Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 905406, 8 pages

http://dx.doi.org/10.1155/2015/905406

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

Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia

Received 24 November 2014; Accepted 20 March 2015

Academic Editor: Ana Carpio

Copyright © 2015 Igor Pažanin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 parameter (representing the ratio of the pipe’s thickness and its length) into the problem and perform asymptotic analysis with respect to . 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.

#### 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 the* Poiseuille 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]:Here stands for the viscosity at atmospheric pressure while 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 the* transformed 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 governing D boundary-value problem. Therefore, inspired by the applications, we introduce the small parameter 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 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 by , is an arbitrary smooth curve given by its natural parametrization. The only constraint we impose on is that it is a generic curve. As a consequence, we can use local Frenet’s basis attached to 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 (.) 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 or . 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 in , denoted by , which serves to define the central curve of the pipe. We suppose that is parameterized by its arc length and denote by its natural parametrization. We also assume that , for every . Since is taken to be generic, it holds , for every . Denoting by the flexion of the curve , we introduce Frenet’s basis in a standard way:

The normal is extended by continuity in points where curvature is zero. We also denote by the torsion of . One of the main goals of this study is to detect the influence of geometric parameters and on the effective flow.

Next, we introduce the small parameter () into the problem and first define an undeformed pipe where 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 as follows: Here stand for the rotated unit vectors with respect to standard Frenet’s normal and binormal (see Figure 1), where the rotation is given by