Bernoulli equation

Bernoulli’s equation

The Bernoulli’s equation shows how the pressure and velocity vary from one point to another within a flowing fluid. It says that the total mechanical energy of the fluid is conserved as it travels from one point to another, but some of this energy can be converted from kinetic to potential energy and its reverse as the fluid flows. In fluid dynamics, which is the study of fluid motion and its associated external forces and internal resistances, the Bernoulli’s equation relates pressure to energy. Bernoulli’s equation states that in an ideal fluid, when the flow is uniform and continuous, the sum of the pressure, kinetic energy, and potential energy of a fluid is constant.

How does it work?

The Bernoulli’s principle states that the sum of the velocity and the kinetic energy of a fluid flowing through a tube is constant. A greater energy associated with pressure in the fluid corresponds to lower pressure and higher energy, because a high pressure means that the fluid can exert higher force. The decrease in pressure when the fluid velocity increases is called the Bernoulli effect.

So, the Bernoulli’s equation states:

Energy due to        Kinetic energy       Potential energy       Energy due to        Kinetic energy      Potential energy pressure at point 1       at point 1                  at point 1        pressure at point 2        at point 2                at point 2

P₁              +          ½𝜌v₁²        +            𝜌gh₁           =              P₂              +        ½𝜌v₂²       +              𝜌gh₂

½𝜌v² = kinetic energy per unit volume

𝜌gh = gravitational potential energy per unit volume

This diagram shows a blood vessel that is narrowed and returning to its normal diameter. In the narrowed area(obstruction of flow) as the diameter decreases, the velocity increases, because the flow is the product of mean velocity and the vessel’s cross-sectional area(which is directly related to the diameter; A= 𝜋r²). If the diameter is reduced by half, the velocity is increased by 4, making kinetic energy increase accordingly. Assuming that the energy is conserved, the increase in kinetic energy results in the decrease of potential energy. When the vessel returns to its original diameter, the kinetic energy will also return back to its original value, thus flow is conserved.

One disadvantage of the Bernoulli’s equation is that it can be complicated for routine clinical use, however, a simplified version (𝝙P=4v²) tells us the difference in pressure between the right ventricle and the right atrium, given that there is no heart valve disorder.

Why is the Bernoulli’s equation important in clinical medicine?

In echocardiography, the Bernoulli’s principle can be applied when interpreting blood flow, which can describe a decrease in localized pressure produced by high flow rate near blockages as mentioned above with an illustration. For clinical medicine, the simplified equation allows for an easy estimation of pressure gradients from velocity.

How was it developed? The Bernoulli’s equation was derived by Daniel Bernoulli(1700-1783) in the 1730s. Daniel Bernoulli was a Swiss mathematician and physicist.

He published a book called Hydrodynamica(1738) where he explains fluid mechanics, with the idea of conservation of energy in mind,

and hydrodynamics. The book describes the nature of hydrodynamic pressure in fluid flow which later became to be known as the Bernoulli’s principle.

References:

Giordano, Nicholas; College Physics: Reasoning and Relationships; 2nd Ed.; 2013; 978-0-8400-5819-5

Rama, Durhaiah D.; Fluid Mechanics and Machinery; 1st Ed.; 2002; 81-224-1386-2

Griffin Brian P.; Manual of Cardiovascular medicine; 4th Ed.; 2013; 978-1-4511-3160-4

Klabunde Richard E.; http://www.cvphysiology.com/Hemodynamics/H012.htm