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Flow in a tube with non-uniform, time-dependent curvature: governing equations and simple examples

Published online by Cambridge University Press:  26 April 2006

D. G. Lynch ,
S. L. Waters  and
T. J. Pedley
D. G. Lynch
Affiliation:
Department of Applied Mathematical Studies, University of Leeds, Leeds, LS2 9JT, UK
S. L. Waters
Affiliation:
Department of Applied Mathematical Studies, University of Leeds, Leeds, LS2 9JT, UK
T. J. Pedley
Affiliation:
Department of Applied Mathematical Studies, University of Leeds, Leeds, LS2 9JT, UK Present address: Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
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Abstract

Motivated by the study of blood flow in the major coronary arteries, which are situated on the outer surface of the pumping heart, we analyse flow of an incompressible Newtonian fluid in a tube whose curvature varies both along the tube and with time. Attention is restricted to the case in which the tube radius is fixed and its centreline moves in a plane. Nevertheless, the governing equations are very complicated, because the natural coordinate system involves acceleration, rotation and deformation of the frame of reference, and their derivation forms a major part of the paper. Then they are applied to two particular, relatively simple examples: a tube of uniform but time-dependent curvature; and a sinuous tube, representing a small-amplitude oscillation about a straight pipe. In the former case the curvature is taken to be small and to vary by a small amount, and the solution is developed as a triple power series in mean curvature ratio δ0, curvature variation ε and Dean number D. In the latter case the Reynolds number is taken to be large and a linearized solution for the perturbation to the flow in the boundary layer at the tube wall is obtained, following Smith (1976a). In each case the solution is taken far enough that the first non-trivial effects of the variable curvature can be determined. Results are presented in terms of the oscillatory wall shear stress distribution and, in the uniform curvature case, the contribution of steady streaming to the mean wall shear stress is calculated. Estimation of the parameters for the human heart indicates that the present results are not directly applicable, but point the way for future work.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 323 , 25 September 1996 , pp. 237 - 265
Copyright
© 1996 Cambridge University Press

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References

Berger, S. A., Talbot, L. & Yao, L. S. 1983 Flow in curved pipes. Ann. Rev. Fluid Mech. 15, 461–512.Google Scholar
Blennerhassett, P. 1976 Secondary motion and the diffusion in unsteady flow in a curved pipe. PhD thesis, Imperial College, London.
Collins, W. M. & Dennis, S. C. R. 1975 The steady motion of a viscous fluid in a curved tube. Q. J. Mech. Appl. Maths 28, 133–156.Google Scholar
Daskopoulos, P. & Lenhoff, A. M. 1989 Flow in curved ducts:bifurcation structure for stationary ducts. J. Fluid Mech. 203, 125–148.Google Scholar
Davies, P. F. 1995 Flow-mediated endothelial mechanotransduction. Physiol. Rev. 75, 519–560.Google Scholar
Dean, W. R. 1927 Note on the motion of fluid in a curved pipe. Phil. Mag. (7) 4, 208–223.Google Scholar
Dean, W. R. 1928 The stream-line motion of fluid in a curved pipe. Phil. Mag. (7), 5, 673–695.Google Scholar
Delfino, A., Moore, J. E. & Meister, J.-J. 1994 Lateral deformation and movement effects on flow through distensible tube models of blood vessels. Biorheology 31, 533–547.Google Scholar
Friedman, M. H. 1993 Arteriosclerosis research using vascular flow models: from 2D branches to compliant replicas. J. Biomech. Engng 115, 595–601.Google Scholar
Giddens, D. P., Zarins, C. K. & Glagov, S. 1993 The role of fluid mechanics in the localization and detection of Atherosclerosis. J. Biomech. Engng 115, 588–594.Google Scholar
Inaba, T. & Murata, S. 1978 Pulsating laminar flow in a sinusoidally curved pipe. Bull. JSME 21, 832–839.Google Scholar
Lyne, W. H. 1971 Unsteady viscous flow in a curved pipe. J. Fluid Mech. 45, 13–31.Google Scholar
McConalogue, D. J. & Srivastava, R. S. 1968 Motion of a fluid in a curved tube. Proc. R. Soc. Lond. A. 307, 37–53.Google Scholar
Moore, J. E., Guggenheim, N., Delfino, A., Doriot, P.-A., Dorsaz, P.-A., Rutishauser, W. & Meister, J.-J. 1994 Preliminary analysis of the effects of blood vessel movement on blood flow patterns in the coronary arteries. J. Biomech. Engng 116, 302–306.Google Scholar
Murata, S., Miyake, Y. & Inaba, T. 1976 Laminar flow in a curved pipe with varying curvature J. Fluid Mech. 73, 735–752.Google Scholar
Pao, Y. C., Lu, J. T. & Ritman, E. L. 1992 Bending and twisting of an in vivo coronary artery at a bifurcation. J. Biomech. 25, 287–295.Google Scholar
Pedley, T. J. 1980 The Fluid Mechanics of Large Blood Vessels. Cambridge University Press.
Pedley, T. J. 1995 High Reynolds number flow in tubes of complex geometry with application to wall shear stress in arteries. Biological Fluid Dynamics, SEB Symp. 49 (ed. C. P. Ellington & T. J. Pedley), London, Company of Biologists.
Singh, M. P. 1974 Entry flow in a curved pipe. J. Fluid Mech. 65, 517–539.Google Scholar
Smith, F. T. 1975 Pulsatile flow in curved pipes. J. Fluid Mech. 71, 15–42.Google Scholar
Smith, F. T. 1976a Fluid flow into a curved pipe. Proc. R. Soc. Lond. A 351, 71–87.Google Scholar
Smith, F. T. 1976b Pipeflows distorted by a non-symmetric indentation or branching. Mathematika 23, 62–83.Google Scholar
Sugawara, M., Kajiya, F., Kitabatake, A. & Matsuo, H. 1989 Blood Flow in the Heart and Large Vessels. Springer.
Yanase, S., Goto, N. & Yamamoto, K. 1989 Dual solutions of the flow through a curved tube. Fluid Dyn. Res. 5, 191–201.Google Scholar
Yao, L. S. & Berger, S. A. 1975 Entry flow in a curved pipe. J. Fluid Mech. 67, 177–196.Google Scholar