# Application of Director Theory to Models of Flow in Straight and Curved Pipes

• Mikaela J Webster

Student thesis: Doctoral ThesisDoctor of Philosophy (PhD)

### Abstract

This thesis is about the application of director theory to modelling fluid flow in pipes
and whether it could be beneficial to the field of cardiovascular modelling. Director
theory was first developed in the field of solid mechanics, primarily by the Cosserat
brothers, but was later applied in fluid dynamics by Green and Naghdi.
Director theory simplifies the solution of the full 3D Navier-Stokes equations in thin
pipe-like geometries with arbitrary variation of thickness and orientation. Instead of
solving the equations pointwise, the theory solves integrated versions of the equations
over the cross-section of the pipe. Some information about the flow properties in the
cross-section are retained by the weighting functions of the velocity directors (vectors)
that depend on the cross-sectional coordinates. It is this property of director theory that
is thought could provide more accuracy than the classical 1D models for cardiovascular
modelling.
This thesis considers the development of director theory in regards to modelling
fluid flow through various pipe geometries of increasing complexity. The accuracy of the
model solutions are then assessed against full 3D computational simulations. The models
considered include straight pipes of constant and varying radius, pipes with constant
curvature of constant and varying radius and pipes of varying curvature.
With the director theory approach, the velocity of the fluid is approximated by an
expansion of directors (vectors dependent on the co-axial coordinate) multiplied by weighting functions (which depend on the cross-sectional coordinates).
The central aim of this thesis is to modernise the presentation of director theory
in the application of fluid mechanics to improve accessibility and help promote future
work in the topic. In addition, a novel approach to deriving the system of equations for
fluid flow in curved pipes is presented. This novel approach is more intuitive within the
field of fluid dynamics as it is derived directly from the Navier-Stokes equations, while
achieving the same end results as the method put forward by Green and Naghdi.
Date of Award 28 Sept 2021 English University of Bristol Alberto M Gambaruto (Supervisor) & Alan R Champneys (Supervisor)

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