Dynamics of Blood Flow
26.3.12
Transport System
A closed double-pump system:
Left side of heart
Lung
Circulation
Right side of heart
Systemic
Circulation
Transport System
Branching of blood vessels
– Ateries branch into arterioles, veins into
venules
Arteries
Arterioles
Heart
Capillaries
Veins
Venules
Volume Flow Rate
The average flow from the heart is the
stroke volume (the volume of blood
ejected in each beat) x number of beats
per second. This is ~ 60 (ml/beat) x 80
(beats/min) = 4800 ml/min
Hagen–Poiseuille law or
Poiseuille law
In fluid dynamics, the Hagen–Poiseuille equation is a physical law
that states that for steady laminar flow of a Newtonian fluid through
a cylindrical tube, the flow rate is directly proportional to the
pressure drop, fourth power of radius of tube and inversely to the
length and viscosity of fluid
or
Where ∆P is the pressure drop
L is the length of pipe,
P1
ƞ is the dynamic viscosity,
Q is the volumetric flow rate and
r is the radius of the pipe
r
P2
L
DP= P1 - P2
Limitations of Poiseuille law
The assumptions of the equation are
• A long rigid cylinder with length much greater than the radius
• Fluid has constant viscosity and is incompressible
• Steady Laminar flow that is not pulsatile and turbulent
• The fluid velocity at the edges of tube is zero
Poiseuille law has certain limitations when applied to circulating blood
in vivo
• Blood vessels are not rigid tubes and are quite distensible so that
their size depends on the blood pressure within them as well as
upon the contraction of smooth muscles in the vessel walls
• Blood is a Non-Newtonian fluid and fluid viscosity is not
constant
• The flow is not steady but pulsatile in most parts of the vascular
bed
Blood volume flow rate Q
Liquid flows along the lumen of a rigid tube from a higher
to lower hydrostatic pressure
In the vascular system, the rate of blood flow
(volume/unit time) is proportional to the hydrostatic
pressure gradient (∆P) across the vessel and inversely
to the resistance (R) offered to its flow
Analogous to Ohms law for electrical circuits (I=V/R) we
can write
Blood flow rate Q = ∆P / R = (Pa-Pv) / R
Vascular Resistance
Vascular resistance is a term used to define the
resistance to flow that must be overcome to push blood
through the circulatory system.
The resistance offered by the peripheral circulation is
known as the systemic vascular resistance (SVR)
The systemic vascular resistance may also be referred
to as the total peripheral resistance
Resistance R
Resistance is dependent on the vessel’s dimensions and
the viscosity of blood
From Poiseuille law,
The resistance decreases rapidly as r increases
R = ΔP/Q = 8 L η / π r4
A narrowing of an artery leads to a large increase in the
resistance to blood flow because of 1/ r4 term
Vasoconstriction (i.e., decrease in blood vessel
diameter) increases SVR, whereas vasodilation
(increase in diameter) decreases SVR
Peripheral resistance can be equated to DC resistance
in electrical circuits
Arrangement of vessels also determines resistance.
When the vessels are arranged in series, the total
resistance to flow through all the vessels is the sum of
individual resistances, whereas when they are arranged
in parallel the reciprocal of the total resistance is the sum
of all the reciprocals of the individual resistance
Less resistance is offered to blood flow when vessels are
arranged in parallel rather than in series
Volume Flow Rate
Often convenient to define a resistance, R
to flow, such that DP=QR
Series
Parallel
R1
R2
R3
DP1
DP2
DP3
DP= DP1 + DP2 + DP3
=QR1+QR2+QR3
=QR
\R=R1+R2+R3
R1,Q1
R2,Q2
Q=Q1+Q2
=DP/R1+DP/R2
=DP/R
\1/R=1/R1+1/R2
Resistances in series add
directly while resistances in
parallel add in reciprocals
Arteries, arterioles, capillaries,
venules and veins are in general
arranged in series with respect to
each other. However, the
vascular supply to the various
organs
and the vessels e.g.
capillaries within an organ are
arranged in parallel
Right and left sides of the heart which
are connected in series. Also seen are
the various systemic organs receiving
blood through parallel arrangement of
vessels
Rate of blood flow
Blood leaves heart at ~ 30 cm/s
In capillaries, flow slows to ~ 1mm/s
– Surprising - continuity should imply higher
flow
Equation of continuity, Bernoulli
effect
a1 and a2 are areas of cross
section and v1 and v2 are
velocities
If cross sectional area is large,
velocity is low and pressure is
high
If cross sectional area of pipe is
small, velocity is high and
pressure is low
Cross sectional area of various
blood vessels
Linear velocity of blood (cm/s)
With cross sectional area of 2.5 cm2 ,linear velocity of
blood in aorta is 22.5cm/s
On the other hand, in capillaries with cross sectional
area of 2500 cm2, linear velocity of blood is simply
0.05cm/s
Linear velocity of blood (cm/s)
Hence aorta has smallest
cross sectional area but the
mean flow velocity is highest
Each capillary is tiny, but
since the overall capillary
bed contains many billions
of vessel, it has total cross
sectional
area
several
hundred times that of the
aortaand hence the mean
blood flow velocity falls
several folds
Vessel cross sectional area versus velocity of blood flow
Vessel cross sectional area vs
velocity of blood flow
To understand the effect of cross sectional area on flow
velocity, a mechanical model has been suggested
Here a series of 1cm diameter balls are depicted as
being pushed down a single tube. The tube branches
into narrower tubes. Each tributary tube has a area of
cross section much smaller than that of the wider tube
Suppose in wide tube each ball moves at 3cm/min . This
means 6 balls leave the wide tube per minuteand enter
narrower tubes
Obviously then these 6 six balls must leave the narrower
tubes per minute. This means each ball is moving at a
slower speed of 1cm/min
Vessel cross sectional area vs
velocity of blood flow
Special features of Blood Flow
Fahreus-Lindqvist Effect:
Relative viscosity of water, serum or plasma is not
altered when they are made to flow through tubes of
different sizes
But the relative viscosity of blood is altered when it
passes through tubes of different sizes i.e. blood flow in
very minute vessels exhibit far less viscous effect than it
does in large vessels. This is called Fahreus-Lindqvist
Effect
This effect is caused by alignment of red blood cells as
they pass through vessels