱₀||_Γ⁻¹∶=lim_𝐲→𝐱0𝑤(𝐲),𝐱₀∈Γ₁,𝐲∈Ω₁.(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 [7–10]. In the analytical investigations [1–3, 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={𝐱∈ℝ²∶0<𝑥₂<ℎ₁∨ℎ₁<𝑥₂<ℎ₂}. 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 Γ_𝑖(𝑖=1,2) (i.e., from the determination of the functions 𝜓_𝑖 describing Γ_𝑖). For Problem (1.1)–(1.4) this scheme can be illustrated by the diagram

𝐯⁰,𝑝⁰,𝜃⁰,𝜓⁰¹,𝜓⁰²⟶𝐯¹,𝑝¹,𝜃¹,𝜓¹¹,𝜓¹²⟶⋯⟶𝐯^𝑚,𝑝^𝑚,𝜃^𝑚,𝜓^𝑚1,𝜓^𝑚2⟶⋯,(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 𝜓_𝑖(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 Π_𝑚(𝑚=1,2) be the strip-like domains

ΠΠ₁∶=𝐱∈ℝ²∶0<𝑥₂<ℎ₁,−∞<𝑥₁<+∞₂,∶=𝐱∈ℝ²∶ℎ₁<𝑥₂<ℎ₂,−∞<𝑥₁<+∞,(3.1) and Π=Π₁∪Π₂ their union. We introduce the space 𝑊^𝛽𝑙,2(Π) of functions 𝐮 on Π with restrictions 𝐮^(𝑚)=𝐮|_Π𝑚 belonging to 𝑊^𝛽𝑙,2(Π_𝑚)(𝑚=1,2) having the finite norms

‖‖𝐮^(𝑚);𝑊^𝛽𝑙,2Π_𝑚‖‖=‖‖‖𝐮^(𝑚)exp𝛽1+𝑥²¹;𝑊^𝑙,2Π_𝑚‖‖‖,(𝑚=1,2),(3.2) where 𝑊^𝑙,2(Π_𝑚) is the usual Sobolev space. The norm in 𝑊^𝛽𝑙,2(Π) is given by

‖‖𝐮;𝑊^𝛽𝑙,2(Π)‖‖=²_𝑚₌₁‖‖‖𝐮_(𝑚)exp𝛽1+𝑥²¹;𝑊^𝑙,2Π_𝑚‖‖‖.(3.3) If 𝛽>0, then elements of 𝑊^𝛽𝑙,2(Π) vanish exponentially as |𝑥₁|→∞, and if 𝛽<0, then elements 𝐮∈𝑊^𝛽𝑙,2(Π) might exponentially increase as |𝑥₁|→∞.

The spaces 𝑊^{𝛽𝑙−1/2,2}(ℝ) of functions defined on ℝ can be introduced analogously. Let 𝑆={𝑥∈Π∶𝑥₁∈ℝ,𝑥₂=ℎ∈{0,ℎ₁,ℎ₂}} be a line. Denote by 𝑊^{𝛽𝑙−1/2,2}(𝑆) the spaces of traces on 𝑆 of functions from 𝑊^𝛽𝑙,2(Π). Then 𝑊^{𝛽𝑙−1/2,2}(ℝ) coincides with 𝑊^{𝛽𝑙−1/2,2}(𝑆), that is, if 𝐮∈𝑊^{𝛽𝑙−1/2,2}(Π), then 𝐮(⋅,ℎ)∈𝑊^{𝛽𝑙−1/2,2}(ℝ).

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={𝐱∈ℝ²∶𝑥₂=𝜀𝜑₀(𝑥₁),−∞<𝑥₁<+∞},𝜑₀∈𝑊^{𝛽𝑙+5/2,2}(ℝ) with 𝑙≥0, 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 𝜀̂,𝑟̂ such that for every 𝜀∈(0,𝜀̂). Problem (1.1)–(1.4) has a unique solution (𝐯,𝑝,𝜃,𝜓₁,𝜓₂)^𝑇. The solution admits the representation 𝐯(𝐱)=𝐯⁰(𝜓𝐱)+𝜀𝐮(𝐱),𝑝(𝐱)=𝑝⁰(𝐱)+𝜀𝑞(𝐱),𝜃(𝐱)=𝜃⁰(𝐱)+𝜀𝜗(𝐱),(4.1)₁𝑥₁=ℎ₁+𝜀Ψ₁𝑥₁,𝜓₂𝑥₁=ℎ₂+𝜀Ψ₂𝑥₁,(4.2) where {𝐯⁰,𝑝⁰,𝜃⁰} are the functions of the basic solution from (A.3)–(A.6), while 𝐔∶=𝐮,𝑞,𝜗,Ψ₁,Ψ₂^𝑇∈𝑊^𝛽𝑙+2,2(Π)²×𝑊^𝛽𝑙+1,2(Π)×𝑊^𝛽𝑙+2,2(Π)×𝑊^{𝛽𝑙+5/2,2}(ℝ)²≡𝒟^𝛽𝑙,2𝑊(Π),(4.3) and the following inequalities hold: ‖‖𝐔;𝒟^𝛽𝑙,2𝑊(Π)‖‖≤𝑟,̂𝜀̂≤const⋅sin²𝛼.(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 𝔑⁻¹ such that

𝔑⁻¹∶ℛ^𝛽𝑙,2𝑊(Π)⟼𝒟^𝛽𝑙,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 ℛ^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 𝒟^𝛽𝑙,2𝑊(Π) (see above). Moreover, one can show that there holds the estimate

‖‖𝔑⁻¹‖‖≤𝐶sin𝛼,(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 𝒟^𝛽𝑙,2𝑊(Π):

𝐔=𝔑⁻¹𝔉(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),Ψ^1(𝑛),Ψ^2(𝑛))^𝑇, it is sufficient to show that the operator 𝔎 is a contraction mapping in a ball of the space 𝒟^𝛽𝑙,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.

Appendix

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)∈ℝ²∶𝑥₂≡0,−∞<𝑥₁<+∞}. We suppose that the (unknown) free interface and the free surface Γ_𝑖 permit the representations Γ_𝑖={𝐱∈ℝ²∶𝑥₂=𝜓_𝑖(𝑥₁)=ℎ_𝑖=const.,−∞<𝑥₁<+∞}, and we are looking for stationary unidirectional flows. Such flows fulfil the assumptions

𝑣₂≡0,𝜕𝑣¹𝜕𝑥₁≡0,𝜕𝜃𝜕𝑥₁≡0.(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

𝜕𝑝−𝜈𝜚∇²𝑣₁=𝜚𝑔sin𝛼,𝜕𝑥₂=−𝜚𝑔cos𝛼,−𝜆∇²𝜃=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:

𝑣⁰¹𝑥₂=⎧⎪⎨⎪⎩𝑔sin𝛼−12𝜈₁𝑥²²+ℎ²−ℎ₁𝑟𝜈₂+ℎ¹𝜈₁𝑥²,0≤𝑥²𝑔sin𝛼−1≤ℎ₁,2𝜈₂ℎ₂−𝑥₂²+12𝜈₂ℎ₂−ℎ₁²+12𝜈₁ℎ²¹+1𝑟𝜈₂ℎ₂−ℎ₁ℎ₁,ℎ¹𝑣≤𝑥₂≤ℎ₂,(A.3)₀₂𝑥₂≡0,with𝑟∶=𝜈¹𝜚₁𝜈₂𝜚₂,(A.4) while

𝑝⁰𝑥₂=ℎ¹ℎ₂−𝑥₂𝜚₁𝑔cos𝛼+ℎ₂−ℎ₁𝜚₂𝑔cos𝛼+𝑝_𝑎,0≤𝑥₂≤ℎ₁,𝜃−𝑥₂𝜚₂𝑔cos𝛼+𝑝_𝑎,ℎ₁≤𝑥₂≤ℎ₂,(A.5)₀𝑥₂=⎧⎪⎪⎨⎪⎪⎩𝜃^𝑎𝜆₂𝜆₂ℎ₁+𝜆₁ℎ₂−ℎ₁+𝜆₁𝜆₂𝜃_𝑎𝑥₂,0≤𝑥₂≤ℎ₁,𝜆₁𝑥₂+ℎ₁𝜆₂−𝜆₁𝜆₂ℎ₁+𝜆₁ℎ₂−ℎ₁+𝜆₁𝜆₂,ℎ₁≤𝑥₂≤ℎ₂.(A.6) The so-called fluxes 𝐹_𝑖(𝑖=1,2) for both liquid layers that are the integrals

𝐹₁=^ℎ1₀𝑣¹⁽¹⁾𝑥₂d𝑥₂,𝐹₂=^ℎ2_ℎ1𝑣¹⁽²⁾𝑥₂d𝑥₂(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

𝐹₁=𝑔sin𝛼13𝜈₁ℎ³¹+12𝑟𝜈₂𝐹₂ℎ²¹ℎ₂−ℎ₁,=𝑔sin𝛼13𝜈₂ℎ₂−ℎ₁³+ℎ₂−ℎ₁12𝜈₁ℎ²¹+1𝑟𝜈₂ℎ₁ℎ₂−ℎ₁.(A.8) Recall that if sin𝛼 is small, then for the fluxes 𝐹_𝑖 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.

Acknowledgment

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