Three cylindrical tubes of 0.5m length are placed co-axially and the central tube is rotated at 5 rpm applying a torque of 6 Nm. Determine the viscosity of oil which fills the space between tubes. Take r1, r2 and r3 as 0.15m, 0.152m and 0.154m.

Step 1: Angular velocity of cylinder 2 (ω):

ω = 2Nπ / 60

Substitute values:

ω = (2 × 5 × π) / 60 = 0.5236 rad/s

Step 2: Tangential velocity of cylinder 2 (u₂):

u₂ = r₂ × ω

Substitute values:

u₂ = 0.152 × 0.5236 = 0.07958 m/s

Step 3: Average radii and thickness of layers:

Average radius for inner layer (r_a):

r_a = (r₁ + r₂) / 2 = (0.15 + 0.152) / 2 = 0.151 m

Average radius for outer layer (r_b):

r_b = (r₂ + r₃) / 2 = (0.152 + 0.154) / 2 = 0.153 m

Thickness of layers:

dy₁ = r₂ – r₁ = 0.002 m

dy₂ = r₃ – r₂ = 0.002 m

Step 4: Torque equation:

T = μ × du / dy₁ × (2πr_aL) × r_a + μ × du / dy₂ × (2πr_bL) × r_b

Substitute values:

6 = μ × (0.07958 / 0.002) × (2π × 0.151 × 0.5) × 0.151 + μ × (0.07958 / 0.002) × (2π × 0.153 × 0.5) × 0.153

Simplify:

6 = μ × (0.07958 / 0.002) × (0.4742 + 0.4803)

μ = 1.038 N·s/m²