Aula Teórica 15

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

escoamento laminar num tubo cilíndico.

Cylindrical pipe• Incompressible and permanent flow:

– Continuity:

0

zuz

Momentum:

zu

zru

rrur

rrzgzp

zuu

ruu

ruu

tu

dtdu zzzz

zzz

rzz

11

rur

rrzgzp 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.

rur

rrzgzp z 10

rCr

zgzp

ru

Crzgzp

rur

rur

rrzgzp

z

z

z

2

2

10

2

2

2

41

41

2

RzgzpC

Crzgzpu

rzgzp

ru

z

z

2

22 1

41

RrR

zgzpuz

Average Velocity, discharge, wall shear stress…

• Wall shear stress:

2

2R

zgzp

rzgzp

ru

Rrrz

zrz

• Discharge:

zgzpR

R

RzgzpR

AQU

8282

2

2

22

22

22

02

422

0 2

22

0

82

22

41

41

22

41

2141

2..

RzgzpRQ

RRzgzpQ

RrrR

zgzpQ

rdrRrR

zgzpQ

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.

00

r

z

uu

The flow in a sink

• Why does the velocity increase towards the center?

ru

rru

zu

zru

rrur

rrrgzpuu

rzuu

ruu

ruu

tu

dtdu r

rzr211

2

• 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.

ru

rru

zu

zru

rrur

rrrgzpuu

rzuu

ruu

ruu

tu

dtdu r

rzr211

2