State Bernoulli’s theorem for steady flow of an incompressible fluid. Derive an expression for Bernoulli’s equation from first principles (Starting from Euler’s Equation) and state the assumptions made for such a derivation.
The correct answer and explanation is:
Correct Answer:
Bernoulli’s Theorem:
For a steady, incompressible, and non-viscous fluid flowing along a streamline, the total mechanical energy (pressure energy, kinetic energy, and potential energy) per unit volume remains constant.
Mathematically: P+12ρv2+ρgh=constantP + \frac{1}{2} \rho v^2 + \rho g h = \text{constant}
Where:
- PP = Pressure energy per unit volume
- ρ\rho = Fluid density
- vv = Fluid velocity
- gg = Acceleration due to gravity
- hh = Height above a reference level
Derivation of Bernoulli’s Equation from Euler’s Equation:
Euler’s Equation for Inviscid Flow Along a Streamline: dPρ+g dz+v dv=0\frac{dP}{\rho} + g\,dz + v\,dv = 0
Where:
- dPρ\frac{dP}{\rho} = Change in pressure head
- g dzg\,dz = Change in gravitational potential head
- v dvv\,dv = Change in kinetic energy head
Integrating each term: ∫dPρ+∫g dz+∫v dv=0\int \frac{dP}{\rho} + \int g\,dz + \int v\,dv = 0 ⇒Pρ+gz+v22=constant\Rightarrow \frac{P}{\rho} + g z + \frac{v^2}{2} = \text{constant}
Multiply through by ρ\rho to get Bernoulli’s Equation: P+12ρv2+ρgz=constantP + \frac{1}{2} \rho v^2 + \rho g z = \text{constant}
Assumptions Made:
- Steady Flow – Properties of the fluid (velocity, pressure) at a point do not change with time.
- Incompressible Fluid – Density ρ\rho is constant.
- Non-viscous Fluid – No energy is lost due to internal friction.
- Flow along a streamline – The equation applies to fluid particles moving along the same path.
- No external work done – No energy is added or extracted (e.g., by pumps or turbines).
Explanation (300 words):
Bernoulli’s theorem is a fundamental principle in fluid dynamics that relates the pressure, velocity, and elevation in a moving fluid. It states that for a steady, incompressible, and frictionless flow along a streamline, the total mechanical energy per unit volume remains constant. This total energy comprises pressure energy (PP), kinetic energy per unit volume (12ρv2\frac{1}{2} \rho v^2), and potential energy per unit volume (ρgh\rho g h).
To derive Bernoulli’s equation from first principles, we start with Euler’s equation for the motion of a fluid particle along a streamline. Euler’s equation is derived from Newton’s Second Law, applied to fluid motion, and expresses the balance of forces on a fluid element due to pressure, gravity, and acceleration. It can be written as: dPρ+g dz+v dv=0\frac{dP}{\rho} + g\,dz + v\,dv = 0
Each term represents a different form of energy per unit mass. When we integrate this equation, we obtain: Pρ+gz+v22=constant\frac{P}{\rho} + g z + \frac{v^2}{2} = \text{constant}
Multiplying by the fluid density ρ\rho transforms the equation into Bernoulli’s form: P+12ρv2+ρgz=constantP + \frac{1}{2} \rho v^2 + \rho g z = \text{constant}
This implies that if the speed of the fluid increases, its pressure must decrease if the height remains constant, and vice versa. Bernoulli’s equation is widely used in applications such as airplane wing design, venturimeters, and pipe flow analysis.
However, its validity relies on several assumptions, such as inviscid flow, no heat or work interactions, and application only along a streamline. If these conditions are not met, corrections or other more complex fluid dynamic equations must be used.