BERNOULLI'S EQUATION FOR REAL FLUIDS

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 in the subject of fluid mechanics, in our recent posts.

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

Before going ahead to find out the Bernoulli’s equation for a real fluid, we will first see the recent posts which will explain the fundamentals and derivation of Euler’s equation of motion and Bernoulli’s equation too.

Let us find the “Euler’s Equation of motion” and “Bernoulli’s equation from Euler’s equation”.

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.

Let us find out the Bernoulli’s equation for real fluid

As we have discussed above various assumptions during deriving the Bernoulli’s equation, such as fluid will be ideal, i.e. inviscid and incompressible. In reality, all real fluid will be viscous and will surely offer some resistance to flow.

Therefore, there must be some losses in fluid flow and we will have to consider these losses also during application of Bernoulli’s equation.

Therefore Bernoulli’s equation for real fluid between two points could be mentioned as here.

We will now find out the Basic principle of venturi meter in the subject of fluid mechanics in our next post.

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