Viscous Fluid Flow in Bifurcating Channels and Pipes

Abstract

In the present thesis the flow of a viscous Newtonian fluid in a bifurcation of thin three-dimensional pipes with a diameter-to-length ratio of order O(epsilon) is studied. The model is based on the steady-state Navier-Stokes equations with pressure conditions on the in- and outflow boundaries. Existence and local uniqueness is proven under the assumption of small data represented by a Reynolds number Re of order O(epsilon). Our aim is to construct an asymptotic expansion in powers of epsilon and Re for the solution of this Navier-Stokes problem. In the first part of the thesis we therefore present a formal method of computing the pressure drop and the flux based on Poiseuille flow. In contrast to the existing literature, we also analyze the influence of the bifurcation geometry on the fluid flow by introducing local Stokes problems in the junction. We show that the solutions of these Stokes problems in the junction of diameter O(M) approximate the solutions of the corresponding Leray problems in the infinite bifurcation up to an error decaying exponentially in M. In the second part of the thesis, the construction of the approximation for the Navier-Stokes solution is presented and its properties are discussed. The approximation is based on the idea of a continuous matching of the Poiseuille velocity to the solution of the junction problem on each pipe-junction interface. The main result of our analysis is the derivation of error estimates for the approximation in powers of epsilon and Re according to the designated approximation accuracy. The obtained results generalize and improve the existing ones in literature. In addition, our results show that Kirchhoff's law of the balancing fluxes has to be corrected in O(epsilon) in order to obtain an adequate error estimate for the gradient of velocity.