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

Authors: Jürgen Socolowsky
Journal: International Journal of Mathematics and Mathematical Sciences
 (2009)

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## 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.


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## Body

## 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 viscositiesνi>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)∈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 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/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 bottomS0 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 interfaceΓ1 separating the two fluid layers and the upper free surface Γ2 admit the parametrizations Γi={x∈ℝ2:x2=ψi(x1),-∞<x1<+∞}(i=1,2), where the functions ψi(i=1,2) are a priori unknown and have to be found. Let hi>0(0<h1<h2) 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=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(1.1)(v·∇)v-ν∇2v+1ϱ∇p=geg,∇·v=0,(v·∇)θ-λ∇2θ=0,
and the boundary and integral conditions are(1.2)v|S0=0,θ|S0=θ0,(1.3)[θ]|Γ1=0,[λ∂θ∂n]|Γ1=0,[v]|Γ1=0,v·n|Γ1-=0,[τ·S(v)n]|Γ1=-b1∂θ∂τ|Γ1-,ddx1ψ1′(x1)1+ψ1′(x1)2=1σ1(θ)[-p+n·S(v)n]|Γ1,lim|x1|→+∞ψ1(x1)=h1,(1.4)(θ-θa+λ2∂θ∂n)|Γ2=0,v·n|Γ2=0,τ·S(v)n|Γ2=-b2∂θ∂τ|Γ2,ddx1ψ2′(x1)1+ψ2′(x1)2=1σ2(θ)(pa-p+n·S(v)n)|Γ2,lim|x1|→+∞ψ2(x1)=h2.
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:(1.5)σi(θ)=ai-biθ(ai,bi>0,i=1,2).
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 a·b we mean the inner product of a,b∈ℝ2,∇=(∂/∂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,(1.6)[w(x0)]|Γ1:=limy→x0w(y)-limx→x0w(x),(x0∈Γ1,y∈Ω1,x∈Ω2),
and the symbol w|Γ1- denotes the limit from below at the interface Γ1; more precisely(1.7)w(x0)|Γ1-:=limy→x0w(y),(x0∈Γ1,y∈Ω1).
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 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(2.1)(v0,p0,θ0,ψ10,ψ20)→(v1,p1,θ1,ψ11,ψ21)→⋯→(vm,pm,θm,ψ1m,ψ2m)→⋯,
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 domainΠ 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(3.1)Π1:={x∈ℝ2:0<x2<h1,-∞<x1<+∞},Π2:={x∈ℝ2:h1<x2<h2,-∞<x1<+∞},
and Π=Π1∪Π2 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(3.2)∥u(m);Wβl,2(Πm)∥=∥u(m)exp(β1+x12);Wl,2(Πm)∥,(m=1,2),
where Wl,2(Πm) is the usual Sobolev space. The norm in Wβl,2(Π) is given by(3.3)∥u;Wβl,2(Π)∥=∑m=12∥u(m)exp(β1+x12);Wl,2(Πm)∥.
If β>0, then elements of Wβl,2(Π) vanish exponentially as |x1|→∞, and if β<0, then elements u∈Wβl,2(Π) might exponentially increase as |x1|→∞.The spacesWβl-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 Wβl-1/2,2(S) the spaces of traces on S of functions from Wβl,2(Π). Then Wβl-1/2,2(ℝ) coincides with Wβl-1/2,2(S), that is, if u∈Wβl-1/2,2(Π), then u(·,h)∈Wβl-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.
LetS0={x∈ℝ2:x2=εφ0(x1),-∞<x1<+∞},φ0∈Wβl+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
(4.1)v(x)=v0(x)+εu(x),p(x)=p0(x)+εq(x),θ(x)=θ0(x)+εϑ(x),(4.2)ψ1(x1)=h1+εΨ1(x1),ψ2(x1)=h2+εΨ2(x1),
where {v0,p0,θ0} are the functions of the basic solution from (A.3)–(A.6), while
(4.3)U:=(u,q,ϑ,Ψ1,Ψ2)T∈[Wβl+2,2(Π)]2×Wβl+1,2(Π)×Wβl+2,2(Π)×[Wβl+5/2,2(ℝ)]2≡𝒟βl,2W(Π),
and the following inequalities hold:
(4.4)∥U;𝒟βl,2W(Π)∥≤r̂,ε̂≤const·sin2α.We would like to present a short sketch of the proof of this theorem by successive approximations. First, the original (perturbed) and unknown flow domainΩ (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(4.5)𝔑-1:ℛβl,2W(Π)↦𝒟βl,2W(Π),
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 ℛβl,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(4.6)∥𝔑-1∥≤Csinα,
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):(4.7)U=𝔑-1𝔉(U)≡𝔎(U),
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.



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