Three flow patterns in the plane are described in words below

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

Three flow patterns in the

xy

plane are described in words below, together with the algebraic form of each. In every case

c

is a positive constant,

x

y

z

are in metres, there is no

z

component and nothing depends on

z

. ((a)) Every arrow points along

+\hat{\mathbf{y}}

, and the arrows are longer the further to the right (the larger

x

) you look:

\mathbf{F} = c\,x\,\hat{\mathbf{y}}

. ((b)) Every arrow points straight away from the

z

axis, with length proportional to the distance from that axis:

\mathbf{F} = c\,(x\,\hat{\mathbf{x}} + y\,\hat{\mathbf{y}})

. ((c)) Every arrow is tangent to the circle through its point centred on the

z

axis, running anticlockwise as seen from

+z

, and all the arrows have the same length:

\mathbf{F} = c\,\dfrac{-y\,\hat{\mathbf{x}} + x\,\hat{\mathbf{y}}}{\sqrt{x^{2}+y^{2}}}

. For each pattern, work out

\nabla\cdot\mathbf{F}

and

(\nabla\times\mathbf{F})_z

at the point

P = (1.0,\ 1.0,\ 0)\ \text{m}

and report their signs, writing

+1

for positive,

0

for zero and

-1

for negative. Give six numbers in the order: divergence of (a), curl of (a), divergence of (b), curl of (b), divergence of (c), curl of (c).

Browse the problem library