Short Answer:

Bernoulli’s equation is obtained from Euler’s equation by applying the assumptions of an inviscid (no viscosity), incompressible fluid and steady flow, and then projecting Euler’s momentum balance along a streamline. The result is the familiar relation that the sum of pressure energy, kinetic energy per unit mass, and potential energy per unit mass is constant along a streamline:

In simple words, starting from Newton’s second law for a fluid (Euler’s equation) and integrating the acceleration term along a streamline gives Bernoulli’s equation. It expresses conservation of mechanical energy per unit mass for ideal steady flow and is widely used in engineering to relate pressure, velocity and elevation.

Detailed Explanation :

Derive Bernoulli’s equation from Euler’s equation

1. Start with Euler’s equation (momentum balance)
   Euler’s equation for an inviscid fluid (no viscous stresses) including body force due to gravity is written in vector form as:

where

•  is fluid density,
•  is the velocity vector,
•  is pressure, and
•  is gravitational acceleration vector (usually  if z is upward).

This equation states that mass times acceleration (left side) equals the net force per unit volume (pressure gradient plus body force).

2. Make standard assumptions for Bernoulli derivation
   To derive the classic Bernoulli equation we assume:

• Flow is steady:  .
• Fluid is incompressible with constant density  .
• Fluid is inviscid (Euler’s equation already assumes this).
• We will first derive the relation along a streamline; later we note the extra condition needed for the relation to hold everywhere.

With steady flow Euler reduces to:

3. Project Euler along a streamline
   Take a small displacement  along a streamline (tangent to velocity), with unit tangent  so that  . The velocity vector is parallel to  ,  . Dot Euler’s equation with   (or equivalently with  ) to get the scalar relation along the streamline:

4. Simplify the convective term
   A key identity for the convective acceleration along the streamline is

Intuitively, this says that the change in kinetic energy per unit mass along the streamline equals the convective acceleration projected along the path. Thus the left-hand side becomes  .

5. Write the pressure and gravity terms
   The pressure term projects to  . For gravity, if we take  as the vertical coordinate (positive upward) and  , then  . So  .
6. Combine and rearrange
   Plugging these into the dotted Euler equation:

Divide through by  :

Rearrange:

Integrate along the streamline to obtain Bernoulli’s equation in the most common form:

Multiplying by   gives the head form used in hydraulics:  .

7. Remarks on domain of validity

• The derivation used steady, inviscid, and incompressible assumptions and was integrated along a streamline. Under these assumptions the result is valid along each streamline.
• If the flow is additionally irrotational (vorticity zero everywhere), the constant is the same for all streamlines, so Bernoulli’s equation holds throughout the flow field (not only along one streamline).
• For unsteady flows, keeping   leads to an unsteady Bernoulli form that contains a time derivative of a velocity potential (if potential exists), i.e.   for potential flows.
• For viscous flows, energy is dissipated by friction; Bernoulli must be modified by adding a head-loss term to account for dissipation.

8. Physical interpretation
   Bernoulli’s equation expresses conservation of mechanical energy per unit mass along a streamline: pressure term  (pressure energy per mass), kinetic term  (kinetic energy per mass), and potential term   (gravitational potential energy per mass). If one term increases the others must decrease so the sum remains constant (neglecting losses).
9. Typical engineering form and use
   Engineers commonly use the head form:

and include a head loss   for real flows:

This modified form is used for pipes, nozzles, and many practical analyses.

Conclusion

Starting from Euler’s momentum equation and assuming steady, incompressible, and inviscid flow, dotting the momentum equation along a streamline and integrating yields Bernoulli’s equation:   constant along a streamline. With the stronger assumption of irrotational flow the constant is global and Bernoulli holds throughout the flow. The equation gives a compact expression of energy conservation for ideal flows and is a cornerstone of practical fluid mechanics.