Short Answer:

Euler’s equation of motion is a fundamental equation in fluid mechanics that expresses the relationship between pressure, velocity, and external forces acting on a fluid in motion. It is derived by applying Newton’s second law of motion to a fluid element, assuming the flow is non-viscous, inviscid, and steady.

In simple words, Euler’s equation states that the rate of change of velocity of a fluid particle is equal to the sum of all forces acting on it, divided by its mass. It forms the basis for deriving Bernoulli’s equation and helps to describe the motion of ideal fluids under the influence of pressure and gravity.

Detailed Explanation:

Euler’s Equation of Motion

The Euler’s equation of motion describes how the velocity of a fluid particle changes under the action of various forces in an ideal, non-viscous flow. It is derived from Newton’s second law, which states that the net force on a particle is equal to the rate of change of its momentum.

This equation is valid for inviscid fluids (fluids without viscosity) and forms the foundation of fluid dynamics. Euler’s equation is used to analyze the motion of fluids in pipelines, nozzles, and open channels and to derive other important equations like Bernoulli’s theorem.

1. Basic Concept

According to Newton’s second law:

For a small fluid element, the forces acting on it are:

1. Pressure forces (acting due to fluid pressure).
2. Body forces (acting due to gravity or other external fields).

When these forces are combined, the total acceleration of the fluid element gives the motion equation, known as Euler’s equation of motion.

2. Derivation of Euler’s Equation of Motion

Let us consider a small fluid element of unit mass moving through a fluid in the x-direction.

• The pressure acting on the left face of the element (at point A) is  .
• The pressure acting on the right face of the element (at point B) is  .

The net pressure force on the element in the x-direction is:

The body force (such as gravity) acting on the element per unit mass in the x-direction is  .

The total acceleration of the fluid particle is:

where   is the velocity component in the x-direction.

Applying Newton’s second law,

and since mass per unit volume is  , we can write:

Dividing through by  :

Similarly, for the y and z directions:

 

These three equations represent the components of Euler’s equation of motion in three-dimensional space.

3. Vector Form of Euler’s Equation

Combining the above three components, we can write Euler’s equation in vector form as:

where,

•  = velocity vector of the fluid,
•  = total acceleration (also called material derivative),
•  = pressure gradient vector,
•  = body force per unit mass (like gravity).

This form is general and applies to fluid motion in all directions.

4. Simplified Form Along a Streamline

For steady flow along a streamline, the acceleration term becomes one-dimensional, and the equation reduces to:

Integrating this equation gives:

This is known as the Bernoulli’s equation, which is derived directly from Euler’s equation of motion. Hence, Bernoulli’s theorem is a special case of Euler’s equation under steady flow conditions.

5. Physical Meaning

The physical meaning of Euler’s equation is that the acceleration of a fluid particle is caused by two factors:

1. Pressure forces – The fluid moves from regions of high pressure to low pressure.
2. Body forces – External forces such as gravity affect the fluid motion by adding or reducing its velocity.

Euler’s equation describes how these two types of forces determine the velocity and pressure distribution within a moving fluid.

6. Assumptions in Euler’s Equation

Euler’s equation is derived under certain assumptions to simplify real fluid motion:

1. The fluid is ideal (non-viscous).
2. The flow is continuous and steady.
3. The fluid is incompressible (density is constant).
4. The flow takes place along a streamline.
5. Body forces are considered (mainly gravity).
6. No energy losses occur due to friction or heat.

These assumptions ensure that the equation applies only to ideal fluid motion.

7. Applications of Euler’s Equation

Euler’s equation is the foundation for several important applications in fluid mechanics:

1. Derivation of Bernoulli’s theorem.
2. Analysis of ideal flow in pipes and nozzles.
3. Determination of pressure variation in a moving fluid.
4. Basis for Navier-Stokes equations when viscosity is included.
5. Used in aerodynamics to describe airflow over wings.

Thus, it forms the theoretical backbone of fluid flow analysis in mechanical and aerospace engineering.

Conclusion

In conclusion, the Euler’s equation of motion expresses the dynamic behavior of a fluid by relating the pressure gradient, velocity, and body forces acting on a fluid particle. Derived from Newton’s second law, it assumes steady, incompressible, and inviscid flow conditions. Euler’s equation serves as a foundation for many advanced fluid flow equations, including Bernoulli’s equation and Navier-Stokes equations. It is essential in understanding the motion of ideal fluids and plays a crucial role in designing and analyzing fluid systems such as turbines, pumps, and pipelines.