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

A velocity field assigns a velocity vector $\mathbf{v}(\mathbf{x},t)$ to each spatial position and time in a continuum. Following a material particle means solving $d\mathbf{x}/dt=\mathbf{v}(\mathbf{x},t)$ with its initial position. The resulting Lagrangian trajectory samples the field at changing positions.

The field's spatial variation is the velocity gradient, whose symmetric part gives local strain rate and whose trace gives velocity divergence. Even when ${\partial }_t\mathbf{v}=0$, a moving particle can have convective acceleration $(\mathbf{v}\cdot \nabla )\mathbf{v}$. A steady flow narrowing through a pipe demonstrates this: fixed sensors give unchanging readings while each passing parcel speeds up. Streamlines are curves tangent to the instantaneous velocity field; they coincide with particle paths only under the appropriate steady-flow conditions.

A velocity field with spatially constant velocity gradient $K$. Differences of fluid velocities satisfy $\mathbf{u}(\mathbf{x}+\mathbf{R})-\mathbf{u}(\mathbf{x})=K\mathbf{R}$, even when the uniform translation $\mathbf{b}$ is nonzero. Incompressibility requires $\mathrm{tr}K=0$.

An affine velocity field with zero uniform translation. It describes the relative motion about a comoving reference point even when the original velocity field has a nonzero spatially uniform component.

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.