Pulsatile Viscous Flows in Elliptical Vessels and Annuli: Solution to the Inverse Problem, with Application to Blood and Cerebrospinal Fluid Flow

Abstract

We consider the fully developed flow of an incompressible Newtonian fluid in a cylindrical vessel with elliptical cross section, and in the annulus between two confocal ellipses. Since flow rate can actually be derived from measurements, we address the inverse problem, namely computing the velocity field associated with a given time-periodic flow rate. We propose a novel numerical strategy, which is nonetheless grounded on several analytical relations and which leads to the solution of systems of ordinary differential equations. We also report numerical results based on measured data for human blood flow in the internal carotid artery, and cerebrospinal fluid flow in the upper cervical region of the human spine. Our method holds promise to be more amenable to implementation than previous ones, based on challenging computation of Mathieu functions, especially for strongly elliptical cross sections. The main goal of this study is to provide an improved source of initial/boundary data, as well as a benchmark solution for pulsatile flows in elliptical sections. In addition to bio-fluid dynamics investigations, the proposed method can be applied to many problems in the biomedical field.

Keywords

  1. pulsatile laminar flow
  2. elliptical vessel
  3. inverse problem
  4. cerebrospinal fluid flow
  5. blood flow

MSC codes

  1. Primary
  2. 76Z05; Secondary
  3. 35Q30
  4. 92C35

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References

1.
F. A. Alhargan, A complete method for the computations of Mathieu characteristic numbers of integer orders, SIAM Rev., 38 (1996), pp. 239--255.
2.
N. Arada, M. Pires, and A. Sequeira, Viscosity effects on flows of generalized Newtonian fluids through curved pipes, Comput. Math. Appl., 53 (2007), pp. 625--646.
3.
K. J. Bathe, H. Zhang, and M. H. Wang, Finite element analysis of incompressible and compressible fluid flows with free surfaces and structural interactions, Comput. Struct., 56 (1995), pp. 193--213.
4.
H. Beira͂o da Veiga, Time periodic solutions of the Navier-Stokes equations in unbounded cylindrical domains---Leray's problem for periodic flows, Arch. Ration. Mech. Anal., 178 (2005), pp. 301--325.
5.
L. C. Berselli, P. Miloro, A. Menciassi, and E. Sinibaldi, Exact solution to the inverse Womersley problem for pulsatile flows in cylindrical vessels, with application to magnetic particle targeting, Appl. Math. Comput., 219 (2013), pp. 5717--5729.
6.
L. C. Berselli and M. Romito, On Leray's problem for almost periodic flows, J. Math. Sci. Univ. Tokyo, 19 (2012), pp. 69--130.
7.
C. D. Bertram, A numerical investigation of waves propagating in the spinal cord and subarachnoid space in the presence of a syrinx, J. Fluids Struct., 25 (2009), pp. 1189--1205.
8.
S. Čanić, J. Tambača, G. Guidoboni, A. Mikelić, C. J. Hartley, and D. Rosenstrauch, Modeling viscoelastic behavior of arterial walls and their interaction with pulsatile blood flow, SIAM J. Appl. Math., 67 (2006), pp. 164--193.
9.
S. Cheng, M. A. Stoodley, J. Wong, S. Hemley, D. F. Fletcher, and L. E. Bilston, The presence of arachnoiditis affects the characteristics of CSF flow in the spinal subarachnoid space: A modelling study, J. Biomech., 45 (2012), pp. 1186--1191.
10.
C. Di Rocco, P. Frassanito, L. Massimi, and S. Peraio, Hydrocephalus and Chiari type I malformation, Childs Nerv. Syst., 27 (2011), pp. 1653--1664.
11.
L. Formaggia, A. Veneziani, and C. Vergara, Flow rate boundary problems for an incompressible fluid in deformable domains: Formulations and solution methods, Comput. Methods Appl. Mech. Engrg., 199 (2010), pp. 677--688.
12.
G. P. Galdi and A. M. Robertson, The relation between flow rate and axial pressure gradient for time-periodic Poiseuille flow in a pipe, J. Math. Fluid Mech., 7 (2005), pp. S215--S223.
13.
L. Greengard and J.-Y. Lee, Accelerating the nonuniform fast Fourier transform, SIAM Rev., 46 (2004), pp. 443--454.
14.
S. Gupta, D. Poulikakos, and V. Kurtcuoglu, Analytical solution for pulsatile viscous flow in a straight elliptic annulus and application to the motion of the cerebrospinal fluid, Phys. Fluids, 20 (2008), 093607.
15.
K. Haddad, O. Ertun\ifmmode ç\else ç\fi, M. Mishra, and A. Delgado, Pulsating laminar fully developed channel and pipe flows, Phys. Rev. E, 81 (2010), 016303.
16.
M. Haslam and M. Zamir, Pulsatile flow in tubes of elliptic cross sections, Ann. Biomed. Eng., 26 (1998), pp. 780--787.
17.
Y. Hoi, B. A. Wasserman, Y. Y. J. Xie, S. S. Najjar, L. Ferruci, E. G. Lakatta, G. Gerstenblith, and D. A. Steinman, Characterization of volumetric flow rate waveforms at the carotid bifurcations of older adults, Physiol. Meas., 31 (2010), pp. 291--302.
18.
D. N. Irani, Cerebrospinal Fluid in Clinical Practice, Saunders, Philadelphia, 2008.
19.
J. Krejza, M. Arkuszewski, S. E. Kasner, J. Weigele, A. Ustymowicz, R. W. Hurst, B. L. Cucchiara, and S. R. Messe, Carotid artery diameter in men and women and the relation to body and neck size, Stroke, 37 (2006), pp. 1103--1105.
20.
M.-C. Lai, Fast direct solver for Poisson equation in a $2$D elliptical domain, Numer. Methods Partial Differential Equations, 20 (2004), pp. 72--81.
21.
A. A. Linninger, M. Xenos, B. Sweetman, S. Ponkshe, X. Guo, and R. Penn, A mathematical model of blood, cerebrospinal fluid and brain dynamics, J. Math. Biol., 59 (2009), pp. 729--759.
22.
F. Loth, M. A. Yardimci, and N. Alperin, Hydrodynamic modeling of cerebrospinal fluid motion within the spinal cavity, J. Biomech. Eng., 123 (2001), pp. 71--79.
23.
N. W. McLachlan, Theory and Application of Mathieu Functions, Clarendon Press, Oxford, UK, 1947.
24.
B. J. Nelson, I. K. Kaliakatsos, and J. J. Abbott, Microrobots for minimally invasive medicine, Annu. Rev. Biomed. Eng., 12 (2010), pp. 55--85.
25.
B. I. Rapoport, J. T. Kedzierski, and R. Sarpeshkar, A glucose fuel cell for implantable brain-machine interfaces, PLoS ONE, 7 (2012), e38436.
26.
Y. V. K. Ravi Kumar, P. S. V. H. N. Krishna Kumari, M. V. Ramana Murthy, and S. Sreenadh, Unsteady peristaltic pumping in a finite length tube with permeable wall, J. Fluids Eng., 132 (2010), 101201.
27.
S. Ray and F. Durst, Semianalytical solutions of laminar fully developed pulsating flows through ducts of arbitrary cross sections, Phys. Fluids, 16 (2004), pp. 4371--4385.
28.
A. M. Robertson, A. Sequeira, and R. G. Owens, Rheological models for blood, in Cardiovascular Mathematics, Model. Simul. Appl. 1, Springer Italia, Milan, 2009, pp. 211--241.
29.
M. B. Robertson, U. Köhler, P. R. Hoskins, and I. Marshall, Flow in elliptical vessels calculated for a physiological waveform, J. Vasc. Res., 38 (2001), pp. 73--82.
30.
K. Rogers, Blood: Physiology and Circulation, Human Body series, Rosen Publishing Group, New York, 2010.
31.
T. Sexl, Uber den von E. G. Richardson entdeckten “Annulareffekt", Z. Phys, 61 (1930), pp. 179--221.
32.
N. Shaffer, B. Martin, and F. Loth, Cerebrospinal fluid hydrodynamics in type I Chiari malformation, Neurol. Res., 33 (2011), pp. 247--260.
33.
J. Shen and L.-L. Wang, On spectral approximations in elliptical geometries using Mathieu functions, Math. Comp., 78 (2009), pp. 815--844.
34.
R. Trip, D. J. Kuik, J. Westerweel, and C. Poelma, An experimental study of transitional pulsatile pipe flow, Phys. Fluids, 24 (2012), 014103.
35.
A. Veneziani and C. Vergara, An approximate method for solving incompressible Navier-Stokes problems with flow rate conditions, Comput. Methods Appl. Mech. Engrg., 196 (2007), pp. 1685--1700.
36.
P. D. Verma, The pulsating viscous flow superposed on the steady laminar motion of incompressible fluid between two co-axial cylinders, Proc. Indian Acad. Sci. Math. Sci., 26 (1960), pp. 447--458.
37.
P. D. Verma, The pulsating viscous flow superposed on the steady laminar motion of incompressible fluid in a tube of elliptic section, Proc. Indian Acad. Sci. Math. Sci., 26 (1960), pp. 282--297.
38.
J. R. Womersley, Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known, J. Physiol., 127 (1955), pp. 553--563.

Information & Authors

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Published In

cover image SIAM Journal on Applied Mathematics
SIAM Journal on Applied Mathematics
Pages: 40 - 59
ISSN (online): 1095-712X

History

Submitted: 21 December 2012
Accepted: 1 October 2013
Published online: 16 January 2014

Keywords

  1. pulsatile laminar flow
  2. elliptical vessel
  3. inverse problem
  4. cerebrospinal fluid flow
  5. blood flow

MSC codes

  1. Primary
  2. 76Z05; Secondary
  3. 35Q30
  4. 92C35

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