Fluid Flow: Bernoulli’s Equation Derivation & Fluid Mechanics

Bernoulli’s Principle is a foundational concept in fluid mechanics, describing how the speed of a fluid relates to its pressure and elevation. This page covers Bernoulli’s equation, its derivation, continuity equation, solved examples, applications, and exam-ready MCQs with detailed FAQs.

What is Bernoulli’s Principle?

• Bernoulli’s principle states that as the velocity of a moving fluid increases, the pressure within the fluid decreases (for steady, incompressible, and non-viscous flow).
• It is based on the law of conservation of energy for a fluid in motion.
• Applications include airplane lift, venturi meters, blood flow, sports, and pipe flow.

Bernoulli’s Equation Formula

Bernoulli’s Equation Derivation (Step-by-Step)

1. Consider fluid flow through a pipe with varying diameter and elevation.
2. Apply the work-energy theorem: Work done = Change in kinetic energy + Change in potential energy.
3. For small fluid volume dV:
   
   • Work done by pressure: dW = (p₁ - p₂)dV
   • Change in kinetic energy: dK = ½ ρdV (v₂² - v₁²)
   • Change in potential energy: dU = ρdVg(h₂ - h₁)
4. By conservation of energy: (p₁ - p₂)dV = ½ ρdV (v₂² - v₁²) + ρdVg(h₂ - h₁)
5. Dividing by dV and rearranging, we get:
   p₁ + ½ ρv₁² + ρgh₁ = p₂ + ½ ρv₂² + ρgh₂

Continuity Equation in Fluid Flow

The continuity equation is based on mass conservation for incompressible, steady fluids:
A₁v₁ = A₂v₂
Where A = area of cross-section, v = fluid velocity at points 1 and 2.
Application: Explains why a narrowing pipe causes fluid to speed up.

Components of Bernoulli’s Equation (Head Terms)

• Pressure Head: p/ρg
• Velocity Head: v²/2g
• Potential Head: h
• Total Head: Pressure Head + Velocity Head + Potential Head = Constant

Applications of Bernoulli’s Principle

• Aeroplane wings: Generate lift by creating a pressure difference above and below the wing.
• Venturi effect: Used in flow meters, carburettors, atomizers.
• Sports: Explains curveballs, soccer kicks, cricket swings.
• Blood flow: Explains pressure/velocity changes in arteries.
• Pipe systems: Analyzes speed and pressure variations in different pipe sections.

Example: Pressure Difference on an Airplane Wing

Frequently Asked Questions (FAQs) about Bernoulli’s Principle and Fluid Flow

1. What is Bernoulli’s principle in simple words?
   It states that for a moving fluid, an increase in speed leads to a decrease in pressure and vice versa.
2. What are the main assumptions in Bernoulli’s equation?
   The fluid is steady, incompressible, non-viscous (ideal), and flows along a streamline.
3. What is the difference between Bernoulli’s equation and the continuity equation?
   Bernoulli’s relates pressure, velocity, and height; continuity equation relates area and velocity for mass conservation.
4. How is Bernoulli’s principle used in aviation?
   Explains how wings create lift by causing faster air over the top surface, reducing pressure above the wing.
5. What is Venturi effect?
   The reduction in fluid pressure that results when a fluid flows through a constricted section (narrowing) of pipe.
6. What is meant by “head” in fluid mechanics?
   Head refers to energy per unit weight of fluid, expressed in terms of height (pressure head, velocity head, and potential head).
7. What is head loss?
   The loss of total head (energy) due to friction or turbulence in pipes or channels, usually calculated using the Darcy-Weisbach equation.
8. Can Bernoulli’s equation be applied to real (viscous) fluids?
   Not directly; real fluids lose energy due to friction, so the equation must be modified to include losses.
9. What are some daily life examples of Bernoulli’s principle?
   Aeroplane flight, curveball in sports, perfume atomizer, blood flow in arteries, river flow.
10. How does fluid pressure change in a narrowing pipe?
    Fluid speeds up and pressure decreases in the narrower section, as per Bernoulli’s equation.
11. What is the formula for velocity of efflux using Bernoulli’s theorem?
    v = √(2gh) for a small opening at the base of a tank (Torricelli’s theorem).
12. Why is Bernoulli’s equation important in engineering?
    It helps design pipe systems, pumps, airplane wings, sports equipment, and medical devices by predicting fluid behavior.

Practice MCQs: Bernoulli’s Principle & Fluid Mechanics

1. According to Bernoulli’s principle, as the speed of a fluid increases, its pressure:
   a) Increases
   b) Decreases
   c) Remains the same
   d) Doubles
2. Bernoulli’s equation is based on the law of:
   a) Conservation of mass
   b) Conservation of energy
   c) Conservation of momentum
   d) Conservation of charge
3. Which of the following is NOT an assumption in Bernoulli’s equation?
   a) Fluid is incompressible
   b) Fluid is non-viscous
   c) Flow is turbulent
   d) Flow is steady
4. The continuity equation for fluid flow is:
   a) A₁v₁ = A₂v₂
   b) p₁v₁ = p₂v₂
   c) ρ₁A₁ = ρ₂A₂
   d) p₁A₁ = p₂A₂
5. What happens to the velocity of a fluid in a pipe when the cross-sectional area decreases?
   a) Increases
   b) Decreases
   c) Remains the same
   d) Becomes zero
6. Which device works on the principle of Bernoulli’s theorem?
   a) Thermometer
   b) Barometer
   c) Venturi meter
   d) Ammeter
7. The "head" in fluid mechanics represents:
   a) Energy per unit charge
   b) Energy per unit weight
   c) Mass per unit area
   d) Pressure per unit time
8. In the Bernoulli’s equation, the term ½ρv² is called:
   a) Pressure head
   b) Velocity head
   c) Potential head
   d) Kinetic energy per unit volume
9. Bernoulli’s principle explains which of the following phenomena?
   a) Boiling point
   b) Lift on airplane wings
   c) Melting ice
   d) Earthquake waves
10. If the fluid in a pipe is at the same height throughout, Bernoulli’s equation reduces to a relationship between:
    a) Only pressures
    b) Pressure and velocity
    c) Only velocities
    d) Only density

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