Oscillatory flow in a tube of time-dependent curvature. Part 1. Perturbation to flow in a stationary curved tube

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Abstract
Motivated by the study of blood flow in the coronary arteries, this paper examines the flow of an incompressible Newtonian fluid in a tube of time-dependent curvature. The flow is driven by an oscillatory pressure gradient with the same dimensionless frequency, α, as the curvature variation. The dimensionless governing parameters of the flow are α, the curvature ratio δ0, a secondary streaming Reynolds number Rs and a parameter Rt representing the time-dependence of curvature. We consider the parameter regime δ0<RtRs and α remain O(1) initially) in which the effect of introducing time-dependent curvature is to perturb the flow driven by an oscillatory pressure gradient in a fixed curved tube. Flows driven by low- and high-frequency pressure gradients are then considered. At low frequency (δ0<Rt<αRs=O(1)). At high frequency (δ0<Rt2Rs. For both the low and high frequency cases, we find the principal corrections introduced by the time-varying curvature to the primary and secondary flows, and hence to the wall shear stress. The physiological application to flow in the coronary arteries is discussed.

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