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Convective Operator

Defined for a Vector Field ${\bf A}$ by $({\bf A}\cdot \nabla)$, where $\nabla$ is the Gradient operator.

Applied in arbitrary orthogonal 3-D coordinates to a Vector Field B, the convective operator becomes

\begin{displaymath}[({\bf A}\cdot \nabla){\bf B}]_j = \sum_{k=1}^3 \left[{{A_k\o...
...rtial q_k}-A_k{\partial h_k\over\partial q_j}}\right)}\right],
\end{displaymath} (1)

where the $h_i$s are related to the Metric Tensors by $h_i=\sqrt{g_{ii}}$. In Cartesian Coordinates,
$({\bf A}\cdot \nabla){\bf B} = \left({A_x{\partial B_x\over\partial x}+A_y{\partial B_x\over\partial y}+A_z {\partial B_x\over\partial z}}\right)\hat {\bf x}$
$\quad +\left({A_x{\partial B_y\over\partial x}+A_y{\partial B_y\over\partial y}+A_z {\partial B_y\over\partial z}}\right)\hat {\bf y}$
$\quad +\left({A_x{\partial B_z\over\partial x}+A_y{\partial B_z\over\partial y}+A_z {\partial B_z\over\partial z}}\right)\hat {\bf z}.$ (2)
In Cylindrical Coordinates,
$({\bf A}\cdot \nabla){\bf B} =\left({A_r{\partial B_r\over \partial r} + {A_\ph...
..._z{\partial B_r\over \partial z} -{A_\phi B_\phi \over r}}\right){\hat {\bf r}}$
$\quad +\left({A_r{\partial B_\phi \over \partial r}+{A_\phi \over r}{\partial B...
... B_\phi \over \partial z}+{A_\phi B_r\over r}}\right){\hat {\boldsymbol{\phi}}}$
$\quad +\left({A_r{\partial B_z\over \partial r}+{A_\phi \over r}{\partial B_z\over \partial \phi }+ A_z{\partial B_z\over \partial z}}\right){\hat {\bf z}}.$ (3)
In Spherical Coordinates,

$({\bf A}\cdot\nabla){\bf B} =\left({A_r{\partial B_r\over\partial r}+{A_\theta\...
...over\partial\phi}-{A_\theta B_\theta + A_\phi B_\phi\over r}}\right)\hat{\bf r}$
$ +\left({A_r{\partial B_\theta\over\partial r}+{A_\theta\over r}{\partial B_\th...
...B_r\over r}-{A_\phi B_\phi\cot\theta\over r}}\right){\hat{\boldsymbol{\theta}}}$
$ +\left({\!A_r{\partial B_\phi\over\partial r}+{A_\theta\over r}{\partial B_\ph...
... r}+{A_\phi B_\theta\cot\theta\over r}}\right)\!{\hat{\boldsymbol{\phi}}}.\quad$ (4)

See also Convective Acceleration, Convective Derivative, Curvilinear Coordinates, Gradient

© 1996-9 Eric W. Weisstein