Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 

 

Slow viscous flow inside a torus--the resistance of small tortuous blood vessels


Author: R. S. Chadwick
Journal: Quart. Appl. Math. 43 (1985), 317-323
MSC: Primary 92A06; Secondary 76D07, 76Z05, 92A09
DOI: https://doi.org/10.1090/qam/814230
MathSciNet review: 814230
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The hydrodynamic resistance of a buckled microvessel in the form of a tightly would helix is approximated by studing the Stokes flow inside a torus. The unidirectional flow is driven by a constant tangential pressure gradient. The solution is obtained by an eigenfunction expansion in toroidal coordinates. The ratio of volume flow carried by the torus to that carried by a straight tube is computed as a function of the vessel radius: coil radius ratio. An asymptotic expansion for this flux ratio is also obtained. The results show that the resistance of a moderately curved vessel is slightly less than the resistance of a straight one, whereas the resistance of a greatly curved vessel is at most $ 3\% $ greater than the straight one.


References [Enhancements On Off] (What's this?)

  • [1] R. F. Potter and A. C. Groom, Capillary diameter and geometry in cardiac and skeletal muscle studied by means of corrosion casts, Microvascular Research 25, 68-84 (1983)
  • [2] A. M. Waxman, Blood vessel growth as a problem in morphogenesis: a physical theory, Microvascular Research 22, 32-42 (1981)
  • [3] W. H. Pell and L. E. Payne, On Stokes flow about a torus, Mathematika 7 (1960), 78–92. MR 0143413, https://doi.org/10.1112/S0025579300001601
  • [4] E. W. Hobson, The theory of spherical and ellipsoidal harmonics, Chelsea Publishing Company, New York, 1955. MR 0064922
  • [5] I. A. Stegun, Legendre functions, in Handbook of Mathematical Functions, National Bureau of Standards Applied Mathematics Series 55, ed. M. Abramovitz and I. A. Stegun, 1968
  • [6] L. W. Schwartz, personal communication
  • [7] J. Larrain and C. F. Bonilla, Theoretical analysis of the pressure drop in the laminar flow of fluid in a coiled pipe, Trans. Soc. Rheol. 14, 135-147 (1970)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 92A06, 76D07, 76Z05, 92A09

Retrieve articles in all journals with MSC: 92A06, 76D07, 76Z05, 92A09


Additional Information

DOI: https://doi.org/10.1090/qam/814230
Article copyright: © Copyright 1985 American Mathematical Society


Brown University The Quarterly of Applied Mathematics
is distributed by the American Mathematical Society
for Brown University
Online ISSN 1552-4485; Print ISSN 0033-569X
© 2017 Brown University
Comments: qam-query@ams.org
AMS Website