Pressure-driven flow in a thin pipe with rough boundary

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Introduction
Laminar fluid flow through pipes appears in various applications (blood circulation, heating/cooling processes, etc.). Experimental studies of the pipe flow go back to 1840s when Poiseuille [32,33] established the relationship between the volumetric efflux rate of fluid from the tube, the driving pressure differential, the tube length and the tube diameter. The distinction between laminar, transitional and turbulent regimes for fluid flows was done by Stokes [37] and later was popularized by Reynolds [34]. In 1886, he also obtained limit equations similar to Poiseuille's law but for flows in thin films [35]. In further experiments done by Nikuradse in 1933 [27], it was shown that ". . . for small Reynolds numbers there is no influence of wall roughness on the flow resistance" and since then for many years, the roughness phenomenon has been traditionally taken into account only in case of turbulent flow [2,12,38,40]. However, in 2000s his experiments were reassessed [17,18] and the importance of considering roughness effects for laminar flows was emphasized [14]. By means of classical analysis, different geometries were analyzed, e.g., detailed velocity and pressure profiles for flows with small Reynolds' numbers in sinusoidal capillaries were obtained numerically in [16]; for creeping flow in pipes of varying radius [36], pressure drop was estimated by using a stream function method; in [39], the Stokes flow through a tube with a bumpy wall was solved through a perturbation in the small amplitude of the three-dimensional bumps.
There are several mathematical approaches to analyze thin pipe flow, e.g., asymptotic expansions with variations (see [20] for the case of helical pipes and [28] for an extended overview of such methods) and two-scale convergence [1,19,26] adapted for thin structures in [22] (see also [23,25,29]). The same techniques appear in analysis of thin film flows ( [3,4,11,24]), where involving surface roughness effects are often connected to problems in sliding or rolling contacts [7,31]. For the case of flow in curved pipes, we refer the reader to [13,21,30].
The present paper studies Stokes flow in a μ-periodic rough pipe Ω εμ of thickness ε with x 3 -length L and an arbitrary transversal geometry, i.e., the representative volume of Ω εμ is a cylinder Q εμ of length μ and arbitrary cross section with the area of order O(ε 2 ).

Geometry
For each z ∈ [0, 1], we denote by Q(z)(⊂ R 2 ) a bounded domain such that a family {Q(z)} z∈(0,1) forms a smooth (actually Lipschitz) pipe Q ⊂ R 3 with the cylindrical (longitudinal) axis z: Fig. 1b) In order for the fluid to pass through the pipe, we assume that there exists α > 0 such that the distance d(∂Q(z), (0, 0, z)) > α for all z ∈ [0, 1]. Let also where int(X) denotes the interior of a set X, and Q min is the largest straight pipe of cross section Q min that is contained in Q: In addition, we impose the periodicity condition Q(0) = Q(1) that allows us to extend Q smoothly along its longitudinal z-axis to infinitely long pipe Q as follows: For small parameters μ, ε 1, we define a smooth rough pipe Ω εμ of the length L ( Fig. 1a):

Problem statement
We consider the Stokes equations for incompressible Newtonian fluid: where 2e(∇U εμ ) = ∇U εμ + (∇U εμ ) t , η is fluid's viscosity and f and p b are external volume force and boundary pressure correspondingly. We assume that the force f = (0, 0, f) acts only in longitudinal x 3direction and both f and p b depend only on variable x 3 -such limitation excludes circular or re-entrant flow in the vicinity of Γ εμ N .

Main result
Under the assumption of small ε, μ, the solution (U εμ , P εμ ) of (1) can be approximated in terms of asymptotic expansions by an effective flow (u λ , p λ ) satisfying the one-dimensional Poiseuille's law: The scalar parameter k λ ≥ 0 arising in (2) takes the micro-geometry of the pipe into account and is built on the solutions of cell problems that depend on mutual ε, μ-ratio.
• PRTP (Proportionally rough thin pipe) In proportional regime, 0 < λ < ∞, the expression for k λ has the form where W 3 is the third velocity component of the solution (W, q) for the Stokes problem in a 3D unit cell Q governed by the unit force e 3 = (0, 0, 1) • HRTP (Homogeneously rough thin pipe) For the very thin regime, λ = 0, the coefficient k 0 can be expressed as follows: where and φ is the solution of the 2D Laplace equation for z-parameterized, z ∈ (0, 1), domain Q(z): ZAMP Flow in thin pipes Page 5 of 20 138 • ROTP (Rapidly oscillating thin pipe) In the very rough regime, λ = ∞, the flow takes place only in the non-rough part of the pipe. In other words, the microstructure of the roughness can be ignored. This regime is characterized by the factor k ∞ that is built on the solution of 2D Laplace equation for a fixed domain Q min :  (2) and obtain an explicit expression for the approximated pressure: However, the analysis provided below can be applied for more general geometries which allow nontrivial dependence k λ = k λ (x 3 ), x 3 ∈ (0, L). The geometrical notation and corresponding results are presented in "Appendix I."

Remark 3.2.
As one can see, by starting from Neumann condition (1c) for the original Stokes problem we come up with Dirichlet pressure condition in the limit equation (2). Such switching in boundary conditions (and in opposite order) is observed also in context of flows in porous media [1,10] and thin domains [9].

Remark 3.3.
Instead of (1), one can also consider the full Navier-Stokes system. As it is expected for laminar flows, the inertial term does not affect the analysis and the approximation results above would still be valid.

Rescaling Ω εμ → Ω μ
First of all, in order to get rid of ε from geometry of the pipe, we change variables ( where

Proportional case ε = λμ
Assume that and let z = x 3 /μ. Since the geometry of the pipe becomes to be 1-periodic in z, we assume the same behavior for functions By substituting (10) into (9) and collecting terms with the same powers of λ, we get the following: (2) Mass equation Equation (9b) gives (3) Neumann condition Equation (9c) gives (4) Dirichlet condition Equation (9d) gives

Analysis of equations.
• Macro-pressure p 0 Equation (11) provides By adding the boundary condition (15), we get • Two-pressure problem Higher-order terms (see (12), (13) and (16)) constitute the system for Due to the linearity of Problem (19), one can write where (W, q) is the 1-periodical in z-direction solution of the cell problem [compare with (4)]: Here, e 3 = (0, 0, 1)-the unit vector in z-direction.
together with expression (3) for k λ -factor is obtained by integrating (14) over Q and using equations (16), (20) together with z-periodicity assumption for u 2 .

Homogeneously rough thin pipe
The case ε μ corresponds to considering the limit λ = ε/μ → 0. Thus, we start with the cell problem (4) and look for the its solution (W, q) in the form Substituting the expansions into (4) gives the following series of equations with different powers of parameter λ.

Momentum equation.
The equations above imply The next order terms satisfy

Rapidly oscillating thin pipe
The fact that in ROTP limit the flow occurs only in no-rough region is expected from a physical point of view. However, its mathematical justification is not trivial and can be of a particular interest. The case ε μ corresponds to λ = μ/ε → 0. So, we replace λ → 1/λ in (4): and look for (W, q) in the form of (21) with respect to λ = μ/ε. Substituting (21) into (31) gives
(ii) The next order terms satisfy (34) and (37). Equation (37) implies By multiplying (34) by ϕ(y, z) = (ϕ 1 , ϕ 2 , 0), such that ϕ = 0 on γ D , 1-periodical with respect to z, and integrating over Q we get The expression above is valid in all Lipschitz domains (such that the divergence theorem holds). Now, let us consider ϕ 1,2 = W 0 1,2 and take into account (38). It gives us the following: that together with (34) additionally gives Moreover, if we consider y / ∈ Q min and z-line passing through the point y, we obtain that the constant (in z) velocity W 0 has to be zero at points of intersection with the boundary γ D that means Since W 0 is a function of y only, (38) provides that ∂W 1 3 /∂z depends also only on y. By periodicity assumption, we get that ∂W 1 3 /∂z must be zero then and thus (iii) The next couple to consider is (35), (39). This system can be analyzed in the same way as at step (ii). Below, we provide the summary of results obtained: (iv) Finally, we come to equation (36) that involves the force term e 3 . By integrating it over Q with a test function ϕ such that ϕ = 0 on γ D and 1-periodic in z, we get (as before the expression above is valid in all Lipschitz domains, i.e., such that divergence theorem holds) By choosing ϕ = ϕ(y) such that ϕ = 0 outside of Q min , we get That is a weak formulation of Remark 4.1. The same results (5), (6) ((7), (8)) for the limit case HRTP (ROTP) can also be obtained by modeling ε, μ-relation in (9) as ε = μ 2 (μ = ε 2 correspondingly) and considering (u εμ , p εμ ) as series with respect to μ (ε) only.

Model example
To illustrate the results above numerically, we consider sinusoidal pipes Ω εμ with representative volume Q: where b > a > 0, a + b = 1, and circular cross section Q(z) of variable radius R min < R < R max . The maximal value of R is preserved to be 1 (R max = 1), whereas R min takes values R min = 0.4, 0.5, 0.6. (Fig. 3 and Table 1). For each value of R min , all three cases (ε/μ = 0, ε/μ ∈ (0, ∞), μ/ε = 0) are considered.  • PRTP case is calculated by means of FEM software COMSOL Multiphysics, and the values for the friction factor k λ are presented in Table 2. • HRTP In this case, the coefficient k 0 is built on the solution of (6) that for circular Q(z) (see (43)) has the form: Thus, has the following values (see Table 3).

• ROTP The flow occurs only in the region
The velocity W 3 is the solution of (8): and according to (7), Table 3).
The comparison between intermediate PRTP and limit cases HRTP and ROTP is presented in Figures 4 and 5, and x-axis is chosen as 1/λ = μ/ε due to better resolution of the graph. As one can see from Fig. 4, the curves k λ = k λ (1/λ) for all values of R min have log shapes and fast rate of convergence k λ → k 0 . According to Fig. 5, k λ allows the linear approximation for 1/λ ∈ (0, 1) and the corresponding convergence k λ → k ∞ has a linear rate in this region.

Justification of the result
The approximation results stated in Sect. 3 are obtained by using asymptotic series with respect to small parameters ε, μ. This analysis is a formal method and needs to be combined with other techniques in order to make conclusions mathematically rigorous.
Both, a priori and error estimates are based on two main methods: that establishes the relation between L 2 -norms of a function and the symmetric part of its gradient; • the Bogovskiǐ operator B : ∇ · v → [v] that recovers (as an equivalence class) an original function v ∈ L 2 (Ω εμ ), v = 0 on Γ εμ D , from its L 2 -divergence ∇ · v and provides the estimate Both constants K, B > 0 are independent on ε, μ. Moreover, the estimate (45) in case of functions v = (v 1 , v 2 , 0) can be improved to For more details and proofs of (44), (45), we refer the reader to Theorems 4.1 and 4.3 in [9] where general thin domains are considered. The case of the rough pipe Ω εμ can be reduced to one in [9] by extending arguments. More precise, due to no-slip boundary condition (1d) one can always prolong U εμ by zero to any straight pipe Ω ε = εY × (0, L) containing Ω εμ . Here, Y ⊂ R 2 is a sufficiently big circle (or square) such that Y ⊃ Q(x 3 ) for any x 3 ∈ (0, L). The comments on Inequality (45 ) can be found in Appendix II (see p. 22).

A priori estimates
There exists a constant C > 0 independent of parameters ε, μ such that the following estimates 1 are valid for the solution (U εμ , P εμ ) of (1). The estimates above are proved in [9] (see Theorem 4.2,4.5) for any smooth enough thin domain Ω ε . The proof is also valid for the rough pipe Ω εμ due to no-slip condition on the rough surface. As one can see, these estimates for U εμ , P εμ do not depend on μ and give corresponding orders of convergence for Problem (1) to System (2) with respect to the pipe thickness ε.

Errors problem statement
After rescaling in (19) to the original variables (y, z, x 3 ) → (x 1 /ε, x 2 /ε, x 3 /μ, x 3 ) and subtracting it from (1), one gets the following system for the error functions Here, Note that all derivatives in expressions for f ε , h ε , g ε are taken with respect to unscaled x 3 only and do not affect cell functions W , q constituting u 2 and p 1 (see (20)). For any test function v such that ∇ · v = 0 in Ω εμ and v = 0 on Γ εμ D , the divergence theorem provides the following weak formulation of (46) with j ε = −∇g ε + f ε − η∇h ε as a source term. Since the functions u 2 , p 1 on the right-hand side of (47) are independent of ε and μ, the next norm estimate holds: ZAMP Flow in thin pipes Page 15 of 20 138