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


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.


  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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Information & Authors


Published In

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


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


  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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