A very significant feature of the Navier–Stokes equations is the presence of convective acceleration: the effect of time independent acceleration of a fluid with respect to space, represented by the nonlinear quantity:
\mathbf{v} \cdot \nabla \mathbf{v},
which may be interpreted either as (\mathbf{v}\cdot\nabla)\,\mathbf{v} or as \mathbf{v}\cdot(\nabla\mathbf{v}), with \nabla \mathbf{v} the tensor derivative of the velocity vector \mathbf{v}. Both interpretations give the same result, independent of the coordinate system — provided \nabla is interpreted as the covariant derivative.[3]
[edit] Interpretation as (v·∇)v
The convection term is often written as
(\mathbf{v} \cdot \nabla) \mathbf{v},
where the advection operator \mathbf{v} \cdot \nabla is used. Usually this representation is preferred because it is simpler than the one in terms of the tensor derivative \nabla \mathbf{v}.[3]
[edit] Interpretation as v·(∇v)
Here \nabla \mathbf{v} is the tensor derivative of the velocity vector, equal in Cartesian coordinates to the component by component gradient. The convection term may, by a vector calculus identity, be expressed without a tensor derivative:[4][5]
\mathbf{v} \cdot \nabla \mathbf{v} = \nabla \left( \frac{\|\mathbf{v}\|^2}{2} \right) + \left( \nabla \times \mathbf{v} \right) \times \mathbf{v}.
The form has use in irrotational flow, where the curl of the velocity (called vorticity) \omega=\nabla \times \mathbf{v} is equal to zero.
Regardless of what kind of fluid is being dealt with, convective acceleration is a nonlinear effect. Convective acceleration is present in most flows (exceptions include one-dimensional incompressible flow), but its dynamic effect is disregarded in creeping flow (also called Stokes flow) .
