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

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<Rt<1 (Rs 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<α<1) the flow is determined by using a sequence of power series expansions (Rs=O(1)). At high frequency (δ0<Rt<1/α2<1) the solution is obtained using matched asymptotic expansions for the region near the wall (Stokes layer) and the region away from the wall in the interior of the pipe. The behaviour of the flow in the interior is then determined at both small and intermediate values of Rs. 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.

[1]  D. Ku,et al.  Pulsatile Flow and Atherosclerosis in the Human Carotid Bifurcation: Positive Correlation between Plaque Location and Low and Oscillating Shear Stress , 1985, Arteriosclerosis.

[2]  R. Schroter,et al.  Atheroma and arterial wall shear - Observation, correlation and proposal of a shear dependent mass transfer mechanism for atherogenesis , 1971, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[3]  A. Guyton,et al.  Textbook of Medical Physiology , 1961 .

[4]  K. Chandran,et al.  Experimental study of physiological pulsatile flow in a curved tube , 1981, Journal of Fluid Mechanics.

[5]  G. Batchelor,et al.  On steady laminar flow with closed streamlines at large Reynolds number , 1956, Journal of Fluid Mechanics.

[6]  S. Berger,et al.  Periodic flows through curved tubes: the effect of the frequency parameter , 1990, Journal of Fluid Mechanics.

[7]  Timothy J. Pedley,et al.  The fluid mechanics of large blood vessels , 1980 .

[8]  T. Mullin,et al.  Oscillatory flow in curved pipes. Part 2. The fully developed case , 1980, Journal of Fluid Mechanics.

[9]  W. H. Lyne,et al.  Unsteady viscous flow in a curved pipe , 1971, Journal of Fluid Mechanics.

[10]  S Glagov,et al.  The role of fluid mechanics in the localization and detection of atherosclerosis. , 1993, Journal of biomechanical engineering.

[11]  J. Tarbell,et al.  An experimental and numerical study of periodic flow in a curved tube , 1980, Journal of Fluid Mechanics.

[12]  S. Berger,et al.  Entry flow in a curved pipe , 1975, Journal of Fluid Mechanics.

[13]  T. Pedley,et al.  Flow in a tube with non-uniform, time-dependent curvature: governing equations and simple examples , 1996, Journal of Fluid Mechanics.

[14]  L. Talbot,et al.  Pulsatile entrance flow in a curved pipe , 1983, Journal of Fluid Mechanics.

[15]  L. Talbot,et al.  Flow in Curved Pipes , 1983 .

[16]  Frank T. Smith,et al.  Pulsatile flow in curved pipes , 1975, Journal of Fluid Mechanics.

[17]  F. Smith Steady motion within a curved pipe , 1976, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[18]  W. R. Dean XVI. Note on the motion of fluid in a curved pipe , 1927 .

[19]  J. Chow,et al.  NUMERICAL SOLUTION FOR FULLY DEVELOPED, LAMINAR PULSATING FLOW IN CURVED TUBES , 1980 .

[20]  High Reynolds number flows with closed streamlines , 1981 .

[21]  M H Friedman,et al.  Arteriosclerosis research using vascular flow models: from 2-D branches to compliant replicas. , 1993, Journal of biomechanical engineering.

[22]  T. Mullin,et al.  Oscillatory flow in curved pipes. Part 1. The developing-flow case , 1980, Journal of Fluid Mechanics.

[23]  M. Singh Entry flow in a curved pipe , 1974, Journal of Fluid Mechanics.

[24]  K. Chandran,et al.  An experimental study of pulsatile flow in a curved tube. , 1979, Journal of biomechanics.