Viscous Drag Torque and Power on a Rotating Shaft

Estimate viscous drag torque, power loss and shaft shear in a concentric annular fluid gap using the exact cylindrical Couette-flow solution.

Exact drag torque

—

N·m, opposing shaft rotation

Enter valid conditions.

Power / heat rate— W

Shaft shear stress— Pa

Surface speed— m/s

Radial clearance— mm

Gap Reynolds, Re_h—

Thin-gap torque, T_thin— N·m

Thin-gap difference—

The gap Reynolds number is an informational screening indicator; it does not by itself establish laminar stability.

How to use

Confirm a concentric rotating shaft, stationary cylindrical housing and Newtonian steady laminar shear model are appropriate.
Enter shaft diameter and the larger housing bore diameter in mm; radial clearance is calculated as h = (D_b − D_a)/2.
Enter positive wetted length in mm, nonnegative speed in rpm, dynamic viscosity at operating temperature in mPa·s, and positive density in kg/m³.
Select Calculate, or edit a field for an immediate update. Select Reset to restore the example.
Read exact drag torque first, then power/heat rate, shear stress, speed, clearance and Re_h.
Use T_thin only as a comparison. If its difference exceeds 5%, disregard the linear thin-gap approximation.
Do not use this model for eccentric, turbulent, Taylor-vortex, non-Newtonian, strongly heated, end-dominated or safety-critical cases requiring a validated bearing or CFD model.

Equations used

With shaft radius a = D_a/2, housing radius b = D_b/2 and radial clearance h = b − a:

ω = 2πN/60; U = ωa

T = 4π μ L ω a² b²/(b² − a²)

P = Tω; τ_a = 2 μ ω b²/(b² − a²)

Tthin = 2π μ L ω a³/h; ΔT = 100(Tthin − T)/T

Reh = ρ U h/μ

T and Tthin are torque magnitudes in N·m; P is steady mechanical loss converted to heat in W; τ_a is shaft shear magnitude in Pa; U is m/s; μ is Pa·s; ρ is kg/m³; and Re_h is dimensionless. Positive N denotes shaft rotation; reported torque acts in the opposite direction.

The exact equation follows the steady azimuthal Navier–Stokes solution v_θ = Ar + B/r with no slip at both concentric surfaces. It assumes constant-property, incompressible Newtonian fluid, stationary outer housing, axisymmetric laminar flow and negligible end, eccentricity, axial-pressure, free-surface and heating effects. The thin-gap equation assumes a linear velocity profile.

References: F. M. White, Viscous Fluid Flow, 3rd ed., cylindrical Couette flow; Engineers Edge equation background.