The analytical method is to solve the Stokes flow field around the rotating object, calculate the surface shear stress, and integrate that stress to obtain torque. For a submerged sphere of radius (R) rotating with angular velocity (\omega) in a Newtonian fluid of viscosity (\mu), the low-Reynolds-number solution gives the resistive torque

[ G=-8\pi\mu\omega R^3 ]

where the negative sign indicates that the hydrodynamic torque opposes the rotation.

In creeping flow, torque is determined entirely by viscous stresses: solve the linear Stokes equations with the rotating surface boundary condition, evaluate the tangential stress at the surface, and integrate its moment over the object.

Why the Stokes-Flow Method Applies

Negligible fluid inertia

The method applies when the Reynolds number is sufficiently small that inertial terms can be neglected relative to viscous terms. The governing momentum equation becomes

[ -\nabla p+\mu\nabla^2\mathbf{v}=0, \qquad \nabla\cdot\mathbf{v}=0 ]

For a rotating sphere in an otherwise quiescent fluid, the flow is axisymmetric and purely azimuthal.

Linear relationship between torque and speed

Because the Stokes equations are linear, the torque is directly proportional to angular velocity. It is also proportional to the fluid viscosity and to the cube of the sphere radius:

[ |G|=8\pi\mu\omega R^3 ]

This proportionality is the rotational analogue of the linear drag law for a translating sphere.

Solving the Rotational Flow Field

Apply the boundary conditions

Use spherical coordinates with the sphere centered at the origin. For a sphere rotating about its axis, the velocity conditions are:

[ v_r=0, \qquad v_\theta=0, \qquad v_\phi(R,\theta)=\omega R\sin\theta ]

The last condition represents the tangential velocity of the sphere surface. Far from the sphere, the fluid is stationary:

[ v_\phi(r,\theta)\rightarrow 0 \quad \text{as} \quad r\rightarrow\infty ]

Determine the azimuthal velocity

Solving the Stokes equations subject to these conditions gives

[ v_\phi(r,\theta)

\frac{\omega R^3\sin\theta}{r^2} ]

The velocity decays as (1/r^2), reflecting the localized nature of the rotational disturbance in creeping flow.

Interpret the flow distribution

The fluid velocity is largest at the sphere’s equator, where (\sin\theta=1), and vanishes at the axis of rotation. The entire flow field is generated by viscous coupling between the rotating surface and the surrounding fluid.

Calculating Surface Shear Stress

Evaluate the tangential stress

For an axisymmetric azimuthal flow, the relevant spherical-coordinate stress component is

[ \tau_{r\phi}

\mu \left( \frac{\partial v_\phi}{\partial r}

\frac{v_\phi}{r} \right) ]

Substituting the velocity field and evaluating at (r=R) gives

[ \tau_{r\phi}(R,\theta)

-3\mu\omega\sin\theta ]

The negative sign shows that the fluid stress opposes the sphere’s rotation.

Account for the torque arm

A surface element at polar angle (\theta) has a perpendicular distance from the rotation axis of

[ R\sin\theta ]

The differential torque therefore equals the tangential force on the surface element multiplied by this moment arm.

Integrating the Torque

Integrate over the sphere

The surface area element is

[ dA=R^2\sin\theta,d\theta,d\phi ]

Thus, the total torque is

[ G

\int_0^{2\pi} \int_0^\pi \tau_{r\phi}(R,\theta) (R\sin\theta) R^2\sin\theta ,d\theta,d\phi ]

Substituting the shear stress gives

[ G

-3\mu\omega R^3 \int_0^{2\pi}d\phi \int_0^\pi\sin^3\theta,d\theta ]

Using

[ \int_0^{2\pi}d\phi=2\pi, \qquad \int_0^\pi\sin^3\theta,d\theta=\frac{4}{3} ]

produces

[ \boxed{G=-8\pi\mu\omega R^3} ]

The required applied torque has the same magnitude and opposite sign:

[ \boxed{G_{\text{applied}}=8\pi\mu\omega R^3} ]

Relate torque to power

The viscous power required to maintain rotation is

[ P=|G|\omega ]

For the rotating sphere,

[ \boxed{P=8\pi\mu\omega^2R^3} ]

This provides a direct estimate of mechanical power dissipated into the fluid.

Extending the Method to Unit Operations

Use rotational resistance coefficients

For other submerged bodies, the same procedure is expressed through a rotational resistance coefficient:

[ G=-\zeta_r\omega ]

For a sphere,

[ \zeta_r=8\pi\mu R^3 ]

The coefficient changes with object shape, confinement, proximity to walls, and the presence of other moving surfaces.

Apply the framework to agitators

In low-speed, highly viscous unit operations, the torque required by an impeller or rotating agitator can be obtained by solving the creeping-flow field or by determining an appropriate resistance coefficient. The resulting torque supports calculations of shear stress, power dissipation, and motor requirements.

Distinguish local and total quantities

The stress distribution describes local fluid loading across the surface. The integrated torque is the global quantity needed for equipment sizing and power calculations.

Understanding the Trade-offs

The low-Reynolds-number assumption is restrictive

The analytical result assumes that fluid inertia is negligible. If the Reynolds number is no longer small, convective inertia changes the velocity and stress fields, and the (G\propto\mu\omega R^3) relationship may no longer be adequate.

The sphere result is not universal

The coefficient (8\pi) applies specifically to an isolated sphere rotating in an unbounded Newtonian fluid. It should not be transferred directly to impellers, cylinders, particles near walls, or confined vessels.

Newtonian behavior is assumed

The derivation assumes constant viscosity and Newtonian constitutive behavior. Polymer solutions, suspensions, biological fluids, and other non-Newtonian materials may require a modified stress model and a different torque-speed relationship.

Boundary effects can be significant

Walls and nearby surfaces alter the velocity field and increase hydrodynamic resistance. In pilot-plant equipment, confinement may therefore require wall-correction factors, numerical Stokes-flow solutions, or experimental calibration.

Making the Right Choice for Your Goal

The analytical sphere solution is a benchmark and a practical first calculation, but the geometry and flow regime must be checked before applying it to equipment.

If your primary focus is estimating torque for an isolated rotating sphere: Use (G=-8\pi\mu\omega R^3), with the sign representing resistance to the imposed rotation.
If your primary focus is sizing a low-speed agitator: Determine the agitator’s geometry-specific rotational resistance rather than applying the sphere coefficient directly.
If your primary focus is predicting local shear: Use the velocity field and surface stress distribution before integrating to obtain total torque.
If your primary focus is operating outside creeping flow: Include fluid inertia and use an appropriate higher-Reynolds-number correlation, numerical model, or experiment.

The governing principle is simple: solve the viscous flow field, evaluate the surface stress, and integrate its moment to obtain the hydrodynamic torque.

Summary Table:

Key aspect	Result or application
Governing model	Linear, creeping-flow Stokes equations
Sphere torque	`G = -8πμωR³`
Rotational resistance	`ζᵣ = 8πμR³` for an isolated sphere
Required power	`P = 8πμω²R³`
Calculation method	Solve flow, evaluate surface stress, and integrate its moment
Main limitations	Shape, walls, inertia, and non-Newtonian fluids require corrections or different models

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