This document discusses blood flow and its properties. It covers topics like cardiovascular physiology, the physical properties of blood including viscosity, steady and oscillatory blood flow models like Poiseuille flow and the Windkessel model. It discusses concepts like entrance length, application of Bernoulli's equation, vascular resistance and branching effects. Unsteady flow models like the Wommersley equations are also covered. The goal is to understand blood flow to improve medical devices and understand cardiovascular disease development.

1. BLOOD FLOW
Barbara Grobelnik
Advisor: dr. Igor Serša

2. Introduction
The study of blood flow CONTENTS
behavior: Cardiovascular physiology
Physical properties of
• Improving the design of implants
blood
(heart valves, artificial heart) and
extra-corporeal flow devices  Viscosity
(blood oxygenators, dialysis Steady blood flow
machines)  Poiseuille’s equation
• Understanding the connection  Entrance effects
between flow characteristics and  Bernoulli’s equation
the development of cardiovascular Oscillatory blood flow
diseases (atherosclerosis,  Windkessel model
thrombosis)
January 2008 Blood Flow
 Wommersley 2

3. Cardiovascular Physiology
• HEART: atrium, ventricles mean diameter number of
[mm] vessels
• BLOOD VESSELS: aorta, aorta 19 - 4.5 1
arteries, arterioles, arteries 4 – 0.15 110.000
capillaries, veinules, veins arterioles 0.05 2.7 ∙106
capillaries 0.008 2.8 ∙109
right ventricle
 lungs  left
atrium
MAIN FUNCTIONS:
• to deliver oxygen and
nutrients to the cells
• to remove cellular wastes
left ventricle  and carbon dioxide
aorta  organs • to maintain organs at a
and tissues  right
atrium constant temperature and pH
January 2008 Blood Flow 3

4. Poiseuille flow
• Steady flow in a rigid cylindrical tube
– Pressure gradient Fp = 2π r ( p1 − p2 )δ r
– Viscous force Fv = −
∂
∂r
(2π rLµ
∂v
∂r
)δ r
The forces are equal and
opposite: p1 − p2
∂ 2 v 1 ∂v
+ + =0
∂r 2 r ∂r µL
p1 − p2
v (r ) = −r 2 + A ln r + B
v(r=R)=0 4µ L
v(r=0)≠∞ p1 − p2
v (r ) = (R 2 − r 2 )
4µ L
L
r R
p1 − p2
r Q = ∫ 2π v(r )rdr = π R 4 volume flow
8µ L
r2
0
v
v=
Q
= R2
p1 − p2 1
= v ( r = 0) =
vmax average
p1 p2 π R2 8µ L 2 2 velocity
January 2008 Blood Flow 4

5. Poiseuille flow - assumptions
• Newtonian fluid 
– in large blood vessels (at high shear rates)
• Laminar flow 
– Reynold’s numbers below the critical value of about
2000

• No slip at the vascular wall
– endothelial cells
x
• Steady flow
– pulsatile flow in arteries
x
• Cylindrical shape
– elliptical shape (veins, pulmonary arteries), taper
x
• Rigid wall
– visco-elastic arterial walls
x
• Fully developed flow
– entrance length; branching points, curved sections
January 2008 Blood Flow 5

6. Physical properties of blood
BLOOD =
plasma + blood cells
( 55%) (45%)
electrolyte Red blood cells (95%)
solution
White blood cells (0.13%)
containing
8% of Platelets (4.9%)
proteins
Reference values
RBC 1 μm PLASMA WHOLE
: BLOOD
density 1035 kg/m 3 1056 kg/m 3
8 μm
viscosit 1.3×10 -3 Pa 3.5 × 10 -3 Pa s
January 2008 Bloody s
Flow 6

7. Viscosity
• Viscosity varies with samples
– variations in species
– variations in proteins and RBC
• Temperature dependent
– decrease with increasing T
In small tubes the blood
• Blood viscosity has a very low value
because of a cell-free zone near
– a non-Newtonian fluid at low the wall.
shear rates (the agreggates of
RBC) Fahraeus-Lindqvist
effect
– a Newtonian fluid above shear
rates of0 50 s -1 dv / dr
τ = τ + Kc
– Casson’s equation
January 2008 Blood Flow 7

8. Fahraeus-Lindqvist Effect
Cell-free marginal layer The Sigma effect theory
model  velocity profile is not
 Core region μc , vc , 0rR- continuous
 small tubes ( N red blood
 region near the wall μ , v , R-r R
Cell-free plasma p p
cells move abreast)
μp , vp
r
μc , vc
R
∆p 1 d  dv   the volume flow is
− =  µr ÷
L r dr  dr  π∆p R 3rewritten
2 µL ∫
Q= r dr
 the volume flow 0
 N concentric laminae,
π R ∆p 1
4
Q= ( 1 − (1 − δ / R)4 (1 − µ p / µc ) ) each of thickness ε
8L µp
π∆p N π∆pR 4 1  ε 
Q= ∑(nε )ε = 8L µ 1 + R ÷
2 µL n =1
3
 
1/μ
1/μ
January 2008 Blood Flow 8

9. Entrance length
• The flow of fluid from a reservoir to a pipe
– flat velocity profile at the entrance point
– the fluid in contact with the wall has zero velocity (‘no slip’)
– retardation due to shearing adjacent to the wall
– boundary layer (where the viscous effects are present)
– acceleration in the core region to maintain the same volume of
flow
– parabolic velocity profile  FULLY DEVELOPED FLOW
d  dv 
Fvisc = µ  ÷ A(r2 − r1 ) viscous force
dr  dr 
 - boundary layer
µU thickness at z
Fvisc = 2 A(r2 − r1 )
δ U - free stream
velocity
U2
Fi = ρ aV = ρ A(r2 − r1 ) inertial force
z
* a=U/t=U/(z/U)
January 2008 Blood Flow 9

10. Entrance length
U2 U
• equating the viscous and inertial force ρ
z
= kµ 2
σ
k – proportionality constant derived from experiments, approximately 0.06
• the boundary layer thickness
µz
δµ
ρU
• the entrance length (when =D/2 the flow Pulsatile flow –
the entrance
becomes fully established)
Uρ length fluctuates
z0 = kD 2
µ
The above derivation is valid only for
the flow originating from a very large
reservoir, where the velocity profile
at the entrance point is relatively flat.
In other cases, the entrance length
is shorter.
January 2008 Blood Flow 10

11. Application of Bernoulli Equation
Bernoulli
equation p + ρ gz + 1 ρ v 2 = const.
2
• Flow trough stenosis • Flow in aneurysms
p1 A1
A1 v1 p2, v2, A2 p1v p2, v2, A2
A1v1 = A2v2 1
– v2 > v1 – v2 < v1
– p 2 < p 1 : caving or closing – p 2 > p 1 : expansion and
of the vessel bursting of the vessel
– decrease in v 2
– reopening of the vessel – caused by the weakening
– fluttering of the arterial wall
January 2008 Blood Flow 11

12. Vacular resistance and branching
• Vascular resistance • Succesive branching:
∆p – Increase in the total cross-
Rv = section area
Q
8µ L
Rv =
– for Poiseuille flow π R4
– major drop in the mean
pressure in arterioles (60
mmHg)
autonomic nervous system
controls muscle tension
arterioles distend or contract
2
v1 nR2
= 2 =d
Mean pressure values [mmHg]: v2 R1
– d A 1= n A 2:
∆p1 nR2 d 2
4
- arteries 100 = 4 =
∆p2 R1 n
- capillaries 30-34 at arterial end,
12-15 at venous end velocity decreases,
n≥2
pressure gradient
average d=1.26 increases
January 2008 Blood Flow 12

13. Turbulent Flow
• Reynolds number
v ρD for flow in rigid straight
Re = critical value Re > 2000 cylindrical pipes
µ
• Flow in the circulatory system is
normally laminar
• Flow in the aorta can destabilize
during the deceleration phase of late
systole
– too short time period for the flow to
become fully turbulent
• Diseased conditions can result in
turbulent blood flow
– vessel narrowing at atherosclerosis,
defective heart valves
– weakening of the wall, progression of the
disease
January 2008 Blood Flow 13

14. Unsteady flow models
• The pressure pulse:
– generated by the contraction of the left ventricle
– travels with a finite speed through the arterial wall
– change in a shape due to interaction with reflected waves
• Windkessel model
– the arteries: a system of interconnected tubes with a
storage capacity
– distensibility Di = dV/dp
– Inflow – Outflow = Rate of Storage A typical pressure pulse curve.
p − pV dV dp
Q (t ) − = = Di b
RS dt dt
systole diastole
SYSTOLE DIASTOLE
Q=Q 0 , 0  t  t s Q=0, t s  t  T
ts T
p(t)  b-(b-p 0 )e -t/a p(t)  e (T-t)/a p0
January 2008 Blood Flow 14

15. Wommersley equations
• The equation for the motion of a
viscous liquid in a cylindrical tube
(general form):
∂ 2 w 1 ∂w 1 ∂p ρ ∂w
+ + =
∂r 2 r ∂r µ ∂z µ ∂t
• Arterial pulse = periodic function iωt
∂p
 the sum of harmonics ∂z ∑
= Ae
The flow velocity pulse and the arterial
pressure pulse (femoral artery of a dog). • The solution:
A * R2  J 0 (α yi 3/ 2 )  iωt
w= 1 − 3/ 2 
e
i µα 2  J 0 (α i ) 
– Wommersley number 
– J 0 (xi 3/2 ) is a Bessel function of the
α = R (ωρ / µ ) first kind of order zero and
complex argument
– y=r/R
January 2008 Blood Flow 15

16. The role of Wommersley number
 - unsteady inertial forces vs. viscous forces
(viscous forces dominate when   1) 10-3    18
The velocity profiles for capillaries aorta
the first four harmonics : 3.34 4.72 5.78 6.67
resulting from the
pressure gradient cos ωt
 Parabolic profile is
not formed
 The laminae near
the wall move first
 Solid mass in the
centre
 Increase in :
flattening of the central
region, reduction of
amplitude and reversal
of flow at the wall
January 2008 Blood Flow 16

17. The sum of harmonics
y=r/R
 Parabolic shape in
the fast systolic rush
 Phase lag between
the pressure gradient
and the movement of
the liquid
 The reversal begins
in the peripheral The time dependence of
laminae (the point of velocity at different distances y.
flow reversal: 25°
after the pressure gradient)
The first four harmonics summed  The peak forward
together with a parabola (representing  Back flow: and backward
the steady forward flow). harmonics are out of velocities: 165
phase and the profile cm/s at 75° 35
is flattened cm/s at 165°
January 2008 Blood Flow 17

18. Conclusion
• What have we learned? • Why am I interested in
blood flow?
future experiment:
dissolving blood clots
under physiological
conditions
PULSATILE FLOW
Artificial heart.
- basic equations of blood
flow
January 2008 Blood Flow 18

19. Non-Newtonian fluid behavior
 Power law fluid  Bingham plastic
 Cassons fluid
Velocity profiles in a round rigid tube.
January 2008 Blood Flow