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<𝜀1, 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.

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


no longer equals to zero due to pressure boundary condition prescribed on Γ𝑝𝜕Ω, (in our case Γ𝑝=Σ𝜀0Σ𝜀. see Figure 1) Here 𝐮 denotes the velocity of the fluid which occupies a bounded domain Ω𝐑3, while 𝐧 stands for the unit outward normal on Γ𝑝. 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 𝐫𝜀(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 𝐫𝜀𝑥1=𝑥1𝑥,𝑎cos1𝜀𝑥,𝑎sin1𝜀,𝑥1[]0,,(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 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 𝐭𝜀𝑥1=𝐫𝜀𝑥1||𝐫𝜀𝑥1||=1𝑎2+𝜀2𝑥𝜀,𝑎sin1𝜀𝑥,𝑎cos1𝜀,𝐧𝜀𝑥1=𝐭𝜀𝑥1||𝐭𝜀𝑥1||=𝑥0,cos1𝜀𝑥,sin1𝜀,𝐛𝜀𝑥1=𝐭𝜀𝑥1×𝐧𝜀𝑥1=1𝑎2+𝜀2𝑥𝑎,𝜀sin1𝜀𝑥,𝜀cos1𝜀.(2.2) It can be easily verified that the curvature (flexion) 𝜅 and the torsion 𝜏 are constant and given by 𝑎𝜅=𝑎2+𝜀2𝜀,𝜏=𝑎2+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: 𝑆𝜀=𝑥𝑥=1,𝑥2,𝑥3𝐑3𝑥1𝑥(0,),𝑥=2,𝑥3=𝜀𝐵(0,)×𝜀𝐵.(2.4) We define the mapping Φ𝜀𝑆𝜀𝐑3 by 𝚽𝜀(𝑥)=𝐫𝜀𝑥1+𝑥2𝐧𝜀𝑥1+𝑥3𝐛𝜀𝑥1,(2.5) and we put Ω𝜀=𝚽𝜀𝑆𝜀.(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 Γ𝜀=𝚽𝜀((0,)×𝜀𝜕𝐵),Σ𝜀𝑖=𝚽𝜀({𝑖}×𝜀𝐵),𝑖=0,.(2.7)

2.2. The Equations

As mentioned before, the fluid inside of the pipe is assumed to be Newtonian so the velocity 𝐮𝜀 and pressure 𝑝𝜀 satisfy the following Navier-Stokes equations: 𝜇Δ𝐮𝜀+𝐮𝜀𝐮𝜀+𝑝𝜀=0inΩ𝜀,div𝐮𝜀=0inΩ𝜀.(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: 𝐮𝜀=0onΓ𝜀,𝐮(2.9)𝜀×𝐭𝜀=0,onΣ𝜀𝑖𝑝,(2.10)𝜀=𝑞𝑖onΣ𝜀𝑖,𝑖=0,.(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, 𝑞<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 Σ𝜀𝑖, while we keep the classical no-slip condition for the velocity on Γ𝜀. 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: 𝑉𝜀=𝐯𝐻1Ω𝜀3div𝐯=0inΩ𝜀,𝐯=0onΓ𝜀,𝐯×𝐭𝜀=0onΣ𝜀𝑖,𝑖=0,.(3.1) The space 𝑉𝜀 is equipped with the norm 𝐯𝑉𝜀=𝐯𝐿2Ω𝜀,(3.2) which is equivalent to the 𝐻1(Ω𝜀)3-norm due to Poincaré's inequality. Now we can write the variational formulation of our problem (2.8)–(2.11): nd𝐮𝜀𝑉𝜀𝜇suchthatΩ𝜀𝐮𝜀𝐯+Ω𝜀𝐮𝜀𝐮𝜀𝐯+𝑞0Σ𝜀0𝐯𝐭𝜀+𝑞Σ𝜀𝐯𝐭𝜀=0𝐯𝑉𝜀.(3.3) Notice that we have eliminated the pressure 𝑝𝜀. Furthermore, it should be observed that replacing 𝑞𝑖 by 𝑞𝑖+𝐶(𝐶=const.)𝑖=0, changes nothing so the relevant quantity is the pressure drop 𝑞0𝑞. As a consequence, the pressure 𝑝𝜀 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 𝜑𝐿2Ω𝜀𝐶1𝜀𝜑𝐿2Ω𝜀,(3.4)𝜑𝐿4Ω𝜀𝐶2𝜀1/4𝜑𝐿2Ω𝜀,(3.5) for all 𝜑𝐻1(Ω𝜀) satisfying 𝜑=0 on Γ𝜀.

Remark 3.2. It is well known that Poincaré’s constant depends on the geometry of the domain Ω𝜀. 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 𝜑𝐻1(Ω𝜀) be such that 𝜑=0 on Γ𝜀. We introduce new functions Θ𝑥(𝑥)=𝜑(𝑧),Ψ1𝑥,𝑦=Θ1,𝜀𝑦,(3.6) with 𝑧=Φ𝜀(𝑥). It is clear that Ψ𝐻1(𝑆), where 𝑆=(0,)×𝐵 and Ψ=0 on Γ=(0,)×𝜕𝐵. Now we extend Ψ by zero on a strip-like domain 𝑆𝑎,𝑏=(0,)×(𝑎,𝑏)2, where (𝑎,𝑏) is chosen such that 𝐵(𝑎,𝑏)2. A simple integration formula yields Ψ𝑥1=,𝑦𝑦2𝑎𝜕Ψ𝜕𝑦2𝑥1,𝜉,𝑦3𝑑𝜉.(3.7) Using the Cauchy-Schwarz inequality, one can easily obtain 𝑆𝑎,𝑏||Ψ𝑥1||,𝑦2𝑑𝑥1𝑑𝑦(𝑏𝑎)22𝑆𝑎,𝑏||||𝜕Ψ𝜕𝑦2𝑥1,𝜉,𝑦3||||2𝑑𝑥1𝑑𝜉𝑑𝑦3.(3.8) Since Ψ=0 on 𝑆𝑎,𝑏𝑆, it follows that Ψ𝐿2(𝑆)𝐶𝜕Ψ𝜕𝑦2𝐿2(𝑆),(3.9) with 𝐶 independent of 𝜀. By a simple change of variables (𝑥=𝜀𝑦), we get 𝑆𝜀||Θ||2𝑑𝑥=𝜀2𝑆||Ψ||2𝑑𝑥1𝑑𝑦,𝑆𝜀||||𝜕Θ𝜕𝑥2||||2𝑑𝑥=𝑆||||𝜕Ψ𝜕𝑦2||||2𝑑𝑥1𝑑𝑦.(3.10) Direct calculation gives detΦ𝜀=𝑔𝜀, where 𝑔𝜀𝑥=1+2𝑎2𝜀2𝑥22𝑎2+𝜀2,(3.11) implying the asymptotic behavior detΦ𝜀=𝑔𝜀=(𝑎/𝜀)1+𝒪(𝜀). In view of that, by simple change of variables, we obtain Ω𝜀||𝜑||2𝑑𝑧=𝑆𝜀||Θ||2𝑔𝜀𝐶𝑑𝑥𝜀𝑆𝜀||Θ||2𝑑𝑥,𝑆𝜀||||𝜕Θ𝜀𝜕𝑥2||||2𝑑𝑥𝑆𝜀||||Θ2𝑔𝜀1𝑔𝜀𝑑𝑥𝐶𝜀Ω𝜀||||𝜑2𝑑𝑧.(3.12) Now from (3.9)–(3.12) we deduce Ω𝜀||𝜑||2𝑑𝑧𝐶𝜀2Ω𝜀||||𝜑2𝑑𝑧,(3.13) implying (3.4). Finally, using the interpolation inequality, the embedding 𝐻1(Ω𝜀)𝐿6(Ω𝜀), and the estimate (3.4) we obtain at once 𝜑𝐿4Ω𝜀𝜑𝐿1/42Ω𝜀𝜑𝐿3/46Ω𝜀𝐶𝜀1/4𝜑𝐿2Ω𝜀.(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 𝜀2𝑞0𝑞<2𝜇29𝐶1𝐶22.(3.15) Then the problem (3.3) admits at least one solution 𝐮𝜀𝑉𝜀. Moreover, such solution is unique in the ball: 𝐵𝜀=𝐯𝑉𝜀𝐯𝐿2Ω𝜀𝜇3𝐶22𝜀,(3.16) where 𝐶1,𝐶2>0 are the constants from Lemma 3.1.

Proof. The idea of the proof is to introduce the mapping 𝑇𝐵𝜀𝐻1(Ω𝜀)3 defined by 𝑇(𝐰)=𝐮, where 𝐮 is the solution of the variational problem: nd𝐮𝑉𝜀𝜇suchthatΩ𝜀𝐮𝐯+Ω𝜀(𝐰)𝐮𝐯+𝑞0Σ𝜀0𝐯𝐭𝜀+𝑞Σ𝜀𝐯𝐭𝜀=0,𝐯𝑉𝜀.(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 𝑎𝑉𝜀×𝑉𝜀𝐑 be the following bilinear form 𝑎(𝐮,𝐯)=𝜇Ω𝜀𝐮𝐯+Ω𝜀(𝐰)𝐮𝐯.(3.18) Using estimate (3.5), we get 𝑎(𝐮,𝐮)𝜇𝐶22𝜀𝐰𝐿2Ω𝜀𝐮2𝐿2Ω𝜀.(3.19) Since 𝐰𝐵𝜀, we conclude 𝑎(𝐮,𝐮)2𝜇3𝐮2𝐿2Ω𝜀,𝐮𝑉𝜀,(3.20) implying that the form 𝑎(,) is elliptic on 𝑉𝜀. Analogously, we can easily obtain that ||||𝑎(𝐮,𝐯)4𝜇3𝐮𝐿2Ω𝜀𝐯𝐿2Ω𝜀.(3.21) Thus, due to Lax-Milgram Lemma, problem (3.17) has a unique solution and 𝑇 is well defined on 𝐵𝜀.
The next step is to prove that 𝑇(𝐵𝜀)𝐵𝜀. Applying (3.4), we obtain ||||𝑞0Σ𝜀0𝐯𝐭𝜀+𝑞Σ𝜀𝐯𝐭𝜀||||=||||Ω𝜀𝑞div0+𝑞𝑞0𝑥1𝐯||||𝑞0𝑞𝐶1𝜀𝜀𝐯𝐿2Ω𝜀.(3.22) From (3.17) and (3.20), it follows that 𝑇(𝐰)𝐿2Ω𝜀=𝐮𝐿2Ω𝜀3𝑞2𝜇0𝑞𝐶1𝜀𝜀.(3.23) Taking into account the condition (3.15), we obtain 𝑇(𝐰)𝐿2Ω𝜀<𝜇3𝐶22𝜀,(3.24) proving that 𝑇(𝐵𝜀)𝐵𝜀.
Now it remains to prove that 𝑇 is a contraction which will enable us to use Banach fixed point theorem. Let 𝐰1,𝐰2𝐵𝜀, 𝐰1𝐰2 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 𝜇Ω𝜀𝑇𝐰1𝐰𝑇2𝐯+Ω𝜀𝐰1𝑇𝐰1𝐰2𝑇𝐰2𝐯=0,𝐯𝑉𝜀.(3.25) Setting 𝐯=𝐮1𝐮2, we obtain 𝜇𝑇𝐰1𝐰𝑇22𝐿2Ω𝜀=Ω𝜀𝐰1𝐰2𝑇𝐰1𝑇𝐰1𝐰𝑇2Ω𝜀𝐰2𝑇𝐰1𝐰𝑇2𝑇𝐰1𝐰𝑇2.(3.26) For the first integral, we have ||||Ω𝜀𝐰1𝐰2𝑇𝐰1𝑇𝐰1𝐰𝑇2||||𝐶22𝜀𝐰1𝐰2𝐿2Ω𝜀𝐰𝑇1𝐿2Ω𝜀𝑇𝐰1𝐰𝑇2𝐿2Ω𝜀𝜇3𝐰1𝐰2𝐿2Ω𝜀𝑇𝐰1𝐰𝑇2𝐿2Ω𝜀.(3.27) Here we used the estimate (3.5) and the fact that 𝑇(𝐰1)𝐵𝜀. For the second integral, we proceed in the similar way and obtain ||||Ω𝜀𝐰2𝑇𝐰1𝐰𝑇2𝑇𝐰1𝐰𝑇2||||𝜇3𝑇𝐰1𝐰𝑇22𝐿2Ω𝜀,(3.28) leading to 𝑇𝐰1𝐰𝑇2𝐿2Ω𝜀12𝐰1𝐰2𝐿2Ω𝜀,(3.29) implying that 𝑇 is a contraction. Now Banach fixed point theorem provides the existence and uniqueness of 𝐮𝜀 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 𝐮𝜀 is unique in some ball 𝐵𝜀 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 𝑝𝐿2(Ω𝜀) 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]).


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