Lesson 5.4: Fluid Dynamics (Euler & Bernoulli Equation, Momentum Equation)

Fluid Dynamics studies fluids in motion under the influence of forces. GATE and PSU exams often test Bernoulli’s principle, Euler’s equation, and momentum-based calculations.

🔹 1. Introduction

• Definition: Fluid Dynamics deals with forces and motion of fluids, including pressure, velocity, and acceleration.

• Applications: Pipe networks, turbines, pumps, flow measurement devices

• Key Concepts:

  1. Steady vs unsteady flow

  2. Laminar vs turbulent flow

  3. Inviscid vs viscous flow (ideal fluid assumption for many problems)

🔹 2. Euler’s Equation of Motion

• Purpose: Relates pressure, velocity, and acceleration in a moving fluid along a streamline.

• Equation (1D along streamline):

dpρ+VdV+gdz=0\frac{dp}{\rho} + V dV + g dz = 0ρdp​+VdV+gdz=0

Where:

• $dp$ = pressure change

• $\rho$ = fluid density

• $V$ = velocity

• $dz$ = elevation change

• Assumptions: Inviscid, incompressible, steady flow

🔹 3. Bernoulli’s Equation

• Derived from Euler’s equation for steady, incompressible, inviscid flow along a streamline.

Pρg+V22g+z=constant\frac{P}{\rho g} + \frac{V^2}{2g} + z = \text{constant}ρgP​+2gV2​+z=constant

Where:

• $P/\rho g$ = pressure head

• V2/2gV^2/2gV2/2g = velocity head

• $z$ = elevation head

• Applications: Flow measurement (Venturi, Pitot tube), pumps, turbines

• Example: Water flows through horizontal pipe with velocity 2 m/s, pressure 200 kPa. Find pressure at a point where velocity is 3 m/s (ignore friction).

P1ρg+V122g=P2ρg+V222g⇒P2≈155kPa\frac{P_1}{\rho g} + \frac{V_1^2}{2g} = \frac{P_2}{\rho g} + \frac{V_2^2}{2g} \Rightarrow P_2 ≈ 155 kPaρgP1​​+2gV12​​=ρgP2​​+2gV22​​⇒P2​≈155kPa

🔹 4. Momentum Equation

• Purpose: Relates force applied to fluid to change in momentum.

∑F=m˙(V2−V1)\sum F = \dot{m} (V_2 – V_1)∑F=m˙(V2​−V1​)

Where:

• m˙=ρQ\dot{m} = \rho Qm˙=ρQ = mass flow rate

• V1,V2V_1, V_2V1​,V2​ = inlet & outlet velocities

• $\sum F$ = net force on fluid

• Applications: Jet propulsion, pipe bends, force on vanes and blades

🔹 5. Solved Examples (PYQ Style)

Example 1 (GATE ME 2016):
Water flows from pipe A (V = 2 m/s, P = 200 kPa) to pipe B (V = 3 m/s). Find P at B using Bernoulli.

Example 2 (PSU Exam):
Fluid jet of 0.1 m³/s strikes a flat plate normally. Velocity = 5 m/s. Determine force on plate using momentum equation.

Example 3:
Calculate pressure head difference in horizontal Venturi meter for given pipe diameters and flow rate.

🔹 6. Practice Exercises

1. Apply Bernoulli’s equation to flow through varying diameter pipe.

2. Calculate force on vanes using momentum equation.

3. Derive Bernoulli’s equation from Euler’s equation along streamline.

4. Solve problems involving pressure and velocity head in open channel flow.

5. Compute thrust on flat and curved plates using momentum principles.

🔹 7. Summary

• Euler’s Equation: Force-velocity-pressure relationship along streamline

• Bernoulli’s Equation: Total head = pressure + velocity + elevation, steady incompressible flow

• Momentum Equation: Force = rate of change of momentum

• Applications: Pipes, pumps, turbines, hydraulic machines, jets