Mecânica dos Fluidos Análise Dimensional e Semalhança

▼❡❝â♥✐❝❛ ❞♦s ❋❧✉✐❞♦s

❆♥á❧✐s❡ ❉✐♠❡♥s✐♦♥❛❧ ❡ ❙❡♠❛❧❤❛♥ç❛

❑❛r❧ P❡t❡r ❇✉rr

✶ ■♥tr♦❞✉çã♦

• ◆❛ ❛✉❧❛ ✺ ❛♣r❡s❡♥t❛♠♦s ❜❛❧❛♥ç♦s ❣❧♦❜❛✐s ❞❡ ♠❛ss❛✱ q✉❛♥t✐❞❛❞❡ ❞❡ ♠♦✈✐♠❡♥t♦ ❧✐♥❡❛r✱

q✉❛♥t✐❞❛❞❡ ❞❡ ♠♦✈✐♠❡♥t♦ ❛♥❣✉❧❛r ❡ ❡♥❡r❣✐❛ ♣❛r❛ ✉♠ ✈♦❧✉♠❡ ❞❡ ❝♦♥tr♦❧❡ ♣❛r❛ ❡st✐♠❛r ♣❛râ♠❡tr♦s ❣❧♦❜❛✐s ❝♦♠♦ ✢✉①♦ ❞❡ ♠❛ss❛✱ ❢♦rç❛✱ ♠♦♠❡♥t♦✱ tr❛❜❛❧❤♦ ❡ tr❛♥s❢❡r❡♥❝✐❛ ❞❡ ❝❛❧♦r✳

• ◆❛ ❛✉❧❛ ✻ ❛♣r❡s❡♥t❛♠♦s ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♣❛r❝✐❛✐s ❜ás✐❝❛s ❞♦ ❡s❝♦❛♠❡♥t♦ ❞❡ ✉♠

✢✉✐❞♦✱ ❡ ❛❧❣✉♠❛s s♦❧✉çõ❡s ♣❛rt✐❝✉❧❛r❡s ❞❡ss❛s ❡q✉❛çõ❡s✳

• ❊ss❡s ❞♦✐s ❝❛♣ít✉❧♦s ❝♦❜r✐r❛♠ t❡❝♥✐❝❛s ❛♥❛❧ít✐❝❛s✱ q✉❡ sã♦ ❧✐♠✐t❛❞❛s ❛ ❣❡♦♠❡tr✐❛s r❛③♦✲

❛✈❡❧♠❡♥t❡ s✐♠♣❧❡s ❡ ❛ ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦ ❜❡♠ ❞❡✜♥✐❞❛s✳ ❆s ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♣❛r❝✐❛✐s ♣♦❞❡♠ s❡r r❡s♦❧✈✐❞❛s ♥✉♠❡r✐❝❛♠❡♥t❡ ♣❛r❛ ❣❡♦♠❡tr✐❛s ♠❛✐s ❝♦♠♣❧❡①❛s✱ ♠❛s q✉❛♥t♦ ♠❛✐♦r ❛ ❝♦♠♣❧❡①✐❞❛❞❡ ❞❛ ❢ís✐❝❛ ❞♦ ❡s❝♦❛♠❡♥t♦ ♠❛✐♦r ♦ ❝✉st♦ ❝♦♠♣✉t❛❝✐♦♥❛❧✳

• P❛r❛ ♣r♦❜❧❡♠❛s r❡❧❛t✐✈❛♠❡♥t❡ ❝♦♠♣❧❡①♦s✱ q✉❡ sã♦ ❛ ♠❛✐♦r ♣❛rt❡ ❞♦s ♣r♦❜❧❡♠❛s ❡♠

♠❡❝â♥✐❝❛ ❞♦s ✢✉✐❞♦s✱ r❡❝♦rr❡♠♦s ❛ t❡❝♥✐❝❛s ❡①♣❡r✐♠❡♥t❛✐s✳ ❖ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦ ❡s❝♦❛♠❡♥t♦ é r❡❣✐str❛❞♦ ❡♠ ❢♦r♠❛ ❞❡ ❞❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s✳ ❚❛✐s ❞❛❞♦s sã♦ ♠✉✐t♦s út❡✐s s❡ ❢♦r❡♠ ❡①♣r❡ss♦s ❞❡ ♠❛♥❡✐r❛ ❝♦♠♣❛❝t❛ ❡ ❡❝♦♥ô♠✐❝❛✳ ❊ss❛s sã♦ ❛s ♠♦t✐✈❛çõ❡s ♣❛r❛ ❛ ❛♥á❧✐s❡ ❞✐♠❡♥s✐♦♥❛❧✳

• ❆ ❛♥á❧✐s❡ ❞✐♠❡♥s✐♦♥❛❧ é ✉♠ ♠ét♦❞♦ ♣❛r❛ r❡❞✉③✐r ♦ ♥ú♠❡r♦ ❡ ❛ ❝♦♠♣❧❡①✐❞❛❞❡ ❞❛s

✈❛r✐á✈❡✐s q✉❡ ❛❢❡t❛♠ ✉♠ ❞❛❞♦ ❢❡♥ô♠❡♥♦ ❢ís✐❝♦✳

✕ ❙❡ ✉♠ ❢❡♥ô♠❡♥♦ ❞❡♣❡♥❞❡ ❞❡ n ✈❛r✐á✈❡✐s ❞✐♠❡♥s✐♦♥❛✐s✱ ❛ ❛♥á❧✐s❡ ❞✐♠❡♥s✐♦♥❛❧

✐rá r❡❞✉③✐r ♦ ♣r♦❜❧❡♠❛ ❛ ❛♣❡♥❛s k ✈❛r✐á✈❡✐s ❛❞✐♠❡♥s✐♦♥❛✐s✱

✕ ♦♥❞❡ ❛ r❡❞✉çã♦ n₋k = 1,2,3♦✉ 4❞❡♣❡♥❞❡ ❞❛ ❝♦♠♣❧❡①✐❞❛❞❡ ❞♦ ♣r♦❜❧❡♠❛✳

✕ ❊♠ ❣❡r❛❧✱ n₋k é ✐❣✉❛❧ ❛♦ ♥ú♠❡r♦ ❞❡ ❞✐♠❡♥sõ❡s ❜ás✐❝❛s✱ ♣r✐♠ár✐❛s ♦✉ ❢✉♥❞❛✲

♠❡♥t❛✐s q✉❡ ❣♦✈❡r♥❛♠ ♦ ♣r♦❜❧❡♠❛✳

✕ ◆❛ ♠❡❝â♥✐❝❛ ❞♦s ✢✉✐❞♦s✱ ❛s q✉❛tr♦ ❞✐♠❡♥sõ❡s ❜ás✐❝❛s ❝♦st✉♠❛♠ s❡r ❡s❝♦❧❤✐❞❛s ❝♦♠♦ s❡♥❞♦ ❛✿

✶✳ ❛ ♠❛ss❛ M✱

✷✳ ♦ ❝♦♠♣r✐♠❡♥t♦ L✱

✸✳ ♦ t❡♠♣♦ T✱

✹✳ ❛ t❡♠♣❡r❛t✉r❛ θ✳

♦✉✱ ❡♠ s✉♠❛✱ ✉♠ s✐st❡♠❛M LT θ✳ ❆s ✈❡③❡s ✉t✐❧✐③❛✲s❡ ✉♠ s✐st❡♠❛ F LT θ✱ ❝♦♠ ❛

❢♦rç❛ s✉❜st✐t✉✐♥❞♦ ❛ ♠❛ss❛✳

• ❆ ❛♥á❧✐s❡ ❞✐♠❡♥s✐♦♥❛❧✱ ❛♣❡s❛r ❞❡ s❡✉ ♦❜❥❡t✐✈♦ s❡❥❛ r❡❞✉③✐r ♦ ♥ú♠❡r♦ ❞❡ ✈❛r✐á✈❡✐s✱ tr❛③

✈ár✐♦s ❜❡♥❡❢í❝✐♦s ❛❞✐❝✐♦♥❛✐s✿

✶✳ ❯♠❛ ❡♥♦r♠❡ ❡❝♦♥♦♠✐❛ ❞❡ t❡♠♣♦ ❡ ❞✐♥❤❡✐r♦✳ P♦r ❡①❡♠♣❧♦✱ s✉♣♦♥❤❛ q✉❡ s❡ s❛✐❜❛ q✉❡ ❛ ❢♦rç❛ F s♦❜r❡ ✉♠ ❝♦r♣♦ ♣❛rt✐❝✉❧❛r ✐♠❡rs♦ ❡♠ ✉♠❛ ❝♦rr❡♥t❡ ❞❡ ✢✉✐❞♦

❞❡♣❡♥❞❡ ❛♣❡♥❛s✿

✕ ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ❝♦r♣♦ L✱

✕ ❞❛ ✈❡❧♦❝✐❞❛❞❡ ❞❛ ❝♦rr❡♥t❡ V✱

✕ ❞❛ ♠❛ss❛ ❡s♣❡❝í✜❝❛ ❞♦ ✢✉✐❞♦ ρ ❡

✕ ❞❛ ✈✐s❝♦s✐❞❛❞❡ ❞♦ ✢✉✐❞♦ µ✱

♦✉ s❡❥❛✱

F =f(L, V, ρ, µ)

❙✉♣♦♥❤❛ q✉❡ ❛ ❣❡♦♠❡tr✐❛ ❡ ❝♦♥❞✐çõ❡s ❞♦ ❡s❝♦❛♠❡♥t♦ s❡❥❛♠ t❛✐s q✉❡ ❛s té❝♥✐❝❛ ❞❛s ❛✉❧❛s ✺ ❡ ✻ ♥ã♦ s❡❥❛♠ ❝❛♣❛③❡s ❞❡ ❞❡t❡r♠✐♥❛r ❛ ❢♦rç❛F✳

❉❡ ♠♦❞♦ ❣❡r❛❧✿

✕ ❞❡♣❡♥❞ê♥❝✐❛ ❡♠ r❡❧❛çã♦ ❛♦ ❝♦♠♣r✐♠❡♥t♦L✱10♣♦♥t♦s ♥❡❝❡ssár✐♦s✱ ♣♦r ❡①❡♠✲

♣❧♦✳

✕ ❞❡♣❡♥❞ê♥❝✐❛ ❡♠ r❡❧❛çã♦ ❛♦ ❝♦♠♣r✐♠❡♥t♦V✱10♣♦♥t♦s ♥❡❝❡ssár✐♦s✱ ♣♦r ❡①❡♠✲

♣❧♦✳

✕ ❞❡♣❡♥❞ê♥❝✐❛ ❡♠ r❡❧❛çã♦ ❛♦ ❝♦♠♣r✐♠❡♥t♦µ✱10♣♦♥t♦s ♥❡❝❡ssár✐♦s✱ ♣♦r ❡①❡♠✲

♣❧♦✳

✕ ❞❡♣❡♥❞ê♥❝✐❛ ❡♠ r❡❧❛çã♦ ❛♦ ❝♦♠♣r✐♠❡♥t♦ρ✱10♣♦♥t♦s ♥❡❝❡ssár✐♦s✱ ♣♦r ❡①❡♠✲

♣❧♦✳

❯♠ t♦t❛❧ ❞❡ 104 ❡①♣❡r✐♠❡♥t♦s✱ ♦ q✉❡ é ♠✉✐t♦ ❝✉st♦s♦✳ ❊♥tr❡t❛♥t♦✱ ❞❛ ❛♥á❧✐s❡

❞✐♠❡♥s✐♦♥❛❧✱ ♣♦❞❡♠♦s r❡❞✉③✐r ❛ ❡q✉❛çã♦ ❛❝✐♠❛ à ❢♦r♠❛ ❡q✉✐✈❛❧❡♥t❡✿

F ρV2

L2 =g

ρV L

µ

♦✉

CF =g(Re),

✐st♦ é✱ ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❢♦rç❛ ❛❞✐♠❡♥s✐♦♥❛❧F/(ρV2

L2

)é ❢✉♥çã♦ ❛♣❡♥❛s ❞♦ ♥ú♠❡r♦

❞❡ ❘❡②♥♦❧❞s ❛❞✐♠❡♥s✐♦♥❛❧ ρV L/µ✳ ❆ ❢✉♥çã♦ g é ♠❛t❡♠❛t✐❝❛♠❡♥t❡ ❞✐❢❡r❡♥t❡ ❞❛

❢✉♥çã♦ f ♦r✐❣✐♥❛❧✱ ♠❛s ❝♦♥t❡♠ ❛s ♠❡s♠❛s ✐♥❢♦r♠❛çõ❡s✳

P❛r❛ ❡st❛❜❡❧❡❝❡rg✱ ❜❛st❛ r❡❛❧✐③r ♦ ❡①♣❡r✐♠❡♥t♦ ♣❛r❛ ❛♣❡♥❛s ✶✵ ✈❛❧♦r❡s ❞❛ ú♥✐❝❛

✈❛r✐á✈❡❧✱ ♦ ♥ú♠❡r♦ ❞❡ ❘❡②♥♦❧❞s✳

✷✳ ❆ ❛♥á❧✐s❡ ❞✐♠❡♥s✐♦♥❛❧ ♥♦s ❛❥✉❞❛ ❛ ♣❡♥s❛r ❡ ♣❧❛♥❡❥❛r ✉♠ ❡①♣❡r✐♠❡♥t♦ ♦✉ ✉♠❛ t❡♦r✐❛✳ ❊❧❛ s✉❣❡r❡✿

✕ ❢♦r♠❛s ❛❞✐♠❡♥s✐♦♥❛✐s ❞❡ ❡s❝r❡✈❡r ❛s ❡q✉❛çõ❡s✱ ✕ ✈❛r✐á✈❡✐s q✉❡ ♣♦❞❡♠ s❡r ❞❡s❝❛rt❛❞❛s✱

❛❧é♠ ❞❡ ♥♦s ❢♦r♥❡❝❡r ✉♠❛ ❜♦❛ ❞♦s❡ ❞❡ ♣❡r❝❡♣çã♦ s♦❜r❡ ❛s ❢♦r♠❛s ❞❛s r❡❧❛çõ❡s ❢ís✐❝❛s q✉❡ ❡st❛♠♦s t❡♥t❛♥❞♦ ❡st✉❞❛r✳

✸✳ ❆ ❛♥á❧✐s❡ ❞✐♠❡♥s✐♦♥❛❧ ♥♦s ❢♦r♥❡❝❡ ❧❡✐s ❞❡ s❡♠❡❧❤❛♥ç❛ q✉❡ ♥♦s ♣❡r♠✐t❡ ❝♦♥✈❡r✲ t❡r ❞❛❞♦s ❞❡ ✉♠ ♠♦❞❡❧♦ ✭♣❡q✉❡♥♦ ❡ ❜❛r❛t♦✮ ❡♠ ✐♥❢♦r♠❛çã♦ ❞❡ ♣r♦❥❡t♦ ❞❡ ✉♠ ♣r♦tót✐♣♦ ✭❣r❛♥❞❡ ❡ ❝❛r♦✮✳

✷ Pr✐♥❝í♣✐♦ ❞❛ ❍♦♠♦❣❡♥❡✐❞❛❞❡ ❉✐♠❡♥s✐♦♥❛❧

❆♦ r❡❛❧✐③❛r♠♦s ❛ r❡❞✉çã♦ ❞♦ ♥ú♠❡r♦ ❞❡ ✈❛r✐á✈❡✐s ❞❡ ✺ ♣❛r❛ ✷ ♥♦ ♣r♦❜❧❡♠❛ ❞❛ ❢♦rç❛ ❞❡ ❛rr❛st♦ ❡♠ ✉♠ ❝♦r♣♦ s♦❜ ❛çã♦ ❞❡ ✉♠❛ ❝♦rr❡♥t❡ ✉♥✐❢♦r♠❡ ✈✐st♦ ❛❝✐♠❛✱ ❡①♣❧♦r❛♠♦s ✉♠❛ r❡❣r❛ q✉❡ é q✉❛s❡ ✉♠ ❛①✐♦♠❛ ❛✉t♦✲❡✈✐❞❡♥t❡ ❡♠ ❢ís✐❝❛✳ ❊ss❛ r❡❣r❛✱ ♦ ♣r✐♥❝í♣✐♦ ❞❛ ❤♦♠♦❣❡♥❡✐❞❛❞❡ ❞✐♠❡♥s✐♦♥❛❧✱ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿

• ❙❡ ✉♠❛ ❡q✉❛çã♦ ❡①♣r✐♠❡ r❡❛❧♠❡♥t❡ ✉♠❛ r❡❧❛çã♦ ❛♣r♦♣r✐❛❞❛ ❡♥tr❡ ❛s ✈❛r✐á✈❡✐s ❡♠ ✉♠

♣r♦❝❡ss♦ ❢ís✐❝♦✱ ❡❧❛ s❡rá ❞✐♠❡♥s✐♦♥❛❧♠❡♥t❡ ❤♦♠♦❣ê♥❡❛❀ ✐st♦ é✱ ❝❛❞❛ ✉♠ ❞❡ s❡✉s t❡r♠♦s ❛❞✐t✐✈♦s t❡rá ❛ ♠❡s♠❛ ❞✐♠❡♥sã♦✳

• ❚♦❞❛s ❛s ❡q✉❛çõ❡s ❞❛ ♠❡❝â♥✐❝❛ t❡ór✐❝❛ sã♦ ❞❡ss❛ ❢♦r♠❛✳ P♦r ❡①❡♠♣❧♦✱ ❝♦♥s✐❞❡r❡ ❛

r❡❧❛çã♦ q✉❡ ❡①♣r❡ss❛ ♦ ❞❡s❧♦❝❛♠❡♥t♦ ❞❡ ✉♠ ❝♦r♣♦ ❡♠ q✉❡❞❛ ❧✐✈r❡

S =S0+V0t+

1 2gt

2

❈❛❞❛ t❡r♠♦ ❞❡ss❛ ❡q✉❛çã♦ é ✉♠ ❞❡s❧♦❝❛♠❡♥t♦✱ ♦✉ ❝♦♠♣r✐♠❡♥t♦✱ t❡♥❞♦ ❞✐♠❡♥sã♦{L_}✳

❆ ❡q✉❛çã♦ é ❞✐♠❡♥s✐♦♥❛❧♠❡♥t❡ ❤♦♠♦❣ê♥❡❛✳ ❖❜s❡r✈❡ t❛♠❜❡♠ q✉❡ q✉❛❧q✉❡r ❝♦♥❥✉♥t♦ ❝♦♥s✐st❡♥t❡ ❞❡ ✉♥✐❞❛❞❡s ♣♦❞❡ s❡r ✉s❛❞♦ ♣❛r❛ ❝❛❧❝✉❧❛r ✉♠ r❡s✉❧t❛❞♦✳

• ❖✉tr♦ ❡①❡♠♣❧♦✿ ❝♦♥s✐❞❡r❡ ❛ ❡q✉❛çã♦ ❞❡ ❇❡r♥♦✉❧❧✐ ♣❛r❛ ❡s❝♦❛♠❡♥t♦ ✐♥❝♦♠♣r❡ssí✈❡❧

p ρ +

1 2V

2

+gz = ❝♦♥st

❈❛❞❛ t❡r♠♦✱ ✐♥❝❧✉✐♥❞♦ ❛ ❝♦♥st❛♥t❡✱ t❡♠ ❞✐♠❡♥sã♦ ❞❡ ✈❡❧♦❝✐❞❛❞❡ ❛♦ q✉❛❞r❛❞♦✱ ♦✉

{L2

T−2_}✳ ❆ ❡q✉❛çã♦ é ❞✐♠❡♥s✐♦♥❛❧♠❡♥t❡ ❤♦♠♦❣ê♥❡❛ ❡ ❢♦r♥❡❝❡ r❡s✉❧t❛❞♦s ❛♣r♦♣r✐❛❞♦s ♣❛r❛ q✉❛❧q✉❡r ❝♦♥❥✉♥t♦ ❝♦♥s✐st❡♥t❡ ❞❡ ✉♥✐❞❛❞❡s✳

• ❊s❝♦❧❤❛ ❞❡ ❱❛r✐á✈❡✐s ❡ P❛râ♠❡tr♦s ❞❡ ❊s❝❛❧❛

❆ ❡q✉❛çã♦ q✉❡ ❡①♣r❡ss❛ ♦ ❞❡s❧♦❝❛♠❡♥t♦ ❞❡ ✉♠ ❝♦r♣♦ ❡♠ q✉❡❞❛ ❧✐✈r❡

S =S0+V0t+

1 2gt

2

❝♦♥t❡♠ ❝✐♥❝♦ ❣r❛♥❞❡③❛s (S, S0, V0, g, t) q✉❡ ♣♦❞❡♠♦s ❞✐✈✐❞✐r✱ ❡♠ ♥♦ss♦ ♣❡♥s❛♠❡♥t♦✱

❡♠ ✈❛r✐á✈❡✐s ❡ ♣❛râ♠❡tr♦s✳

✕ ❆s ✈❛r✐á✈❡✐s sã♦ ❛s q✉❛♥t✐❞❛❞❡s q✉❡ ❞❡s❡❥❛♠♦s ♣❧♦t❛r✱ ❛ s❛í❞❛ ❜ás✐❝❛ ❞♦ ❡①♣❡r✐✲ ♠❡♥t♦ ♦✉ t❡♦r✐❛✳ ◆♦ ❝❛s♦ ❞❛ ❡q✉❛çã♦ ❛❝✐♠❛✱ S ✈❡rs✉s t✳

✕ ❖s ♣❛râ♠❡tr♦s sã♦ ❛q✉❡❧❛s ❣r❛♥❞❡③❛s ❝✉❥♦ ❡❢❡✐t♦ s♦❜r❡ ❛s ✈❛r✐á✈❡✐s ❞❡s❡❥❛♠♦s ❝♦♥❤❡❝❡r✳ ◆♦ ❝❛s♦ ❞❛ ❡q✉❛çã♦ ❛❝✐♠❛✱ S0✱ V0 ❡ g✳

✕ P❛r❛ ❛❞✐♠❡♥s✐♦♥❛❧✐③❛r ♥♦ss♦s r❡s✉❧t❛❞♦s✱ ♣r❡❝✐s❛♠♦s s❛❜❡r q✉❛♥t❛s ❞✐♠❡♥sõ❡s ❡stã♦ ❝♦♥t✐❞❛s ❡♥tr❡ ♥♦ss❛s ✈❛r✐á✈❡✐s ❡ ♣❛râ♠❡tr♦s✳

◆♦ ♣r♦❜❧❡♠❛ ❞❡ q✉❡❞❛ ❧✐✈r❡✱ t❡♠♦s ❛♣❡♥❛s ❞✉❛s ❞✐♠❡♥sõ❡s✱ ❝♦♠♣r✐♠❡♥t♦{L_}❡

t❡♠♣♦ {T_}✳ ❊s❝r❡✈❛ ❛ ❞✐♠❡♥sã♦ ❞❡ ❝❛❞❛ ❣r❛♥❞❡③❛✿

{S_}=_{S0}={L}

{t_}=_{T_}

{V0}={LT− 1

} {g_}=_{LT−2

}

❊♥tr❡ ♦s ♥♦ss♦s ♣❛râ♠❡tr♦s s❡❧❡❝✐♦♥❛♠♦s ❞♦✐s ♣❛r❛ s❡r❡♠ ♣❛râ♠❡tr♦s ❞❡ ❡s❝❛❧❛✱ ✉s❛❞♦s ♥❛ ❞❡✜♥✐çã♦ ❞❡ ✈❛r✐á✈❡✐s ❛❞✐♠❡♥s✐♦♥❛✐s✳ ❆q✉❡❧❡s q✉❡ r❡st❛r❡♠ s❡rã♦ ♦s ♣❛râ♠❡tr♦s ❜ás✐❝♦s✱ ❝✉❥♦ ❡❢❡✐t♦ q✉❡r❡♠♦s ♠♦str❛r✳ ❊ss❛s ❡s❝♦❧❤❛s ✐rã♦ ❛❢❡t❛r ❛♣❡♥❛s ❛ ❢♦r♠❛ ❞❛ ❛♣r❡s❡♥t❛çã♦ ❞♦s r❡s✉❧t❛❞♦s✱ ♠❛s ♥ã♦ ♦ ❝♦♥t❡ú❞♦✳

P❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❡ q✉❡❞❛ ❧✐✈r❡✱ ❡♥tr❡ ♦s três ♣❛râ♠❡tr♦s✱ s❡❧❡❝✐♦♥❛♠♦s ❞♦✐s q✉❛✐sq✉❡r ♣❛r❛ s❡r❡♠ ♣❛râ♠❡tr♦s ❞❡ ❡s❝❛❧❛✳ ▲♦❣♦ t❡♠♦s três ♦♣çõ❡s✿

✶✳ P❛râ♠❡tr♦s ❞❡ ❡s❝❛❧❛ S0 ❡ V0✿ ♦ ❡❢❡✐t♦ ❞❛ ❣r❛✈✐❞❛❞❡ g✳ ❉❡s❧♦❝❛♠❡♥t♦ ❡

t❡♠♣♦ ❛❞✐♠❡♥s✐♦♥❛✐s✿

S∗ ₌ S

S0

t∗ ₌ V0t

S0

❋♦r♠❛ ❛❞✐♠❡♥s✐♦♥❛❧ ❞❛ ❡q✉❛çã♦ ❞♦ ❞❡s❧♦❝❛♠❡♥t♦ ♣❛r❛ q✉❡❞❛ ❧✐✈r❡✿

S∗ _{= 1 +t}∗₊1

2α(t

∗₎2

❝♦♠ α = gS0

V2 0

❍á ✉♠ ú♥✐❝♦ ♣❛râ♠❡tr♦s ❛❞✐♠❡♥s✐♦♥❛❧α q✉❡ ♥❡ss❡ ❝❛s♦ ♠♦str❛ ♦ ❡❢❡✐t♦ ❞❛

❣r❛✈✐❞❛❞❡✳

✷✳ P❛râ♠❡tr♦s ❞❡ ❡s❝❛❧❛ V0 ❡ g✿ ♦ ❡❢❡✐t♦ ❞♦ ❞❡s❧♦❝❛♠❡♥t♦ ✐♥✐❝✐❛❧ S0✳ ❉❡s❧♦❝❛✲

♠❡♥t♦ ❡ t❡♠♣♦ ❛❞✐♠❡♥s✐♦♥❛✐s✿

S∗ ₌ Sg

V2 0

t∗ ₌ gt

V0

❋♦r♠❛ ❛❞✐♠❡♥s✐♦♥❛❧ ❞❛ ❡q✉❛çã♦ ❞♦ ❞❡s❧♦❝❛♠❡♥t♦ ♣❛r❛ q✉❡❞❛ ❧✐✈r❡✿

S∗ _=α+t∗₊ 1

2(t

∗₎2

❝♦♠ α= gS0

V2 0

◆♦✈❛♠❡♥t❡✱ t❡♠♦s ♦ ú♥✐❝♦ ♣❛râ♠❡tr♦s ❛❞✐♠❡♥s✐♦♥❛❧αq✉❡ ♥❡ss❡ ❝❛s♦ ♠♦str❛

♦ ❡❢❡✐t♦ ❞♦ ❞❡s❧♦❝❛♠❡♥t♦ ✐♥✐❝✐❛❧✳

✸✳ P❛râ♠❡tr♦s ❞❡ ❡s❝❛❧❛S0 ❡g✿ ♦ ❡❢❡✐t♦ ❞❛ ✈❡❧♦❝✐❞❛❞❡ ✐♥✐❝✐❛❧V0✳ ❉❡s❧♦❝❛♠❡♥t♦

❡ t❡♠♣♦ ❛❞✐♠❡♥s✐♦♥❛✐s✿

S∗ ₌ S

S0

t∗ _=t

g S0

1/2

❋♦r♠❛ ❛❞✐♠❡♥s✐♦♥❛❧ ❞❛ ❡q✉❛çã♦ ❞♦ ❞❡s❧♦❝❛♠❡♥t♦ ♣❛r❛ q✉❡❞❛ ❧✐✈r❡✿

S∗ _{= 1 +√}1

αt

∗₊1

2(t

∗₎2 ❝♦♠ _α= gS0

V2 0

❍á ✉♠ ú♥✐❝♦ ♣❛râ♠❡tr♦s ❛❞✐♠❡♥s✐♦♥❛❧α q✉❡ ♥❡ss❡ ❝❛s♦ ♠♦str❛ ♦ ❡❢❡✐t♦ ❞❛

✈❡❧♦❝✐❞❛❞❡ ✐♥✐❝✐❛❧✳

✸ ❖ ❚❡♦r❡♠❛

π

❞❡ ❇✉❝❦✐♥❣❤❛♠

• ❊①✐st❡♠ ✈ár✐♦s ♠ét♦❞♦s ♣❛r❛ r❡❞✉③✐r ✉♠ ♥ú♠❡r♦ ❞❡ ✈❛r✐á✈❡✐s ❞✐♠❡♥s✐♦♥❛✐s ❛ ✉♠

♥ú♠❡r♦ ♠❡♥♦r ❞❡ ❣r✉♣♦s ❛❞✐♠❡♥s✐♦♥❛✐s✳

• ❖ ❡sq✉❡♠❛ ❞❛❞♦ ❢♦✐ ♣r♦♣♦st♦ ❡♠ ✶✾✶✹ ♣♦r ❇✉❝❦✐♥❣❤❛♠✱ ❡ é ❝♦♥❤❡❝✐❞♦ ❤♦❥❡ ❝♦♠♦

❚❡♦r❡♠❛ π ❞❡ ❇✉❝❦✐♥❣❤❛♠✳

• ❖ ♥♦♠❡ π ✈❡♠ ❞❛ ♥♦t❛çã♦ ♠❛t❡♠át✐❝❛Π✱ ♣❛r❛ r❡♣r❡s❡♥t❛r ✉♠ ♣r♦❞✉t♦ ❞❡ ✈❛r✐á✈❡✐s✳

❖s ❣r✉♣♦s ❛❞✐♠❡♥s✐♦♥❛✐s ❡♥❝♦♥tr❛❞♦s ❛ ♣❛rt✐r ❞♦ t❡♦r❡♠❛ sã♦ ♣r♦❞✉t♦s ❞❡ ♣♦t❡♥❝✐❛ ❞❡♥♦t❛❞♦s ♣♦r Π1,Π2,Π3✱ ❡t❝✳

• ❆ ♣r✐♠❡✐r❛ ♣❛rt❡ ❞♦ t❡♦r❡♠❛ π ❞❡t❡r♠✐♥❛ ❛ r❡❞✉çã♦ ❞❡ ✈❛r✐á✈❡✐s ❡s♣❡r❛❞❛✿

❙❡ ✉♠ ♣r♦❝❡ss♦ ❢ís✐❝♦ s❛t✐s❢❛③ ♦ ♣r✐♥❝í♣✐♦ ❞❛ ❤♦♠♦❣♥❡✐❞❛❞❡ ❞✐♠❡♥s✐♦♥❛❧ ❡ ❡♥✈♦❧✈❡ n

✈❛r✐á✈❡✐s ❞✐♠❡♥s✐♦♥❛✐s✱ ❡❧❡ ♣♦❞❡ s❡r r❡❞✉③✐❞♦ ❛ ✉♠❛ r❡❧❛çã♦ ❡♥tr❡ ❛♣❡♥❛s k ✈❛r✐á✈❡✐s

❛❞✐♠❡♥s✐♦♥❛✐s Π➫s✳ ❆ r❡❞✉çã♦ j = n ₋k é ✐❣✉❛❧ ❛♦ ♥ú♠❡r♦ ♠á①✐♠♦ ❞❡ ✈❛r✐á✈❡✐s

q✉❡ ♥ã♦ ❢♦r♠❛♠ ✉♠ π ❡♥tr❡ s✐ ♠❡s♠❛s✱ s❡♥❞♦ s❡♠♣r❡ ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛♦ ♥ú♠❡r♦ ❞❡

❞✐♠❡♥sõ❡s q✉❡ ❞❡s❝r❡✈❡♠ ❛s ✈❛r✐á✈❡✐s✳

P♦r ❡①❡♠♣❧♦✱ ♥♦ ❝❛s♦ ❡s♣❡❝í✜❝♦ ❞❛ ❢♦rç❛ s♦❜r❡ ✉♠ ❝♦r♣♦ ✐♠❡rs♦✱ ♦ ♣r♦❜❧❡♠❛ ❛♣r❡s❡♥t❛ ✺ ✈❛r✐á✈❡✐s✱F✱L✱U✱ρ❡µ✱ ❞❡s❝r✐t❛s ♣♦r três ❞✐♠❡♥sõ❡s_{M LT_}✳ ▲♦❣♦✱ n= 5❡j = 3✳

P♦rt❛♥t♦✱ ♣♦❞❡♠♦s r❡❞✉③✐r ♦ ♣r♦❜❧❡♠❛ ❛ k= 2 ❛❞✐♠❡♥s✐♦♥❛✐s✱ Π1 =CF ❡Π2 =Re✳

• ❆ s❡❣✉♥❞❛ ♣❛rt❡ ❞♦ t❡♦r❡♠❛ ♠♦str❛ ❝♦♠♦ ❡♥❝♦♥tr❛r ♦s ❛❞✐♠❡♥s✐♦♥❛✐s✱ ✉♠ ❞❡ ❝❛❞❛ ✈❡③✿

❊♥❝♦♥tr❛❞❛ ❛ r❡❞✉çã♦ j✱ s❡❧❡❝✐♦♥❛♠♦s ❡♥tã♦ j ✈❛r✐á✈❡✐s ❞❡ ❡s❝❛❧❛ q✉❡ ♥ã♦ ❢♦r♠❡♠

✉♠ ❣r✉♣♦ ❛❞✐♠❡♥s✐♦♥❛❧ Π ❡♥tr❡ ❡❧❛s ♠❡s♠❛s✳ ❈❛❞❛ ❣r✉♣♦ ♦✉ ✈❛r✐á✈❡❧ ❛❞✐♠❡♥s✐♦♥❛❧ Π s❡rá ✉♠ ♣r♦❞✉t♦ ❞❡ ♣♦t❡♥❝✐❛s ❞❡ss❛s j ✈❛r✐á✈❡✐s ❡ ✉♠❛ ✈❛r✐á✈❡❧ ❛❞✐❝✐♦♥❛❧✱ à q✉❛❧ é

❛tr✐❜✉✐❞♦ q✉❛❧q✉❡r ❡①♣♦❡♥t❡ ♥ã♦✲♥✉❧♦ ❝♦♥✈❡♥✐❡♥t❡✳ ❈❛❞❛ ❣r✉♣♦ ♦✉ ✈❛r✐á✈❡❧ ❛❞✐♠❡♥s✐✲ ♦♥❛❧ Π ❛ss✐♠ ❡♥❝♦♥tr❛❞❛ é ✐♥❞❡♣❡♥❞❡♥t❡✳

P♦r ❡①❡♠♣❧♦✱ ❝♦♥s✐❞❡r❡ q✉❡ ♦ ♣r♦❝❡ss♦ ❡♥✈♦❧✈❡ ❝✐♥❝♦ ✈❛r✐á✈❡✐s

ν1 =f(ν2, ν3, ν4, ν5)

❙✉♣♦♥❤❛ q✉❡ ❡①✐st❛♠ três ❞✐♠❡♥sõ❡s {M LT_} ❡✱ ❛♣ós ✐♥s♣❡çã♦✱ ❞❡t❡r♠✐♥❡♠♦s q✉❡✱

❞❡ ❢❛t♦✱ j = 3✳ ❊♥tã♦✱ k = 5₋3 = 2 ❡✱ ❞♦ t❡♦r❡♠❛✱ ❡s♣❡r❛♠♦s ❞♦✐s ❡ ❛♣❡♥❛s ❞♦✐s

❣r✉♣♦s ❛❞✐♠❡♥s✐♦♥❛✐s Π✳ ❙❡♣❛r❡ três ✈❛r✐á✈❡✐s ❝♦♥✈❡♥✐❡♥t❡s q✉❡ ♥ã♦ ❢♦r♠❡♠ ✉♠Π✱ ❡

s✉♣♦♥❤❛ q✉❡ ❡❧❛s s❡❥❛♠ ν2, ν3 ❡ν4✳ ❡♥tã♦✱ ♦s ❞♦✐s ❣r✉♣♦s Π t❡rã♦ ❛ ❢♦r♠❛✿

Π1 =(ν2)a(ν3)b(ν4)cν1 =M 0

L0

T0

Π2 =(ν2)a(ν3)b(ν4)cν5 =M 0

L0

T0

■❣✉❛❧❛♥❞♦ ♦ ❡①♣♦❡♥t❡ ❞❛s ✈ár✐❛s ❞✐♠❡♥sõ❡s✱ ♦ t❡♦r❡♠❛ ♥♦s ❣❛r❛♥t❡ ✈❛❧♦r❡s ú♥✐❝♦s ♣❛r❛

a, b ❡c✳

• ❘♦t❡✐r♦✿

✶✳ ▲✐st❡ ❡ ❝♦♥t❡ ❛s n ✈❛r✐á✈❡✐s ❡♥✈♦❧✈✐❞❛s ♥♦ ♣r♦❜❧❡♠❛✳ ❙❡ ❢❛❧t❛r ✉♠❛ ✈❛r✐á✈❡❧

✐♠♣♦rt❛♥t❡✱ ❛ ❛♥á❧✐s❡ ❞✐♠❡♥s✐♦♥❛❧ ✐rá ❢❛❧❤❛r✳

✷✳ ▲✐st❡ ❛ ❞✐♠❡♥sã♦ ❞❡ ❝❛❞❛ ✈❛r✐á✈❡❧ ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ s✐st❡♠❛{M LT θ_}♦✉_{F LT θ_}✳

✸✳ ❊♥❝♦♥tr❡ j✳ ■♥✐❝✐❛❧♠❡♥t❡✱ ❡s❝♦❧❤❛ j ✐❣✉❛❧ ❛♦ ♥ú♠❡r♦ ❞❛s ❞✐❢❡r❡♥t❡s ❞✐♠❡♥sõ❡s

♣r❡s❡♥t❡s✱ ❡ ♣r♦❝✉r❡j ✈❛r✐á✈❡✐s q✉❡ ♥ã♦ ❢♦r♠❡♠ ✉♠ ❣r✉♣♦ ❛❞✐♠❡♥s✐♦♥❛❧Π ❡♥tr❡

s✐✳ ❙❡ ♥ã♦ ❢♦r ♣♦ssí✈❡❧✱ r❡❞✉③❛j ❞❡ ✶ ❡ ♣r♦❝✉r❡ ♥♦✈❛♠❡♥t❡✳

✹✳ ❙❡❧❡❝✐♦♥❡j ♣❛râ♠❡tr♦s ❞❡ ❡s❝❛❧❛ q✉❡ ♥ã♦ ❢♦r♠❡♠ ✉♠ ❣r✉♣♦ ❛❞✐♠❡♥s✐♦♥❛❧Π❡♥tr❡

s✐✳ ❈❡rt✐✜q✉❡✲s❡ q✉❡ s❡❥❛♠ s❛t✐s❢❛tór✐♦s ❡✱ s❡ ♣♦sí✈❡❧✱ t❡♥❤❛♠ ❛❧❣✉♠❛ ❣❡♥❡r❛❧✐✲ ❞❛❞❡✱ ♣♦r q✉❡ ❡❧❡s ✐rã♦ ❛♣❛r❡❝❡r ❡♠ t♦❞♦s ♦s ❣r✉♣♦s ❛❞✐♠❡♥s✐♦♥❛✐s Π✳ ❊s❝♦❧❤❛

❛ ♠❛ss❛ ❡s♣❡❝í✜❝❛✱ ✉♠❛ ✈❡❧♦❝✐❞❛❞❡ ♦✉ ✉♠ ❝♦♠♣r✐♠❡♥t♦✳

✺✳ ❆❣r❡❣✉❡ ✉♠❛ ✈❛r✐á✈❡❧ ❛❞✐❝✐♦♥❛❧ ás j ✈❛r✐á✈❡✐s r❡♣❡t✐t✐✈❛s✱ ❡ ❢♦r♠❡ ✉♠ ♣r♦❞✉t♦

❞❡ ♣♦t❡♥❝✐❛s✳ ❆❧❣❡❜r✐❝❛♠❡♥t❡✱ ❡♥❝♦♥tr❡ ♦s ❡①♣♦❡♥t❡s q✉❡ t♦r♠❛♠ ♦ ♣r♦❞✉t♦ ❛❞✐♠❡♥s✐♦♥❛❧✳ ❋❛ç❛ ✐ss♦ s❡qü❡♥❝✐❛❧♠❡♥t❡✱ ❛❝r❡s❝❡♥t❛♥❞♦ ✉♠❛ ♥♦✈❛ ✈❛r✐á✈❡❧ ❞❡ ❝❛❞❛ ✈❡③✱ ❡ ✈♦❝❡ ✐rá ❡♥❝♦♥tr❛r t♦❞♦s ♦s k = n ₋j ❣r✉♣♦s ❛❞✐♠❡♥s✐♦♥❛✐s Π

❞❡s❡❥❛❞♦s✳

✻✳ ❊s❝r❡✈❛ ❛ ❢✉♥çã♦ ❛❞✐♠❡♥s✐♦♥❛❧ ✜♥❛❧✱ ❡ ❝♦♥✜r❛ ♦ s❡✉ tr❛❜❛❧❤♦ ♣❛r❛ t❡r ❝❡rt❡③❛ q✉❡ t♦❞♦s ♦s ❣r✉♣♦s ❛❞✐♠❡♥s✐♦♥❛✐s Π s❡❥❛♠ r❡❛❧♠❡♥t❡ ❛❞✐♠❡♥s✐♦♥❛✐s✳

❊①❡♠♣❧♦ ✶

❈♦♥s✐❞❡r❡ q✉❡ ❛ ❢♦rç❛F s♦❜r❡ ✉♠ ❝♦r♣♦ ♣❛rt✐❝✉❧❛r ✐♠❡rs♦ ❡♠ ✉♠❛ ❝♦rr❡♥t❡ ❞❡ ✢✉✐❞♦ ❞❡✲

♣❡♥❞❡ ❛♣❡♥❛s ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ❝♦r♣♦L✱ ❞❛ ✈❡❧♦❝✐❞❛❞❡ ❞❛ ❝♦rr❡♥t❡U✱ ❞❛ ♠❛ss❛ ❡s♣❡❝í✜❝❛ ρ ❡ ❞❛ ✈✐s❝♦s✐❞❛❞❡ ❞♦ ✢✉✐❞♦ µ✳ ❉❡t❡r♠✐♥❡ ♦s ❣r✉♣♦s ❛❞✐♠❡♥s✐♦♥❛✐s Π✳

❊①❡♠♣❧♦ ✷

❘❡❞✉③❛ ❛ r❡❧❛çã♦ ❞♦ ❝♦r♣♦ ❡♠ q✉❡❞❛ ❧✐✈r❡ ❛ ✉♠❛ ❢✉♥çã♦ ❞❡ ✈❛r✐á✈❡✐s ❛❞✐♠❡♥s✐♦♥❛✐s✳ P♦r q✉❡ ❡①✐st❡♠ três ❢♦r♠✉❧❛çõ❡s ❞✐❢❡r❡♥t❡s❄

❊①❡♠♣❧♦ ✸

❊♠ ❜❛✐①❛s ✈❡❧♦❝✐❞❛❞❡s ✭❡s❝♦❛♠❡♥t♦ ❧❛♠✐♥❛r✮✱ ❛ ✈❛③ã♦ ✈♦❧✉♠étr✐❝❛Q❛tr❛✈és ❞❡ ✉♠ t✉❜♦

❞❡ ♣❡q✉❡♥♦ ❞✐â♠❡tr♦ é ✉♠❛ ❢✉♥çã♦ ❛♣❡♥❛s ❞♦ r❛✐♦ ❞♦ t✉❜♦ R✱ ❞❛ ✈✐s❝♦s✐❞❛❞❡ ❞♦ ✢✉✐❞♦ µ

❡ ❞❛ q✉❡❞❛ ❞❡ ♣r❡ssã♦ ♣♦r ✉♥✐❞❛❞❡ ❞❡ ❝♦♠♣r✐♠❡♥t♦ dp/dx✳ ❯s❛♥❞♦ ♦ t❡♦r❡♠❛ π✱ ❡♥❝♦♥tr❡

✉♠❛ r❡❧❛çã♦ ❛❞✐♠❡♥s✐♦♥❛❧ ❛♣r♦♣r✐❛❞❛✳

❊①❡♠♣❧♦ ✹

❈♦♥s✐❞❡r❡ q✉❡ ❛ ❞❡✢❡①ã♦ δ ♥❛ ♣♦♥t❛ ❞❡ ✉♠❛ ✈✐❣❛ ❡♠ ❜❛❧❛♥ç♦ é ✉♠❛ ❢✉♥çã♦ ❞♦ ❝❛rr❡✲

❣❛♠❡♥t♦ ♥❛ ♣♦♥t❛ P✱ ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ✈✐❣❛ L✱ ❞♦ ♠♦♠❡♥t♦ ❞❡ ✐♥ér❝✐❛ I✱ ❞♦ ♠ó❞✉❧♦ ❞❡

❡❧❛st✐❝✐❞❛❞❡ E✱ ✐st♦ é✱ δ = f(P, L, I, E)✳ ❘❡❡s❝r❡✈❛ ❡ss❛ ❢✉♥çã♦ ♥❛ ❢♦r♠❛ ❞✐♠❡♥s✐♦♥❛❧✱ ❡

❝♦♠❡♥t❡ s♦❜r❡ s✉❛ ❝♦♠♣❧❡①✐❞❛❞❡ ❡ s♦❜r❡ ♦ ✈❛❧♦r ♣❡❝✉❧✐❛r ❞❡ j✳

✹ ❆❞✐♠❡♥s✐♦♥❛❧✐③❛çã♦ ❞❛s ❊q✉❛çõ❡s ❇ás✐❝❛s

• ❯♠❛ ♦✉tr❛ té❝♥✐❝❛ ❡✜❝❛③ é ❛❜♦r❞❛r ❛s ❡q✉❛çõ❡s ❞❛ ❛✉❧❛ ✻ s♦❜ ♦ ❡♥❢♦q✉❡ ❞❛ ❛♥á❧✐s❡

❞✐♠❡♥s✐♦♥❛❧✳

• ❊♠❜♦r❛ ❡♠ ❣❡r❛❧ ♥ã♦ ♣♦ss❛♠♦s r❡s♦❧✈❡r ❡ss❛s ❡q✉❛çõ❡s✱ ❡❧❛s ✐rã♦ r❡✈❡❧❛r ♣❛râ♠❡tr♦s

❛❞✐♠❡♥s✐♦♥❛✐s ❜ás✐❝♦s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ♦ ♥ú♠❡r♦ ❞❡ ❘❡②♥♦❧❞s✱ ♥❛s s✉❛ ❢♦r♠❛ ❡ ♣♦s✐çõ❡s ❛♣r♦♣r✐❛❞❛s✱ ❢♦r♥❡❝❡♥❞♦ ✐♥❞í❝✐♦s ❞❡ q✉❛♥❞♦ ♦s t❡r♠♦s ♠✉❧t✐♣❧✐❝❛❞♦s ♣♦r ❡ss❡s ❛❞✐♠❡♥s✐♦♥❛✐s ♣♦❞❡rã♦ s❡r ❞❡s♣r❡③í✈❡✐s✳

• ❆s ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦ t❛♠❜❡♠ ♣♦❞❡♠ s❡r ❛❞✐♠❡♥s✐♦♥❛❧✐③❛❞❛s✳

• ❱❛♠♦s ❛♣❧✐❝❛r ❡ss❛ té❝♥✐❝❛ ♣❛r❛ ❛s ❡q✉❛çõ❡s ❞❡ ❣♦✈❡r♥♦ ❞❡ ✉♠ ❡s❝♦❛♠❡♥t♦ ✐♥❝♦♠✲

♣r❡ssí✈❡❧ ❞❡ ✉♠ ✢✉✐❞♦ ◆❡✇t♦♥✐❛♥♦ ❝♦♠ ✈✐s❝♦s✐❞❛❞❡ ❝♦♥st❛♥t❡✿ ✕ ❊q✉❛çã♦ ❞❡ ❝♦♥t✐♥✉✐❞❛❞❡✿

~

∇ ·V~ = 0

✕ q✉❛♥t✐❞❛❞❡ ❞❡ ♠♦✈✐♠❡♥t♦✿

ρd~V

dt =ρ~g−∇~p+µ∇

2_~

V

✕ ❙✉♣❡r❢í❝✐❡ só❧✐❞❛✿

~ V = 0

✕ ❊♥tr❛❞❛ ♦✉ s❛í❞❛✿

~

V , p❝♦♥❤❡❝✐❞❛s

✕ ❙✉♣❡r❢í❝✐❡ ❧✐✈r❡✿

w= dη

dt, p=pa+ρgz−σ(R

−1

x +R−

1

y )❡♠ z=η

• ❆s ❡q✉❛çõ❡s ❛❝✐♠❛ ❝♦♥t❡♠ ❛s três ❞✐♠❡♥sõ❡s ❜ás✐❝❛s M✱ L ❡T✳

• ❚♦❞❛s ❛ ✈❛r✐á✈❡✐s p, ~V , x, y, z ❡ t ♣♦❞❡♠ s❡r ❛❞✐♠❡♥s✐♦♥❛❧✐③❛❞❛s ✉s❛♥❞♦✲s❡ ❛ ♠❛ss❛

❡s♣❡❝í✜❝❛ ❡ ❞✉❛s ❝♦♥st❛♥t❡s ❞❡ r❡❢❡rê♥❝✐❛ q✉❡ ❞❡✈❡♠ s❡r ❝❛r❛❝t❡r✐st✐❝❛s ❞♦ ❡s❝♦❛♠❡♥t♦ ♣❛rt✐❝✉❧❛r✿

✈❡❧♦❝✐❞❛❞❡ ❞❡ r❡❢❡rê♥❝✐❛ =U

❝♦♠♣r✐♠❡♥t♦ ❞❡ r❡❢❡rê♥❝✐❛=L

P♦r ❡①❡♠♣❧♦✱ U ♣♦❞❡ s❡r ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❡ ❡♥tr❛❞❛ ♦✉ ❞❡ s❛✐❞❛ ❡ L ♦ ❞✐â♠❡tr♦ ❞❡ ✉♠

❝♦r♣♦ ✐♠❡rs♦ ❡♠ ✉♠❛ ❝♦rr❡♥t❡✳

• ❉❡✜♥❛ ❛❣♦r❛ ❛s ✈❛r✐á✈❡✐s ❛❞✐♠❡♥s✐♦♥❛✐s r❡❧❡✈❛♥t❡s✱ ❞❡♥♦t❛♥❞♦✲❛s ♣♦r ✉♠ ❛st❡r✐s❝♦✿

~ V∗ ₌ V~

U x∗ ₌ x

L, y

∗ ₌ y

L, z

∗ ₌ z

L

t∗ ₌ tU

L p∗ ₌ p+ρgz

ρU2

• ❈♦♠♦ ρ, U ❡ L sã♦ t♦❞❛s ❝♦♥st❛♥t❡s✱ ❛s ❞❡r✐✈❛❞❛s q✉❡ ❛♣❛r❡❝❡♠ ♥❛ ❡q✉❛çõ❡s ❞❡ ❣♦✲

✈❡r♥♦ ❞♦ ❡s❝♦❛♠❡♥t♦ ❡ ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦ ♣♦❞❡♠ s❡r ♣♦st❛s ♥❛ ❢♦r♠❛ ❛❞✐♠❡♥s✐♦♥❛❧ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❞✐♠❡♥s✐♦♥❛✐s✳ P♦r ❡①❡♠♣❧♦✿

∂u ∂x =

∂U u∗

∂Lx∗ =

U L

∂u∗

∂x∗

• ❆❞✐♠❡♥s✐♦♥❛❧✐③❛♥❞♦ ❛s ❡q✉❛çõ❡s ❞❡ ❣♦✈❡r♥♦ ❡ ❛s ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦ ♦❜t❡♠♦s✿

✕ ❊q✉❛çã♦ ❞❡ ❝♦♥t✐♥✉✐❞❛❞❡✿

~

∇∗_·V~∗ _{= 0}

✕ q✉❛♥t✐❞❛❞❡ ❞❡ ♠♦✈✐♠❡♥t♦✿

ρd~V

∗

dt∗ =−

~

∇∗_p∗₊ µ

ρU L∇

∗2_V~∗

✕ ❙✉♣❡r❢í❝✐❡ só❧✐❞❛✿

~ V∗ _{= 0}

✕ ❊♥tr❛❞❛ ♦✉ s❛í❞❛✿

~

V∗_{, p}∗ ❝♦♥❤❡❝✐❞❛s

✕ ❙✉♣❡r❢í❝✐❡ ❧✐✈r❡✿

w∗ ₌ dη ∗

dt∗, p

∗ ₌ pa

ρU2 +

gL U2z

∗₋ σ

ρU2

L(R

−1

x +R−

1

y )❡♠ z∗ =η∗

• ❊ss❛s ❡q✉❛çõ❡s r❡✈❡❧❛♠ ✉♠ t♦t❛❧ ❞❡ q✉❛tr♦ ♣❛râ♠❡tr♦s ❛❞✐♠❡♥s✐♦♥❛✐s✱ ✉♠ ♥❛ ❡q✉❛çã♦

❞❛ q✉❛♥t✐❞❛❞❡ ❞❡ ♠♦✈✐♠❡♥t♦ ❡ três ♥❛ ❝♦♥❞✐çã♦ ❞❡ ❝♦♥t♦r♥♦ ❞❡ ♣r❡ssã♦ ♥❛ s✉♣❡r❢í❝✐❡ ❧✐✈r❡✳

• P❛râ♠❡tr♦s ❛❞✐♠❡♥s✐♦♥❛✐s✿

✕ ◆❛ ❡q✉❛çã♦ ❞❡ ❝♦♥t✐♥✉✐❞❛❞❡ ♥ã♦ ❤á ♣❛râ♠❡tr♦s ❛❞✐♠❡♥s✐♦♥❛✐s✳

✕ ◆❛ ❡q✉❛çã♦ ❞❛ q✉❛♥t✐❞❛❞❡ ❞❡ ♠♦✈✐♠❡♥t♦ t❡♠♦s ✉♠ ♣❛râ♠❡tr♦s✱ ♦ ♥ú♠❡r♦ ❞❡ ❘❡②♥♦❧❞s✿

Re= ρU L

µ

✕ ❆s ❝♦♥❞✐çõ❡s ❞❡ ♥ã♦ ❡s❝♦rr❡❣❛♠❡♥t♦ ❡ ❞❡ ❡♥tr❛❞❛ ❡ s❛í❞❛ ♥ã♦ ❝♦♥t❡♠ ♣❛râ♠❡tr♦s✳ ✕ ❛ ❝♦♥❞✐çã♦ ❞❡ ♣r❡ssã♦ ♥❛ s✉♣❡r❢í❝✐❡ ❧✐✈r❡ ❝♦♥t❡♠ três✿

◆ú♠❡r♦ ❞❡ ❊✉❧❡r :Eu= pa

ρU

◆ú♠❡r♦ ❞❡ ❋r♦✉❞❡:F r= U

2

gL

◆ú♠❡r♦ ❞❡ ❲❡❜❡r :W e= ρU

2

L σ

• P❛râ♠❡tr♦s ❞❡ ❝♦♠♣r❡ss✐❜✐❧✐❞❛❞❡✿

◆ú♠❡r♦ ❞❡ ▼❛❝❤:M a= U

a

❘❛③ã♦ ❡♥tr❡ ❝❛❧♦r❡s ❡s♣❡❝í✜❝♦s:k = cp

cv

♦♥❞❡ a é ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ s♦♠ ♥♦ ✢✉✐❞♦ ❛ ✉♠❛ ❞❛❞❛ ♣r❡ssã♦ ❡ t❡♠♣❡r❛t✉r❛✳

• ❊s❝♦❛♠❡♥t♦s ❖s❝✐❧❛tór✐♦s✿

◆ú♠❡r♦ ❞❡ ❙tr♦✉❤❛❧:St= ωL

U

✺ ▼♦❞❡❧♦s

❆s ❝♦♥❞✐çõ❡s ❞❡ ❡s❝♦❛♠❡♥t♦ ❡♠ ✉♠ t❡st❡ ❞❡ ♠♦❞❡❧♦ sã♦ ❝♦♠♣❧❡t❛♠❡♥t❡ s❡♠❛❧❤❛♥t❡s às ❞♦ ♣r♦tót✐♣♦ s❡ t♦❞♦s ♦s ♣❛râ♠❡tr♦s ❛❞✐♠❡♥s✐♦♥❛✐s r❡❧❡✈❛♥t❡s tê♠ ♦s ♠❡s♠♦s ✈❛❧♦r❡s ❝♦rr❡s✲ ♣♦♥❞❡♥t❡s ♣❛r❛ ♦ ♠♦❞❡❧♦ ❡ ♣❛r❛ ♦ ♣r♦tót✐♣♦✳

✺✳✶ ❙❡♠❡❧❤❛♥ç❛ ●❡♦♠étr✐❝❛

❯♠ ♠♦❞❡❧♦ ❡ ✉♠ ♣r♦tót✐♣♦ sã♦ ❣❡♦♠❡tr✐❝❛♠❡♥t❡ s❡♠❡❧❤❛♥t❡s s❡ ❡ s♦♠❡♥t❡ s❡ t♦❞❛s ❛s ❞✐♠❡♥sõ❡s ❞♦ ❝♦r♣♦ ♥❛s três ❝♦♦r❞❡♥❛❞❛s t❡♠ ❛ ♠❡s♠❛ r❛③ã♦ ❞❡ ❡s❝❛❧❛ ❧✐♥❡❛r✳

P♦♥t♦s ❤♦♠♦❧♦❣♦s ♣♦ss✉❡♠ ❛ ♠❡s♠❛ ❧♦❝❛❧✐③❛çã♦ r❡❧❛t✐✈❛✳ ❆ s❡♠❡❧❤❛♥ç❛ ❣❡♦♠étr✐❝❛ r❡q✉❡r q✉❡ t♦❞♦s ♦s ♣♦♥t♦s ❤♦♠♦❧♦❣♦s s❡❥❛♠ r❡❧❛❝✐♦♥❛❞♦s ♣❡❧❛ ♠❡s♠❛ r❛③ã♦ ❞❡ ❡s❝❛❧❛ ❧✐♥❡❛r✳

✺✳✷ ❙❡♠❡❧❤❛♥ç❛ ❈✐♥❡♠át✐❝❛

❆ s❡♠❡❧❤❛♥ç❛ ❝✐♥❡♠át✐❝❛ r❡q✉❡r q✉❡ ♦ ♠♦❞❡❧♦ t❡♥❤❛♠ ❛ ♠❡s♠❛ r❛③ã♦ ❞❡ ❡s❝❛❧❛ ❞❡ ❝♦♠✲ ♣r✐♠❡♥t♦ ❡ ❛ ♠❡s♠❛ r❛③ã♦ ❞❡ ❡s❝❛❧❛ ❞❡ t❡♠♣♦s✳

❖s ♠♦✈✐♠❡♥t♦s ❞❡ ❞♦✐s s✐st❡♠❛s sã♦ ❝✐♥❡♠❛t✐❝❛♠❡♥t❡ s❡♠❡❧❤❛♥t❡s s❡ ♣❛rtí❝✉❧❛s ❤♦♠ó✲ ❧♦❣❛s ❛t✐♥❣❡♠ ♣♦♥t♦s ❤♦♠♦❧♦❣♦s ❡♠ t❡♠♣♦s ❤♦♠♦❧♦❣♦s

❆ ❡q✉✐✈❛❧❡♥❝✐❛ ❞❡ ❡s❝❛❧❛s ❞❡ ❝♦♠♣r✐♠❡♥t♦ s✐♠♣❧❡s♠❡♥t❡ ✐♠♣❧✐❝❛ s❡♠❡❧❤❛♥ç❛ ❣❡♦♠étr✐❝❛✱ ♠❛s ❛ ❡q✉✐✈❛❧❡♥❝✐❛ ❞❡ ❡s❝❛❧❛s ❞❡ t❡♠♣♦s r❡q✉❡r ❝♦♥s✐❞❡r❛çõ❡s ❞✐♥â♠❝❛s ❛❞✐❝✐♦♥❛✐s✱ t❛✐s ❝♦♠♦ ❡q✉✐✈❛❧❡♥❝✐❛ ❞♦ ♥ú♠❡r♦ ❞❡ ❘❡②♥♦❧❞s ❡ ❞❡ ▼❛❝❤✳

✺✳✸ ❙❡♠❡❧❤❛♥ç❛ ❉✐♥â♠✐❝❛

❆ s❡♠❡❧❤❛♥ç❛ ❞✐♥â♠✐❝❛ ❡①✐st❡ q✉❛♥❞♦ ♦ ♠♦❞❡❧♦ ❡ ♦ ♣r♦tót✐♣♦ t❡♠ ❛s ♠❡s♠❛s r❛③õ❡s ❞❡ ❡s❝❛❧❛ ❞❡ ❝♦♠♣r✐♠❡♥t♦✱ ❡s❝❛❧❛ ❞❡ t❡♠♣♦ ❡ ❡s❝❛❧❛ ❞❡ ❢♦rç❛✳ ◆♦✈❛♠❡♥t❡ ❛ s❡♠❡❧❤❛♥ç❛ ❣❡♦✲ ♠étr✐❝❛ é ✉♠ ♣r✐♠❡✐r♦ r❡q✉✐s✐t♦✳ ❆ s❡♠❡❧❤❛♥ç❛ ❞✐♥â♠✐❝❛ ❡①✐st✐rá✱ s✐♠✉❧t❛♥❡❛♠❡♥t❡✱ ❝♦♠ ❛ s❡♠❡❧❤❛♥ç❛ ❝✐♥❡♠át✐❝❛✱ s❡ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ♣r❡ssã♦ ❡ ❞❡ ❢♦rç❛ ❞♦ ♠♦❞❡❧♦ ❡ ❞♦ ♣r♦tót✐♣♦ ❢♦r❡♠ ✐❞❡♥t✐❝♦s✳ ■ss♦ ❡st❛rá ❣❛r❛♥t✐❞♦ s❡✿

✶✳ P❛r❛ ❡s❝♦❛♠❡♥t♦ ❝♦♠♣r❡ssí✈❡❧✱ ♦ ♥ú♠❡r♦ ❞❡ ❘❡②♥♦❧❞s✱ ♦ ♥ú♠❡r♦ ❞❡ ▼❛❝❤ ❡ ❛ r❛③ã♦ ❡♥tr❡ ❝❛❧♦r❡s ❡s♣❡❝í✜❝♦s ❞♦ ♠♦❞❡❧♦ ❡ ♣r♦tót✐♣♦ ❢♦r❡♠ ❝♦rr❡s♣♦♥❞❡♥t❡♠❡♥t❡ ✐❣✉❛✐s✳ ✷✳ P❛r❛ ❡s❝♦❛♠❡♥t♦ ✐♥❝♦♠♣r❡ssí✈❡❧✿

• ❙❡♠ s✉♣❡r❢í❝✐❡ ❧✐✈r❡✿ ♦s ♥ú♠❡r♦s ❞❡ ❘❡②♥♦❧❞s ❞♦ ♠♦❞❡❧♦ ❡ ❞♦ ♣r♦tót✐♣♦ ❢♦r❡♠

✐❣✉❛✐s✳

• ❈♦♠ s✉♣❡r❢í❝✐❡ ❧✐✈r❡✿ ♦ ♥ú♠❡r♦ ❞❡ ❘❡②♥♦❧❞s✱ ♦ ♥ú♠❡r♦ ❞❡ ❋r♦✉❞❡ ❡ ✭s❡ ♥❡❝❡ssár✐♦✮

♦ ♥ú♠❡r♦ ❞❡ ❲❡❜❡r ❡ ♦ ♥ú♠❡r♦ ❞❡ ❝❛✈✐t❛çã♦ ❞♦ ♠♦❞❡❧♦ ❡ ❞♦ ♣r♦tót✐♣♦ ❢♦r❡♠ ❝♦rr❡s♣♦♥❞❡♥t❡♠❡♥t❡ ✐❣✉❛✐s✳

✺✳✹ ❉✐s❝r❡♣â♥❝✐❛s ♥♦s t❡st❡s ❝♦♠ á❣✉❛ ❡ ❛r

❆ s❡♠❡❧❤❛♥ç❛ ❞✐♥â♠✐❝❛ ♣❡r❢❡✐t❛ é ♠❛✐s s♦♥❤♦ q✉❡ r❡❛❧✐❞❛❞❡✱ ♣♦r q✉❡ ❛s ❡q✉✐✈❛❧❡♥❝✐❛s ❡①❛t❛s ❞♦ ♥ú♠❡r♦ ❞❡ ❘❡②♥♦❧❞s ❡ ❞♦ ♥ú♠❡r♦ ❞❡ ❋r♦✉❞❡ só ♣♦❞❡♠ s❡r ❛t✐♥❣✐❞❛s ♣♦r ❛❧t❡r❛çõ❡s ❞rást✐❝❛s ♥❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ✢✉✐❞♦✱ ❛♦ ♣❛ss♦ q✉❡ ♦s t❡st❡s sã♦✱ ♥❛ ✈❡r❞❛❞❡✱ r❡❛❧✐③❛❞♦s s✐♠♣❧❡s♠❡♥t❡ ❝♦♠ á❣✉❛ ❡ ❛r✱ ♦s ✢✉✐❞♦s ♠❛✐s ❜❛r❛t♦s ❞✐s♣♦♥í✈❡✐s✳