🎓 Lesson 3
D2
Streamlines, Pathlines, and Streaklines
Streamlines, pathlines, and streaklines are three different ways to visualize how fluid moves — like drawing snapshots, time-lapse videos, or smoke trails in water or air.
🎯 Learning Objectives
- ✓ Explain the physical distinction among streamlines, pathlines, and streaklines using velocity field analysis
- ✓ Analyze a given 2D velocity field to derive and plot streamlines and determine whether flow is steady or unsteady
- ✓ Apply time-dependent integration to compute a pathline for a specified initial position and velocity function
- ✓ Design a streakline visualization experiment for a laboratory-scale flow system
📖 Why This Matters
In mining ventilation, slurry transport in tailings pipelines, or explosive gas dispersion modeling after blasting, engineers must interpret how fluids move—not just where pressure acts (statics), but *how* mass transports over time. Misinterpreting a smoke test (streakline) as a particle trajectory (pathline) can lead to flawed hazard zone predictions or inefficient duct design. Mastering these three line types builds the foundation for CFD validation, tracer studies, and safe airflow management in underground mines.
📘 Core Principles
Streamlines reflect the 'frozen' snapshot of flow direction—computed by solving dy/dx = v/u (in 2D) at a fixed time. Pathlines require integrating particle velocity over time: dx/dt = u(x,y,z,t), dy/dt = v(x,y,z,t), etc., with initial conditions. Streaklines are more complex: they demand solving the same ODEs but for *all* particles that passed through a given injection point at times τ ≤ t—making them inherently time-dependent and non-analytic except in simple cases. In steady flow (velocity independent of time), all three coincide; in unsteady flow (e.g., blast-induced air surges, pulsating slurry pumps), they diverge significantly—a critical insight for transient safety analysis.
📐 Streamline Equation (2D Cartesian)
For a 2D velocity field u(x,y,t), v(x,y,t), the streamline equation at fixed time t₀ is derived from the condition that the velocity vector is tangent to the curve. It yields an ordinary differential equation solvable via separation or exact methods.
💡 Worked Example
Problem: Given a steady 2D velocity field: u(x,y) = 2y m/s, v(x,y) = 4x m/s. Find the streamline passing through point (1,1).
1.
Step 1: Write dy/dx = v/u = (4x)/(2y) = 2x/y
2.
Step 2: Separate variables: y dy = 2x dx
3.
Step 3: Integrate both sides: ∫y dy = ∫2x dx → y²/2 = x² + C
4.
Step 4: Apply initial condition (x=1, y=1): (1)²/2 = (1)² + C → C = −0.5
5.
Step 5: Solve for implicit form: y² = 2x² − 1
Answer:
The streamline is y² = 2x² − 1, a hyperbola passing through (1,1). Since velocity has no time dependence, this is also the pathline and streakline.
🏗️ Real-World Application
At the Bingham Canyon Mine (Utah), transient CFD modeling of post-blast ventilation airflow used streakline visualization to map the 30-second dispersion envelope of NO₂ gas from detonation. Engineers injected virtual tracers at blast-hole collars every 0.5 s and rendered streaklines to define the ‘breathing zone’ boundary for re-entry protocols—revealing that peak gas concentration lagged behind the leading streakline by 8–12 s due to unsteady acceleration effects. This directly informed revised lockout-tagout timing per MSHA Part 46 requirements.