Mathematical modelling of flow through shunts: application to patent ductus arteriosus and side-to-side anastomosis
Author(s)
Setchi, Adriana
Type
Thesis or dissertation
Abstract
This thesis develops mathematical models for the flow through shunts
in the human body in two particular parameter regimes: flows of large
Womersley number, i.e. ones dominated by high-frequency terms, and
flows of low Reynolds number, e.g. of high viscosity. The first regime can
be extended to flows with the properties that the Womersley number is of
order bigger than the square root of the Reynolds number. The geometries
that are considered in this work are idealised and therefore enable for the
mathematical understanding of some fascinating, complicated, and not-very-
well-understood problems in fluid dynamics.
Analytical solutions are derived for high-frequency flow in three different
idealised geometries. These rely on solving Laplaces equation for all linearly
independent steady solutions in each particular geometry, and providing
a suitable time-dependent behaviour through the choice of boundary
conditions. The analysis uses complex potential theory, Schwarz-
Christoffel transformations, conformal mappings and Fourier series. The
solutions are applied to study the hemodynamics in the vicinity of a patent
ductus arteriosus (PDA): a shunt between the aorta and pulmonary artery
in some adults. Of particular interest are the distributions of velocity and
pressure in the two arteries, as well as the shear stress at the cardiovascular
walls, during a cardiac cycle in asymptomatic patients. Different
hypotheses are tested by introducing different boundary conditions. The
main results are based on the assumption that the flow in asymptomatic
PDA patients is similar to the flow in healthy adults.
The thesis also considers the low-Reynolds-number flow in a two-dimensional
geometry of a shunt between two vessels. An analytical solution is derived
by constructing piece-wise continuous functions that solve the biharmonic
equation. The method uses the orthogonality properties of Papkovich-
Fadle eigenfunctions. This work is applied to model the flow distribution
in a side-to-side anastomosis.
in the human body in two particular parameter regimes: flows of large
Womersley number, i.e. ones dominated by high-frequency terms, and
flows of low Reynolds number, e.g. of high viscosity. The first regime can
be extended to flows with the properties that the Womersley number is of
order bigger than the square root of the Reynolds number. The geometries
that are considered in this work are idealised and therefore enable for the
mathematical understanding of some fascinating, complicated, and not-very-
well-understood problems in fluid dynamics.
Analytical solutions are derived for high-frequency flow in three different
idealised geometries. These rely on solving Laplaces equation for all linearly
independent steady solutions in each particular geometry, and providing
a suitable time-dependent behaviour through the choice of boundary
conditions. The analysis uses complex potential theory, Schwarz-
Christoffel transformations, conformal mappings and Fourier series. The
solutions are applied to study the hemodynamics in the vicinity of a patent
ductus arteriosus (PDA): a shunt between the aorta and pulmonary artery
in some adults. Of particular interest are the distributions of velocity and
pressure in the two arteries, as well as the shear stress at the cardiovascular
walls, during a cardiac cycle in asymptomatic patients. Different
hypotheses are tested by introducing different boundary conditions. The
main results are based on the assumption that the flow in asymptomatic
PDA patients is similar to the flow in healthy adults.
The thesis also considers the low-Reynolds-number flow in a two-dimensional
geometry of a shunt between two vessels. An analytical solution is derived
by constructing piece-wise continuous functions that solve the biharmonic
equation. The method uses the orthogonality properties of Papkovich-
Fadle eigenfunctions. This work is applied to model the flow distribution
in a side-to-side anastomosis.
Date Issued
2011
Online Publication Date
2012-08-17T14:16:51Z
Date Awarded
2012-08
Advisor
Siggers, Jennifer
Mestel, Jonathan
Parker, Kim
Sponsor
Engineering and Physical Sciences Research Council ; Imperial College London
Publisher Department
Bioengineering
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)