Applet: A rotating sphere indicating the presence of curl
The rotation of the sphere, with its center fixed at a point $(x,y,z)$ indicates the presence of nonzero curl of the vector field giving the velocity of fluid flow. The curl at $(x,y,z)$ is represented by the green arrow. The length of the curl vector indicates the rotation speed. The direction of the curl vector is determined by the rotation direction using the “right hand rule”: curl finger of right hand in direction of rotation and your thumb will indicate the direction of the curl vector. To see the rotation of the sphere, hold your mouse cursor over the graph. (If you double-click, the animation will stop; double-click again to restart the animation.)
Even though the sphere is moved off the axis of the rotation of the fluid, it rotates in the same direction as the general fluid rotation. Notice that the arrows continue to get longer as one moves away from the axis around which the fluid is rotating. For this reason, the fluid flow pushes the sphere more strongly on the side away from this axis, causing the sphere to spin in the same direction and speed as before. The general rotation of the flow also contributes to the sphere's spinning, as it causes the fluid to push against the sphere for a greater distance on the side away from the fluid's axis of rotation.
This vector field is $\dlvf(x,y,z)= (-y,x-z,y).$ Its curl is \begin{align*} \curl \dlvf(x,y,z) &= \left(\pdiff{}{y}y-\pdiff{}{z}(x-z), \pdiff{}{z}(-y)- \pdiff{}{x}y, \pdiff{}{x}(x-z) - \pdiff{}{y}(-y) \right)\\ &=(2,0,2), \end{align*} which is a constant independent of point $(x,y,z)$ and always points at an angle halfway between the positive $x$ and $z$ axes. For this particular vector field, the rotation of the sphere is independent of its location.
Applet file: curlmovedsphere.m
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