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.

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Now we will go ahead to understand the fundamentals
and derivation of Euler’s equation of motion of a fluid, in the subject of
fluid mechanics, with the help of this post.

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**Euler’s
Equation of motion**

Euler’s equation of motion of an ideal
fluid, for a steady flow along a stream line, is basically a relation between velocity,
pressure and density of a moving fluid. Euler’s equation of motion is based on
the basic concept of Newton’s second law of motion.

When fluid will be in motion, there will
be following forces associated as mentioned here.

1. Pressure force

2. Gravity force

3. Friction force due to viscosity

4. Force due to turbulence force

5. Force due to compressibility

In Euler’s equation of motion, we will
consider the forces due to gravity and pressure only. Other forces will be
neglected.

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**Assumptions
**

Euler’s equation of motion is based on
the following assumptions as mentioned here

1. The fluid is non-viscous. Frictional losses
will be zero

2. The fluid is homogeneous and incompressible.

3. Fluid flow is steady, continuous and
along the streamline.

4. Fluid flow velocity is uniform over the
section

5. Only gravity force and pressure force
will be under consideration.

Let us consider that fluid is flowing
from point A to point B and we have considered here one very small cylindrical
section of this fluid flow of length dS and cross-sectional area dA as
displayed here in following figure.

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*
Let us think about the forces acting on the
cylindrical element *

*Let us think about the forces acting on the cylindrical element*

Pressure force PdA, in the direction of
fluid flow

Pressure
force [P + (∂P/∂S) dS] dA, in the opposite
direction of fluid flow

Weight of
fluid element (ρ g dA dS)

Image: Force on a fluid element

Let us consider that θ is the angle
between the direction of fluid flow and the line of action of weight of the
fluid element.

As we have mentioned above that Euler’s
equation of motion is based on the basic concept of Newton’s second law of
motion. Therefore, we can write here following equation as mentioned here

Net force over the fluid element in the
direction of S = Mass of the fluid element x acceleration in the direction S.

Above equation is termed as Euler’s
equation of motion.

We will now derive the Bernoulli’s equation from Euler’s Equation of motions, in the subject of fluid mechanics, in our next post.

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We will now derive the Bernoulli’s equation from Euler’s Equation of motions, in the subject of fluid mechanics, in our next post.

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

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**
Reference:**

Fluid mechanics, By R. K. Bansal