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, R s . 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 ∈ 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 R s 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.

Linear and nonlinear dynamics of pulsatile channel flow

“…Thomas et al (2011) have found neutrally stable conditions for pulsating pipe flow that could be used as starting points for nonlinear simulations; these conditions, however, correspond to essentially oscillating flow with a weak steady component, thus complicating their continuation toward physiological conditions. Furthermore, pulsatile flow through curved pipes (Siggers & Waters 2008) probably sustains even more complex dynamics.…”

The effect of in-plane arterial curvature on blood flow and oxygen transport in arterio-venous fistulae

“…1) can be leveraged to manipulate flow and oxygen transport patterns within AVF. Whilst flow in curved pipes has been studied extensively before, [25][26][27][28] we believe the concept of leveraging arterial curvature to manipulate flow in AVF to be novel.…”

Thin-film flow in helically-wound rectangular channels of arbitrary torsion and curvature

“…The study of flows in curved geometries has been motivated by flows in rivers and pipes (Thomson 1876(Thomson , 1877Dean 1927Dean , 1928, the circulatory system (Lynch et al 1996;Siggers & Waters 2005, 2008, and the cochlea (Manoussaki & Chadwick 2000). The impact of Dean's work was such that these flows are often termed "Dean flow".…”