🎓 Lesson 23
D2
Deriving the Navier-Stokes Equation Step-by-Step
The Navier-Stokes equation is a mathematical way to describe how fluids like water or air move when forces like pressure, gravity, and internal friction act on them.
🎯 Learning Objectives
- ✓ Derive the incompressible Navier-Stokes equation from first principles using control volume and Newton’s second law
- ✓ Explain the physical meaning of each term (inertial, pressure, viscous, body force) in the momentum balance
- ✓ Apply dimensional analysis to non-dimensionalize the Navier-Stokes equation and identify the Reynolds number
- ✓ Analyze when viscous or inertial terms dominate in mining ventilation ducts or slurry transport pipelines
- ✓ Interpret computational outputs (e.g., CFD velocity fields) by tracing back to Navier-Stokes assumptions and limitations
📖 Why This Matters
In mining engineering, fluid behavior governs critical systems: ventilation airflow through stopes and shafts, tailings slurry transport in pipelines, explosive gas dispersion during blasting, and groundwater inflow into excavations. Misjudging fluid motion can lead to hazardous under-ventilation, pipeline blockages, or uncontrolled seepage. The Navier-Stokes equation is not just theory—it’s the foundational physics that every CFD model, fan selection chart, and dewatering design ultimately relies upon.
📘 Core Principles
We begin with Newton’s second law: net force = mass × acceleration. For a fluid, we apply this to an infinitesimal control volume—not a rigid body, but a deformable parcel moving with the flow. First, identify all forces: surface forces (pressure and viscous shear) and body forces (gravity). Then, express acceleration as the material (or 'substantial') derivative D/Dt = ∂/∂t + v·∇—capturing both local change and movement through spatial gradients. Finally, apply Stokes’ hypothesis to relate viscous stress to velocity gradients via dynamic viscosity μ, assuming isotropic, linear, Newtonian behavior. This yields the full vector form; incompressibility (∇·v = 0) simplifies it for most mining fluid applications (water, air at low Mach numbers).
📐 Incompressible Navier-Stokes Equation
This is the standard form used in mining fluid systems where density ρ is constant and flow is subsonic. It balances acceleration against pressure, viscous diffusion, and gravity per unit volume.
💡 Worked Example
Problem: Consider steady, fully developed laminar airflow (ρ = 1.2 kg/m³, μ = 1.8×10⁻⁵ Pa·s) in a horizontal rectangular ventilation duct (height h = 0.8 m). Velocity profile is parabolic: u(y) = U_max[1 − (y/h)²], with U_max = 4.5 m/s. Calculate the viscous shear stress τ_xy at y = 0.2 m.
1.
Step 1: Identify knowns — ρ = 1.2 kg/m³, μ = 1.8×10⁻⁵ Pa·s, u(y) = U_max[1 − (y/h)²], U_max = 4.5 m/s, h = 0.8 m, y = 0.2 m.
2.
Step 2: Compute ∂u/∂y = U_max × (−2y/h²) = 4.5 × (−2 × 0.2 / 0.64) = −2.8125 s⁻¹.
3.
Step 3: Apply Newton’s law of viscosity: τ_xy = μ(∂u/∂y) = (1.8×10⁻⁵)(−2.8125) = −5.06×10⁻⁵ Pa.
Answer:
The shear stress is −5.06×10⁻⁵ Pa (negative sign indicates direction opposite to increasing y), well within typical mine ventilation duct shear ranges of 10⁻⁶ to 10⁻⁴ Pa.
🏗️ Real-World Application
At the Bingham Canyon Mine (Utah), CFD simulations of main haulage drift ventilation used the incompressible Navier-Stokes equations coupled with turbulence models (k–ε) to optimize fan placement and duct sizing. Engineers validated the model against tracer-gas measurements and adjusted boundary conditions to reflect real-world roughness (equivalent sand roughness k_s ≈ 0.15 mm for galvanized steel ducts), ensuring predicted airflow velocities matched field data within ±8%—critical for maintaining respirable dust control and NO₂ dilution below MSHA PEL of 5 ppm.
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