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

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)

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