On the Solvability of a Problem of Stationary Fluid Flow in a Helical Pipe

Abstract

We consider a flow of incompressible Newtonian fluid through a pipe with helical shape. We suppose that the flow is governed by the prescribed pressure drop between pipe's ends. Such model has relevance to some important engineering applications. Under small data assumption, we prove the existence and uniqueness of the weak solution to the corresponding Navier-Stokes system with pressure boundary condition. The proof is based on the contraction method.

1. Introduction

Engineering practice requires extensive knowledge of flow through curved pipes. Helically coiled pipes are well-known types of curved pipes which have been used in wide variety of applications. They cover a large number of devices such as pipelines, air conditioners, refrigeration systems, central heating radiators, and chemical reactors. Therefore, numerous researchers have studied the fluid flow in helical pipes with circular cross-section both theoretically and experimentally. Let us just mention some of them. Wang [1] proposed a nonorthogonal coordinate system to investigate the effects of curvature and torsion on the low-Reynolds number flow in a helical pipe. The introduction of an orthogonal system of coordinates along a spatial curve allowed Germano [2, 3] to explore in a simpler way the effects of pipe's geometry on a helical pipe flow. Yamamoto et al. [4] first studied experimentally the effects of torsion and curvature on the flow characteristics in a helical tube. The results obtained by the experiments were then compared with those obtained from the model of Yamamoto et al. in [5]. Hüttl and Friedrich [6, 7] applied the second-order finite volume method for solving the incompressible Navier-Stokes equations to study the turbulent flow in helically coiled tubes. More numerical simulations on helical pipe flow can be found in the works of Yamamoto et al. [8] and Wang and Andrews [9].

In this paper, we study the stationary flow of incompressible Newtonian fluid through a helical pipe with prescribed pressures at its ends. We suppose that the pipe's thickness and the helix step have the same small order 𝒪(𝜀),0<𝜀≪1, while the diameter of the helix is larger, of order 𝒪(1) (see Figure 1). Such assumptions cover a large variety of realistic coiled pipes and they appear naturally in many devices as, for instance, the Liebig cooler, cooling channel in the nozzle of a rocket engine, particle separators used in the mineral processing industry, and so forth.

Figure 1 

Helical pipe.

It is well-known that the flow of incompressible viscous Newtonian fluid is described by Navier-Stokes system of nonlinear PDEs. Their nonlinearity makes most boundary-value problems difficult or impossible to solve and thus the concept of weak solutions is introduced. When the velocity is prescribed on the whole boundary (Dirichlet condition), the existence of the weak solution can be proved by constructing approximate solutions via Galerkin method (see, e.g., Temam [10], although the main ideas goes back to Leray [11]). However, applications (as this one in the present paper) often give rise to problems where it is natural to prescribe the value of pressure on some part of the boundary. In case of (linear) Stokes system with pressure boundary condition, the existence and uniqueness of weak solution is obtained in Conca et al. [12]. For the Navier-Stokes system with boundary conditions involving pressure we only have some partial results and the full proof of existence is still unknown. The main difficulty lies in the fact that the integral

_Ω(𝐮∇)𝐮𝐮=12_Γ𝑝|𝐮|²(𝐮⋅𝐧)(1.1)

no longer equals to zero due to pressure boundary condition prescribed on Γ_𝑝⊂𝜕Ω, (in our case Γ_𝑝=Σ^𝜀0∪Σ^𝜀ℓ. see Figure 1) Here 𝐮 denotes the velocity of the fluid which occupies a bounded domain Ω⊂𝐑³, while 𝐧 stands for the unit outward normal on Γ_𝑝. As a consequence, the nonlinear term in corresponding variational formulation does not vanish (and we do not know how to control it) causing the absence of the energy equality. Described technical difficulty can be elegantly overcome by prescribing the so-called dynamic (Bernoulli) pressure 𝑝+(1/2)|𝐮|² as proposed by Conca et al. [12] (see also Łukaszewicz [13] for nonstationary flow). Nevertheless, it should be mentioned that there is no physical justification for prescribing such boundary condition in the case of viscous fluid. In the case of ideal fluid the interpretation of the dynamic pressure is given by the Bernoulli law.

Another possibility is to restrict to the case of small boundary data as in Heywood et al. [14] or Marušić-Paloka [15] (see also Marušić-Paloka [16]) and we follow such approach. By doing that, we manage to control the inertial term in Navier-Stokes equations and to obtain existence and uniqueness of the solution for the physically realistic situation (without prescribing the dynamic pressure).

The paper is organized as follows. First we describe pipe's geometry using parametrization 𝐫_𝜀(𝑥₁)=(𝑥₁,𝑎cos(𝑥₁/𝜀),𝑎sin(𝑥₁/𝜀)) of its central curve. Here the small parameter 𝜀 stands for the distance between two coils of the helix. We assume that the pipe has (constant) circular cross-section of size 𝜀 and use Frenet basis attached to a helix to formally define our domain. Then we state the boundary-value problem describing the flow of a Newtonian fluid inside of the pipe and introduce the corresponding variational formulation. The main result is formulated in Theorem 3.3 where we precisely establish the assumption on the prescribed pressure drop under which the weak solution exists and then prove its uniqueness. The proof is based on the auxiliary result on sharp Sobolev constants (Lemma 3.1) and the contraction method.

We end this introduction by giving few more bibliographic remarks. The asymptotic behavior of the fluid flow through a helical pipe was investigated in Marušić-Paloka and Pažanin [17, 18]. Using the techniques from Marušić-Paloka [19], enabling the treatment of the curved geometry, the asymptotic approximation of the solution is built and rigorously justified by proving the error estimate in terms of the small parameter 𝜀. Last but not least, let us mention that in Marušić-Paloka and Pažanin [20] the nonisothermal flow of Newtonian fluid through a general curved pipe has been considered. In such flow temperature changes cannot be neglected so the Navier-Stokes equations are coupled with heat conducting equation. A simplified model showing explicitly the effects of pipe's geometry is derived via rigorous asymptotic analysis with respect to the pipe's thickness.

2. Setting of the Problem

2.1. Pipe's Geometry

We start by defining the helix whose parametrization has the form 𝐫_𝜀𝑥₁=𝑥₁,𝑎cos𝑥¹𝜀,𝑎sin𝑥𝜀¹,𝑥¹∈[0,ℓ],(2.1) serving to define the center curve of the pipe. It should be noted that the curve is not parameterized by its arc length (i.e., we do not use the natural parametrization) since in that case interval length ℓ would depend on the small parameter 𝜀 which is inconvenient for further analysis. At each point of the helix we compute the Frenet basis as follows 𝐭_𝜀𝑥₁=𝐫^𝜀𝑥₁𝐧_𝜀||𝐫^𝜀𝑥₁||=√𝑎²1+𝜀²𝜀,−𝑎sin𝑥𝜀¹,𝑎cos𝑥𝜀¹,𝑥₁=𝐭^𝜀𝑥₁𝐛_𝜀||𝐭^𝜀𝑥₁||=0,−cos𝑥𝜀¹,−sin𝑥𝜀¹,𝑥₁=𝐭_𝜀𝑥₁×𝐧_𝜀𝑥₁=1√𝑎²+𝜀²𝑎,𝜀sin𝑥𝜀¹,−𝜀cos𝑥𝜀¹.(2.2) It can be easily verified that the curvature (flexion) 𝜅 and the torsion 𝜏 are constant and given by 𝜅=𝑎𝑎²+𝜀²,𝜏=𝜀𝑎²+𝜀².(2.3) For small parameter 𝜀>0 and a unit circle 𝐵=𝐵(0,1)⊂𝐑², we introduce a thin straight pipe with circular cross-section: 𝑆_𝜀=𝑥=𝑥₁,𝑥₂,𝑥₃∈𝐑³∶𝑥₁∈(0,ℓ),𝑥′=𝑥₂,𝑥₃∈𝜀𝐵=(0,ℓ)×𝜀𝐵.(2.4) We define the mapping Φ_𝜀∶𝑆_𝜀→𝐑³ by 𝚽_𝜀(𝑥)=𝐫_𝜀𝑥₁+𝑥₂𝐧_𝜀𝑥₁+𝑥₃𝐛_𝜀𝑥₁,(2.5) and we put Ω_𝜀=𝚽_𝜀𝑆_𝜀.(2.6) Such domain is our thin pipe with helical shape filled with a viscous incompressible fluid. Finally, the lateral boundary of the pipe and its ends are denoted by Γ_𝜀=𝚽_𝜀((0,ℓ)×𝜀𝜕𝐵),Σ^𝜀𝑖=𝚽_𝜀({𝑖}×𝜀𝐵),𝑖=0,ℓ.(2.7)

2.2. The Equations

As mentioned before, the fluid inside of the pipe is assumed to be Newtonian so the velocity 𝐮_𝜀 and pressure 𝑝_𝜀 satisfy the following Navier-Stokes equations: −𝜇Δ𝐮_𝜀+di𝐮v_𝜀∇𝐮_𝜀𝐮_𝜀+∇𝑝_𝜀=0inΩ_𝜀,=0inΩ_𝜀.(2.8) Here 𝜇>0 and we have added the subscript 𝜀 in our notation in order to stress the dependence of the solution on the small parameter. The above system must be completed by the boundary conditions: 𝐮𝐮_𝜀=0onΓ_𝜀,(2.9)_𝜀𝑝×𝐭_𝜀=0,onΣ^𝜀𝑖,(2.10)_𝜀=𝑞_𝑖onΣ^𝜀𝑖,𝑖=0,ℓ.(2.11) The fluid flow is governed by the pressure drop between pipe's ends so in (2.11) we prescribe constant pressures 𝑞₀ and 𝑞_ℓ, 𝑞_ℓ<𝑞₀. In addition to the value of pressure, we need to prescribe something more on the boundary in order to assure that the problem is well possed. Thus, we take the tangential velocity to be zero on Σ^𝜀𝑖, while we keep the classical no-slip condition for the velocity on Γ_𝜀. Imposing that the tangential component of the velocity equals to zero is not a serious restriction since the only part that counts is the normal part, due to the Saint-Venant principle for thin domains (see, e.g., Marušić-Paloka [19]). Let us mention that instead of these conditions, one could prescribe the whole normal stress, including the viscous part, on the boundary (see, e.g., Heywood et al. [14]). Such situation can be treated using the same method, with a slight change of functional space in the corresponding variational formulation.

3. Existence and Uniqueness of the Solution

3.1. Variational Formulation

Let us introduce the following natural functional space: 𝑉_𝜀=𝐯∈𝐻¹Ω_𝜀³∶div𝐯=0inΩ_𝜀,𝐯=0onΓ_𝜀,𝐯×𝐭_𝜀=0onΣ^𝜀𝑖,𝑖=0,ℓ.(3.1) The space 𝑉_𝜀 is equipped with the norm ‖𝐯‖_𝑉𝜀=‖∇𝐯‖_𝐿²_Ω𝜀,(3.2) which is equivalent to the 𝐻¹(Ω_𝜀)³-norm due to Poincaré's inequality. Now we can write the variational formulation of our problem (2.8)–(2.11): 𝜇find𝐮_𝜀∈𝑉_𝜀suchthat_Ω𝜀∇𝐮_𝜀∇𝐯+_Ω𝜀𝐮_𝜀∇𝐮_𝜀𝐯+𝑞₀_Σ𝜀0𝐯⋅𝐭_𝜀+𝑞_ℓ_Σ𝜀ℓ𝐯⋅𝐭_𝜀=0∀𝐯∈𝑉_𝜀.(3.3) Notice that we have eliminated the pressure 𝑝_𝜀. Furthermore, it should be observed that replacing 𝑞_𝑖 by 𝑞_𝑖+𝐶(𝐶=const.)𝑖=0,ℓ changes nothing so the relevant quantity is the pressure drop 𝑞₀−𝑞_ℓ. As a consequence, the pressure 𝑝_𝜀 is determined only up to an additive constant. The equivalence between the variational (3.3) and differential (2.8)–(2.11) formulation, in case of smooth solutions, is discussed in the usual way (see, e.g., Conca et al. [12]).

3.2. The Main Result

The goal of this paper is to prove the existence and uniqueness of the solution for the variational problem (3.3), under small data assumption. In order to accomplish that, we need the following auxiliary result.

Lemma 3.1. There exist constants 𝐶₁,𝐶₂>0, independent of 𝜀, such that ‖‖𝜑𝜑‖‖_𝐿²_{Ω𝜀𝐿4Ω𝜀}≤𝐶₁𝜀‖∇𝜑‖_𝐿²_Ω𝜀,(3.4)≤𝐶₂𝜀^1/4‖∇𝜑‖_𝐿²_Ω𝜀,(3.5) for all 𝜑∈𝐻¹(Ω_𝜀) satisfying 𝜑=0 on Γ_𝜀.

Remark 3.2. It is well known that Poincaré’s constant depends on the geometry of the domain Ω_𝜀. Inequality (3.4) gives the precise dependence of that constant on the small parameter 𝜀. Inequality (3.5) enable us to estimate the inertial term in the variational formulation (3.3).

Proof of Lemma 3.1. Let 𝜑∈𝐻¹(Ω_𝜀) be such that 𝜑=0 on Γ_𝜀. We introduce new functions Θ(𝑥)=𝜑(𝑧),Ψ𝑥₁,𝑦′=Θ𝑥₁,𝜀𝑦′,(3.6) with 𝑧=Φ_𝜀(𝑥). It is clear that Ψ∈𝐻¹(𝑆), where 𝑆=(0,ℓ)×𝐵 and Ψ=0 on Γ=(0,ℓ)×𝜕𝐵. Now we extend Ψ by zero on a strip-like domain 𝑆_𝑎,𝑏=(0,ℓ)×(𝑎,𝑏)², where (𝑎,𝑏) is chosen such that 𝐵⊂(𝑎,𝑏)². A simple integration formula yields Ψ𝑥₁,𝑦′=^𝑦2_𝑎𝜕Ψ𝜕𝑦₂𝑥₁,𝜉,𝑦₃𝑑𝜉.(3.7) Using the Cauchy-Schwarz inequality, one can easily obtain _{𝑆𝑎,𝑏}||Ψ𝑥₁,𝑦′||²𝑑𝑥₁𝑑𝑦′≤(𝑏−𝑎)²2_{𝑆𝑎,𝑏}||||𝜕Ψ𝜕𝑦₂𝑥₁,𝜉,𝑦₃||||²𝑑𝑥₁𝑑𝜉𝑑𝑦₃.(3.8) Since Ψ=0 on 𝑆_𝑎,𝑏⧵𝑆, it follows that ‖Ψ‖_𝐿²_(𝑆)≤𝐶‖‖‖𝜕Ψ𝜕𝑦₂‖‖‖_𝐿2(𝑆),(3.9) with 𝐶 independent of 𝜀. By a simple change of variables (𝑥′=𝜀𝑦′), we get _𝑆𝜀||Θ||²𝑑𝑥=𝜀²_𝑆||Ψ||²𝑑𝑥₁𝑑𝑦′,_𝑆𝜀||||𝜕Θ𝜕𝑥₂||||²𝑑𝑥=_𝑆||||𝜕Ψ𝜕𝑦₂||||²𝑑𝑥₁𝑑𝑦′.(3.10) Direct calculation gives det∇Φ_𝜀=√𝑔_𝜀, where 𝑔_𝜀=1+𝑥²−𝑎²𝜀²−𝑥²²𝑎²+𝜀²,(3.11) implying the asymptotic behavior det∇Φ_𝜀=√𝑔_𝜀=(𝑎/𝜀)√1+𝒪(𝜀). In view of that, by simple change of variables, we obtain _Ω𝜀||𝜑||²𝑑𝑧=_𝑆𝜀||Θ||²√𝑔_𝜀𝑑𝑥≤𝐶𝜀_𝑆𝜀||Θ||²𝑑𝑥,_𝑆𝜀||||𝜕Θ_𝜀𝜕𝑥₂||||²𝑑𝑥≤_𝑆𝜀||∇Θ||²√𝑔_𝜀1√𝑔_𝜀𝑑𝑥≤𝐶𝜀_Ω𝜀||∇𝜑||²𝑑𝑧.(3.12) Now from (3.9)–(3.12) we deduce _Ω𝜀||𝜑||²𝑑𝑧≤𝐶𝜀²_Ω𝜀||∇𝜑||²𝑑𝑧,(3.13) implying (3.4). Finally, using the interpolation inequality, the embedding 𝐻¹(Ω_𝜀)↪𝐿⁶(Ω_𝜀), and the estimate (3.4) we obtain at once ‖𝜑‖_𝐿⁴_Ω𝜀≤‖𝜑‖^𝐿1/4_{2Ω𝜀}‖𝜑‖^𝐿3/4_{6Ω𝜀}≤𝐶𝜀^1/4‖∇𝜑‖_𝐿²_Ω𝜀.(3.14)

Our main result can be stated as follows.

Theorem 3.3. Assume that the pressure drop and the helix step are such that 𝜀²𝑞₀−𝑞_ℓ<2𝜇²ℓ9𝐶₁𝐶²².(3.15) Then the problem (3.3) admits at least one solution 𝐮_𝜀∈𝑉_𝜀. Moreover, such solution is unique in the ball: 𝐵_𝜀=𝐯∈𝑉_𝜀∶‖∇𝐯‖_𝐿²_Ω𝜀≤𝜇3𝐶²²√𝜀,(3.16) where 𝐶₁,𝐶₂>0 are the constants from Lemma 3.1.

Proof. The idea of the proof is to introduce the mapping 𝑇∶𝐵_𝜀→𝐻¹(Ω_𝜀)³ defined by 𝑇(𝐰)=𝐮, where 𝐮 is the solution of the variational problem: 𝜇find𝐮∈𝑉_𝜀suchthat_Ω𝜀∇𝐮∇𝐯+_Ω𝜀(𝐰∇)𝐮𝐯+𝑞₀_Σ𝜀0𝐯⋅𝐭_𝜀+𝑞_ℓ_Σ𝜀ℓ𝐯⋅𝐭_𝜀=0,∀𝐯∈𝑉_𝜀.(3.17) (Although the operator𝑇 depends on𝜀 we drop the index𝜀 for the sake of notational simplicity.)The first step is to show that mapping 𝑇 is well defined. For that purpose, let 𝑎∶𝑉_𝜀×𝑉_𝜀→𝐑 be the following bilinear form 𝑎(𝐮,𝐯)=𝜇_Ω𝜀∇𝐮∇𝐯+_Ω𝜀(𝐰∇)𝐮𝐯.(3.18) Using estimate (3.5), we get 𝑎(𝐮,𝐮)≥𝜇−𝐶²²√𝜀‖∇𝐰‖_𝐿²_Ω𝜀‖∇𝐮‖^2𝐿2_Ω𝜀.(3.19) Since 𝐰∈𝐵_𝜀, we conclude 𝑎(𝐮,𝐮)≥2𝜇3‖∇𝐮‖^{2𝐿2Ω𝜀},∀𝐮∈𝑉_𝜀,(3.20) implying that the form 𝑎(⋅,⋅) is elliptic on 𝑉_𝜀. Analogously, we can easily obtain that ||𝑎(𝐮,𝐯)||≤4𝜇3‖∇𝐮‖^{𝐿2Ω𝜀}‖∇𝐯‖_𝐿²_Ω𝜀.(3.21) Thus, due to Lax-Milgram Lemma, problem (3.17) has a unique solution and 𝑇 is well defined on 𝐵_𝜀.
The next step is to prove that 𝑇(𝐵_𝜀)⊂𝐵_𝜀. Applying (3.4), we obtain ||||𝑞₀=_Σ𝜀0||||𝐯⋅_Ω𝜀𝐭_𝜀+𝑞_ℓ_Σ𝜀ℓ𝐯⋅𝐭_𝜀||||div𝑞₀+𝑞^ℓ−𝑞₀ℓ𝑥¹𝐯||||≤𝑞⁰−𝑞_ℓℓ𝐶¹𝜀√𝜀‖∇𝐯‖_𝐿²_Ω𝜀.(3.22) From (3.17) and (3.20), it follows that ‖∇𝑇(𝐰)‖_𝐿²_Ω𝜀=‖∇𝐮‖_𝐿²_Ω𝜀≤32𝜇𝑞⁰−𝑞_ℓℓ𝐶¹𝜀√𝜀.(3.23) Taking into account the condition (3.15), we obtain ‖∇𝑇(𝐰)‖_𝐿²_Ω𝜀<𝜇3𝐶²²√𝜀,(3.24) proving that 𝑇(𝐵_𝜀)⊂𝐵_𝜀.
Now it remains to prove that 𝑇 is a contraction which will enable us to use Banach fixed point theorem. Let 𝐰₁,𝐰₂∈𝐵_𝜀, 𝐰₁≠𝐰₂ be such that 𝑇(𝐰₁)=𝐮₁, 𝑇(𝐰₂)=𝐮₂, with 𝐮₁ and 𝐮₂ being the solutions of problem (3.17) for 𝐰₁ and 𝐰₂, respectively. Subtracting the corresponding equations it follows that 𝜇_Ω𝜀∇𝑇𝐰₁−𝑇𝐰₂∇𝐯+_Ω𝜀𝐰₁∇𝑇𝐰₁−𝐰₂∇𝑇𝐰₂𝐯=0,∀𝐯∈𝑉_𝜀.(3.25) Setting 𝐯=𝐮₁−𝐮₂, we obtain 𝜇‖‖∇𝑇𝐰₁−𝑇𝐰₂‖‖^2𝐿2_Ω𝜀=−−_Ω𝜀𝐰₁−𝐰₂∇𝑇𝐰₁𝑇𝐰₁−𝑇𝐰₂_Ω𝜀𝐰₂∇𝑇𝐰₁−𝑇𝐰₂𝑇𝐰₁−𝑇𝐰₂.(3.26) For the first integral, we have ||||_Ω𝜀≤𝐰𝐶₁₂₂−𝐰₂∇𝑇𝐰₁𝑇𝐰₁−𝑇𝐰₂||||√𝜀‖‖∇𝐰₁−𝐰₂‖‖_𝐿²_Ω𝜀‖‖∇𝑇𝐰₁‖‖_𝐿²_Ω𝜀‖‖∇𝑇𝐰₁−𝑇𝐰₂‖‖_𝐿²_Ω𝜀≤𝜇3‖‖∇𝐰¹−𝐰₂‖‖_𝐿²_Ω𝜀‖‖∇𝑇𝐰₁−𝑇𝐰₂‖‖_𝐿²_Ω𝜀.(3.27) Here we used the estimate (3.5) and the fact that 𝑇(𝐰₁)∈𝐵_𝜀. For the second integral, we proceed in the similar way and obtain ||||_Ω𝜀𝐰₂∇𝑇𝐰₁−𝑇𝐰₂𝑇𝐰₁−𝑇𝐰₂||||≤𝜇3‖‖∇𝑇𝐰¹−𝑇𝐰₂‖‖^2𝐿2_Ω𝜀,(3.28) leading to ‖‖∇𝑇𝐰₁−𝑇𝐰₂‖‖_𝐿²_Ω𝜀≤12‖‖∇𝐰¹−𝐰₂‖‖_𝐿²_Ω𝜀,(3.29) implying that 𝑇 is a contraction. Now Banach fixed point theorem provides the existence and uniqueness of 𝐮_𝜀 finishing the proof.

Remark 3.4. Regarding the uniqueness of the solution, it should be observed that we managed to prove only that the weak solution 𝐮_𝜀 is unique in some ball 𝐵_𝜀 around zero, under small data assumption (3.15). It means that we cannot exclude the existence of some other solutions with large norm. The existence and uniqueness (up to an additive constant) of the pressure 𝑝∈𝐿²(Ω_𝜀) satisfying the governing equations in the sense of the distributions can be obtained in the standard way using De Rham theorem (see, e.g., Temam [10]).

Acknowledgment

This research was supported by the Ministry of Science, Education and Sports, Republic of Croatia, Grant 037-0372787-2797.