Collar Bearing – Torque to Overcome Viscous Shear (Annular Film) For a collar (annular) bearing with outer radius R1 and inner radius R2, lubricant viscosity mu, angular speed omega, and film thickness h, the viscous torque resisting rotation is T = (pi * mu * omega / (2 * h)) * (R1^4 − R2^4).

Introduction:
Viscous losses in collar bearings influence heating and power requirements. The torque expression is derived by summing shear contributions from concentric annular rings across the lubricated area.

Given Data / Assumptions:

• Annular contact, inner radius R2, outer radius R1.
• Newtonian lubricant of viscosity mu, uniform film thickness h.
• No slip at boundaries; rigid-body rotation at angular speed omega.

Concept / Approach:

Local tangential velocity at radius r is v = omega * r. Shear rate across film is dv/dy = v / h = (omega * r) / h; shear stress tau = mu * (omega * r) / h. Differential torque is shear force times radius integrated over the annulus.

Step-by-Step Solution:

Verification / Alternative check:

Dimensional analysis: [mu] Ns/m^2, multiply by omega/h (1/s / m) and R^4 (m^4) gives Nm, confirming torque units.

Why Other Options Are Wrong:

Option B uses area moment (R^2) rather than the correct r^3 integrand. Option C adds an extra omega factor (power, not torque). Option D uses a sum instead of the difference in fourth powers. Option E has wrong radius exponent and missing constants.

Common Pitfalls:

Assuming uniform shear independent of r; forgetting that torque weighting introduces r^3 in the integral; mixing footstep (solid disk) and collar (annulus) formulas.

Final Answer:

T = (pi * mu * omega / (2 * h)) * (R1^4 − R2^4)