Fundamentals of Haemodynamics - Computational Hemodynamics: Theory, Modelling and Applications - page 74

by F x is greater than the pressure acting by

F +∆ . The pressure gradient for this

xx

case is a negative value (e.g. dp d − ) since it is decreasing from left to right. This

is given the term “favourable pressure gradient” in fluid dynamics. Conversely if
pressure is increasing from its initial position on the left to its latter position on the
right, then the pressure gradient has a positive value (e.g.

/

dp d + ) and is given the
term “adverse pressure gradient”. If the fluid is static, then the pressure on opposing
faces is the same and the pressure gradient is zero.

/

4.5.3 Laminar and Turbulent Flow

Let us imagine that fluid is made up of individual clusters of fluid particles moving,
sliding, and mixing together. If the fluid moves sufficiently slow without much mix-
ing between fluid particles, then the flow is considered laminar. If there is intense
and chaotic mixing between the fluid particles then the flow is considered turbulent.
To what extent of mixing do we consider whether a fluid is laminar or turbulent?
For flow in pipes the dimensionless Reynolds number, Re is used to help establish
this flow regime. The Reynolds number is a ratio of the inertial force, produced by
the flow in motion, to the viscosity of the fluid.

ρ
µ

UD

inertia force

h

Re = =

viscous force

where r is density, D h is the hydraulic diameter, U is the fluid velocity, and μ is the
viscosity. Physically the inertia force can be viewed as the ability of the fluid to
move and deform freely while the viscosity acts to hold the fluid together, resist-
ing any deformation. The inertia force is contributed by larger blood vessels, fluid
density, or high flow velocity ( ρ, U, D h ) which leads to increased fluid movement
and high energy flows that are turbulent. A greater viscosity will hold the fluid to-
gether preventing significant deformation, leading to laminar flow behaviour. If we
assume arteries as circular cylinders, having diameter of D h , then the flow regime is
indicated by the following Reynolds number ranges

• Re < 2000 the flow is laminar
• 2000 < Re < 4000 the flow is transitional
• Re > 4000 the flow is turbulent

The Re number range that classify whether a flow is laminar or turbulent also de-
pends on any disturbances and roughness that are present in the vessel and flow. For
example the presence of rough vessel walls or a stenosis can disrupt the orderly and
regular flow pattern of an otherwise laminar flow, and trigger the flow into turbu-
lence at lower Re numbers than expected.
If we were to inject dye into the middle of a flow stream in a pipe, and observe
its path, the dye streak for each flow regime would look like those in Fig. 4.6 . The