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Bernoulli's Equation

Physics Class 12 - Fluid Dynamics

P1 + 1/2 rho v1^2 + rho g h1 = P2 + 1/2 rho v2^2 + rho g h2

Point 1

Point 2

P2 (Pressure at Point 2)

196500.00 Pa

Assumptions

  • Steady, incompressible flow
  • Non-viscous (frictionless) fluid
  • rho = 1000 kg/m3 (water)
  • g = 9.8 m/s2

Understanding Bernoulli's Equation

Definition: Bernoulli's equation describes the conservation of energy in fluid flow, named after Swiss mathematician Daniel Bernoulli (1700-1782). It states that for an ideal, incompressible, non-viscous fluid flowing steadily, the total mechanical energy (pressure energy + kinetic energy + potential energy) remains constant along a streamline. This fundamental principle of fluid dynamics explains how fluid velocity and pressure are related and how energy transforms between different forms in flowing fluids.

Mathematical Derivation:

Starting from Newton's second law applied to fluid elements along a streamline, we can derive Bernoulli's equation. Consider a fluid element moving from point 1 to point 2:

Work done by pressure = Change in KE + Change in PE

P1A1ds1 - P2A2ds2 = 1/2m(v2 squared - v1 squared) + mg(h2 - h1)

Using dm = rhoA1ds1 = rhoA2ds2 (incompressible)

P1 + 1/2 rho v1 squared + rho g h1 = P2 + 1/2 rho v2 squared + rho g h2 = constant

Each term has dimensions of pressure (Pa = N/m squared). Dividing by rho g gives the head form with dimensions of length.

The Bernoulli Equation:

P1 + 1/2 rho v1 squared + rho g h1 = P2 + 1/2 rho v2 squared + rho g h2

Alternative Forms:

Per unit weight: P/rho g + v squared/2g + h = constant (Head)

Per unit mass: P/rho + 1/2 v squared + gh = constant

Per unit volume: P + 1/2 rho v squared + rho g h = constant

Worked Example 1: Water flows through a horizontal pipe. At point 1 (diameter 20 cm), pressure is 200 kPa and velocity is 3 m/s. At point 2 (diameter 10 cm), find the pressure.

Continuity: A1v1 = A2v2
v2 = v1(A1/A2) = v1(d1 squared/d2 squared) = 3(20 squared/10 squared) = 12 m/s

Bernoulli (horizontal, h1 = h2 = 0):
P1 + 1/2 rho v1 squared = P2 + 1/2 rho v2 squared
200,000 + 1/2(1000)(9) = P2 + 1/2(1000)(144)
204,500 = P2 + 72,000
P2 = 132,500 Pa = 132.5 kPa

Worked Example 2: A tank filled with water has a small hole 5 m below the water surface. Find the efflux velocity.

Point 1: Surface (v1 approx 0, h1 = 5 m, P1 = atmospheric)
Point 2: Hole (h2 = 0, P2 = atmospheric)

P1 + 1/2 rho v1 squared + rho g h1 = P2 + 1/2 rho v2 squared + rho g h2
Atmospheric cancels: 1/2 rho v2 squared = rho g h
v2 = sqrt(2gh) = sqrt(2 x 9.8 x 5) = sqrt(98) = 9.9 m/s

This is Torricelli's theorem: v = sqrt(2gh)

Worked Example 3: A Pitot tube measures stagnation pressure of 150 kPa in water flowing at 10 m/s. What is the static pressure? (rho = 1000 kg/m cubed)

At stagnation point: v2 = 0, so all kinetic converts to pressure
P stagnation = P static + 1/2 rho v squared
150,000 = P static + 1/2(1000)(100)
P static = 150,000 - 50,000 = 100 kPa

Physical Interpretation - The Three Heads:

  • Pressure Head (P/rho g): The height a fluid column would rise due to pressure energy. Measured in meters of fluid.
  • Velocity Head (v squared/2g): The height the fluid would reach if converted entirely to potential energy. Also called dynamic head.
  • Elevation Head (h): The height above a reference datum. Potential energy per unit weight.

Key Insights from Bernoulli's Principle:

  • In a horizontal pipe (h1 = h2), if velocity increases, pressure decreases. This is the basis for many devices.
  • When fluid flows from wider to narrower pipe, velocity increases and pressure decreases (Venturi effect).
  • Total mechanical energy along a streamline remains constant (assuming no losses).
  • Where speed is high, pressure is low (and vice versa). This is the fundamental inverse relationship.
  • The pressure difference creates lift on airfoils and enables many flow instruments.

Real-World Applications:

  • Aircraft Wing Lift: The curved upper surface of an airplane wing forces air to travel faster over the top than underneath. According to Bernoulli, faster flow means lower pressure above the wing, creating an upward lift force that supports the aircraft.
  • Venturi Meters: These flow measurement devices use a constricted pipe section. The pressure drop across the constriction, measured by manometers, indicates flow rate. The formula Q = A2 sqrt[2(P1-P2)/rho] follows directly from Bernoulli.
  • Carburetors and Atomizers: Air flowing through a constriction (venturi) creates low pressure that draws up fuel. This mixes air and fuel for combustion in engines and spray guns.
  • Hydroelectric Power Plants: Water flowing through pipes from a dam (high elevation) to turbines converts pressure and potential energy to kinetic energy. The penstock design uses Bernoulli principles to maximize energy extraction.
  • Spray Paint Guns: High-velocity air creates low pressure over a liquid reservoir, drawing paint up and atomizing it into fine droplets for even coating.
  • Blood Flow and Aneurysms: In arteries narrowed by plaque, blood velocity increases and pressure drops, which can cause further damage. Understanding this helps diagnose vascular conditions.
  • Drafting in Racing: Race cars and cyclists position themselves behind others to enter the low-pressure wake, reducing drag and improving fuel efficiency.
  • Gravity-Fed Water Systems: Water towers use elevation to create pressure (P = rho g h). The flow rate from taps depends on height difference via Bernoulli's equation.

Continuity Equation Connection:

A1v1 = A2v2 (Conservation of mass for incompressible flow)
This must be used together with Bernoulli to solve pipe flow problems

Limitations and Assumptions:

  • Incompressible Flow: Valid for liquids and low-speed gases (Mach less than 0.3). For compressible flow at high speeds, density changes matter.
  • Steady Flow: Properties don't change with time at any point. No pulsatile or turbulent variations.
  • Frictionless Flow: Neglects viscous effects. Real fluids lose energy due to friction (minor losses in pipes).
  • No Energy Added/Removed: Doesn't account for pumps, turbines, or heat transfer. Extended Bernoulli includes these terms.
  • Along a Streamline: The energy equation applies only along the same streamline, not between different streamlines.

Common Mistakes to Avoid:

  • Using Wrong Reference Point: Height h must be measured from the same datum for both points. Inconsistent datums cause errors.
  • Ignoring Continuity: In pipes, mass conservation (A1v1 = A2v2) must be satisfied along with Bernoulli. Velocity cannot be arbitrary.
  • Applying to Wrong Flow Types: Bernoulli applies along streamlines, not across them. Don't use it for flows with significant mixing.
  • Forgetting Pressure Reference: Pressures are often gauge pressures, not absolute. Ensure consistency throughout the problem.
  • Overlooking Viscosity: Real fluids have viscosity. Bernoulli overestimates velocities and underestimates pressures in viscous flows.
  • Wrong Height Sign: Increasing height means increasing potential energy. Going up (h2 greater than h1) decreases pressure if velocity is constant.
  • Using Wrong Fluid Density: rho must be the actual fluid density. Air has rho approx 1.2 kg/m cubed, water has rho approx 1000 kg/m cubed.

Extended Bernoulli's Equation (with losses and pumps):

P1/rho g + v1 squared/2g + h1 + h pump - h loss = P2/rho g + v2 squared/2g + h2

Where h pump adds energy and h loss accounts for friction and minor losses.