Fluid mechanics and laws of fluid dynamics are important for understanding blood flow and pressure in the cardiovascular system. Blood pressure and velocity are directly related through equations of continuity and Bernoulli's principle. Blood vessel resistance also impacts this relationship, such that increased resistance decreases flow velocity. Transient ischemic attacks occur when blood flow is temporarily blocked, potentially causing speech or limb weakness on the side of the brain affected. Loss of speech and left-sided weakness can result from vertebrobasilar insufficiency reducing blood supply to critical areas of the brainstem and left frontal lobe. Precise relationships govern how blood pressure, flow, and vascular resistance interact according to the principles of fluid mechanics.

1. Fluid mechanics
Biophysics case (5)
Pbl (G5)

2. Objectives
1. Fluid Mechanics And Laws
2. Blood Pressure
3. Blood Velocity
4. Relationship Between Blood Pressure And Blood Velocity
5. Relationship Between Blood Vessels And Blood Pressure
6. Transient Ischemic Attack (Tia)
7. Relationship Between Loss Of Speech Production And Left Sided Weakness
8. Relationship Between Blood Pressure And Blood Velocity According To Laws Of Fluid
Mechanics

3. Fluid
Fluid is a substance that has no fixed shape and yields
easily to external pressure; a gas or (especially) a liquid.
the two constituents of Fluid Mechanics are:
1- static Fluids
2- dynamic Fluid

4. Static Fluid :
Example :
Calculating Pressure A 60-kg person’s two feet cover an area of 500 cm2
A- Determine the pressure exerted by the two feet on the ground.
B- If the person stands on one foot, what will be the pressure under that foot?
A- The pressure on the ground exerted by the two feet is
P= F/A = mg /A = (60kg)(9.8m/s2
)/(0.050 m2
)= 12 × 103
N/m2
B- If the person stands on one foot, the force is still equal to the person’s weight, but
the area will be half as much, so the pressure will be twice as much:
24 × 103
N/ m2

5. Hydrostatic pressure
Hydrostatic pressure is the pressure that is exerted by a fluid at
equilibrium at a given point within the fluid, due to the force of gravity.
Hydrostatic pressure increases in proportion to depth measured from
the surface because of the increasing weight of fluid exerting downward
force from above.
Formula :
Unit: N/m2

6. The Difference in Pressure
Is how the pressure changes over a small change in depth (∆h) within a fluid
∆P = P2
- P1
+ pg (h2
- h1
)
Or
∆P = pg ∆h

7. Example:
Pressure at a faucet The surface of the water in a storage tank is 30m above a water
faucet in the kitchen of a house. Calculate the difference in water pressure between the
faucet and the surface of the water tank.
Answer : we assume the atmospheric pressure at the surface of the water in the storage
tank is the same as at the faucet . So, the water pressure difference between the faucet
and the surface of water in the tank is
∆P = pg ∆h = (1.0 × 103
kg/m3
)(9.8m/s2
)(30m)= 2.9 × 105
N/m2

8. Dynamics Fluid Laws
● The Equation of continuity :
Consider a situation where an incompressible fluid completely
fills a channel such as a pipe. Then if more fluid enters one
end of the channel, an equal amount must leave the other end.
Flow rate Q : the volume of fluid flowing past a point in a
channel per unit time .

9. Dynamics Fluid Laws
● The Equation of continuity :
it can be written also as :
Q=Av
v= the velocity
A= cross section area

10. Dynamics Fluid Laws
● The Equation of continuity :
If an incompressible fluid enters one end of a channel at a rate Q , it must leave the
other end at a rate Q2 , which is the same . Q1
= Q2
A1
v1
=A2
v2
that
means Av is constant; if at some point A decreases, v must increase .

11. Example :The Equation of continuity :
A water pipe leading up to a hose has a radius of 0.01 m. water leaves the hose at a
rate 5 x 10-5
m3
per second .
A) Find the velocity of the water in the pipe.
B) The hose has a radius of 0.005 m . what is the velocity of the water in the hose ?

12. Bernoulli’s Equation

13. Special cases of Bernoulli flow:

14. Bernoulli’s Principle
States; when the velocity of a fluid is high,
the pressure is low, and when the velocity is
low, the pressure is high.

15. An example of real life:

16. Blood viscosity
Blood viscosity is a measurement of the thickness and stickiness of a patient’s blood.
This important hemodynamic biomarker determines the amount of friction against the
blood vessels, the degree to which the heart must work, and the quantity of oxygen
delivery to the tissues and organs. It is a direct measure of the “flow ability” of blood
and is modifiable with existing naturopathic therapies.
Water and plasma are considered newtonian fluids. This means that their viscosity
remains the same whether they are flowing fast or slowly. Whole blood, on the other
hand, is a non-newtonian fluid, and its viscosity changes with its velocity. This point
becomes important clinically when monitoring blood viscosity.

17. Factors affecting blood viscosity
Five primary factors determine blood viscosity. These include
Hematocrit,
erythrocyte deformability,
plasma viscosity,
erythrocyte aggregation,
temperature.

18. Blood
Blood is a non-Newtonian fluid. Its viscosity changes depending on how much stress is
placed on it. Blood in veins is pretty thick, but move it around and the deep in the veins.
It’s a so-called “shear-thinning” liquid–the more blood is agitated the less viscous it
becomes.

19. Blood Pressure
A person's arterial blood pressures are usually measured with the help of an inflatable
cuff wrapped around the upper arm at the level of heart.
Physicians express blood pressures in torr, where 1 torr is the pressure exerted by a
column of mercury 1 mm high; it is equivalent to 133 Pa. The torr was formerly referred
to as the "millimeter of mercury", abbreviated mm Hg. The unit is named after
Evangelista Torricelli (1608-1647), the Italian Physicist who invented the barometer,
which measures atmospheric pressure. Average atmospheric pressure is 760 torr. In a
healthy person the systolic and diastolic blood pressures are, respectively, about 120
and 80 torr.

20. The relationship between blood vessel and
blood pressure
From physics when fluid moves inside the pipe that fluid will create
force on the walls of that pipe and that will create pressure.
In the same analogous way when blood on the blood vessel that blood
will create force on the walls of that blood vessel and that will create
blood pressure.

22. DEFINITION OF TRANSIENT ISCHEMIC
ATTACK
A transient ischemic attack (TIA) is a stroke that lasts only a few
minutes. It happens when the blood supply to part of the brain is
briefly blocked. Symptoms of a TIA are like other stroke symptoms,
but do not last as long.

23. CAUSES OF TIA
Loss of blood supply to portions of the brain may occur for a variety of
reasons. A blood vessel can become blocked, and blood supply to a
part of the brain is lost, or a blood vessel can leak blood into the brain
(brain hemorrhage). Most commonly however, the blood vessel is
blocked. The blockage can be caused by a blood clot that forms in the
blood vessel (thrombosis) or it can be caused by a clot or debris that
floats downstream

24. SYMPTOMS OF TIA
● Numbness or weakness, especially on one side of the body
● Confusion or trouble speaking or understanding speech
● Trouble seeing in one or both eyes
● Difficulty walking
● Dizziness
● Loss of balance or coordination
Most symptoms of a TIA disappear within an hour, although they
may last for up to 24 hours. Because you cannot tell if these
symptoms are from a TIA or a stroke, you should go to the hospital
right away.

25. TREATMENT FOR TIA
Blood thinners reduce the risk of heart attack and stroke by reducing the formation
of blood clots in your arteries and veins. You may also take a blood thinner if you
have
● An abnormal heart rhythm called atrial fibrillation
● Heart valve surgery
● Congenital heart defects
There are two main types of blood thinners. Anticoagulants, such as heparin or
warfarin (also called Coumadin), work on chemical reactions in your body to
lengthen the time it takes to form a blood clot. Antiplatelet drugs, such as aspirin,
prevent blood cells called platelets from clumping together to form a clot.

26. Relationship between speech and left sided weakness.
● The part of brain in the left frontal lobe ( Broca’s area) is the language-dominant
hemisphere has been shown to significantly affect use of spontaneous speech and
motor speech control.
● One side of the brain controls the opposite side of the body, if a stroke occurs in
the brain's right side, the left side of the body and the left side of the face will be
affected, which causes paralysis on the left side of the body.

27. What causes loss of speech and left sided weekness.
● The vertebral arteries are branches of the subclavian (upper extremity) arteries.
They arise, one on each side of the body, enter the skull via the hole at the base of
the skull called the foramen magnum.
● Inside the skull, the two vertebral arteries join up to form the basilar artery. The
basilar artery supplies arterial branches to the brain stem, cerebella and occipital
lobes, which control the autonomic nervous system.

28. Vertebrobasilar insufficiency. (VBI)
● It is a condition characterized by poor blood flow to the posterior (back) portion of
the brain, which is fed by the basilar artery.
● Blockage of these arteries occurs over time due to build-up of plaque.
● When there is a blockage, according to Bernoulli equation, velocity increases and
pressure decreases and blood is unable to supply oxygen and nutrient to brain and
a person loses ability to speak.

29. ● The vertebrobasilar arteries supply oxygen and glucose to the parts of the brain
responsible for consciousness, vision, coordination, balance and many other
essential functions. Both restricted blood flow and the complete blockage of it for
brain cells.
● So , when VBI or TIA occurs it affects the left side of the brain causing loss of
speech and left sided weakness.

30. SYMPTOMS of VBI
The symptoms of VBI vary depending on the severity of the condition. Some symptoms may last for a few
minutes, and some may become permanent. Common symptoms of VBI include:
● Loss of vision in part or all of both eyes
● Double vision
● Numbness or tingling
● Nausea and vomiting
● Slurred speech
● Loss of coordination, dizziness or confusion
● A drop attack — sudden generalized weakness

31. TREATMENT
Your doctor can recommend several different treatment options depending on the severity of your symptoms.
They will also recommend lifestyle changes, including:
● quitting smoking if you smoke
● changing your diet to control cholesterol levels
● losing weight if you’re overweight or obese
● becoming more active
-----------------------------------------------------------------------------------------
● Endovascular repair
● CT or MRI scans to look at the vessels at the back of your brain
● magnetic resonance angiography (MRA)
● blood tests to evaluate clotting ability
● an echocardiogram
● an X-ray of your arteries, which is called an angiogram

34. Relationship between blood pressure, blood flow & vascular
resistance.
● Blood pressure : is the pressure of the blood in the circulatory system.
● Blood flow: is the volume of blood flowing through a vessel, organ, or the entire
circulation in a given period.
● Vascular resistance: Systemic vascular resistance (SVR) refers to the resistance to
blood flow offered by all of the systemic vasculature, excluding the pulmonary
vasculature. Mechanisms that cause vasoconstriction increase SVR, and those
mechanisms that cause vasodilation decrease SVR. Although SVR is primarily
determined by changes in blood vessel diameters, changes in blood viscosity also
affect SVR.

35. It is clear that the higher the pressure exerted by the heart, the faster blood
will flow. This is an example of a direct or proportional relationship between
two quantities.
There is also another factor which controls the blood flow rate, and it is the
resistance of the blood vessels to blood flow. This resistance is simply due
to the width of the vessels - it's hard to push a lot of blood through a thin
tube! Thus, we have an inverse relationship between blood vessel resistance
and the blood flow rate - the higher the resistance, the slower the flow rate.
The relationship is expressed as follows:
Now let's see if this makes any sense in numbers. The usual pressure difference between the
left and right ventricles is about 100 mmHg. The normal cardiac output (the blood flow in the
above equation) is about 5 liters/minute. The total peripheral resistance is about 20
(mmHg*min/liters).

36. Stroke and Aneurysms
● An aneurysm is an excessive enlargement (point) of an artery caused by weakness
in the arterial wall.
● Stroke is the sudden death of brain cells due to lack of oxygen, caused by blockage
of blood flow or Aneurysm, which ruptures the blood vessel
● If A1 is the area at the constriction then,
A1>A2
From equation of continuity,
A1v1 = A2v2, hence speed decreases at aneurysm
● If P1 is the pressure at the sight, the from bernoulli’s principle, pressure increases.
Hence, resulting in rupturing.

37. Vessel Bifurcation
● Division of a blood vessel into two smaller and symmetrical vessels

38. Assume that,
● The bifurcation is symmetric, so the flows in the two daughter vessels are identical
(u1 = u2).
The blood is:
● incompressible, that is the density ρ of the blood (mass per unit volume)
● is constant, and inviscid, that is there are no viscous forces.
● steady , that is, it does not change with time
Find the mass flux in (mass per unit time). In this case it is m0 in the parent artery. Find
the mass flux out. In this case it is the sum of the two (identical) fluxes ˙m1 and ˙m1
out of the daughter arteries. The flux in must equal the flux out (this is mass
conservation). In this case ˙m0 = 2 ˙m1, which we can use to find the velocity u1 in the
daughter arteries.

39. Mass conservation implies,
For division of a large artery into many small arteries,
Q = Q1
+Q2
…...Qn
Q = NQ1
Av = NA1
v1

41. References
Book :
PHYSICS , third edition , 1988 , KANE.JOSEPH and STERNHEIM . MORTON , page 319- 320 , 324 -325

42. References
Websites:
● https://medlineplus.gov/transientischemicattack.html
● https://medlineplus.gov/bloodthinners.html
● https://www.khanacademy.org/science/physics/fluids/fluid-dynamics/a/what-is-v
olume-flow-rate
● http://math.arizona.edu/~maw1999/blood/pressure.html
● http://www.cvphysiology.com/Blood%20Pressure/BP021
● http://classes.midlandstech.edu/carterp/Courses/bio211/chap19/chap19.html
● https://www.merriam-webster.com/dictionary/blood%20pressure

43. Thank You !