Asymptotic analysis of the fluid flow with a pressure-dependent viscosity in a system of thin pipes
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Asymptotic analysis of the fluid flow with a pressure-dependent viscosity in a system of thin pipes Eduard Maruši´c-Paloka1 · Igor Pažanin1
Received: 10 March 2016 / Accepted: 21 April 2016 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2016
Abstract We consider the incompressible fluid with a pressure-dependent viscosity flowing through a multiple pipe system. The viscosity–pressure relation is given by the Barus law commonly used in the engineering applications. Assuming that the ratio between pipes thickness and its length is small, we propose a rigorous asymptotic approach based on the concept of the transformed pressure. As a result, we obtain new macroscopic model describing the effective behavior of the fluid in the system. In particular, the generalized version of the Kirchhoff’s law is derived giving the explicit formula for the junction pressure. The error estimate for the asymptotic approximation is also provided. Mathematical analysis presented here can be applied to a general viscosity–pressure relation satisfied by other empiric laws. Keywords Pipe network · Pressure-dependent viscosity · Transformed pressure · Kirchhoff’s law · Asymptotic analysis Mathematics Subject Classification 35B40 · 35Q35 · 76M45
1 Introduction In his celebrated work, Stokes (1845) suggested that the viscosity of many liquids can depend on the pressure. Since then, numerous experimental investigations (see e.g., Binding et al. 1989; Goubert et al. 2001; Del Gaudio and Behrens 2009) confirmed that, as the pressure is increased by orders of magnitude, the variations of the viscosity with pressure should be taken into account while the flow is still incompressible. For that reason, incompressible fluid flow with a pressure-dependent viscosity has been extensively studied in recent years both in
Communicated by Raphaèle Herbin.
B 1
Igor Pažanin [email protected] Department of Mathematics, Faculty of Science, University of Zagreb, Bijeniˇcka 30, 10000 Zagreb, Croatia
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E. M.-Paloka, I. Pažanin
the engineering and mathematical literature (see e.g., Malek et al. 2002; Renardy 2003; Kalogirou et al. 2011; Housiadas et al. 2015). The viscosity–pressure relation is most commonly described by the Barus law Barus (1893) stating that the viscosity increases exponentially with pressure. The exponential law rules out the possibility of deriving analytical solutions (even for simple flows) and that is why it has been avoided in theoretical studies throughout the literature. Another reason is that not much has been done in proving the well posedness of the corresponding boundary-value problems. Recently, the authors of the present paper made some progress in both directions, we refer the reader to Maruši´c-Paloka and Pažanin (2013); Maruši´c-Paloka (2014). In this paper, we consider a multiple pipe system filled with incompressible fluid (with viscosity satisfying Barus law). We are strongly motivated by the fact that, in real-life situations (see e.g., Swamee and Sharma 2008), several thin pipes are interconne
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