Repositório UFAL: Força e torque de radiação sobre uma partícula viscoelástica em um fluido ideal

  

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❆❧❛❣♦❛s

■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛

❏♦sé P❡r❡✐r❛ ▲❡ã♦ ◆❡t♦

  

❋♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛

✈✐s❝♦❡❧ást✐❝❛ ❡♠ ✉♠ ✢✉✐❞♦ ✐❞❡❛❧

4 [nJ] 5

  2 4 ky

  3 2

  2 1

  4 4 2 kx 2 4

❋♦rç❛ ❞❡ r❛❞✐❛çã♦ tr❛♥s✈❡rs❛❧ ❞❡ ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ❡♠ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡

♣♦❧✐❡t✐❧❡♥♦✳

  

▼❛❝❡✐ó ✲ ❇r❛s✐❧

❙❡t❡♠❜r♦ ✲ ✷✵✶✺

  

❏♦sé P❡r❡✐r❛ ▲❡ã♦ ◆❡t♦

❋♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛

✈✐s❝♦❡❧ást✐❝❛ ❡♠ ✉♠ ✢✉✐❞♦ ✐❞❡❛❧

  ❚❡s❡ ❛♣r❡s❡♥t❛❞❛ ♥♦ ■♥st✐t✉t♦ ❞❡ ❋í✲ s✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❆❧❛❣♦❛s✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♥❡❝❡ssár✐♦ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ❉♦✉t♦r ❡♠ ❋ís✐❝❛✳

  ❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ●❧❛✉❜❡r ❚✳ ❙✐❧✈❛ ▼❛❝❡✐ó ✲ ❇r❛s✐❧

  ❙❡t❡♠❜r♦ ✲ ✷✵✶✺

  ❈❛t❛❧♦❣❛çã♦ ♥❛ ❢♦♥t❡ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❆❧❛❣♦❛s ❇✐❜❧✐♦t❡❝❛ ❈❡♥tr❛❧ ❉✐✈✐sã♦ ❞❡ ❚r❛t❛♠❡♥t♦ ❚é❝♥✐❝♦

  ❇✐❜❧✐♦t❡❝ár✐♦ ❘❡s♣♦♥sá✈❡❧✿ ❱❛❧t❡r ❞♦s ❙❛♥t♦s ❆♥❞r❛❞❡

  ▲✹✸✼❢ ▲❡ã♦ ◆❡t♦✱ ❏♦sé P❡r❡✐r❛✳ ❋♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ✈✐s❝♦❡❧ást✐❝❛ ❡♠ ✉♠

  ✢✉✐❞♦ ✐❞❡❛❧ ✴ ❏♦sé P❡r❡✐r❛ ▲❡ã♦ ◆❡t♦✳ ✕ ✷✵✶✺✳ ✾✽ ❢✳✿ ✐❧✳ ❖r✐❡♥t❛❞♦r✿ ●❧❛✉❜❡r ❚✳ ❙✐❧✈❛✳

  ❚❡s❡ ✭❞♦✉t♦r❛❞♦ ❡♠ ❋ís✐❝❛✮ ✕ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❆❧❛❣♦❛s✳ ■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛✳ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ❋ís✐❝❛✳ ▼❛❝❡✐ó✱ ✷✵✶✺✳

  ❇✐❜❧✐♦❣r❛✜❛✿ ❢✳ ✾✵ ✕ ✾✽✳ ✶✳ ❋♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛✳ ✷✳ ❚♦rq✉❡ ❞❡ r❛❞✐❛çã♦✳ ✸✳ ❆❝úst✐❝❛✳ ✹✳

  ❊s♣❛❧❤❛♠❡♥t♦ ❛❝úst✐❝♦✳ ✺✳ ❱✐s❝♦❡❧❛st✐❝✐❞❛❞❡✳ ✻✳ ▼❛♥✐♣✉❧❛çã♦ ❞❡ ♣❛rtí❝✉❧❛s✳ ■✳ ❚ít✉❧♦✳

  ❈❉❯✿ ✺✸✵✳✶✹✺✳✻✿✺✸✾✳✶✷

  ❉❡❞✐❝♦ ❡st❛ t❡s❡ ❛ ♠✐♥❤❛ ❡s♣♦s❛ ▲✐❞✐❛♥❡ ❖♠❡♥❛✱ ❛♦s ♠❡✉s P❛✐s ■❧s♦♥ ▲❡ã♦ ❡ ▼❛r✐❛

  ❞❡ ❋át✐♠❛ ❡ ❛♦s ♠❡✉s ✐r♠ã♦s✱ ■❧s♦♥ ❏r✳✱ ■❡❧s♦♥✱ ■✈❡♥s ❡ ❋❡r♥❛♥❞❛✳

  

❆❣r❛❞❡❝✐♠❡♥t♦s

  ❆ ❉❡✉s ♣♦r t❡r s✐❞♦ ♠✐♥❤❛ ❢♦rt❛❧❡③❛ ❡ s✉st❡♥t♦ ♥❛s ❤♦r❛s ❞✐❢í❝❡✐s ❞✉r❛♥t❡ ❡st❛ ❡t❛♣❛ ❞❛ ♠✐♥❤❛ ✈✐❞❛✳

  ❆ ♠✐♥❤❛ ❡s♣♦s❛ ▲✐❞✐❛♥❡ ❖♠❡♥❛✱ ♣♦r ❡st❛r ❛♦ ♠❡✉ ❧❛❞♦ ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s ♠❡ ✐♥❝❡♥t✐✈❛♥❞♦✱ ♣❡❧♦ ♦ s❡✉ ❛♠♦r ✐♥❝♦♥❞✐❝✐♦♥❛❧ ❡ ♣♦r t♦❞❛ ❛ s✉❛ ♣❛❝✐ê♥❝✐❛ ❡ ❝♦♠♣r❡❡♥sã♦ ♥❡st❛ ❡t❛♣❛ tã♦ ❞✐❢í❝✐❧✳

  ❆♦s ♠❡✉s ♣❛✐s ■❧s♦♥ ❡ ❋át✐♠❛✱ ♣♦r t♦❞♦ ♦ s❛❝r✐❢í❝✐♦✱ ❞❡❞✐❝❛çã♦✱ ❡ ♣♦r ❛❜❞✐❝❛r❡♠ ❞❡ ♣r♦❥❡t♦s ♣❡ss♦❛✐s ♣❛r❛ q✉❡ ❡✉ t✐✈❡ss❡ ❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ ❡st✉❞❛r✳ ❊♥✜♠✱ ♦❜r✐❣❛❞♦ ♣♦r t❡r ♠❡ ❞❛❞♦ ❛ ❡❞✉❝❛çã♦ ❡ ❞✐r❡❝✐♦♥❛♠❡♥t♦ q✉❡ ❢♦r❛♠ ✐♠♣r❡s❝✐♥❞í✈❡✐s ♣❛r❛ ❛ ♠✐♥❤❛ ❢♦r♠❛çã♦✳

  ❆♦s ♠❡✉s ✐r♠ã♦s ■❧s♦♥ ❏r✱ ■❡❧s♦♥ ▲❡ã♦✱ ■✈❡♥s ▲❡ã♦ ❡ ■❧③❛ ❋❡r♥❛♥❞❛ ♣♦r t♦❞♦ ❝♦♠♣❛✲ ♥❤❡✐r✐s♠♦✱ ✜❞❡❧✐❞❛❞❡ ❡ ♣♦r s❡♠♣r❡ s❡r❡♠ ❡①❡♠♣❧♦s ❞❡ ❞❡t❡r♠✐♥❛çã♦ ❡ ✉♥✐ã♦ ♥❛ ♠✐♥❤❛ ✈✐❞❛✳

  ❆ ●r❛♥❞❡ ❢❛♠í❧✐❛✿ ❉✳ ❈❧❛✉❞❡t❡✱ ❉✳ ❙♦❝♦rr♦✱ ▲✐❞②❛♥❡ ■❣♥á❝✐♦✱ ■r✐s ▲✐♠❛✱ ●❛❜r✐❡❧❛ ❆③❡✈❡❞♦✱ ❘❡❣✐♥❛ ❚♦❧❡❞♦ ❡ ▼❛r❝♦s ❆♥❞ré ♣♦r t♦❞❛ ❛ ✉♥✐ã♦✱ ❛♣♦✐♦ ❡ ✐♥❝❡♥t✐✈♦✳ ❆❧é♠ ❞❛s ❞❛t❛s ✐♠♣♦rt❛♥t❡s✱ ♠♦♠❡♥t♦s ❡s♣❡❝✐❛✐s✱ q✉❡ ❡st❛♠♦s t♦❞♦s ❥✉♥t♦s ❢❛③❡♥❞♦ s❡♠♣r❡ ❜♦❛s ❝♦♥❢r❛t❡r♥✐③❛çõ❡s✱ ❝♦♠ ❜♦❛s ❝♦♥✈❡rs❛s ❡ r✐s❛❞❛s✳ ◆ã♦ ♣♦❞❡r✐❛ ❞❡✐①❛r ❞❡ ❛❣r❛❞❡❝❡r ❛♦s ♠❡✉s s♦❜r✐♥❤♦s✿ ▲❡♦♥❛r❞♦ ■❣♥á❝✐♦✱ ▼❛r✐❛ ▲ét✐❝✐❛ ❡ ■❛♥ ❘❛❢❛❡❧ ♣❡❧❛ ❛❧❡❣r✐❛ ❡ ❡♥❡r❣✐❛ ❞❡ s❡♠♣r❡✳

  ❆ t♦❞♦s ♠❡✉s ❢❛♠✐❧✐❛r❡s q✉❡ ❡st✐✈❡r❛♠ ♣r❡s❡♥t❡s ❞✉r❛♥t❡ ❛ ♠✐♥❤❛ ❢♦r♠❛çã♦✳ ❆♦ Pr♦❢✳ ●❧❛✉❜❡r ❚✳ ❙✐❧✈❛ ♣♦r ❛❝r❡❞✐t❛r ♥♦ ♠❡✉ tr❛❜❛❧❤♦✱ ♣❡❧♦ s❡✉ ❛♣♦✐♦✱ ❞❡❞✐❝❛çã♦

  ❡ ❞✐s♣♦s✐çã♦ ❡♠ ❛❥✉❞❛r✳ ❉✐s♣♦♥❞♦ ❞❡ ❤♦r❛s ❞❡ ♦r✐❡♥t❛çã♦ ♣❛r❛ ❛ ❝♦♥❝r❡t✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳ ❆♦ ❣r✉♣♦ t✉❝❛♥♦ ❛③✉❧✱ q✉❡ ❥á sã♦ ♠✐♥❤❛ ❢❛♠í❧✐❛✿ ❆♥❞ré ▼♦✉r❛ ✭●❛❜❡❤✮✱ ❊♠♠❛♥✉❡❧❧❡

  ❙❛t✐❦♦✱ ❍❡♥r✐q✉❡ ▲♦♣❡s ✭●❛❧♦ ❈❡❣♦✮ ❡ ❚❛♠✐r②s ▼❛❝❡❞♦✳ P❡❧♦s ♠♦♠❡♥t♦s ❞❡ ❞❡s❝♦♥tr❛çã♦✱ q✉❡ ♥ã♦ ❢♦r❛♠ ♣♦✉❝♦s✱ ♣❡❧❛ ❛♠✐③❛❞❡ ❞❡ ❝❛❞❛ ✉♠ ❞❡ ✈♦❝ês✳ P♦r ❡st❛r❡♠ s❡♠♣r❡ ♣r❡s❡♥t❡s✱ ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ ♠♦♠❡♥t♦ q✉❡ ❝❛❞❛ ✉♠ ❡st❛ ♣❛ss❛♥❞♦✳

  ❆♦s ❛♠✐❣♦s ❋r❡❞❡r✐❝♦ P❛ss♦s✱ ❆❧❡① ❈♦st❛ ❡ ❆♥❞❡rs♦♥ ❇❛♥❞❡✐r❛ ♣♦r ❡st❛r❡♠ s❡♠♣r❡ ❞✐s♣♦st♦s ❛ ❛❥✉❞❛r✱ ♣♦r t❡r ❛♣r❡♥❞✐❞♦ ♠✉✐t♦ ❝♦♠ ✈♦❝ês ❡ ♣❡❧♦s ✐♥❝❡♥t✐✈♦s ❞❛❞♦s ❞✉r❛♥t❡ t♦❞❛ ❛ ♠✐♥❤❛ ❢♦r♠❛çã♦✳

  ❆♦s ❛♠✐❣♦s ▲✉✐③ ❊❞✉❛r❞♦✱ ❱❛❧♠✐r♦ ❍♦r❛ ❡ ❘♦❣ér✐♦ ▼❛t✐❛s q✉❡ t✐✈❡ ❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ ❝♦♥✈✐✈❡r ❡ ❛♣r❡♥❞❡r ♠✉✐t♦✳

  ❛

  ❆♦s ❛♠✐❣♦s Pr♦❢✳ ❲❛♥❞❡❛r❧❡② ❉✐❛s ❡ Pr♦❢ ✳ ❙♦❝♦rr♦ ❙❡✐①❛s ♣❡❧♦ ❝♦♠♣❛♥❤❡✐r✐s♠♦✱ ♣❡❧❛s ❜♦❛s ❝♦♥✈❡rs❛s ❡ ♣❡❧♦s ✐♥❝❡♥t✐✈♦s q✉❡ ❢♦r❛♠ ❡ss❡♥❝✐❛✐s ♥❡st❛ ❡t❛♣❛✳

  ❆♦s ❛♠✐❣♦s ❞♦ ●❆❋✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦ Pr♦❢✳ ❆♥❞ré ❇❛❣❣✐♦ ♣♦r t♦❞❛ ❝♦❧❛❜♦r❛çã♦✳ ❆♦s ♣r♦❢❡ss♦r❡s ❞❛ ❜❛♥❝❛✱ ♣♦r ❛❝❡✐t❛r ♣❛rt✐❝✐♣❛r ❞❡st❡ ♠♦♠❡♥t♦ ✐♠♣♦rt❛♥t❡ ♥❛ ♠✐♥❤❛

  ❢♦r♠❛çã♦ ❡ ❝♦♥tr✐❜✉✐r ❝♦♠ ❡st❡ tr❛❜❛❧❤♦✳ ❆ t♦❞♦s ♦s ❛♠✐❣♦s ❞♦ ■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛✱ ❡♠ ❡s♣❡❝✐❛❧✱ ❛♦s ❛♠✐❣♦s ❞❛ ❇❛t❝❛✈❡r♥❛✳ ❆ t♦❞♦ ❝♦r♣♦ ❞♦❝❡♥t❡ ❞♦ ■❋✱ ♣♦r ❝♦♥tr✐❜✉í❞♦ s✐❣♥✐✜❝❛t✐✈❛♠❡♥t❡ ♥❛ ♠✐♥❤❛ ❢♦r♠❛çã♦

  ❛❝❛❞ê♠✐❝❛✳ ❊♠ ❡s♣❡❝✐❛❧ ❛♦s ♣r♦❢❡ss♦r❡s✱ ❈r✐só❣♦♥♦ ❙✐❧✈❛ ❡ ❑❧é❜❡r ❙❡rr❛✳ ❆ ❈❆P❊❙ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦ ❞✉r❛♥t❡ ♦ ❉♦✉t♦r❛❞♦✳

  ✏❆♣❡s❛r ❞♦s ♥♦ss♦s ❞❡❢❡✐t♦s✱ ♣r❡❝✐s❛♠♦s ❡♥①❡r❣❛r q✉❡ s♦♠♦s ♣ér♦❧❛s ú♥✐❝❛s ♥♦ t❡❛tr♦ ❞❛ ✈✐❞❛ ❡ ❡♥t❡♥❞❡r q✉❡ ♥ã♦ ❡①✐st❡♠

  ♣❡ss♦❛s ❞❡ s✉❝❡ss♦ ♦✉ ♣❡ss♦❛s ❢r❛❝❛ss❛❞❛s✳ ❖ q✉❡ ❡①✐st❡ sã♦ ♣❡ss♦❛s q✉❡ ❧✉t❛♠ ♣❡❧♦s s❡✉s s♦♥❤♦s ♦✉ ❞❡s✐st❡♠ ❞❡❧❡s✳✑

  ❆✉❣✉st♦ ❈✉r②

  

❘❡s✉♠♦

  ❖ ❡st✉❞♦ ❞♦s ❢❡♥ô♠❡♥♦s ❞❡ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ t❡♠ ❛tr❛í❞♦ ✉♠ ❡♥♦r♠❡ ✐♥t❡r❡ss❡ ❞❛ ❝♦♠✉♥✐❞❛❞❡ ❝✐❡♥tí✜❝❛✱ ❞❡✈✐❞♦ ❛ ❛♣❧✐❝❛çõ❡s ❞❡ss❡s ❢❡♥ô♠❡♥♦s ❡♠ ♠❛♥✐♣✉❧❛çã♦ ❞❡ ♣❛rtí❝✉❧❛s s❡♠ ❝♦♥t❛t♦✳ ◆❡st❡ tr❛❜❛❧❤♦✱ r❡❛❧✐③❛♠♦s ✉♠❛ ❛♥á❧✐s❡ t❡ór✐❝❛ ❞❛ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ❡①❡r❝✐❞♦ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ✈✐s❝♦❡❧ást✐❝❛ ❤♦♠♦❣ê♥❡❛ ♥♦ ❧✐♠✐t❡ ❞♦ ❡s♣❛❧❤❛♠❡♥t♦ ❞❡ ❘❛②❧❡✐❣❤ ✭♦ r❛✐♦ ❞❛ ♣❛rtí❝✉❧❛ é ♠✉✐t♦ ♠❡♥♦r q✉❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ♦♥❞❛ ✐♥❝✐❞❡♥t❡✮ ♣♦r ✉♠❛ ♦♥❞❛ ❝♦♠ ❣❡♦♠❡tr✐❛ ❛r❜✐trár✐❛✳ ◆♦ss♦ ❡st✉❞♦ ❜❛s❡✐❛✲s❡ ♥❛ ❡①♣❛♥sã♦ ❞❡ ♦♥❞❛s ♣❛r❝✐❛✐s ❡♠ ❝♦♦r❞❡♥❛❞❛s ❡s❢ér✐❝❛s ❞❛s ♦♥❞❛s ✐♥❝✐❞❡♥t❡ ❡ ❡s♣❛❧❤❛❞❛✳ ◆❡ss❡ ❝♦♥t❡①t♦✱ ❛ ❢♦rç❛ ❡ ♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ sã♦ ♦❜t✐❞♦s ❛♥❛❧✐t✐❝❛♠❡♥t❡ ❡♠ t❡r♠♦s ❞❡ ✉♠❛ sér✐❡ ✐♥✜♥✐t❛ q✉❡ ❡♥✈♦❧✈❡ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❡①♣❛♥sã♦ ❞❛s ♦♥❞❛s ❡s♣❛❧❤❛❞❛ ❡ ✐♥❝✐❞❡♥t❡✳ ❆ss✉♠✐♠♦s q✉❡ ❛ ♣❛rtí❝✉❧❛ s❡ ❝♦♠♣♦rt❛ ❝♦♠♦ ✉♠ só❧✐❞♦ ✈✐s❝♦❡❧ást✐❝♦ ❧✐♥❡❛r✱ q✉❡ ♦❜❡❞❡❝❡ ♦ ♠♦❞❡❧♦ ❞❡ ❑❡❧✈✐♥✲❱♦✐❣t ❢r❛❝✐♦♥ár✐♦✳ ❋ór♠✉❧❛s ❛♥❛❧ít✐❝❛s ♣❛r❛ ❛ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ sã♦ ♦❜t✐❞❛s ❝♦♥s✐❞❡r❛♥❞♦ ✉♠❛ ❛♣r♦①✐♠❛çã♦ ❞❡ ❜❛✐①❛ ❡ ❞❡ ❛❧t❛ ❢r❡q✉ê♥❝✐❛ ♥♦ ♠♦❞❡❧♦ ✈✐s❝♦❡❧ást✐❝♦✳ ❆ t❡♦r✐❛ ❞❡s❡♥✈♦❧✈✐❞❛ é ✉s❛❞❛ ♣❛r❛ ❞❡s❝r❡✈❡r ❛ ✐♥t❡r❛çã♦ ❞❡ ♦♥❞❛s ❛❝úst✐❝❛s ✭♦♥❞❛ ♣❧❛♥❛ ♣r♦❣r❡ss✐✈❛✱ ♦♥❞❛ ♣❧❛♥❛ ❡st❛❝✐♦♥ár✐❛✱ ❢❡✐①❡s ❞❡ ❇❡ss❡❧ ❞❡ ♦r❞❡♠ ③❡r♦ ❡ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✮ ❝♦♠ ♣❛rtí❝✉❧❛s ❞❡ ♣♦❧✐❡t✐❧❡♥♦ ❞❡ ❜❛✐①❛ ❡ ❞❡ ❛❧t❛ ❞❡♥s✐❞❛❞❡✳ ❖s ♥♦ss♦s r❡s✉❧t❛❞♦s ♠♦str❛♠ q✉❡ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❛①✐❛❧ ♣♦❞❡ s❡r ♥❡❣❛t✐✈❛ ✭✐st♦ é✱ ❡♠ ♦♣♦s✐çã♦ à ❞✐r❡çã♦ ❞❡ ♣r♦♣❛❣❛çã♦ ❞❛ ♦♥❞❛✮ q✉❛♥❞♦ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ❝♦♥❞✐çã♦ ❡♥✈♦❧✈❡♥❞♦ ♦s ♣❛râ♠❡tr♦s ❢ís✐❝♦s ❞❛ ♣❛rtí❝✉❧❛ é s❛t✐s❢❡✐t❛✳ ❚♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ♥❡❣❛t✐✈♦ ❞❡✈✐❞♦ ❛ ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ t❛♠❜é♠ ♣♦❞❡ ♦❝♦rr❡r q✉❛♥❞♦ ❛ ♠❡s♠❛ ❝♦♥❞✐çã♦ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ♥❡❣❛t✐✈❛ ❢♦r ❛t❡♥❞✐❞❛✳ ◆♦t❛✈❡❧♠❡♥t❡✱ ❡st❛ é ❛ ♣r✐♠❡✐r❛ ✈❡③ q✉❡ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ♥❡❣❛t✐✈❛ é ♣r❡✈✐st❛ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❤♦♠♦❣ê♥❡❛ ♥♦ r❡❣✐♠❡ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❘❛②❧❡✐❣❤✳ ❆❧é♠ ❞✐ss♦✱ ❛ ❡st❛❜✐❧✐❞❛❞❡ tr❛♥s✈❡rs❛❧ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❣❡r❛❞❛ ♣❡❧♦ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ t❛♠❜é♠ é ✐♥✈❡st✐❣❛❞❛✳ ▼♦str❛♠♦s q✉❡ ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ tr❛t♦r ✸❉ ❝♦♠♣❧❡t♦ ❛t✉❛ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡ ♣♦❧✐❡t✐❧❡♥♦ ❞❡ ❛❧t❛ ❞❡♥s✐❞❛❞❡ ✭P❊❆❉✮✳ ◆❛ ❛♥á❧✐s❡ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❣❡r❛❞❛ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ✈✐s❝♦❡❧ást✐❝❛ ♣♦r ✉♠❛ ♦♥❞❛ ♣❧❛♥❛ ❡st❛❝✐♦♥ár✐❛✱ ❞❡s✈✐♦s r❡❧❡✈❛♥t❡s s✉r❣✐r❛♠ ❡♠ ❝♦♠♣❛r❛çã♦ ❝♦♠ ♣❛rtí❝✉❧❛ só❧✐❞❛ ❡❧ást✐❝❛✳ ❆ ❛♠♣❧✐t✉❞❡ ❞❛ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ s♦❜r❡ ✉♠❛ P❊❆❉ ❞❡s❝r✐t❛ ♣❡❧♦ ♠♦❞❡❧♦ ✈✐s❝♦❡❧ást✐❝♦ ❛♣r❡s❡♥t❛ ✉♠ ❝♦♠♣♦rt❛♠❡♥t♦ ❞✐❢❡r❡♥t❡ ✭❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ♥❡❣❛t✐✈❛✮ ❝♦♠♣❛r❛❞♦s ❝♦♠ ♦s ♦✉tr♦s ♠❛t❡r✐❛✐s ✭só❧✐❞❛ ❡❧ást✐❝❛ ❡ ✢✉✐❞❛ ❝♦♠ ❛❜s♦rçã♦ ❧♦♥❣✐t✉❞✐♥❛❧✮ ❞❡✈✐❞♦ ❛ ♦♥❞❛ ♣❧❛♥❛ ♣r♦❣r❡ss✐✈❛ ❡ ❢❡✐①❡s ❞❡ ❇❡ss❡❧✳ P♦r ✜♠✱ ❛❝r❡❞✐t❛♠♦s q✉❡ ❡st❡ ❡st✉❞♦ ♣♦❞❡ ❛❥✉❞❛r ❛ ♠❡❧❤♦r❛r ❛✐♥❞❛ ♠❛✐s ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ❞✐s♣♦s✐t✐✈♦s ❞❡ ❧❡✈✐t❛çã♦ ❛❝úst✐❝❛✱ ♠❛♥✐♣✉❧❛çã♦ ❞❡ ♣❛rtí❝✉❧❛s ❡♠ ❛❝✉st♦✢✉í❞✐❝❛ ❡ ♣✐♥ç❛s ❛❝úst✐❝❛s✳

  P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❋♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛✱ ❚♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦✱ ❊s♣❛❧❤❛✲ ♠❡♥t♦ ❛❝úst✐❝♦✱ ❱✐s❝♦❡❧❛st✐❝✐❞❛❞❡✱ ▼❛♥✐♣✉❧❛çã♦ ❞❡ ♣❛rtí❝✉❧❛s✳

  

❆❜str❛❝t

  ❚❤❡ st✉❞② ♦❢ ❛❝♦✉st✐❝ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ❛♥❞ t♦rq✉❡ ♣❤❡♥♦♠❡♥❛ ❤❛s ❛ttr❛❝t❡❞ ❛♥ ❡♥♦r♠♦✉s ✐♥t❡r❡st ♦❢ t❤❡ s❝✐❡♥t✐✜❝ ❝♦♠♠✉♥✐t②✱ ❞✉❡ t♦ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡s❡ ♣❤❡♥♦♠❡♥❛ ✐♥ ♥♦♥❝♦♥✲ t❛❝t ♣❛rt✐❝❧❡s ♠❛♥✐♣✉❧❛t✐♦♥✳ ■♥ t❤✐s ✇♦r❦✱ ✇❡ ♣❡r❢♦r♠ ❛ t❤❡♦r❡t✐❝❛❧ ❛♥❛❧②s✐s ♦❢ ❛❝♦✉st✐❝ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ❛♥❞ t♦rq✉❡ ❡①❡rt❡❞ ♦♥ ❛ ❤♦♠♦❣❡♥❡♦✉s ✈✐s❝♦❡❧❛st✐❝ ♣❛rt✐❝❧❡ ✐♥ t❤❡ ❘❛②❧❡✐❣❤ s❝❛tt❡r✐♥❣ ❧✐♠✐t ✭t❤❡ ♣❛rt✐❝❧❡ r❛❞✐✉s ✐s ♠✉❝❤ s♠❛❧❧❡r t❤❛♥ t❤❡ ✐♥❝✐❞❡♥t ✇❛✈❡❧❡♥❣t❤✮ ❜② ❛ ✇❛✈❡ ✇✐t❤ ❛r❜✐tr❛r② ❣❡♦♠❡tr②✳ ❖✉r st✉❞② ✐s ❜❛s❡❞ ♦♥ t❤❡ ♣❛rt✐❛❧✲✇❛✈❡ ❡①♣❛♥s✐♦♥ ✐♥ s♣❤❡✲ r✐❝❛❧ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ✐♥❝✐❞❡♥t ❛♥❞ s❝❛tt❡r❡❞ ✇❛✈❡s✳ ■♥ t❤✐s ❝♦♥t❡①t✱ t❤❡ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ❛♥❞ t♦rq✉❡ ❛r❡ ♦❜t❛✐♥❡❞ ❛♥❛❧②t✐❝❛❧❧② ✐♥ t❡r♠s ♦❢ ❛♥ ✐♥✜♥✐t❡ s❡r✐❡s ✇❤✐❝❤ ✐♥✈♦❧✈❡s t❤❡ s❝❛t✲ t❡r✐♥❣ ❛♥❞ ✐♥❝✐❞❡♥t ❡①♣❛♥s✐♦♥ ❝♦❡✣❝✐❡♥ts✳ ❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ♣❛rt✐❝❧❡ ❜❡❤❛✈❡s ❛s ❛ ❧✐♥❡❛r ✈✐s❝♦❡❧❛st✐❝ s♦❧✐❞✱ ✇❤✐❝❤ ♦❜❡②s t❤❡ ❢r❛❝t✐♦♥❛❧ ❑❡❧✈✐♥✲❱♦✐❣t ♠♦❞❡❧✳ ❆♥❛❧②t✐❝❛❧ ❡①♣r❡ss✐♦♥s ❢♦r t❤❡ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ❛♥❞ t♦rq✉❡ ❛r❡ ♦❜t❛✐♥❡❞ ❝♦♥s✐❞❡r✐♥❣ t❤❡ ❧♦✇✲ ❛♥❞ ❤✐❣❤✲❢r❡q✉❡♥❝② ❛♣♣r♦①✐♠❛t✐♦♥ ✐♥ t❤❡ ✈✐s❝♦❡❧❛st✐❝ ♠♦❞❡❧✳ ❚❤❡ ❞❡✈❡❧♦♣❡❞ t❤❡♦r② ✐s ✉s❡❞ t♦ ❞❡s❝r✐❜❡ t❤❡ ✐♥t❡r❛❝t✐♦♥ ♦❢ ❛❝♦✉st✐❝ ✇❛✈❡s ✭tr❛✈❡❧✐♥❣ ❛♥❞ st❛♥❞✐♥❣ ♣❧❛♥❡ ✇❛✈❡s✱ ❛♥❞ ③❡r♦ ❛♥❞ ✜rst✲♦r❞❡r ❇❡ss❡❧ ❜❡❛♠s✮ ✇✐t❤ ❛ ❧♦✇✲❛♥❞ ❤✐❣❤✲❞❡♥s✐t② ♣♦❧②❡t❤②❧❡♥❡ ♣❛rt✐❝❧❡✳ ❖✉r r❡s✉❧ts s❤♦✇ t❤❛t t❤❡ ❛①✐❛❧ ❛❝♦✉st✐❝ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ♠✐❣❤t ❜❡❝♦♠❡ ♥❡❣❛t✐✈❡ ✭✐✳❡✳ ✐♥ ♦♣♣♦s✐t✐♦♥ t♦ t❤❡ ✇❛✈❡ ♣r♦♣❛❣❛t✐♦♥ ❞✐r❡❝t✐♦♥✮ ✇❤❡♥ ❛ ❝❡rt❛✐♥ ❝♦♥❞✐t✐♦♥ ✐♥✈♦❧✈✐♥❣ t❤❡ ♣❤②s✐❝❛❧ ♣❛r❛♠❡t❡rs ♦❢ t❤❡ ♣❛rt✐❝❧❡ ✐s s❛t✐s✜❡❞✳ ◆❡❣❛t✐✈❡ ❛❝♦✉st✐❝ r❛❞✐❛t✐♦♥ t♦rq✉❡ ❞✉❡ ❛ ❜❡❛♠ ♦❢ ✜rst✲♦r❞❡r ❇❡ss❡❧ ♠❛② ❛❧s♦ ♦❝❝✉r ✇❤❡♥ t❤❡ s❛♠❡ ❝♦♥❞✐t✐♦♥ ♦❢ ♥❡❣❛t✐✈❡ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ✐s ♠❡t✳ ❘❡♠❛r❦❛❜❧②✱ t❤✐s ✐s t❤❡ ✜rst t✐♠❡ t❤❛t ♥❡❣❛t✐✈❡ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ✐s ♣r❡❞✐❝t❡❞ ♦♥ ❛ ❤♦♠♦❣❡♥❡♦✉s ♣❛rt✐❝❧❡ ✐♥ t❤❡ ❘❛②❧❡✐❣❤ s❝❛tt❡r✐♥❣ r❡❣✐♠❡✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ st❛❜✐❧✐t② ♦❢ t❤❡ tr❛♥s✈❡rs❡ ❛❝♦✉st✐❝ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ❣❡♥❡r❛t❡❞ ❜② ❛ ❇❡ss❡❧ ❜❡❛♠ ✐s ❛❧s♦ ✐♥✈❡st✐❣❛t❡❞✳ ❲❡ s❤♦✇ ❛ ❢✉❧❧ ✸❉ tr❛❝t♦r ❇❡ss❡❧ ✈♦rt❡① ❜❡❛♠ ❛❝t✐♥❣ ♦♥ t❤❡ ❤✐❣❤✲❞❡♥s✐t② ♣♦❧②❡t❤②❧❡♥❡ ✭❍❉P❊✮✳ ■♥ t❤❡ ❛♥❛❧②s✐s ♦❢ ❛❝♦✉st✐❝ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ❣❡♥❡r❛t❡❞ ♦♥ ❛ ✈✐s❝♦❡❧❛st✐❝ ♣❛rt✐❝❧❡ ❜② ❛ st❛♥❞✐♥❣ ♣❧❛♥❡ ✇❛✈❡✱ r❡❧❡✈❛♥t ❞❡✈✐❛t✐♦♥s ❛r♦s❡ ✐♥ ❝♦♠♣❛r✐s♦♥ ✇✐t❤ t❤❡ s♦❧✐❞ ❡❧❛st✐❝ ♠♦❞❡❧ ❢♦r t❤❡ ♣❛rt✐❝❧❡s✳ ❚❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ❛♥❞ t♦rq✉❡ ♦♥ ❛ ❍❉P❊ ❞❡s❝r✐❜❡❞ ❜② t❤❡ ✈✐s❝♦❡❧❛st✐❝ ♠♦❞❡❧ ❜❡❤❛✈❡s ❞✐✛❡r❡♥t❧② ✭♥❡❣❛t✐✈❡ r❛❞✐❛t✐♦♥ ❢♦r❝❡✮ ❝♦♠♣❛r❡❞ ✇✐t❤ ♦t❤❡r ♠❛t❡r✐❛❧s ✭s♦❧✐❞ ❡❧❛st✐❝ ❛♥❞ ❝♦♠♣r❡ss✐❜❧❡ ✢✉✐❞ ♣❛rt✐❝❧❡✮ ❞✉❡ t♦ tr❛✈❡❧✐♥❣ ♣❧❛♥❡ ✇❛✈❡ ❛♥❞ ❇❡ss❡❧ ❜❡❛♠s✳ ❋✐♥❛❧❧②✱ ✇❡ ❜❡❧✐❡✈❡ t❤❛t t❤✐s st✉❞② ♠❛② ❤❡❧♣ ❢✉rt❤❡r ❡♥❤❛♥❝❡ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❛❝♦✉st✐❝ ❧❡✈✐t❛t✐♦♥✱ ♣❛rt✐❝❧❡ ❤❛♥❞❧✐♥❣ ✐♥ ❛❝♦✉st♦✢✉✐❞s✱ ❛♥❞ ❛❝♦✉st✐❝❛❧ t✇❡❡③❡rs ❞❡✈✐❝❡s✳

  ❑❡②✇♦r❞s✿ ❆❝♦✉st✐❝ r❛❞✐❛t✐♦♥ ❢♦r❝❡✱ ❆❝♦✉st✐❝ r❛❞✐❛t✐♦♥ t♦rq✉❡✱ ❆❝♦✉st✐❝ s❝❛tt❡r✐♥❣✱ ❱✐s❝♦❡❧❛st✐❝✐t②✱ P❛rt✐❝❧❡ ♠❛♥✐♣✉❧❛t✐♦♥✳

  

▲✐st❛ ❞❡ sí♠❜♦❧♦s

  ❙í♠❜♦❧♦ ◆♦♠❡ ❯♥✐❞❛❞❡s✭❙✳■✳✮ 3

  ρ ❞❡♥s✐❞❛❞❡ ❞♦ ✢✉✐❞♦ ✐❞❡❛❧ kg · m 3

  ρ ❞❡♥s✐❞❛❞❡ kg · m m

  ♠❛ss❛ kg c ✈❡❧♦❝✐❞❛❞❡ ❞♦ s♦♠ ♥♦ ✢✉✐❞♦ ✐❞❡❛❧ m/s c

  ✈❡❧♦❝✐❞❛❞❡ ❞♦ s♦♠ ❧♦♥❣✐t✉❞✐♥❛❧ m/s c

  s

  ✈❡❧♦❝✐❞❛❞❡ ❞♦ s♦♠ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦ m/s τ t❡♠♣♦ ❞❡ r❡❧❛①❛çã♦ ❧♦♥❣✐t✉❞✐♥❛❧ s τ

  s

  t❡♠♣♦ ❞❡ r❡❧❛①❛çã♦ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦ s a r❛✐♦ ❞❛ ♣❛rtí❝✉❧❛ m w

  ✈❡t♦r ❞❡s❧♦❝❛♠❡♥t♦ m v ✈❡❧♦❝✐❞❛❞❡ ❞❛ ♣❛rtí❝✉❧❛ m/s

  ǫ t❡♥s♦r ❞❡❢♦r♠❛çã♦

  σ t❡♥s♦r ❞❡ t❡♥sõ❡s ❡♠ só❧✐❞♦s

  ♣ ♣r❡ssã♦ ❛❝úst✐❝❛ Pa 3 L ❞❡♥s✐❞❛❞❡ ▲❛❣r❛♥❣✐❛♥❛ ❛❝úst✐❝❛ J/m 2

  φ /s

  ♣♦t❡♥❝✐❛❧ ❞❡ ✈❡❧♦❝✐❞❛❞❡ m 2 ψ 1, s 2, s

  /s ✱ ψ ♣♦t❡♥❝✐❛✐s ❡s❝❛❧❛r❡s ❞❡ ❉❡❜②❡ m 1

  ω ❢r❡q✉ê♥❝✐❛ ❛♥❣✉❧❛r s 1

  λ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ♦♥❞❛ m 1

  ❦ ♥ú♠❡r♦ ❞❡ ♦♥❞❛ m 1

  ❦ ♥ú♠❡r♦ ❞❡ ♦♥❞❛ ❧♦♥❣✐t✉❞✐♥❛❧ m 1

  ❦ s ♥ú♠❡r♦ ❞❡ ♦♥❞❛ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦ m a nm ❝♦❡✜❝✐❡♥t❡ ❢♦r♠❛ ❞♦ ❢❡✐①❡ s n

  ❝♦❡✜❝✐❡♥t❡ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❡s❝❛❧❛r b n n n ✱c ✱d ❝♦❡✜❝✐❡♥t❡s ❞❡ ❡①♣❛♥sã♦

  ε ♣❛râ♠❡tr♦ ❞❡ ❡s❝❛❧❛ ❡♠ t❡r♠♦s ❞♦ t❛♠❛♥❤♦ ❞❛ ♣❛rtí❝✉❧❛

  ε j ♣❛râ♠❡tr♦ ❞❡ ❡s❝❛❧❛ ❡♠ t❡r♠♦s t❡♠♣♦ ❞❡ r❡❧❛①❛çã♦ ϕ

  â♥❣✉❧♦ ❛③✐♠✉t❛❧ θ

  â♥❣✉❧♦ r❛❞✐❛❧ j n (1) ❢✉♥çã♦ ❞❡ ❇❡ss❡❧ ❡s❢ér✐❝❛ ❞❡ ♦r❞❡♠ ♥ h n (2) ❢✉♥çã♦ ❞❡ ❍❛♥❦❡❧ ❡s❢ér✐❝❛ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ♥ h n m ❢✉♥çã♦ ❞❡ ❍❛♥❦❡❧ ❡s❢ér✐❝❛ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ♥ Y n m

  ❤❛r♠ô♥✐❝♦s ❡s❢ér✐❝♦s P n ❢✉♥çõ❡s ❛ss♦❝✐❛❞❛s ❞❡ ▲❡❣❡♥❞r❡ ❞❡ ♦r❞❡♠ ♥

  r❛❞

  F ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ N

  Y , Y , Y x y z ❝♦♠♣♦♥❡♥t❡s ❛❞✐♠❡♥s✐♦♥❛✐s ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦

  r❛❞

  N t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ N · m

  N x , N y , N z ❝♦♠♣♦♥❡♥t❡s ❛❞✐♠❡♥s✐♦♥❛✐s ❞♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ 1

  ♦♣❡r❛❞♦r ❣r❛❞✐❡♥t❡ m ∇ 1

  ∇· ♦♣❡r❛❞♦r ❞✐✈❡r❣❡♥t❡ m

  − 1

  ∇× 2 ♦♣❡r❛❞♦r r♦t❛❝✐♦♥❛❧ m 2

  ∇ ♦♣❡r❛❞♦r ▲❛♣❧❛❝✐❛♥♦ m r

  ✈❡t♦r ♣♦s✐çã♦ ♠ (x, y, z)

  ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s (r, θ, φ)

  ❝♦♦r❞❡♥❛❞❛s ❡s❢ér✐❝❛s (e , e , e ) x y z ❝♦♠♣♦♥❡♥t❡s ❝❛rt❡s✐❛♥❛s ✉♥✐trát✐❛s n

  ✈❡t♦r ♥♦r♠❛❧ à s✉♣❡r❢í❝✐❡ n

  ❝

  í♥❞✐❝❡ ❞❡ r❡❢r❛çã♦ ❝♦♠♣❧❡①♦ n R í♥❞✐❝❡ ❞❡ r❡❢r❛çã♦ ❞♦ ♠❛t❡r✐❛❧

  α ˜ ❢✉♥çã♦ ❞❡ ❛❜s♦rçã♦ ❛❞✐♠❡♥s✐♦♥❛❧

  ❘❡ ♣❛rt❡ r❡❛❧

  ■♠ ♣❛rt❡ ✐♠❛❣✐♥ár✐❛

  √ ✐ = −1

  ✉♥✐❞❛❞❡ ✐♠❛❣✐♥ár✐❛

  ▲✐st❛ ❞❡ ❚❛❜❡❧❛s

   ✳ ✳ ✳ ✳ ✳ ✺✶

   ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺

   ✳ ✳ ✳ ✼✽

  ▲✐st❛ ❞❡ ❋✐❣✉r❛s

  

  

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

  

  

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼

  

  

  

  

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

  

  

  

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺

  

  

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺

  

  

  

  

  ✳ ✳ ✳ ✳ ✹✻

  

  

  

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹

  

  

  

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼

  

  

  

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶

  

  

  

  ✼✻

  

  

  

  ✼✼

  

  

  

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵

  

  

  

  

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶

  

  

  

  ✳ ✳ ✳ ✳ ✳ ✽✷

  

  

  

  

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸

  

  

  

  

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✺

  

  

  

  

  ✳ ✳ ✳ ✳ ✽✻

  ❙✉♠ár✐♦

   ✶✻

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷

   ✷✹

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶

   ✹✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵

   ✺✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻

   ✻✽

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