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For a constant-mass density incompressible flow with no imposed body force, the two-dimensional Navier-Stokes equation isA conservative body force, if retained, may be incorporated into a modified pressure. Using scales $L,U_0,L/U_0$ and $\rho U_0^2$ givesHere $\nu =\mu /\rho$ is the molecular kinematic viscosity. At high Reynolds number, inertia and the pressure gradient dominate on the outer length scale, while molecular viscous diffusion is small there. Thin boundary layers, narrow shear flows and fine turbulent scales can still have sufficiently large velocity gradients for viscous effects to matter. High Reynolds number permits instability and turbulence, but does not alone prove that every flow is turbulent. Molecular viscous dissipation remains essential at the small end of the turbulent cascade.

A planar turbulent mixing layer develops between two parallel streams of different mean velocities. Their shear flow generates turbulence that transports streamwise linear momentum across the interface. A thin-layer eddy viscosity model with ${\nu }_T=C\Delta U\delta (x)$ has thickness proportional to downstream distance. For $\Delta U/U_c\ll 1$, its leading mean velocity transition is an error function.

A useful ideal magnetohydrodynamics reduction has $\mathbf{u}=\mathbf{v}(z,t)+ax{\mathbf{e}}_y$, $\mathbf{B}=\mathbf{B}(z,t)$ and $\rho =\rho (z,t)$. The constant background shear flow supplies $a{v}_x$ in the axial momentum equation and $a{B}_x$ in axial magnetic induction. The solenoidal constraint and MHD induction equation together make $B_z$ constant.

A renovating shear flow holds a planar shear flow fixed for an interval $T$, then chooses a fresh orientation independently of preceding intervals. Each interval acts on a material line by a rotated copy of the displayed deformation gradient. The orientation law must be specified: independent uniformly distributed orientations define the isotropic model, whereas a biased orientation law generally gives another Lyapunov exponent.

This three-layer shear flow has $U=\Delta Uz/h$ and density ${\rho }_0$ between $z=\pm h/2$. Above and below, the velocity is the corresponding constant $\pm \Delta U/2$, and the mass density is ${\rho }_0\mp \Delta \rho /2$. The density drops and vorticity jumps coincide at both interfaces. Its hydrodynamic stability involves coupled gravity-vorticity interface waves, unlike a profile with uniform shear extending through all three layers.

A uniform-density inviscid shear flow with $U=U_{0}z/d$ for $|z|<d$ and $U=\pm U_0$ outside has two vorticity jumps but no walls. Decaying normal modes satisfy $w^{\prime \prime }-k^{2}w=0$ in each region. Continuity of $w$ and pressure matching at a piecewise-linear shear interface gives a two-by-two determinant whose diagonal entries are $1-2kd\pm 2kdc/U_0$ and off-diagonal entries $e^{-2kd}$. Its vanishing gives the displayed dispersion relation. The Kelvin-Helmholtz instability band is $0<kd<{\alpha }_s$, where $2{\alpha }_s-1=e^{-2{\alpha }_s}$ has its unique positive root above $1/2$, approximately $0.63923227$. Short waves decouple the two edges and remain neutral; long waves approach the vortex sheet instability with $c=\pm i{U}_0$.

With weak kinematic viscosity $\nu$, zero buoyancy diffusion and locally unchanged plane internal gravity wave polarization, the displacement pseudomomentum of an internal gravity wave obeys $P_t+{\partial }_z(c_{gz}P)=-\nu K^{2}P$. For constant wavenumbers and a stationary wave envelope, $P\propto |\hat{\sigma }{|}^2$ proves the displayed amplitude decay. Decay is along the group velocity, so $c_{gz}<0$ describes a source above the receiving level. In a slowly varying shear flow, use wave action and the changing intrinsic frequency rather than treating $P/|\hat{\sigma }{|}^2$ as constant.

Use vorticity to distinguish the local spin of a fluid element from the motion of its center along a pathline. Circular streamlines can occur both with and without vorticity: the two velocity fields below even have the same circulation around the unit circle.

For a continuously differentiable velocity field $\mathbf{u}=(u_x,u_y,u_z)$, vorticity is the curl vectorEach partial derivative divides velocity by distance, so vorticity has units ${\mathrm{s}}^{-1}$. In a planar velocity field independent of $z$, only ${\omega }_z={\partial }_x{u}_y-{\partial }_y{u}_x$ survives. Positive ${\omega }_z$ points along $+{\mathbf{e}}_z$: the right-hand rule associates it with counterclockwise spin viewed from positive $z$ toward the plane. ^[ref]

The local rotational angular velocity is half the vorticity, ${\mathbf{\Omega }}_{\mathrm{spin}}=\mathbf{\omega }/2$. To see the factor, split the velocity gradient $L_{ij}={\partial }_j{u}_i$ into its symmetric rate-of-strain tensor $E=(L+L^T)/2$ and antisymmetric spin tensor $W=(L-L^T)/2$. A nearby displacement $\delta \mathbf{x}$ has relative velocity $L\delta \mathbf{x}$; its rotational part satisfies $W\delta \mathbf{x}=(\mathbf{\omega }/2)\times \delta \mathbf{x}$. This is a proof outline of the local spin interpretation. The rate-of-strain tensor can also change the angle of a particular material line, so that line's angular velocity need not equal $\mathbf{\omega }/2$.

For solid-body rotation, $\mathbf{u}=(-\Omega y,\Omega x,0)$ gives $\mathbf{\omega }=(0,0,2\Omega )$ and ${\mathbf{\Omega }}_{\mathrm{spin}}=(0,0,\Omega )$. Taking $\Omega =1{\mathrm{s}}^{-1}$, the vorticity is $2{\mathbf{e}}_z{\mathrm{s}}^{-1}$, although the angular velocity is only $1{\mathbf{e}}_z{\mathrm{s}}^{-1}$. For the potential line vortex, $\mathbf{u}=K{\mathbf{e}}_{\theta }/r$, the planar curl is $r^{-1}{\partial }_r(r{u}_{\theta })=r^{-1}{\partial }_{r}K=0$ wherever $r>0$. Its circular pathlines therefore coexist with zero local spin.

This distinction makes vorticity useful for recognizing shear flow as well as swirling motion; taking the curl of the Euler equations also removes a pressure gradient and leads to the vorticity equation. A historical landmark is Hermann von Helmholtz's 1858 paper on fluid vortex motion, an early study devoted to vortex dynamics. ^[ref]