Section 1.4: Fluid Dynamics

This section studies how fluids move and how pressure, velocity, and cross-sectional area relate. We cover continuity equation and Bernoulli’s principle.

Key Equations & Concepts:
  • Continuity Equation: \( A_1 v_1 = A_2 v_2 \) (conservation of mass)
  • Bernoulli’s Equation: \( P + \frac{1}{2}\rho v^2 + \rho g h = \text{constant} \)
  • Flow speed increases as cross-sectional area decreases.
  • Pressure decreases where flow speed increases (inviscid fluid).
  • Applications: Venturi effect, pipe flow, airplane lift (simplified).

Example 1

Water flows through a pipe with area 0.05 m² at 2 m/s. The pipe narrows to 0.02 m². Find the flow speed in the narrow section.

Using continuity: \( A_1 v_1 = A_2 v_2 \)

\( 0.05 \cdot 2 = 0.02 \cdot v_2 \Rightarrow v_2 = 5 \text{ m/s} \)

Example 2

Fluid flows horizontally in a pipe. At section 1, speed = 3 m/s, pressure = 150 kPa. At section 2, area halves. Find new pressure (neglect height change, \( \rho = 1000 \text{ kg/m³} \)).

Speed at section 2: \( v_2 = 6 \text{ m/s} \) (continuity)

Bernoulli: \( P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2 \)

\( 150{,}000 + 0.5 \cdot 1000 \cdot 3^2 = P_2 + 0.5 \cdot 1000 \cdot 6^2 \)

\( 150{,}000 + 4500 = P_2 + 18{,}000 \Rightarrow P_2 = 136{,}500 \text{ Pa} \)

Practice Problems

  1. Water flows from a 0.1 m² pipe at 1 m/s into a 0.05 m² section. Find new speed.
  2. Fluid flows horizontally. Speed doubles from 2 m/s to 4 m/s. How does pressure change? \( \rho = 1000 \text{ kg/m³} \)
  3. A horizontal pipe narrows, causing speed to increase from 3 m/s to 6 m/s. Initial pressure = 120 kPa. Find final pressure.
  4. A garden hose nozzle reduces area from 0.02 m² to 0.005 m². Flow speed at wide section = 1 m/s. Find speed at nozzle.
  5. Explain the Venturi effect with a real-world example.
  6. A pipe with varying height: section 1 at 2 m, section 2 at 5 m. Speed constant. Find pressure difference.
  7. Water flows past a horizontal venturi tube. Section 1: 2 m/s, 200 kPa. Section 2: 4 m/s. Find pressure at section 2.
  8. Calculate flow rate (m³/s) if pipe radius = 0.1 m, speed = 3 m/s.
  9. A fluid flows through two sections, speeds 1.5 m/s and 3 m/s. Cross-sectional area of first section = 0.06 m². Find area of second section.
  10. Explain why pressure decreases when fluid speed increases in a pipe.
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