Aula Teórica 17

Equações em coordenadas cilíndricas. Força centrífuga, força de coriolis e

escoamento laminar num tubo cilíndico.

Cylindrical pipe• Incompressible and permanent flow:

– Continuity:

0

z

uz

Momentum:

z

u

zr

u

rr

ur

rrz

gzp

z

uu

r

uu

r

uu

t

u

dt

du zzzzz

zzr

zz

11

r

ur

rrz

gzp z 10

Boundary Conditions: Velocity is zero on the pipe surface. The velocity profile is symmetrical with maximum velocity (zero gradient and thus null stress) at the centerline.

• The constant must be zero because at the centerline the shear stress (and thus the velocity gradient) must be zero.

r

ur

rrz

gzp z 10

r

Cr

z

gzp

r

u

Cr

z

gzp

r

ur

r

ur

rrz

gzp

z

z

z

2

2

10

2

2

2

4

1

4

12

Rz

gzpC

Crz

gzpu

r

z

gzp

r

u

z

z

2

22 1

4

1

R

rR

z

gzpuz

Average Velocity, discharge, wall shear stress…

• Wall shear stress:

2

2R

z

gzp

r

z

gzp

r

u

Rrrz

zrz

• Discharge:

z

gzpR

R

RzgzpR

A

QU

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2

2

22

22

22

0

2

422

0 2

22

0

8

2

22

4

1

4

1

22

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1

214

1

2..

Rz

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z

gzpQ

R

rrR

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gzpQ

rdrR

rR

z

gzpQ

rdrnudAnuQ

R

R

R

A

Maximum velocity is proportional to the radius square and so is the average velocity. The discharge is proportional radius fourth, as a consequence.

Solid Rotation

• There is equilibrium between centrifugal force and radial pressure gradient.

• The centrifugal force is in fact an inertia force, since it arises from the advective term of the equations, i.e. from the convective acceleration.

0

0

r

z

u

u

The flow in a sink

• Why does the velocity increase towards the center?

r

u

rr

u

z

u

zr

u

rr

ur

rrr

gzpuu

rz

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r

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r

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t

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dt

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2112

• When the radius tends to zero, generating very high tangential velocity that in fact is limited by frictionruur 1

• What is the rotation sense?

Hurricane

Generation mechanism

• It is caused by a depression e.g. associated to strong local heating that decreases the air density.

• Air with low density generates lower pressure. Generating the radial flow.

• The earth rotation generates tangential velocity that increases towards the center of the hurricane.

• This tangential velocity generates increases the radial pressure gradient. At the soil level the tangential velocity is lower because of friction and a radial velocity directed to the center is created. On the top for the same reason an outward velocity is created.

The Hurricane eye

• At the center viscous dissipation is very high (increases with the invers of the radius square) and thus we have solid rotation, with velocity lower than in the periphery of the nucleus. The pressure is low and consequently air from the top can descend.

r

u

rr

u

z

u

zr

u

rr

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rrr

gzpuu

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r

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r

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t

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dt

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2112