On the Viscous Flow Through a Long Pipe With Non-Constant Pressure Boundary Condition

1 Introduction

The use of pipe networks in the transportation of fluid is essential in industrial and engineering applications. Such networks consist of several elements (pipes, hydraulic pumps, valves, etc.) that are interconnected to transport a fluid from the supply site to the demand locations. A typical example of such structure would be the water distribution network in a city area. The optimal design of a pipe network and its elements is a very challenging task and, thus, has been the case of study and interest of many researchers. We refer the reader to Akpan et al. [1], Costa et al. [2], Larock et al. [3], Raoni et al. [4], Sarbu and Ostafe [5] and the large list of references provided therein.

To understand the behavior of the fluid flow through a water supply network, one should start with a simple pipeline system, i.e. a single pipe transporting water from one reservoir to another. It is well known that the stationary Navier-Stokes system, which describes the viscous flow in pipes with impermeable walls governed by the prescribed pressure drop between the pipe’s ends, has a solution in the form of the Hagen-Poiseuille flow. In case of the pipe with a constant circular cross section, it reads

According to Landau and Lifchitz [6], the average of the above formula was empirically found by G. Hagen (1839) and, independently, by J. L. M. Poiseuille (1840). Its theoretical derivation in the form (1) is due to G. G. Stokes (1845). The engineering approach to the pipe flow is mainly based on the Hagen-Poiseuille formula (1), although it provides an exact solution only in the case of laminar flow through one straight pipe with a constant cross section and with prescribed constant pressures on both ends of the pipe. If the flow is time dependent, or the pipe has a variable cross section or is curved, or we consider a multiple pipe system, the appropriate versions of the Hagen-Poiseuille formula can be derived in the case of long (or thin) pipes, but it represents only a zero-order approximation of the solution. Therefore, these formulae need to be corrected by lower-order terms, leading to more accurate asymptotic approximations. We refer the reader to the papers by Marušić-Paloka and Pažanin [7], [8], [9], [10], Nazarov and Piletskas [[11], [12]], and Panasenko and Piletskas [13], [14], [15].

In the present paper, we intend to study the incompressible viscous flow through a long pipe with a constant circular cross section. By long we mean that the length of the pipe L is much larger than the diameter of the pipe’s cross section. To simplify the notation, we assume that the radius of the pipe’s cross section equals to 1. Such pipes naturally appear in numerous engineering applications such as pipelines, heat exchangers, chemical reactors, etc. Traditionally, the asymptotic analysis of the fluid flow is done with respect to the small parameter representing the ratio between the pipe’s radius and its length. Thus, we denote by ε=1L the small parameter. If (x′, y′, z′) are the physical variables, we rescale them and define x = x′/L, y = y′/L, and z = z′/L. In new variables, our problem is posed in a pipe with length 1 and radius ε. We denote by ( U(x′, y′, z′), P(x′, y′, z′)) the fluid velocity and the pressure, respectively, and by ( u^ε(x, y, z), P^ε(x, y, z)) those same functions in the new variables.

We assume that the fluid flow inside the pipe is governed by a prescribed pressure drop between the pipe’s ends. The choice of the pressure boundary condition is directly motivated by the water supply system, as described above. In those systems, the hydraulic components such as pumps for supplying the pressure are required to overcome static head and losses, whereas valves are employed to control the rate, direction, and pressure of the water flow. In view of that, we prescribe the pressure on both ends of the pipe, but we assume that the pressure on the left end is not constant. To be more precise, we study the pressure boundary condition

and aim to replace it by an effective condition governing the macroscopic flow. Taking into account the previous discussion, it is clear that we cannot hope for an exact solution of the corresponding boundary value problem. Still, it is reasonable to believe that the Hagen-Poiseuille formula provides at least a zero-order approximation. In fact, the expected result would be that the Hagen-Poiseuille approximation is given by taking the mean value of p₀ and prescribing that as the (constant) pressure on the right end of the pipe. However, the analysis in the sequel suggests that the right choice of the pressure to be prescribed on the left end (pipe’s entry) is not equal to the mean value of p₀, see (13). Such result is rather unexpected to us and, to our knowledge, cannot be found in the existing literature. Moreover, in view of the above-described applications, it is natural to raise the question on how to choose the optimal value of p₀, such that the pressure drop (and, thus, the velocity of the fluid) is maximal. It turns out that the pressure drop inside the pipe can be significantly increased by applying non-constant, more localised pressure on the entry of the pipe. We believe that this particular finding could be instrumental for optimal designing of pump-pipe systems naturally appearing in water distribution networks.

2 The Problem and the Main Result

As explained in the Introduction, we consider a pipe defined by

Denoting uε=(uxε,uyε,uzε) and taking a function p₀ defined on a unit circle B(0, 1) ⊂ R² and a constant p₁, we study the following system:

So, we prescribe the pressure on both ends of the pipe, with the pressure on the left end being non-constant. The existence and uniqueness result for the solution of the above problem can be found in the paper by Conca et al. [16] (see also Heywood et al. [17] for the Navier-Stokes system). Because of the pressure boundary condition (5)₂, the problem (2)–(5) cannot be solved explicitly, i.e. the Hagen-Poiseuille formula does not give an exact solution anymore. Nevertheless, the Hagen-Poiseuille formula gives a zero-order approximation of the solution. As indicated in the Introduction, the expected result would be that the Hagen-Poiseuille approximation is given by taking the mean value of p₀ and prescribing that as the (constant) pressure on the pipe’s entry. Thus, we take

with

Taking this into account, now we expect to have the approximation of the solution in the form (hereinafter ( i, j, k) denotes the standard Cartesian basis)

with

Indeed, plugging the asymptotic expansion of the form

into (2), after collecting equal powers of ε, we arrive at

where ξ=(ξ2,ξ3)=(yε,zε), divξV=∂V2∂ξ2+∂V3∂y3, Δy′V=∂2V∂ξ22+∂2V∂ξ32, ∇ξΦ=∂Φ∂ξ2j+∂Φ∂ξ3k, for a scalar function Φ and a vector function V = (V¹, V², V³). In view of the boundary conditions (3)–(5), the above system can be solved by taking P¹ = P¹(x) and

leading to (7)–(8).

However, our analysis yields the conclusion that the right choice of the constant d to be prescribed on the left end is given by

which is not equal to p0¯, unless p₀ is a constant.

3 The Boundary Layer Corrector

We start from our approximation (7):

with

and try to correct it in order to satisfy the pressure boundary condition on the left end of the pipe. To accomplish that, we add the boundary layer corrector in the form

where those functions are defined by a boundary value problem posed in an infinite tube G = (0, +∞) × B(0, 1). The main idea is to pick them in a way that

As a consequence,

is going to meet the boundary condition (5)₂ for x = 0. In view of that, the boundary layer corrector satisfies the following problem:

Here, the subscript ξ denotes the differential operators taken with respect to ξ = (ξ₁, ξ₂, ξ₃). The problem (16)–(18) is a typical boundary layer-type problem, and it has a unique solution that vanishes exponentially as ξ₁ ⟶ +∞. As a matter of fact, the velocity B vanishes, whilst the pressure b only exponentially decays to a constant, i.e.

The above assertion can be proved following standard arguments (see, e.g. Galdi [18]). The fact that the pressure decays to a constant is the key point since the approximation P⁰(x) + b(x/ε, y/ε, z/ε) does not satisfy the boundary condition on the right end x = 1 anymore. Indeed, now we have

Consequently, we have to change the right boundary condition satisfied by P⁰ so we impose

Thus, the new P⁰ now takes the form

It is essential to observe that our discussion makes no sense if b_∞ = 0. Therefore, in the following, we are going to prove that this is not the case, unless p₀ is a constant.

Taking the form (8) of w, we can compute ∫B(0,1)w=π8μ so that

4 Optimal Pressure

Motivated by the applications (in particular, by the pump-pipe systems), in this section we want to derive the optimal boundary pressure p₀, such that the pressure drop (and thus the velocity) is maximal (for a given mean pressure). More precisely, let Q > 0 be a given number. We look for a function g such that ∫B(0,1)g(r,φ)rdrdφ=Q and

First of all, notice that we can find the upper bound for the above integral:

Because the function w is given by w(r)=14μ(1−r2), it attains its maximum for r = 0, i.e. in the middle of the pipe. Thus, we obviously have to look for a function concentrated in the middle of the pipe. It is easy to compute that

where δ stands for Dirac measure concentrated in the centre of the circle B(0, 1). So, for any delta sequence in Y={g∈L1(B(0,1)):∫B(0,1)g=Q}, i.e. any sequence g_n ∈ Y such that

We have

Thus,

The maximum is not attained in set Y because δ ∉ Y.

Finally, we notice that for constant boundary pressure gc=Qπ, we get

which is, in fact, the half of the maximal value. It means that the pressure drop can be significantly increased by applying non-constant, more localised, pressure on the entry of the pipe.

5 Conclusion

This paper reports an analytical investigation of the viscous fluid flow through a long pipe governed by the pressure drop between a pipe’s ends. In the analytical treatments of such problems, prescribing constant pressures on both ends of the pipe is usual throughout the literature. Motivated by real-life applications, namely, water supply pipe networks, here we assume that the pressure p₀ on the pipe’s entry is variable. This makes the problem more challenging from a mathematical point of view, because the boundary layer phenomena appear, see the section on The Boundary Layer Corrector. Our approach is based on a formal asymptotic analysis with respect to the small parameter ε=1L, where L is the pipe’s length. The main result suggests that the effective condition on the pipe’s entry is given by (13), i.e. it is not equal to the mean value of p₀, as one would expect. To our knowledge, this is a completely new result that cannot be found in the existing literature.

In addition, we also address the optimisation problem, which is important in the context of the design of hydraulic pumps being a part of the pipe networks. Namely, we aimed to find an optimal pressure such that the pressure drop (and, consequently, the fluid velocity) is maximal, for a given mean pressure. The mathematical analysis conducted in the section on Optimal Pressure shows that the pressure drop inside the pipe can be significantly increased by applying more localised (variable) pressure on the left end of the pipe. This result represents the second novelty of this paper. To conclude, we believe that both results attempt to improve the known engineering practice connected to the optimal design of urban water supply pipe networks.