BERNOULLI'S EQUATION DERIVATION FROM EULER'S EQUATION

We were discussing the basic concept of streamline and equipotential line, dimensional homogeneity, Buckingham pi theorem, difference between model and prototype, basic principle of similitude i.e. types of similarity, various forces acting on moving fluid and model laws or similarity laws and in the subject of fluid mechanics, in our recent posts. 

Now we will go ahead to find out the Bernoulli’s equation from Euler’s equation of motion of a fluid, in the subject of fluid mechanics, with the help of this post.

Before going ahead, we will first see the recent post which will explain the fundamentals and derivation of Euler’s equation of motion. We will find out now the Bernoulli’s equation from Euler’s equation of motion of a fluid.

Euler’s Equation of motion

Bernoulli’s equation from Euler’s equation of motion could be derived by integrating the Euler’s equation of motion.

According to Bernoulli’s theorem.....

In an incompressible, ideal fluid when the flow is steady and continuous, the sum of pressure energy, kinetic energy and potential energy will be constant along a stream line.

Assumptions

Assumptions made for deriving the Bernoulli’s equation from Euler’s equation of motion is as mentioned here.

1. Fluid is ideal, i.e. inviscid and incompressible.

2. Fluid flow is steady, one-dimensional and uniform

3. Fluid flow is irrational

4. Forces which are considered are only pressure force and gravity force. Rest forces acting on fluid are neglected.

Let us recall the Euler’s equation of motion. We will integrate the Euler’s equation of motion in order to secure the Bernoulli’s equation.

Above equation is termed as Bernoulli’s equation.

We will now find out the Bernoulli’s equation for real fluid in the subject of fluid mechanics in our next post.

Do you have any suggestions? Please write in comment box.

Reference:

Fluid mechanics, By R. K. Bansal

Image Courtesy: Google

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