Vorticity is not merely a diagnostic computed from a velocity field. In a viscous fluid it is transported, stretched, tilted, and diffused according to its own evolution equation.

For an incompressible Newtonian fluid of constant density and kinematic viscosity $\nu$, Navier-Stokes is

$$\frac{\partial\mathbf v}{\partial t} +(\mathbf v\cdot\nabla)\mathbf v =-\frac1\rho\nabla p +\nu\nabla^2\mathbf v +\mathbf f_b.$$

Assume the body force is conservative, so its curl vanishes. Define vorticity by

$$\boldsymbol\omega=\nabla\times\mathbf v.$$

Take the curl of the momentum equation

The curl of a gradient is zero, so the pressure term disappears:

$$\nabla\times\nabla p=\mathbf0.$$

For constant $\nu$,

$$\nabla\times(\nu\nabla^2\mathbf v) =\nu\nabla^2\boldsymbol\omega.$$

Use the identity

$$({\mathbf v}\cdot\nabla)\mathbf v =\nabla\left(\frac12|\mathbf v|^2\right) -\mathbf v\times\boldsymbol\omega.$$

Taking its curl removes the gradient term. For incompressible flow,

$$\nabla\cdot\mathbf v=0,$$

and because the divergence of a curl is zero,

$$\nabla\cdot\boldsymbol\omega=0.$$

The resulting vorticity equation is

$$\boxed{ \frac{\partial\boldsymbol\omega}{\partial t} +(\mathbf v\cdot\nabla)\boldsymbol\omega =(\boldsymbol\omega\cdot\nabla)\mathbf v +\nu\nabla^2\boldsymbol\omega }.$$

Using the material derivative,

$$\boxed{ \frac{D\boldsymbol\omega}{Dt} =(\boldsymbol\omega\cdot\nabla)\mathbf v +\nu\nabla^2\boldsymbol\omega }.$$

Each term has a distinct physical role.

Advection

The material derivative

$$\frac{D\boldsymbol\omega}{Dt} =\frac{\partial\boldsymbol\omega}{\partial t} +(\mathbf v\cdot\nabla)\boldsymbol\omega$$

follows a moving fluid parcel. The advective part transports existing vorticity through space with the flow.

Advection moves vorticity; by itself it does not create or destroy its value for a parcel.

Vortex stretching and tilting

The term

$$\boxed{(\boldsymbol\omega\cdot\nabla)\mathbf v}$$

changes vorticity because the velocity varies along the direction of the vorticity vector.

It can

• increase or decrease vorticity magnitude by stretching or compressing vortex lines;
• change vorticity direction by tilting those lines.

This mechanism is intrinsically important in three-dimensional flow.

Worked stretching example

Consider the incompressible extensional velocity field

$$\mathbf v= -\frac a2x,\hat{\mathbf x} -\frac a2y,\hat{\mathbf y} +az,\hat{\mathbf z},$$

where $a>0$ is constant. Its divergence is

$$-\frac a2-\frac a2+a=0.$$

Suppose a fluid parcel carries vorticity aligned with $z$,

$$\boldsymbol\omega=\omega\hat{\mathbf z},$$

and neglect viscosity locally. Then

$$({\boldsymbol\omega}\cdot\nabla)\mathbf v =\omega\frac{\partial\mathbf v}{\partial z} =a\omega\hat{\mathbf z}.$$

Therefore

$$\frac{D\omega}{Dt}=a\omega.$$

The solution along the parcel is

$$\boxed{\omega(t)=\omega_0e^{at}}.$$

Axial stretching amplifies the vorticity while transverse compression preserves incompressible volume.

Viscous diffusion

The term

$$\boxed{\nu\nabla^2\boldsymbol\omega}$$

spreads vorticity from regions of strong concentration toward neighboring regions, just as viscosity diffuses momentum.

This is why vorticity generated near a no-slip wall can penetrate into the surrounding fluid, and why an ideal sharp vortex sheet becomes smoothed by viscosity.

Two-dimensional flow

For a planar flow with no $z$ dependence,

$$\mathbf v=(u,v,0),$$

and

$$\boldsymbol\omega=\omega_z\hat{\mathbf z}.$$

Because the velocity has no variation along $z$,

$$({\boldsymbol\omega}\cdot\nabla)\mathbf v =\omega_z\frac{\partial\mathbf v}{\partial z} =\mathbf0.$$

The stretching term disappears, leaving

$$\boxed{ \frac{D\omega_z}{Dt} =\nu\nabla^2\omega_z }.$$

Thus two-dimensional vorticity is advected and viscously diffused but not stretched.

If viscosity is also negligible,

$$\boxed{\frac{D\omega_z}{Dt}=0}.$$

Each fluid parcel then carries its scalar vorticity unchanged.

Connection to Stokes' first problem

For the unsteady parallel shear flow

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

the vorticity is

$$\omega_z=-\frac{\partial u}{\partial y}.$$

The velocity satisfies

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

Differentiating with respect to $y$ gives

$$\boxed{\omega_t=\nu\omega_{yy}}.$$

The transient shear layer can therefore be read equivalently as momentum diffusion or vorticity diffusion away from the wall.

What is absent from this form

The equation above assumes constant density. In variable-density flow, taking the curl of the pressure acceleration can produce an additional baroclinic term proportional to

$$\nabla\rho\times\nabla p.$$

That mechanism can generate vorticity when pressure and density surfaces are not aligned.

For constant-density incompressible Newtonian flow, however, the central picture is compact:

vorticity is carried by the flow, changed by three-dimensional stretching and tilting, and smoothed by viscosity.