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Curl

The curl of the vector ${\bf A}$ is

\begin{displaymath}
\nabla \times {\bf A} = \left ({\partial \over \partial x},...
...partial z}\right )\times \left
(A_{x}, A_{y}, A_{z}\right ),
\end{displaymath}

or
\begin{displaymath}
\nabla \times {\bf A} = \left ({\partial A_{z}\over \partia...
...\partial x} - {\partial A_{x}\over \partial y}\right ){\bf k}.
\end{displaymath} (1.21)

Thus, $\nabla \times {\bf v}$ represents the amount of curl or twist in a flow. This is illustrated in the next example.

Example 1. .9Imagine a stick in a flow given by ${\bf v} = \left ( v_{x}(x,y),
y_{y}(x,y), 0\right )$. How does it rotate? The angle $\theta $ to a fixed direction can increase in time $\delta t$ due to a spatially varying flow. Figure 1.14 illustrates this.

Figure 1.14: A flow $v_{y}(x,y)$ causes a stick to rotate since the distance moved by the right hand side exceeds the distance moved by the left hand side.
\includegraphics [bb=75 170 468 305,clip]{fundfig14.ps}

The right hand side moves a distance $v_{y}(x+\delta x, y)\delta t$ while the left hand side moves a distance $v_{y}(x,y)\delta t$. Thus the change in angle $\delta \theta $ is given by

\begin{displaymath}
\tan \delta \theta \approx {\Delta\over \delta x},
\end{displaymath}

where

\begin{displaymath}
\Delta = \left (v_{y}(x + \delta x, y) - v_{y}(x,y)\right )\delta t.
\end{displaymath}

Assuming the $\delta \theta $ is small we may approximate the tangent function by its small argument expansion. Thus,

\begin{displaymath}
\tan \delta \theta \approx \delta \theta.
\end{displaymath}

Using these expressions we may manipulate the equations to get the rate of which the angle is changing as

\begin{displaymath}
{\delta \theta \over \delta t} \approx {v_{y}(x+\delta x, y...
...(x,y)\over \delta x} \approx {\partial v_{y}\over \partial x}.
\end{displaymath}

Now a similar effect occurs due to the velocity component $v_{x}$. This time a larger value of $v_{x}(x, y+\delta y)$ than $v_{x}(x,y)$ causes the angle to decrease in time and

\begin{displaymath}
{\delta \theta\over \delta t} \approx -{v_{x}(x, y+\delta y...
...(x,y)\over \delta y} \approx -{\partial v_{x}\over \partial y}
\end{displaymath}

Thus, the net effect of both velocity components is t rotate the stick by an amount proportional to

\begin{displaymath}
{\partial v_{y}\over \partial x} - {\partial v_{x}\over \partial y}.
\end{displaymath}

This is the ${\bf k}$ component of $\nabla \times {\bf v}$.

The curl of the velocity is given the special name, vorticity. Thus, the vorticity, $\omega$ is

\begin{displaymath}
\omega = \nabla \times {\bf v}.
\end{displaymath} (1.22)

The amount of curl or twist in a magnetic field is given by $\nabla
\times {\bf B}$ and is proportional to the electric current, ${\bf j}$.
\begin{displaymath}
{\bf j} = {\nabla \times {\bf B}\over \mu}.
\end{displaymath} (1.23)


next up previous
Next: The Laplacian Operator, Up: Curl , , Triple Previous: Curl , , Triple
Prof. Alan Hood
2000-11-06