Steady flows in pipes with finite curvature

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Abstract
Motivated by blood flow in a curved artery, we consider fluid flow through a uniformly curved pipe, driven by a steady pressure gradient. Previous studies have typically considered pipes with asymptotically small curvature in which only the centrifugal effect is retained in the governing equations. Here we consider flows in pipes of finite curvature, and determine the effects of both the centrifugal and Coriolis forces on the flow. The flow behavior is governed by two dimensionless parameters: the curvature, δ, and the Dean number, D. Asymptotic solutions are developed for D⪡1 (using regular perturbation expansions) and D⪢1 (using matched asymptotic expansions and the Pohlhausen method). For intermediate values of D we use a pseudospectral code to obtain solutions. For values of D greater than a critical value, Dc, multiple steady solutions to the governing equations exist; we determine how Dc and the form of the solutions depends on δ. The results indicate that while many of the qualitative features of steady curved-pipe flows are captured by the asymptotic results, the effects of finite curvature can lead to substantial quantitative differences, e.g., in the wall shear stress distribution. The physiological implications of the results for blood flow in arteries are discussed.

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