Section 6.7: Fluid Dynamics
Fluid dynamics studies the motion of fluids and the forces acting on them. Key concepts include viscosity, flow rate, and pressure differences that drive fluid motion.
Flow Rate
The volume flow rate is the volume of fluid passing through a cross-section per unit time:
\[
Q = A v
\]
where:
\( Q \) = flow rate (m³/s)
\( A \) = cross-sectional area (m²)
\( v \) = fluid velocity (m/s)
Viscosity and Resistance
Viscosity describes the internal friction of a fluid. According to Poiseuille’s law for laminar flow through a cylindrical pipe:
\[
Q = \frac{\pi r^4 \Delta P}{8 \eta L}
\]
where:
\( r \) = pipe radius (m)
\( \Delta P \) = pressure difference (Pa)
\( \eta \) = dynamic viscosity (Pa·s)
\( L \) = length of pipe (m)
Example 1: Flow Rate Calculation
Water flows through a pipe of area 0.02 m² at 3 m/s. Find the flow rate.
\( Q = A v = 0.02 \times 3 = 0.06 \, \text{m³/s} \)
Flow rate is 0.06 m³/s.
Example 2: Viscosity Effect
A 1 m long pipe of radius 0.01 m carries oil with viscosity \( \eta = 0.5 \, \text{Pa·s} \). Pressure difference is 2000 Pa. Find the flow rate (laminar).
Using Poiseuille’s law:
\( Q = \frac{\pi (0.01)^4 (2000)}{8 (0.5)(1)} \approx 1.57 \times 10^{-5} \, \text{m³/s} \)
Flow rate is approximately 1.57 × 10⁻⁵ m³/s.
Practice Problems
- A pipe of radius 0.05 m carries water with velocity 2 m/s. Calculate flow rate.
- Water flows through a hose of 0.03 m² at 4 m/s. Find volume flow rate.
- Oil with viscosity 0.4 Pa·s flows through a 0.5 m long pipe of radius 0.01 m with ΔP = 1000 Pa. Find Q.
- Explain how increasing viscosity affects flow rate in a horizontal pipe.
- Compare flow rates if pipe radius is doubled while pressure difference and viscosity remain the same.