Fundamentals of Haemodynamics - Computational Hemodynamics: Theory, Modelling and Applications - page 83

however, does not include a viscosity term, which means it neglects any frictional
losses, (e.g. neglects the influence of viscosity and the fluid is considered an ideal
fluid). Referring to Eq. 4.18, we see that as the contribution from the pressure en-
ergy increases, the kinetic energy must decrease to maintain energy conservation
(here we have neglected the influence of potential energy since changes in h are
small in this example). In fact the kinetic and pressure energy can be inter-converted
so that total energy remains unchanged.
The Bernoulli equation predicts flow velocity and explains the presence of local
high-velocity jets found in arterial disease states that are characterized by stenosis.
Consider steady flow in a stenosed artery with a fluid that has constant density
and negligible friction loss. In the regions before and after the stenosis, the flow
streamlines are parallel which and the pressure is constant, neglecting effects of
gravity over the small difference in height, h . As the flow enters the stenosis, the
cross-sectional area decreases forcing streamlines to converge and become closer
together. The flow velocity increases, and the pressure decreases. This produces a
local acceleration and jet, and a high wall shear stress in the stenosed area.
As the flow exits the stenosis, the streamlines now diverge as the cross-sectional
area increases; the flow velocity decreases, and the pressure increases. This pres-
sure gradient is positive in that

dP dx > and is known as an adverse pressure gra-
dient. Close to the wall where the flow is slowest, an adverse pressure gradient can
slow the velocity as far as to zero and even a negative (reverse) flow. When this
occurs the flow is said to be separated from the surface which significantly modifies
the pressure distribution along the wall surface.
Let us quantify the relationship between pressure, and velocity during the flow
through a stenosis. Mass conservation tells us that as the artery diameter decreases
the velocity increases to maintain a constant flow rate by an inverse proportion to
the square power, 2

/ 0

1 u ∝ . This means if the artery is occluded by half due to
cholesterol build-up and atherosclerosis, the velocity is increased by four times.
Since the kinetic energy is proportional by the square power to the velocity, 2

k ∝
there is a 16 fold increase in the kinetic energy. As a result, this energy can be inter
converted with the pressure energy (when we neglect the potential energy) which
means that pressure decreases by 16 fold. After the stenosis, the kinetic energy re-
turns to its original value when we consider no viscous effects. Bernoulli's equation
assumes a laminar flow and that the distance between the two points is short enough
so that viscous losses can be neglected. In reality viscous effects, and the likelihood
of turbulence, will reduce the post-stenosis total energy.
The effects of the potential energy become relevant when we consider the blood
flow throughout the body from feet up to the head (Fig. 4.14a ). If we consider a con-
stant kinetic energy, in that the blood is flowing at similar velocity, then the change
in pressure is related to the potential energy by

P −= − P ρ gy ( y )

(4.19)

head feet head feet