Aula teórica 11

Rotação do tipo corpo sólido.Princípios de Conservação

(massa, momentum e energia)

Rotating Fluid

http://www.youtube.com/watch?v=gCZL3mZAQos&feature=related

http://www.youtube.com/watch?v=YNC2_vtNtyk

Rotação do tipo corpo sólido

Angular velocity:

Aceleration:

Pressure Gradient must balance the inertia force:

Integração da Equação

• Integrating fist in r (in an horizontal plane)

• Differentiating in “z”

• Where p0 is the pressure at the free surface. In fact at r=0 and z=0

The free surface is a parabola

• The position of the free surface can be found using volume conservation or knowing that:

220 2

rgzpp

The over-elevation

• Applied to the point located on the glass surface, at the same vertical coordinate as the free surface at the rotation axis gives:

g

Rh

rghpp

2

2

1

22

2200

• The over-elevation above the still level is h/2.

The Tea Cup paradox

• Is the effect that moves tea leaves laying on the cup bottom to the cup axis when one rotates the tea inside the cup, using a spoon.

• This case differs from the “solid rotation” because the velocity of the tea on the cup bottom is zero. As a consequence the pressure gradient necessary for balancing the centrifugal force is higher at the surface and lower at the bottom. See details at http://en.wikipedia.org/wiki/Tea_leaf_paradox

Conservation of momentum (Quantidade de movimento)

• The force exerted by the jet on the ball is equal to the force exerted by the ball on the jet (contrary sign) and balances ball weight.