# Development of Bernoulli's equation

• intervoxel
This is because the same pressure would be acting at every instant on all sides of the fluid particles. Therefore, the partial differential can be replaced by the total derivative in Eq. (1.28)In summary, according to the book, the changes of pressure as a function of time cannot accelerate a fluid particle because the same pressure would be acting on all sides of the particle at every instant. This means that the partial differential in the equation (1.28) can be replaced by the total derivative.
intervoxel
My book says:

$\frac{\partial V}{\partial s}\frac{ds}{dt}=-\frac{1}{\rho}\frac{dP}{ds}-g\frac{dz}{ds}$ (1.28)

The changes of pressure as a function of time cannot accelerate a fluid particle. This is because the same pressure would be acting at every instant on all sides of the fluid particles. Therefore, the partial differential can be replaced by the total derivative in Eq. (1.28)

$V\frac{dV}{ds}=-\frac{1}{\rho}\frac{dP}{ds}-g\frac{dz}{ds}$

I can't understand the explanation. Please, help me.

The changes of pressure as a function of time cannot accelerate a fluid particle.

You need pressure to change as a function of space (position) to impart acceleration. That is you must have a a pressure difference at the same time.

## 1. What is Bernoulli's equation?

Bernoulli's equation is a fundamental principle in fluid mechanics that describes the relationship between pressure, velocity, and elevation in a fluid flow.

## 2. Who developed Bernoulli's equation?

Bernoulli's equation was developed by the Swiss mathematician and physicist, Daniel Bernoulli, in the 18th century.

## 3. What are the assumptions made in Bernoulli's equation?

Bernoulli's equation assumes that the fluid flow is steady, incompressible, and has a constant density. It also assumes that there is no friction or energy loss within the system.

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

Bernoulli's equation has many practical applications, including calculating airspeed in aviation, designing water distribution systems, and understanding the flow of blood in the human body.

## 5. How is Bernoulli's equation derived?

Bernoulli's equation is derived from the conservation of energy principle, where the sum of the kinetic, potential, and flow energies of a fluid remains constant along a streamline. This leads to the equation P + 1/2ρv^2 + ρgh = constant, where P is pressure, ρ is density, v is velocity, g is the acceleration due to gravity, and h is elevation.

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