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.
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)
A. M. Waxman, Blood vessel growth as a problem in morphogenesis: a physical theory, Microvascular Research 22, 32–42 (1981)
- W. H. Pell and L. E. Payne, On Stokes flow about a torus, Mathematika 7 (1960), 78–92. MR 143413, DOI https://doi.org/10.1112/S0025579300001601
- E. W. Hobson, The theory of spherical and ellipsoidal harmonics, Chelsea Publishing Company, New York, 1955. MR 0064922
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
L. W. Schwartz, personal communication
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)
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)
A. M. Waxman, Blood vessel growth as a problem in morphogenesis: a physical theory, Microvascular Research 22, 32–42 (1981)
L. E. Payne and W. H. Pell, On Stokes flow about a torus, Mathematika 7, 78–92 (1960)
E. W. Hobson, The theory of spherical and ellipsoidal harmonics, Chelsea Publishing, New York, 1955
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
L. W. Schwartz, personal communication
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
Article copyright:
© Copyright 1985
American Mathematical Society
