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Unformatted text preview: Curl, Laplacian and all that! Asim Gangopadhyaya We all have seen a problem where a river has a differential velocity on its surface. The velocity is parallel to the shore everywhere and it has maximum magnitude at the center and symmetrically goes down on both sides as moves toward shore. The curl of the velocity field is a vector field pointing down on (into the water) one side of the river and coming out on the other. Willium Kolada of PHYS 351 asked whether Curl of the velocity field in a tube would form circles - I said yes! Let us follow this answer a little further and see what we get. Model: Let us consider a vertical tube of radius R through which a very viscous fluid is moving upward. Let us align the z-axis to be along the axis of the tube pointing up. For simplicity, assume the velocity to be v k at the center and linearly goes to zero at the inner surface of the tube. Thus, the velocity field is given by: ~v ( , , z ) = v 1- R k . (1) Curl of a vector field ~v...
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- Spring '09