# Asymptotic Analysis of the Curved-Pipe Flow with a Pressure-Dependent Viscosity Satisfying Barus Law

Authors: Igor Pažanin
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

Curved-pipe flows have been the subject of many theoretical investigations due to their importance in various applications. The goal of this paper is to study the flow of incompressible fluid with a pressure-dependent viscosity through a curved pipe with an arbitrary central curve and constant circular cross section. The viscosity-pressure dependence is described by the well-known Barus law extensively used by the engineers. We introduce the small parameterε (representing the ratio of the pipe’s thickness and its length) into the problem and perform asymptotic analysis with
respect to ε. The main idea is to rewrite the governing problem using the appropriate transformation and then to compute the asymptotic solution using curvilinear coordinates and two-scale asymptotic expansion. Applying the inverse transformation, we derive the asymptotic approximation of the flow clearly showing the influence of pipe’s distortion and viscosity-pressure dependence on the
effective flow.


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## Body

## 1. Introduction

Curved-pipe flows have gained much attention over past years due to their importance in numerous industrial and engineering applications. Air conditioners, refrigeration systems, central heating radiators, and chemical reactors are only few examples of devices where we can find different types of curved pipes. From the theoretical point of view, curved-pipe flows are interesting due to the appearance of secondary flows caused by the effects of the pipe’s distortion. Therefore, when analyzing such problems, the main attempt is to detect the effects of the pipe’s geometry on the velocity and pressure distribution through the pipe. The engineering approach to the curved-pipe flows is often based on thePoiseuille formula providing an exact solution only in case of stationary, laminar Newtonian flow through a straight pipe with constant cross section. However, if the pipe is curved, such formula only gives an approximation of the solution with low order of accuracy. In view of that, the Poiseuille flow has to be corrected by the lower-order term which contains the effects of pipe’s curvedness leading to a more accurate approximation.In his celebrated work, Stokes [1] suggested that the viscosity of the fluid can depend on the pressure. Since then, numerous researchers confirmed that, especially at high values of pressure, the variations of the viscosity with pressure should be taken into account while the flow is still incompressible. For that reason, the problem of incompressible fluid flow with a pressure-dependent viscosity is very attractive and has been extensively studied in recent years, mostly in the engineering literature (see, e.g., [2–6]). There exist several ways to describe the viscosity-pressure relation. Among all, the most famous one is, without any doubt, the Barus law [7]:(1)μ(p)=μ0eβp.Here μ0 stands for the viscosity at atmospheric pressure while β>0 is the pressure-viscosity coefficient. Barus formula has been extensively used throughout the engineering literature.Motivated by the above discussion, the aim of this paper is to study the incompressible fluid with a pressure-dependent viscosity obeying Barus law and flowing through a curved pipe with constant circular cross section. Introducing the viscosity-pressure dependence (1) into the Navier-Stokes system completely changes the nature of the system making it very challenging from the mathematical point of view. It brings the additional nonlinearity to the system and the flow becomes non-Newtonian. The main difficulty lies in the fact that we cannot treat pressure as we did in the classical, Newtonian case so we need to change our approach. In view of that, our strategy consists of the following three steps:(1)
rewriting the governing system by replacing the original pressure with a new,transformed pressure;(2)
finding the solution of thetransformed system satisfied by the velocity and a new pressure;(3)
reconstructing the effective pressure by applying the inverse transformation.Naturally, it is not reasonable to expect that we will succeed to find the exact solution of the governing3D boundary-value problem. Therefore, inspired by the applications, we introduce the small parameter ε into the system (denoting the ratio between pipe’s thickness and its length) and consider the flow in a pipe which is either very thin or very long. By doing that, we are in position to perform the asymptotic analysis as ε→0 and to build the asymptotic approximation of the flow with high order of accuracy.Along with the viscosity-pressure dependence, our aim is to treat as general domain as possible. Thus, we assume that the pipe’s central curve, denoted byγ, is an arbitrary smooth curve given by its natural parametrization. The only constraint we impose on γ is that it is a generic curve. As a consequence, we can use local Frenet’s basis attached to γ and use the curvilinear coordinates to formally define our domain. An efficient technique for handling curved geometries has been proposed in some of our previous works (see, e.g., [8–10]) and we employ it here to construct the asymptotic solution of the transformed system (Step 2). It enables us to explicitly compute the terms from the two-scale asymptotic expansion and to detect the effects we seek for. Indeed, after applying the inverse transformation, we obtain the asymptotic approximation for the velocity and pressure explicitly acknowledging the effects of pressure-dependent viscosity and the pipe’s curvedness. By taking those effects into account, we believe that the obtained result is very relevant with regard to numerical simulations and could improve the known engineering practice. We should also mention that the presented approach can be generalized to a case of a general viscosity-pressure relation, as commented in the concluding section.We conclude the introduction by providing more bibliographic remarks on the subject. Curved-pipe flow in case of constant viscosity (μ=const.) has been extensively investigated for various liquids and regime of flows; see, for example, [11–15]. In case of pressure-dependent viscosity, to our knowledge there are no analytical results on the curved-pipe flows. Analytical solutions have been reported only in some simplified situations like unidirectional and plane-parallel flows and under the assumption of the linear law μ(p)=βp or μ(p)=μ0(1+βp). We refer the reader to [16–20]. However, a year ago, the straight-pipe flow has been successfully addressed by Marušić-Paloka and Pažanin [21] in the case of exponential law (1). The flow through a specific helical pipe frequently appearing in the applications has been analyzed this year in [22]. The goal of the present paper is to extend the analysis presented in [21, 22] to a general framework, that is, the case of general curved pipe.


## 2. Position of the Problem

### 2.1. The Pipe’s Geometry

In this section we formally describe the complex pipe’s geometry. As emphasized in Introduction, we want to address the case of a general curved pipe with circular cross section. In view of that, we introduce a generic curve inR3, denoted by γ, which serves to define the central curve of the pipe. We suppose that γ is parameterized by its arc length x1∈[0,l] and denote by π∈C3([0,l];R3) its natural parametrization. We also assume that π′(x1)≠0, for every x1∈[0,l]. Since γ is taken to be generic, it holds π′′(x1)>0, for every x1∈[0,l]. Denoting by κ(x1)=π′′(x1) the flexion of the curve γ, we introduce Frenet’s basis in a standard way: (2)t=π′(thetangent),n=1κt′(thenormal),b=t×n(thebinormal).The normaln is extended by continuity in points where curvature is zero. We also denote by τ(x1)=-b′(x1) the torsion of γ. One of the main goals of this study is to detect the influence of geometric parameters κ and τ on the effective flow.Next, we introduce the small parameterε (0<ε≪1) into the problem and first define an undeformed pipe (3)Tε={x=(x1,x2,x3)∈R3:x1∈(0,l),x~=(x2,x3)∈εB},where B=B(0,1)=y∈R2:|y|<1 is the unit circle. Now we have to choose the appropriate parametrization to define our curved pipe. The best way to do this is to introduce the mapping Φεα:Tε→R3 as follows: (4)Φεα(x)=π(x1)+x2nα(x1)+x3bα(x1).Here (5)nα(x1)=cos⁡α(x1)n(x1)+sinα(x1)b(x1),bα(x1)=-sinα(x1)n(x1)+cos⁡α(x1)b(x1)stand for the rotated unit vectors with respect to standard Frenet’s normal and binormal (see Figure 1), where the rotation is given by(6)α(x1)=-∫x0x1‍τ(ξ)dξ+α0,(x0,α0arbitraryconstants).Figure 1
The reference system.Observe that by puttingα≡0 we get classical Frenet’s system which is most usually employed in describing the curved geometries. However, since we analyze the problem in the pipe with circular cross section (the most common one in the engineering applications), it is plausible to use this particular reference system in which the domain’s cross section possesses no rotation with respect to the tangent vector t. Let us mention that it was originally introduced by Germano in the 80s for treating the classical Newtonian flow through a helically coiled pipe (see [23, 24]).We are now in position to define our three-dimensional domain representing thin (or long) curved pipe with an arbitrary central curveγ and circular cross section. It is given by (7)Ωεα=Φεα(Tε).Finally, we denote byΣεi=Φεα({i}×εB), i=0,l, the pipe’s ends and by Γεα=Φεα((0,l)×ε∂B) the lateral boundary.Remark 1.
From the strictly mathematical point of view, we have to ensure the local injectivity of the parametrizationΦεα. It can be accomplished by assuming that ε is sufficiently small; namely, εκL∞(0,l)<1/2 (see (29)).


### 2.2. The Governing System

We suppose that the pipeΩεα is filled with incompressible fluid with a viscosity depending on pressure. For such fluids, the stress tensor can be taken as (8)T=-pI+2μ(p)D(u),where D(u)=(1/2)∇u+(∇u)T stands for the symmetric part of the velocity gradient. We can always assume that the Reynolds number is not too large and neglect the inertial term in the original Navier-Stokes system. In view of that, the flow in Ωεα is going to be governed by the following system for the unknown velocity uε and pressure pε:(9)-div2μpεDuε+∇pε=0inΩεα,(10)divuε=0inΩεα.As mentioned before, we use the well-known Barus law to describe the viscosity-pressure dependence:(11)μ(p)=μ0eβp,μ0,β=const.>0.The above system should be completed by the appropriate boundary conditions. Since our aim is to study a real-life situation, we assume that the flow is governed by the prescribed pressure drop between pipe’s ends. On the lateral part of the boundary we prescribe the classical no-slip boundary condition for the velocity:(12)uε=0on⁡Γεα,uε×t=0,pε=pion⁡Σεi,i=0,l.Here p0 and pl are the given constant pressures (p0>pl). The well-posedness of the above problem has been recently established by Marušić-Paloka [25]. Our goal is to find the asymptotic behavior of the flow, as the thickness ε→0.


## 2.1. The Pipe’s Geometry

In this section we formally describe the complex pipe’s geometry. As emphasized in Introduction, we want to address the case of a general curved pipe with circular cross section. In view of that, we introduce a generic curve inR3, denoted by γ, which serves to define the central curve of the pipe. We suppose that γ is parameterized by its arc length x1∈[0,l] and denote by π∈C3([0,l];R3) its natural parametrization. We also assume that π′(x1)≠0, for every x1∈[0,l]. Since γ is taken to be generic, it holds π′′(x1)>0, for every x1∈[0,l]. Denoting by κ(x1)=π′′(x1) the flexion of the curve γ, we introduce Frenet’s basis in a standard way: (2)t=π′(thetangent),n=1κt′(thenormal),b=t×n(thebinormal).The normaln is extended by continuity in points where curvature is zero. We also denote by τ(x1)=-b′(x1) the torsion of γ. One of the main goals of this study is to detect the influence of geometric parameters κ and τ on the effective flow.Next, we introduce the small parameterε (0<ε≪1) into the problem and first define an undeformed pipe (3)Tε={x=(x1,x2,x3)∈R3:x1∈(0,l),x~=(x2,x3)∈εB},where B=B(0,1)=y∈R2:|y|<1 is the unit circle. Now we have to choose the appropriate parametrization to define our curved pipe. The best way to do this is to introduce the mapping Φεα:Tε→R3 as follows: (4)Φεα(x)=π(x1)+x2nα(x1)+x3bα(x1).Here (5)nα(x1)=cos⁡α(x1)n(x1)+sinα(x1)b(x1),bα(x1)=-sinα(x1)n(x1)+cos⁡α(x1)b(x1)stand for the rotated unit vectors with respect to standard Frenet’s normal and binormal (see Figure 1), where the rotation is given by(6)α(x1)=-∫x0x1‍τ(ξ)dξ+α0,(x0,α0arbitraryconstants).Figure 1
The reference system.Observe that by puttingα≡0 we get classical Frenet’s system which is most usually employed in describing the curved geometries. However, since we analyze the problem in the pipe with circular cross section (the most common one in the engineering applications), it is plausible to use this particular reference system in which the domain’s cross section possesses no rotation with respect to the tangent vector t. Let us mention that it was originally introduced by Germano in the 80s for treating the classical Newtonian flow through a helically coiled pipe (see [23, 24]).We are now in position to define our three-dimensional domain representing thin (or long) curved pipe with an arbitrary central curveγ and circular cross section. It is given by (7)Ωεα=Φεα(Tε).Finally, we denote byΣεi=Φεα({i}×εB), i=0,l, the pipe’s ends and by Γεα=Φεα((0,l)×ε∂B) the lateral boundary.Remark 1.
From the strictly mathematical point of view, we have to ensure the local injectivity of the parametrizationΦεα. It can be accomplished by assuming that ε is sufficiently small; namely, εκL∞(0,l)<1/2 (see (29)).


## 2.2. The Governing System

We suppose that the pipeΩεα is filled with incompressible fluid with a viscosity depending on pressure. For such fluids, the stress tensor can be taken as (8)T=-pI+2μ(p)D(u),where D(u)=(1/2)∇u+(∇u)T stands for the symmetric part of the velocity gradient. We can always assume that the Reynolds number is not too large and neglect the inertial term in the original Navier-Stokes system. In view of that, the flow in Ωεα is going to be governed by the following system for the unknown velocity uε and pressure pε:(9)-div2μpεDuε+∇pε=0inΩεα,(10)divuε=0inΩεα.As mentioned before, we use the well-known Barus law to describe the viscosity-pressure dependence:(11)μ(p)=μ0eβp,μ0,β=const.>0.The above system should be completed by the appropriate boundary conditions. Since our aim is to study a real-life situation, we assume that the flow is governed by the prescribed pressure drop between pipe’s ends. On the lateral part of the boundary we prescribe the classical no-slip boundary condition for the velocity:(12)uε=0on⁡Γεα,uε×t=0,pε=pion⁡Σεi,i=0,l.Here p0 and pl are the given constant pressures (p0>pl). The well-posedness of the above problem has been recently established by Marušić-Paloka [25]. Our goal is to find the asymptotic behavior of the flow, as the thickness ε→0.


## 3. Asymptotic Modeling

### 3.1. Transformed System

Substituting the Barus law into the momentum equation (9) leads to (13)0=-div2μ0eβpεD(uε)+∇pε=-μ0eβpεΔuε-2μ0βeβpεD(uε)∇pε+∇pε.We divide it byμ0eβpε to obtain(14)-Δuε+1μ0e-βpε∇pε=2βD(uε)∇pε.The form of the above equation suggests introducing a new function, denoted byqε, such that(15)1μ0e-βpε∇pε=∇qε.Here and in the sequel we will call it thetransformed pressure. It is obvious that (15) will be satisfied if we take(16)qε=1βμ0e-βq0-e-βpε,q0∈Rarbitrary.By a simple calculation we deduce(17)∇pε=μ0eβpε∇qε=μ0e-βq0-βμ0qε∇qε.Consequently, (14) becomes (18)-Δuε+∇qε=2βμ0e-αq0-βμ0qεD(uε)∇qε.The above procedure yields the following system satisfied by the velocity uε and the transformed pressure qε (we will refer to it as thetransformed system):(19)-Δuε+∇qε=2βμ0e-βq0-βμ0qεDuε∇qεinΩεα,(20)divuε=0inΩεα,(21)uε=0on⁡Γεα,(22)uε×t=0,qε=1βμ0e-βq0-e-βpion⁡Σεi,i=0,l.The obtained transformed system is in the form of a nonlinear Stokes-like system (with nonlinearity appearing on the right-hand side in (19)) and it will be the subject of our investigation in the sequel. More precisely, after writing the transformed system in curvilinear coordinates (xi), we will construct its asymptotic solution via two-scale asymptotic technique. The liberty in choice of parameter q0 (see (16)) will enable us to control the nonlinearity in a way that it does not contribute to the macroscopic model. Indeed, we can always choose q0 small enough such that(23)limq0→-∞βμ0e-βq0-βμ0qε=0.It is essential to be aware that, throughout the whole process,q0 plays the role of anauxiliary parameter. It means that, by choosing q0 such that (23) holds, we do not impose any additional constraints since it turns out that the effective pressure will not depend on the parameter q0 at all (see Section 3.4.).


### 3.2. Transformed System in Curvilinear Coordinates

#### 3.2.1. Tools from Differential Geometry

In order to write the transformed equations (19)-(20) in the curvilinear coordinates (xi), we need to introduce some advanced notions from differential geometry. We begin by introducing thecovariant basis defined as (24)ai(x)=∂Φεα∂xi(x),i=1,2,3.Using the fact thatα′=-τ together with(25)t′=κn,n′=-κt+τb,b′=-τn,it is straightforward to obtain(26)a1=(1-κ(eα·x~))t,a2=cos⁡αn+sinαb,a3=-sinαn+cos⁡αb.To make the complex notation more compact, here and in the sequel we introduce(27)eα=(cos⁡α,-sinα),eα⊥=(sinα,cos⁡α)and denote by · the standard scalar product in R2. We also denote B=tnb implying (28)∇Φεα=B1-κ(eα·x~)000cos⁡α-sinα0sinαcos⁡α.Observe that(29)det⁡∇Φεα=1-κ(eα·x~).We employed this fact to assure the local injectivity of the parametrizationΦεα (see Remark 1).The covariant basis is complemented with thecontravariant basis given by the relation (30)ai·aj=δij,i,j=1,2,3.In our case it reads(31)a1=11-κ(eα·x~)t,a2=cos⁡αn+sinαb,a3=-sinαn+cos⁡αb.Finally,Christoffel’s symbols are defined as (32)Γjki=ai·∂ak∂xj,i,j,k=1,2,3.Its fundamental property is that they are symmetric in lower indices; that is,Γjki=Γkji. We leave the reader to confirm that nonzero Christoffel’s symbols in our setting are given by (33)Γ111=-κ′(eα·x~)+κτ(eα⊥·x~)1-κ(eα·x~),Γ121=Γ211=-κcos⁡α1-κ(eα·x∗),Γ131=Γ311=κsinα1-κ(eα·x~),Γ112=κ1-κ(eα·x~)cos⁡α,Γ113=-κ1-κ(eα·x~)sinα.Now we are going to establish the asymptotic behavior of the above quantities needed in the sequel. Having in mind thatx2,x3=O(ε), from (29), we first conclude that det⁡∇Φεα=O(1). Next, we have(34)∇Φεα-1=1+εκ(eα·y)000cos⁡αsinα0-sinαcos⁡αBT+O(ε2),where y=(y2,y3), yi=xi/ε, i=2,3. Here we used the fact that the vectors of the contravariant basis are, in fact, the rows of (∇Φεα)-1. Finally, for Christoffel’s symbols we obtain(35)Γ111=O(ε),Γ121=Γ211=-κcos⁡α+O(ε),Γ131=Γ311=κsinα+O(ε),Γ112=κcos⁡α+O(ε),Γ113=-κsinα+O(ε).


#### 3.2.2. The Equations in Curvilinear Coordinates

In [10], the reader can find detailed derivation of the following formulae for differential operators in curvilinear coordinates:(36)∇sT∘Φεα=∇Φεα-T∇ST,(∇v)∘Φεα=∇Φεα-T∂Vk∂xlk,l-VjΓj∇Φεα-1,Γi=Γjkij,k,(Δv)∘Φεα=∇Φεα-T∂∂xi∂Vk∂xlk,l-VjΓj-∂Vk∂xlk,l-VjΓjΓ^i-Γ^iT∂Vk∂xlk,l-VjΓj∇Φεα-1ai,Γ^i=Γikjj,k.HereS=s∘Φεα denotes a scalar field, while V=v∘Φεα stands for a vector field (Vi=V·ai are the corresponding covariant components). It is important to be aware that, in the above formulae, summation is taken over repeated indices (Einstein summation convention).Rather complicated but straightforward calculations based on (34)–(36) provide us with the transformed equations written in the curvilinear coordinates. We write only the terms of order 1 and ε; that is, we neglect the terms with higher powers of ε. For notational simplicity, we omit the nonlinear terms in the momentum equations since those terms will not contribute to the macroscopic model. The equations read (37)-ΔVε1+κcos⁡α∂Vε1∂x2-∂Vε2∂x1-∂Vε2∂x1+sinα∂Vε3∂x1-∂Vε1∂x3+κ2-2κVε1-κ′Vε2+κ(eα·x~)·3∂2Vε1∂x12+Δx′Vε1-κcos⁡α∂Vε1∂x2+κsinα∂Vε1∂x3+3∂Vε3∂x1-3κ′Vε2-3κ1+cos⁡α∂Vε2∂x1+κ2-6κVε1+(1+κ(eα·x~))∂Qε∂x1=0,(38)-cos⁡αΔVε2-sinαΔVε3+2κ∂Vε1∂x1-κcos⁡2α∂Vε2∂x2-κsin2α∂Vε3∂x3+κ′Vε1+κ2sin2α∂Vε2∂x3+∂Vε3∂x2-κ2Vε2+2κeα·x~·cos⁡α∂2Vε2∂x12-sinα∂2Vε3∂x12+2κ∂Vε1∂x1-κcos⁡2α∂Vε2∂x2-κsin2α∂Vε3∂x3+κ2sin2α∂Vε2∂x3+∂Vε3∂x2+κ′Vε1-κ2Vε2+cos⁡α∂Qε∂x2-sinα∂Qε∂x3=0,(39)-sinαΔVε2+cos⁡αΔVε3-κcos⁡2α∂Vε3∂x2+κsin2α∂Vε2∂x3+κ2sin2α∂Vε3∂x3-∂Vε2∂x2+κτVε1+2κeα·x~·sinα∂2Vε2∂x12+cos⁡α∂2Vε3∂x12-κcos⁡2α∂Vε3∂x2+κsin2α∂Vε2∂x3+κ2sin2α∂Vε3∂x3-∂Vε2∂x2+κτVε1+sinα∂Qε∂x2-cos⁡α∂Qε∂x3=0,(40)∂Vε1∂x1-κVε2+∂Vε2∂x2+∂Vε3∂x3+2κ(eα·x~)∂Vε1∂x1-κVε2=0,with (41)Uε=uε∘Φεα=Vε1t+Vε2n+Vε3b,Qε=qε∘Φεα.


### 3.3. Asymptotic Solution of the Transformed Problem

Now we apply the two-scale asymptotic technique on the transformed problem (37)–(40) posed in Tε=(0,l)×εB. More precisely, we expand the unknowns Uε and Qε in powers of small parameter ε as follows:(42)Vεix=ε2V0ix1,x2ε,x3ε+ε3V1ix1,x2ε,x3ε+⋯,i=1,2,3,Qε(x)=Q0x1,x2ε,x3ε+εQ1x1,x2ε,x3ε+⋯.Plugging the above expansions in (37)–(40), after collecting the terms with equal powers of ε we are going to obtain the recursive sequence of problems now posed in the ε-independent domain T=(0,l)×B. In view of that, we introduce the rescaled variable y=(y2,y3)=x2/ε,x3/ε and use the following notation for partial differential operators: (43)∇y=∂∂y2j+∂∂y3k,ΔyV=∂2V∂y22+∂2V∂y32,divyV=∂V2∂y2+∂V3∂y3,V=V1i+V2j+V3k.First, let us note that substituting (42) in momentum equations (38)-(39) implies that ∇yQ0=0; that is, the lowest-order approximation for the transformed pressure depends only on the variable x1 going along the pipe. That was to be expected due to the small pipe’s thickness. Next, we deduce the problem satisfied by the zero-order approximation for the velocity:(44)1:-ΔyV0+dQ0dx1,∂Q1∂y2,∂Q1∂y3=0inT,ε: divyV0=0inT,ε2:∂V01∂x1+divyV1=0inT,V0=0on⁡Γ=(0,l)×∂B,Q0(i)=1βμ0e-βq0-e-βpii=0,l.Observe that, at this stage, the angleα is eliminated from the system by adding/substracting (38) and (39) multiplied by cos⁡α and sinα. Consequently, the system (44) will yield the solution in the following form:(45)V01(y)=14βμ0le-βp0-e-βpl(y2-1),Q0(x1)=1βμ0e-βq0-e-βp0+e-βp0-e-βplβμ0lx1,V02=V03=0,Q1=Q1(x1).Seeking for the effects of the pipe’s distortion, we continue the computation and try to construct the velocity corrector. TheO(ε) term from (37) provides the following equation for the first component V11:(46)ε:-ΔyV11+κcos⁡α∂V01∂y2-κsinα∂V01∂y3+κ(eα·y)ΔyV01+κ(eα·y)dQ0dx1+dQ1dx1=0.Taking into account (45), we arrive at(47)-ΔyV11=κ2βμ0le-βp0-e-βplcos⁡αy2-sinαy3-dQ1dx1x1inT,V11=0on⁡Γ.SinceQ1=0 for x1=0,l, the above system can be solved by putting Q1=0 and(48)V11x1,y=κx116βμ0le-βp0-e-βpl·(1-y2)cos⁡αy2-sinαy3.By a simple integration (passing to polar coordinates), one can easily check that∫B‍V11=0. It means that the pipe’s curvature, appearing in the correction for the tangential velocity, affects only the flow profile but not the mean flow.Finally, from (38)–(40) we deduce the problem for the remaining two components:(49)ε:-ΔyV~1+∇yQ2=0inT,ε2: divyV1=0inT,V~1=0on⁡Γ,where V~1=(V12,V13). It is well-known that the above system has (unique) trivial solution so we do not observe the desired effects here.To conclude this section we write the asymptotic solution of the transformed problem (19)–(22). The velocity part is given by(50)uεapprox(z)=Vε(x),z=Φεα(x),Vεx=ε2V01+ε3V11t=e-βpl-e-βp04βμ0lε2-x22-x32·1-κ(x1)4cos⁡αx2-sinαx3tε(x1).For the transformed pressure we obtain(51)qεapprox(z)=Qε(x),z=Φε(x),Qε(x)=Q0(x1)=1βμ0e-βq0-e-βp0+e-βp0-e-βplβμ0lx1.


### 3.4. Back to the Original Problem

Now we apply the inverse transformation, that is, we reconstruct the original pressure from (16):(52)pεapprox=1βln⁡1e-βq0-βμ0qεapprox.In view of (51), we deduce(53)pεapprox(z)=Pε(x),z=Φε(x),Pε(x)=1βln⁡1e-βp0+(e-βpl-e-βp0/l)x1.This represents our asymptotic approximation for the effective pressure. Obviously, it is well-defined sincep0>pl implying e-αpl>e-αp0. Next, notice that the effective pressure does not depend on q0 at all. This means that, throughout the whole process, the parameter q0 has been just an auxiliary parameter; that is, it was justified to choose the parameter q0 such that (23) holds. If we compare pεapprox with the approximation for straight-pipe flow derived in [21], we do not observe any effects coming due to the pipe’s distortion. Those effects could be retrieved by computing lower-order terms in the pressure expansion. However, those terms would serve only for the convergence proof; that is, they would not influence the effective flow.For the velocity approximation, we have(54)uεapprox(z)=Vε(x),z=Φεα(x),Vεx=e-βpl-e-βp04βμ0l(ε2-x22-x32)·1-κ(x1)4(eα·x~)tε(x1)and we can clearly detect the effects of the pipe’s curvedness. Indeed, though its main part remains in thePoiseuille form (in direction tangential to the central line of the pipe—note that the tangent is not constant vector), the flow profile is not perfectly parabolic any more and it is being corrected. More precisely, the tangential corrector is proportional to the curvature of the pipe’s central curve, and that should be acknowledged in the precise analysis of the curved-pipes flow.Finally, to emphasize the importance of the obtained result, let us compare it with the one from [12] derived for Newtonian (i.e., the constant viscosity) case. The differences coming due to the effects of the pressure-dependent viscosity are particularly observed in the approximation for the pressure (53). Instead of simple linear function Qε(x1)=(pl-p0/l)x1+p0, here we obtain a new asymptotic solution clearly acknowledging the viscosity-pressure dependence of the Barus law. The velocity approximation (54) also feels the effects of the variable viscosity. Indeed, the constant term multiplying the approximation is consistent with the viscosity obeying the Barus law and that was to be expected. The main difference with respect to the Newtonian case is that the velocity is exclusively in the tangential direction; that is, no effects of the central curve’s flexion and torsion appear in the direction of the normal or binormal.


## 3.1. Transformed System

Substituting the Barus law into the momentum equation (9) leads to (13)0=-div2μ0eβpεD(uε)+∇pε=-μ0eβpεΔuε-2μ0βeβpεD(uε)∇pε+∇pε.We divide it byμ0eβpε to obtain(14)-Δuε+1μ0e-βpε∇pε=2βD(uε)∇pε.The form of the above equation suggests introducing a new function, denoted byqε, such that(15)1μ0e-βpε∇pε=∇qε.Here and in the sequel we will call it thetransformed pressure. It is obvious that (15) will be satisfied if we take(16)qε=1βμ0e-βq0-e-βpε,q0∈Rarbitrary.By a simple calculation we deduce(17)∇pε=μ0eβpε∇qε=μ0e-βq0-βμ0qε∇qε.Consequently, (14) becomes (18)-Δuε+∇qε=2βμ0e-αq0-βμ0qεD(uε)∇qε.The above procedure yields the following system satisfied by the velocity uε and the transformed pressure qε (we will refer to it as thetransformed system):(19)-Δuε+∇qε=2βμ0e-βq0-βμ0qεDuε∇qεinΩεα,(20)divuε=0inΩεα,(21)uε=0on⁡Γεα,(22)uε×t=0,qε=1βμ0e-βq0-e-βpion⁡Σεi,i=0,l.The obtained transformed system is in the form of a nonlinear Stokes-like system (with nonlinearity appearing on the right-hand side in (19)) and it will be the subject of our investigation in the sequel. More precisely, after writing the transformed system in curvilinear coordinates (xi), we will construct its asymptotic solution via two-scale asymptotic technique. The liberty in choice of parameter q0 (see (16)) will enable us to control the nonlinearity in a way that it does not contribute to the macroscopic model. Indeed, we can always choose q0 small enough such that(23)limq0→-∞βμ0e-βq0-βμ0qε=0.It is essential to be aware that, throughout the whole process,q0 plays the role of anauxiliary parameter. It means that, by choosing q0 such that (23) holds, we do not impose any additional constraints since it turns out that the effective pressure will not depend on the parameter q0 at all (see Section 3.4.).


## 3.2. Transformed System in Curvilinear Coordinates

### 3.2.1. Tools from Differential Geometry

In order to write the transformed equations (19)-(20) in the curvilinear coordinates (xi), we need to introduce some advanced notions from differential geometry. We begin by introducing thecovariant basis defined as (24)ai(x)=∂Φεα∂xi(x),i=1,2,3.Using the fact thatα′=-τ together with(25)t′=κn,n′=-κt+τb,b′=-τn,it is straightforward to obtain(26)a1=(1-κ(eα·x~))t,a2=cos⁡αn+sinαb,a3=-sinαn+cos⁡αb.To make the complex notation more compact, here and in the sequel we introduce(27)eα=(cos⁡α,-sinα),eα⊥=(sinα,cos⁡α)and denote by · the standard scalar product in R2. We also denote B=tnb implying (28)∇Φεα=B1-κ(eα·x~)000cos⁡α-sinα0sinαcos⁡α.Observe that(29)det⁡∇Φεα=1-κ(eα·x~).We employed this fact to assure the local injectivity of the parametrizationΦεα (see Remark 1).The covariant basis is complemented with thecontravariant basis given by the relation (30)ai·aj=δij,i,j=1,2,3.In our case it reads(31)a1=11-κ(eα·x~)t,a2=cos⁡αn+sinαb,a3=-sinαn+cos⁡αb.Finally,Christoffel’s symbols are defined as (32)Γjki=ai·∂ak∂xj,i,j,k=1,2,3.Its fundamental property is that they are symmetric in lower indices; that is,Γjki=Γkji. We leave the reader to confirm that nonzero Christoffel’s symbols in our setting are given by (33)Γ111=-κ′(eα·x~)+κτ(eα⊥·x~)1-κ(eα·x~),Γ121=Γ211=-κcos⁡α1-κ(eα·x∗),Γ131=Γ311=κsinα1-κ(eα·x~),Γ112=κ1-κ(eα·x~)cos⁡α,Γ113=-κ1-κ(eα·x~)sinα.Now we are going to establish the asymptotic behavior of the above quantities needed in the sequel. Having in mind thatx2,x3=O(ε), from (29), we first conclude that det⁡∇Φεα=O(1). Next, we have(34)∇Φεα-1=1+εκ(eα·y)000cos⁡αsinα0-sinαcos⁡αBT+O(ε2),where y=(y2,y3), yi=xi/ε, i=2,3. Here we used the fact that the vectors of the contravariant basis are, in fact, the rows of (∇Φεα)-1. Finally, for Christoffel’s symbols we obtain(35)Γ111=O(ε),Γ121=Γ211=-κcos⁡α+O(ε),Γ131=Γ311=κsinα+O(ε),Γ112=κcos⁡α+O(ε),Γ113=-κsinα+O(ε).


### 3.2.2. The Equations in Curvilinear Coordinates

In [10], the reader can find detailed derivation of the following formulae for differential operators in curvilinear coordinates:(36)∇sT∘Φεα=∇Φεα-T∇ST,(∇v)∘Φεα=∇Φεα-T∂Vk∂xlk,l-VjΓj∇Φεα-1,Γi=Γjkij,k,(Δv)∘Φεα=∇Φεα-T∂∂xi∂Vk∂xlk,l-VjΓj-∂Vk∂xlk,l-VjΓjΓ^i-Γ^iT∂Vk∂xlk,l-VjΓj∇Φεα-1ai,Γ^i=Γikjj,k.HereS=s∘Φεα denotes a scalar field, while V=v∘Φεα stands for a vector field (Vi=V·ai are the corresponding covariant components). It is important to be aware that, in the above formulae, summation is taken over repeated indices (Einstein summation convention).Rather complicated but straightforward calculations based on (34)–(36) provide us with the transformed equations written in the curvilinear coordinates. We write only the terms of order 1 and ε; that is, we neglect the terms with higher powers of ε. For notational simplicity, we omit the nonlinear terms in the momentum equations since those terms will not contribute to the macroscopic model. The equations read (37)-ΔVε1+κcos⁡α∂Vε1∂x2-∂Vε2∂x1-∂Vε2∂x1+sinα∂Vε3∂x1-∂Vε1∂x3+κ2-2κVε1-κ′Vε2+κ(eα·x~)·3∂2Vε1∂x12+Δx′Vε1-κcos⁡α∂Vε1∂x2+κsinα∂Vε1∂x3+3∂Vε3∂x1-3κ′Vε2-3κ1+cos⁡α∂Vε2∂x1+κ2-6κVε1+(1+κ(eα·x~))∂Qε∂x1=0,(38)-cos⁡αΔVε2-sinαΔVε3+2κ∂Vε1∂x1-κcos⁡2α∂Vε2∂x2-κsin2α∂Vε3∂x3+κ′Vε1+κ2sin2α∂Vε2∂x3+∂Vε3∂x2-κ2Vε2+2κeα·x~·cos⁡α∂2Vε2∂x12-sinα∂2Vε3∂x12+2κ∂Vε1∂x1-κcos⁡2α∂Vε2∂x2-κsin2α∂Vε3∂x3+κ2sin2α∂Vε2∂x3+∂Vε3∂x2+κ′Vε1-κ2Vε2+cos⁡α∂Qε∂x2-sinα∂Qε∂x3=0,(39)-sinαΔVε2+cos⁡αΔVε3-κcos⁡2α∂Vε3∂x2+κsin2α∂Vε2∂x3+κ2sin2α∂Vε3∂x3-∂Vε2∂x2+κτVε1+2κeα·x~·sinα∂2Vε2∂x12+cos⁡α∂2Vε3∂x12-κcos⁡2α∂Vε3∂x2+κsin2α∂Vε2∂x3+κ2sin2α∂Vε3∂x3-∂Vε2∂x2+κτVε1+sinα∂Qε∂x2-cos⁡α∂Qε∂x3=0,(40)∂Vε1∂x1-κVε2+∂Vε2∂x2+∂Vε3∂x3+2κ(eα·x~)∂Vε1∂x1-κVε2=0,with (41)Uε=uε∘Φεα=Vε1t+Vε2n+Vε3b,Qε=qε∘Φεα.


## 3.2.1. Tools from Differential Geometry

In order to write the transformed equations (19)-(20) in the curvilinear coordinates (xi), we need to introduce some advanced notions from differential geometry. We begin by introducing thecovariant basis defined as (24)ai(x)=∂Φεα∂xi(x),i=1,2,3.Using the fact thatα′=-τ together with(25)t′=κn,n′=-κt+τb,b′=-τn,it is straightforward to obtain(26)a1=(1-κ(eα·x~))t,a2=cos⁡αn+sinαb,a3=-sinαn+cos⁡αb.To make the complex notation more compact, here and in the sequel we introduce(27)eα=(cos⁡α,-sinα),eα⊥=(sinα,cos⁡α)and denote by · the standard scalar product in R2. We also denote B=tnb implying (28)∇Φεα=B1-κ(eα·x~)000cos⁡α-sinα0sinαcos⁡α.Observe that(29)det⁡∇Φεα=1-κ(eα·x~).We employed this fact to assure the local injectivity of the parametrizationΦεα (see Remark 1).The covariant basis is complemented with thecontravariant basis given by the relation (30)ai·aj=δij,i,j=1,2,3.In our case it reads(31)a1=11-κ(eα·x~)t,a2=cos⁡αn+sinαb,a3=-sinαn+cos⁡αb.Finally,Christoffel’s symbols are defined as (32)Γjki=ai·∂ak∂xj,i,j,k=1,2,3.Its fundamental property is that they are symmetric in lower indices; that is,Γjki=Γkji. We leave the reader to confirm that nonzero Christoffel’s symbols in our setting are given by (33)Γ111=-κ′(eα·x~)+κτ(eα⊥·x~)1-κ(eα·x~),Γ121=Γ211=-κcos⁡α1-κ(eα·x∗),Γ131=Γ311=κsinα1-κ(eα·x~),Γ112=κ1-κ(eα·x~)cos⁡α,Γ113=-κ1-κ(eα·x~)sinα.Now we are going to establish the asymptotic behavior of the above quantities needed in the sequel. Having in mind thatx2,x3=O(ε), from (29), we first conclude that det⁡∇Φεα=O(1). Next, we have(34)∇Φεα-1=1+εκ(eα·y)000cos⁡αsinα0-sinαcos⁡αBT+O(ε2),where y=(y2,y3), yi=xi/ε, i=2,3. Here we used the fact that the vectors of the contravariant basis are, in fact, the rows of (∇Φεα)-1. Finally, for Christoffel’s symbols we obtain(35)Γ111=O(ε),Γ121=Γ211=-κcos⁡α+O(ε),Γ131=Γ311=κsinα+O(ε),Γ112=κcos⁡α+O(ε),Γ113=-κsinα+O(ε).


## 3.2.2. The Equations in Curvilinear Coordinates

In [10], the reader can find detailed derivation of the following formulae for differential operators in curvilinear coordinates:(36)∇sT∘Φεα=∇Φεα-T∇ST,(∇v)∘Φεα=∇Φεα-T∂Vk∂xlk,l-VjΓj∇Φεα-1,Γi=Γjkij,k,(Δv)∘Φεα=∇Φεα-T∂∂xi∂Vk∂xlk,l-VjΓj-∂Vk∂xlk,l-VjΓjΓ^i-Γ^iT∂Vk∂xlk,l-VjΓj∇Φεα-1ai,Γ^i=Γikjj,k.HereS=s∘Φεα denotes a scalar field, while V=v∘Φεα stands for a vector field (Vi=V·ai are the corresponding covariant components). It is important to be aware that, in the above formulae, summation is taken over repeated indices (Einstein summation convention).Rather complicated but straightforward calculations based on (34)–(36) provide us with the transformed equations written in the curvilinear coordinates. We write only the terms of order 1 and ε; that is, we neglect the terms with higher powers of ε. For notational simplicity, we omit the nonlinear terms in the momentum equations since those terms will not contribute to the macroscopic model. The equations read (37)-ΔVε1+κcos⁡α∂Vε1∂x2-∂Vε2∂x1-∂Vε2∂x1+sinα∂Vε3∂x1-∂Vε1∂x3+κ2-2κVε1-κ′Vε2+κ(eα·x~)·3∂2Vε1∂x12+Δx′Vε1-κcos⁡α∂Vε1∂x2+κsinα∂Vε1∂x3+3∂Vε3∂x1-3κ′Vε2-3κ1+cos⁡α∂Vε2∂x1+κ2-6κVε1+(1+κ(eα·x~))∂Qε∂x1=0,(38)-cos⁡αΔVε2-sinαΔVε3+2κ∂Vε1∂x1-κcos⁡2α∂Vε2∂x2-κsin2α∂Vε3∂x3+κ′Vε1+κ2sin2α∂Vε2∂x3+∂Vε3∂x2-κ2Vε2+2κeα·x~·cos⁡α∂2Vε2∂x12-sinα∂2Vε3∂x12+2κ∂Vε1∂x1-κcos⁡2α∂Vε2∂x2-κsin2α∂Vε3∂x3+κ2sin2α∂Vε2∂x3+∂Vε3∂x2+κ′Vε1-κ2Vε2+cos⁡α∂Qε∂x2-sinα∂Qε∂x3=0,(39)-sinαΔVε2+cos⁡αΔVε3-κcos⁡2α∂Vε3∂x2+κsin2α∂Vε2∂x3+κ2sin2α∂Vε3∂x3-∂Vε2∂x2+κτVε1+2κeα·x~·sinα∂2Vε2∂x12+cos⁡α∂2Vε3∂x12-κcos⁡2α∂Vε3∂x2+κsin2α∂Vε2∂x3+κ2sin2α∂Vε3∂x3-∂Vε2∂x2+κτVε1+sinα∂Qε∂x2-cos⁡α∂Qε∂x3=0,(40)∂Vε1∂x1-κVε2+∂Vε2∂x2+∂Vε3∂x3+2κ(eα·x~)∂Vε1∂x1-κVε2=0,with (41)Uε=uε∘Φεα=Vε1t+Vε2n+Vε3b,Qε=qε∘Φεα.


## 3.3. Asymptotic Solution of the Transformed Problem

Now we apply the two-scale asymptotic technique on the transformed problem (37)–(40) posed in Tε=(0,l)×εB. More precisely, we expand the unknowns Uε and Qε in powers of small parameter ε as follows:(42)Vεix=ε2V0ix1,x2ε,x3ε+ε3V1ix1,x2ε,x3ε+⋯,i=1,2,3,Qε(x)=Q0x1,x2ε,x3ε+εQ1x1,x2ε,x3ε+⋯.Plugging the above expansions in (37)–(40), after collecting the terms with equal powers of ε we are going to obtain the recursive sequence of problems now posed in the ε-independent domain T=(0,l)×B. In view of that, we introduce the rescaled variable y=(y2,y3)=x2/ε,x3/ε and use the following notation for partial differential operators: (43)∇y=∂∂y2j+∂∂y3k,ΔyV=∂2V∂y22+∂2V∂y32,divyV=∂V2∂y2+∂V3∂y3,V=V1i+V2j+V3k.First, let us note that substituting (42) in momentum equations (38)-(39) implies that ∇yQ0=0; that is, the lowest-order approximation for the transformed pressure depends only on the variable x1 going along the pipe. That was to be expected due to the small pipe’s thickness. Next, we deduce the problem satisfied by the zero-order approximation for the velocity:(44)1:-ΔyV0+dQ0dx1,∂Q1∂y2,∂Q1∂y3=0inT,ε: divyV0=0inT,ε2:∂V01∂x1+divyV1=0inT,V0=0on⁡Γ=(0,l)×∂B,Q0(i)=1βμ0e-βq0-e-βpii=0,l.Observe that, at this stage, the angleα is eliminated from the system by adding/substracting (38) and (39) multiplied by cos⁡α and sinα. Consequently, the system (44) will yield the solution in the following form:(45)V01(y)=14βμ0le-βp0-e-βpl(y2-1),Q0(x1)=1βμ0e-βq0-e-βp0+e-βp0-e-βplβμ0lx1,V02=V03=0,Q1=Q1(x1).Seeking for the effects of the pipe’s distortion, we continue the computation and try to construct the velocity corrector. TheO(ε) term from (37) provides the following equation for the first component V11:(46)ε:-ΔyV11+κcos⁡α∂V01∂y2-κsinα∂V01∂y3+κ(eα·y)ΔyV01+κ(eα·y)dQ0dx1+dQ1dx1=0.Taking into account (45), we arrive at(47)-ΔyV11=κ2βμ0le-βp0-e-βplcos⁡αy2-sinαy3-dQ1dx1x1inT,V11=0on⁡Γ.SinceQ1=0 for x1=0,l, the above system can be solved by putting Q1=0 and(48)V11x1,y=κx116βμ0le-βp0-e-βpl·(1-y2)cos⁡αy2-sinαy3.By a simple integration (passing to polar coordinates), one can easily check that∫B‍V11=0. It means that the pipe’s curvature, appearing in the correction for the tangential velocity, affects only the flow profile but not the mean flow.Finally, from (38)–(40) we deduce the problem for the remaining two components:(49)ε:-ΔyV~1+∇yQ2=0inT,ε2: divyV1=0inT,V~1=0on⁡Γ,where V~1=(V12,V13). It is well-known that the above system has (unique) trivial solution so we do not observe the desired effects here.To conclude this section we write the asymptotic solution of the transformed problem (19)–(22). The velocity part is given by(50)uεapprox(z)=Vε(x),z=Φεα(x),Vεx=ε2V01+ε3V11t=e-βpl-e-βp04βμ0lε2-x22-x32·1-κ(x1)4cos⁡αx2-sinαx3tε(x1).For the transformed pressure we obtain(51)qεapprox(z)=Qε(x),z=Φε(x),Qε(x)=Q0(x1)=1βμ0e-βq0-e-βp0+e-βp0-e-βplβμ0lx1.


## 3.4. Back to the Original Problem

Now we apply the inverse transformation, that is, we reconstruct the original pressure from (16):(52)pεapprox=1βln⁡1e-βq0-βμ0qεapprox.In view of (51), we deduce(53)pεapprox(z)=Pε(x),z=Φε(x),Pε(x)=1βln⁡1e-βp0+(e-βpl-e-βp0/l)x1.This represents our asymptotic approximation for the effective pressure. Obviously, it is well-defined sincep0>pl implying e-αpl>e-αp0. Next, notice that the effective pressure does not depend on q0 at all. This means that, throughout the whole process, the parameter q0 has been just an auxiliary parameter; that is, it was justified to choose the parameter q0 such that (23) holds. If we compare pεapprox with the approximation for straight-pipe flow derived in [21], we do not observe any effects coming due to the pipe’s distortion. Those effects could be retrieved by computing lower-order terms in the pressure expansion. However, those terms would serve only for the convergence proof; that is, they would not influence the effective flow.For the velocity approximation, we have(54)uεapprox(z)=Vε(x),z=Φεα(x),Vεx=e-βpl-e-βp04βμ0l(ε2-x22-x32)·1-κ(x1)4(eα·x~)tε(x1)and we can clearly detect the effects of the pipe’s curvedness. Indeed, though its main part remains in thePoiseuille form (in direction tangential to the central line of the pipe—note that the tangent is not constant vector), the flow profile is not perfectly parabolic any more and it is being corrected. More precisely, the tangential corrector is proportional to the curvature of the pipe’s central curve, and that should be acknowledged in the precise analysis of the curved-pipes flow.Finally, to emphasize the importance of the obtained result, let us compare it with the one from [12] derived for Newtonian (i.e., the constant viscosity) case. The differences coming due to the effects of the pressure-dependent viscosity are particularly observed in the approximation for the pressure (53). Instead of simple linear function Qε(x1)=(pl-p0/l)x1+p0, here we obtain a new asymptotic solution clearly acknowledging the viscosity-pressure dependence of the Barus law. The velocity approximation (54) also feels the effects of the variable viscosity. Indeed, the constant term multiplying the approximation is consistent with the viscosity obeying the Barus law and that was to be expected. The main difference with respect to the Newtonian case is that the velocity is exclusively in the tangential direction; that is, no effects of the central curve’s flexion and torsion appear in the direction of the normal or binormal.


## 4. Concluding Remarks

In the previous section, we formally derived an asymptotic model describing the flow of incompressible fluid with a pressure-dependent viscosity through a thin (or long) curved pipe. We assumed that the pressure-dependent viscosity obeyed Barus formula (most commonly used in the engineering community) and worked with physically relevant Dirichlet boundary conditions. Since the pipe’s central curve is assumed to be a general (generic) curve, the applicability of the obtained result is broad. Moreover, by obtaining the explicit expressions (53)-(54) for the pressure and velocity distribution, we believe that the result is very relevant with regard to numerical simulations. The effects of pipe’s distortion are clearly detected and the difference between classical Newtonian and non-Newtonian flow is established as well.From the rigorous mathematical point of view, we should link our formally obtained solution with the original solution by proving some kind of convergence result. It can be accomplished by evaluating the difference between those two solutions in the appropriate rescaled norm. Though rigorous justification is out of scope of the present paper, let us comment on this as well. Sincedet⁡∇Φεα=O(1), it is easy to confirm that the appropriate norm is, in fact, ·ε=ε-1·L2(Ωεα). Thus, the idea is to derive the error estimates (expressed in this norm) for the asymptotic solution of the transformed problem and then to use the continuity of the inverse transformation. The first part can be done using techniques from [10, 12] developed for curved geometries (see also [8, 9]), while the second part is straightforward.To conclude, it is important to emphasize that the presented approach can be extended to a case of general viscosity-pressure relationp↦μ(p) satisfied by Barus law and other empiric laws. To assure the well-posedness (see [25]), we only have to impose some technical conditions addressing the behavior of the viscosity for large negative pressures being, obviously, irrelevant from the physical point of view. Instead of (16), we simply introduce qε as B(pε)=∫q0pε‍(dξ/μξ) and apply the same procedure as presented above. We refer the reader to [21] for details.



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*Source: 905406-2015-03-26.xml*