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Asymptotic analysis of the fluid flow with a pressure-dependent viscosity in a system of thin pipes


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.

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Fig. 1


  1. Same principles of analysis can be applied in case of multiple junctions as well.

  2. We also perform the matching procedure on the interface between exterior and interior layer.

  3. Technical conditions fulfilled by Barus and other empiric laws that serve for the existence proof (see Marušić-Paloka 2014).


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The authors have been supported by the Croatian Science Foundation (project 3955: Mathematical modeling and numerical simulations of processes in thin or porous domains).

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Correspondence to Igor Pažanin.

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Communicated by Raphaèle Herbin.

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Marušić-Paloka, E., Pažanin, I. Asymptotic analysis of the fluid flow with a pressure-dependent viscosity in a system of thin pipes. Comp. Appl. Math. 37, 297–305 (2018).

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  • Pipe network
  • Pressure-dependent viscosity
  • Transformed pressure
  • Kirchhoff’s law
  • Asymptotic analysis

Mathematics Subject Classification

  • 35B40
  • 35Q35
  • 76M45