Velocity field ($\mathbf{u}(\mathbf{x},t)$)

A velocity field assigns a velocity vector to every position and time in a moving continuum. Its spatial derivatives describe local deformation, rotation, and volume change.

A secondary flow is a usually smaller velocity component superposed on a primary flow selected by symmetry or a simplified force balance. Examples include meridional circulation around a rotating sphere, or transverse circulation in a curved channel. Its direction depends on the forcing mechanism: elastic normal stresses and inertial effects can produce opposite circulations.

The velocity gradient tensor is $A_{ij}={\partial }_j{u}_i$. Its symmetric part is the rate-of-strain tensor $S$ and its antisymmetric part describes local rotation. For incompressible flow, $\mathrm{tr}A=\mathrm{tr}S=0$. The sign convention for a cubic invariant must be specified explicitly.

The vertical velocity is the component of a fluid velocity field along the selected upward vertical direction. In incompressible flow, impermeable horizontal boundaries impose $w=0$. Eliminating pressure often yields a closed hydrodynamic stability equation for this component.

An axisymmetric flow is invariant under rotations about one fixed axis, so scalar fields and cylindrical-coordinate velocity components do not depend on the azimuthal angle.

For a smooth axisymmetric flow, write $u_z(r,z)=w(z)+O(r^2)$ near the axis. The incompressible flow condition is $r^{-1}{\partial }_r(r{u}_r)+{\partial }_z{u}_z=0$. Integrating radially and excluding a singular line source at the axis gives $u_r=-r{w}^{\prime }(z)/2+O(r^3)$. The signed axial velocity is essential: specifying only its magnitude does not fix the sign of the radial velocity.