Viscosity does more than create a shear stress at one instant. In an unsteady flow, viscous stresses transport momentum across neighboring fluid layers, progressively spreading a velocity disturbance through the fluid.

Consider an incompressible Newtonian fluid with parallel velocity

$$\boxed{\mathbf v=u(y,t),\hat{\mathbf x}}.$$

Assume there is no axial pressure gradient and no body-force component in the $x$ direction.

Reduction of Navier-Stokes

Because the velocity has no $y$ component and $u$ is independent of $x$,

$$({\mathbf v}\cdot\nabla)\mathbf v=\mathbf0.$$

The flow can still be unsteady, so

$$\frac{\partial\mathbf v}{\partial t} =\frac{\partial u}{\partial t}\hat{\mathbf x}.$$

The viscous term is

$$\nabla^2\mathbf v =\frac{\partial^2u}{\partial y^2}\hat{\mathbf x}.$$

The $x$ component of the incompressible Navier-Stokes equation therefore becomes

$$\rho\frac{\partial u}{\partial t} =\mu\frac{\partial^2u}{\partial y^2}.$$

Using the kinematic viscosity

$$\boxed{\nu=\frac{\mu}{\rho}},$$

we obtain

$$\boxed{ \frac{\partial u}{\partial t} =\nu\frac{\partial^2u}{\partial y^2} }.$$

This is a diffusion equation for velocity, or equivalently for momentum per unit mass.

What kinematic viscosity means dynamically

The dimensions of kinematic viscosity are

$$[\nu]=\mathrm{m^2/s}.$$

It therefore plays the same mathematical role for momentum diffusion that thermal diffusivity plays for temperature diffusion and mass diffusivity plays for concentration diffusion.

A larger $\nu$ spreads a velocity disturbance through the fluid more rapidly.

Dimensional reasoning gives the characteristic penetration distance

$$\boxed{\delta\sim\sqrt{\nu t}}.$$

Equivalently, viscous communication across a distance $L$ takes a time of order

$$\boxed{t_\nu\sim\frac{L^2}{\nu}}.$$

The quadratic length dependence matters: communicating wall motion across twice the distance takes roughly four times as long when all else is equal.

Why this is momentum diffusion

For a Newtonian fluid,

$$\tau_{xy}=\mu\frac{\partial u}{\partial y}.$$

If one fluid layer is moving faster than a neighboring layer, viscosity creates shear stress. Spatial changes in that shear stress produce forces that accelerate slower layers and decelerate faster ones.

The result is a smoothing of velocity differences, represented mathematically by

$$\nu u_{yy}.$$

Momentum is not destroyed by this smoothing. It is redistributed through the fluid, while viscous deformation dissipates mechanical energy into internal energy.

Relation to steady Couette flow

Suppose fluid lies between two plates separated by $H$. If one plate starts moving, the velocity profile does not become linear everywhere immediately. Wall momentum first penetrates only a distance of order

$$\sqrt{\nu t}.$$

Only after a time comparable to

$$t_\nu\sim\frac{H^2}{\nu}$$

can viscosity communicate the wall motion across the entire gap and allow the flow to approach its steady Couette profile.

The steady solution and the transient diffusion process therefore answer different questions:

• steady Couette flow describes the final spatial balance;
• viscous momentum diffusion describes how that balance is established in time.

Worked scale estimate

For water near room conditions, take approximately

$$\nu=1.0\times10^{-6},\mathrm{m^2/s}.$$

After

$$t=4.0,\mathrm s,$$

the characteristic viscous penetration distance is

$$\delta\sim\sqrt{(1.0\times10^{-6})(4.0)}$$

$$=2.0\times10^{-3},\mathrm m.$$

Thus

$$\boxed{\delta\sim2.0,\mathrm{mm}}.$$

A wall velocity change has strongly influenced only a millimeter-scale region after a few seconds in this scaling sense.

For a gap of thickness

$$H=1.0,\mathrm{cm},$$

the viscous diffusion time is approximately

$$t_\nu\sim\frac{(0.010)^2}{1.0\times10^{-6}} =\boxed{100,\mathrm s}.$$

Scope

The simple equation

$$u_t=\nu u_{yy}$$

applies when the geometry and velocity field eliminate convective acceleration and when axial pressure/body-force terms are absent or treated separately. More general unsteady flows retain advection, multidimensional diffusion, pressure gradients, or other forcing.

Within its scope, the equation exposes a fundamental physical interpretation of viscosity: kinematic viscosity is a momentum diffusivity.