Modified Reynolds equation for steady flow through a curved pipe

A modified Reynolds equation governing the steady flow of a fluid with low Reynolds number through a curvilinear, narrow tube, with its derivation from Stokes equations through asymptotic methods is presented. The channel considered may have large curvature and torsion. Approximations of the velocity and the pressure of the fluid inside the channel are constructed by artificially imposing appropriate boundary conditions at the inlet and the outlet. A justification for the approximations is provided along with a comparison with a simpler case.


Introduction
Fluid flow through narrow tubes have a wide range of applications including industrial, vehicular and biological systems. This has generated interest in the mathematical study of such flows and various results have been produced based on different cases, configurations and assumptions, see for example [12,15] and references therein. A vast majority of the existing works are about straight pipes. Straight narrow pipes with varying cross-sections are considered in [11]. Curvilinear pipes with torsion and curvature have been studied in [9], although a fixed cross-section is assumed throughout the length of the pipe. However, in [9], the effects of the curvature and torsion can only be seen in the second term of the asymptotic approximation as the leading terms are free of these effects. The Reynolds equation is widely used to describe flow through thin channels and its derivation from the Stokes equations can be found in [1,2].
We consider steady flows through thin pipes having a variable geometry of the cross-section while allowing large longitudinal curvature and torsion. We aim to construct an analogue of the Reynolds equation that can be successfully used for this general situation. Certainly, there are some natural restrictions, for example, on the longitudinal curvature.
Our main task is to derive a one-dimensional Reynolds equation for the pressure and an equation for the longitudinal component of the velocity resulting in a modified Poiseuille flow. To construct these terms, we need two boundary conditions which are usually prescribed through pressure and/or flux. The discrepancy is represented by the boundary layer terms that are significant near the end cross-sections. This is the reason that we choose artificial boundary conditions which allow us to easily estimate the discrepancy.
1.1. Nondimensionalization of the problem. Let us consider a curvilinear pipe of length L with varying channel diameter and non-circular cross-section. Let us denote the mean radius by H which satisfies H = Lh. We assume that the pipe is narrow or in other words, h ≪ 1 is a small parameter. Let us denote the steady state velocity and the kinematic pressure of the flowing fluid by V and P respectively. These satisfy the Navier-Stokes system −ν∆ X V + (V · ∇ X )V + ∇ X P = 0, −∇ X · V = 0, complemented by suitable boundary conditions and where ν is the kinematic viscosity of the fluid flowing through the tube and X signifies the position vector in suitable units of length. Let F be the flux through a cross-section. We introduce new dimensionless variables for our problem: In terms of the new dimensionless quantities and the Reynolds number Re := F L νH 2 , the Stokes system takes the form For the article, we assume a small Reynolds number so that we are able to discard the convective term.

1.2.
Formulation of the problem. Let c denote the arc-length parameterized centre curve for the pipe in consideration with s ∈ [0, 1] being the arc-length parameter. Let hR > 0 give the distance of the interior boundary of the pipe from c(s) along a direction which is perpendicular to the centre curve at s.
Let us consider the regions Ω h = {x(r, θ, s) : η = h −1 r, 0 ≤ r < hR(θ, s), θ ∈ [0, 2π), s ∈ (0, 1)}, The region Ω h is assumed to be locally Lipschitz and each transversal cross-section, hence ω(s) as well, to be a star domain for every s.. The goal is to find an asymptotic approximation of the solution of the Stokes problem supplemented with appropriate boundary conditions at the cross-sections s = 0 and s = 1.

1.3.
Results. The primary highlight of this article is the derivation of a modified Reynolds equation −∂ s (G(s)∂ s p 0 (s)) = 0, s ∈ (0, 1), for flow through a curved pipe. Here, the function G, defined as depends on the geometry of the pipe and p 0 is the leading term in the formal asymptotic expansion of p h as stated in (34). Defining β(η, θ, s) as the scale factor Ω h e 1 (θ, s) corresponding to the longitudinal parameter s and ∇ ‡ as the gradient operator on ω(s), the function Ψ is obtained as the solution of −∇ ‡ · β∇ ‡ Ψ = 2 in ω(s), Ψ = 0 on ∂ω(s).
The modified Reynolds equation takes into consideration a relatively wide range of curvature as well as variation of the diameter of the pipe as characterized by the following restrictions: for some positive δ. In particular, the first inequality implies Our approach in this article is to treat these geometrical quantities as separate parameters in the beginning and choosing the optimal orders by the end of the analysis. The new equation covers the case of smooth curvature (order 0 w.r.t h) and nearly constant radius (order 1 w.r.t h) of the pipe as well which is achieved by replacing the right hand sides in (4) by a constant independent of h.
Based on the modified Reynolds equation, under the said assumptions and provided the appropriate boundary conditions, we proceed to construct an approximation . Accordingly, we obtain the representation We finally prove that the error terms {v h rem , p h rem } in (5) admit the bounds thereby justifying the approximate solution for any positive δ.

Geometry and notations
The ambient three dimensional space is taken to have a canonical Cartesian coordinate system. The initial direction of the curve is assumed to be along the third coordinate direction, i.e., c ′ (0) = (0, 0, 1) T . The vector quantities for the problem are described using the coordinate frame consisting of the triplet {e 1 (θ, s), e 2 (θ, s), c ′ (s)}, depicted in Figure 1, where e i are obtained by solving with the initial conditions e 1 (θ, 0) = (cos θ, sin θ, 0) T and e 2 (θ, 0) = (− sin θ, cos θ, 0) T .
The parameter θ ∈ [0, 2π) signifies the direction from a point on c, along the plane perpendicular to c ′ at that point, with respect to some reference direction. Throughout this article, vectors are denoted in bold while their components along e 1 , e 2 and c ′ are given by the corresponding letter with subscripts 1, 2 and 3 respectively.
Note that in the absence of curvature, {r, θ, s} are cylindrical coordinates for the tube.
Clearly, R must be positive and sufficiently smooth and we define Also as a result, Let ∇ • be the two dimensional gradient operator on a cross-section, i.e., Let ∇ ‡ and ∆ ‡ denote the components of the gradient operator and the Laplacian respectively, on a cross-section in terms of the scaled parameters η := h −1 r and θ, i.e., corresponding to the parameter s and used the fact that Then due to (10), (11) and (12), we have (13) |∂ s β| ≤ cλ and |∂ 2 s β| ≤ c(λ * + h −1/2 λ 3/2 ). The physical restriction on the curvature such that hc ′′ (s) · e 1 (θ, s) < R(θ, s) −1 for all (θ, s) ∈ [0, 2π) × [0, 1] eliminates the possibility of the pipe curving into itself. Moreover, to ensure the validity of the asymptotic procedure followed in this article, we must additionally assume For integration over the cross sections, we have the area element dσ(r, θ) = rdrdθ.
On the other hand, the volume element is given in the new coordinates as dx(r, θ, s) = β(r, θ, s)dσ(r, θ)ds.

Let us denote the velocity vector component wise as
The components of the quantities in (1) along any cross-section ω(s) of the pipe satisfy On the other hand, (1) results in the following equation for the direction along the length of the pipe: In the above and henceforth, ‡ in the subscript of the vector symbols denote their respective projections onto the cross-sectional plane.
In order to ensure uniqueness of the asymptotic solution of the above problem, we intend to impose additional artificial conditions at the ends of the pipe. We shall argue in the next sections that a prescribed flux at the inlet and an ambient (possibly atmospheric) pressure condition at the outlet are sufficient for our purpose.

Model problems and estimates
In this section, we present the estimates related to some model problems that we rely upon in the asymptotic procedure. We use similar notations for function spaces as in [14], which include standard notations for Sobolev spaces. In particular, the function spaces denoted by bold letters represent the corresponding space of vector/tensor valued functions of the appropriate dimension.
3.1. Stokes system. We first consider a modified Stokes problem on the two-dimensional domain ω(s). We present the relevant estimates in the theorem that follows.

The solutions admit the estimate
whereq is q averaged over ω(s).
Furthermore, if f ∈ L 2 (ω(s)), g ∈ H 1 (ω(s)) and h ∈ H 3/2 (∂ω(s)), then u ∈ H 2 (ω(s)) and q ∈ H 1 (ω(s)) satisfy Proof. We accept (20) without proof as it is a standard estimate for generalised Stokes systems, see e.g. [4]. that can be applied to this case owing to the boundedness of the parameter β. In order to obtain (21), we rewrite (19) as Using the boundedness of β and (12), we have the following estimate due to results in [14].
We have the following corollary as a consequence of the above theorem.
3.2. The elliptic system. The next theorem provides us the estimates for the model problem for scalar functions that appear in the asymptotic procedure. The results are standard (see e.g. [7]) and hence the proof is omitted.
3.3. The divergence equation. In this subsection, we consider the divergence equation for two different cases of a curvilinear pipe having a variable cross-section. The divergence equation frequently appears in the study of flows and hence is an important auxiliary problem, see [3,6]. For the case of thin tubular domains, in the previous works starting with [10], coordinate dilation and uniform scaling of the transversal velocity components were sufficient to derive the specific estimates. See also [13]. The presence of curvature complicates our case, therefore, position dependent scaling involving the curvature dependent scale factor β is introduced to tackle this problem.
Firstly, we present a Lemma about the divergence equation in a thin curvilinear pipe Ω h laving length 1.
Then there exists a (non unique) solution w ∈ H 1 (Ω h ) of the divergence equation which obeys the estimate for some constant C independent of f and h.
Proof. Noting the fact that ∇ ‡ β = −hc ′′ and in accordance with the scaled parameter η = h −1 r, we introduce the scaled function Thus, we have . Clearly, the terms within the brackets in the last equality represent the polar form of the divergence of a vector field defined in a straight cylinder. As a result, we can say that w satisfies (29) if and only ifw :=ŵ 1η +ŵ 2θ +ŵ 3ŝ (likewise forη,θ andŝ being the unit vectors corresponding to cylindrical coordinates) satisfies the system divw = βf in Ξ, Here Ξ is a cylinder with a straight axis and given as For the function βf, we have that Thus the compatibility condition is met by βf.
Therefore, by a classical result on the divergence equation (see [5]) in a fixed Lipschitz domain, we havew ∈ H 1 (Ξ) ⇒ŵ ∈ H 1 (Ω h ) and for a constant C independent of the data, the estimate Owing to the bounds (10) and (12), the above leads us to (30).
We present another lemma on the divergence equation restricted to a length of a pipe that is comparable to the thickness of the pipe. The estimate in this case is modified as compared to that in Lemma 1 due to the differing aspect ratio of the segment of the curvilinear pipe in question. Let us consider (29) and (28) restricted to the domain Ω h end : Then there exists a (non unique) solution w ∈ H 1 (Ω h end ) of the divergence equation which obeys the estimate Proof. We introduce the scaled parameters η = h −1 r and τ = h −1 s and the scaled functionŵ = βw ‡ + w 3 c ′ . The rest of the proof follows the steps in the proof of Lemma 1 and we get the required estimate.

Formal asymptotic procedure
Let us consider the asymptotic Ansätze: Having an O(h −1 ) velocity still results in an O(h) flux through the cross-sections so that ignoring the convective term in the Navier-Stokes equations can still be justified. The first step of matching coefficients of the leading order of h in (15), (17) and (3) produces the following system of equations: For the solution of (37), due to (24), we have the estimate Applying Corollary 2 and using (12) and (13), we obtain the additional estimates We also need the boundedness of the functions Ψ and Ψ −1 to proceed further and hence we present the following proposition. Proposition 1. There exist constants C 1 , C 2 > 0 dependent on the domain ω such that Proof. For a bounded domain ω, let us consider a general elliptic operator The coefficients a ij are real-valued from L ∞ (ω) and satisfy for some µ 1 , µ 2 > 0. Let G = G(x, y) be the Green's function for L with the homogeneous boundary condition on ∂ω. Then, G > 0 and for x, y ∈ ω, for |x − y| ≤ 1 2 dist(y, ∂ω). By results in [8], it is sufficient to verify this for the Laplacian for which it is known. Here, C 1 and C 2 are positive constants that depend only on µ 1 and µ 2 . The function Ψ is represented as where G now represents the Green's function for the operator β −1 ∇ ‡ · β∇ ‡ . The required estimate follows from (41)  Now we consider the next step in the asymptotic procedure, that is to compare the coefficients of the next order terms. We have in ω(s), v 2 ‡ = 0 ‡ on ∂ω(s). Owing to the zero boundary conditions for v 2 ‡ and Ψ, we get the compatibility condition for this problem Ψ(η, θ, s)dσ(η, θ)∂ s p 0 (s)).
which motivates the imposition of the boundary flux condition We can also argue to impose the condition The mixed boundary problem stated in (44), (45) and (46) has the solution This leads to where we have used (40) and the fact that ∂ s Ψ L 2 (∂ω(s)) ≤ c( ∂ s Ψ H 1 (∂ω(s)) ). As a result of the above along with (38), (39) and (40), (36) gives us Here and henceforth, · shall denote the usual norm on L 2 (Ω h ) and all constants c will be of the form c = C(|F 0 | + |p 0 per |) where C is independent of F 0 , p 0 per . Consequently, by (20), since ∂ s v 1 3 ∈ L 2 (ω(s)) and v 1 3 ∈ L 2 (ω(s)) ⊂ H −1 (ω(s)), denoting the average of p 2 over the cross-section byp 2 we have for the solution of (43), and similarly by (22)

Boundary conditions at the ends
In order to solve the Stokes problem, we need to specify appropriate boundary conditions at the inlet and the outlet. We consider the domain Ω h to be an arbitrarily chosen segment of a much larger pipe in which the fluid is injected at one end and it flows out at the other. Such conditions at the end cross-sections are extremely difficult to model reasonably hence we restrict ourselves to the chosen segment, possibly far away from the ends. Imposing artificial boundary conditions at the ends of the chosen segment gives rise to the boundary layer phenomena near those ends. It brings about a quick variability near the end cross-sections in the solution {v h , p h } of the problem. Although, from a practical point of view, it is absurd to expect such quick variability at arbitrarily chosen portions of the full pipe.
The function of the boundary layer terms in the solutions is to reduce the discrepancy in the artificial boundary conditions. We want to impose such boundary conditions which make the discrepancy as small as possible. One can of course formulate elaborate sets of conditions to achieve this. We however, opt for the simpler way of preparing the boundary data in accordance to our approximations. We take the traces of our approximate fields at the end cross-sections and use them as the boundary data. Thus we reduce the discrepancy at the boundaries to zero while also diminishing the error estimates.

5.1.
Boundary conditions on the cross-section ω h (0). We note that the components of the Ansätze (34) at the point s = 0 are completely determined by the data of the problem. Indeed, according to the boundary condition (45) we have In this case, the right-hand side of the continuity equation, ∂ s Ψ(η, θ, 0)∂ s p 0 (0) + Ψ(η, θ, 0)∂ 2 s p 0 (0), can be evaluated by using the boundary condition (45) and the differential equation (44).
Thus, we should take the boundary conditions 5.2. Boundary conditions on the cross-section ω h (1). Since the fluxes through the cross-sections do not change, the expressions (so called velocities of pseudodeformations) generated by the ansatz (34), , 1), j = 1, 2, can be also evaluated by using the problem's data. The term h −3 p 0 (1) is essentially (h −2 times) larger than the other in (52). Therefore, we can take as one of the boundary conditions on the cross-section ω h (1) with p h per = h −3 p 0 per . We emphasize that the pressure itself can be taken in the boundary condition since it does not appear in the Green formula for the Stokes system alone. Further, we complement (53) by the following conditions: Due to the continuity equation (2) and relation (54), we have 0 = ∇ x · v h (r, θ, 1) = β −1 ∂ s v h 3 (r, θ, 1) and hence the boundary conditions (53) and (54) lead to a constant pressure on the cross-section ω h (1). 6. Estimates of the asymptotical remainder terms in the Stokes problem.
Recall that the solution (v h , p h ) of the problem (1) where we take the approximate solution to be and v 1 3 and v 2 ‡ are solutions of (35) and (43) respectively. Inclusion of the term v 2 ‡ makes the approximate velocity divergence-free inside the channel away from the ends. Moreover, the order of magnitude of |v 2 ‡ | grows closer to that of the leading term h −1 v 1 3 with increasing λ as is evident from (47)   In order to derive estimates of the error terms, we require an approximate velocity that is divergence-free in the entire domain including near the ends. Hence, to compensate for the error in the divergence of v v v h near the inlet and the outlet, we consider w which satisfies in Ω h ,

The compatibility condition for this problem is satisfied as
Therefore one could apply Lemma 1 to get the corresponding estimates for the vector field w. However, the estimates can be improved upon by observing that the right hand side vanishes in most of the domain.
. Let us denote the inner product in L 2 (Ω h ) by (·, ·). The discrepancy in (1) is . Rearranging the terms with respect to orders of h, we have . Applying (35) to the above, we obtain Substituting the expression for F h , the equation above can be written as Integrating by parts again and using the fact thatṽ h is divergence-free, we get Let us now estimate the terms in (57) by using the previously derived estimates. At this point, we shall assume that the parameters λ, γ, λ * and γ * are such that γ = O(λ), γ * = (λ * ), and λ * = O(h −1/2 λ 3/2 ).
With the help of (47) and (12), the first term in the right hand side of (57) can be estimated as Using (47) to estimate the second term, we have We use (48) to get Then due to (49) and (12), we have Finally, (56) gives us Also, by Friedrichs's inequality, for the curved cylinder Ω h . Thus, for the discrepancy in the velocity, we arrive at the estimate . Thus it is safe to conclude that the approximation of velocity is justified for a λ which is O(h −1+2δ ) for any δ > 0.
Let us now estimate the discrepancy in the approximation of pressure. Let us denote the average of a scalar field over Ω h by placing a bar over the corresponding symbol. Consider the velocity fieldw such that Clearly, the compatibility condition (28) is satisfied. Then, integration by parts and (53) result in Similar steps as before lead us to . Using (58) and Lemma 1, we estimate the first term on the right hand side of (59) as Due to (56), for the next term, we have Once again by Lemma 1,(47) and (12), we get Then due to (49) and (12), we have Thus, for the discrepancy in the pressure, we get , once again we see that the approximate pressure upto a constant is justified for a λ which is O(h −1+2δ ) for any δ > 0.
To summarize, we have shown that (6) and (7) hold under the assumptions (4) thereby justifying our asymptotic approximations.

The case of O(1) curvature
The more conventional method to tackle the problem in the case of a mildly curving pipe, where we assume that c is a smooth function whose derivatives are bounded independently of h, would be to expand the scale factor β as 1−hηc ′′ ·e 1 , instead of keeping it as a parameter for the asymptotic procedure. Let us compare the results obtained by our method in this case with those obtained with the conventional method as mentioned. For this case, the assumptions on the geometry of the centre curve are such that |c ′′′ | ≤ ch 0 , |c ′′′′ | ≤ ch 0 , |∂ s R| ≤ ch 0 and |∂ 2 s R| ≤ ch 0 . The first inequality above implies that the curvature has the restriction |c ′′ | ≤ ch 0 .
Combining the above, the claim is proved.