Velocity Field

Velocity Field: The velocity field \( \mathbf{u} = (u, v, w) \) represents the fluid flow components along the \(x\), \(y\), and \(z\) axes, respectively.

Components: Here, \(u\) is the zonal velocity, \(v\) is the meridional velocity, and \(w\) is the vertical velocity.

\[ \nabla \cdot \mathbf{u} = \sum_{i=1}^{3} \frac{\partial u_i}{\partial x_i} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} \]

Divergence Meaning: The divergence operator quantifies the net flux out of a region of space, representing the expansion or contraction of the fluid in that region.

Local Field and Velocity Vector: The local velocity vector represents the speed and direction of a fluid particle at any given location.