# Different forms of Bernoulli's equation

• Jak243
In summary, the conversation discusses a version of the Bernoulli equation for blood flow in the aorta, which includes a flow acceleration term and a viscous friction term. The origin of the flow acceleration term is explained as a correction for unsteady state flow, while the viscous friction term accounts for drag at the walls of the conduit. The conversation also mentions the derivation of this form of the Bernoulli equation from conservation of energy.
Jak243
I was reading some textbooks on doppler echo for medicine and came across this version of the bernoulli equation for blood flow in the aorta. $$P_{1} - P_{2}= 1/2 \rho (v{_{2}}^{2}- v{_{1}}^{2}) + \rho \int_{1}^{2} \frac{\overrightarrow{dv}}{dt}\cdot \overrightarrow{ds} + R(\overrightarrow{v})$$ They call the second term on the rhs the flow acceleration and the last term on the rhs is the viscous friction. Can someone help explain the origin of the flow acceleration term and also offer a derivation of this form of the bernoulli equation from conservation of energy. Most simple derivations of the bernoulli equation only yield the familiar C = 1/2 ρ v^2 + ρgh + p

Jak243 said:
I was reading some textbooks on doppler echo for medicine and came across this version of the bernoulli equation for blood flow in the aorta. $$P_{1} - P_{2}= 1/2 \rho (v{_{2}}^{2}- v{_{1}}^{2}) + \rho \int_{1}^{2} \frac{\overrightarrow{dv}}{dt}\cdot \overrightarrow{ds} + R(\overrightarrow{v})$$ They call the second term on the rhs the flow acceleration and the last term on the rhs is the viscous friction. Can someone help explain the origin of the flow acceleration term and also offer a derivation of this form of the bernoulli equation from conservation of energy. Most simple derivations of the bernoulli equation only yield the familiar C = 1/2 ρ v^2 + ρgh + p
The simple version you wrote is for a steady state flow in which the fluid velocity at any location in the fluid is not changing as a function of time. In the first equation, the 2nd term on the rhs is a correction to allow for the velocity to be changing with time. Google "unsteady state Bernoulli equation." The third term corrects for viscous drag at the walls of the conduit. In blood flow applications, particularly capillaries, the term can dominate. Google "viscous flow in a pipe."

Have you looked into trying to derive this yourself at all? I mean, just from looking at the terms you ought to be able to get a sense of what it is trying to do. The Bernoulli equation is essentially an energy balance that usually deals with energy due to static pressure, kinetic energy (dynamic pressure), and sometimes gravity. Clearly this neglects gravity, but the second RHS term ought to look pretty familiar to you in terms of an energy balance, and the last term is basically a catchall term for viscous terms.

## 1. What is Bernoulli's equation?

Bernoulli's equation is a fundamental equation in fluid dynamics that describes the relationship between the pressure, velocity, and height of a fluid moving in a steady flow.

## 2. What are the different forms of Bernoulli's equation?

There are three main forms of Bernoulli's equation: the classical form, the integral form, and the differential form. The classical form is the most commonly used and is expressed as P1 + (1/2)ρv1^2 + ρgh1 = P2 + (1/2)ρv2^2 + ρgh2, where P is pressure, ρ is density, v is velocity, g is acceleration due to gravity, and h is height. The integral and differential forms are used for more complex flow situations.

## 3. How are the different forms of Bernoulli's equation derived?

The classical form of Bernoulli's equation can be derived from the principle of conservation of energy by considering the work done by pressure, gravity, and kinetic energy on a fluid element. The integral form is derived by applying the divergence theorem to the classical form, while the differential form is derived using the continuity equation and Euler's equation of motion.

## 4. What are the limitations of Bernoulli's equation?

Bernoulli's equation is only valid for ideal fluids, which are non-viscous, incompressible, and have steady flow. It also assumes no energy losses due to friction or heat transfer. Additionally, it is only applicable to streamline flow, where the velocity and pressure are constant along a streamline.

## 5. How is Bernoulli's equation used in real-world applications?

Bernoulli's equation is used in various engineering applications, such as designing aircraft wings, calculating water flow rates in pipes, and understanding the flow of blood in the human body. It is also used in weather forecasting and studying ocean currents. In all these applications, Bernoulli's equation helps in predicting changes in pressure, velocity, and height of fluids in motion.

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