Cardiovascular System, Biological Fluids, Renal Function (CFR) CFR.10 CVS4 – The Bernouilli Principle in the CVS

Bernoulli Principle in the CVS

Learning Outcomes

State the Bernoulli Principle.
Write the Bernoulli equation and outline any assumptions made in its use.
Recall conservation of energy and the work-energy principle.
Explain how aneurysm development can be explained using the Bernoulli and continuity principles and Laplace principle
Explain (using the Bernoulli equation) the role of gravity in blood circulation.
Explain fluid entrainment with reference to the Bernoulli equation and outline some clinical examples of its use.

Bernoulli Effect (Principles)

Bernoulli's Equation states that the work done on a fluid as it moves from one place to another is equal to the change in its mechanical energy.
$\Delta W = \Delta KE + \Delta PE$ (work energy relation)

Continuity Equation and Force

When a liquid flows through a pipe with varying diameters, its velocity changes. This is described by the Continuity Equation: $A1v1 = A2v2$
The change in velocity implies acceleration $(a)$ , indicating a net force $(F = ma)$ acting on the liquid.
This force results from a pressure difference within the liquid $\Delta P$

Bernoulli Equation

The Bernoulli Equation expresses the conservation of energy for fluids:

$P + \frac{1}{2} \rho v^2 + \rho gh = Constant$

Where:

$P$ is the external pressure.
$\frac{1}{2} \rho v^2$ is the kinetic energy per unit volume.
$\rho gh$ is the fluid pressure (potential energy).

Therefore, the sum of pressure, kinetic energy per unit volume, and potential energy per unit volume is constant throughout the fluid.

Assumptions for Bernoulli's Equation

Bernoulli’s Equation can only be used given the following assumptions:

Fluid is incompressible (i.e. its density does not change).
Fluid is non-viscous (i.e. no appreciable frictional effects).
Fluid flow is streamline (laminar)
Flow velocity at a fixed point does not change rapidly (i.e. steady state conditions apply).
Bernoulli’s Equation is only an approximation

Bernoulli Effect

$P + \rho gh + \frac{1}{2} \rho v^2 = Constant$

If Velocity Increases → Pressure Decreases
If Velocity Decreases → Pressure Increases

This is known as the Bernoulli Effect.

Aneurysm Development

An aneurysm is a swelling in a blood vessel which may rupture – can be fatal.
These usually occur due to a weakness in the wall of the vessel which ‘bubbles’ due to the fluid pressure within the vessel.
They can occur throughout the body and may have several forms.

Aneurysm Growth

Once an aneurysm starts, it will continue to grow until:

It cannot grow anymore because of an obstruction (bone or tissue obstruction or stent).
The elastic tension in the vessel wall is large enough to contain it.
It bursts.

The fact that the aneurysm continues to grow is a direct consequence of the Continuity Principle and the Bernoulli Principle.

Mechanism of Aneurysm Growth

Blood flows through an artery, and fluid pressure acts perpendicular to the surface.
A region of weakness in the artery wall exists.
The fluid pressure causes the vessel wall to swell in the region of weakness.
As the cross-sectional area gets bigger, the velocity decreases (Continuity Equation).
As the velocity decreases, the fluid pressure increases (Bernoulli Principle).
The increased fluid pressure causes the aneurysm to swell even more.
This cycle repeats.

Cross-sectional area gets even bigger.
Blood velocity decreases even further.
Fluid pressure increases more.
Causes blood vessel to expand more.
Area gets even bigger.
etc…

Once the aneurysm starts, it will rapidly expand & grow uncontrollably and could rupture with fatal consequences.

Detected aneurysms may be treated by clamping the blood vessel or using a stent.

Stents

A stent is a cylindrical ‘tube’ which can be placed within the lumen of the blood vessel at the location of the aneurysm.

This will contain the blood (allowing it to flow through it), but stop the fluid pressure from acting directly on the vessel wall where the aneurism exists.
This image shows a stent being used for a blocked blood vessel.

Transient Ischaemic Attack

High blood velocity past the constriction causes low BP in left vertebral artery. A reverse (downward) blood flow can then result: so-called Subclavian Steal Syndrome results in TIA.

Continuity Eq.: $AV = Constant$ , Area decreases, Velocity increases.
Bernoulli Eq.: $P + \rho gh + \frac{1}{2} \rho v^2 = Constant$ , Velocity increases, Pressure decreases.

Bernoulli’s Principle and Vessel Disease

Vessels in circulatory system suffers one of two fates:

Aneurysm
Obstruction due to clot

Continuity Eq.: Area x velocity = Constant
Bernoulli's Equation: $P + \rho gh + \frac{1}{2} \rho v^2 = Constant$

Role of Gravity in Blood Pressure

Can reinterpret role of gravity with reference to Bernoulli Equation (for fixed velocity):

$P + \rho gh = Constant$

$\Delta P = \rho g \Delta h$

Role of Gravity in Circulation

Blood will gather in the lower body because of gravity, and it is necessary to get it back to the heart through the veins.

Consider the pressures within 3 main arteries (brain, heart, feet) when the body is in a reclined position:

Brain: 13.1 kPa
Heart: 13.3 kPa
Feet: 13.2 kPa

Consider the same person standing:

Brain (B): $PB \approx 9.3 kPa$ , height: $hB$
Heart (H): $PH \approx 13.3 kPa$ , height: $hH$
Feet (F): $P_F \approx 26.8 kPa$

Effect of External Accelerations

The value of $g$ is critical, and if the body is exposed to external accelerations (of magnitude $a m/s^2$ ), then the pressure in each of the arteries will change $[by \rho(g \pm a)h$ if the acceleration is in the up or down direction] – this is directly analogous to Effective Weight.

Normal circulation is severely affected when the body is exposed to external accelerations - and this can lead to a number of problems.

Example Calculation: Maximum Vertical Acceleration

When the human body is accelerated vertically, blood pressure in the brain will drop. Determine the maximum vertical acceleration that a human can withstand before losing consciousness

That is, determine the acceleration that would reduce the blood pressure in the brain to zero. Assume a typical systolic pressure of 16 kPa and that the base of the brain is 20 cm above the top of the heart. (Blood has a density of about 1035 kg/m3).

Solution:

$P{head} = P{heart} - \rho g'h$

As the pressure at the brain is assumed to be zero:

$P_{heart} = \rho g'h$

$g' = \frac{P_{heart}}{\rho h} = \frac{16000}{(1035)(0.2)} = 77.3 m/s^2 \approx 8g$

The person experiences a force of 8g.

Venous Return and Varicose Veins

Blood will gather in the lower body because of gravity, and it is necessary to get it back to the heart through the veins.

Normally (e.g., during walking) muscle action helps return venous blood from the legs one-way valves in blood vessels passing through muscle help in venous return.

One-way valves keep blood from flowing backward
Defective valves means pooling of blood in leg veins (Varicose Veins)
Varicose veins start because of problems with the blood vessel valves

Fluid Entrainment

Under static conditions with no air flow through the manometer system so that all parts has same height level are at atmospheric pressure

If high pressure air supplied. the center manometer has a higher level because of rapid air flow results lowered pressure in the constriction by Bernoulli effect

Fluid Entrainment (Venturi effect):

The use of the Bernoulli Principle to draw a second fluid into the initial fluid is called entrainment.

As the main fluid flows from the wider tube into the constricted tube, its flow speed increases because the cross- sectional area has decreased (Continuity Principle). Consequently, the fluid pressure in the constricted region decreases.

If the vertical tube is connected to a reservoir of other fluid, then this will be ‘entrained’ into the main fluid provided the pressure in the constricted region is lower than that in the fluid reservoir.

Venturi Mask

The process is based on the Bernoulli effect in which there is a relative reduction in pressure associated with the higher oxygen velocity

$P + \rho gh + \frac{1}{2} \rho v^2 = Constant$

Aspirators

If the pressure decreases below normal atmospheric pressure, then air will be ‘sucked in’ through the vertical open tube.

Vacuum Aspirator provides suction for surgical or dental applications.

A vacuum aspirator pump is essentially a pipe with a narrowing in it. As air flows through that narrowing, it speeds up and its pressure drops,

A tiny opening in the side of the narrowing allows water or air to enter the high-speed flow

$P + \rho gh + \frac{1}{2} \rho v^2 = Constant$