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TECHNICAL PAPERS

Helical Flow Around Arterial Bends for Varying Body Mass

[+] Author and Article Information
L. Zabielski, A. J. Mestel

Department of Mathematics, Imperial College of Science, Technology and Medicine, 180 Queen’s Gate, SW7 2BZ London, United Kingdom

J Biomech Eng 122(2), 135-142 (Oct 28, 1999) (8 pages) doi:10.1115/1.429635 History: Received January 14, 1999; Revised October 28, 1999
Copyright © 2000 by ASME
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References

Caro,  C. G., Fitz-Gerald,  J. M., and Schroter,  R. C., 1971, “Atheroma and Arterial Wall Shear: Observation, Correlation and Proposal of a Shear Dependent Mass Transfer Mechanism of Atherogenesis,” Proc. R. Soc. London, Ser. B, 177, pp. 109–159.
Caro,  C. G., Doorly,  D. J., Tarnawski,  M., Scott,  K. T., Long,  Q., and Dumoulin,  C. L., 1996, “Non-planar Curvature and Branching of Arteries and Non-planar-type of Flow,” Proc. R. Soc. London, Ser. A, 452, pp. 185–197.
Zabielski,  L., and Mestel,  A. J., 1998, “Steady Flow in a Helically Symmetric Pipe,” J. Fluid Mech., 370, pp. 297–320.
Zabielski,  L., and Mestel,  A. J., 1998, “Unsteady Blood Flow in a Helically Symmetric Pipe,” J. Fluid Mech., 370, pp. 321–345.
Childress, S., Landman, M., and Strauss, H., 1989, “Steady Motion With Helical Symmetry at Large Reynolds Number,” in: Proc. IUTAM Symp. on Topological Fluid Dynamics, H. K. Moffatt and A. Tsinober, eds., Cambridge University Press, pp. 216–224.
Dean,  W. R., 1928, “The Streamline Motion of Fluid in a Curved Pipe,” Philos. Mag., 30, pp. 673–695.
Lyne,  W. H., 1970, “Unsteady Viscous Flow in a Curved Pipe,” J. Fluid Mech., 45, pp. 13–31.
Schmidt-Nielsen, K., 1984, Scaling: Why Is Animal Size So Important? Cambridge University Press.
Chandran,  K. B., 1993, “Flow Dynamics in the Human Aorta,” ASME J. Biomech. Eng., 115, pp. 611–616.
Stahl,  W. R., 1968, “Scaling of Respiratory Variables in Mammals,” J. Appl. Phys., 22, pp. 453–460.
Pedley, T. J., 1978, “The Fluid Mechanics of Circulatory Systems,” Comparative Physiology—Water, Ions and Fluid Mechanics, Schmidt-Nielsen, Bolis, and Maddrell, eds., pp. 283–301.
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Smith,  F. T., 1976, “Steady Motion in a Curved Pipe,” Proc. R. Soc. London, Ser. A, 347, pp. 345–370.
Parker, K. H., 1998, private communication, Imperial College, London.
McDonald, D. A., 1990, Blood Flow in Arteries, The Camelot Press Ltd., Southampton, United Kingdom.
Kilner,  P. J., Yanz,  G. Z., Mohiaddin,  R. H., Firmin,  D. N., and Longmore,  D. B., 1993, “Helical and Retrograde Secondary Flow Patterns in the Aortic Arch Studied by Three-Dimensional Magnetic Resonance Velocity Mapping,” Circulation, 5, pp. 2235–2247.
Axel, L., McLean, M., Tarnawski, M., Doorly, D. J., Dumoulin, C. L., and Caro, C. G., 1996, “Magnetic Resonance Imaging of Helical Flows,” Proc. Society of Magnetic Resonance 4th Scientific Meeting, New York.

Figures

Grahic Jump Location
A helical pipe of radius a, pitch 2πa=2π/ε, and radius of curvature b=2.5a, appropriate for the aortic arch 9
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(a) Measured velocity profile from Parker 15, and (b) pressure gradient px(t) inferred from Eq. (15). Velocity is to be scaled with α−1Rsν/a.
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Flow patterns inside a helical bend with ε=1, b=2.5, at various stages of the cardiac cycle as indicated on the far left. Flow conditions correspond to the aortic arch of a 20 kg dog. The inside of the bend is to the left. The middle column portrays contours of the down-pipe velocity ν. The left column shows the streamlines of the cross-pipe flow Ψ, and contours of the down-pipe vorticity ξ are drawn in the right-hand column. The boundary layers on the top and bottom separate near the inside of the bend and complex structures with regions of back flow are thrown into the mainstream.
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Flow just after separation inside a planar (toroidal) bend with ε=100, b=2.5, and mass 3 kg (rabbit). From left to right: Ψ, ν, and ξ. The boundary layers on the top and bottom collide and separate symmetrically near the inner bend on the left, leading to strong shear in the core.
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Separated flow inside a stretched helical bend ε=0.5, b=2.5, for a mass of 60 kg (man)
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Computed flux, Q, with ε=1, b=2.5, for a dog of mass 20 kg
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A typical time dependence, σ̃H(t), of the down-pipe component of the wall shear rate inside a helical bend ε=1, b=2.5, and mass 20 kg
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Spatial distribution w(x) of the down-pipe shear in nondimensional units to be multiplied by σ̄H(t) [s−1]. For a helical bend ε=1, b=2.5: ––– dog, mass 20 kg; –⋅ – rat, mass 0.6 kg. For a helical bend ε=0.5, b=2.5a: —— man, mass 50 kg. x measures angle around the pipe cross section starting from the outer bend toward the top of the pipe.
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Wall shear on the boundary of a helical bend ε=1, b=2.5. flow conditions corresponding to the canine aortic arch with mass 20 kg. The direction and magnitude of the total shear stress are shown from two angles at the indicated times. The pipe is viewed from on top (left) and from the bottom (right).

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