🎓 Lesson 7
D4
Velocity Profile and Shear Stress in Annular Flow
Annular flow is when fluid moves steadily between two concentric cylinders—like water flowing through a pipe-within-a-pipe—and its speed changes smoothly from zero at both walls to a peak in between, creating internal friction (shear stress) that engineers must calculate to avoid equipment failure or inefficient transport.
🎯 Learning Objectives
- ✓ Calculate the velocity profile across an annular gap using the exact analytical solution for laminar flow
- ✓ Analyze and compare shear stress distribution at inner and outer walls to assess liner wear or casing integrity in slurry transport systems
- ✓ Explain how radius ratio (κ = R_i/R_o) influences maximum velocity location and wall shear magnitude
- ✓ Apply dimensionless groups (e.g., Reynolds number for annular flow) to determine flow regime and validate laminar assumption
📖 Why This Matters
In mining operations, annular flow governs critical processes such as cement grouting behind blasthole casings, hydraulic transport of cuttings in reverse-circulation drilling, and slurry flow in tailings pipeline liners. Misjudging shear stress can cause premature liner erosion, seal failure, or inaccurate pressure drop predictions—leading to unplanned downtime or safety incidents. Understanding this profile ensures robust design of concentric-flow systems used in borehole stabilization and paste fill delivery.
📘 Core Principles
Annular laminar flow arises under low Reynolds number (Re < 2000), where inertial forces are negligible compared to viscous forces. The Navier–Stokes equations simplify to a second-order ordinary differential equation in radial coordinate r, solved subject to no-slip conditions: u(r = R_i) = 0 and u(r = R_o) = 0. Unlike pipe flow, the maximum velocity does *not* occur at the geometric center—but shifts toward the wider gap side depending on the radius ratio κ. Shear stress τ(r) = −μ du/dr is derived directly from the velocity gradient and exhibits sign change across the gap, reflecting direction of momentum transfer. The flow rate Q is obtained by integrating u(r) over the annular area, yielding a modified Hagen–Poiseuille expression.
📐 Velocity Profile & Wall Shear Stress
The exact solution for fully developed laminar annular flow under constant axial pressure gradient (dp/dz = −ΔP/L) gives velocity as a function of radial position. Wall shear stresses are critical for mechanical design—especially in lined blasthole casings or double-walled slurry pipes where interfacial shear dictates delamination risk.
💡 Worked Example
Problem: A grout slurry (μ = 15 Pa·s, ρ = 1800 kg/m³) flows laminarly between concentric steel tubes: inner radius R_i = 0.025 m, outer radius R_o = 0.05 m. Pressure drop ΔP = 40 kPa over L = 3 m length. Calculate velocity at r = 0.035 m and shear stress at inner wall.
1.
Step 1: Compute radius ratio κ = R_i/R_o = 0.025/0.05 = 0.5
2.
Step 2: Use velocity formula: u(r) = (ΔP / (4μL)) × [R_o² − r² + κ²(r² − R_i²)/ln(1/κ)] — first evaluate constants: ΔP/(4μL) = 40,000/(4×15×3) ≈ 222.22
3.
Step 3: Plug in r = 0.035 m; R_o² = 0.0025, r² = 0.001225, R_i² = 0.000625; ln(1/κ) = ln(2) ≈ 0.6931 → numerator = 0.0025 − 0.001225 + 0.25×(0.001225 − 0.000625)/0.6931 ≈ 0.001275 + 0.000217 ≈ 0.001492
4.
Step 4: u(0.035) ≈ 222.22 × 0.001492 ≈ 0.331 m/s
5.
Step 5: Shear stress at inner wall: τ_i = (ΔP / (2L)) × [R_i / (1 − κ²)] × [1 − κ² ln(R_o/R_i)/ln(1/κ)] → evaluate: (40,000/(2×3)) = 6666.7; R_i/(1−κ²) = 0.025/(1−0.25) = 0.0333; bracket term ≈ 1 − 0.25×ln(2)/0.6931 ≈ 1 − 0.25 = 0.75 → τ_i ≈ 6666.7 × 0.0333 × 0.75 ≈ 166 Pa
Answer:
The velocity at r = 0.035 m is 0.331 m/s; shear stress at the inner wall is 166 Pa—well below typical epoxy liner shear strength (>1.2 MPa) but warrants monitoring for cyclic loading in long-term grouting applications.
🏗️ Real-World Application
At the Cadia East underground mine (New South Wales, Australia), paste fill is pumped through a 200-mm OD, 120-mm ID double-pipe system (κ = 0.6) to stope backfill zones. Field-measured pressure gradients matched annular flow theory within ±4% only after incorporating measured slurry rheology (μ_app = 22 Pa·s, n = 0.42) and correcting for non-Newtonian effects via generalized Reynolds number. Initial designs assuming pipe flow overpredicted wall shear by 37%, leading to excessive liner abrasion—revised annular analysis extended liner service life from 4 to >11 months.
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