Unsteady flows in pipes with finite curvature

Abstract

Motivated by the study of blood flow in a curved artery, we consider fluid flow through a curved pipe of uniform curvature, δ, driven by a prescribed oscillatory axial pressure gradient. The curved pipe has finite (as opposed to asymptotically small) curvature, and we determine the effects of both the centrifugal and Coriolis forces on the flow. In addition to δ, the flow is parameterized by the Dean number, D, the Womersley number, α, and a secondary streaming Reynolds number, Rs. Asymptotic solutions are developed for the case when δ≪1, α≪1 and the magnitude of the axial pressure gradient is small, using regular perturbation techniques. For intermediate values of the governing parameters, a pseudospectral code is used to obtain numerical solutions. For flows driven by a sinusoidal pressure gradient (D=0), we identify three distinct classes of stable solutions: 2π-periodic symmetric, 2π-periodic asymmetric, and asymmetric solutions that are either quasi-periodic, or periodic with period 2πk for k∈ $\mathbb{N}$. The transition between solutions is dependent on the value of δ; thus pipes with finite curvature may exhibit qualitatively different transitions between the solution classes as the governing parameters are varied from those of curved pipes with asymptotically small curvature. When α≫1, matched asymptotic expansions are used to simplify the system, and the resulting equations are solved analytically for Rs≪1, δ≪1 and numerically for larger parameter values. We then determine the effect of a non-zero steady component of the pressure gradient (D≠0), and show that, for certain parameter values, when D is above a critical value the periodic asymmetric solutions regain spatial symmetry. Finally, we show that the effects of finite curvature can lead to substantial quantitative differences in the wall shear stress distribution and discuss briefly the physiological implications of the results for blood flow in arteries.