Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 63, pp. 1–17. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.63 CURVED-PIPE FLOW WITH BOUNDARY CONDITIONS INVOLVING BERNOULLI PRESSURE TVRTKO DOREŠIĆ, IGOR PAŽANIN Abstract. In this article, we study the steady-state flow of the incompressible viscous fluid through a thin distorted pipe with an arbitrary central curve. We prescribe the inflow and outflow boundary conditions involving the Bernoulli pressure with a given pressure drop. Using the multiscale expansion technique with respect to the pipe’s thickness, we construct the higher-order asymptotic approximation of the flow given by the explicit formulae for the velocity and pressure. We also perform a detailed error analysis justifying the usage of the proposed solution and indicating its order of accuracy. 1. Introduction Curved-pipe flows have been studied analytically for many years due to its obvi- ous practical importance. The pioneering work is due to Dean [5] back in 1927 who first used the perturbation techniques to investigate the fluid flow through a curved pipe with circular cross-section. Having in mind that thin (or long) pipes naturally appear in numerous applications, it is no surprise that in the last two decades sev- eral results have been reported that propose new asymptotic models describing the effective flows through thin pipe-like domains. The models for steady-state flow of a Newtonian fluid through 3D curved pipes have been proposed and justified in [15, 17], whereas a system of thin pipes has been studied in [13, 16]. The non- Newtonian, micropolar fluid has been investigated in [6] and [7] in 2D domains such as a periodically constricted tubes and curvilinear channels. The 3D cases includ- ing a curved pipe and a multiple pipe system have been studied in [1, 20]. The corresponding rigorous results for non-steady flows have been reported in [18, 19] for a Newtonian fluid flowing through a system of thin pipes, while a curved-pipe flow has been addressed in [3]. The analysis of the non-steady micropolar fluid flow can be found in [2, 21]. The flow of the incompressible viscous Newtonian fluid is described by the nonlin- ear Navier-Stokes system and the concept of weak solutions is naturally introduced. When the inflows and outflows are described by the given velocity, the existence of the weak solution can be proved using the standard Galerkin method (see e.g. 2020 Mathematics Subject Classification. 35C20, 35Q35, 76M45. Key words and phrases. Newtonian fluid; Bernoulli pressure boundary condition; curved pipe; asymptotic analysis. ©2024. This work is licensed under a CC BY 4.0 license. Submitted May 15, 2024. Published October 22, 2024. 1 2 T. DOREŠIĆ, I. PAŽANIN EJDE-2024/63 Temam [22]). From the perspective of applications, prescribing boundary condi- tions involving pressure is evidently more plausible. However, in such a setting, it is much more difficult to establish the solvability of the corresponding variational fomulation. One usually needs to restrict to the case of small boundary data in order to control the inertial term in Navier-Stokes equations (see e.g. [10, 16]). The another approach would be to prescribe the dynamic boundary condition, namely the Bernoulli pressure Ψ = p+ 1 2 |u| 2, as in [4]. Quite recently, in [12], the authors considered the steady-state flow of a viscous fluid through a finite unde- formed pipe with inflow and outflow boundary conditions involving the Bernoulli pressure. Using the method of Leray [14], they succeeded to prove the existence of the weak solution for arbitrary data and its uniqueness for small data. Relying on this result, the asymptotic approximation of the flow through a network of thin straight pipes has been rigorously derived in [11]. Inspired by these works, in this article we investigate the flow through a thin curved pipe subjected to inflow and outflow boundary conditions involving the Bernoulli pressure with a given pressure drop. Since the Bernoulli pressure represents an important quantity for moderately high Reynolds number, we seek for the inertial effects on the effective flow along with the effects of pipe’s distortion. This article is organized as follows. In Section 2, we formally describe the geom- etry of the curved pipe with an arbitrary central curve and a circular cross-section. To do so, we choose the so-called Germano’s frame of reference (introduced in [8, 9]) in which the domain’s cross-section posseses no rotation with respect to the tangent vector. The flow is assumed to be governed by the Navier-Stokes equations endowed with the no-slip boundary condition for the velocity prescribed on the pipe’s lateral boundary. In Section 3, using the approach from [17, 20], we write the differential operators in curvilinear coordinates transforming the governing problem to an un- deformed pipe. Motivated by the applications, we assume that the ratio between pipe’s thickness and its length is small (and denoted by ε) meaning that we are con- sidering the fluid flow in a pipe which is either thin or long Thus, in Section 4, we employ the two-scale asymptotic expansion in powers of ε and construct the asymp- totic approximation for the velocity and pressure up to a second–order. By doing that, we are able to capture not only the effects of the pipe’s geometry, but also the effects of the additional inertial term appearing due to the Bernoulli pressure inflow-outflow boundary conditions. It should be emphasized that the asymptotic approximation is provided in the explicit form and, therefore, it can be used as an useful check for numerical simulations. Lastly, to justify the usage of the proposed asymptotic solution and provide its order of accuracy, in Section 5, we prove the error estimates in suitable norms by using functional analysis techniques. 2. Formulation of the problem 2.1. The domain. Let Ωα ε denote our thin domain representing a curved pipe characterized by a smooth central curve γ and a circular cross-section. The curve γ is taken to be a generic curve in R3, parameterized by its arc length x1 ∈ [0, l]. We denote the corresponding natural parametrization as π ∈ C3([0, l];R3), satisfying π′(x1) ̸= 0 for every x1 ∈ [0, l]. To formally describe Ωα ε , we start by defining a straight pipe with a circular cross-section. Introducing the small positive parameter ε ≪ 1 and a unit circle EJDE-2024/63 CURVED-PIPE FLOW WITH BERNOULLI PRESSURE 3 B = B(0, 1) ⊂ R2, a thin straight pipe is given by Ωε = {x = (x1, x2, x3) ∈ R3 : x1 ∈ ⟨0, l⟩, x∗ := (x2, x3) ∈ εB} . We now introduce the appropriate frame of reference attached to the curve γ and pass from Ωε to Ωα ε via the appropriate parametrization. Firstly, at each point π(x1) of the curve γ, we introduce the standard Frenet’s basis t = π′, n = 1 κ t′, b = t× n , where t is the tangent, n the normal and b the binormal. The flexion of the curve γ is given by κ(x1) = |π′′(x1)|, whereas the torsion is denoted by τ(x1) = −|b′(x1)|. The Frenet’s basis (t,n,b) satisfies the system t′ = κn, n′ = −κt+ τb, b′ = −τn . (2.1) Figure 1. Pipe’s cross-section with a frame of reference attached. Let us now introduce the rotated unit vectors (see Figure 1) nα(x1) = cosα(x1)n(x1) + sinα(x1)b(x1) , bα(x1) = − sinα(x1)n(x1) + cosα(x1)b(x1) , with rotation given by α(x1) = − ∫ x1 x0 τ(ξ)dξ + α0, where x0 and α0 are arbitrary constants. Defining the mapping Φα ε : Ωε → R3 by Φα ε (x) = π(x1) + x2 nα(x1) + x3 bα(x1) , 4 T. DOREŠIĆ, I. PAŽANIN EJDE-2024/63 and putting Ωα ε = Φα ε (Ωε) , we obtain our curved pipe with the central curve γ and circular cross section εB. Note that the variable x1 follows the central curve of the pipe, while x∗ = (x2, x3) describes its cross-section. The local injectivity of Φα ε can be easily established assuming ε is sufficiently small (see [20] for details). Finally, by Γα ε = Φα ε (⟨0, l⟩ × ε ∂B) , Σi ε = Φα ε ({i} × εB), i = 0, l , we denote the pipe’s lateral boundary and its ends, respectively. 2.2. Governing system. As explained in the introduction, we study the steady- state flow of the incompressible viscous fluid in a thin curved pipe Ωα ε . The usual no- slip boundary condition for the velocity is imposed on the pipe’s lateral boundary Γα ε , while we prescribe the dynamic boundary condition involving the Bernoulli pressure on the pipe’s ends Σi ε. To close the problem, we take the tangential velocity to be zero on Σi ε, which 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. [15]). Therefore, the governing system reads −ν∆uε + (uε · ∇)uε +∇pε = 1 ε2 f in Ωα ε , divuε = 0 in Ωα ε , uε = 0 on Γα ε , uε × t = 0 on Σi ε, i = 0, l , −ν∂tuε · t+ (pε + 1 2 |uε|2) = ci ε2 on Σi ε, i = 0, l . (2.2) where ν is a positive constant (the viscosity of the fluid), uε is the velocity, pε is the pressure, ∂tg = ∇g · t denotes the tangential derivative of g (with respect to the curve γ), ci are some constants and f ∈ L2(Ωα ε ) is a given function (external force). It should be noted that, using the boundary condition (2.2)4, and the incom- pressibility equation, it follows that −ν∂tuε ·t|Σi ε = 0. Now, employing the identity 1 2 (∇u2 ε) = uε · (∇uε) t = n∑ k=1 uk∇uk , the system (2.2) can be rewritten as −ν∆uε + (uε · ∇)uε − uε · (∇uε) t +∇Ψε = 1 ε2 f in Ωα ε , divuε = 0 in Ωα ε , uε = 0 on Γα ε , uε × t = 0 on Σi ε, i = 0, l, Ψε = pi on Σi ε, i = 0, l . (2.3) In (2.3), a new quantity has been naturally introduced, namely the Bernoulli pres- sure given by Ψε = pε + 1 2 |uε|2 , EJDE-2024/63 CURVED-PIPE FLOW WITH BERNOULLI PRESSURE 5 while pi denotes the constants ci ε2 . Using the idea from [12], it is straightforward to prove that problem (2.3) admits at least one weak solution which is unique under the small data assumption (see [11] for details). The goal of this article is to investigate the asymptotic behavior of the fluid flow described by (2.3) via rigorous asymptotic analysis with respect to the small parameter ε. 3. Curvilinear coordinates To ensure comprehensive understanding, we provide a succinct summary of for- mulating the problem (2.3) in curvilinear coordinates (xi). We follow the procedure in [17, 20] and begin by introducing the necessary geometric tools. We define the covariant basis as the gradient of the function mapping Φα ε , namely, ai(x) := ∂Φα ε ∂xi (x) . Taking into account the Frenet system (2.1), we deduce a1 = (1− κ (eα · x∗)) t , where x = (x1,x∗) , eα = (cosα,− sinα) , e⊥α = (sinα, cosα) . Note that in computing the vector a1, we used the fact that α′ = τ . The contravariant basis is defined as the dual to the covariant basis, i.e. ai ·aj = δij . We thus have a1 = 1 1− κ (eα · x∗) t , a2 = cosαn+ sinαb , a3 = − sinαn+ cosαb . It should be observed that ∇Φα ε = [ a1 a2 a3 ] , (∇Φα ε ) −1 =  ( a1 )t( a2 )t( a3 )t  . Christoffel’s symbols are defined as Γi jk = ai · ∂ak ∂xj , being symmetric in lower indices. For the non-zero ones, we obtain Γ1 11 = − ( κ′(eα · x∗) + κτ(e⊥α · x∗) ) 1− κ(eα · x∗) = O(ε) Γ1 12 = Γ1 21 = − κ cosα 1− κ(eα · x∗) = −κ cosα+O(ε) , Γ1 13 = Γ1 31 = κ sinα 1− κ(eα · x∗) = κ sinα+O(ε) Γ2 11 = κ (1− κ(eα · x∗)) cosα = κ cosα+O(ε) , Γ3 11 = −κ (1− κ(eα · x∗)) sinα = −κ sinα+O(ε) . Moreover, we use the following formulae (see e.g. [20, Appendix] for its proof). (∇s)T ◦Φα ε = (∇Φα ε ) −T (∇S)T , (3.1) (∇v) ◦Φα ε = (∇Φα ε ) −T ([∂V k ∂xl ] k,l − V jΓj ) (∇Φα ε ) −1 , Γi = [Γi jk]j,k , (3.2) 6 T. DOREŠIĆ, I. PAŽANIN EJDE-2024/63 (∆v) ◦Φα ε = (∇Φα ε ) −T ( ∂ ∂xi ([∂V k ∂xl ] k, l − V j Γj ) − ([∂V k ∂xl ] k, l − V j Γj ) Γ̂i − Γ̂T i ([∂V k ∂xl ] k, l − V j Γj )) (∇Φα ε ) −1ai , Γ̂i = [Γj ik]j,k , (3.3)( (rotv) ◦Φα ε ) × c = (∇Φα ε ) −T ( rotV × (∇Φα ε ) −1c ) , c ∈ R3 , (3.4) for the scalar field S = s ◦ Φα ε and the vector field V = v ◦ Φα ε . In the above identities, the summation is taken over the repeated indices, while V i = V · ai denote the contravariant components. Introducing Vε = uε ◦Φα ε = V 1 ε a 1 + V 2 ε a 2 + V 3 ε a 3 , Pε = Ψε ◦Φα ε , (3.5) we now proceed with expressing each differential operator occurring in equations (2.3) in curvilinear coordinates. Let λ := 1− κ(eα · x∗). The second-order term −ν∆uε takes the form (−ν∆uε) ◦Φα ε = −ν [ ∆V 1 ε + κ cosα (∂V 1 ε ∂x2 − ∂V 2 ε ∂x1 ) + κ sinα (∂V 3 ε ∂x1 − ∂V 1 ε ∂x3 ) + ∂ ∂x1 ( V 3 ε κ sinα− V 2 ε κ cosα ) − κ2V 1 ε ] a1 − ν ( 1 λ2 − 1 )[∂2V 1 ε ∂x21 + ∂ ∂x1 ( V 3 ε κ sinα− V 2 ε κ cosα ) − κ cosα (∂V 1 ε ∂x2 + V 2 ε ∂x1 ) + κ sinα (∂V 1 ε ∂x3 − ∂V 3 ε ∂x1 ) − 2κ2V 1 ε ] a1 − ν λ2 [ ∆V 2 ε + ∂ ∂x1 ( V 1 ε κ cosα ) − κ cosα (∂V 2 ε ∂x2 + V 2 ε κ cosα− V 3 ε κ sinα− ∂V 1 ε ∂x1 ) + κ sinα ∂V 2 ε ∂x3 ] a2 + ν ( 1 λ2 − 1 )[∂2V 2 ε ∂x22 + ∂2V 2 ε ∂x23 ] a2 − ν λ2 [ ∆V 3 ε − ∂ ∂x1 ( V 1 ε κ sinα ) − κ cosα ∂V 3 ε ∂x2 + κ sinα (∂V 3 ε ∂x3 + V 2 ε κ cosα− V 3 ε κ sinα− ∂V 1 ε ∂x1 )] a3 + ν ( 1 λ2 − 1 )[∂2V 3 ε ∂x22 + ∂2V 3 ε ∂x23 ] a3 The terms including the pressure and the external force are transformed as follows ∇ψε ◦Φα ε = ∂Pε ∂x1 a1 + ∂Pε ∂x2 a2 + ∂Pε ∂x3 a3 , f ◦Φα ε = f1λa 1 + (f2 cosα+ f3 sinα)a 2 + (f3 cosα− f2 sinα)a 3 . To rewrite the inertial terms (uε · ∇)uε and uε · (∇uε) t, we start by writing the velocity gradient as ∇uε ◦Φα ε EJDE-2024/63 CURVED-PIPE FLOW WITH BERNOULLI PRESSURE 7 =  1 λ2 (∂V 1 ε ∂x1 − V 2 ε κ cosα+ V 3 ε κ sinα ) ∂V 1 ε ∂x2 + V 1 ε κ cosα ∂V 1 ε ∂x3 − V 1 ε κ sinα 1 λ2 (∂V 2 ε ∂x1 + V 1 ε κ cosα ) ∂V 2 ε ∂x2 ∂V 2 ε ∂x3 1 λ2 (∂V 3 ε ∂x1 − V 1 ε κ sinα ) ∂V 3 ε ∂x2 ∂V 3 ε ∂x3  Therefore, the term (uε · ∇)uε becomes[V 1 ε λ3 (∂V 1 ε ∂x1 − V 2 ε κ cosα+ V 3 ε κ sinα ) + ( V 2 ε cosα− V 3 ε sinα ) (∂V 1 ε ∂x2 + V 1 ε κ cosα ) + ( V 2 ε sinα+ V 3 ε cosα ) (∂V 1 ε ∂x3 − V 1 ε κ sinα )] a1 + [V 1 ε λ3 (∂V 2 ε ∂x1 + V 1 ε κ cosα ) + ( V 2 ε cosα− V 3 ε sinα ) ∂V 2 ε ∂x2 + ( V 2 ε sinα+ V 3 ε cosα ) ∂V 2 ε ∂x3 ] a2 + [V 1 ε λ3 (∂V 3 ε ∂x1 − V 1 ε κ sinα ) + ( V 2 ε cosα− V 3 ε sinα ) ∂V 3 ε ∂x2 + ( V 2 ε sinα+ V 3 ε cosα ) ∂V 3 ε ∂x3 ] a3 . Similarly, the term uε · (∇uε) t becomes[V 1 ε λ3 (∂V 1 ε ∂x1 − V 2 ε κ cosα+ V 3 ε κ sinα ) + 1 λ2 ( V 2 ε cosα− V 3 ε sinα ) (∂V 2 ε ∂x1 + V 1 ε κ cosα ) + 1 λ2 ( V 2 ε sinα+ V 3 ε cosα ) (∂V 3 ε ∂x1 − V 1 ε κ sinα )] a1 + [V 1 ε λ (∂V 1 ε ∂x2 + V 1 ε κ cosα ) + ( V 2 ε cosα− V 3 ε sinα ) ∂V 2 ε ∂x2 + ( V 2 ε sinα+ V 3 ε cosα ) ∂V 3 ε ∂x2 ] a2 + [V 1 ε λ (∂V 1 ε ∂x3 − V 1 ε κ sinα ) + ( V 2 ε cosα− V 3 ε sinα ) ∂V 2 v e ∂x3 + ( V 2 ε sinα+ V 3 ε cosα ) ∂V 3 ε ∂x3 ] a3 . Finally, taking the trace in the expression for the velocity gradient gives divv ◦Φα ε = 1 λ2 (∂V 1 ε ∂x1 − V 2 ε κ cosα+ V 3 ε κ sinα ) + ∂V 2 ε ∂x2 + ∂V 3 ε ∂x3 . (3.6) 4. Asymptotic expansion To construct the approximation of the solution, we now expand the unknown velocity components and the pressure in the two-scale asymptotic expansions in powers of ε, namely, V i ε (x) = V i 0 ( x1, x2 ε , x3 ε ) + εV i 1 ( x1, x2 ε , x3 ε ) + ε2V i 2 ( x1, x2 ε , x3 ε ) + . . . Pε(x) = 1 ε2 P0(x1) + 1 ε P1 ( x1, x2 ε , x3 ε ) + P2 ( x1, x2 ε , x3 ε ) + . . . (4.1) The idea is to use the derived expressions of differential operators in curvilinear coordinates (see Section 3), substitute the expansions (4.1) and collect the terms with equal powers of ε. By doing that, we shall obtain the recursive sequence of linear problems that, combining with the boundary conditions, we aim to solve explicitly. 8 T. DOREŠIĆ, I. PAŽANIN EJDE-2024/63 The O(1/ε2)-order term. Let us introduce the dilated variable y∗ = x∗/ε capturing the fast changes of the solution on the pipe’s cross-section. In the sequel, by Ω = ⟨0, l⟩ × B and Γ = ⟨0, l⟩ × ∂B, we denote the corresponding rescaled regions and employ the following notation for the formal partial differential operators ∇y∗ = ( ∂ ∂y2 , ∂ ∂y3 ) , ∆y∗V = ∂2V ∂y22 + ∂2V ∂y23 , divy∗ V = ∂V 2 ∂y2 + ∂V 3 ∂y3 , V = (V 1, V 2, V 3). By collecting terms of order O ( 1 ε2 ) in (2.3)1, we arrive at − ν(∆y∗V 1 0 a 1 +∆y∗V 2 0 a 2 +∆y∗V 3 0 a 3) + ∂P0 ∂x1 a1 + ∂P1 ∂y2 a2 + ∂P1 ∂x3 a3 = f1λa1 + (cosαf2 + sinαf3)a 2 + (− sinαf2 + cosαf3)a 3 in Ω , Taking into account the incompressibility condition (divy∗ V0 = 0 in Ω), the zero boundary condition for the velocity on Γ and the pressure boundary condition P0(i) = pi, (i = 0, l), we deduce that V 1 0 (y∗) = 2 π ( 1− |y∗|2 ) F ∗ 0 , V 2 0 = 0, V 3 0 = 0. P0(x1) = −8ν π F ∗ 0 x1 + ∫ x1 0 f1(ξ) dξ + p0. P1(x1,y∗) = f2(x1) (eα(x1)y∗) + f3(x1) ( e⊥α (x1)y∗ ) , where F ∗ 0 = π 8µℓ ( p0 − pl + ∫ l 0 f1(ξ)dξ ) is a constant. As expected, we obtained the zero-order approximation for the velocity given as the Poiseuille-type solution with no effects of the pipe’s distortion and inertial terms. Thus, we aim to construct the higher-order correctors. The O(1/ε)-order term. Grouping the terms of order O ( 1 ε ) from the momentum equation, we obtain the following equation satisfied by the first velocity component − ν ( ∆y∗V 1 1 + κ cosα ∂V 1 0 ∂y2 − κ sinα ∂V 1 0 ∂y3 + κ (eα · y∗)∆y∗V 1 0 ) + κ (eα · y∗) ∂P0 ∂x1 + ∂P1 ∂x1 = 0 , leading to the problem −ν∆y∗V 1 1 = y2H1(x1) + y3H2(x2) in Ω , V 1 1 = 0 on Γ , where H1(x1) = − cosα (4νκ π F ∗ 0 + κf1 + ∂f2 ∂x1 − τf3 ) − sinα ( τf2 + ∂f3 ∂x1 ) , H2(x1) = sinα (4νκ π F ∗ 0 + κf1 + ∂f2 ∂x1 − τf3 ) − cosα ( τf2 + ∂f3 ∂x1 ) . Following [20], we obtain V 1 1 = 1 8ν ( 1− |y∗|2 ) (y2H1(x1) + y3H2(x1)) . EJDE-2024/63 CURVED-PIPE FLOW WITH BERNOULLI PRESSURE 9 It should be observed that the corrector V 1 1 feels the effects of the pipe’s distortion through the explicit appearance of the flexion κ and torsion τ . For the remaining two velocity components, we obtain − ν ( ∆y∗V 2 1 − κ cosα ∂V 2 0 ∂y2 + κ sinα ∂V 2 0 ∂y3 ) + ∂P2 ∂y2 − V 1 0 ∂V 1 0 ∂y2 + ( V 2 0 sinα+ V 3 0 cosα ) (∂V 2 0 ∂y3 − ∂V 3 0 ∂y2 ) = 0 in Ω , − ν ( ∆y∗V 3 1 − κ cosα ∂V 3 0 ∂y2 + κ sinα ∂V 3 0 ∂y3 ) + ∂P2 ∂y3 − V 1 0 ∂V 1 0 ∂y3 + ( V 2 0 cosα− V 3 0 sinα ) (∂V 3 0 ∂y2 − ∂V 2 0 ∂y3 ) = 0, in Ω . In view of the zero-order approximation obtained above, the problem satisfied by (V 2 1 , V 3 1 ) becomes −ν∆y∗V 2 1 + ∂P2 ∂y2 − V 1 0 ∂V 1 0 ∂y2 = 0 in Ω , −ν∆y∗V 3 1 + ∂P2 ∂y3 − V 1 0 ∂V 1 0 ∂y3 = 0 in Ω , ∂V 2 1 ∂y2 + ∂V 3 1 ∂y3 = 0 in Ω, V 2 1 = V 3 1 = 0 on Γ . implying the solution is given by V 2 1 = V 3 1 = 0, P2(y∗) = 2 π2 (F ∗ 0 ) 2 ( 1− |y∗|2 )2 . The computed first-order corrector still contains no contribution from the inertial term originating from the imposed Bernoulli pressure at the pipe’s ends. Therefore, we need to continue the computation. The O(1)-order term. Grouping the O(1)-terms from the momentum equation yields the following equation for the first component − ν (∂2V 1 0 ∂x21 +∆y∗V 1 2 + κ cosα ∂V 1 1 ∂y2 − κ sinα ∂V 1 1 ∂y3 − κ2V 1 0 ) − νκ (eα · y∗) ( ∆y∗V 1 1 − κ cosα ∂V 1 0 ∂y2 + κ sinα ∂V 1 0 ∂y3 ) − νκ2(eα · y∗) 2∆y∗V 1 0 + κ2(eα · y∗) 2 ∂P0 ∂x1 + κ (eα · y∗) ∂P1 ∂x1 + ∂P2 ∂x1 = 0 in Ω . If we apply the derived expressions for the zero and the first-order approximation and include the no-slip condition, we arrive at ∆y∗V 1 2 = A1y 2 2 +A2y2y3 +A3y 2 3 +A4 in Ω , V 1 2 = 0 on Γ , (4.2) where A1 = 11H1 8ν κ cosα− H2 8ν κ sinα− 2κ2F ∗ 0 π − κ2 cos2 α 4F ∗ 0 π + κ2 ν cos2 αf1 10 T. DOREŠIĆ, I. PAŽANIN EJDE-2024/63 + κ ν [ f ′2 cos 2 α+ f ′3 cosα sinα+ τf2 cosα sinα− τf3 cos 2 α ] , A2 = 5H2 4ν κ cosα− 5H1 4ν κ sinα+ 8κF ∗ 0 π κ sinα cosα− 2κ2 ν cosα sinαf1 + κ ν [ − 2 cosα sinαf ′2 + (cos2 α− sin2 α)f ′3 + (cos2 α− sin2 α)τf2 + 2 cosα sinατf3 ] , A3 = H1 8ν κ cosα− 11H2 8ν κ sinα− 2κ2F ∗ 0 π − κ2 sin2 α 4F ∗ 0 π + κ2 ν sin2 αf1 + κ ν [ f ′2 sin 2 α− f ′3 cosα sinα− τf2 cosα sinα− τf3 sin 2 α ] , A4 = −κ cosαH1 8ν + κ sinα H2 8ν + 2 κ2F ∗ 0 π . As in [21], the system (4.2) can be explicitly solved as V 1 2 (x1,y∗) = ( |y∗|2 − 1 ) [ B1y 2 2 +B2y2y3 +B3y 2 3 +B4 ] , where B1 = 7A1 −A3 96 , B2 = A2 12 , B3 = 7A3 −A1 96 , B4 = A4 4 + A1 +A3 32 . Finally, in the remaining two components of the momentum equation, we deduce the contribution of the inertial terms. Indeed, collecting the O(1) terms in the momentum equation, we obtain − ν ( ∆y∗V 2 2 − κ cosα ∂V 2 1 ∂y2 + κ sinα ∂V 2 1 ∂y3 + ( 2κ ∂V 1 0 ∂x1 + κ′V 1 0 ) cosα+ κτV 1 0 sinα ) + ∂P3 ∂y2 + ( V 1 0 )2 κ cosα− ( V 1 0 )2 κ cosα− κ(eα · y∗)V 1 0 ∂V 1 0 ∂y2 = V 1 1 ∂V 1 0 ∂y2 − V 1 0 ∂V 1 1 ∂y2 = 0 in Ω , − ν ( ∆y∗V 3 2 + κ sinα ∂V 3 1 ∂y3 − κ cosα ∂V 3 1 ∂y2 − ( 2κ ∂V 1 0 ∂x1 + κ′V 1 0 ) sinα+ κτV 1 0 cosα ) + ∂P3 ∂y3 − ( V 1 0 )2 κ sinα+ ( V 1 0 )2 κ sinα− κ(eα · y∗)V 1 0 ∂V 1 0 ∂y3 − V 1 1 ∂V 1 0 ∂y3 − V 1 0 ∂V 1 1 ∂y3 = 0 in Ω , which, since V 2 1 = V 3 1 = 0 and V 1 0 = V 1 0 (y∗), reduces to the problem − ν ( ∆y∗V 2 2 + κ′V 1 0 cosα+ κτV 1 0 sinα ) + ∂P3 ∂y2 − κ(eα · y∗)V 1 0 ∂V 1 0 ∂y2 − V 1 1 ∂V 1 0 ∂y2 − V 1 0 ∂V 1 1 ∂y2 = 0 in Ω , − ν ( ∆y∗V 3 2 − κ′V 1 0 sinα+ κτV 1 0 cosα ) + ∂P3 ∂y3 − κ(eα · y∗)V 1 0 ∂V 1 0 ∂y3 − V 1 1 ∂V 1 0 ∂y3 − V 1 0 ∂V 1 1 ∂y3 = 0 in Ω , ∂V 1 1 ∂x1 + ( κ′ (eα · y∗) + κτ ( e⊥α · y∗ )) V 1 0 + ∂V 2 2 ∂y2 + ∂V 3 3 ∂y3 = 0 in Ω , EJDE-2024/63 CURVED-PIPE FLOW WITH BERNOULLI PRESSURE 11 V 2 2 = V 3 2 = 0 on Γ . To simplify the proof layout, we split the desired solution in two particular solutions V 2 2 := V 2 2 + Ṽ 2 2 , V 3 2 := V 3 2 + Ṽ 3 2 , P3 := P 3 + P̃3 , which satisfy −ν ( ∆y∗V 2 2 + κ′V 1 0 cosα+ κτV 1 0 sinα ) + ∂P 3 ∂y2 = 0 in Ω , −ν ( ∆y∗V 3 2 − κ′V 1 0 sinα+ κτV 1 0 cosα ) + ∂P 3 ∂y3 = 0 in Ω , ∂V 1 1 ∂x1 + ( κ′ (eα · y∗) + κτ ( e⊥α · y∗ )) V 1 0 + ∂V 2 2 ∂y2 + ∂V 3 3 ∂y3 = 0 in Ω , V 2 2 = V 3 2 = 0 on Γ , (4.3) and −ν∆y∗ Ṽ 2 2 + ∂P3 ∂y2 − κ(eα · y∗)V 1 0 ∂V 1 0 ∂y2 − V 1 1 ∂V 1 0 ∂y2 − V 1 0 ∂V 1 1 ∂y2 = 0 in Ω , −ν∆y∗ Ṽ 3 2 + ∂P3 ∂y3 − κ(eα · y∗)V 1 0 ∂V 1 0 ∂y3 − V 1 1 ∂V 1 0 ∂y3 − V 1 0 ∂V 1 1 ∂y3 = 0 in Ω , ∂Ṽ 2 2 ∂y2 + ∂Ṽ 3 3 ∂y3 = 0 in Ω , Ṽ 2 2 = Ṽ 3 2 = 0 on Γ . (4.4) When comparing with the second order corrector from [21], the difference is ex- actly in the particular solutions Ṽ 2 2 , Ṽ 3 2 and P̃3 which satisfy (4.4). Moreover, by expanding (4.4), we obtain ∆y∗ Ṽ 2 2 − 1 ν ∂P̃3 ∂y2 = 8(F ∗ 0 ) 2κ ν2π2 (1− |y∗|2)(y2 cosα− y3 sinα)y2 − ∂ ∂y2 ( F ∗ 0 4ν2π ( 1− |y∗|2 )2 (y2H1(x1) + y3H2(x1)) ) , ∆y∗ Ṽ 3 2 − 1 ν ∂P̃3 ∂y3 = 8(F ∗ 0 ) 2κ ν2π2 (1− |y∗|2)(y2 cosα− y3 sinα)y3 − ∂ ∂y3 ( F ∗ 0 4ν2π ( 1− |y∗|2 )2 (y2H1(x1) + y3H2(x1)) ) , ∂Ṽ 2 2 ∂y2 + ∂Ṽ 3 3 ∂y3 = 0 in Ω , Ṽ 2 2 = Ṽ 3 2 = 0 on Γ . (4.5) In view of the above, we seek for the solution of the form V 2 2(x1,y∗) = (1− |y∗|2) ( B5y 2 2 +B6y2y3 +B7y 2 3 +B8 ) , V 3 2(x1,y∗) = (1− |y∗|2) ( B9y 2 2 +B10y2y3 +B11y 2 3 +B12 ) , P 3(x1,y∗) =M1y 3 2 +M2y 2 3 +M3y2y 2 3 +M4y 2 2y3 +M5y2 +M6y3 , 12 T. DOREŠIĆ, I. PAŽANIN EJDE-2024/63 Ṽ 2 2 (x1,y∗) = (1− |y∗|2) ( C1y 4 2 + C2y 3 2y3 + C3y 2 2y 2 3 + C4y2y 3 3 + C5y 4 3 + C6y 2 2 + C7y2y3 + C8y 2 3 + C9 ) , Ṽ 3 2 (x1,y∗) = (1− |y∗|2) ( D1y 4 3 +D2y 3 3y2 +D3y 2 3y 2 2 +D4y3y 3 2 +D5y 4 2 +D6y 2 3 +D7y3y2 +D8y 2 2 +D9 ) , P̃3(x1,y∗) = F ∗ 0 4ν2π ( 1− |y∗|2 )2 (y2H1(x1) + y3H2(x1)) + L1y 5 2 + L2y 4 2y3 + L3y 3 2y 2 3 + L4y 2 2y 3 3 + L5y2y 4 3 + L6y 5 3 + L7y 3 2 + L8y 2 2y3 + L9y2y 2 3 + L10y 3 3 such that V 2 2, V 3 2 and P 3 satisfy (4.3), and Ṽ 2 2 , Ṽ 3 2 , P̃3 satisfy (4.5). Denoting a := 2ν π (κ′ cosα+ κτ sinα)F ∗ 0 , b := 2ν π (κτ cosα− κ′ sinα)F ∗ 0 , c := 8(F ∗ 0 ) 2κ ν2π2 cosα , d := −8(F ∗ 0 ) 2κ ν2π2 sinα , after tedious but straightforward calculation, we obtain the sought coefficients: B5 = −a 96 , B6 = b 24 , B7 = −5a 96 , B8 = a 96 , B9 = −5b 96 , B10 = a 24 , B11 = −b 96 , B12 = b 96 , M1 = ν −a 4 , M2 = ν −b 4 , M3 = ν −a 4 , M4 = ν −b 4 , M5 = ν 5a 6 , M6 = ν 5b 6 , C1 = −c 1152 , C2 = d 192 , C3 = −c 144 , C4 = d 192 , C5 = −7c 1152 , C6 = 5c 1152 , C7 = −d 64 , C8 = 23c 1152 , C9 = 7c 1152 , D1 = −d 1152 , D2 = c 192 , D3 = −d 144 , D4 = c 192 , D5 = −7d 1152 , D6 = 5d 1152 , D7 = −c 64 , D8 = 23d 1152 , D9 = 7d 1152 , L1 = ν 5c 24 , L2 = ν 5d 24 , L3ν = ν 5c 12 , L4 = ν 5d 12 , L5 = ν 5c 24 , L6 = ν 5d 24 , L7 = ν −3c 8 , L8 = ν −3d 8 , L9 = ν −3c 8 , L10 = ν −3d 8 To conclude, the second-order velocity corrector takes the form V2 = ( V 1 2 , V 2 2 , V 3 2 ) = ( V 1 2 , V 2 2 + Ṽ 2 2 , V 3 2 + Ṽ 3 2 ) . 5. Error analysis 5.1. Asymptotic approximation. Summarizing the calculation performed in Sec- tion 4, we write the asymptotic approximation of the solution to the problem (2.3) as νε(z) := Vapprox ε (x), Vapprox ε (x) = V 1 0 (x∗ ε ) a1 + εV 1 1 ( x1, x∗ ε ) a1 + ε2V2 ( x1, x∗ ε ) , (5.1) EJDE-2024/63 CURVED-PIPE FLOW WITH BERNOULLI PRESSURE 13 qε(z) := P approx ε (x) , P approx ε (x) = 1 ε2 P0(x1) + 1 ε P1 ( x1, x∗ ε ) + P2 ( x1, x∗ ε ) , (5.2) for z := Φα ε (x). In the following, we aim to evaluate the difference between the original solution (which cannot be found) and the approximate one (given by (5.1)– (5.2)) in a suitable functional norm. 5.2. A priori estimates. Before proving the a priori estimate for the velocity, we first recall some technical results which can be proved in a standard manner by taking into account the dependence of the domain on the small parameter ε (see [15], [16] for details). Throughout Section 5, C > 0 denotes a generic constant independent of ε. Lemma 5.1. Poincare’s inequality holds, ∥φ∥L2(Ωα ε ) ≤ Cε∥∇φ∥L2(Ωα ε ) , for all φ ∈ H1(Ωα ε ) such that φ = 0 on Γα ε . Lemma 5.2. Let K ∈ L2 0(Ω α ε ). Then the problem divφε = K in Ωα ε , φε = 0 on ∂Ωα ε admits a solution satisfying ∥∇φε∥L2(Ωα ε ) ≤ C ε ∥K∥L2(Ωα ε ) . We now prove the a priori estimate for the velocity which we need in the error analysis. Proposition 5.3. Let (uε,Ψε) be the solution of (2.3). Then there exists a con- stant C such that ∥∇uε∥L2(Ωα ε ) ≤ C. (5.3) Proof. Employing uε as a test function in the momentum equation gives ν ∫ Ωα ε |∇uε|2 = p0 ∫ Σ0 ε uε · t(0)− pl ∫ Σl ε uε · t(l)− ∫ Ωα ε (uε · ∇)uε · uε + ∫ Ωα ε uε · (∇uε) t · uε + 1 ε2 ∫ Ωα ε fuε Using Lemma 5.1 and the fact that |Ωα ε | = O(ε2), we have∣∣∣p0 ∫ Σ0 ε uε · t(0)− pl ∫ Σl ε uε · t(l) ∣∣∣ = ∣∣∣ ∫ Ωα ε div (( p0 + pl − p0 l x1 ) uε )∣∣∣ ≤ Cε2∥∇uε∥L2(Ωα ε ) . Similarly, 1 ε2 ∣∣ ∫ Ωα ε fuε ∣∣ ≤ 1 ε2 ∥f∥L2(Ωα ε )∥uε∥L2(Ωα ε ) ≤ C∥∇uε∥L2(Ωα ε ) . 14 T. DOREŠIĆ, I. PAŽANIN EJDE-2024/63 For the remaining terms, by straight forward calculation, we deduce (uε · ∇)uε · uε − uε · (∇uε) t · uε = ( (uε · ∇)uε − uε · (∇uε) t ) · uε = ([ 3∑ i=1 ui∂iu1 3∑ i=1 ui∂iu2 3∑ i=1 ui∂iu3 ] − [ 3∑ i=1 ui∂1ui 3∑ i=1 ui∂2ui 3∑ i=1 ui ∂3ui ]) · uε = 0 . implying ∫ Ωα ε (uε · ∇)uε · uε − ∫ Ωα ε uε · (∇uε) t · uε = 0 . Collecting the above, we deduce (5.3). □ 5.3. Error estimates. We are now in the position to formulate and prove the main result of this section providing the order of accuracy of the proposed asymptotic solution. Since we work in a thin-domain setting, we express the error estimates in the rescaled norm |Ωα ε |−1/2∥ · ∥L2(Ωα ε ). The goal is to derive the satisfactory error estimates acknowledging the contributions of the pipe’s distortion and the Bernoulli pressure on the effective flow. Theorem 5.4. The following estimates hold: |Ωα ε |−1/2∥uε − νε∥L2(Ωα ε ) ≤ Cε3 , (5.4) |Ωα ε |−1/2∥ψε − qε∥L2(Ωα ε )/R ≤ Cε . (5.5) where νε and qε are given by (5.1)–(5.2).. Proof. The pair (νε, qε) satisfies −ν∆νε + (νε · ∇)νε − νε · (∇νε) t +∇qε = 1 ε2 f +Eε in Ωα ε , div νε = πε in Ωα ε , νε = 0 on Γα ε , where the remainder ∥Eε∥L∞(Ωα ε ) ≤ Cε and thus, since |Ωα ε | = O(ε2), we obtain ∥Eε∥L2(Ωα ε ) = O(ε2). Analogously, we have ∥πε∥L2(Ωα ε ) = O(ε3). Denoting the differences Rε = uε − νε, rε = pε − qε , we obtain − ν∆Rε +∇rε + (Rε · ∇)uε + (νε · ∇)Rε − ( Rε(∇uε) t + νε (∇Rε) t ) = −Eε in Ωα ε , divRε = −πε in Ωα ε , Rε = 0 on Γα ε . (5.6) Let us now introduce the test function dε as the solution of the auxiliary problem divdε = rε in Ωα ε , dε = 0 in ∂Ωα ε . EJDE-2024/63 CURVED-PIPE FLOW WITH BERNOULLI PRESSURE 15 Since the pressure is determined up to an additive constant, we can suppose ∫ Ωα ε rε = 0, so the problem admits the solution satisfying ∥∇dε∥L2(Ωα ε ) ≤ C ε ∥rε∥L2(Ωα ε ) , (5.7) because of Lemma 5.2. Using dε as the test-function in (5.6) yields ∥rε∥2L2(Ωα ε ) = ν ∫ Ωα ε ∇Rε∇dε + ∫ Ωα ε Eε dε − ∫ Ωα ε [ (Rε · ∇)uε + (νε · ∇)Rε − ( Rε(∇uε) t + νε (∇Rε) t )] dε . The first two integrals are estimated using (5.7) and the fact that ∥Eε∥L2(Ωα ε ) = O(ε2) and ∥πε∥L2(Ωα ε ) = O(ε3):∣∣ ∫ Ωα ε ∇Rε∇dε ∣∣ ≤ ∥∇Rε∥L2(Ωα ε )∥∇dε∥L2(Ωα ε ) ≤ C ε ∥∇Rε∥L2(Ωα ε )∥rε∥L2(Ωα ε ) ,∣∣ ∫ Ωα ε Eε dε ∣∣ ≤ ∥Eε∥L2(Ωα ε )∥dε∥L2(Ωα ε ) ≤ Cε2∥rε∥L2(Ωα ε ) . To estimate the remaining terms, we will need the special case of the Gagliardo- Nirenberg interpolation inequality ∥f∥L4 ≤ C∥f∥1/4L2 ∥∇f∥3/4L2 . (5.8) From this, (5.3), and the Poincare’s inequality, we obtain∣∣ ∫ Ωα ε (Rε · ∇)uεdε ∣∣ ≤ C∥∇uε∥L2(Ωα ε )∥Rε∥L4(Ωα ε )∥dε∥L4(Ωα ε ) ≤ Cε1/2∥∇Rε∥L2(Ωα ε )∥∇dε∥L2(Ωα ε ) ≤ C ε1/2 ∥∇Rε∥L2(Ωα ε )∥rε∥L2(Ωα ε ) ,∣∣ ∫ Ωα ε (νε · ∇)Rε dε ∣∣ ≤ ∥∇Rε∥L2(Ωα ε )∥νε∥L4∥dε∥L4(Ωα ε ) ≤ Cε1/2∥∇Rε∥L2(Ωα ε )∥∇dε∥L2(Ωα ε ) ≤ C ε1/2 ∥∇Rε∥L2(Ωα ε )∥rε∥L2(Ωα ε ) . In the same way, for the last two terms, we have∣∣ ∫ Ωα ε ( Rε(∇uε) t + νε (∇Rε) t ) dε ∣∣ ≤ C ε1/2 ∥∇Rε∥L2(Ωα ε )∥rε∥L2(Ωα ε ) . Collecting these estimates, we obtain ∥rε∥L2(Ωα ε ) ≤ C ε ∥∇Rε∥L2(Ωα ε ) . (5.9) Now we go back to the momentum equation (5.6) and use Rε as a test function. As a result, we obtain ν∥∇Rε∥2L2(Ωα ε ) = ∫ Ωα ε rε divRε − ∫ Ωα ε EεRε + ∫ Ωα ε [( Rε · ∇ ) uε + ( νε · ∇ ) Rε − ( Rε(∇uε) t + νε(∇Rε) t )] Rε . 16 T. DOREŠIĆ, I. PAŽANIN EJDE-2024/63 Using Lemma 5.1, the a priori estimate (5.3), (5.8) and (5.9), we obtain∣∣ ∫ Ωα ε rε divRε ∣∣ = ∣∣ ∫ Ωα ε rεπε ∣∣ ≤ ∥rε∥L2(Ωα ε )∥πε∥L2(Ωα ε ) ≤ Cε2∥∇Rε∥L2(Ωα ε )∣∣ ∫ Ωα ε EεRε ∣∣ ≤ ∥Eε∥L2(Ωα ε )∥Rε∥L2(Ωα ε ) ≤ Cε3∥∇Rε∥L2(Ωα ε ) ,∣∣ ∫ Ωα ε (Rε · ∇)uε Rε ∣∣ ≤ C ∥∇uε∥L2(Ωα ε )∥Rε∥2L4(Ωα ε ) ≤ Cε1/2∥∇Rε∥2L2(Ωα ε ) ,∣∣ ∫ Ωα ε (νε · ∇)RεRε ∣∣ ≤ ∥∇Rε∥L2(Ωα ε )∥νε∥L4∥Rε∥L4(Ωα ε ) ≤ Cε1/2∥∇Rε∥2L2(Ωα ε ) ,∣∣ ∫ Ωα ε ( Rε(∇uε) t + νε (∇Rε) t ) Rε ∣∣ ≤ Cε1/2∥∇Rε∥2L2(Ωα ε ) . Putting the above estimates together yields ν∥∇Rε∥2L2(Ωα ε ) ≤ Cε1/2∥∇Rε∥2L2(Ωα ε ) + Cε2∥∇Rε∥L2(Ωα ε ) . Using Young’s inequality we can estimate the second term on the right-hand side as ε2∥∇Rε∥L2(Ωα ε ) ≤ ε3 2 + ε 2 ∥∇Rε∥2L2(Ωα ε ) leading to ∥∇Rε∥L2(Ωα ε ) ≤ C ε3 . This proves (5.4), whereas the estimate for the pressure (5.5) then follows directly from (5.9). □ Acknowledgements. This research was supported by the Croatian Science Foun- dation under the projects MultiFM IP-2019-04-1140 and AsyAn IP-2022-10-1091. References [1] M. Beneš, I. Pažanin; Effective flow of incompressible micropolar fluid through a system of thin pipes, Acta Appl. Math., 143(1) (2016), 29–43. [2] M. Beneš, I. Pažanin, M. Radulović; Rigorous derivation of the asymptotic model describing a nonsteady micropolar fluid flow through a thin pipe, Comput. Math. Appl., 76 (2018), 2035-2060. [3] G. Castineira, E. Marušić-Paloka, I. Pažanin, J. M. 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Radulović; Asymptotic analysis of the nonsteady micropolar fluid flow through a curved pipe, Appl. Anal., 99 (2020), 2045–2092. [22] R. Temam; Navier-Stokes equations, Vol II, North-Holland, New York, 1977. Tvrtko Dorešić Department of Mathematics, Faculty of Science, University of Zagreb, Croatia Email address: tvrtko.doresic@math.hr Igor Pažanin Department of Mathematics, Faculty of Science, University of Zagreb, Croatia Email address: pazanin@math.hr 1. Introduction 2. Formulation of the problem 2.1. The domain 2.2. Governing system 3. Curvilinear coordinates 4. Asymptotic expansion 5. Error analysis 5.1. Asymptotic approximation 5.2. A priori estimates 5.3. Error estimates Acknowledgements References