When viscous forces dominate fluid inertia, the Navier-Stokes equations simplify dramatically. This regime is called creeping flow or Stokes flow.

For a characteristic speed $U$ and length $L$, the Reynolds number is

$$\boxed{Re=\frac{\rho UL}{\mu}}.$$

It compares inertial and viscous effects.

Scaling the momentum equation

For steady incompressible flow, the inertial term scales as

$$\rho(\mathbf v\cdot\nabla)\mathbf v \sim\rho\frac{U^2}{L},$$

while the viscous term scales as

$$\mu\nabla^2\mathbf v \sim\mu\frac{U}{L^2}.$$

Their ratio is

$$\frac{\rho U^2/L}{\mu U/L^2} =\frac{\rho UL}{\mu}=Re.$$

Therefore, when

$$\boxed{Re\ll1},$$

convective inertia is small compared with viscous momentum transport.

For a steady flow where unsteady inertia is also absent, Navier-Stokes reduces to the Stokes equations:

$$\boxed{-\nabla p+\mu\nabla^2\mathbf v+\rho\mathbf g=\mathbf0},$$

$$\boxed{\nabla\cdot\mathbf v=0}.$$

If gravity is unimportant or absorbed into a modified pressure, this becomes

$$\boxed{-\nabla p+\mu\nabla^2\mathbf v=\mathbf0}.$$

The pressure gradient and viscous stresses balance almost directly; appreciable fluid acceleration is not needed to transmit forces through the flow.

Why the equations become linear

The full Navier-Stokes equations contain the nonlinear term

$$({\mathbf v}\cdot\nabla)\mathbf v.$$

Removing that term makes the steady Stokes equations linear in $\mathbf v$ and $p$.

Consequently, if two velocity-pressure fields solve two Stokes-flow problems with compatible boundary conditions and forcing, linear combinations of those solutions can also be used to construct solutions. Superposition becomes available again.

This linearity is one reason low-Reynolds-number hydrodynamics has a distinctive structure.

Reversibility of steady Stokes flow

Suppose every imposed boundary velocity and driving force is reversed. Because the Stokes equations are linear and contain no quadratic inertial term, the velocity field reverses as well while following the same geometric pattern.

This is often called kinematic reversibility of creeping flow.

It does not mean that viscosity disappears or that no energy is dissipated. Viscous stresses still convert mechanical work into internal energy. It means that the instantaneous relation between forcing and velocity is linear when inertia is negligible.

At finite Reynolds number, inertial terms break this exact reversal symmetry.

No inertial wake is required

At moderate or large Reynolds number, flow around a body can separate and form an asymmetric wake whose momentum deficit contributes strongly to drag.

In creeping flow, the dominant resistance instead comes from viscous stresses and pressure acting throughout a smooth low-inertia disturbance around the body. The physics of drag is therefore qualitatively different from high-Reynolds-number pressure drag.

Worked regime check

A small particle moves through water at

$$U=1.0\times10^{-5},\mathrm{m/s}$$

with characteristic diameter

$$L=2.0\times10^{-5},\mathrm m.$$

Take

$$\rho=1000,\mathrm{kg/m^3},$$

$$\mu=1.0\times10^{-3},\mathrm{Pa,s}.$$

Then

$$Re=\frac{(1000)(1.0\times10^{-5})(2.0\times10^{-5})} {1.0\times10^{-3}} =\boxed{2.0\times10^{-4}}.$$

Because this is far below unity, viscous effects dominate inertia strongly and a steady Stokes-flow model is appropriate.

By contrast, increasing size or speed by several orders of magnitude can move the same fluid into a regime where inertia can no longer be neglected.

Scope

The steady Stokes equations require both

• sufficiently small Reynolds number so convective inertia is negligible;
• a situation where unsteady inertia is also negligible.

If the flow changes rapidly in time, the term $\rho,\partial\mathbf v/\partial t$ can matter even when convective inertia is weak. That regime is described by unsteady Stokes flow, not the steady equations above.

Creeping flow is central to the motion of small particles, droplets, microorganisms, aerosols, microfluidic devices, porous media, and many other systems where characteristic lengths or speeds are small enough that viscosity overwhelms inertia.