Curl

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

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

or

$$\nabla \times {\bf A} = \left ({\partial A_{z}\over \partia... ...\partial x} - {\partial A_{x}\over \partial y}\right ){\bf k}.$$ (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.

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

$$\tan \delta \theta \approx {\Delta\over \delta x},$$

where

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

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

$$\tan \delta \theta \approx \delta \theta.$$

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

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

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

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

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

$${\partial v_{y}\over \partial x} - {\partial v_{x}\over \partial y}.$$

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

$$\omega = \nabla \times {\bf v}.$$ (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, ${\bf j}$.

$${\bf j} = {\nabla \times {\bf B}\over \mu}.$$ (1.23)