It is importatnt

1. Dr Patrick Geoghegan
Caro 2011 “The Mechanics of Circulation –
2nd Edition” Cambridge Press Chapters 4,
5, 10, 12
FEA/CFD for
Biomedical Engineering
Week 5/6: Biofluid
Mechanics (Arteries)

2. Revision

3. Characteristics of blood flow in humans
• Pulsating flow, repetition from 1 to 3 beats/sec
• Not necessarily laminar flow
• Short entrance lengths
• Branching
• Reynolds numbers usually below 2500, non-turbulent flow
• Very complicated flow patterns

4. Arteries and veins in the body

5. Physical dimensions of arteries and veins
Data taken from Caro et al. (1974)

6. Velocity parameters in arteries and veins

7. • A force acting on the surface
of a fluid can be resolved into
normal and shear forces.
• Stress = Force / Area
• Normal stress = is the fluid
pressure
• Unit is N/m²
What you may have forgot…….

8. • A fluid will exert a normal force on any boundary it is in contact
with. Since these boundaries may be large and the force may
differ from place to place it is convenient to work in terms of
pressure, 𝐩, which is the force per unit area.
Fluid Pressure

9. If we take the pressure to be atmospheric (pA) at
the surface of the tank then p at any depth is
This is Hydrostatic pressure
Hydrostatic Pressure
𝑝1𝐴 − 𝑝2𝐴 − 𝜌𝑔ℎ𝐴 = 0
𝑝1 − 𝑝2 = 𝜌𝑔ℎ Eq (1)
𝑝 = 𝑝𝐴 + 𝜌𝑔ℎ Eq (2)

10. • Equation can be used to measure
pressure difference
• two regions, containing different
density fluid
• Connected to the two arms of a U-
tube containing a liquid of known
density
• the levels of the liquid in the two
arms will differ, by a height h.
• The pressure difference between
the two regions is then
immediately given by the equation
to be ρgh,
• ρ is the liquid density.
Hydrostatic Pressure
𝑝1 − 𝑝2 = 𝜌𝑔ℎ

11. • Viscous flows: Flows in which the frictional effects are
significant.
• Inviscid flow regions: In many flows of practical interest, there
are regions (typically regions not close to solid surfaces) where
viscous forces are negligibly small compared to inertial or
pressure forces.
Viscous versus Inviscid Regions of Flow
The flow of an originally uniform fluid
stream over a flat plate
The regions of viscous flow (next to the
plate on both sides) and inviscid flow
(away from the plate).

12. Fundamental law - Conservation of Momentum
Firstly:
Momentum is simply mass x velocity
Conservation of Momentum:
Total momentum in a system remains the
same, though some objects may gain
momentum and others lose it, these
balance out. This is true without
exception
We often see small objects coming to a
dead stop- a car braking for example.
• It might seem that the momentum is
completely lost.
• In fact it’s transferred to the road, part of
the Earth, such a massive object that
the increase in velocity is imperceptible.
• Some of the kinetic energy is transferred
to the Earth, and some dissipated as
heat in the brakes

13. Conservation of Momentum
Fundamental Law
The rate of change of momentum
of a body is equal to the resultant
force acting upon the body, and
takes place in the direction of that
force. We can write this as:
𝐹 = ሶ
𝑚 𝑢2 − 𝑢1
• F: the force required to
accelerate the fluid in the
direction of flow.
• ሶ
𝑚 : mass flow rate
• u2:exit velocity from control
volume
• u1:entrance velocity to control
volume
https://www.youtube.com/watch?v=uW0nPporLA0

14. Conservation of Mass
ሶ
𝑚 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 𝜌1𝐴1𝑢1 = 𝜌2𝐴2𝑢2
A: Cross-sectional Area of the pipe
u: Mean velocity of the fluid flow
For incompressible flow ρ1 = ρ2 = ρ
Therefore 𝐴1𝑢1 = 𝐴2𝑢2
Fundamental Law of fluid flow: The mass of the fluid must be conserved

15. Bernoulli Equation
Bernoulli’s equation is one of the most important/useful
equations in fluid mechanics.
• Bernoulli Equation:
– An approximate relation between pressure, velocity, and
elevation
– Valid in regions of steady, incompressible flow where net
frictional forces are negligible.

16. Bernoulli Equation
𝑝1
𝜌
+
1
2
𝑢1
2
+ 𝑔ℎ1 =
𝑝2
𝜌
+
1
2
𝑢2
2
+ 𝑔ℎ2
Flow
Energy
Kinetic
Energy
Potential
Energy
𝑝1 +
1
2
𝜌𝑢1
2
+ 𝜌𝑔ℎ1 = 𝑝2 +
1
2
𝜌𝑢2
2
+ 𝜌𝑔ℎ2

17. Bernoulli Equation
Bernoulli’s equation has some restrictions in its applicability, they
are:
• Flow is steady;
• Density is constant (which also means the fluid is
incompressible);
• Friction losses are negligible.
• The equation relates the states at two points along a single
streamline, (not conditions on two different streamlines).
All these conditions are impossible to satisfy at any instant in
time! Fortunately for many real situations where the conditions
are approximately satisfied, the equation gives very good results.

18. Bernoulli Equation
1. Steady flow: The Bernoulli equation is applicable to steady flow.
2. Frictionless flow: Every flow involves some friction, no matter how small, and
frictional effects may or may not be negligible.
3. No shaft work: The Bernoulli equation is not applicable in a flow section that
involves a pump, turbine, fan, or any other machine or impeller since such
devices destroy the streamlines and carry out energy interactions with the fluid
particles. When these devices exist, the energy equation should be used
instead.
4. Incompressible flow: Density is taken constant in the derivation of the
Bernoulli equation. The flow is incompressible for liquids and also by gases at
Mach numbers less than about 0.3.
5. No heat transfer: The density of a gas is inversely proportional to
temperature, and thus the Bernoulli equation should not be used for flow
sections that involve significant temperature change such as heating or cooling
sections.
6. Flow along a streamline: The Bernoulli equation is applicable along a
streamline. However, when a region of the flow is irrotational and there is
negligibly small vorticity in the flow field, the Bernoulli equation becomes
applicable across streamlines as well.

19. Shear Force
• Because a fluid cannot resist the deformation force, it
moves, it flows under the action of the force. Its shape will
change continuously as long as the force is applied.
• The deformation is caused by shearing forces which act
tangentially to a surface.
• The force 𝐅 acting tangentially on a rectangular (solid lined)
element 𝐀𝐁𝐃𝐂.
• This is a shearing force and produces the (dashed lined)
parallelogram element 𝐀’𝐁’𝐃𝐂.