Water in a wide shallow channel flows with the steady velocity field v = (6.0 ,x)

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Fluids & Continuum Physics · Kinematics, conservation laws and ideal flow

Electricity, Magnetism and Electromagnetism, Ch Divergence, curl and the local form of the field laws

Problem

Water in a wide shallow channel flows with the steady velocity field

\mathbf{v} = (6.0\,x)\,\hat{\mathbf{y}},

where

x

y

and

z

are Cartesian coordinates in metres, the coefficient

6.0

is in

\mathrm{s^{-1}}

and

\mathbf{v}

comes out in

\mathrm{m\,s^{-1}}

. All the streamlines are straight lines parallel to the

y

axis. ((a)) Compute

\nabla\cdot\mathbf{v}

. ((b)) Compute

\nabla\times\mathbf{v}

, and state the sense in which a small paddle wheel with its axis along

z

turns, seen from the

+z

side.

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