Derive Bernoulli's Equation From Euler's Equation Of Motion

Deriving Bernoulli's Equation from Euler's Equation of Motion

This article details the derivation of Bernoulli's equation from Euler's equation of motion for an inviscid, incompressible fluid flowing along a streamline. Understanding this derivation provides a deeper appreciation for the limitations and applications of Bernoulli's principle. This process involves several steps, each building upon the fundamental principles of fluid mechanics.

What is Euler's Equation of Motion?

Euler's equation is a fundamental equation in fluid dynamics that describes the motion of an inviscid (frictionless) fluid. It's a simplified form of the Navier-Stokes equations, neglecting viscous effects. The equation states:

ρ(∂v/∂t + v⋅∇v) = -∇p + ρg

where:

• ρ is the fluid density
• v is the fluid velocity vector
• t is time
• p is the pressure
• g is the acceleration due to gravity

Simplifying Assumptions for Derivation

To derive Bernoulli's equation, we make several crucial simplifying assumptions:

• Incompressible flow: The fluid density (ρ) remains constant.
• Steady flow: The fluid velocity doesn't change with time (∂v/∂t = 0).
• Irrotational flow: The fluid flow is characterized by zero vorticity (curl v = 0). This implies the existence of a velocity potential.
• Flow along a streamline: We consider the equation along a single streamline.

Step-by-Step Derivation

1. Steady-State Euler Equation: With steady flow (∂v/∂t = 0), Euler's equation simplifies to:

   ρ(v⋅∇v) = -∇p + ρg
   

2. Streamline Consideration: Along a streamline, the convective acceleration term (v⋅∇v) can be rewritten as:

   v⋅∇v = (1/2)∇(v²)
   

   This is a vector identity valid for streamline flow.

3. Substituting and Rearranging: Substituting the above into the simplified Euler equation gives:

   ρ(1/2)∇(v²) = -∇p + ρg
   

   Rearranging the terms:

   ∇p + ρg + (1/2)ρ∇(v²) = 0
   

4. Integrating along a Streamline: Since we are considering flow along a streamline, we can integrate the equation along that streamline:

   ∫∇p ⋅ dl + ∫ρg ⋅ dl + ∫(1/2)ρ∇(v²) ⋅ dl = 0
   

   where dl is a differential element along the streamline.

5. Simplification of Integrals: The integrals simplify significantly:

   • ∫∇p ⋅ dl = p₂ - p₁ (pressure difference between two points along the streamline)
   • ∫ρg ⋅ dl = ρg(z₂ - z₁) (change in potential energy due to gravity, assuming z is the vertical coordinate)
   • ∫(1/2)ρ∇(v²) ⋅ dl = (1/2)ρ(v₂² - v₁²) (change in kinetic energy)

6. Bernoulli's Equation: Combining these simplified integrals, we obtain Bernoulli's equation:

   p₁ + (1/2)ρv₁² + ρgz₁ = p₂ + (1/2)ρv₂² + ρgz₂
   

   This equation states that the sum of pressure energy, kinetic energy, and potential energy remains constant along a streamline for an inviscid, incompressible, steady, and irrotational flow.

Applications and Limitations

Bernoulli's equation is a powerful tool with numerous applications in various fields including aerodynamics, hydraulics, and meteorology. However, it's crucial to remember its limitations due to the simplifying assumptions made during its derivation. Real-world flows are often viscous, unsteady, and rotational, rendering Bernoulli's equation an approximation at best. Nevertheless, it provides valuable insights and a fundamental understanding of fluid flow behavior in many practical scenarios. Remember to always consider the limitations when applying Bernoulli's equation to real-world problems.