Simulation du bullage dans un photobioréacteur

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Submitted on 1 Feb 2019

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Simulation du bullage dans un photobioréacteur

Wenbiao Jiang

To cite this version:

Wenbiao Jiang. Simulation du bullage dans un photobioréacteur. Autre. Université Paris-Saclay, 2018. Français. �NNT : 2018SACLC086�. �tel-02000282�

NNT

:

2018SA

CLC086

Simulation of bubbling in a

photobioreactor

Th `ese de doctorat de l’Universit ´e Paris-Saclay pr ´epar ´ee `a CentraleSup ´elec Ecole doctorale n◦_{579 Sciences m ´ecaniques et ´energ ´etiques, mat ´eriaux et}

g ´eosciences (SMEMaG)

Sp ´ecialit ´e de doctorat : G ´enie des proc ´ed ´es

Th `ese pr ´esent ´ee et soutenue `a Gif-sur-Yvette, le 5 d ´ecembre 2018, par

M. W

ENBIAO

J

IANG

Composition du Jury :

M. HAVET Jean-Louis

Professeur, GPIP, CNAM Paris Pr ´esident Mme. CHARCOSSET Catherine

DR CNRS, LAGEP, Universit ´e Lyon 1 Rapporteur Mme. CHARTON Sophie

Chercheur Senior HdR, CEA Marcoule Rapporteur M. DALMAZZONE Didier

Professeur, UCP, ENSTA Paris Tech Examinateur

M. POZZOBON Victor

Ing ´enieur de Recherche, Dr., LGPM, CentraleSup ´elec Co-directeur de th `ese M. PUEL Franc¸ois

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✐①

❈♦♥t❡♥ts

❉❡❝❧❛r❛t✐♦♥ ♦❢ ❆✉t❤♦rs❤✐♣ ✐✐✐ ✶ ■♥tr♦❞✉❝t✐♦♥ ✶ ✷ ❙t❛t❡ ♦❢ ❛rt ✸ ✷✳✶ ▼✐❝r♦❛❧❣❛❧ ❝✉❧t✐✈❛t✐♦♥ s②st❡♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✷✳✷ P❤♦t♦❜✐♦r❡❛❝t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷✳✸ ●❛s✲❧✐q✉✐❞ ✢♦✇ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✹ ▼❛ss tr❛♥s❢❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✹✳✶ ▼❛ss tr❛♥s❢❡r ✐♥ ❛ ♣❤♦t♦❜✐♦r❡❛❝t♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✹✳✷ ❉✐✛✉s✐♦♥ ✐♥ ❛ s✐♥❣❧❡ ♣❤❛s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✹✳✸ ❉✐✛✉s✐♦♥ ❜❡t✇❡❡♥ ❣❛s ❛♥❞ ❧✐q✉✐❞ ♣❤❛s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✺ ▲✐❣❤t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✻ ❈❡❧❧ ❣r♦✇t❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✻✳✶ ❑✐♥❡t✐❝ ♠♦❞❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✻✳✷ ▼❛ss ❜❛❧❛♥❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ▼❛ss ❜❛❧❛♥❝❡ ♦❢ ❝❡❧❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ▼❛ss ❜❛❧❛♥❝❡ ♦❢ s✉❜str❛t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ▼❛ss ❜❛❧❛♥❝❡ ♦❢ ♣r♦❞✉❝ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✻✳✸ P♦♣✉❧❛t✐♦♥ ❜❛❧❛♥❝❡ ❡q✉❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✼ ❚❤❡ ✐♥✢✉❡♥❝❡ ♦❢ ❜✉❜❜❧❡ ❜❡❤❛✈✐♦rs ♦♥ ❝❡❧❧ ❣r♦✇t❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✽ ❚❤❡ ❝♦♠♣❧❡t❡ ♠♦❞❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✾ ❖❜❥❡❝t✐✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸ ❊①♣❡r✐♠❡♥t❛❧ st✉❞② ✸✸ ✸✳✶ Pr❡❧✐♠✐♥❛r② r❡✢❡❝t✐♦♥ ♦♥ t❤❡ r♦❧❡ ♦❢ ✇❡tt❛❜✐❧✐t② ✐♥ ❜✉❜❜❧❡ ❢♦r♠❛t✐♦♥ ❛♥❞ ❞❡t❛❝❤♠❡♥t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✷ ❲♦r❦✐♥❣ ♣r✐♥❝✐♣❧❡ ♦❢ s❤❛❞♦✇❣r❛♣❤② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✳✸ ❊①♣❡r✐♠❡♥t❛❧ s❡t✉♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✸✳✹ ■♠❛❣❡ ♣r♦❝❡ss✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✸✳✺ ■♠❛❣❡ ❛♥❛❧②s✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✸✳✺✳✶ ❈♦✉♥t✐♥❣ ❜✉❜❜❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✸✳✺✳✷ ❇✉❜❜❧✐♥❣ ❢r❡q✉❡♥❝② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✸✳✺✳✸ ❇✉❜❜❧❡ ✈♦❧✉♠❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✸✳✺✳✹ ❇✉❜❜❧❡ s❤❛♣❡ ❢❛❝t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✸✳✻ ❈♦♥❝❧✉s✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✹ ◆✉♠❡r✐❝❛❧ st✉❞② ✺✾ ✹✳✶ ■♥tr♦❞✉❝t✐♦♥ t♦ ❱❖❋ ♠❡t❤♦❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✹✳✶✳✶ ❱♦❧✉♠❡ ❢r❛❝t✐♦♥ ❢✉♥❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ❉❡✜♥✐t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ❚❡♠♣♦r❛❧ ❡✈♦❧✉t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ◆♦r♠❛❧ ✈❡❝t♦r ♦❢ t❤❡ ✐♥t❡r❢❛❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶

① ✹✳✶✳✷ ▼♦♠❡♥t✉♠ ❡q✉❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✹✳✶✳✸ ■♥t❡r❢❛❝❡ r❡❝♦♥str✉❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✹✳✶✳✹ Pr♦s ❛♥❞ ❝♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✹✳✷ ■♥tr♦❞✉❝t✐♦♥ t♦ ❖♣❡♥❋❖❆▼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✹✳✸ ✷❉ s✐♠✉❧❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✹✳✸✳✶ ◆✉♠❡r✐❝❛❧ ❝♦♥✜❣✉r❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✹✳✸✳✷ ◆✉♠❡r✐❝❛❧ t❡sts ♦♥ ❈♦✉r❛♥t ♥✉♠❜❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ✹✳✸✳✸ ◆✉♠❡r✐❝❛❧ t❡sts ♦♥ ♠❡s❤ s✐③❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ ✹✳✸✳✹ Pr❡❧✐♠✐♥❛r② ✈❛❧✐❞❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✹✳✹ ✸❉ s✐♠✉❧❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✹✳✹✳✶ ◆✉♠❡r✐❝❛❧ ❝♦♥✜❣✉r❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✹✳✹✳✷ ❱✐s✉❛❧ ✈❛❧✐❞❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼ ✹✳✹✳✸ ❇✉❜❜❧❡ ✈♦❧✉♠❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷ ✹✳✹✳✹ ❇✉❜❜❧❡ s❤❛♣❡ ❡✈♦❧✉t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸ ✹✳✺ ❈♦♥❝❧✉s✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✼ ✺ Pr❡♣❛r✐♥❣ ❛ ❝♦♠♣❧❡t❡ ♠♦❞❡❧ ✽✾ ✺✳✶ ◗✉❛s✐✲st❡❛❞② st❛t❡ ♦❢ t✇♦✲♣❤❛s❡ ✢♦✇ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✾ ✺✳✷ ❙❝❛❧❛r tr❛♥s♣♦rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✶ ✺✳✸ ▲❛❣r❛♥❣✐❛♥ tr❛♥s♣♦rt ❛♥❞ P■❱ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✸ ✺✳✸✳✶ ❲♦r❦✐♥❣ ♣r✐♥❝✐♣❧❡ ♦❢ P■❱ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✸ ✺✳✸✳✷ P■❱ s❡t✉♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✹ ✺✳✸✳✸ ❈r♦ss✲❝♦rr❡❧❛t✐♦♥ ♦❢ ✐♠❛❣❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✹ ✺✳✸✳✹ P♦t❡♥t✐❛❧ ❛♣♣❧✐❝❛t✐♦♥ ✐♥ t❤❡ ❢♦❧❧♦✇✲✉♣ st✉❞② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺ ✺✳✹ ▲✐❣❤t ♠♦❞❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺ ✺✳✺ ❈♦♥❝❧✉s✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✻ ✻ ●❡♥❡r❛❧ ❝♦♥❝❧✉s✐♦♥s ✾✼ ❆ ❘❡♣r❡s❡♥t❛t✐✈❡ st✉❞✐❡s ♦♥ ❣❛s✲❧✐q✉✐❞ ✢♦✇ ✐♥ ❛ r❡❛❝t♦r ✾✾ ❇ ▼❛t❧❛❜ ❝♦❞❡ ❢♦r ✐♠❛❣❡ ♣r♦❝❡ss✐♥❣ ❛♥❞ ❛♥❛❧②s✐s ✶✵✸ ❇✐❜❧✐♦❣r❛♣❤② ✶✵✼

①✐

▲✐st ♦❢ ❋✐❣✉r❡s

✶✳✶ ❲♦r❧❞ ❡♥❡r❣② ❝♦♥s✉♠♣t✐♦♥✱ ✐❧❧✉str❛t✐♦♥ ❢r♦♠ t❤❡ ❇P ❙t❛t✐st✐❝❛❧ ❘❡✈✐❡✇ ♦❢ ❲♦r❧❞ ❊♥❡r❣② ❏✉♥❡ ✷✵✶✼✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✷✳✶ ■❧❧✉str❛t✐♦♥ ♦❢ ▼✐❝r♦❛❧❣❛❡ ♣r♦❝❡ss ❢r♦♠ ❬✷✻❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✷✳✷ ❆ s❝❤❡♠❛t✐❝ ❞✐❛❣r❛♠ ♦❢ r❛❝❡✇❛② ♣♦♥❞✱ ✐❧❧✉str❛t✐♦♥ ❢r♦♠ ❬✷✼❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✷✳✸ ❆ s❝❤❡♠❛t✐❝ ❞✐❛❣r❛♠ ♦❢ ❞✐✛❡r❡♥t t②♣❡s ♦❢ ♣❤♦t♦❜✐♦r❡❛❝t♦r✿ ✭❛✮ st✐rr❡❞ t❛♥❦ ❬✷✽❪✱ ✭❜✮ ❛✐r❧✐❢t ❬✷✾❪✱ ✭❝✮ ❜✉❜❜❧❡ ❝♦❧✉♠♥ ❬✸✵❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✷✳✹ ❱❛r✐♦✉s t②♣❡s ♦❢ ♣❤♦t♦❜✐♦r❡❛❝t♦rs✿ ✭❛✮ ❜✉❜❜❧❡ ❝♦❧✉♠♥✱ ✭❜✮ ❛✐r❧✐❢t✱ ✭❝✮ ❤♦r✐③♦♥t❛❧ t✉❜✉❧❛r✱ ✭❞✮ ✢❛t✲♣❧❛t❡✱ ✭❡✮ st✐rr❡❞ t❛♥❦✱ ✐❧❧✉str❛t✐♦♥ ❢r♦♠ ❬✸✽❪✳ ✻ ✷✳✺ ❱❡❧♦❝✐t② ✜❡❧❞ ❛♥❞ ❣❛s ❤♦❧❞✲✉♣ ❢r♦♠ ❊✉❧❡r✐❛♥✲❊✉❧❡r✐❛♥ s✐♠✉❧❛t✐♦♥s✱ ✐❧✲ ❧✉str❛t✐♦♥ ❢r♦♠ ❬✹✽❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✻ ❙✐♠✉❧❛t✐♦♥ ♦❢ ❣❛s✲❧✐q✉✐❞ ✢♦✇ ❜② ❱♦❧✉♠❡ ♦❢ ❋❧✉✐❞ ♠❡t❤♦❞✱ ✐❧❧✉str❛t✐♦♥ ❢r♦♠ ❬✺✽❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✼ ❚❤❡ ✈❡❧♦❝✐t② ✜❡❧❞ ♦❢ ❧✐q✉✐❞ ♣❤❛s❡ ❢r♦♠ ❊✉❧❡r✐❛♥✲▲❛❣r❛♥❣✐❛♥ s✐♠✉❧❛t✐♦♥✱ ✐❧❧✉str❛t✐♦♥ ❢r♦♠ ❬✹✺❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✽ ❇✉❜❜❧❡ s✇❛r♠ s✐♠✉❧❛t❡❞ ❜② ❊✉❧❡r✐❛♥✲▲❛❣r❛♥❣✐❛♥ ♠♦❞❡❧✱ ✐❧❧✉str❛t✐♦♥ ❢r♦♠ ❬✺✾❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✾ ❚❤❡ ❜✉❜❜❧❡s ❝❛♣t✉r❡❞ ❜② s❤❛❞♦✇❣r❛♣❤② t❡❝❤♥✐q✉❡✱ ✐❧❧✉str❛t✐♦♥ ❢r♦♠ ❬✼✹❪✳ ✶✷ ✷✳✶✵ ❚❤❡ ❣❛s ❝♦♥❝❡♥tr❛t✐♦♥ ♠❡❛s✉r❡❞ ❜② ✐♥❢r❛✲r❡❞ t❡❝❤♥✐q✉❡✱ ✐❧❧✉str❛t✐♦♥ ❢r♦♠ ❬✻✶❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✶✶ ✭❛✮ ✢♦✇ ♣❛tt❡r♥s✱ ✭❜✮ ✢♦✇✲♣❛tt❡r♥ ♠❛♣✱ ✐❧❧✉str❛t✐♦♥ ❢r♦♠ ❬✼✺❪✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✶✷ ❈❤❛✐♥ ♦❢ ♠❛ss tr❛♥s❢❡r ✐♥ ❛ ♣❤♦t♦❜✐♦r❡❛❝t♦r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✶✸ ❚❤❡ ♣r♦❝❡ss ♦❢ ❞✐✛✉s✐♦♥✱ ✐❧❧✉str❛t✐♦♥ ❢r♦♠ ❲✐❦✐♣❡❞✐❛ ❛rt✐❝❧❡ ❉✐✛✉s✐♦♥✳ ✶✺ ✷✳✶✹ ❯♥st❡❛❞② st❛t❡✿ ❝♦♥❝❡♥tr❛t✐♦♥ ✐s ❞❡♣❡♥❞❡♥t ♦❢ t✐♠❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✶✺ ❚❤❡ ✈♦❧✉♠❡ ❡❧❡♠❡♥t ✉s❡❞ t♦ ❞❡❞✉❝❡ ❋✐❝❦✬s s❡❝♦♥❞ ❧❛✇ ♦❢ ❞✐✛✉s✐♦♥✳ ✳ ✳ ✶✻ ✷✳✶✻ ◆❡r♥st✬s ✜❧♠ t❤❡♦r②✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✶✼ ❚✇♦ ✜❧♠ ♠♦❞❡❧✿ ✜❧♠s ❡①✐st ✐♥ ❜♦t❤ ♣❤❛s❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✶✽ P❤♦t♦s②♥t❤❡s✐s✿ ❧✐❣❤t r❡❛❝t✐♦♥s ❛♥❞ ❞❛r❦ r❡❛❝t✐♦♥s✱ ✐❧❧✉str❛t✐♦♥ ❢r♦♠ ❬✽✺❪✳ ✷✵ ✷✳✶✾ ▲✐❣❤t✲r❡s♣♦♥s❡ ❝✉r✈❡✿ t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ r❛t❡ ♦❢ ♣❤♦t♦s②♥t❤❡s✐s ❛♥❞ ❧✐❣❤t ✐♥t❡♥s✐t②✱ ✐❧❧✉str❛t✐♦♥ ❢r♦♠ ❬✽✼❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✷✵ ❚❤❡ ✈❛r✐❛t✐♦♥ ♦❢ s✉❜str❛t❡ ❝♦♥❝❡♥tr❛t✐♦♥ ✇✐t❤ t✐♠❡ ✐♥ ❛ ❝❧♦s❡❞ s②st❡♠✳ ✷✺ ✷✳✷✶ µ✲s♣❛❝❡ ✐s ❞✐✈✐❞❡❞ ✐♥t♦ N s✉❜s♣❛❝❡s✱ ❡❛❝❤ s✉❜s♣❛❝❡ ❤♦❧❞s ❛ ❝♦♥st❛♥t µi✱ ❛♥❞ µi ✐♥❝r❡❛s❡s ❢r♦♠ ❧❡❢t t♦ r✐❣❤t ✐♥ t❤❡ ✐❧❧✉str❛t✐♦♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✷✷ ❋♦r µi < µ❡♥✈✱ ❝❡❧❧s ✏✢♦✇✑ ✐♥ µ✲s♣❛❝❡ ❢r♦♠ s♠❛❧❧❡r µi t♦ ❧❛r❣❡r µi+1❀ ❢♦r µj > µ❡♥✈✱ ❝❡❧❧s ❛r❡ ✏tr❛♥s♣♦rt❡❞✑ ✐♥ µ✲s♣❛❝❡ ❢r♦♠ ❧❛r❣❡r µj t♦ s♠❛❧❧❡r µj−1✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✷✸ ▼✐❝r♦❛❧❣❛❡ ❛r❡ ❛❜s♦r❜❡❞ ♦♥ t❤❡ ❜✉❜❜❧❡ s✉r❢❛❝❡✱ ✐❧❧✉str❛t✐♦♥ ❢r♦♠ ❬✶✵✻❪✳ ✷✽ ✷✳✷✹ ❚❤❡ ❜❛s✐❝ str✉❝t✉r❡ ♦❢ t❤❡ ❝♦♠♣❧❡t❡ ♠♦❞❡❧✳ ❆♥ ❡❧❧✐♣s❡ r❡♣r❡s❡♥ts ❛ ♠♦❞❡❧ ❤❛♥❞❧✐♥❣ ❛ s♣❡❝✐✜❝ ♣r♦❜❧❡♠✱ ❛♥❞ ❛ r❡❝t❛♥❣❧❡ r❡♣r❡s❡♥ts t❤❡ ✐♥♣✉ts r❡q✉✐r❡❞ ❜② ❛ ♠♦❞❡❧✱ ♦r ♦✉t♣✉ts ②✐❡❧❞❡❞ ❢r♦♠ ❛ ♠♦❞❡❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵

①✐✐ ✸✳✶ ❆ ❜✉❜❜❧❡ t❤❛t ❞♦❡s ♥♦t s♣r❡❛❞ ❜❡②♦♥❞ t❤❡ ♦r✐✜❝❡ ❡❞❣❡✳ ✭❛✮ ❚❤❡ t❤r❡❡ s✉r❢❛❝❡ str❡ss❡s σ ❛❝t ♦♥ t❤❡ tr✐♣❧❡ ❧✐♥❡ ❧♦❝❛t❡❞ ❛t t❤❡ ♦r✐✜❝❡ ❡❞❣❡✳ α✱ β ❛r❡ t❤❡ ❛♥❣❧❡s ❜❡t✇❡❡♥ t❤❡ ✐♥t❡r❢❛❝❡s✱ ❛♥❞ β ✐s s❡t t♦ ❜❡ π/2✳ ✭❜✮ ❚❤❡ ❣❛s✲❧✐q✉✐❞ s✉r❢❛❝❡ str❡ss σGL ❞♦❡s ♥♦t ❛♣♣❡❛r ✐♥ t❤❡ ❢♦r❝❡s ❛❝t✐♥❣ ♦♥ t❤❡ ❜✉❜❜❧❡ ❛s ❛ ✇❤♦❧❡✳ ❚❤❡ ❧✐q✉✐❞✲s♦❧✐❞ s✉r❢❛❝❡ str❡ss σH LS ♣✉❧❧s t❤❡ ❜✉❜❜❧❡ ♦✉t✇❛r❞s ❛♥❞ ♠❛② ♠❛❦❡ t❤❡ ❜✉❜❜❧❡ s♣r❡❛❞ ♦✈❡r t❤❡ ❤♦r✐③♦♥t❛❧ s✉r❢❛❝❡✳ ❚❤❡ ❣❛s✲s♦❧✐❞ s✉r❢❛❝❡ str❡ss σV GS r❡s✐sts t❤❡ ❜✉♦②❛♥❝② ❢♦r❝❡ FB✳ ✸✹ ✸✳✷ ❚❤❡ ♠❡t❤♦❞ ❢♦r ❝❛❧❝✉❧❛t✐♥❣ ❜✉❜❜❧❡ ✈♦❧✉♠❡✳ ✭❛✮ ΩV ✐s t❤❡ ♠❛①✐♠✉♠ ❣❛s ✈♦❧✉♠❡ t❤❛t ❝❛♥ ❜❡ s✉♣♣♦rt❡❞ ❜② t❤❡ ✈❡rt✐❝❛❧ ❣❛s✲s♦❧✐❞ ✐♥t❡r❢❛❝❡✳ ✭❜✮ ❉✉r✐♥❣ ❜✉❜❜❧❡ s♣r❡❛❞✐♥❣✱ Ωh ✐s t❤❡ ❣❛s ✈♦❧✉♠❡ t❤❛t ✐s ❤❡❧❞ ❜② t❤❡ ❤♦r✐③♦♥t❛❧ s♦❧✐❞ s✉r❢❛❝❡✱ ✐ts ♠❛①✐♠✉♠ ✈❛❧✉❡ ✐s ΩH = (Ωh)♠❛①✳ ✭❝✮ ❚❤❡ ❜✉❜❜❧❡ ✈♦❧✉♠❡ ✐s ❝♦♥s✐❞❡r❡❞ ❛s ❛ s✉♠ ♦❢ t❤❡ ♣r❡✈✐♦✉s t✇♦ ♠❛①✐♠✉♠ ✈♦❧✉♠❡s✳ ◆♦t✐❝✐♥❣ t❤❛t t❤❡ ♦✈❡r❧❛♣♣✐♥❣ ✈♦❧✉♠❡ r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ❞❛r❦ r❡❣✐♦♥ ❛♣♣❡❛rs t✇✐❝❡ ✐♥ t❤❡ t♦t❛❧ ✈♦❧✉♠❡✱ t❤✉s t❤❡ t❤❡♦r❡t✐❝❛❧ ❜✉❜❜❧❡ ✈♦❧✉♠❡ ✐s ❛ ❧✐tt❧❡ ❧❛r❣❡r t❤❛♥ t❤❡ ❛❝t✉❛❧ ♦♥❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✸ ❚❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❜✉♦②❛♥❝② ❢♦r❝❡ Fh B❛♥❞ t❤❡ ❛❞❤❡s✐✈❡ ❢♦r❝❡ Fa ✈❡rs✉s s♣r❡❛❞✐♥❣ r❛❞✐✉s r ❢♦r t❤❡ ❤♦r✐③♦♥t❛❧ ♣❛rt✳ δ = 0 ✐♥❞✐❝❛t❡s Fh B= Fa✱ δ < 0 ✐♥❞✐❝❛t❡s FBh < Fa✳ ❆♥❞ r♠❛① ✐s t❤❡ ♠❛①✐♠✉♠ s♣r❡❛❞✐♥❣ r❛❞✐✉s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✹ ❋✉♥❝t✐♦♥ ΩH = ε h_{(1−cos θH)}3 sin3 θH 2+3 cos θH−cos3_θH i1 2 ❢♦r ❞✐✛❡r❡♥t κ✳ ❚❤❡ ❝✉r✈❡s ❛r❡ s❧✐❣❤t❧② st❛❣❣❡r❡❞ ❢♦r ❜❡tt❡r ✈✐s✐❜✐❧✐t② ♦❢ t❤❡ ✢❛t t❛✐❧s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✸✳✺ ▼❛①✐♠✉♠ ❜✉❜❜❧❡ ✈♦❧✉♠❡ Ω ✈❡rs✉s ✇❡tt❛❜✐❧✐t② θH✿ ❛ ❝♦♠♣❛r✐s♦♥ ❜❡✲ t✇❡❡♥ t❤❡ t❤❡♦r❡t✐❝❛❧ ♣r❡❞✐❝t✐♦♥ ✭✸✳✷✷✮ ❛♥❞ t❤❡ ❡①♣❡r✐♠❡♥t❛❧ ❞❛t❛ ❢r♦♠ ▲✐♥ ❡t ❛❧✳ ❬✶✶✹❪✳ ❚❤❡ t❤❡♦r② ❛❧✐❣♥s ✇❡❧❧ ✇✐t❤ t❤❡ ❡①♣❡r✐♠❡♥ts ❢♦r t✇♦ ♦r✐✜❝❡ s✐③❡s✱ R1= 0.275♠♠ ❛♥❞ R2= 0.125♠♠✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✳✻ ❆ ❧✐❣❤t r❛② tr❛✈❡❧✐♥❣ ✐♥ ❧✐q✉✐❞ ✇✐❧❧ ❜❡ r❡❢r❛❝t❡❞ ✇❤❡♥ ✐t ❛rr✐✈❡s ❛t t❤❡ ❣❛s✲❧✐q✉✐❞ ✐♥t❡r❢❛❝❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✸✳✼ ❆♥ ❛♥♥✉❧❛r✲❧✐❦❡ s❤❛❞♦✇ ✐s ❢♦r♠❡❞ ♦♥ t❤❡ r❡❝♦r❞✐♥❣ ♣❧❛♥❡ ✇❤❡♥ ❛ ❧✐❣❤t ❜❡❛♠ tr❛✈❡❧✐♥❣ ✐♥ ❧✐q✉✐❞ ♠❡❡t ❛ ❣❛s ❜✉❜❜❧❡ ❞✉❡ t♦ t❤❡ r❡❢r❛❝t✐♦♥✳ ✳ ✳ ✳ ✹✶ ✸✳✽ ❙❝❤❡♠❛t✐❝ s❦❡t❝❤ ♦❢ t❤❡ ❡①♣❡r✐♠❡♥t❛❧ s❡t✉♣✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✸✳✾ ❚❤❡ ✾ ♥♦③③❧❡s ❛t t❤❡ ❜♦tt♦♠ ♦❢ t❤❡ r❡❛❝t♦r ✭t❤❡ s✐❞❡ ❝♦♥♥❡❝t✐♥❣ ✇✐t❤ ❣❛s s♦✉r❝❡✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✸✳✶✵ ❇✉❜❜❧❡ ♠❡t❡r ♠❡t❤♦❞ t♦ ♠❡❛s✉r❡ ❣❛s ✢♦✇ r❛t❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✸✳✶✶ ■♠❛❣❡ ♣r♦❝❡ss✐♥❣✳ ✭❛✮ r❛✇ ✐♠❛❣❡✱ ✭❜✮ ❜✐♥❛r✐③❡❞ ✐♠❛❣❡✱ ✭❝✮ ✇❤✐t❡ ③♦♥❡s ❡❧✐♠✐♥❛t❡❞✱ ✭❞✮ ✐♥✈❡rt❡❞ ✐♠❛❣❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✸✳✶✷ ❚❤❡ ❤✐st♦❣r❛♠ ♦❢ ❋✐❣✳ ✸✳✶✶✭❛✮✱ ❜✐♥ ✇✐❞t❤✿ ✶✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✸✳✶✸ ❈♦✉♥t✐♥❣ ❜✉❜❜❧❡s ❛t ❱❋❈ ✶%✳ ✭❛✮ t❤❡ ♣r♦❜❡ ♣♦✐♥t ✭r❡❞ ♣♦✐♥t✮✱ ✭❜✮ ♣✐①❡❧ ✐♥t❡♥s✐t② ♦❢ ♣r♦❜❡ ♣♦✐♥t ♦✈❡r ❛♥ ✐♠❛❣❡ s❡q✉❡♥❝❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✸✳✶✹ ❚❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ ✭❛✮ ❜✉❜❜❧❡ ♥✉♠❜❡r ❛♥❞ ✭❜✮ ❜✉❜❜❧✐♥❣ ❢r❡q✉❡♥❝② ❢♦r t❤❡ tr✐♣❧✐❝❛t❡ r❡❝♦r❞✐♥❣s ❛t ❱❋❈ ✶ %✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✸✳✶✺ ❇✉❜❜❧✐♥❣ ❢r❡q✉❡♥❝② ❢♦r ❛❧❧ t❤❡ ✢♦✇ r❛t❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✸✳✶✻ ❚❤❡ ❤♦r✐③♦♥t❛❧ s❡❣♠❡♥t❛t✐♦♥ ♠❡t❤♦❞ t♦ ❝❛❧❝✉❧❛t❡ ❜✉❜❜❧❡ ✈♦❧✉♠❡✳ ✳ ✳ ✳ ✹✾ ✸✳✶✼ ✭❛✮ s❤❛♣❡ ❛♥❞ ✭❜✮ ✈♦❧✉♠❡ ♦❢ ❛ ❥✉st ❞❡t❛❝❤❡❞ ❜✉❜❜❧❡ ❛t ❞✐✛❡r❡♥t ♠♦♠❡♥ts ✭❱❋❈ ✶%✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✸✳✶✽ ✭❛✮ t❤❡ t❡♠♣♦r❛❧ ❞✐str✐❜✉t✐♦♥ ♦❢ ❜✉❜❜❧❡ ✈♦❧✉♠❡✱ ✭❜✮ ❛ s♠❛❧❧ ❜✉❜❜❧❡ ♦❝❝✉rr❡❞ ❛t t = 10.1 s ❛t st❡❛❞② st❛t❡✱ ✭❝✮ ❤✐st♦❣r❛♠ ♦❢ ❜✉❜❜❧❡ ✈♦❧✉♠❡ ❛t st❡❛❞② st❛t❡✱ ❜✐♥ ✇✐❞t❤✿ ✸ ✭❱❋❈ ✶%✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✸✳✶✾ ❚❤❡ t❡♠♣♦r❛❧ ❞✐str✐❜✉t✐♦♥ ✭❧❡❢t ❝♦❧✉♠♥✮ ❛♥❞ ❤✐st♦❣r❛♠ ✭r✐❣❤t ❝♦❧✉♠♥✮ ♦❢ ❜✉❜❜❧❡ ✈♦❧✉♠❡ ❛t ❱❋❈ ✸%✲✾%✱ ❜✐♥ ✇✐❞t❤✿ ✸✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷

①✐✐✐ ✸✳✷✵ ❚❤❡ t❡♠♣♦r❛❧ ❞✐str✐❜✉t✐♦♥ ✭❧❡❢t ❝♦❧✉♠♥✮ ❛♥❞ ❤✐st♦❣r❛♠ ✭r✐❣❤t ❝♦❧✉♠♥✮ ♦❢ ❜✉❜❜❧❡ ✈♦❧✉♠❡ ❛t ❱❋❈ ✶✷%✲✸✵%✱ ❜✐♥ ✇✐❞t❤✿ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✸✳✷✶ ❚❤❡ ♠❡❛♥ ❛♥❞ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥ ♦❢ ❜✉❜❜❧❡ ✈♦❧✉♠❡s ❛t st❡❛❞② st❛t❡✳ ✳ ✺✹ ✸✳✷✷ ✭❛✮ ❛ ❥✉st ❞❡t❛❝❤❡❞ ❜✉❜❜❧❡ ❛t st❡❛❞② st❛t❡✱ ❛♥❞ ✭❜✮ ✐ts ❡✈♦❧✉t✐♦♥ ♦❢ ❛s♣❡❝t r❛t✐♦ ❛♥❞ ❝✐r❝✉❧❛r✐t② ✭❱❋❈ ✶%✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✸✳✷✸ ❚❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ ❛s♣❡❝t r❛t✐♦ ❛♥❞ ❝✐r❝✉❧❛r✐t② ♦❢ ❛ ♣♦♣✉❧❛t✐♦♥ ♦❢ ❜✉❜❜❧❡s ❛t st❡❛❞② st❛t❡ ✭❱❋❈ ✶%✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✸✳✷✹ ❚❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ ❛s♣❡❝t r❛t✐♦ ❛♥❞ ❝✐r❝✉❧❛r✐t② ♦❢ ❛ ♣♦♣✉❧❛t✐♦♥ ♦❢ ❜✉❜❜❧❡s ❛t st❡❛❞② st❛t❡ ✭❱❋❈ ✸✵%✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✸✳✷✺ ✭❛✮ ❛♥ ❡❧♦♥❣❛t❡❞ ❜✉❜❜❧❡✱ ✭❜✮ ❢❛❧s❡ ❜✉❜❜❧❡ ❝♦❛❧❡s❝❡♥❝❡✱ ✭❝✮ ❛✉t❤❡♥t✐❝ ❜✉❜❜❧❡ ❝♦❛❧❡s❝❡♥❝❡✳ ✭❱❋❈ ✸✵%✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✹✳✶ ❚❤❡ ✈❛❧✉❡ ♦❢ α ✐♠♣❧✐❡s t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t✇♦ ✢✉✐❞s ✐♥ t❤❡ ❝♦♠♣✉t❛✲ t✐♦♥❛❧ ❞♦♠❛✐♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✹✳✷ ❈♦♠♣❛r✐s♦♥ ♦❢ ❛❧❣♦r✐t❤♠s ❢♦r ✐♥t❡r❢❛❝❡ r❡❝♦♥str✉❝t✐♦♥✳ ✭❛✮ ❛❝t✉❛❧ ✐♥t❡r✲ ❢❛❝❡✱ ✭❜✮ ♣✐❡❝❡✇✐s❡ ❧✐♥❡❛r ✐♥t❡r❢❛❝❡ ❝❛❧❝✉❧❛t✐♦♥ ✭P▲■❈✮✱ ✭❝✮ s✐♠♣❧❡ ❧✐♥❡ ✐♥t❡r❢❛❝❡ ❝❛❧❝✉❧❛t✐♦♥ ✭❙▲■❈✮✴❞♦♥♦r✲❛❝❝❡♣t♦r ✭❉✲❆✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✹✳✸ ❚✇♦ st❡♣s ♦❢ P▲■❈✿ ✶✳ t❤❡ ♥♦r♠❛❧ ❞✐r❡❝t✐♦♥ ✐s ❛♣♣r♦①✐♠❛t❡❞ ❜② ∇α✱ ✷✳ ♣❛r❛❧❧❡❧ tr❛♥s♣♦rt ♦❢ t❤❡ ✐♥t❡r❢❛❝❡ t♦ s❛t✐s❢② t❤❡ ✈♦❧✉♠❡ ❢r❛❝t✐♦♥✳ ✳ ✳ ✳ ✳ ✻✸ ✹✳✹ ❇✉❜❜❧❡ ❜r❡❛❦❛❣❡ ❛♥❞ ❞❡❢♦r♠❛t✐♦♥ ❝❛❧❝✉❧❛t❡❞ ✐♠♣❧✐❝✐t❧② ❜② ❱❖❋ ♠❡t❤♦❞✱ ✐❧❧✉str❛t✐♦♥s ❢r♦♠ ♦✉r s✐♠✉❧❛t✐♦♥s ✭s❝❛❧❡✿ ❡①t❡r✐♦r ❞✐❛♠❡t❡r ♦❢ t❤❡ ♥♦③③❧❡ ✐s ✸ ♠♠✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✹✳✺ ❙✐♠✉❧❛t✐♦♥ ❜② ✐♥t❡r❋♦❛♠✿ ✸❉ r✐s✐♥❣ ❜✉❜❜❧❡ ❛♥❞ ✈❡❧♦❝✐t② ✜❡❧❞ ❛r♦✉♥❞ ✐t✱ ✐❧❧✉str❛t✐♦♥ ❢r♦♠ ❬✶✺✽❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✹✳✻ ❙✐♠✉❧❛t✐♦♥ ❜② ✐♥t❡r❋♦❛♠✿ ✷❉ r✐s✐♥❣ ❜✉❜❜❧❡✱ ✐❧❧✉str❛t✐♦♥ ❢r♦♠ ❬✶✺✾❪✳ ✳ ✳ ✻✺ ✹✳✼ ❚❤❡ ❞✐♠❡♥s✐♦♥s ♦❢ ✷❉ s✐♠✉❧❛t✐♦♥s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ✹✳✽ ❚❤❡ ✐♥✢✉❡♥❝❡ ♦❢ ❈♦✉r❛♥t ♥✉♠❜❡r ♦♥ ❣❛s ❤♦❧❞✲✉♣✱ ♠❡s❤ s✐③❡ ∆x = 1 ♠♠✳ ✻✽ ✹✳✾ ✷❉ s✐♠✉❧❛t✐♦♥s ✇✐t❤ ❞✐✛❡r❡♥t ♠❡s❤ s✐③❡s✱ ❛t ♣❤②s✐❝❛❧ t✐♠❡ t = 1 s✳ ✭❛✮ ∆x = 0.25ds✱ ✭❜✮ ∆x = 0.5ds✱ ✭❝✮ ∆x = ds✳ ds ✐s t❤❡ ✐♥♥❡r ❞✐❛♠❡t❡r ♦❢ t❤❡ s♣❛r❣❡r✳ ✭❙❝❛❧❡✿ ✇✐❞t❤ ♦❢ t❤❡ ♥✉♠❡r✐❝❛❧ ❞♦♠❛✐♥ ✐s ✷✽✻ ♠♠✳✮ ✳ ✳ ✳ ✳ ✻✾ ✹✳✶✵ ❚✇♦ ❜✉❜❜❧❡s ✐♥ ❞✐✛❡r❡♥t s❤❛♣❡ r❡❣✐♠❡✳ ❈♦♥❞✐t✐♦♥s✿ Co♠❛① = 0.75✱ ∆x = 0.5 ♠♠✱ ✐♥❥❡❝t✐♥❣ ❣❛s ✈❡❧♦❝✐t② vg = 0.2 ♠✴s✳ ❚❤❡ ❡q✉✐✈❛❧❡♥t ❞✐❛♠❡t❡r d1≈ 1.5 ♠♠✱ d2≈ 4.5 ♠♠✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ✹✳✶✶ ❙❤❛♣❡ r❡❣✐♠❡ ♠❛♣ ❢♦r ❜✉❜❜❧❡s ❛♥❞ ❞r♦♣s ❬✶✻✻❪✱ ✐❧❧✉str❛t✐♦♥ ❢r♦♠ ❬✶✼✵❪✳ ✼✷ ✹✳✶✷ ❚❤❡ ♠✐♥✐✲r❡❛❝t♦r ✐♥ ✸❉ s✐♠✉❧❛t✐♦♥s✳ ✭◆✉♠❡r✐❝❛❧ ❞♦♠❛✐♥✿ 14 × 14 × 20 ♠♠✱ ❡①t❡r✐♦r ❞✐❛♠❡t❡r ♦❢ t❤❡ ♥♦③③❧❡✿ ✸ ♠♠✳✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✹✳✶✸ ❚❤❡ ❜♦tt♦♠ ✈✐❡✇ ♦❢ t❤❡ ♠❡s❤❡❞ ❝♦♠♣✉t❛t✐♦♥❛❧ ❞♦♠❛✐♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻ ✹✳✶✹ ❉②♥❛♠✐❝ ♠❡s❤ ✐s ✏❛❝t✐✈❛t❡❞✑ ❛s s♦♦♥ ❛s t❤❡ ♠❡s❤ ♠❡❡t t❤❡ ✐♥t❡r❢❛❝❡✱ ✐❧❧✉str❛t✐♦♥ ❢r♦♠ s✐♠✉❧❛t✐♦♥ ✇✐t❤ ❱❋❈ ✸%✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻ ✹✳✶✺ ❈♦♠♣❛r✐s♦♥ ♦❢ ❜✉❜❜❧❡ ❡✈♦❧✉t✐♦♥ ✐♥ s✐♠✉❧❛t✐♦♥s ❛♥❞ ✐♥ ❡①♣❡r✐♠❡♥ts✳ ❯♣✲ ♣❡r ✐♠❛❣❡s ❛r❡ ❢r♦♠ s✐♠✉❧❛t✐♦♥s✱ ❧♦✇❡r ✐♠❛❣❡s ❛r❡ ❢r♦♠ ❡①♣❡r✐♠❡♥ts✱ t❤❡② ❤❛✈❡ t❤❡ s❛♠❡ s❝❛❧❡✿ ✸ ♠♠ ❂ ✶✷✻ ♣✐①❡❧s✳ ❚❤❡ ♥♦③③❧❡ ❞✐❛♠❡t❡r ✐♥ s✐♠✉❧❛t✐♦♥s ✐s t❤❡ s❛♠❡ ❛s t❤❡ ♦♥❡ ✐♥ ❡①♣❡r✐♠❡♥ts✱ ✐✳❡✳ ✸ ♠♠✳ ❚✐♠❡ ✐♥t❡r✈❛❧ ❜❡t✇❡❡♥ ✐♠❛❣❡s ✐♥ ✉♣♣❡r ❛♥❞ ❧♦✇❡r r♦✇ ✐s s✐♠✐❧❛r✳ ❱❋❈✿ ✶%✳ ✳ ✼✼ ✹✳✶✻ ❈♦♠♣❛r✐s♦♥ ♦❢ ❜✉❜❜❧❡ ❡✈♦❧✉t✐♦♥ ✐♥ s✐♠✉❧❛t✐♦♥s ❛♥❞ ✐♥ ❡①♣❡r✐♠❡♥ts✳ ❯♣✲ ♣❡r ✐♠❛❣❡s ❛r❡ ❢r♦♠ s✐♠✉❧❛t✐♦♥s✱ ❧♦✇❡r ✐♠❛❣❡s ❛r❡ ❢r♦♠ ❡①♣❡r✐♠❡♥ts✱ t❤❡② ❤❛✈❡ t❤❡ s❛♠❡ s❝❛❧❡✿ ✸ ♠♠ ❂ ✶✷✻ ♣✐①❡❧s✳ ❚❤❡ ♥♦③③❧❡ ❞✐❛♠❡t❡r ✐♥ s✐♠✉❧❛t✐♦♥s ✐s t❤❡ s❛♠❡ ❛s t❤❡ ♦♥❡ ✐♥ ❡①♣❡r✐♠❡♥ts✱ ✐✳❡✳ ✸ ♠♠✳ ❚✐♠❡ ✐♥t❡r✈❛❧ ❜❡t✇❡❡♥ ✐♠❛❣❡s ✐♥ ✉♣♣❡r ❛♥❞ ❧♦✇❡r r♦✇ ✐s s✐♠✐❧❛r✳ ❱❋❈✿ ✸%✳ ✳ ✼✽

①✐✈ ✹✳✶✼ ❈♦♠♣❛r✐s♦♥ ♦❢ ❜✉❜❜❧❡ ❡✈♦❧✉t✐♦♥ ✐♥ s✐♠✉❧❛t✐♦♥s ❛♥❞ ✐♥ ❡①♣❡r✐♠❡♥ts✳ ❯♣✲ ♣❡r ✐♠❛❣❡s ❛r❡ ❢r♦♠ s✐♠✉❧❛t✐♦♥s✱ ❧♦✇❡r ✐♠❛❣❡s ❛r❡ ❢r♦♠ ❡①♣❡r✐♠❡♥ts✱ t❤❡② ❤❛✈❡ t❤❡ s❛♠❡ s❝❛❧❡✿ ✸ ♠♠ ❂ ✶✷✻ ♣✐①❡❧s✳ ❚❤❡ ♥♦③③❧❡ ❞✐❛♠❡t❡r ✐♥ s✐♠✉❧❛t✐♦♥s ✐s t❤❡ s❛♠❡ ❛s t❤❡ ♦♥❡ ✐♥ ❡①♣❡r✐♠❡♥ts✱ ✐✳❡✳ ✸ ♠♠✳ ❚✐♠❡ ✐♥t❡r✈❛❧ ❜❡t✇❡❡♥ ✐♠❛❣❡s ✐♥ ✉♣♣❡r ❛♥❞ ❧♦✇❡r r♦✇ ✐s s✐♠✐❧❛r✳ ❱❋❈✿ ✻%✳ ✳ ✼✽ ✹✳✶✽ ❈♦♠♣❛r✐s♦♥ ♦❢ ❜✉❜❜❧❡ ❡✈♦❧✉t✐♦♥ ✐♥ s✐♠✉❧❛t✐♦♥s ❛♥❞ ✐♥ ❡①♣❡r✐♠❡♥ts✳ ❯♣✲ ♣❡r ✐♠❛❣❡s ❛r❡ ❢r♦♠ s✐♠✉❧❛t✐♦♥s✱ ❧♦✇❡r ✐♠❛❣❡s ❛r❡ ❢r♦♠ ❡①♣❡r✐♠❡♥ts✱ t❤❡② ❤❛✈❡ t❤❡ s❛♠❡ s❝❛❧❡✿ ✸ ♠♠ ❂ ✶✷✻ ♣✐①❡❧s✳ ❚❤❡ ♥♦③③❧❡ ❞✐❛♠❡t❡r ✐♥ s✐♠✉❧❛t✐♦♥s ✐s t❤❡ s❛♠❡ ❛s t❤❡ ♦♥❡ ✐♥ ❡①♣❡r✐♠❡♥ts✱ ✐✳❡✳ ✸ ♠♠✳ ❚✐♠❡ ✐♥t❡r✈❛❧ ❜❡t✇❡❡♥ ✐♠❛❣❡s ✐♥ ✉♣♣❡r ❛♥❞ ❧♦✇❡r r♦✇ ✐s s✐♠✐❧❛r✳ ❱❋❈✿ ✾%✳ ✳ ✼✾ ✹✳✶✾ ❈♦♠♣❛r✐s♦♥ ♦❢ ❜✉❜❜❧❡ ❡✈♦❧✉t✐♦♥ ✐♥ s✐♠✉❧❛t✐♦♥s ❛♥❞ ✐♥ ❡①♣❡r✐♠❡♥ts✳ ❯♣✲ ♣❡r ✐♠❛❣❡s ❛r❡ ❢r♦♠ s✐♠✉❧❛t✐♦♥s✱ ❧♦✇❡r ✐♠❛❣❡s ❛r❡ ❢r♦♠ ❡①♣❡r✐♠❡♥ts✱ t❤❡② ❤❛✈❡ t❤❡ s❛♠❡ s❝❛❧❡✿ ✸ ♠♠ ❂ ✶✷✻ ♣✐①❡❧s✳ ❚❤❡ ♥♦③③❧❡ ❞✐❛♠❡t❡r ✐♥ s✐♠✉❧❛t✐♦♥s ✐s t❤❡ s❛♠❡ ❛s t❤❡ ♦♥❡ ✐♥ ❡①♣❡r✐♠❡♥ts✱ ✐✳❡✳ ✸ ♠♠✳ ❚✐♠❡ ✐♥t❡r✈❛❧ ❜❡t✇❡❡♥ ✐♠❛❣❡s ✐♥ ✉♣♣❡r ❛♥❞ ❧♦✇❡r r♦✇ ✐s s✐♠✐❧❛r✳ ❱❋❈✿ ✶✷%✳ ✼✾ ✹✳✷✵ ❈♦♠♣❛r✐s♦♥ ♦❢ ❜✉❜❜❧❡ ❡✈♦❧✉t✐♦♥ ✐♥ s✐♠✉❧❛t✐♦♥s ❛♥❞ ✐♥ ❡①♣❡r✐♠❡♥ts✳ ❯♣✲ ♣❡r ✐♠❛❣❡s ❛r❡ ❢r♦♠ s✐♠✉❧❛t✐♦♥s✱ ❧♦✇❡r ✐♠❛❣❡s ❛r❡ ❢r♦♠ ❡①♣❡r✐♠❡♥ts✱ t❤❡② ❤❛✈❡ t❤❡ s❛♠❡ s❝❛❧❡✿ ✸ ♠♠ ❂ ✶✷✻ ♣✐①❡❧s✳ ❚❤❡ ♥♦③③❧❡ ❞✐❛♠❡t❡r ✐♥ s✐♠✉❧❛t✐♦♥s ✐s t❤❡ s❛♠❡ ❛s t❤❡ ♦♥❡ ✐♥ ❡①♣❡r✐♠❡♥ts✱ ✐✳❡✳ ✸ ♠♠✳ ❚✐♠❡ ✐♥t❡r✈❛❧ ❜❡t✇❡❡♥ ✐♠❛❣❡s ✐♥ ✉♣♣❡r ❛♥❞ ❧♦✇❡r r♦✇ ✐s s✐♠✐❧❛r✳ ❱❋❈✿ ✷✵%✳ ✽✵ ✹✳✷✶ ❈♦♠♣❛r✐s♦♥ ♦❢ ❜✉❜❜❧❡ ❡✈♦❧✉t✐♦♥ ✐♥ s✐♠✉❧❛t✐♦♥s ❛♥❞ ✐♥ ❡①♣❡r✐♠❡♥ts✳ ❯♣✲ ♣❡r ✐♠❛❣❡s ❛r❡ ❢r♦♠ s✐♠✉❧❛t✐♦♥s✱ ❧♦✇❡r ✐♠❛❣❡s ❛r❡ ❢r♦♠ ❡①♣❡r✐♠❡♥ts✱ t❤❡② ❤❛✈❡ t❤❡ s❛♠❡ s❝❛❧❡✿ ✸ ♠♠ ❂ ✽✵ ♣✐①❡❧s✳ ❚❤❡ ♥♦③③❧❡ ❞✐❛♠❡t❡r ✐♥ s✐♠✉❧❛t✐♦♥s ✐s t❤❡ s❛♠❡ ❛s t❤❡ ♦♥❡ ✐♥ ❡①♣❡r✐♠❡♥ts✱ ✐✳❡✳ ✸ ♠♠✳ ❚✇♦ ❜✉❜✲ ❜❧❡s ❞✐❞ ♥♦t ✜♥❛❧❧② ♠❡r❣❡ ✐♥ ❜♦t❤ s✐♠✉❧❛t✐♦♥s ❛♥❞ ❡①♣❡r✐♠❡♥ts✳ ❚✐♠❡ ✐♥t❡r✈❛❧ ❜❡t✇❡❡♥ ✐♠❛❣❡s ✐♥ ✉♣♣❡r ❛♥❞ ❧♦✇❡r r♦✇ ✐s s✐♠✐❧❛r✳ ❱❋❈✿ ✸✵%✳ ✽✵ ✹✳✷✷ ❈♦♠♣❛r✐s♦♥ ♦❢ ❜✉❜❜❧❡ ❡✈♦❧✉t✐♦♥ ✐♥ s✐♠✉❧❛t✐♦♥s ❛♥❞ ✐♥ ❡①♣❡r✐♠❡♥ts✳ ❯♣✲ ♣❡r ✐♠❛❣❡s ❛r❡ ❢r♦♠ s✐♠✉❧❛t✐♦♥s✱ ❧♦✇❡r ✐♠❛❣❡s ❛r❡ ❢r♦♠ ❡①♣❡r✐♠❡♥ts✱ t❤❡② ❤❛✈❡ t❤❡ s❛♠❡ s❝❛❧❡✿ ✸ ♠♠ ❂ ✽✵ ♣✐①❡❧s✳ ❚❤❡ ♥♦③③❧❡ ❞✐❛♠❡t❡r ✐♥ s✐♠✉❧❛t✐♦♥s ✐s t❤❡ s❛♠❡ ❛s t❤❡ ♦♥❡ ✐♥ ❡①♣❡r✐♠❡♥ts✱ ✐✳❡✳ ✸ ♠♠✳ ❚✇♦ ❜✉❜✲ ❜❧❡s ✜♥❛❧❧② ♠❡r❣❡❞ ✐♥ s✐♠✉❧❛t✐♦♥s✱ ♥♦t ✐♥ ❡①♣❡r✐♠❡♥ts✳ ❚✐♠❡ ✐♥t❡r✈❛❧ ❜❡t✇❡❡♥ ✐♠❛❣❡s ✐♥ ✉♣♣❡r ❛♥❞ ❧♦✇❡r r♦✇ ✐s s✐♠✐❧❛r✳ ❱❋❈✿ ✸✵%✳ ✳ ✳ ✳ ✳ ✳ ✽✶ ✹✳✷✸ ❈♦♠♣❛r✐s♦♥ ♦❢ ❜✉❜❜❧❡ ✈♦❧✉♠❡s ♦❜t❛✐♥❡❞ ❢r♦♠ s✐♠✉❧❛t✐♦♥s ✭Vs✐♠✮ ❛♥❞ ❡①♣❡r✐♠❡♥ts ✭V✐♠❣✮✳ ❚❤❡ ✈❡rt✐❝❛❧ ❜❛r ✐s t❤❡ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥✳ ❚❤❡ ❝✉r✈❡ ❢♦r s✐♠✉❧❛t✐♦♥s ❛r❡ s❧✐❣❤t❧② s❤✐❢t❡❞ t♦ t❤❡ r✐❣❤t t♦ ❜❡tt❡r ❞✐st✐♥✲ ❣✉✐s❤ t❤❡ t✇♦ ❝✉r✈❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷ ✹✳✷✹ ❊✈♦❧✉t✐♦♥ ♦❢ ❜✉❜❜❧❡ s❤❛♣❡ ❢❛❝t♦rs✳ ❚❤❡ ✐♠❛❣❡s ❤❛✈❡ t❤❡ s❛♠❡ s❝❛❧❡ ✿ ✸ ♠♠ ❂ ✶✷✻ ♣✐①❡❧s✳ ❱❋❈✿ ✶%✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹ ✹✳✷✺ ❊✈♦❧✉t✐♦♥ ♦❢ ❜✉❜❜❧❡ s❤❛♣❡ ❢❛❝t♦rs✳ ❚❤❡ ✐♠❛❣❡s ❤❛✈❡ t❤❡ s❛♠❡ s❝❛❧❡ ✿ ✸ ♠♠ ❂ ✶✷✻ ♣✐①❡❧s✳ ❱❋❈✿ ✸%✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹ ✹✳✷✻ ❊✈♦❧✉t✐♦♥ ♦❢ ❜✉❜❜❧❡ s❤❛♣❡ ❢❛❝t♦rs✳ ❚❤❡ ✐♠❛❣❡s ❤❛✈❡ t❤❡ s❛♠❡ s❝❛❧❡ ✿ ✸ ♠♠ ❂ ✶✷✻ ♣✐①❡❧s✳ ❱❋❈✿ ✻%✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✺ ✹✳✷✼ ❊✈♦❧✉t✐♦♥ ♦❢ ❜✉❜❜❧❡ s❤❛♣❡ ❢❛❝t♦rs✳ ❚❤❡ ✐♠❛❣❡s ❤❛✈❡ t❤❡ s❛♠❡ s❝❛❧❡ ✿ ✸ ♠♠ ❂ ✶✷✻ ♣✐①❡❧s✳ ❱❋❈✿ ✾%✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✺ ✹✳✷✽ ❊✈♦❧✉t✐♦♥ ♦❢ ❜✉❜❜❧❡ s❤❛♣❡ ❢❛❝t♦rs✳ ❚❤❡ ✐♠❛❣❡s ❤❛✈❡ t❤❡ s❛♠❡ s❝❛❧❡ ✿ ✸ ♠♠ ❂ ✶✷✻ ♣✐①❡❧s✳ ❱❋❈✿ ✶✷%✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✻ ✹✳✷✾ ❊✈♦❧✉t✐♦♥ ♦❢ ❜✉❜❜❧❡ s❤❛♣❡ ❢❛❝t♦rs✳ ❚❤❡ ✐♠❛❣❡s ❤❛✈❡ t❤❡ s❛♠❡ s❝❛❧❡ ✿ ✸ ♠♠ ❂ ✶✷✻ ♣✐①❡❧s✳ ❱❋❈✿ ✷✵%✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✻ ✹✳✸✵ ❊✈♦❧✉t✐♦♥ ♦❢ ❜✉❜❜❧❡ s❤❛♣❡ ❢❛❝t♦rs✳ ❚❤❡ ✐♠❛❣❡s ❤❛✈❡ t❤❡ s❛♠❡ s❝❛❧❡ ✿ ✸ ♠♠ ❂ ✽✵ ♣✐①❡❧s✳ ❱❋❈✿ ✸✵%✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✼

①✈ ✺✳✶ ❚❤❡ ❝②❧✐♥❞r✐❝❛❧ r❡❛❝t♦r ✉s❡❞ ❢♦r t❤❡ ✐♥✈❡st✐❣❛t✐♦♥ ♦❢ ❤②❞r♦❞②♥❛♠✐❝s✳ ✳ ✳ ✽✾ ✺✳✷ ❚❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ ✈♦❧✉♠❡✲❛✈❡r❛❣❡❞ ♠♦♠❡♥t✉♠✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✶ ✺✳✸ ❲❤❛t t❤❡ ♣❤♦t♦❜✐♦r❡❛❝t♦r ❧♦♦❦ ❧✐❦❡ ✵✳✶ s ❛❢t❡r tr❛❝❡r ✐♥❥❡❝t✐♦♥✳ ●r❡❡♥✿ ❜✉❜❜❧❡ ✐♥t❡r❢❛❝❡✱ ❜❧✉❡✿ tr❛❝❡r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✷ ✺✳✹ ❚❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ ♥♦r♠❛❧✐③❡❞ ✈❛r✐❛♥❝❡✳ ❚❤❡ st❛rt t✐♠❡ ✐s t❤❡ ♠♦♠❡♥t t❤❛t tr❛❝❡r ✐s ✐♥❥❡❝t❡❞✱ ❛♥❞ t❤❡ ❧✐q✉✐❞ ❛❧r❡❛❞② r❡❛❝❤❡s q✉❛s✐✲st❡❛❞② st❛t❡ ❛t t❤❛t ♠♦♠❡♥t✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✸ ✺✳✺ ❙❝❤❡♠❛t✐❝ s❦❡t❝❤ ♦❢ ❛ P■❱ s②st❡♠✱ ✐❧❧✉str❛t✐♦♥ ❢r♦♠ ❬✶✽✸❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✹

①✈✐✐

▲✐st ♦❢ ❚❛❜❧❡s

✷✳✶ ❆ ❝♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ ♦♣❡♥ ❛♥❞ ❝❧♦s❡❞ ♠✐❝r♦❛❧❣❛❧ ❝✉❧t✐✈❛t✐♦♥ s②st❡♠s✱ ❜❛s❡❞ ♦♥ ❬✸✷❪ ❬✸✸❪ ❬✸✹❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✷✳✷ ❆ ❝♦♠♣❛r✐s♦♥ ♦❢ ✈❛r✐♦✉s t②♣❡s ♦❢ P❇❘s✱ ❜❛s❡❞ ♦♥ ❬✸✶❪ ❬✸✺❪ ❬✸✻❪ ❬✸✼❪✳ ✳ ✳ ✻ ✷✳✸ ❱❛r✐♦✉s st✉❞✐❡s t❤❛t ❛♣♣❧✐❡s ❞✐✛❡r❡♥t ♠✉❧t✐♣❤❛s❡ ♠♦❞❡❧s t♦ ✐♥✈❡st✐❣❛t❡ ❣❛s✲❧✐q✉✐❞ ✢♦✇ ✐♥ ❛ r❡❛❝t♦r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷✳✹ ❉✐✛❡r❡♥t ❦✐♥❡t✐❝ ♠♦❞❡❧s ❬✶✵✵❪ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸✳✶ ❈❛❧✐❜r❛t✐♦♥ ♦❢ ❱❋❈✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✸✳✷ ▼❡❛♥ ❜✉❜❜❧✐♥❣ ❢r❡q✉❡♥❝② ❛t st❡❛❞② st❛t❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✹✳✶ ❈♦♠♣❛r✐s♦♥ ♦❢ ❱❖❋ ❛♥❞ ❊✉❧❡r✐❛♥✲▲❛❣r❛♥❣✐❛♥ ♠♦❞❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✹✳✷ ❈♦♥✜❣✉r❛t✐♦♥s ❢♦r ✷❉ s✐♠✉❧❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✹✳✸ ❇♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❢♦r ✷❉ s✐♠✉❧❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ✹✳✹ ❈♦♥✜❣✉r❛t✐♦♥s ❢♦r ✸❉ s✐♠✉❧❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺ ✺✳✶ ◆✉♠❡r✐❝❛❧ s❡t✉♣ ❢♦r t❤❡ ❝②❧✐♥❞r✐❝❛❧ r❡❛❝t♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✵ ❆✳✶ ❆ s✉♠♠❛r② ♦❢ r❡♣r❡s❡♥t❛t✐✈❡ ❧✐t❡r❛t✉r❡ ❛✐♠✐♥❣ t♦ ✐♥✈❡st✐❣❛t❡ ❣❛s✲❧✐q✉✐❞ ✢♦✇ ✐♥ ❛ ❜✉❜❜❧❡ ❝♦❧✉♠♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✾

①✐①

▲✐st ♦❢ ❆❜❜r❡✈✐❛t✐♦♥s

✷❉ ❚✇♦✲❞✐♠❡♥s✐♦♥❛❧ ✸❉ ❚❤r❡❡✲❞✐♠❡♥s✐♦♥❛❧ ❈❋❉ ❈♦♠♣✉t❛t✐♦♥❛❧ ❋❧✉✐❞ ❉②♥❛♠✐❝s ❈❋▲ ❈♦✉r❛♥t✲❋r✐❡❞r✐❝❤s✲▲❡✇② ❈P❯ ❈❡♥tr❛❧ Pr♦❝❡ss✐♥❣ ❯♥✐t ❈❙❋ ❈♦♥t✐♥✉✉♠ ❙✉r❢❛❝❡ ❋♦r❝❡ ❈✳❱✳ ❈♦♥tr♦❧ ❱♦❧✉♠❡ ❉✲❆ ❉♦♥♦r✲❆❝❝❡♣t♦r ❉❖▼ ❉✐s❝r❡t❡ ❖r❞✐♥❛t❡s ▼❡t❤♦❞ ❊✲❊ ❊✉❧❡r✐❛♥✲❊✉❧❡r✐❛♥ ❊✲▲ ❊✉❧❡r✐❛♥✲▲❛❣r❛♥❣✐❛♥ ❍❙▼ ❍♦r✐③♦♥t❛❧ ❙❡❣♠❡♥t❛t✐♦♥ ▼❡t❤♦❞ ■❆ ■♥t❡rr♦❣❛t✐♦♥ ❆r❡❛ ▲❉❆ ▲❛s❡r ❉♦♣♣❧❡r ❆♥❡♠♦♠❡tr② ▲❊❉ ▲✐❣❤t✲❡♠✐tt✐♥❣ ❞✐♦❞❡ ▲■❋ ▲❛s❡r✲■♥❞✉❝❡❞ ❋❧✉♦r❡s❝❡♥❝❡ ▼❈▼ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞ ❖♣❡♥❋❖❆▼ ❖♣❡♥ ❙♦✉r❝❡ ❋✐❡❧❞ ❖♣❡r❛t✐♦♥ ❛♥❞ ▼❛♥✐♣✉❧❛t✐♦♥ P❇❊ P♦♣✉❧❛t✐♦♥ ❇❛❧❛♥❝❡ ❊q✉❛t✐♦♥ P❇❘ P❤♦t♦❜✐♦r❡❛❝t♦r P❈ P❡rs♦♥❛❧ ❈♦♠♣✉t❡r P❉❊ P❛rt✐❛❧ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥ P■❱ P❛rt✐❝❧❡ ■♠❛❣❡ ❱❡❧♦❝✐♠❡tr② P▲■❈ P✐❡❝❡✇✐s❡ ▲✐♥❡❛r ■♥t❡r❢❛❝❡ ❈❛❧❝✉❧❛t✐♦♥ ❘❚❊ ❘❛❞✐❛t✐✈❡ ❚r❛♥s❢❡r ❊q✉❛t✐♦♥ ❙▲■❈ ❙✐♠♣❧❡ ▲✐♥❡❛r ■♥t❡r❢❛❝❡ ❈❛❧❝✉❧❛t✐♦♥ ❱❋❈ ❱♦❧✉♠❡tr✐❝ ❋❧♦✇ ❈♦♥tr♦❧❧❡r ❱❖❋ ❱♦❧✉♠❡ ❖❢ ❋❧✉✐❞

①①✐

▲✐st ♦❢ ❙②♠❜♦❧s

❙②♠❜♦❧ ◆❛♠❡ ❯♥✐ts A ❛r❡❛ ♠2 [A] ♠❛tr✐① r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛♥ ♦♣❡r❛t♦r ✲ AR ❛s♣❡❝t r❛t✐♦ ✶ [b] ❦♥♦✇♥ ❝♦❧✉♠♥ ✈❡❝t♦r ✲ C ♠❛ss ❝♦♥❝❡♥tr❛t✐♦♥ ❦❣·♠−3 c ❣❡♥❡r❛❧ ❝♦❡✣❝✐❡♥t ✲ Co ❈♦✉r❛♥t ♥✉♠❜❡r ✶ D ❞✐❧✉t✐♦♥ r❛t❡ s−1 D ❞✐✛✉s✐✈✐t② ♠2_·s−1 d ❞✐❛♠❡t❡r ♠ e ❜❛s❡ ♦❢ t❤❡ ♥❛t✉r❛❧ ❧♦❣❛r✐t❤♠ ✲ ❊♦ ❊öt✈ös ♥✉♠❜❡r ✶ F ✐♥❞✐❝❛t♦r ❢✉♥❝t✐♦♥ ✶ F ❣❡♥❡r❛❧ ❢♦r❝❡ ◆ ✭❦❣·♠·s−2_✮ f_❡①t ❢♦r❝❡ ♣❡r ✉♥✐t ♠❛ss ♠·s−2 fC ❢♦r❝❡ ♣❡r ✉♥✐t ❧❡♥❣t❤ ◆·♠−1 fσ ❢♦r❝❡ ♣❡r ✉♥✐t ✈♦❧✉♠❡ ❦❣·♠−2·s−2 f ❣❡♥❡r❛❧ ❢♦r❝❡ ♣❡r ✉♥✐t ♠❛ss✴✈♦❧✉♠❡✴❧❡♥❣t❤ ◆·❦❣−1✱ ◆·♠−3 ♦r ◆·♠−1 f ❢r❡q✉❡♥❝② ❍③ ✭s−1✮ f_❝✐r❝ ❝✐r❝✉❧❛r✐t② ✶ G ❣❡♥❡r❛❧ ❢✉♥❝t✐♦♥ ✶ H ❞✐♠❡♥s✐♦♥❧❡ss ❍❡♥r② ❝♦❡✣❝✐❡♥t ✶ h ❤❡✐❣❤t ♠ I ♣✐①❡❧ ✐♥t❡♥s✐t② ✲ I s♣❡❝✐✜❝ r❛❞✐❛t✐✈❡ ✐♥t❡♥s✐t② ❲·♠−2·sr−1_·❍③−1 i ❣❡♥❡r❛❧ ✐♥t❡❣❡r ✲ J ♠❛ss ✢✉① ❦❣·♠−2_·s−1 j ❣❡♥❡r❛❧ ✐♥t❡❣❡r ✲ je ❡♠✐ss✐♦♥ ❝♦❡✣❝✐❡♥t ♠·s−3·sr−1 k ❣❡♥❡r❛❧ ✐♥t❡❣❡r ✲ ka ❣❡♥❡r❛❧ ❝♦❡✣❝✐❡♥t ✲ k, kg, kl ♠❛ss tr❛♥s❢❡r ❝♦❡✣❝✐❡♥t ♠·s−1 Ks s✉❜str❛t❡ ❝♦♥❝❡♥tr❛t✐♦♥ ✇❤❡♥ µ = 0.5µ♠❛① ❦❣·♠−3 Kg, Kl ♦✈❡r❛❧❧ ♠❛ss tr❛♥s❢❡r ❝♦❡✣❝✐❡♥t ♠·s−1 l ❣❡♥❡r❛❧ ✐♥t❡❣❡r ✲ M ♠❛ss ❦❣ m ❣❡♥❡r❛❧ ✐♥t❡❣❡r ✲ ▼♦ ▼♦rt♦♥ ♥✉♠❜❡r ✶ N ❣❡♥❡r❛❧ ✐♥t❡❣❡r ✲ N t♦t❛❧ ❝❡❧❧ ♥✉♠❜❡r ✲ n r❡❢r❛❝t✐✈❡ ✐♥❞❡① ✶

①①✐✐ n ❛♠♦✉♥t ♦❢ s✉❜st❛♥❝❡ ♠♦❧ ~n ♥♦r♠❛❧ ✈❡❝t♦r ❛t ✐♥t❡r❢❛❝❡ (≈ ∇αl) ♠−1 ˆ n ✉♥✐t ♥♦r♠❛❧ ✈❡❝t♦r ❛t ✐♥t❡r❢❛❝❡ ✶ nc ❝❡❧❧ ♥✉♠❜❡r ❞❡♥s✐t② ✐♥ t♦t❛❧ s♣❛❝❡ ✲ O tr✉♥❝❛t✐♦♥ ❡rr♦r ✲ P ♣r♦❞✉❝t ❝♦♥❝❡♥tr❛t✐♦♥ ❦❣·♠−3 P r❛t❡ ♦❢ ♣❤♦t♦s②♥t❤❡s✐s ✲ P ♣❡r✐♠❡t❡r ♠ p ♠♦♠❡♥t✉♠ ❦❣·♠·s−1 p ✈♦❧✉♠❡✲❛✈❡r❛❣❡❞ ♠♦♠❡♥t✉♠ ❦❣·♠−2·s−1 p ♣r❡ss✉r❡ P❛ ✭❦❣·♠−1_·s−2✮ Q ✈♦❧✉♠❡tr✐❝ ✢♦✇ r❛t❡ ♠3_·s−1 q q✉❛❞r❛t✉r❡ ✇❡✐❣❤t ✲ rX ❝❡❧❧ ❣r♦✇t❤ r❛t❡ ❦❣·♠−3·s−1 rD ❝❡❧❧ ❞❡❛t❤ r❛t❡ ❦❣·♠−3·s−1 rS s✉❜str❛t❡ ❝♦♥s✉♠♣t✐♦♥ r❛t❡ ❦❣·♠−3·s−1 r ✭❛s s✉❜s❝r✐♣t✮ s♣❛t✐❛❧ ❝♦♦r❞✐♥❛t❡s ♠ r r❛❞✐✉s ♠ ~r ♣♦s✐t✐♦♥ ♠ R r❛❞✐✉s ♠ R r❛♥❞♦♠ ♥✉♠❜❡r ✲ ❘❡ ❘❡②♥♦❧❞s ♥✉♠❜❡r ✶ S s✉❜str❛t❡ ❝♦♥❝❡♥tr❛t✐♦♥ ❦❣·♠−3 s s♦✉r❝❡ t❡r♠ ✲ ˆ s ❣❡♥❡r❛❧ ♥♦r♠❛❧✐③❡❞ ✈❡❝t♦r ♠ sc ✈❛r✐❛t✐♦♥ ♦❢ ❝❡❧❧ ♥✉♠❜❡r ❞❡♥s✐t② ✐♥ t♦t❛❧ s♣❛❝❡ ✲ t t✐♠❡ s T t❤❡r♠♦❞②♥❛♠✐❝ t❡♠♣❡r❛t✉r❡ ❑ u ✈❡❧♦❝✐t② ♠·s−1 V ✈♦❧✉♠❡ ♠3 v ✈❡❧♦❝✐t② ✐♥ P■❱ ♠·s−1 vg ✐♥❥❡❝t✐♥❣ ❣❛s ✈❡❧♦❝✐t② ♠·s−1 w ✇♦r❦ ♣❡r ✉♥✐t ❛r❡❛ ❏·♠−2 ✭❦❣·s−2✮ w ✇✐❞t❤ ♠ X ❝❡❧❧ ❝♦♥❝❡♥tr❛t✐♦♥ ❦❣·♠−3 [x] ✉♥❦♥♦✇♥ ❝♦❧✉♠♥ ✈❡❝t♦r ✲ Y ②✐❡❧❞ ❝♦❡✣❝✐❡♥t ✶ x, y, z s♣❛t✐❛❧ ❝♦♦r❞✐♥❛t❡s ♠ α ❝♦♥t❛❝t ❛♥❣❧❡ r❛❞ αl ✈♦❧✉♠❡ ❢r❛❝t✐♦♥ ♦❢ ❧✐q✉✐❞ ✶ β ❝♦♥t❛❝t ❛♥❣❧❡ r❛❞ βe ❡①t✐♥❝t✐♦♥ ❝♦❡✣❝✐❡♥t ♠−1 δ ❞✐✛❡r❡♥❝❡ ✲ ∆ ❞✐✛❡r❡♥❝❡ ✲ ∆x✭✐♥ ❝❤❛♣t❡r ✹✮ ♠❡s❤ s✐③❡ ♠ ∆r ❞✐s♣❧❛❝❡♠❡♥t ♠ ε ❣❡♥❡r❛❧ ❝♦❡✣❝✐❡♥t ✲ ǫ ✈❛r✐❛t✐♦♥ ♦❢ ❝❡❧❧ ♥✉♠❜❡r ❞❡♥s✐t② ✐♥ ♣❤②s✐❝❛❧ s♣❛❝❡ ✲ θ✭✐♥ ❝❤❛♣t❡r ✷✮ ❝♦♥t❛❝t ❛♥❣❧❡ r❛❞

①①✐✐✐ θ✭✐♥ ❝❤❛♣t❡r ✺✮ ❛③✐♠✉t❤❛❧ ❛♥❣❧❡ ✐♥ s♣❤❡r✐❝❛❧ ❝♦♦r❞✐♥❛t❡s r❛❞ ζ ✈❡❧♦❝✐t② ✐♥ ❛♥ ✐♥t❡r♥❛❧ s♣❛❝❡ ✲ η ❞②♥❛♠✐❝ ✈✐s❝♦s✐t② P❛·s ✭❦❣·♠−1_·s−1_✮ κ ❝✉r✈❛t✉r❡ ♠−1 κa ❛❜s♦r♣t✐♦♥ ❝♦❡✣❝✐❡♥t ♠−1 Λ ❛ ❝♦♥st❛♥t ❜❡t✇❡❡♥ ✵ ❛♥❞ ✶ ✲ λ ❍❡♥r② ❝♦❡✣❝✐❡♥t P❛·♠3·♠♦❧−1 µa ❛tt❡♥✉❛t✐♦♥ ❝♦❡✣❝✐❡♥t ♠−1 µ s♣❡❝✐✜❝ ❝❡❧❧ ❣r♦✇t❤ r❛t❡ s−1 ν ❦✐♥❡♠❛t✐❝ ✈✐s❝♦s✐t② ♠2·s−1 ξ ❝♦♦r❞✐♥❛t❡s ♦❢ ❛♥ ✐♥t❡r♥❛❧ s♣❛❝❡ ✲ π P✐ ✲ ρ ❞❡♥s✐t② ❦❣·♠−3 σ s✉r❢❛❝❡ t❡♥s✐♦♥ ◆·♠−1 ✭❦❣·s−2✮ σs s❝❛tt❡r✐♥❣ ❝♦❡✣❝✐❡♥t ♠−1 τ ♦♣t✐❝❛❧ ♣❛t❤ ❧❡♥❣t❤ ♠ Φ ❝r♦ss✲❝♦rr❡❧❛t✐♦♥ ✲ φ ♣♦❧❛r ❛♥❣❧❡ ✐♥ s♣❤❡r✐❝❛❧ ❝♦♦r❞✐♥❛t❡s r❛❞ χ ❝❡❧❧ ♥✉♠❜❡r ❞❡♥s✐t② ✐♥ ♣❤②s✐❝❛❧ s♣❛❝❡ ✲ Ψ s❝❛tt❡r✐♥❣ ♣❤❛s❡ ❢✉♥❝t✐♦♥ ✲ ψ ❣❡♥❡r❛❧ s❝❛❧❛r ✜❡❧❞ ✲ Ω ✈♦❧✉♠❡ ♠3 Ω✭❛s dΩ✮ s♦❧✐❞ ❛♥❣❧❡ sr ∇ ♥❛❜❧❛ ♦♣❡r❛t♦r✱ ∇ =∂x∂ ,∂y∂ ,∂z∂  ♠−1

①①✈

P❤②s✐❝❛❧ ❈♦♥st❛♥ts

❆✐r ❞❡♥s✐t② ρ_❛✐r=✶✳✶✾ ❦❣·♠−3 ❆✐r ❞②♥❛♠✐❝ ✈✐s❝♦s✐t② µ_❛✐r= 1.80_{× 10}−5 P❛·s ❆✐r ♠♦❧❛r ♠❛ss M_❛✐r= 2.90_{× 10}−2 _{❦❣·♠♦❧}−1 ❆✐r✲✇❛t❡r s✉r❢❛❝❡ t❡♥s✐♦♥ σ0 = 7.20× 10−2 ◆·♠−1 ❆t♠♦s♣❤❡r✐❝ ♣r❡ss✉r❡ p_❛t♠= 1.00_{× 10}5 P❛ ●r❛✈✐t❛t✐♦♥❛❧ ❛❝❝❡❧❡r❛t✐♦♥ g =✾✳✽✵✻✻✺ ♠·s−2 ●❛s ❝♦♥st❛♥t R ₌✽✳✸✶✹✹✺✾✽ ❏·♠♦❧−1_·❑−1 ❘♦♦♠ t❡♠♣❡r❛t✉r❡ ❞✉r✐♥❣ ❡①♣❡r✐♠❡♥ts T0 =✷✾✸ ❑ ❲❛t❡r ❞❡♥s✐t② ρ_✇❛t❡r=✾✾✽ ❦❣·♠−3 ❲❛t❡r ❞②♥❛♠✐❝ ✈✐s❝♦s✐t② µ_✇❛t❡r= 1.00× 10−3 _P❛·s

①①✈✐✐

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✶

❈❤❛♣t❡r ✶

■♥tr♦❞✉❝t✐♦♥

❆s t❤❡ ✐♥❞✉str✐❛❧✐③❛t✐♦♥ ♣r♦❝❡ss s♣❡❡❞ ✉♣✱ ✇❡ ❞❡♠❛♥❞ ♠♦r❡ ❛♥❞ ♠♦r❡ ❡♥❡r❣② t♦ ❢✉❧✜❧❧ t❤❡ ❡❝♦♥♦♠✐❝ ❞❡✈❡❧♦♣♠❡♥t✱ ❛s s❤♦✇♥ ✐♥ ❋✐❣✳ ✶✳✶ ❢r♦♠ t❤❡ r❡♣♦rt ♦❢ P❡tr♦❧❡✉♠ ❬✶❪✳ ▼❡❛♥✇❤✐❧❡✱ ✇❡ ❝❛♥ s❡❡ ❢r♦♠ ❋✐❣✳ ✶✳✶t❤❛t t❤❡ ❧❛r❣❡st ❢r❛❝t✐♦♥ ♦❢ ❝♦♥s✉♠❡❞ ❡♥❡r❣② ✐s st✐❧❧ ❢♦ss✐❧ ❢✉❡❧✱ ❛♥❞ ✐t s❡❡♠s t❤❛t t❤❡ s✐t✉❛t✐♦♥ ✇✐❧❧ ❝♦♥t✐♥✉❡ t♦ ❜❡ s♦ ❢♦r ❞❡❝❛❞❡s ❬✷❪✳ ❍♦✇❡✈❡r✱ ❢♦ss✐❧ ❢✉❡❧ ✐s ❛ ♥♦♥✲r❡♥❡✇❛❜❧❡ r❡s♦✉r❝❡✱ ✐♥ ♦t❤❡r ✇♦r❞s✱ ✐t ❞♦❡s ♥♦t r❡♥❡✇ ✐ts❡❧❢ ❛t ❛ s✉✣❝✐❡♥t r❛t❡ ❢♦r s✉st❛✐♥❛❜❧❡ ❡❝♦♥♦♠✐❝ ❡①tr❛❝t✐♦♥ ✐♥ ♠❡❛♥✐♥❣❢✉❧ ❤✉♠❛♥ t✐♠❡✲❢r❛♠❡s ❬✸❪✳ ❲♦rs❡ st✐❧❧✱ t❤❡ ❝♦♠❜✉st✐♦♥ ♦❢ ❢♦ss✐❧ ❢✉❡❧ ❡♠✐ts ❡♥♦r♠♦✉s ❛♠♦✉♥ts ♦❢ ❝❛r❜♦♥ ❞✐♦①✐❞❡✱ ❛ ❣r❡❡♥❤♦✉s❡ ❣❛s✱ ✐♥t♦ t❤❡ ❛t♠♦s♣❤❡r❡✳ ❆❜♦✉t ❤❛❧❢ ♦❢ t❤❡ ❡♠✐tt❡❞ ❝❛r❜♦♥ ❞✐♦①✐❞❡ r❡♠❛✐♥s ✐♥ t❤❡ ❛t♠♦s♣❤❡r❡✱ ✇❤✐❝❤ r❡s✉❧ts ✐♥ t❤❡ ❣❧♦❜❛❧ ✇❛r♠✐♥❣ ❛♥❞ ♦t❤❡r ❝❧✐♠❛t❡ ♣r♦❜❧❡♠s ❬✹❪✳ World consumption

Million tonnes oil equivalent

0 15 16 14 13 12 11 10 09 08 07 06 05 04 03 02 01 00 99 98 97 96 95 94 93 92 91 14000 13000 12000 11000 10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 Coal Hydroelectricity Nuclear energy Natural gas Oil Renewables

World primary energy consumption grew by 1.0% in 2016, well below the 10-year average of 1.8% and the third consecutive year at or below 1%. As was the case in 2015, growth was below average in all regions except Europe & Eurasia. All fuels except oil and nuclear power grew at below-average rates. Oil provided the largest increment to energy consumption at 77 million tonnes of oil equivalent (mtoe), followed by natural gas (57 mtoe) and renewable power (53 mtoe).

❋✐❣✉r❡ ✶✳✶✿ ❲♦r❧❞ ❡♥❡r❣② ❝♦♥s✉♠♣t✐♦♥✱ ✐❧❧✉str❛t✐♦♥ ❢r♦♠ t❤❡ ❇P ❙t❛t✐st✐❝❛❧ ❘❡✈✐❡✇ ♦❢ ❲♦r❧❞ ❊♥❡r❣② ❏✉♥❡ ✷✵✶✼✳ ❚♦ ♠✐t✐❣❛t❡ t❤❡ ❝❧✐♠❛t❡ ❝❤❛♥❣❡✱ ❈❖2 ❝❛♣t✉r❡ ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ t♦ ❢♦ss✐❧ ❢✉❡❧ ♣♦✇❡r ♣❧❛♥ts✱ ❢✉❡❧ ♣r♦❝❡ss✐♥❣ ♣❧❛♥ts ❛♥❞ ♦t❤❡r ✐♥❞✉str✐❛❧ ♣❧❛♥ts✳ ❚❤❡r❡ ❛r❡ t❤r❡❡ ♠❛✐♥ s②s✲ t❡♠s ❢♦r ❝❛♣t✉r✐♥❣ ❈❖2✿ ♣♦st✲❝♦♠❜✉st✐♦♥ ❝❛♣t✉r❡✱ ♣r❡✲❝♦♠❜✉st✐♦♥ ❝❛♣t✉r❡ ❛♥❞ ♦①②✲ ❢✉❡❧ ❝♦♠❜✉st✐♦♥ ❝❛♣t✉r❡ ❬✺❪✳ ❍❡r❡ ✇❡ ❢♦❝✉s ♠❛✐♥❧② ♦♥ t❤❡ ♣♦st✲❝♦♠❜✉st✐♦♥ ❝❛♣t✉r❡✱