By Atul Sharma

Advent to Computational Fluid Dynamics: improvement, program and research is an introductory textbook overlaying computational fluid dynamics (CFD). the 1st half introduces the basics of the subject, overlaying fluid dynamics, warmth move and numerical tools for CFD. half considers a unique physics established strategy for CFD for cartesian geometry on a uniform grid, masking unsteady computational warmth conduction and warmth convection. half 3 introduces a standard arithmetic established method for CFD for cartesian geometry on a non-uniform grid, and at last half 4 covers CFD for complicated geometry on a curvilinear grid. CFD improvement is roofed intimately with precise specialise in the implementation info, that are confirmed with figures, animations, and pseudo-code. CFD software and research is confirmed with difficulties and codes written in open-source software program. Powerpoint slides and animations also are supplied on an accompanying web site.

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Atul Sharma. © Author, 2017. Published by Athena Academic Ltd. and John Wiley & Sons Ltd.

Tr♦❞✉❝t✐♦♥ t♦ ❈❋❉ ✶✸ ❝❤r♦♥✐③❡❞ ♠♦✈✐❡ ♦❢ t❤❡ ✢♦✇✲♣r♦♣❡rt✐❡s ❛♥❞ ❡♥❣✐♥❡❡r✐♥❣✲♣❛r❛♠❡t❡rs✱ ✇✐t❤ ❛ t✐♠❡✲✇✐s❡ ✈❛r✐❛t✐♦♥ ♦❢ ♣✐❝t✉r❡ ❛♥❞ s②♠❜♦❧ ✭♦✈❡r t❤❡ ❧✐♥❡ ♣❧♦t✮✱ r❡s♣❡❝t✐✈❡❧②✳ ❙❝✐❡♥t✐✜❝✲✐♥✈❡st✐❣❛t✐♦♥ ❝♦rr❡s♣♦♥❞s t♦ ❛ st♦r② ❛❜♦✉t t❤❡ ✐♥t❡r✲ ❡st✐♥❣ ✢✉✐❞✲❞②♥❛♠✐❝s ♣❤❡♥♦♠❡♥♦♥ ✭✢♦✇✲s❡♣❛r❛t✐♦♥✱ ✇❛❦❡✲❢♦r♠❛t✐♦♥✱ ❛♥❞ ✈♦rt❡①✲s❤❡❞❞✐♥❣✮ ❡♥❝♦✉♥t❡r❡❞ ✐♥ t❤✐s ♣r♦❜❧❡♠✳ ❲❤❡r❡❛s✱ t❤❡ ❡♥❣✐♥❡❡r✐♥❣✲✐♥✈❡st✐❣❛t✐♦♥ ❝♦rr❡s♣♦♥❞s t♦ ❛ ♠❡t❤♦❞♦❧♦❣② ❢♦r t❤❡ str✉❝✲ t✉r❛❧ ❞❡s✐❣♥ ♦❢ t❤❡ ♣✐❧❧❛rs ❛❢t❡r ♦❜t❛✐♥✐♥❣ t❤❡ t✐♠❡ ✇✐s❡ ✈❛r②✐♥❣ ❧✐❢t ❛♥❞ ❞r❛❣ ❢♦r❝❡s✳ ❆ s❝✐❡♥t✐✜❝ ❛s ✇❡❧❧ ❛s ❡♥❣✐♥❡❡r✐♥❣ ✐♥✈❡st✐❣❛t✐♦♥ ❝♦rr❡✲ s♣♦♥❞s t♦ ✐❞❡♥t✐❢②✐♥❣ ❛ ❝♦rr❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❢♦r❝❡s ❛♥❞ ✢♦✇ str✉❝✲ t✉r❡ ❛s ✇❡❧❧ ❛s t❤❡ ✢✉✐❞✲❞②♥❛♠✐❝s ♣❤❡♥♦♠❡♥♦♥✳ ❚❤❡ r♦❧❡ ♦❢ ❛ ❈❋❉ s♦❢t✇❛r❡ ✐s ♥♦t ♦♥❧② ❧✐♠✐t❡❞ t♦ ❝r❡❛t✐♥❣ s❝✐❡♥t✐✜❝❛❧❧②✲❡①❝✐t✐♥❣ ✢✉✐❞ ❞②♥❛♠✐❝ ♠♦✈✐❡ ♦❢ ✢♦✇✲♣r♦♣❡rt✐❡s✱ ❜✉t ❛❧s♦ t♦ ❝r❡❛t❡ ❡♥❣✐♥❡❡r✐♥❣✲ r❡❧❡✈❛♥t ♠♦✈✐❡ ♦❢ ❡♥❣✐♥❡❡r✐♥❣✲♣❛r❛♠❡t❡rs✱ ❢♦r ❛ ✉♥✐✜❡❞ ❝❛✉s❡✲❛♥❞✲ ❡✛❡❝t st✉❞② ♦❢ ✈❛r✐♦✉s ❤❡❛t ❛♥❞ ✢✉✐❞ ✢♦✇ s✐t✉❛t✐♦♥s✳ ❚❤❡ ❛❜♦✈❡ ❝❛✉s❡✲❛♥❞✲❡✛❡❝t st✉❞② ❢♦r t❤❡ ❝②❧✐♥❞❡r ✐s ❛❧s♦ r❡❧❡✈❛♥t ❢♦r t❤❡ ✢♦✇ ❛❝r♦ss ❛ ❝❛r ❛♥❞ ❛❧s♦ ♦✈❡r ❛♥ ❛✐r❝r❛❢t✳ ✶✳✶✳✷ ❆♥❛❧♦❣② ✇✐t❤ ❛ ❱✐❞❡♦✲❈❛♠❡r❛ ✶✳✶✳✷✳✶ ❈❋❉ ❛s ❛ ❋✐♥✐t❡✲❘❡s♦❧✉t✐♦♥ ❱✐rt✉❛❧ ❱✐❞❡♦✲❈❛♠❡r❛ ❆❢t❡r ❞✐s❝✉ss✐♥❣ t❤❡ r♦❧❡ ♦❢ ❈❋❉ ❜❛s❡❞ s❝✐❡♥t✐✜❝❛❧❧②✲❡①❝✐t✐♥❣ ❛♥❞ ❡♥❣✐♥❡❡r✐♥❣✲r❡❧❡✈❛♥t ♠♦✈✐❡s ♦♥ t❤❡ ✐♥✈❡st✐❣❛t✐♦♥ ♦❢ ❛ ❤❡❛t✲tr❛♥s❢❡r ❛♥❞ ❛ ✢✉✐❞✲❞②♥❛♠✐❝s ♣r♦❜❧❡♠✱ ❛♥ ❛♥❛❧♦❣② ❜❡t✇❡❡♥ ❛ r❡❛❧ ❛♥❞ ✈✐rt✉❛❧ ✭❢r♦♠ ❛ ❈❋❉ s♦❢t✇❛r❡✮ ✈✐❞❡♦✲❝❛♠❡r❛ ✐s ♣r❡s❡♥t❡❞✳ ❖♥❡ ♥❡❡❞s t♦ ❢♦❝✉s t♦ ❛ s♣❛t✐❛❧ r❡❣✐♦♥ ♦❢ ✐♥t❡r❡st t♦ ❝r❡❛t❡ ❛ ♠♦✈✐❡ ✉s✐♥❣ ❛ ✈✐❞❡♦✲❝❛♠❡r❛✳ ❚❤✐s ✐s ❛❧s♦ tr✉❡ ✐♥ ❈❋❉ ❛♥❞ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ r❡❣✐♦♥ ✐♥✈♦❧✈✐♥❣ ❤❡❛t ❛♥❞✴♦r ✢✉✐❞ ✢♦✇ − ❝❛❧❧❡❞ ❛s ❝♦♠♣✉t❛t✐♦♥❛❧ ❞♦♠❛✐♥✳ ❲✐t❤ ❛ ✈✐❞❡♦✲ ❝❛♠❡r❛✱ t❤❡ s✐③❡ ♦❢ t❤❡ ❢♦❝✉s❡❞✲r❡❣✐♦♥ ✐s ❝♦♥str❛✐♥t ❜② ✐ts ❢♦❝✉s✐♥❣ s♣❡❝✐✜❝❛t✐♦♥✳ ❲❤❡r❡❛s✱ ✐♥ ❈❋❉✱ t❤❡ s✐③❡ ♦❢ ❝♦♠♣✉t❛t✐♦♥❛❧✲❞♦♠❛✐♥ ❛r❡ t❛❦❡♥ s✉❝❤ t❤❛t ❛ ♣❤②s✐❝❛❧❧② r❡❧❡✈❛♥t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ ❛t t❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ❞♦♠❛✐♥✳ ❆ ♣✐❝t✉r❡ t❛❦❡♥ ❢r♦♠ ❛ ✈✐❞❡♦ ❝❛♠❡r❛ ✐s ♠❛❞❡ ✉♣ ♦❢ t❤❡ ✐♠❛❣❡✴✲ ❝♦❧♦r ❝❛♣t✉r❡❞ ❛t ❛ ❧❛r❣❡ ♥✉♠❜❡r ♦❢ ❞✐s❝r❡t❡ ♣✐①❡❧s✳ ❆ ♣✐❝t✉r❡ ❢r♦♠ ❛ ❈❋❉ s♦❢t✇❛r❡ ✐s ♠❛❞❡ ✉♣ ♦❢ t❤❡ ❝♦❧♦rs ❜❛s❡❞ r❡♣r❡s❡♥t❛t✐♦♥ ✭♦❢ ❛ ✢♦✇✲♣r♦♣❡rt②✮ ❛t t❤❡ ❞✐s❝r❡t❡ ♣♦✐♥ts ✐♥ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❞♦♠❛✐♥❀ t❤❡ ♣♦✐♥ts ❛r❡ ❝❛❧❧❡❞ ❛s ❣r✐❞✲♣♦✐♥ts✳ ❚❤❡ ❝♦❧♦rs ❛t t❤❡ ❣r✐❞✲♣♦✐♥ts ❝♦r✲ r❡s♣♦♥❞ t♦ t❤❡ ♥✉♠❜❡rs ❝♦♠♣✉t❡❞ ❢♦r t❤❡ ❛ss♦❝✐❛t❡❞ ✢♦✇✲♣r♦♣❡rt✐❡s✱ ✶✹ ✶✳ ■♥tr♦❞✉❝t✐♦♥ ♠❛♣♣❡❞ ❜② ❛ ❣r❛♣❤✐❝❛❧ s♦❢t✇❛r❡✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ✢♦♦❞❡❞ ❣r❡②✲❝♦❧♦r ❝♦♥t♦✉r ✇✐t❤ ❛ ❝♦❧♦r✲❜❛r ✐s s❤♦✇♥ ✐♥ ❋✐❣✳ ✶✳✷✭❛✮✲✭❞✮ ❢♦r t❡♠♣❡r❛t✉r❡✱ ❋✐❣✳ ✶✳✸✭❛✶✮✲✭❞✶✮ ❢♦r ✈♦rt✐❝✐t②✱ ❛♥❞ ❋✐❣✳ ✶✳✸✭❛✷✮✲✭❞✷✮ ❢♦r ♣r❡ss✉r❡✳ ❚❤❡ ✜❣✉r❡s ❝♦rr❡s♣♦♥❞ t♦ ❛ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❣r✐❞✲♣♦✐♥ts ✭❛♥❛❧♦❣♦✉s t♦ ♣✐①❡❧s✮ ❛s 4000 ❢♦r t❡♠♣❡r❛t✉r❡✱ 67500 ❢♦r ✈♦rt✐❝✐t②✱ ❛♥❞ 32250 ❢♦r ♣r❡ss✉r❡✳ ❚❤❡ q✉❛❧✐t② ♦❢ ❛ ♣✐❝t✉r❡ ♦❜t❛✐♥❡❞ ❢r♦♠ ❛ ❝❛♠❡r❛ ❞❡♣❡♥❞s ♦♥ t❤❡ ♥✉♠❜❡r ♦❢ ♣✐①❡❧s✳ ❲❤❡r❡❛s✱ t❤❡ s♣❛t✐❛❧ ❛❝❝✉r❛❝② ♦❢ t❤❡ r❡s✉❧ts ✭♣r❡s❡♥t❡❞ ❛s ❛ ♣✐❝t✉r❡✮ ♦❜t❛✐♥❡❞ ❢r♦♠ ❛ ❈❋❉ s♦❢t✇❛r❡ ❞❡♣❡♥❞s ♦♥ t❤❡ ♥✉♠❜❡r ♦❢ ❣r✐❞✲♣♦✐♥ts✳ ❙♣❛t✐❛❧ r❡s♦❧✉t✐♦♥ ❢♦r ❛ ♣✐❝t✉r❡ ✐s ❡q✉❛❧ t♦ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ♣✐①❡❧s ✐♥ ❛ ✈✐❞❡♦✲❝❛♠❡r❛ ❛♥❞ t❤❡ ♥✉♠❜❡r ♦❢ ❣r✐❞✲♣♦✐♥ts ✐♥ ❈❋❉✳ ❚❤❡ r❡s♦❧✉t✐♦♥ ✐s ✉♥✐❢♦r♠ ✐♥ ❛ ✈✐❞❡♦ ❝❛♠❡r❛ ❛♥❞ ♠♦st❧② ♥♦♥✲✉♥✐❢♦r♠ ✐♥ ❈❋❉✳ ❚❤✐s ✐s ❜❡❝❛✉s❡ t❤❡ ❣r✐❞✲♣♦✐♥ts ❛r❡ ♥♦♥✲✉♥✐❢♦r♠❧② s♣❛❝❡❞ t♦ ❡✣❝✐❡♥t❧② ❝❛♣t✉r❡ t❤❡ s♣❛t✐❛❧❧② ♥♦♥✲✉♥✐❢♦r♠ ❛❝t✐♦♥✴❞②♥❛♠✐❝s ♦❢ ✢♦✇ ✐♥s✐❞❡ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❞♦♠❛✐♥✳ ❆ ♠♦✈✐❡ ✐s ♠❛❞❡ ❜② ❛ s❡r✐❡s ♦❢ ♣✐❝t✉r❡s ♦❜t❛✐♥❡❞ ❛t ❛ t✐♠❡ ✐♥t❡r✈❛❧ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❢r❛♠❡ r❛t❡ ♦❢ t❤❡ ✈✐❞❡♦ ❝❛♠❡r❛✱ ❛♥❞ t♦ t❤❡ t✐♠❡✲st❡♣ ✐♥ ❝❛s❡ ♦❢ ❛ ❈❋❉ s♦❢t✇❛r❡✱ ❝❛❧❧❡❞ ❛s t❡♠♣♦r❛❧ r❡s♦❧✉t✐♦♥✳ ❚❤❡ s♣❛t✐❛❧ ❛♥❞ t❤❡ t❡♠♣♦r❛❧ r❡s♦✲ ❧✉t✐♦♥s ❝♦♥tr♦❧s t❤❡ q✉❛❧✐t② ♦❢ t❤❡ ❣r❛♣❤✐❝❛❧ r❡s✉❧ts✱ ❥✉❞❣❡❞ ❜② t❤❡ ✈✐s✉❛❧✲❛♣♣❡❛r❛♥❝❡ ✐♥ ❛ ✈✐❞❡♦✲❝❛♠❡r❛ ❛♥❞ t❤❡ ❛❝❝✉r❛❝② ✐♥ ❛ ❈❋❉ s♦❢t✲ ✇❛r❡✳ ❚❤❡ ❛♥❛❧♦❣② ♣r❡s❡♥t❡❞ ❛❜♦✈❡ ❛♣♣❧✐❡s ♥♦t ♦♥❧② t♦ ❈❋❉ ❜✉t ❛❧s♦ ❛♣♣❧✐❝❛❜❧❡ t♦ ❝♦♠♣✉t❛t✐♦♥❛❧✲s♦❧✐❞✲❞②♥❛♠✐❝s ❛s ✇❡❧❧ ❛s ♠❛♥② ♦t❤❡r ♣❤②s✐❝s ❜❛s❡❞ s✐♠✉❧❛t✐♦♥ s♦❢t✇❛r❡ ✐♥ s❝✐❡♥❝❡ ❛♥❞ ❡♥❣✐♥❡❡r✐♥❣✳ ❋♦r ❛❧♠♦st ❛❧❧ t❤❡ ✢✉✐❞ ✢♦✇ ❛♥❞ ❤❡❛t✲tr❛♥s❢❡r s✐t✉❛t✐♦♥s✱ ❈❋❉ ♣r♦✈✐❞❡s ❛ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ ✇❤✐❝❤ ✐s ❛♣♣r♦①✐♠❛t❡✱ ❛♥❞ ❛t ❝❡rt❛✐♥ s♣❛t✐❛❧ ❛♥❞ t❡♠♣♦r❛❧ ❝♦♦r❞✐♥❛t❡s✳ ❚❤✉s✱ ❈❋❉ ✐s ❛ ✜♥✐t❡ r❡s♦❧✉t✐♦♥ ✈✐rt✉❛❧ ✈✐❞❡♦ ❝❛♠❡r❛✳ ✶✳✶✳✷✳✷ ❆♥❛❧②t✐❝❛❧ ❙♦❧✉t✐♦♥ ❛s ❛♥ ■♥✜♥✐t❡✲❘❡s♦❧✉t✐♦♥ ❱✐rt✉❛❧ ❱✐❞❡♦✲❈❛♠❡r❛ ❆♥❛❧②t✐❝❛❧ s♦❧✉t✐♦♥ ❢♦r ❛♥ ✉♥st❡❛❞② ♣r♦❜❧❡♠ ✐s ❛♥ ✐♥✜♥✐t❡ r❡s♦❧✉t✐♦♥ ✈✐rt✉❛❧ ✈✐❞❡♦ ❝❛♠❡r❛✳ ❚❤❡ ❣♦✈❡r♥✐♥❣ ❡q✉❛t✐♦♥s ✐♥ ✢✉✐❞✲❞②♥❛♠✐❝s ❛♥❞ ❤❡❛t✲tr❛♥s❢❡r ❛r❡ P❛rt✐❛❧ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✭P❉❊s✮ − ✇✐t❤ ❞❡✲ ♣❡♥❞❡♥t ✈❛r✐❛❜❧❡s ❛s ✢♦✇✲♣r♦♣❡rt✐❡s✱ ❛♥❞ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ❛s s♣❛✲ t✐❛❧ ❛♥❞ t❡♠♣♦r❛❧ ❝♦♦r❞✐♥❛t❡s✳ ❚❤❡ ❛♥❛❧②t✐❝❛❧ s♦❧✉t✐♦♥ ♦❢ t❤❡ P❉❊s ✐s ❛ ❢✉♥❝t✐♦♥❛❧ r❡❧❛t✐♦♥s❤✐♣ ♦❢ t❤❡ ❞❡♣❡♥❞❡♥t ✐♥ t❡r♠s ♦❢ t❤❡ ✐♥❞❡♣❡♥✲ ❞❡♥t ✈❛r✐❛❜❧❡s✳ ❚❤❡ s♦❧✉t✐♦♥ r❡s✉❧ts ✐♥ ✢♦✇✲♣r♦♣❡rt✐❡s ❛t ❛♥② ✈❛❧✉❡ ♦❢ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈❋❉ ✶✺ s♣❛t✐❛❧ ❛♥❞ t❡♠♣♦r❛❧ ❝♦♦r❞✐♥❛t❡s − ✐♥✜♥✐t❡ r❡s♦❧✉t✐♦♥ ✈✐rt✉❛❧ ✈✐❞❡♦ ❝❛♠❡r❛✳ ❯♥❢♦rt✉♥❛t❡❧②✱ t❤❡ ❛♥❛❧②t✐❝❛❧ ✐s ♥♦t ②❡t ♣♦ss✐❜❧❡ ❢♦r ❛ ❣❡♥❡r❛❧ ✢✉✐❞✲ ❞②♥❛♠✐❝s ♣r♦❜❧❡♠ ❛♥❞ ✐s ♦♥❡ ♦❢ t❤❡ ♠♦st ✐♠♣♦rt❛♥t ✉♥r❡s♦❧✈❡❞ ♣r♦❜✲ ❧❡♠ ✐♥ s❝✐❡♥❝❡ ❛♥❞ ❡♥❣✐♥❡❡r✐♥❣✳ ❚❤❡ ❛❞✈❡♥t ♦❢ t❤❡ ❣❡♥❡r❛❧✲♣✉r♣♦s❡ ❛♥❛❧②t✐❝❛❧ s♦❧✉t✐♦♥ ✇✐❧❧ ❧❡❛❞ t♦ ❝❧♦s✉r❡ ♦❢ ❛❧❧ t❤❡ ❈❋❉ s♦❢t✇❛r❡ ❝♦♠✲ ♣❛♥✐❡s ❛s ✇❡❧❧ ❛s t❤❡ ❡①✐st❡♥❝❡ ♦❢ t❤✐s ❝♦✉rs❡✳ ✶✳✷ ❈❋❉✿ ❲❤② t♦ st✉❞②❄ ❉❡✈❡❧♦♣♠❡♥ts ✐♥ t❤❡ ❡♥❣✐♥❡❡r✐♥❣ s✐♠✉❧❛t✐♦♥ ♠❡t❤♦❞♦❧♦❣② ❤❛s ❧❡❞ t♦ ❛ ❞r❛st✐❝ ✐♠♣r♦✈❡♠❡♥ts ✐♥ t❤❡ ❞❡s✐❣♥ ♠❡t❤♦❞♦❧♦❣② ❢♦r ❛ ♣r♦❞✉❝t✳ ◆♦✇✱ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ❝r❡❛t❡ ✈✐rt✉❛❧ ♣r♦❞✉❝t ✐♥ ❛ ❝♦♠♣✉t❡r ✭✉s✐♥❣ ❛ s♦❧✐❞ ♠♦❞❡❧❡r✮✱ ❛♥❛❧②③❡ ✐ts ♣❡r❢♦r♠❛♥❝❡✱ s✉❣❣❡st s♦♠❡ ❞❡s✐❣♥ ♠♦❞✐✜❝❛t✐♦♥s✱ ❛♥❞ ❝♦♥✜r♠ ❡♥❤❛♥❝❡♠❡♥t ✐♥ ♣❡r❢♦r♠❛♥❝❡ ♦r r❡❞✉❝t✐♦♥ ✐♥ ❝♦st✳ ■t ♣r♦✈✐❞❡s ❛♥ ❡①❝❡❧❧❡♥t ♣❧❛t❢♦r♠ ❢♦r ✐♥♥♦✈❛t✐♦♥s ✐♥ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ t❤❡ ♥❡✇ ♣r♦❞✉❝ts✳ ❲❤❡r❡❛s✱ ❢♦r t❤❡ ❡①✐st✐♥❣ ♣r♦❞✉❝ts✱ ✐t ❧❡❛❞s t♦ ❛ tr❡♠❡♥❞♦✉s ✐♠♣r♦✈❡♠❡♥ts ✐♥ t❤❡ ♣❡r❢♦r♠❛♥❝❡ ❛♥❞ ❡✣❝✐❡♥❝②✳ ◆♦✇ ❛ ❞❛②s✱ ❛ ❝♦♠♣✉t❡r s✐♠✉❧❛t✐♦♥ ❛♥❞ ❛♥❛❧②s✐s ❤❛✈❡ ❜❡❝♦♠❡ ❛♥ ✐♥t❡❣r❛❧ ♣❛rt ♦❢ ❛♥ ❡♥❣✐♥❡❡r✐♥❣ st✉❞②✳ ❈❋❉ ❤❛s ❧❡❞ t♦ ❛ ♠♦r❡ s❝✐❡♥t✐✜❝ ❛s ❝♦♠♣❛r❡❞ t♦ ❡♠♣✐r✐❝❛❧ ❛♣✲ ♣r♦❛❝❤ ✐♥ t❤❡ ❞❡s✐❣♥ ❛♥❞ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❛ ❣❧♦❜❛❧❧② ❝♦♠♣❡t✐t✐✈❡ ♣r♦❞✲ ✉❝t✳ ❙❝✐❡♥t✐sts ❛s ✇❡❧❧ ❛s ❡♥❣✐♥❡❡rs ✇❛♥t t♦ ✉s❡ ❈❋❉ ❛s ❛ ❞❡s✐❣♥ ❛♥❞ ♦♣t✐♠✐③❛t✐♦♥ t♦♦❧✳ ❍♦✇❡✈❡r✱ ❞✉❡ t♦ t❤❡ ✉♥❛✛♦r❞❛❜❧❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦st✱ ♣r❡s❡♥t❧② ✐t ✐s ♠♦r❡ ❝♦♠♠♦♥❧② ✉s❡❞ ❛s ❛ ♣♦✇❡r❢✉❧ ❛♥❛❧②s✐s t♦♦❧✳ ❈❋❉ ✐s ✉s❡❞ ❢♦r ❛♥ ❛♥❛❧②s✐s ❜❛s❡❞ ✐♠♣r♦✈❡♠❡♥ts ✐♥ t❤❡ ❞❡s✐❣♥ ♦❢ ❛ ✢✉✐❞✲❞②♥❛♠✐❝s ❛♥❞ ❤❡❛t✲tr❛♥s❢❡r s②st❡♠s ✐♥ ✈❛r✐♦✉s ✐♥❞✉str✐❡s❀ s✉❝❤ ❛s ❛❡r♦s♣❛❝❡✱ ❛✉t♦♠♦❜✐❧❡✱ t✉r❜♦✲♠❛❝❤✐♥❡r②✱ ❝❤❡♠✐❝❛❧✱ ❡❧❡❝tr✐❝❛❧✱ ❡❧❡❝✲ tr♦♥✐❝s✱ ❜✐♦♠❡❞✐❝❛❧✱ ❡t❝✳ ■♥ ❢✉t✉r❡✱ ❈❋❉ ✇✐❧❧ ❜❡ ♠♦r❡ ♦❢t❡♥ ✉s❡❞ ❛s ❛ ❞❡s✐❣♥ ❛♥❞ ♦♣t✐♠✐③❛t✐♦♥ t♦♦❧✳ ■♥ ❛❝❛❞❡♠✐❝s✱ ❈❋❉ ✐s t❛✉❣❤t ❛s ❛♥ ✉♥❞❡r❣r❛❞✉❛t❡ ❡❧❡❝t✐✈❡ ❛♥❞ ♣♦st❣r❛❞✉❛t❡ ❝♦r❡ ❝♦✉rs❡ ✐♥ ❆❡r♦s♣❛❝❡✱ ❈❤❡♠✐❝❛❧✱ ❛♥❞ ▼❡❝❤❛♥✐❝❛❧ ❜r❛♥❝❤❡s ♦❢ ❊♥❣✐♥❡❡r✐♥❣✳ ❋✉rt❤❡r♠♦r❡✱ ✐t t❛✉❣❤t ❛s ❛♥ ✐♥t❡r❞✐s❝✐✲ ♣❧✐♥❛r② ❝♦✉rs❡ ✐♥ t❤❡ ❈✐✈✐❧✱ ▼❡t❛❧❧✉r❣✐❝❛❧✱ ❊❧❡❝tr✐❝❛❧✱ ❛♥❞ ❇✐♦♠❡❞✐❝❛❧ ❊♥❣✐♥❡❡r✐♥❣✳ ❚❤❡ ✐♥❝r❡❛s✐♥❣ ✐♠♣♦rt❛♥❝❡ ♦❢ ❈❋❉ ❞❡✈❡❧♦♣♠❡♥t✱ ❛♣✲ ♣❧✐❝❛t✐♦♥✱ ❛♥❞ ❛♥❛❧②s✐s ✐♥ t❤❡ ✈❛r✐♦✉s ✐♥❞✉str✐❡s ❛s ✇❡❧❧ ❛s r❡s❡❛r❝❤ ♦r❣❛♥✐③❛t✐♦♥s✱ ❛❧♦♥❣ ✇✐t❤ t❤❡ ❧❛❝❦ ♦❢ tr❛✐♥❡❞ ♠❛♥♣♦✇❡r ✐♥ t❤✐s s✉❜✲ ❥❡❝t✱ ❤❛s ❣r❡❛t❧② ✐♥❝r❡❛s❡❞ t❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ t❤✐s ❝♦✉rs❡✳ ✶✻ ✶✳ ■♥tr♦❞✉❝t✐♦♥ ❋✐❣✉r❡ ✶✳✹✿ P❤②s✐❝❛❧ ❛♥❞ ♠❛t❤❡♠❛t✐❝❛❧ ❛♣♣r♦❛❝❤❡s ♦❢ ❈❋❉ ❞❡✈❡❧♦♣♠❡♥t✳ ✶✳✸ ◆♦✈❡❧t②✱ ❙❝♦♣❡✱ ❛♥❞ P✉r♣♦s❡ ♦❢ t❤✐s ❇♦♦❦ ❋r♦♠ t❤❡ ✢✉✐❞✲❞②♥❛♠✐❝s ❛♥❞ ❤❡❛t✲tr❛♥s❢❡r ❝♦✉rs❡✱ ✇❡ ❧❡❛r♥ t♦ ❞❡✲ r✐✈❡ ❣♦✈❡r♥✐♥❣ P❛rt✐❛❧ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✭P❉❊s✮ ❜② ❛♣♣❧②✐♥❣ t❤❡ ♠❛ss✱ ♠♦♠❡♥t✉♠✱ ❛♥❞ ❡♥❡r❣② ❝♦♥s❡r✈❛t✐♦♥ ❧❛✇s t♦ ❛♥ ❡❧❡♠❡♥✲ ❈♦♥tr♦❧✲❱♦❧✉♠❡ ✭❈❱✮✳ ❲❤❡r❡❛s✱ ✐♥ ❈❋❉✱ t❤❡ ❣♦✈❡r♥✲ ✐♥❣ P❉❊s ❛r❡ ♠❛t❤❡♠❛t✐❝❛❧❧② ♦♣❡r❛t❡❞ t♦ ❞❡r✐✈❡ ❛ s②st❡♠ ♦❢ ▲✐♥❡❛r ❆❧❣❡❜r❛✐❝ ❊q✉❛t✐♦♥s ✭▲❆❊s✮− ❝❛❧❧❡❞ ❛s ❞✐s❝r❡t✐③❛t✐♦♥ ♠❡t❤♦❞❀ s✉❝❤ ❛s✱ ❋✐♥✐t❡ ❱♦❧✉♠❡ ▼❡t❤♦❞ ✭❋❱▼✮✳ ❚❤❡ s②st❡♠ ♦❢ ▲❆❊s r❡s✉❧t✐♥❣ t❛❧ ✢✉✐❞✴s♦❧✐❞ ❢r♦♠ t❤❡ ❛❧❣❡❜r❛✐❝ ❢♦r♠✉❧❛t✐♦♥ ❛❝ts ❛s t❤❡ ❣♦✈❡r♥✐♥❣ ❡q✉❛t✐♦♥s ❢♦r ❈❋❉✱ ❛♥❞ ❝♦♥s✐st ♦❢ t❤❡ ✢♦✇✲♣r♦♣❡rt✐❡s ✭✈❡❧♦❝✐t②✱ ♣r❡ss✉r❡✱ ❛♥❞ t❡♠✲ ♣❡r❛t✉r❡✮ ❛s t❤❡ ✉♥❦♥♦✇♥ ✜❡❧❞ ✈❛r✐❛❜❧❡s✳ s♦❧✈❡❞ ❜② ❛ s♦❧✉t✐♦♥ ♠❡t❤♦❞♦❧♦❣②✳ ❚❤❡ s②st❡♠ ♦❢ ▲❆❊s ❛r❡ ❚❤❡ P❉❊ ❜❛s❡❞ ❋❱▼ ❛♥❞ t❤❡ ♠❛t❤❡♠❛t✐❝❛❧ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ ▲❆❊s ❜❛s❡❞ s♦❧✉t✐♦♥ ♠❡t❤♦❞♦❧♦❣②✱ ✉s❡❞ ✐♥ ❛❧♠♦st ❛❧❧ t❤❡ ❜♦♦❦s ✐♥ ❈❋❉✱ ✐s ❝❛❧❧❡❞ ❤❡r❡ ❛s ❛ ♠❛t❤❡♠❛t✐❝❛❧ ❛♣♣r♦❛❝❤ ♦❢ ❈❋❉ ❞❡✈❡❧♦♣♠❡♥t❀ s❤♦✇♥ ✐♥ ❋✐❣✳ ✶✳✹✳ ❚❤❡ ♥♦✈❡❧t② ✐♥ t❤✐s ❜♦♦❦ ✐s ❛ ♣❤②s✐❝❛❧ ❛♣♣r♦❛❝❤ ♦❢ ❈❋❉ ❞❡✈❡❧✲ ♦♣♠❡♥t ✱ ♣r♦♣♦s❡❞ ❤❡r❡ ❛♥❞ ♣r♦❜❛❜❧② ♥♦t ❢♦✉♥❞ ✐♥ ❛♥② ♦❢ t❤❡ ❜♦♦❦s ♦♥ ❈❋❉✳ ❚❤❡ ❛❜♦✈❡ P❉❊ ❜❛s❡❞ ❋❱▼ ❧❡❛❞s t♦ ❛ q✉❡st✐♦♥✿ ❲❤② ❞♦ ✇❡ s✇✐t❝❤ ❢r♦♠ t❤❡ ❛❧❣❡❜r❛✐❝ ❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ♣❤②s✐❝❛❧ ❧❛✇s t♦ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠✉❧❛t✐♦♥ t♦ t❤❡♥ ❣♦ ❜❛❝❦ t♦ t❤❡ ❛❧❣❡❜r❛✐❝ ❢♦r♠✉❧❛t✐♦♥❄ ❲❤② ❞♦♥✬t ✇❡ ❦❡❡♣ t❤❡ ❛❧❣❡❜r❛✐❝ ❢♦r♠✉❧❛t✐♦♥✱ ❛♥❞ ❝♦♥✈❡rt t❤❡ ❝♦♥t✐♥✲ ✉♦✉s ✈❛r✐❛❜❧❡s ✐♥ t❤❡ ♣❤②s✐❝❛❧ ❧❛✇s t♦ ❛♣♣r♦♣r✐❛t❡ ❞✐s❝r❡t❡ ✈❛r✐❛❜❧❡s ✉s✐♥❣ ❝❡rt❛✐♥ ❛♣♣r♦①✐♠❛t✐♦♥s❄ ❚❤✐s ❧❡❞ t♦ ♣❤②s✐❝❛❧ ❧❛✇s ❜❛s❡❞ ✭P❉❊ ❢r❡❡✮ ❋❱▼✱ s❤♦✇♥ ✐♥ ❋✐❣✳ ✶✳✹✳ ❚❤❡ ✜❣✉r❡ ❛❧s♦ s❤♦✇s t❤❛t t❤❡ ♣❤②s✐✲ ❝❛❧ ❛♣♣r♦❛❝❤ ♦❢ ❈❋❉ ❞❡✈❡❧♦♣♠❡♥t ❝♦♥s✐st ♦❢ ❛ ✢✉① ✭♣❤②s✐❝❛❧ q✉❛♥t✐t②✮ ❜❛s❡❞ s♦❧✉t✐♦♥ ♠❡t❤♦❞♦❧♦❣②✳ ❚❤❡ ♣r❡s❡♥t ❜♦♦❦ ✐s ♦♥ ♣❤②s✐❝❛❧ ❧❛✇ ❜❛s❡❞ ❋❱▼✱ ❛♥❞ ♣r❡s❡♥ts ❜♦t❤ t❤❡ ✢✉① ❛♥❞ ❝♦❡✣❝✐❡♥t ♦❢ ▲❆❊s ❜❛s❡❞ s♦✲ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈❋❉ ✶✼ ❧✉t✐♦♥ ♠❡t❤♦❞♦❧♦❣②✳ ❋✉rt❤❡r ❞❡t❛✐❧s ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ♠❡t❤♦❞♦❧♦❣✐❡s ✇✐❧❧ ❜❡ ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ♥❡①t ❝❤❛♣t❡r✳ ❲✐t❤ ❛ ♠♦t✐✈❛t✐♦♥ t♦ ✐♥❝r❡❛s❡ ❛s ✇❡❧❧ ❛s ❡♥r✐❝❤ t❤❡ ❧❡❛r♥✐♥❣ ♦❢ t❤✐s s✉❜❥❡❝t ✇❤✐❝❤ s❤♦✉❧❞ ❧❡❛❞ t♦ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❛ ❝✉st♦♠✐③❡❞ ❈❋❉ s♦❢t✇❛r❡ ❛s ❛ ♣r♦❞✉❝t − − ❜❡ ❛ ❝r❡❛t♦r t❤❛♥ ❛ ♣✐❡❝❡ ♦❢ ❝r❡❛t✐♦♥ t❤❡ ♣❤②s✐❝❛❧ ❛♣♣r♦❛❝❤ ❡✈♦❧✈❡❞ ✐♥❞❡♣❡♥❞❡♥t❧② ♦✈❡r ♠② t❡❛❝❤✐♥❣ ❛♥❞ 18 12 ②❡❛rs ♦❢ ②❡❛rs ♦❢ r❡s❡❛r❝❤ ❡①♣❡r✐❡♥❝❡ ✐♥ ❈❋❉✳ ❍♦✇❡✈❡r✱ ♠✉❝❤ ❧❛t❡r✱ t❤❡ ❣❡♥❡r❛❧ ✐❞❡❛ ♦❢ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠✉❧❛t✐♦♥ ❢r❡❡ ❞✐s❝r❡t❡ ❛❧❣❡❜r❛✐❝ ❢♦r♠✉❧❛t✐♦♥ ✇❛s ❢♦✉♥❞ ✐♥ s♦♠❡ r❡s❡❛r❝❤ ♣❛♣❡rs ✭r❡❝❡♥t❧② ❜② ❚♦♥t✐✱ ✷✵✶✹✮✳ ■♥ t❤❡ ✐♥s✐❣❤t❢✉❧ ♣❛♣❡r✱ ❚♦♥t✐ ✭✷✵✶✹✮ ❤❛s st❛t❡❞✿ ✏ ❚❤❡ ✏ ❡①❛❝t✑ s♦❧✉t✐♦♥✱ t❤❛t ✐s ♣r♦♠✐s❡❞ ❜② t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠✉✲ ❧❛t✐♦♥✱ ✐s ❤❛r❞❧② ❡✈❡r ❛tt❛✐♥❡❞ ✐♥ ♣r❛❝t✐❝❡✳ ❇② ❝♦♥tr❛st✱ t❤❡ ❣r❡❛t t❡❝❤✲ ♥♦❧♦❣✐❝❛❧ ♣r♦❣r❡ss ♦❢ ♦✉r ❞❛②s ✐s ♠❛❞❡ ♣♦ss✐❜❧❡ ❜② t❤❡ ❢❛❝t ♦❢ ❜❡✐♥❣ ❛❜❧❡ t♦ ❤❛✈❡ ❛♥ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥✱ ✐♥ ♣❛rt✐❝✉❧❛r ❛ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ t♦ t❤❡ ♣r♦❜❧❡♠s ♣♦s❡❞ ❜② t❤❡ t❡❝❤♥✐q✉❡✳ ❋♦r ♦✉r ❝✉❧t✉r❡✱ ❢♦r♠❡❞ ♦♥ t❤❡ ♠♦❞❡❧ ♦❢ ✐♥✜♥✐t❡s✐♠❛❧ ❛♥❛❧②s✐s✱ t❤❡ t❡r♠ ✏ ❛♣♣r♦①✐♠❛t❡✑ s♦✉♥❞s ❧✐❦❡ ✏ ✐♠♣❡r❢❡❝t✳✑ ❍♦✇❡✈❡r✱ ✇❡ ♠✉st ♥♦t ❢♦r❣❡t t❤❛t t❤❡ ♦❜❥❡❝t✐✈❡ ♦❢ ❛ ♥✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥ ♦❢ ♣❤②s✐❝❛❧ ♣r♦❝❡ss❡s ✐s t❤❡ ❛❣r❡❡♠❡♥t ✇✐t❤ t❤❡ ❡①♣❡r✐♠❡♥t❛❧ ♠❡❛s✉r❡♠❡♥ts ❛♥❞ ♥♦t t❤❡ ❝♦♥✈❡r❣❡♥❝❡ t♦ ❛♥ ❛♥❛❧②t✐❝❛❧ s♦❧✉t✐♦♥✱ ✉s✉❛❧❧② ♥♦t ❛tt❛✐♥❛❜❧❡✳ ■♥ ❛❞❞✐t✐♦♥✱ t❤❡ r❡q✉❡st ♦❢ r❡❞✉❝✐♥❣ t❤❡ ❡rr♦r ♦❢ ❛♥ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥ ❞♦❡s ♥♦t ♠❡❛♥ ♠❛❦❡ t❤❡ ❡rr♦r ✏ ❛r❜✐tr❛r✐❧② s♠❛❧❧✑ ✱ ❛s r❡q✉✐r❡❞ ❜② t❤❡ ♣r♦❝❡ss ♦❢ ❧✐♠✐t✱ ❜✉t ♠❛❦✐♥❣ t❤❡ ❡rr♦r s♠❛❧❧❡r t❤❛♥ ❛ ♣r❡❛ss✐❣♥❡❞ t♦❧❡r❛♥❝❡✳ ❚❤❡ ♥♦t✐♦♥ ♦❢ ♣r❡❝✐s✐♦♥ ♦❢ ❛ ♠❡❛s✉r✐♥❣ ❞❡✈✐❝❡ ♣❧❛②s t❤❡ s❛♠❡ r♦❧❡ ❛s t❤❡ ♥♦t✐♦♥ ♦❢ t♦❧❡r❛♥❝❡ ✐♥ ♠❛♥✉❢❛❝t✉r✐♥❣ ❛♥❞ t❤❡ ♥♦t✐♦♥ ♦❢ ❡rr♦r ✐♥ ♥✉♠❡r✐❝❛❧ ❛♥❛❧②s✐s✳✑ ❙✐♥❝❡ t❤❡ ❛❞✈❡♥t ♦❢ ❝♦♠♣✉t❡r✱ t❤❡r❡ ✐s ❛♥ ❡♥♦r♠♦✉s ❞❡✈❡❧♦♣♠❡♥t ✐♥ t❤❡ ❞✐s❝r❡t❡ ♠❛t❤❡♠❛t✐❝s✳ ❍♦✇❡✈❡r✱ ✐❢ ✇❡ ❝❤❛♥❣❡ ♦✉r ❤❛❜✐ts ♦❢ ❝♦♥s✐❞❡r✐♥❣ t❤❡ ❞✐✛❡r❡♥t✐❛❧✴❝♦♥t✐♥✉♦✉s ❢♦r♠✉❧❛t✐♦♥ ❛s ✐♥❞✐s♣❡♥s❛❜❧❡ ❛♥❞ ✇♦r❦ ♦♥ t❤❡ ❞✐s❝r❡t❡ ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ❝♦♥t✐♥✉♦✉s ♠❛t❤❡♠❛t✐❝s✱ ✐t ✇✐❧❧ ❧❡❛❞ t♦ ❛♥ ✐♥❝r❡❛s❡ ❛s ✇❡❧❧ ❛s ❡♥r✐❝❤♠❡♥t ✐♥ t❤❡ r❛♥❣❡ ♦❢ ❛♣♣❧✐✲ ❝❛t✐♦♥s ✭❉❡s❜r✉♥ ❡t ❛❧✳✱ ✷✵✵✺✮✳ ❚❤✐s ✇✐❧❧ ❛❧s♦ ❧❡❛❞ t♦ ❛ ♠✉❝❤ ❜❡tt❡r ✉♥❞❡rst❛♥❞✐♥❣ ❛s ✇❡❧❧ ❛s ❞❡✈❡❧♦♣♠❡♥ts ♦❢ t❤❡ s✐♠✉❧❛t✐♦♥ ♠❡t❤♦❞♦❧♦❣② ✐♥ t❤❡ ✈❛r✐♦✉s ❜r❛♥❝❤❡s ♦❢ s❝✐❡♥❝❡ ❛♥❞ ❡♥❣✐♥❡❡r✐♥❣✳ ❚❤❡r❡ ❛r❡ s❡✈❡r❛❧ ❝❧❛ss✐❝s ❛♥❞ ❣❡♥❡r❛❧ ♣✉r♣♦s❡ ❜♦♦❦s ♦♥ ❈❋❉ s✉❝❤ ❛s t❤❡ ❜♦♦❦ ❜② ❘♦❛❝❤❡ ✭✶✾✼✻✮✱ P❡②r❡t ❛♥❞ ❚❛②❧♦r ✭✶✾✽✸✮✱ ❏❛❧✉r✐❛ ❛♥❞ ❚♦rr❛♥❝❡ ✭✶✾✽✻✮✱ ❋❧❡t❝❤❡r ✭✶✾✽✽✮✱ ❍✐r❝❤ ✭✶✾✽✾✱ ✶✾✾✵✮✱ ❛♥❞ ❍♦✛♠❛♥♥ ❛♥❞ ❈❤✐❛♥❣ ✭✷✵✵✵✮✳ ❆♠♦♥❣ t❤❡ s❡✈❡r❛❧ ✈❡r② ❣♦♦❞ ❜♦♦❦s ♦♥ ❈❋❉✱ ✶✽ ✶✳ ■♥tr♦❞✉❝t✐♦♥ ✇♦✉❧❞ ❧✐❦❡ t♦ ♠❡♥t✐♦♥ s❡✈❡♥ ♦❢ t❤❡♠✿ ✜rst✱ ❜② P❛t❛♥❦❛r ✭✶✾✽✵✮✱ ❛ ❝❧❛ss✐❝❛❧ ❛♥❞ ♣r♦❜❛❜❧② t❤❡ ♠♦st r❡❢❡rr❡❞ ❜♦♦❦❀ s❡❝♦♥❞✱ ❜② ❆♥❞❡rs♦♥✱ ❚❛♥♥❡❤✐❧❧ ❛♥❞ P❧❡t❝❤❡r ✭✶✾✽✹✮✱ ❛♥♦t❤❡r ❝❧❛ss✐❝❛❧ ❛♥❞ ♣r♦❜❛❜❧② t❤❡ ✜rst ❝♦♠♣r❡❤❡♥s✐✈❡ ❜♦♦❦❀ t❤✐r❞✱ ❜② ❆♥❞❡rs♦♥ ✭✶✾✾✺✮✱ t❤❡ ❜♦♦❦ ✇✐t❤ ♣r♦❜❛❜❧② t❤❡ ❜❡st ✇r✐t✐♥❣ st②❧❡❀ ❢♦✉rt❤✱ ❜② ❱❡rst❡❡❣ ❛♥❞ ▼❛❧❛❧❛s❡❦❡r❛ ✭✶✾✾✺✮✱ ❛♥ ❡①❝❡❧❧❡♥t ❜♦♦❦s ❢♦r ❜❡❣✐♥♥❡rs❀ ✜❢t❤✱ ❜② ❋❡r③✐❣❡r ❛♥❞ P❡r✐❝ ✭✷✵✵✷✮✱ ♣r♦❜❛❜❧② t❤❡ ♠♦st ❝♦♠♣r❡❤❡♥s✐✈❡ ❜♦♦❦❀ s✐①t❤✱ ❡❞✐t❡❞ ❜② ▼✉✲ r❛❧✐❞❤❛r ❛♥❞ ❙✉♥❞❛r❛r❛❥❛♥ ✭✷✵✵✸✮✱ ❛♥♦t❤❡r ❝♦♠♣r❡❤❡♥s✐✈❡ ❜♦♦❦ ✇✐t❤ s✉✣❝✐❡♥t ❞❡t❛✐❧s ❢♦r ❛ ❜❡❣✐♥♥❡r❀ ❛♥❞ s❡✈❡♥t❤✱ ❜② ❉❛t❡ ✭✷✵✵✺✮✱ ❛♥ ♦✉tst❛♥❞✐♥❣ ❜♦♦❦ ❛♥❞ ♣r♦❜❛❜❧② ✇✐t❤ t❤❡ ❧❛r❣❡st ♥✉♠❜❡r ♦❢ ❡①❡r❝✐s❡ ♣r♦❜❧❡♠s✳ ❚❤❡ ❛✉t❤♦r ♦❢ t❤✐s ❜♦♦❦ ✐s ❤✐❣❤❧② ✐♥s♣✐r❡❞ ❜② t❤❡ ✜rst✱ t❤✐r❞ ❛♥❞ ✜❢t❤ ❜♦♦❦❀ ❛♥❞ ❤❛s ❝♦♥tr✐❜✉t❡❞ ❛ ❝❤❛♣t❡r ♦♥ t❤❡ ❋❱▼ ✭❙❤❛r♠❛ ❛♥❞ ❊s✇❛r❛♥✱ ✷✵✵✸✮ ✐♥ ❛ r❡✈✐s❡❞ ❡❞✐t✐♦♥ ♦❢ t❤❡ s✐①t❤ ❜♦♦❦✱ ✉s✐♥❣ t❤❡ ♠❛t❤❡♠❛t✐❝s ❜❛s❡❞ ❛♣♣r♦❛❝❤ ♦❢ ❈❋❉ ❞❡✈❡❧♦♣♠❡♥t ✭❋✐❣✳ ✶✳✹✮✳ ❚❤❡ s❝♦♣❡ ♦❢ t❤✐s ❜♦♦❦ ✐s ♠❛✐♥❧② ♦♥ ❈❋❉ ❞❡✈❡❧♦♣♠❡♥t t❤❛♥ ♦♥ ❈❋❉ ❛♣♣❧✐❝❛t✐♦♥s ❛♥❞ ❛♥❛❧②s✐s❀ ❤♦✇❡✈❡r✱ t❤❡ ❛♣♣❧✐❝❛t✐♦♥s ❛♥❞ ❛♥❛❧✲ ②s✐s ❛r❡ ♣r❡s❡♥t❡❞ ❝♦♥❝✐s❡❧② ✐♥ t❤❡ ❡①❛♠♣❧❡ ❛♥❞ ❡①❡r❝✐s❡ ♣r♦❜❧❡♠s✳ ❚❤✐s ✐s ♠♦t✐✈❛t❡❞ ❜② t❤❡ ❢❛❝t t❤❛t ✐t ✐s ♠♦r❡ ❝❤❛❧❧❡♥❣✐♥❣ ❛♥❞ s❛t✐s❢②✲ ✐♥❣ t♦ ❧❡❛r♥ t♦ ❞❡✈❡❧♦♣ t❤❛♥ t♦ ✉s❡ ❛ ♣r♦❞✉❝t✳ ❙✐♥❝❡ t❤❡ ♣r❡s❡♥t ❜♦♦❦ ✐s ❢♦r ❛♥ ✐♥tr♦❞✉❝t♦r② ❝♦✉rs❡ ♦♥ ❈❋❉✱ t❤❡ ❈❋❉ ❞❡✈❡❧♦♣♠❡♥t ✐s ❧✐♠✲ ✐t❡❞ t♦ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ❢♦r ❛ ❈❛rt❡s✐❛♥ ❛s ✇❡❧❧ ❛s ❝♦♠♣❧❡① ❣❡♦♠❡tr②❀ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❣❡♦♠❡tr✐❡s ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ♥❡①t ❝❤❛♣t❡r✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ ❜♦♦❦ ✐s ♠♦st❧② ♦♥ ♥✉♠❡r✐❝❛❧✲♠❡t❤♦❞ ❛s ❝♦♠♣❛r❡❞ t♦ ♥✉♠❡r✐❝❛❧ ❛♥❛❧②s✐s ❛s♣❡❝t ♦❢ ❈❋❉✳ ❚❤❡ ♣r❡s❡♥t ❜♦♦❦ ✐s ♦♥ ✐♥❝♦♠♣r❡ss✐❜❧❡✲✢♦✇✱ ❝♦♠♠♦♥❧② ❡♥❝♦✉♥t❡r❡❞ ✐♥ t❤❡ ❈❤❡♠✐❝❛❧ ❛♥❞ ▼❡❝❤❛♥✐❝❛❧ ❊♥❣✐♥❡❡r✐♥❣❀ ❤♦✇❡✈❡r✱ ✐t ❝❛♥ ❜❡ ❡①✲ t❡♥❞❡❞ t♦ t❤❡ ❝♦♠♣r❡ss✐❜❧❡ ✢♦✇ ❢♦r t❤❡ ❆❡r♦s♣❛❝❡ ❊♥❣✐♥❡❡r✐♥❣✳ ❋✉r✲ t❤❡r♠♦r❡✱ t❤❡ ♣r❡s❡♥t ❜♦♦❦ ❞❡❛❧s ✇✐t❤ ◆❡✇t♦♥✐❛♥✲✢✉✐❞s✱ ❝♦♠♠♦♥❧② ❡♥❝♦✉♥t❡r❡❞ ✐♥ ▼❡❝❤❛♥✐❝❛❧ ❛♥❞ ❆❡r♦s♣❛❝❡ ❡♥❣✐♥❡❡r✐♥❣❀ ❤♦✇❡✈❡r✱ ✐t ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ♥♦♥✲◆❡✇t♦♥✐❛♥ ✢✉✐❞s ❢♦r t❤❡ ❈❤❡♠✐❝❛❧ ❊♥❣✐♥❡❡r✲ ✐♥❣✳ ❋❱▼ ✐s t❤❡ ❞✐s❝r❡t✐③❛t✐♦♥ ♠❡t❤♦❞ ✇❤✐❝❤ ✐s ✉s❡❞ ✐♥ t❤❡ ♠♦st ♦❢ t❤❡ ♣r❡s❡♥t ❞❛② ❈❋❉ s♦❢t✇❛r❡❀ t❤✉s✱ t❤❡ ♣r❡s❡♥t ❜♦♦❦ ✐s ♠❛✐♥❧② ♦♥ ❋❱▼✳ ❍♦✇❡✈❡r✱ ❋✐♥✐t❡ ❉✐✛❡r❡♥❝❡ ▼❡t❤♦❞ ✭❋❉▼✮ ✐s ❛❧s♦ ♣r❡s❡♥t❡❞ ❤❡r❡ ❛s ❛ t♦♣✐❝ ✐♥ t❤❡ ❝❤❛♣t❡r ♦♥ ❡ss❡♥t✐❛❧s ♦❢ ♥✉♠❡r✐❝❛❧ ♠❡t❤♦❞s✳ ❍❡r❡✱ t❤❡ ♥✉♠❡r✐❝❛❧ ♠❡t❤♦❞♦❧♦❣② ✐s ♣r❡s❡♥t❡❞ ♠♦st❧② ❢♦r ✷❉ ♣r♦❜✲ ❧❡♠s ✭✇❤✐❝❤ ❝❛♥ ❜❡ ❡❛s✐❧② ❡①t❡♥❞❡❞ t♦ ✸❉✮ ❛♥❞ ❢♦r ✉♥st❡❛❞② ✢♦✇ ❛♥❞ ❤❡❛t✲tr❛♥s❢❡r✳ ❚❤❡ ♣r❡s❡♥t s✉❜❥❡❝t ✐♥✈♦❧✈❡s ❧❡❛r♥✐♥❣ t♦ ❞❡✈❡❧♦♣ ❛♥❞ ♣r♦♣❡r❧② ✉s❡ ❛ ♣r♦❞✉❝t − ❛ ❈❋❉ s♦❢t✇❛r❡✳ ❚❤❡ ❞❡t❛✐❧s ❢♦r t❤❡ ❈❋❉ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈❋❉ ✶✾ ❋✐❣✉r❡ ✶✳✺✿ ▼✐♥❞ ♠❛♣ ❢♦r t❤❡ ♣r❡s❡♥t ❜♦♦❦ ❞❡✈❡❧♦♣♠❡♥t ♣r❡s❡♥t❡❞ ❤❡r❡ ✐s t❤❡ ♠❛✐♥ str✉❝t✉r❡ ♦❢ t❤❡ ❈❋❉ s♦❢t✲ ✇❛r❡❀ ❤♦✇❡✈❡r✱ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ♠❡t❤♦❞♦❧♦❣② ❢♦r ❝❡rt❛✐♥ ✜♥✐s❤✐♥❣ t♦✉❝❤❡s ✐♥ t❤❡ s♦❢t✇❛r❡ ✭s✉❝❤ ❛s ❣r❛♣❤✐❝❛❧ ✉s❡r ✐♥t❡r❢❛❝❡ ❞❡✈❡❧♦♣♠❡♥t ❢♦r ✉s❡r ✐♥♣✉t ❛♥❞ ♦✉t♣✉t✮ ❛r❡ ♥♦t ✐♥❝❧✉❞❡❞ ❤❡r❡✳ ❚❤✉s✱ ❛ r❡❛❞❡r s❤♦✉❧❞ ❜❡ ❛❜❧❡ t♦ ❞❡✈❡❧♦♣ ❈❋❉ ❝♦❞❡s ✇❤✐❝❤ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ❛ ❈❋❉ s♦❢t✇❛r❡✳ ❆ ♠✐♥❞✲♠❛♣ ❢♦r t❤❡ ♠❛t❡r✐❛❧ ❝♦✈❡r❡❞ ✐♥ t❤✐s ❜♦♦❦ ✐s ❣✐✈❡♥ ✐♥ ❋✐❣✳ ✶✳✺❀ ✐♥s♣✐r❡❞ ❜② ❆♥❞❡rs♦♥ ✭✶✾✾✺✮✳ ❚❤❡ ♦❜❥❡❝t✐✈❡ ♦❢ t❤✐s ♠✐♥❞ ♠❛♣ ✐s t♦ s❤♦✇ ②♦✉ ❛ ❜✐❣❣❡r ♣✐❝t✉r❡ ♦♥ ❤♦✇ t❤❡ ♠❛t❡r✐❛❧ ✢♦✇s ✐♥ ❛ ❧♦❣✐❝❛❧ ❢❛s❤✐♦♥✱ ❜❡❢♦r❡ ❣♦✐♥❣ ♦♥ t♦ t❤❡ ❞❡t❛✐❧s ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❤❛♣t❡rs✳ ❋✉rt❤❡r♠♦r❡✱ ❧♦❝❛❧✐③❡❞ ♠✐♥❞ ♠❛♣ ✐s ✐♥❝❧✉❞❡❞ ✐♥ ❛❧❧ t❤❡ ❝❤❛♣t❡rs ❛❢t❡r t❤❡ ✜rst t✇♦ ✐♥tr♦❞✉❝t♦r② ❝❤❛♣t❡rs − t♦ ❞❡♠♦♥str❛t❡ t❤❡ ✢♦✇ ♦❢ ✐❞❡❛s ✐♥ ❡❛❝❤ ❝❤❛♣t❡r✳ ❋✐❣✉r❡ ✶✳✺ s❤♦✇s t❤❛t t❤❡ ❜♦♦❦ st❛rts ✇✐t❤ P❛rt ■ ♦♥ ■♥tr♦❞✉❝t✐♦♥ ❛♥❞ ❊ss❡♥t✐❛❧s✱ ❞✐s❝✉ss❡❞ ✐♥ t❤❡ ♣r❡s❡♥t ❛♥❞ t❤❡ ♥❡①t t❤r❡❡ ❝❤❛♣t❡rs✳ ❚❤❡ ❣❡♥❡r❛❧ ✐♥tr♦❞✉❝t✐♦♥ ♦❢ ❈❋❉ ✐s ♣r❡s❡♥t❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r✱ ❛♥❞ t❤❡ ♥❡①t ❝❤❛♣t❡r ♣r❡s❡♥ts ❛♥ ♦✈❡r✈✐❡✇ ♦❢ ❈❋❉ ❞❡✈❡❧♦♣♠❡♥t✱ ❛♣♣❧✐❝❛t✐♦♥✱ ❛♥❞ ❛♥❛❧②s✐s✳ ❚❤❡r❡❛❢t❡r✱ t❤❡ t✇♦ ❝❤❛♣t❡rs ♣r❡s❡♥ts t❤❡ ❛❜s♦❧✉t❡❧② ❡ss❡♥t✐❛❧s ♦❢ t✇♦ ♣r❡r❡q✉✐s✐t❡ s✉❜❥❡❝ts ✭❢♦r ❛ ❝♦✉rs❡ ♦♥ ❈❋❉✮✿ ✢✉✐❞✲❞②♥❛♠✐❝s ❛♥❞ ❤❡❛t✲tr❛♥s❢❡r ✐♥ ❈❤❛♣t❡r ✸✱ ❛♥❞ ✷✵ ✶✳ ■♥tr♦❞✉❝t✐♦♥ ♥✉♠❡r✐❝❛❧✲♠❡t❤♦❞s ✐♥ ❈❤❛♣t❡r ✹✳ ❆❢t❡r ❜✉✐❧❞✐♥❣ t❤❡ ❢♦✉♥❞❛t✐♦♥ ♦❢ t❤❡ ❝♦✉rs❡ ✐♥ P❛rt ■✱ ❋✐❣✳ ✶✳✺ s❤♦✇s t❤❛t t❤❡ ❜♦♦❦ st❛rts ✇✐t❤ t❤❡ P❛rt ■■ ♦♥ ❈❋❉ ❢♦r ❛ ❈❛rt❡s✐❛♥ ●❡♦♠❡tr②✱ ❛♥❞ ❡♥❞s ✇✐t❤ P❛rt ■■■ ♦♥ ❈❋❉ ❢♦r ❛ ❈♦♠♣❧❡① ●❡♦♠❡tr②✳ ❋♦r t❤❡ P❛rt ■■✱ ✐t ✐s ❢❡❧t ❜❡tt❡r t♦ ✐♥tr♦❞✉❝❡ t❤❡ ❈❋❉ ❞❡✈❡❧♦♣♠❡♥t ✇✐t❤ t❤❡ ✢✉① ❜❛s❡❞ s♦❧✉t✐♦♥✲♠❡t❤♦❞♦❧♦❣② ✭❢♦r ❛ ❜❡tt❡r ❛♣♣r❡❝✐❛t✐♦♥ ♦❢ t❤❡ ♣❤②s✐❝❛❧ ❧❛✇ ❜❛s❡❞ ❋❱▼✮✱ ❢♦❧❧♦✇❡❞ ❜② t❤❡ ❝♦❡✣❝✐❡♥t ♦❢ ▲❆❊s ❜❛s❡❞ ♠❡t❤♦❞♦❧♦❣② ✭❋✐❣✳ ✶✳✹✮✳ ❚❤❡ P❛rt ■■ ♣r❡s❡♥ts ❈❋❉ ❞❡✈❡❧♦♣♠❡♥t ❢♦r ❡①♣❧✐❝✐t ❛s ✇❡❧❧ ❛s ✐♠♣❧✐❝✐t ♠❡t❤♦❞✱ ✉♥✐❢♦r♠ ❛s ✇❡❧❧ ❛s ♥♦♥✲✉♥✐❢♦r♠ ❣r✐❞✱ ❛♥❞ st❛❣❣❡r❡❞ ❛s ✇❡❧❧ ❛s ❝♦✲❧♦❝❛t❡❞ ❣r✐❞✳ ❆♥② ❞❡✈❡❧♦♣♠❡♥t ❛❝t✐✈✐t② ✐♥✈♦❧✈❡s ❧❡❛r♥✐♥❣ t♦ ❞❡✈❡❧♦♣ ❛s ✇❡❧❧ ❛s t❡st t❤❡ ❞✐✛❡r❡♥t ❝♦♠♣♦♥❡♥ts✱ ❜❡❢♦r❡ t❤❡✐r ❛ss❡♠❜❧② t♦ ♦❜t❛✐♥ t❤❡ ✜♥❛❧ ♣r♦❞✉❝t✳ ❆s t❤❡ ♣r♦❞✉❝t ❡♥✈✐s❛❣❡❞ ❤❡r❡ ✐s ❛ ❈❋❉ s♦❢t✇❛r❡✱ t❤✐s ❜♦♦❦ ✐s str✉❝t✉r❡❞ ✇✐t❤ ❛ ♠♦❞✉❧❡✲❜②✲♠♦❞✉❧❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ t❤❡ ♥✉♠❡r✐✲ ❝❛❧ ❢♦r♠✉❧❛t✐♦♥✳ ■t ♣r❡s❡♥ts t❤❡ ♠❡t❤♦❞♦❧♦❣② ♦❢ ❞❡✈❡❧♦♣✐♥❣ ❛s ✇❡❧❧ ❛s t❡st✐♥❣ t❤❡ ❞✐✛❡r❡♥t ❝♦♠♣♦♥❡♥ts✴♠♦❞✉❧❡s✴s♦❧✈❡rs ♦❢ t❤❡ s♦❢t✇❛r❡✿ ❞✐❢✲ ❢✉s✐♦♥ ❛♥❞ ❛❞✈❡❝t✐♦♥✱ ❛♥❞ t❤❡✐r ❝♦♠❜✐♥❛t✐♦♥ ❛s ❝♦♥✈❡❝t✐♦♥❀ ♣r❡s❡♥t❡❞ ❛s s❡♣❛r❛t❡ ❝❤❛♣t❡rs ♦❢ P❛rt ■■✳ ❚❤✉s✱ ❋✐❣✳ ✶✳✺ s❤♦✇s t❤❛t P❛rt ■■ ♦❢ t❤✐s ❜♦♦❦ ❝♦♥s✐sts ♦❢ ❝❤❛♣t❡rs ✇✐t❤ t❤❡ t✐t❧❡ ❛s ❝♦♠♣✉t❛t✐♦♥❛❧ ❤❡❛t ❝♦♥❞✉❝t✐♦♥ ✭❢♦r ❞✐✛✉s✐♦♥✮✱ ❝♦♠♣✉t❛t✐♦♥❛❧ ❤❡❛t ❛❞✈❡❝t✐♦♥✱ ❝♦♠♣✉t❛✲ t✐♦♥❛❧ ❤❡❛t ❝♦♥✈❡❝t✐♦♥✱ ❛♥❞ ❝♦♠♣✉t❛t✐♦♥❛❧ ✢✉✐❞✲❞②♥❛♠✐❝s✳ ❚❤❡ ❝♦♠✲ ♣✉t❛t✐♦♥❛❧ ✢✉✐❞✲❞②♥❛♠✐❝s ❝♦rr❡s♣♦♥❞s t♦ ◆❛✈✐❡r✲❙t♦❦❡s s♦❧✈❡r ❛s t❤❡ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ ❞✐✛✉s✐♦♥ ❛♥❞ ❛❞✈❡❝t✐♦♥ s♦❧✈❡r✱ ❛❧♦♥❣ ✇✐t❤ ❛ s♦✉r❝❡ t❡r♠✳ ❚❤❡ ♣❤②s✐❝❛❧ ❡①❛♠♣❧❡ ♦❢ t❤❡ ✐♥❞✐✈✐❞✉❛❧ s♦❧✈❡r ✐s ♠♦r❡ ❝♦♠✲ ♠♦♥❧② ❡♥❝♦✉♥t❡r❡❞ ✐♥ ❤❡❛t✲tr❛♥s❢❡r ❜✉t ♥♦t ✐♥ ✢✉✐❞✲❞②♥❛♠✐❝s✳ ❚❤✉s✱ t❤❡ ❞✐✛✉s✐♦♥✱ ❛❞✈❡❝t✐♦♥ ❛♥❞ ❝♦♥✈❡❝t✐♦♥ s♦❧✈❡r ✭♣r❡s❡♥t❡❞ ✐♥ ❈❤❛♣t❡r ✺✲✼ ♦❢ P❛rt ■■✮ ❛r❡ ❝♦♥s✐❞❡r❡❞ ❤❡r❡ ❢♦r ❤❡❛t✲tr❛♥s❢❡r❀ ❛❧t❤♦✉❣❤ t❤❡② ❛r❡ ❛❧s♦ ❡♥❝♦✉♥t❡r❡❞ ✭✐♥ ❝♦♠❜✐♥❛t✐♦♥✮ ✐♥ ✢✉✐❞✲❞②♥❛♠✐❝s✳ P❛rt ■■ ✐s ♦♥ ✉♥✐❢♦r♠ ❛s ✇❡❧❧ ❛s ♥♦♥✲✉♥✐❢♦r♠ ❈❛rt❡s✐❛♥ ❣r✐❞✱ ❛♥❞ P❛rt ■■■ ✐s ♦♥ t❤❡ ❝✉r✈✐❧✐♥❡❛r ❣r✐❞❀ ✇✐t❤ t❤❡ ❛♣♣r♦♣r✐❛t❡ ❣r✐❞ ❣❡♥❡r❛t✐♦♥ ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ✜rst ❝❤❛♣t❡r ♦❢ P❛rt ■■ ❛♥❞ ■■■✳ P❛rt ■■■ ♦♥ ❈❋❉ ❢♦r ❝♦♠♣❧❡① ❣❡♦♠❡tr② st❛rts ✇✐t❤ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❤❡❛t ❝♦♥❞✉❝t✐♦♥ ❛♥❞ ❡♥❞s ✇✐t❤ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ✢✉✐❞✲❞②♥❛♠✐❝s✳ ❋♦r ❈❋❉ ❞❡✈❡❧♦♣♠❡♥t✱ t❤❡r❡ ❛r❡ ✜✈❡ ♠❛✐♥ st❡♣s✿ ✜rst✱ ❣r✐❞✲ ❣❡♥❡r❛t✐♦♥❀ s❡❝♦♥❞✱ ✜♥✐t❡ ✈♦❧✉♠❡ ♠❡t❤♦❞❀ t❤✐r❞✱ s♦❧✉t✐♦♥ ♠❡t❤♦❞✲ ♦❧♦❣②❀ ❢♦✉rt❤✱ ❝♦❞❡ ❞❡✈❡❧♦♣♠❡♥t❀ ❛♥❞ ✜❢t❤✱ t❡st✐♥❣✳ ❚❤❡ ✜rst ❝❤❛♣t❡r ♦❢ P❛rt ■■ ❛s ✇❡❧❧ ❛s P❛rt ■■■ ♦❢ t❤✐s ❜♦♦❦ ♣r❡s❡♥ts t❤❡ ✜rst st❡♣✳ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈❋❉ ✷✶ ❚❤❡ r❡♠❛✐♥✐♥❣ st❡♣s ❛r❡ ❞✐s❝✉ss❡❞ ✐♥ t❤❡ s❡♣❛r❛t❡ s❡❝t✐♦♥s ❛♥❞ t❤❡ ❡①❛♠♣❧❡ ♣r♦❜❧❡♠s ♦❢ t❤❡ ✈❛r✐♦✉s ❝❤❛♣t❡rs ♦❢ P❛rt ■■ ❛♥❞ ■■■✳ ❚❤❡ ❝♦♠♣✉t❡r✲♣r♦❣r❛♠s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❢♦✉rt❤ st❡♣ ✐s ♣r❡s❡♥t❡❞ ❤❡r❡ ❢♦r t❤❡ ❜❛s✐❝ ♠♦❞✉❧❡s ♦♥ t❤❡ ❤❡❛t ❝♦♥❞✉❝t✐♦♥✱ ❛❞✈❡❝t✐♦♥✱ ❛♥❞ ❝♦♥✲ ✈❡❝t✐♦♥ ✐♥ ❈❤❛♣t❡r ✺✲✼❀ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ t❤❡ ✢✉✐❞✲❞②♥❛♠✐❝s✳ ❚❤❡ ♣r♦❣r❛♠s ❛r❡ ✇r✐tt❡♥ ✉s✐♥❣ ❙❝✐❧❛❜ − ❛♥ ♦♣❡♥✲s♦✉r❝❡ s♦❢t✇❛r❡ ❢♦r ♥✉✲ ♠❡r✐❝❛❧ ❝♦♠♣✉t❛t✐♦♥s✱ ✇✐t❤ ❛♥ ✐♥✲❜✉✐❧t ❣r❛♣❤✐❝❛❧ ✉s❡r ✐♥t❡r❢❛❝❡✳ ❈❋❉ ❛♣♣❧✐❝❛t✐♦♥ ❛♥❞ ❛♥❛❧②s✐s ❛r❡ ♣r❡s❡♥t❡❞ ❜② ❝❡rt❛✐♥ ❝❛r❡❢✉❧❧② ❞❡s✐❣♥❡❞ ❡①❛♠♣❧❡ ❛♥❞ ❡①❡r❝✐s❡ ♣r♦❜❧❡♠s✳ ❲✐t❤ t❤✐s t❤r❡❡✲♣r♦♥❣❡❞ ❛♣♣r♦❛❝❤ ♦❢ ❞❡✈❡❧♦♣♠❡♥t✱ ❛♣♣❧✐❝❛t✐♦♥✱ ❛♥❞ ❛♥❛❧②s✐s✱ t❤❡ ♣✉r♣♦s❡ ♦❢ t❤✐s ❜♦♦❦ ✐s t❤❛t t❤❡ r❡❛❞❡r ✇✐❧❧ ❧❡❛r♥ t♦ ❛♣♣r❡❝✐❛t❡ t❤❡ t❤❡♦r② ❛s ✇❡❧❧ ❛s ✉t✐❧✐③❡ ✐t ❢♦r t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❛ ♥❡✇ ❈❋❉ s♦❢t✇❛r❡✱ ♦r ❛♥ ❡①✐st✐♥❣ ♦♥❡ ❝❛♥ ❜❡ ✉s❡❞ ✐♥t❡❧❧✐❣❡♥t❧② ❢♦r t❤❡ ❈❋❉ ❛♣♣❧✐❝❛t✐♦♥ ❛♥❞ ❛♥❛❧②s✐s✳ ❚❤✉s✱ ❛ r❡❛❞❡r ✇✐❧❧ ❜❡ ✜r♠❧② s❡t ♦♥ t❤❡ ♣❛t❤ ♦❢ ❜❡❝♦♠✐♥❣ ❛ ❈❋❉ ❊①♣❡rt✳ ✷ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈❋❉✿ ❉❡✈❡❧♦♣♠❡♥t✱ ❆♣♣❧✐❝❛t✐♦♥✱ ❛♥❞ ❆♥❛❧②s✐s ❈♦♠♣✉t❛t✐♦♥❛❧ ✢✉✐❞ ❞②♥❛♠✐❝s ✐♥✈♦❧✈❡s ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❛ s♦❢t✇❛r❡✱ ✐ts ❛♣♣❧✐❝❛t✐♦♥ ❢♦r ❛ ✢✉✐❞✲❞②♥❛♠✐❝s ♣r♦❜❧❡♠ ✭t♦ ♦❜t❛✐♥ t❤❡ s❝✐❡♥t✐✜❝❛❧❧②✲ ❡①❝✐t✐♥❣ ❛s ✇❡❧❧ ❛s ❡♥❣✐♥❡❡r✐♥❣✲r❡❧❡✈❛♥t r❡s✉❧ts✮✱ ❛♥❞ ❛♥❛❧②s✐s ♦❢ t❤❡ r❡s✉❧ts ✭❢♦r ❛ ✉♥✐✜❡❞ ❝❛✉s❡✲❛♥❞✲❡✛❡❝t st✉❞②✮✳ ❆ ❞❡t❛✐❧❡❞ ♦✈❡r✈✐❡✇ ♦♥ t❤❡ ❞❡✈❡❧♦♣♠❡♥t✱ ❛♣♣❧✐❝❛t✐♦♥✱ ❛♥❞ ❛♥❛❧②s✐s ❛s♣❡❝ts ♦❢ ❈❋❉ ✐s ♣r❡✲ s❡♥t❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r✱ t♦ ❛♥s✇❡r ❤♦✇ ❈❋❉ ✇♦r❦s❄ ✷✳✶ ❈❋❉ ❉❡✈❡❧♦♣♠❡♥t ❈❋❉ ❞❡✈❡❧♦♣♠❡♥t ✐♥✈♦❧✈❡s ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❝♦♠♣✉t❡r✲♣r♦❣r❛♠s ✭❛❧s♦ ❝❛❧❧❡❞ ❛s ❝♦❞❡s ✇❤✐❝❤ ❧❡❛❞ t♦ ❛ s♦❢t✇❛r❡✮ ❛s ✇❡❧❧ ❛s t❡st✐♥❣✳ ❚❤❡ ❞❡✲ ✈❡❧♦♣♠❡♥t ❢♦❧❧♦✇s ❛ ♥✉♠❡r✐❝❛❧ ♠❡t❤♦❞♦❧♦❣② ✇❤✐❝❤ ❝♦♥s✐sts ♦❢ ✜✈❡ ♠❛✐♥ st❡♣s✿ ✜rst✱ ❣r✐❞✲❣❡♥❡r❛t✐♦♥❀ t❤✐r❞✱ s♦❧✉t✐♦♥✲♠❡t❤♦❞♦❧♦❣②❀ s❡❝♦♥❞✱ ❞✐s❝r❡t✐③❛t✐♦♥✲♠❡t❤♦❞❀ ❢♦✉rt❤✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❡♥❣✐♥❡❡r✐♥❣✲ ♣❛r❛♠❡t❡rs❀ ❛♥❞ ✜❢t❤✱ t❡st✐♥❣✳ ❚❤❡② ❛r❡ s❤♦✇♥ s❝❤❡♠❛t✐❝❛❧❧② ✐♥ ❋✐❣✳ ✷✳✶ ❛♥❞ ♣r❡s❡♥t❡❞ ✐♥ s❡♣❛r❛t❡ s✉❜s❡❝t✐♦♥s ❜❡❧♦✇✳ ❈❋❉ ❞❡✈❡❧♦♣♠❡♥t ❤❛s ❜❡❡♥ ♠❛✐♥❧② ❝♦♥tr✐❜✉t❡❞ ❜② t✇♦ ❞✐✛❡r❡♥t t②♣❡s ♦❢ ♣❡♦♣❧❡✿ ✢✉✐❞✲❞②♥❛♠✐❝✐sts ❛♥❞ ❝♦♠♣✉t❡r✲s❝✐❡♥t✐sts✳ ❋❧✉✐❞✲ ❞②♥❛♠✐❝✐sts ❞❡✈❡❧♦♣❡❞ ❛ ♥❡✇ ♦r ❛♥ ✐♠♣r♦✈❡❞ ♠❛t❤❡♠❛t✐❝❛❧✲♠♦❞❡❧ ❛♥❞✴♦r ❛ ♥✉♠❡r✐❝❛❧✲♠❡t❤♦❞♦❧♦❣② ✇❤✐❝❤ ❝❛♥ ❣✐✈❡ ❛❝❝✉r❛t❡ r❡s✉❧ts ✐♥ ❛ ❧❡❛st ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦st✳ ❲❤❡r❡❛s✱ ❝♦♠♣✉t❡r✲s❝✐❡♥t✐sts ❞❡✈❡❧✲ ♦♣❡❞ ❛ ♥❡✇ ✭♦r ❛♥ ✐♠♣r♦✈❡❞✮ ❝♦♠♣✉t❡r✲s②st❡♠ ❛♥❞✴♦r ♣r♦❣r❛♠♠✐♥❣✲ ♣❛r❛❞✐❣♠ t♦ r❡❞✉❝❡ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦st✳ ❚❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ✷✸ Introduction to Computational Fluid Dynamics: Development, Application and Analysis, First Edition.

Tr♦❞✉❝t✐♦♥ t♦ ❈❋❉ ✶✸ ❝❤r♦♥✐③❡❞ ♠♦✈✐❡ ♦❢ t❤❡ ✢♦✇✲♣r♦♣❡rt✐❡s ❛♥❞ ❡♥❣✐♥❡❡r✐♥❣✲♣❛r❛♠❡t❡rs✱ ✇✐t❤ ❛ t✐♠❡✲✇✐s❡ ✈❛r✐❛t✐♦♥ ♦❢ ♣✐❝t✉r❡ ❛♥❞ s②♠❜♦❧ ✭♦✈❡r t❤❡ ❧✐♥❡ ♣❧♦t✮✱ r❡s♣❡❝t✐✈❡❧②✳ ❙❝✐❡♥t✐✜❝✲✐♥✈❡st✐❣❛t✐♦♥ ❝♦rr❡s♣♦♥❞s t♦ ❛ st♦r② ❛❜♦✉t t❤❡ ✐♥t❡r✲ ❡st✐♥❣ ✢✉✐❞✲❞②♥❛♠✐❝s ♣❤❡♥♦♠❡♥♦♥ ✭✢♦✇✲s❡♣❛r❛t✐♦♥✱ ✇❛❦❡✲❢♦r♠❛t✐♦♥✱ ❛♥❞ ✈♦rt❡①✲s❤❡❞❞✐♥❣✮ ❡♥❝♦✉♥t❡r❡❞ ✐♥ t❤✐s ♣r♦❜❧❡♠✳ ❲❤❡r❡❛s✱ t❤❡ ❡♥❣✐♥❡❡r✐♥❣✲✐♥✈❡st✐❣❛t✐♦♥ ❝♦rr❡s♣♦♥❞s t♦ ❛ ♠❡t❤♦❞♦❧♦❣② ❢♦r t❤❡ str✉❝✲ t✉r❛❧ ❞❡s✐❣♥ ♦❢ t❤❡ ♣✐❧❧❛rs ❛❢t❡r ♦❜t❛✐♥✐♥❣ t❤❡ t✐♠❡ ✇✐s❡ ✈❛r②✐♥❣ ❧✐❢t ❛♥❞ ❞r❛❣ ❢♦r❝❡s✳ ❆ s❝✐❡♥t✐✜❝ ❛s ✇❡❧❧ ❛s ❡♥❣✐♥❡❡r✐♥❣ ✐♥✈❡st✐❣❛t✐♦♥ ❝♦rr❡✲ s♣♦♥❞s t♦ ✐❞❡♥t✐❢②✐♥❣ ❛ ❝♦rr❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❢♦r❝❡s ❛♥❞ ✢♦✇ str✉❝✲ t✉r❡ ❛s ✇❡❧❧ ❛s t❤❡ ✢✉✐❞✲❞②♥❛♠✐❝s ♣❤❡♥♦♠❡♥♦♥✳ ❚❤❡ r♦❧❡ ♦❢ ❛ ❈❋❉ s♦❢t✇❛r❡ ✐s ♥♦t ♦♥❧② ❧✐♠✐t❡❞ t♦ ❝r❡❛t✐♥❣ s❝✐❡♥t✐✜❝❛❧❧②✲❡①❝✐t✐♥❣ ✢✉✐❞ ❞②♥❛♠✐❝ ♠♦✈✐❡ ♦❢ ✢♦✇✲♣r♦♣❡rt✐❡s✱ ❜✉t ❛❧s♦ t♦ ❝r❡❛t❡ ❡♥❣✐♥❡❡r✐♥❣✲ r❡❧❡✈❛♥t ♠♦✈✐❡ ♦❢ ❡♥❣✐♥❡❡r✐♥❣✲♣❛r❛♠❡t❡rs✱ ❢♦r ❛ ✉♥✐✜❡❞ ❝❛✉s❡✲❛♥❞✲ ❡✛❡❝t st✉❞② ♦❢ ✈❛r✐♦✉s ❤❡❛t ❛♥❞ ✢✉✐❞ ✢♦✇ s✐t✉❛t✐♦♥s✳ ❚❤❡ ❛❜♦✈❡ ❝❛✉s❡✲❛♥❞✲❡✛❡❝t st✉❞② ❢♦r t❤❡ ❝②❧✐♥❞❡r ✐s ❛❧s♦ r❡❧❡✈❛♥t ❢♦r t❤❡ ✢♦✇ ❛❝r♦ss ❛ ❝❛r ❛♥❞ ❛❧s♦ ♦✈❡r ❛♥ ❛✐r❝r❛❢t✳ ✶✳✶✳✷ ❆♥❛❧♦❣② ✇✐t❤ ❛ ❱✐❞❡♦✲❈❛♠❡r❛ ✶✳✶✳✷✳✶ ❈❋❉ ❛s ❛ ❋✐♥✐t❡✲❘❡s♦❧✉t✐♦♥ ❱✐rt✉❛❧ ❱✐❞❡♦✲❈❛♠❡r❛ ❆❢t❡r ❞✐s❝✉ss✐♥❣ t❤❡ r♦❧❡ ♦❢ ❈❋❉ ❜❛s❡❞ s❝✐❡♥t✐✜❝❛❧❧②✲❡①❝✐t✐♥❣ ❛♥❞ ❡♥❣✐♥❡❡r✐♥❣✲r❡❧❡✈❛♥t ♠♦✈✐❡s ♦♥ t❤❡ ✐♥✈❡st✐❣❛t✐♦♥ ♦❢ ❛ ❤❡❛t✲tr❛♥s❢❡r ❛♥❞ ❛ ✢✉✐❞✲❞②♥❛♠✐❝s ♣r♦❜❧❡♠✱ ❛♥ ❛♥❛❧♦❣② ❜❡t✇❡❡♥ ❛ r❡❛❧ ❛♥❞ ✈✐rt✉❛❧ ✭❢r♦♠ ❛ ❈❋❉ s♦❢t✇❛r❡✮ ✈✐❞❡♦✲❝❛♠❡r❛ ✐s ♣r❡s❡♥t❡❞✳ ❖♥❡ ♥❡❡❞s t♦ ❢♦❝✉s t♦ ❛ s♣❛t✐❛❧ r❡❣✐♦♥ ♦❢ ✐♥t❡r❡st t♦ ❝r❡❛t❡ ❛ ♠♦✈✐❡ ✉s✐♥❣ ❛ ✈✐❞❡♦✲❝❛♠❡r❛✳ ❚❤✐s ✐s ❛❧s♦ tr✉❡ ✐♥ ❈❋❉ ❛♥❞ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ r❡❣✐♦♥ ✐♥✈♦❧✈✐♥❣ ❤❡❛t ❛♥❞✴♦r ✢✉✐❞ ✢♦✇ − ❝❛❧❧❡❞ ❛s ❝♦♠♣✉t❛t✐♦♥❛❧ ❞♦♠❛✐♥✳ ❲✐t❤ ❛ ✈✐❞❡♦✲ ❝❛♠❡r❛✱ t❤❡ s✐③❡ ♦❢ t❤❡ ❢♦❝✉s❡❞✲r❡❣✐♦♥ ✐s ❝♦♥str❛✐♥t ❜② ✐ts ❢♦❝✉s✐♥❣ s♣❡❝✐✜❝❛t✐♦♥✳ ❲❤❡r❡❛s✱ ✐♥ ❈❋❉✱ t❤❡ s✐③❡ ♦❢ ❝♦♠♣✉t❛t✐♦♥❛❧✲❞♦♠❛✐♥ ❛r❡ t❛❦❡♥ s✉❝❤ t❤❛t ❛ ♣❤②s✐❝❛❧❧② r❡❧❡✈❛♥t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ ❛t t❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ❞♦♠❛✐♥✳ ❆ ♣✐❝t✉r❡ t❛❦❡♥ ❢r♦♠ ❛ ✈✐❞❡♦ ❝❛♠❡r❛ ✐s ♠❛❞❡ ✉♣ ♦❢ t❤❡ ✐♠❛❣❡✴✲ ❝♦❧♦r ❝❛♣t✉r❡❞ ❛t ❛ ❧❛r❣❡ ♥✉♠❜❡r ♦❢ ❞✐s❝r❡t❡ ♣✐①❡❧s✳ ❆ ♣✐❝t✉r❡ ❢r♦♠ ❛ ❈❋❉ s♦❢t✇❛r❡ ✐s ♠❛❞❡ ✉♣ ♦❢ t❤❡ ❝♦❧♦rs ❜❛s❡❞ r❡♣r❡s❡♥t❛t✐♦♥ ✭♦❢ ❛ ✢♦✇✲♣r♦♣❡rt②✮ ❛t t❤❡ ❞✐s❝r❡t❡ ♣♦✐♥ts ✐♥ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❞♦♠❛✐♥❀ t❤❡ ♣♦✐♥ts ❛r❡ ❝❛❧❧❡❞ ❛s ❣r✐❞✲♣♦✐♥ts✳ ❚❤❡ ❝♦❧♦rs ❛t t❤❡ ❣r✐❞✲♣♦✐♥ts ❝♦r✲ r❡s♣♦♥❞ t♦ t❤❡ ♥✉♠❜❡rs ❝♦♠♣✉t❡❞ ❢♦r t❤❡ ❛ss♦❝✐❛t❡❞ ✢♦✇✲♣r♦♣❡rt✐❡s✱ ✶✹ ✶✳ ■♥tr♦❞✉❝t✐♦♥ ♠❛♣♣❡❞ ❜② ❛ ❣r❛♣❤✐❝❛❧ s♦❢t✇❛r❡✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ✢♦♦❞❡❞ ❣r❡②✲❝♦❧♦r ❝♦♥t♦✉r ✇✐t❤ ❛ ❝♦❧♦r✲❜❛r ✐s s❤♦✇♥ ✐♥ ❋✐❣✳ ✶✳✷✭❛✮✲✭❞✮ ❢♦r t❡♠♣❡r❛t✉r❡✱ ❋✐❣✳ ✶✳✸✭❛✶✮✲✭❞✶✮ ❢♦r ✈♦rt✐❝✐t②✱ ❛♥❞ ❋✐❣✳ ✶✳✸✭❛✷✮✲✭❞✷✮ ❢♦r ♣r❡ss✉r❡✳ ❚❤❡ ✜❣✉r❡s ❝♦rr❡s♣♦♥❞ t♦ ❛ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❣r✐❞✲♣♦✐♥ts ✭❛♥❛❧♦❣♦✉s t♦ ♣✐①❡❧s✮ ❛s 4000 ❢♦r t❡♠♣❡r❛t✉r❡✱ 67500 ❢♦r ✈♦rt✐❝✐t②✱ ❛♥❞ 32250 ❢♦r ♣r❡ss✉r❡✳ ❚❤❡ q✉❛❧✐t② ♦❢ ❛ ♣✐❝t✉r❡ ♦❜t❛✐♥❡❞ ❢r♦♠ ❛ ❝❛♠❡r❛ ❞❡♣❡♥❞s ♦♥ t❤❡ ♥✉♠❜❡r ♦❢ ♣✐①❡❧s✳ ❲❤❡r❡❛s✱ t❤❡ s♣❛t✐❛❧ ❛❝❝✉r❛❝② ♦❢ t❤❡ r❡s✉❧ts ✭♣r❡s❡♥t❡❞ ❛s ❛ ♣✐❝t✉r❡✮ ♦❜t❛✐♥❡❞ ❢r♦♠ ❛ ❈❋❉ s♦❢t✇❛r❡ ❞❡♣❡♥❞s ♦♥ t❤❡ ♥✉♠❜❡r ♦❢ ❣r✐❞✲♣♦✐♥ts✳ ❙♣❛t✐❛❧ r❡s♦❧✉t✐♦♥ ❢♦r ❛ ♣✐❝t✉r❡ ✐s ❡q✉❛❧ t♦ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ♣✐①❡❧s ✐♥ ❛ ✈✐❞❡♦✲❝❛♠❡r❛ ❛♥❞ t❤❡ ♥✉♠❜❡r ♦❢ ❣r✐❞✲♣♦✐♥ts ✐♥ ❈❋❉✳ ❚❤❡ r❡s♦❧✉t✐♦♥ ✐s ✉♥✐❢♦r♠ ✐♥ ❛ ✈✐❞❡♦ ❝❛♠❡r❛ ❛♥❞ ♠♦st❧② ♥♦♥✲✉♥✐❢♦r♠ ✐♥ ❈❋❉✳ ❚❤✐s ✐s ❜❡❝❛✉s❡ t❤❡ ❣r✐❞✲♣♦✐♥ts ❛r❡ ♥♦♥✲✉♥✐❢♦r♠❧② s♣❛❝❡❞ t♦ ❡✣❝✐❡♥t❧② ❝❛♣t✉r❡ t❤❡ s♣❛t✐❛❧❧② ♥♦♥✲✉♥✐❢♦r♠ ❛❝t✐♦♥✴❞②♥❛♠✐❝s ♦❢ ✢♦✇ ✐♥s✐❞❡ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❞♦♠❛✐♥✳ ❆ ♠♦✈✐❡ ✐s ♠❛❞❡ ❜② ❛ s❡r✐❡s ♦❢ ♣✐❝t✉r❡s ♦❜t❛✐♥❡❞ ❛t ❛ t✐♠❡ ✐♥t❡r✈❛❧ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❢r❛♠❡ r❛t❡ ♦❢ t❤❡ ✈✐❞❡♦ ❝❛♠❡r❛✱ ❛♥❞ t♦ t❤❡ t✐♠❡✲st❡♣ ✐♥ ❝❛s❡ ♦❢ ❛ ❈❋❉ s♦❢t✇❛r❡✱ ❝❛❧❧❡❞ ❛s t❡♠♣♦r❛❧ r❡s♦❧✉t✐♦♥✳ ❚❤❡ s♣❛t✐❛❧ ❛♥❞ t❤❡ t❡♠♣♦r❛❧ r❡s♦✲ ❧✉t✐♦♥s ❝♦♥tr♦❧s t❤❡ q✉❛❧✐t② ♦❢ t❤❡ ❣r❛♣❤✐❝❛❧ r❡s✉❧ts✱ ❥✉❞❣❡❞ ❜② t❤❡ ✈✐s✉❛❧✲❛♣♣❡❛r❛♥❝❡ ✐♥ ❛ ✈✐❞❡♦✲❝❛♠❡r❛ ❛♥❞ t❤❡ ❛❝❝✉r❛❝② ✐♥ ❛ ❈❋❉ s♦❢t✲ ✇❛r❡✳ ❚❤❡ ❛♥❛❧♦❣② ♣r❡s❡♥t❡❞ ❛❜♦✈❡ ❛♣♣❧✐❡s ♥♦t ♦♥❧② t♦ ❈❋❉ ❜✉t ❛❧s♦ ❛♣♣❧✐❝❛❜❧❡ t♦ ❝♦♠♣✉t❛t✐♦♥❛❧✲s♦❧✐❞✲❞②♥❛♠✐❝s ❛s ✇❡❧❧ ❛s ♠❛♥② ♦t❤❡r ♣❤②s✐❝s ❜❛s❡❞ s✐♠✉❧❛t✐♦♥ s♦❢t✇❛r❡ ✐♥ s❝✐❡♥❝❡ ❛♥❞ ❡♥❣✐♥❡❡r✐♥❣✳ ❋♦r ❛❧♠♦st ❛❧❧ t❤❡ ✢✉✐❞ ✢♦✇ ❛♥❞ ❤❡❛t✲tr❛♥s❢❡r s✐t✉❛t✐♦♥s✱ ❈❋❉ ♣r♦✈✐❞❡s ❛ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ ✇❤✐❝❤ ✐s ❛♣♣r♦①✐♠❛t❡✱ ❛♥❞ ❛t ❝❡rt❛✐♥ s♣❛t✐❛❧ ❛♥❞ t❡♠♣♦r❛❧ ❝♦♦r❞✐♥❛t❡s✳ ❚❤✉s✱ ❈❋❉ ✐s ❛ ✜♥✐t❡ r❡s♦❧✉t✐♦♥ ✈✐rt✉❛❧ ✈✐❞❡♦ ❝❛♠❡r❛✳ ✶✳✶✳✷✳✷ ❆♥❛❧②t✐❝❛❧ ❙♦❧✉t✐♦♥ ❛s ❛♥ ■♥✜♥✐t❡✲❘❡s♦❧✉t✐♦♥ ❱✐rt✉❛❧ ❱✐❞❡♦✲❈❛♠❡r❛ ❆♥❛❧②t✐❝❛❧ s♦❧✉t✐♦♥ ❢♦r ❛♥ ✉♥st❡❛❞② ♣r♦❜❧❡♠ ✐s ❛♥ ✐♥✜♥✐t❡ r❡s♦❧✉t✐♦♥ ✈✐rt✉❛❧ ✈✐❞❡♦ ❝❛♠❡r❛✳ ❚❤❡ ❣♦✈❡r♥✐♥❣ ❡q✉❛t✐♦♥s ✐♥ ✢✉✐❞✲❞②♥❛♠✐❝s ❛♥❞ ❤❡❛t✲tr❛♥s❢❡r ❛r❡ P❛rt✐❛❧ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✭P❉❊s✮ − ✇✐t❤ ❞❡✲ ♣❡♥❞❡♥t ✈❛r✐❛❜❧❡s ❛s ✢♦✇✲♣r♦♣❡rt✐❡s✱ ❛♥❞ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ❛s s♣❛✲ t✐❛❧ ❛♥❞ t❡♠♣♦r❛❧ ❝♦♦r❞✐♥❛t❡s✳ ❚❤❡ ❛♥❛❧②t✐❝❛❧ s♦❧✉t✐♦♥ ♦❢ t❤❡ P❉❊s ✐s ❛ ❢✉♥❝t✐♦♥❛❧ r❡❧❛t✐♦♥s❤✐♣ ♦❢ t❤❡ ❞❡♣❡♥❞❡♥t ✐♥ t❡r♠s ♦❢ t❤❡ ✐♥❞❡♣❡♥✲ ❞❡♥t ✈❛r✐❛❜❧❡s✳ ❚❤❡ s♦❧✉t✐♦♥ r❡s✉❧ts ✐♥ ✢♦✇✲♣r♦♣❡rt✐❡s ❛t ❛♥② ✈❛❧✉❡ ♦❢ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈❋❉ ✶✺ s♣❛t✐❛❧ ❛♥❞ t❡♠♣♦r❛❧ ❝♦♦r❞✐♥❛t❡s − ✐♥✜♥✐t❡ r❡s♦❧✉t✐♦♥ ✈✐rt✉❛❧ ✈✐❞❡♦ ❝❛♠❡r❛✳ ❯♥❢♦rt✉♥❛t❡❧②✱ t❤❡ ❛♥❛❧②t✐❝❛❧ ✐s ♥♦t ②❡t ♣♦ss✐❜❧❡ ❢♦r ❛ ❣❡♥❡r❛❧ ✢✉✐❞✲ ❞②♥❛♠✐❝s ♣r♦❜❧❡♠ ❛♥❞ ✐s ♦♥❡ ♦❢ t❤❡ ♠♦st ✐♠♣♦rt❛♥t ✉♥r❡s♦❧✈❡❞ ♣r♦❜✲ ❧❡♠ ✐♥ s❝✐❡♥❝❡ ❛♥❞ ❡♥❣✐♥❡❡r✐♥❣✳ ❚❤❡ ❛❞✈❡♥t ♦❢ t❤❡ ❣❡♥❡r❛❧✲♣✉r♣♦s❡ ❛♥❛❧②t✐❝❛❧ s♦❧✉t✐♦♥ ✇✐❧❧ ❧❡❛❞ t♦ ❝❧♦s✉r❡ ♦❢ ❛❧❧ t❤❡ ❈❋❉ s♦❢t✇❛r❡ ❝♦♠✲ ♣❛♥✐❡s ❛s ✇❡❧❧ ❛s t❤❡ ❡①✐st❡♥❝❡ ♦❢ t❤✐s ❝♦✉rs❡✳ ✶✳✷ ❈❋❉✿ ❲❤② t♦ st✉❞②❄ ❉❡✈❡❧♦♣♠❡♥ts ✐♥ t❤❡ ❡♥❣✐♥❡❡r✐♥❣ s✐♠✉❧❛t✐♦♥ ♠❡t❤♦❞♦❧♦❣② ❤❛s ❧❡❞ t♦ ❛ ❞r❛st✐❝ ✐♠♣r♦✈❡♠❡♥ts ✐♥ t❤❡ ❞❡s✐❣♥ ♠❡t❤♦❞♦❧♦❣② ❢♦r ❛ ♣r♦❞✉❝t✳ ◆♦✇✱ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ❝r❡❛t❡ ✈✐rt✉❛❧ ♣r♦❞✉❝t ✐♥ ❛ ❝♦♠♣✉t❡r ✭✉s✐♥❣ ❛ s♦❧✐❞ ♠♦❞❡❧❡r✮✱ ❛♥❛❧②③❡ ✐ts ♣❡r❢♦r♠❛♥❝❡✱ s✉❣❣❡st s♦♠❡ ❞❡s✐❣♥ ♠♦❞✐✜❝❛t✐♦♥s✱ ❛♥❞ ❝♦♥✜r♠ ❡♥❤❛♥❝❡♠❡♥t ✐♥ ♣❡r❢♦r♠❛♥❝❡ ♦r r❡❞✉❝t✐♦♥ ✐♥ ❝♦st✳ ■t ♣r♦✈✐❞❡s ❛♥ ❡①❝❡❧❧❡♥t ♣❧❛t❢♦r♠ ❢♦r ✐♥♥♦✈❛t✐♦♥s ✐♥ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ t❤❡ ♥❡✇ ♣r♦❞✉❝ts✳ ❲❤❡r❡❛s✱ ❢♦r t❤❡ ❡①✐st✐♥❣ ♣r♦❞✉❝ts✱ ✐t ❧❡❛❞s t♦ ❛ tr❡♠❡♥❞♦✉s ✐♠♣r♦✈❡♠❡♥ts ✐♥ t❤❡ ♣❡r❢♦r♠❛♥❝❡ ❛♥❞ ❡✣❝✐❡♥❝②✳ ◆♦✇ ❛ ❞❛②s✱ ❛ ❝♦♠♣✉t❡r s✐♠✉❧❛t✐♦♥ ❛♥❞ ❛♥❛❧②s✐s ❤❛✈❡ ❜❡❝♦♠❡ ❛♥ ✐♥t❡❣r❛❧ ♣❛rt ♦❢ ❛♥ ❡♥❣✐♥❡❡r✐♥❣ st✉❞②✳ ❈❋❉ ❤❛s ❧❡❞ t♦ ❛ ♠♦r❡ s❝✐❡♥t✐✜❝ ❛s ❝♦♠♣❛r❡❞ t♦ ❡♠♣✐r✐❝❛❧ ❛♣✲ ♣r♦❛❝❤ ✐♥ t❤❡ ❞❡s✐❣♥ ❛♥❞ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❛ ❣❧♦❜❛❧❧② ❝♦♠♣❡t✐t✐✈❡ ♣r♦❞✲ ✉❝t✳ ❙❝✐❡♥t✐sts ❛s ✇❡❧❧ ❛s ❡♥❣✐♥❡❡rs ✇❛♥t t♦ ✉s❡ ❈❋❉ ❛s ❛ ❞❡s✐❣♥ ❛♥❞ ♦♣t✐♠✐③❛t✐♦♥ t♦♦❧✳ ❍♦✇❡✈❡r✱ ❞✉❡ t♦ t❤❡ ✉♥❛✛♦r❞❛❜❧❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦st✱ ♣r❡s❡♥t❧② ✐t ✐s ♠♦r❡ ❝♦♠♠♦♥❧② ✉s❡❞ ❛s ❛ ♣♦✇❡r❢✉❧ ❛♥❛❧②s✐s t♦♦❧✳ ❈❋❉ ✐s ✉s❡❞ ❢♦r ❛♥ ❛♥❛❧②s✐s ❜❛s❡❞ ✐♠♣r♦✈❡♠❡♥ts ✐♥ t❤❡ ❞❡s✐❣♥ ♦❢ ❛ ✢✉✐❞✲❞②♥❛♠✐❝s ❛♥❞ ❤❡❛t✲tr❛♥s❢❡r s②st❡♠s ✐♥ ✈❛r✐♦✉s ✐♥❞✉str✐❡s❀ s✉❝❤ ❛s ❛❡r♦s♣❛❝❡✱ ❛✉t♦♠♦❜✐❧❡✱ t✉r❜♦✲♠❛❝❤✐♥❡r②✱ ❝❤❡♠✐❝❛❧✱ ❡❧❡❝tr✐❝❛❧✱ ❡❧❡❝✲ tr♦♥✐❝s✱ ❜✐♦♠❡❞✐❝❛❧✱ ❡t❝✳ ■♥ ❢✉t✉r❡✱ ❈❋❉ ✇✐❧❧ ❜❡ ♠♦r❡ ♦❢t❡♥ ✉s❡❞ ❛s ❛ ❞❡s✐❣♥ ❛♥❞ ♦♣t✐♠✐③❛t✐♦♥ t♦♦❧✳ ■♥ ❛❝❛❞❡♠✐❝s✱ ❈❋❉ ✐s t❛✉❣❤t ❛s ❛♥ ✉♥❞❡r❣r❛❞✉❛t❡ ❡❧❡❝t✐✈❡ ❛♥❞ ♣♦st❣r❛❞✉❛t❡ ❝♦r❡ ❝♦✉rs❡ ✐♥ ❆❡r♦s♣❛❝❡✱ ❈❤❡♠✐❝❛❧✱ ❛♥❞ ▼❡❝❤❛♥✐❝❛❧ ❜r❛♥❝❤❡s ♦❢ ❊♥❣✐♥❡❡r✐♥❣✳ ❋✉rt❤❡r♠♦r❡✱ ✐t t❛✉❣❤t ❛s ❛♥ ✐♥t❡r❞✐s❝✐✲ ♣❧✐♥❛r② ❝♦✉rs❡ ✐♥ t❤❡ ❈✐✈✐❧✱ ▼❡t❛❧❧✉r❣✐❝❛❧✱ ❊❧❡❝tr✐❝❛❧✱ ❛♥❞ ❇✐♦♠❡❞✐❝❛❧ ❊♥❣✐♥❡❡r✐♥❣✳ ❚❤❡ ✐♥❝r❡❛s✐♥❣ ✐♠♣♦rt❛♥❝❡ ♦❢ ❈❋❉ ❞❡✈❡❧♦♣♠❡♥t✱ ❛♣✲ ♣❧✐❝❛t✐♦♥✱ ❛♥❞ ❛♥❛❧②s✐s ✐♥ t❤❡ ✈❛r✐♦✉s ✐♥❞✉str✐❡s ❛s ✇❡❧❧ ❛s r❡s❡❛r❝❤ ♦r❣❛♥✐③❛t✐♦♥s✱ ❛❧♦♥❣ ✇✐t❤ t❤❡ ❧❛❝❦ ♦❢ tr❛✐♥❡❞ ♠❛♥♣♦✇❡r ✐♥ t❤✐s s✉❜✲ ❥❡❝t✱ ❤❛s ❣r❡❛t❧② ✐♥❝r❡❛s❡❞ t❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ t❤✐s ❝♦✉rs❡✳ ✶✻ ✶✳ ■♥tr♦❞✉❝t✐♦♥ ❋✐❣✉r❡ ✶✳✹✿ P❤②s✐❝❛❧ ❛♥❞ ♠❛t❤❡♠❛t✐❝❛❧ ❛♣♣r♦❛❝❤❡s ♦❢ ❈❋❉ ❞❡✈❡❧♦♣♠❡♥t✳ ✶✳✸ ◆♦✈❡❧t②✱ ❙❝♦♣❡✱ ❛♥❞ P✉r♣♦s❡ ♦❢ t❤✐s ❇♦♦❦ ❋r♦♠ t❤❡ ✢✉✐❞✲❞②♥❛♠✐❝s ❛♥❞ ❤❡❛t✲tr❛♥s❢❡r ❝♦✉rs❡✱ ✇❡ ❧❡❛r♥ t♦ ❞❡✲ r✐✈❡ ❣♦✈❡r♥✐♥❣ P❛rt✐❛❧ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✭P❉❊s✮ ❜② ❛♣♣❧②✐♥❣ t❤❡ ♠❛ss✱ ♠♦♠❡♥t✉♠✱ ❛♥❞ ❡♥❡r❣② ❝♦♥s❡r✈❛t✐♦♥ ❧❛✇s t♦ ❛♥ ❡❧❡♠❡♥✲ ❈♦♥tr♦❧✲❱♦❧✉♠❡ ✭❈❱✮✳ ❲❤❡r❡❛s✱ ✐♥ ❈❋❉✱ t❤❡ ❣♦✈❡r♥✲ ✐♥❣ P❉❊s ❛r❡ ♠❛t❤❡♠❛t✐❝❛❧❧② ♦♣❡r❛t❡❞ t♦ ❞❡r✐✈❡ ❛ s②st❡♠ ♦❢ ▲✐♥❡❛r ❆❧❣❡❜r❛✐❝ ❊q✉❛t✐♦♥s ✭▲❆❊s✮− ❝❛❧❧❡❞ ❛s ❞✐s❝r❡t✐③❛t✐♦♥ ♠❡t❤♦❞❀ s✉❝❤ ❛s✱ ❋✐♥✐t❡ ❱♦❧✉♠❡ ▼❡t❤♦❞ ✭❋❱▼✮✳ ❚❤❡ s②st❡♠ ♦❢ ▲❆❊s r❡s✉❧t✐♥❣ t❛❧ ✢✉✐❞✴s♦❧✐❞ ❢r♦♠ t❤❡ ❛❧❣❡❜r❛✐❝ ❢♦r♠✉❧❛t✐♦♥ ❛❝ts ❛s t❤❡ ❣♦✈❡r♥✐♥❣ ❡q✉❛t✐♦♥s ❢♦r ❈❋❉✱ ❛♥❞ ❝♦♥s✐st ♦❢ t❤❡ ✢♦✇✲♣r♦♣❡rt✐❡s ✭✈❡❧♦❝✐t②✱ ♣r❡ss✉r❡✱ ❛♥❞ t❡♠✲ ♣❡r❛t✉r❡✮ ❛s t❤❡ ✉♥❦♥♦✇♥ ✜❡❧❞ ✈❛r✐❛❜❧❡s✳ s♦❧✈❡❞ ❜② ❛ s♦❧✉t✐♦♥ ♠❡t❤♦❞♦❧♦❣②✳ ❚❤❡ s②st❡♠ ♦❢ ▲❆❊s ❛r❡ ❚❤❡ P❉❊ ❜❛s❡❞ ❋❱▼ ❛♥❞ t❤❡ ♠❛t❤❡♠❛t✐❝❛❧ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ ▲❆❊s ❜❛s❡❞ s♦❧✉t✐♦♥ ♠❡t❤♦❞♦❧♦❣②✱ ✉s❡❞ ✐♥ ❛❧♠♦st ❛❧❧ t❤❡ ❜♦♦❦s ✐♥ ❈❋❉✱ ✐s ❝❛❧❧❡❞ ❤❡r❡ ❛s ❛ ♠❛t❤❡♠❛t✐❝❛❧ ❛♣♣r♦❛❝❤ ♦❢ ❈❋❉ ❞❡✈❡❧♦♣♠❡♥t❀ s❤♦✇♥ ✐♥ ❋✐❣✳ ✶✳✹✳ ❚❤❡ ♥♦✈❡❧t② ✐♥ t❤✐s ❜♦♦❦ ✐s ❛ ♣❤②s✐❝❛❧ ❛♣♣r♦❛❝❤ ♦❢ ❈❋❉ ❞❡✈❡❧✲ ♦♣♠❡♥t ✱ ♣r♦♣♦s❡❞ ❤❡r❡ ❛♥❞ ♣r♦❜❛❜❧② ♥♦t ❢♦✉♥❞ ✐♥ ❛♥② ♦❢ t❤❡ ❜♦♦❦s ♦♥ ❈❋❉✳ ❚❤❡ ❛❜♦✈❡ P❉❊ ❜❛s❡❞ ❋❱▼ ❧❡❛❞s t♦ ❛ q✉❡st✐♦♥✿ ❲❤② ❞♦ ✇❡ s✇✐t❝❤ ❢r♦♠ t❤❡ ❛❧❣❡❜r❛✐❝ ❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ♣❤②s✐❝❛❧ ❧❛✇s t♦ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠✉❧❛t✐♦♥ t♦ t❤❡♥ ❣♦ ❜❛❝❦ t♦ t❤❡ ❛❧❣❡❜r❛✐❝ ❢♦r♠✉❧❛t✐♦♥❄ ❲❤② ❞♦♥✬t ✇❡ ❦❡❡♣ t❤❡ ❛❧❣❡❜r❛✐❝ ❢♦r♠✉❧❛t✐♦♥✱ ❛♥❞ ❝♦♥✈❡rt t❤❡ ❝♦♥t✐♥✲ ✉♦✉s ✈❛r✐❛❜❧❡s ✐♥ t❤❡ ♣❤②s✐❝❛❧ ❧❛✇s t♦ ❛♣♣r♦♣r✐❛t❡ ❞✐s❝r❡t❡ ✈❛r✐❛❜❧❡s ✉s✐♥❣ ❝❡rt❛✐♥ ❛♣♣r♦①✐♠❛t✐♦♥s❄ ❚❤✐s ❧❡❞ t♦ ♣❤②s✐❝❛❧ ❧❛✇s ❜❛s❡❞ ✭P❉❊ ❢r❡❡✮ ❋❱▼✱ s❤♦✇♥ ✐♥ ❋✐❣✳ ✶✳✹✳ ❚❤❡ ✜❣✉r❡ ❛❧s♦ s❤♦✇s t❤❛t t❤❡ ♣❤②s✐✲ ❝❛❧ ❛♣♣r♦❛❝❤ ♦❢ ❈❋❉ ❞❡✈❡❧♦♣♠❡♥t ❝♦♥s✐st ♦❢ ❛ ✢✉① ✭♣❤②s✐❝❛❧ q✉❛♥t✐t②✮ ❜❛s❡❞ s♦❧✉t✐♦♥ ♠❡t❤♦❞♦❧♦❣②✳ ❚❤❡ ♣r❡s❡♥t ❜♦♦❦ ✐s ♦♥ ♣❤②s✐❝❛❧ ❧❛✇ ❜❛s❡❞ ❋❱▼✱ ❛♥❞ ♣r❡s❡♥ts ❜♦t❤ t❤❡ ✢✉① ❛♥❞ ❝♦❡✣❝✐❡♥t ♦❢ ▲❆❊s ❜❛s❡❞ s♦✲ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈❋❉ ✶✼ ❧✉t✐♦♥ ♠❡t❤♦❞♦❧♦❣②✳ ❋✉rt❤❡r ❞❡t❛✐❧s ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ♠❡t❤♦❞♦❧♦❣✐❡s ✇✐❧❧ ❜❡ ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ♥❡①t ❝❤❛♣t❡r✳ ❲✐t❤ ❛ ♠♦t✐✈❛t✐♦♥ t♦ ✐♥❝r❡❛s❡ ❛s ✇❡❧❧ ❛s ❡♥r✐❝❤ t❤❡ ❧❡❛r♥✐♥❣ ♦❢ t❤✐s s✉❜❥❡❝t ✇❤✐❝❤ s❤♦✉❧❞ ❧❡❛❞ t♦ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❛ ❝✉st♦♠✐③❡❞ ❈❋❉ s♦❢t✇❛r❡ ❛s ❛ ♣r♦❞✉❝t − − ❜❡ ❛ ❝r❡❛t♦r t❤❛♥ ❛ ♣✐❡❝❡ ♦❢ ❝r❡❛t✐♦♥ t❤❡ ♣❤②s✐❝❛❧ ❛♣♣r♦❛❝❤ ❡✈♦❧✈❡❞ ✐♥❞❡♣❡♥❞❡♥t❧② ♦✈❡r ♠② t❡❛❝❤✐♥❣ ❛♥❞ 18 12 ②❡❛rs ♦❢ ②❡❛rs ♦❢ r❡s❡❛r❝❤ ❡①♣❡r✐❡♥❝❡ ✐♥ ❈❋❉✳ ❍♦✇❡✈❡r✱ ♠✉❝❤ ❧❛t❡r✱ t❤❡ ❣❡♥❡r❛❧ ✐❞❡❛ ♦❢ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠✉❧❛t✐♦♥ ❢r❡❡ ❞✐s❝r❡t❡ ❛❧❣❡❜r❛✐❝ ❢♦r♠✉❧❛t✐♦♥ ✇❛s ❢♦✉♥❞ ✐♥ s♦♠❡ r❡s❡❛r❝❤ ♣❛♣❡rs ✭r❡❝❡♥t❧② ❜② ❚♦♥t✐✱ ✷✵✶✹✮✳ ■♥ t❤❡ ✐♥s✐❣❤t❢✉❧ ♣❛♣❡r✱ ❚♦♥t✐ ✭✷✵✶✹✮ ❤❛s st❛t❡❞✿ ✏ ❚❤❡ ✏ ❡①❛❝t✑ s♦❧✉t✐♦♥✱ t❤❛t ✐s ♣r♦♠✐s❡❞ ❜② t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠✉✲ ❧❛t✐♦♥✱ ✐s ❤❛r❞❧② ❡✈❡r ❛tt❛✐♥❡❞ ✐♥ ♣r❛❝t✐❝❡✳ ❇② ❝♦♥tr❛st✱ t❤❡ ❣r❡❛t t❡❝❤✲ ♥♦❧♦❣✐❝❛❧ ♣r♦❣r❡ss ♦❢ ♦✉r ❞❛②s ✐s ♠❛❞❡ ♣♦ss✐❜❧❡ ❜② t❤❡ ❢❛❝t ♦❢ ❜❡✐♥❣ ❛❜❧❡ t♦ ❤❛✈❡ ❛♥ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥✱ ✐♥ ♣❛rt✐❝✉❧❛r ❛ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ t♦ t❤❡ ♣r♦❜❧❡♠s ♣♦s❡❞ ❜② t❤❡ t❡❝❤♥✐q✉❡✳ ❋♦r ♦✉r ❝✉❧t✉r❡✱ ❢♦r♠❡❞ ♦♥ t❤❡ ♠♦❞❡❧ ♦❢ ✐♥✜♥✐t❡s✐♠❛❧ ❛♥❛❧②s✐s✱ t❤❡ t❡r♠ ✏ ❛♣♣r♦①✐♠❛t❡✑ s♦✉♥❞s ❧✐❦❡ ✏ ✐♠♣❡r❢❡❝t✳✑ ❍♦✇❡✈❡r✱ ✇❡ ♠✉st ♥♦t ❢♦r❣❡t t❤❛t t❤❡ ♦❜❥❡❝t✐✈❡ ♦❢ ❛ ♥✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥ ♦❢ ♣❤②s✐❝❛❧ ♣r♦❝❡ss❡s ✐s t❤❡ ❛❣r❡❡♠❡♥t ✇✐t❤ t❤❡ ❡①♣❡r✐♠❡♥t❛❧ ♠❡❛s✉r❡♠❡♥ts ❛♥❞ ♥♦t t❤❡ ❝♦♥✈❡r❣❡♥❝❡ t♦ ❛♥ ❛♥❛❧②t✐❝❛❧ s♦❧✉t✐♦♥✱ ✉s✉❛❧❧② ♥♦t ❛tt❛✐♥❛❜❧❡✳ ■♥ ❛❞❞✐t✐♦♥✱ t❤❡ r❡q✉❡st ♦❢ r❡❞✉❝✐♥❣ t❤❡ ❡rr♦r ♦❢ ❛♥ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥ ❞♦❡s ♥♦t ♠❡❛♥ ♠❛❦❡ t❤❡ ❡rr♦r ✏ ❛r❜✐tr❛r✐❧② s♠❛❧❧✑ ✱ ❛s r❡q✉✐r❡❞ ❜② t❤❡ ♣r♦❝❡ss ♦❢ ❧✐♠✐t✱ ❜✉t ♠❛❦✐♥❣ t❤❡ ❡rr♦r s♠❛❧❧❡r t❤❛♥ ❛ ♣r❡❛ss✐❣♥❡❞ t♦❧❡r❛♥❝❡✳ ❚❤❡ ♥♦t✐♦♥ ♦❢ ♣r❡❝✐s✐♦♥ ♦❢ ❛ ♠❡❛s✉r✐♥❣ ❞❡✈✐❝❡ ♣❧❛②s t❤❡ s❛♠❡ r♦❧❡ ❛s t❤❡ ♥♦t✐♦♥ ♦❢ t♦❧❡r❛♥❝❡ ✐♥ ♠❛♥✉❢❛❝t✉r✐♥❣ ❛♥❞ t❤❡ ♥♦t✐♦♥ ♦❢ ❡rr♦r ✐♥ ♥✉♠❡r✐❝❛❧ ❛♥❛❧②s✐s✳✑ ❙✐♥❝❡ t❤❡ ❛❞✈❡♥t ♦❢ ❝♦♠♣✉t❡r✱ t❤❡r❡ ✐s ❛♥ ❡♥♦r♠♦✉s ❞❡✈❡❧♦♣♠❡♥t ✐♥ t❤❡ ❞✐s❝r❡t❡ ♠❛t❤❡♠❛t✐❝s✳ ❍♦✇❡✈❡r✱ ✐❢ ✇❡ ❝❤❛♥❣❡ ♦✉r ❤❛❜✐ts ♦❢ ❝♦♥s✐❞❡r✐♥❣ t❤❡ ❞✐✛❡r❡♥t✐❛❧✴❝♦♥t✐♥✉♦✉s ❢♦r♠✉❧❛t✐♦♥ ❛s ✐♥❞✐s♣❡♥s❛❜❧❡ ❛♥❞ ✇♦r❦ ♦♥ t❤❡ ❞✐s❝r❡t❡ ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ❝♦♥t✐♥✉♦✉s ♠❛t❤❡♠❛t✐❝s✱ ✐t ✇✐❧❧ ❧❡❛❞ t♦ ❛♥ ✐♥❝r❡❛s❡ ❛s ✇❡❧❧ ❛s ❡♥r✐❝❤♠❡♥t ✐♥ t❤❡ r❛♥❣❡ ♦❢ ❛♣♣❧✐✲ ❝❛t✐♦♥s ✭❉❡s❜r✉♥ ❡t ❛❧✳✱ ✷✵✵✺✮✳ ❚❤✐s ✇✐❧❧ ❛❧s♦ ❧❡❛❞ t♦ ❛ ♠✉❝❤ ❜❡tt❡r ✉♥❞❡rst❛♥❞✐♥❣ ❛s ✇❡❧❧ ❛s ❞❡✈❡❧♦♣♠❡♥ts ♦❢ t❤❡ s✐♠✉❧❛t✐♦♥ ♠❡t❤♦❞♦❧♦❣② ✐♥ t❤❡ ✈❛r✐♦✉s ❜r❛♥❝❤❡s ♦❢ s❝✐❡♥❝❡ ❛♥❞ ❡♥❣✐♥❡❡r✐♥❣✳ ❚❤❡r❡ ❛r❡ s❡✈❡r❛❧ ❝❧❛ss✐❝s ❛♥❞ ❣❡♥❡r❛❧ ♣✉r♣♦s❡ ❜♦♦❦s ♦♥ ❈❋❉ s✉❝❤ ❛s t❤❡ ❜♦♦❦ ❜② ❘♦❛❝❤❡ ✭✶✾✼✻✮✱ P❡②r❡t ❛♥❞ ❚❛②❧♦r ✭✶✾✽✸✮✱ ❏❛❧✉r✐❛ ❛♥❞ ❚♦rr❛♥❝❡ ✭✶✾✽✻✮✱ ❋❧❡t❝❤❡r ✭✶✾✽✽✮✱ ❍✐r❝❤ ✭✶✾✽✾✱ ✶✾✾✵✮✱ ❛♥❞ ❍♦✛♠❛♥♥ ❛♥❞ ❈❤✐❛♥❣ ✭✷✵✵✵✮✳ ❆♠♦♥❣ t❤❡ s❡✈❡r❛❧ ✈❡r② ❣♦♦❞ ❜♦♦❦s ♦♥ ❈❋❉✱ ✶✽ ✶✳ ■♥tr♦❞✉❝t✐♦♥ ✇♦✉❧❞ ❧✐❦❡ t♦ ♠❡♥t✐♦♥ s❡✈❡♥ ♦❢ t❤❡♠✿ ✜rst✱ ❜② P❛t❛♥❦❛r ✭✶✾✽✵✮✱ ❛ ❝❧❛ss✐❝❛❧ ❛♥❞ ♣r♦❜❛❜❧② t❤❡ ♠♦st r❡❢❡rr❡❞ ❜♦♦❦❀ s❡❝♦♥❞✱ ❜② ❆♥❞❡rs♦♥✱ ❚❛♥♥❡❤✐❧❧ ❛♥❞ P❧❡t❝❤❡r ✭✶✾✽✹✮✱ ❛♥♦t❤❡r ❝❧❛ss✐❝❛❧ ❛♥❞ ♣r♦❜❛❜❧② t❤❡ ✜rst ❝♦♠♣r❡❤❡♥s✐✈❡ ❜♦♦❦❀ t❤✐r❞✱ ❜② ❆♥❞❡rs♦♥ ✭✶✾✾✺✮✱ t❤❡ ❜♦♦❦ ✇✐t❤ ♣r♦❜❛❜❧② t❤❡ ❜❡st ✇r✐t✐♥❣ st②❧❡❀ ❢♦✉rt❤✱ ❜② ❱❡rst❡❡❣ ❛♥❞ ▼❛❧❛❧❛s❡❦❡r❛ ✭✶✾✾✺✮✱ ❛♥ ❡①❝❡❧❧❡♥t ❜♦♦❦s ❢♦r ❜❡❣✐♥♥❡rs❀ ✜❢t❤✱ ❜② ❋❡r③✐❣❡r ❛♥❞ P❡r✐❝ ✭✷✵✵✷✮✱ ♣r♦❜❛❜❧② t❤❡ ♠♦st ❝♦♠♣r❡❤❡♥s✐✈❡ ❜♦♦❦❀ s✐①t❤✱ ❡❞✐t❡❞ ❜② ▼✉✲ r❛❧✐❞❤❛r ❛♥❞ ❙✉♥❞❛r❛r❛❥❛♥ ✭✷✵✵✸✮✱ ❛♥♦t❤❡r ❝♦♠♣r❡❤❡♥s✐✈❡ ❜♦♦❦ ✇✐t❤ s✉✣❝✐❡♥t ❞❡t❛✐❧s ❢♦r ❛ ❜❡❣✐♥♥❡r❀ ❛♥❞ s❡✈❡♥t❤✱ ❜② ❉❛t❡ ✭✷✵✵✺✮✱ ❛♥ ♦✉tst❛♥❞✐♥❣ ❜♦♦❦ ❛♥❞ ♣r♦❜❛❜❧② ✇✐t❤ t❤❡ ❧❛r❣❡st ♥✉♠❜❡r ♦❢ ❡①❡r❝✐s❡ ♣r♦❜❧❡♠s✳ ❚❤❡ ❛✉t❤♦r ♦❢ t❤✐s ❜♦♦❦ ✐s ❤✐❣❤❧② ✐♥s♣✐r❡❞ ❜② t❤❡ ✜rst✱ t❤✐r❞ ❛♥❞ ✜❢t❤ ❜♦♦❦❀ ❛♥❞ ❤❛s ❝♦♥tr✐❜✉t❡❞ ❛ ❝❤❛♣t❡r ♦♥ t❤❡ ❋❱▼ ✭❙❤❛r♠❛ ❛♥❞ ❊s✇❛r❛♥✱ ✷✵✵✸✮ ✐♥ ❛ r❡✈✐s❡❞ ❡❞✐t✐♦♥ ♦❢ t❤❡ s✐①t❤ ❜♦♦❦✱ ✉s✐♥❣ t❤❡ ♠❛t❤❡♠❛t✐❝s ❜❛s❡❞ ❛♣♣r♦❛❝❤ ♦❢ ❈❋❉ ❞❡✈❡❧♦♣♠❡♥t ✭❋✐❣✳ ✶✳✹✮✳ ❚❤❡ s❝♦♣❡ ♦❢ t❤✐s ❜♦♦❦ ✐s ♠❛✐♥❧② ♦♥ ❈❋❉ ❞❡✈❡❧♦♣♠❡♥t t❤❛♥ ♦♥ ❈❋❉ ❛♣♣❧✐❝❛t✐♦♥s ❛♥❞ ❛♥❛❧②s✐s❀ ❤♦✇❡✈❡r✱ t❤❡ ❛♣♣❧✐❝❛t✐♦♥s ❛♥❞ ❛♥❛❧✲ ②s✐s ❛r❡ ♣r❡s❡♥t❡❞ ❝♦♥❝✐s❡❧② ✐♥ t❤❡ ❡①❛♠♣❧❡ ❛♥❞ ❡①❡r❝✐s❡ ♣r♦❜❧❡♠s✳ ❚❤✐s ✐s ♠♦t✐✈❛t❡❞ ❜② t❤❡ ❢❛❝t t❤❛t ✐t ✐s ♠♦r❡ ❝❤❛❧❧❡♥❣✐♥❣ ❛♥❞ s❛t✐s❢②✲ ✐♥❣ t♦ ❧❡❛r♥ t♦ ❞❡✈❡❧♦♣ t❤❛♥ t♦ ✉s❡ ❛ ♣r♦❞✉❝t✳ ❙✐♥❝❡ t❤❡ ♣r❡s❡♥t ❜♦♦❦ ✐s ❢♦r ❛♥ ✐♥tr♦❞✉❝t♦r② ❝♦✉rs❡ ♦♥ ❈❋❉✱ t❤❡ ❈❋❉ ❞❡✈❡❧♦♣♠❡♥t ✐s ❧✐♠✲ ✐t❡❞ t♦ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ❢♦r ❛ ❈❛rt❡s✐❛♥ ❛s ✇❡❧❧ ❛s ❝♦♠♣❧❡① ❣❡♦♠❡tr②❀ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❣❡♦♠❡tr✐❡s ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ♥❡①t ❝❤❛♣t❡r✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ ❜♦♦❦ ✐s ♠♦st❧② ♦♥ ♥✉♠❡r✐❝❛❧✲♠❡t❤♦❞ ❛s ❝♦♠♣❛r❡❞ t♦ ♥✉♠❡r✐❝❛❧ ❛♥❛❧②s✐s ❛s♣❡❝t ♦❢ ❈❋❉✳ ❚❤❡ ♣r❡s❡♥t ❜♦♦❦ ✐s ♦♥ ✐♥❝♦♠♣r❡ss✐❜❧❡✲✢♦✇✱ ❝♦♠♠♦♥❧② ❡♥❝♦✉♥t❡r❡❞ ✐♥ t❤❡ ❈❤❡♠✐❝❛❧ ❛♥❞ ▼❡❝❤❛♥✐❝❛❧ ❊♥❣✐♥❡❡r✐♥❣❀ ❤♦✇❡✈❡r✱ ✐t ❝❛♥ ❜❡ ❡①✲ t❡♥❞❡❞ t♦ t❤❡ ❝♦♠♣r❡ss✐❜❧❡ ✢♦✇ ❢♦r t❤❡ ❆❡r♦s♣❛❝❡ ❊♥❣✐♥❡❡r✐♥❣✳ ❋✉r✲ t❤❡r♠♦r❡✱ t❤❡ ♣r❡s❡♥t ❜♦♦❦ ❞❡❛❧s ✇✐t❤ ◆❡✇t♦♥✐❛♥✲✢✉✐❞s✱ ❝♦♠♠♦♥❧② ❡♥❝♦✉♥t❡r❡❞ ✐♥ ▼❡❝❤❛♥✐❝❛❧ ❛♥❞ ❆❡r♦s♣❛❝❡ ❡♥❣✐♥❡❡r✐♥❣❀ ❤♦✇❡✈❡r✱ ✐t ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ♥♦♥✲◆❡✇t♦♥✐❛♥ ✢✉✐❞s ❢♦r t❤❡ ❈❤❡♠✐❝❛❧ ❊♥❣✐♥❡❡r✲ ✐♥❣✳ ❋❱▼ ✐s t❤❡ ❞✐s❝r❡t✐③❛t✐♦♥ ♠❡t❤♦❞ ✇❤✐❝❤ ✐s ✉s❡❞ ✐♥ t❤❡ ♠♦st ♦❢ t❤❡ ♣r❡s❡♥t ❞❛② ❈❋❉ s♦❢t✇❛r❡❀ t❤✉s✱ t❤❡ ♣r❡s❡♥t ❜♦♦❦ ✐s ♠❛✐♥❧② ♦♥ ❋❱▼✳ ❍♦✇❡✈❡r✱ ❋✐♥✐t❡ ❉✐✛❡r❡♥❝❡ ▼❡t❤♦❞ ✭❋❉▼✮ ✐s ❛❧s♦ ♣r❡s❡♥t❡❞ ❤❡r❡ ❛s ❛ t♦♣✐❝ ✐♥ t❤❡ ❝❤❛♣t❡r ♦♥ ❡ss❡♥t✐❛❧s ♦❢ ♥✉♠❡r✐❝❛❧ ♠❡t❤♦❞s✳ ❍❡r❡✱ t❤❡ ♥✉♠❡r✐❝❛❧ ♠❡t❤♦❞♦❧♦❣② ✐s ♣r❡s❡♥t❡❞ ♠♦st❧② ❢♦r ✷❉ ♣r♦❜✲ ❧❡♠s ✭✇❤✐❝❤ ❝❛♥ ❜❡ ❡❛s✐❧② ❡①t❡♥❞❡❞ t♦ ✸❉✮ ❛♥❞ ❢♦r ✉♥st❡❛❞② ✢♦✇ ❛♥❞ ❤❡❛t✲tr❛♥s❢❡r✳ ❚❤❡ ♣r❡s❡♥t s✉❜❥❡❝t ✐♥✈♦❧✈❡s ❧❡❛r♥✐♥❣ t♦ ❞❡✈❡❧♦♣ ❛♥❞ ♣r♦♣❡r❧② ✉s❡ ❛ ♣r♦❞✉❝t − ❛ ❈❋❉ s♦❢t✇❛r❡✳ ❚❤❡ ❞❡t❛✐❧s ❢♦r t❤❡ ❈❋❉ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈❋❉ ✶✾ ❋✐❣✉r❡ ✶✳✺✿ ▼✐♥❞ ♠❛♣ ❢♦r t❤❡ ♣r❡s❡♥t ❜♦♦❦ ❞❡✈❡❧♦♣♠❡♥t ♣r❡s❡♥t❡❞ ❤❡r❡ ✐s t❤❡ ♠❛✐♥ str✉❝t✉r❡ ♦❢ t❤❡ ❈❋❉ s♦❢t✲ ✇❛r❡❀ ❤♦✇❡✈❡r✱ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ♠❡t❤♦❞♦❧♦❣② ❢♦r ❝❡rt❛✐♥ ✜♥✐s❤✐♥❣ t♦✉❝❤❡s ✐♥ t❤❡ s♦❢t✇❛r❡ ✭s✉❝❤ ❛s ❣r❛♣❤✐❝❛❧ ✉s❡r ✐♥t❡r❢❛❝❡ ❞❡✈❡❧♦♣♠❡♥t ❢♦r ✉s❡r ✐♥♣✉t ❛♥❞ ♦✉t♣✉t✮ ❛r❡ ♥♦t ✐♥❝❧✉❞❡❞ ❤❡r❡✳ ❚❤✉s✱ ❛ r❡❛❞❡r s❤♦✉❧❞ ❜❡ ❛❜❧❡ t♦ ❞❡✈❡❧♦♣ ❈❋❉ ❝♦❞❡s ✇❤✐❝❤ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ❛ ❈❋❉ s♦❢t✇❛r❡✳ ❆ ♠✐♥❞✲♠❛♣ ❢♦r t❤❡ ♠❛t❡r✐❛❧ ❝♦✈❡r❡❞ ✐♥ t❤✐s ❜♦♦❦ ✐s ❣✐✈❡♥ ✐♥ ❋✐❣✳ ✶✳✺❀ ✐♥s♣✐r❡❞ ❜② ❆♥❞❡rs♦♥ ✭✶✾✾✺✮✳ ❚❤❡ ♦❜❥❡❝t✐✈❡ ♦❢ t❤✐s ♠✐♥❞ ♠❛♣ ✐s t♦ s❤♦✇ ②♦✉ ❛ ❜✐❣❣❡r ♣✐❝t✉r❡ ♦♥ ❤♦✇ t❤❡ ♠❛t❡r✐❛❧ ✢♦✇s ✐♥ ❛ ❧♦❣✐❝❛❧ ❢❛s❤✐♦♥✱ ❜❡❢♦r❡ ❣♦✐♥❣ ♦♥ t♦ t❤❡ ❞❡t❛✐❧s ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❤❛♣t❡rs✳ ❋✉rt❤❡r♠♦r❡✱ ❧♦❝❛❧✐③❡❞ ♠✐♥❞ ♠❛♣ ✐s ✐♥❝❧✉❞❡❞ ✐♥ ❛❧❧ t❤❡ ❝❤❛♣t❡rs ❛❢t❡r t❤❡ ✜rst t✇♦ ✐♥tr♦❞✉❝t♦r② ❝❤❛♣t❡rs − t♦ ❞❡♠♦♥str❛t❡ t❤❡ ✢♦✇ ♦❢ ✐❞❡❛s ✐♥ ❡❛❝❤ ❝❤❛♣t❡r✳ ❋✐❣✉r❡ ✶✳✺ s❤♦✇s t❤❛t t❤❡ ❜♦♦❦ st❛rts ✇✐t❤ P❛rt ■ ♦♥ ■♥tr♦❞✉❝t✐♦♥ ❛♥❞ ❊ss❡♥t✐❛❧s✱ ❞✐s❝✉ss❡❞ ✐♥ t❤❡ ♣r❡s❡♥t ❛♥❞ t❤❡ ♥❡①t t❤r❡❡ ❝❤❛♣t❡rs✳ ❚❤❡ ❣❡♥❡r❛❧ ✐♥tr♦❞✉❝t✐♦♥ ♦❢ ❈❋❉ ✐s ♣r❡s❡♥t❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r✱ ❛♥❞ t❤❡ ♥❡①t ❝❤❛♣t❡r ♣r❡s❡♥ts ❛♥ ♦✈❡r✈✐❡✇ ♦❢ ❈❋❉ ❞❡✈❡❧♦♣♠❡♥t✱ ❛♣♣❧✐❝❛t✐♦♥✱ ❛♥❞ ❛♥❛❧②s✐s✳ ❚❤❡r❡❛❢t❡r✱ t❤❡ t✇♦ ❝❤❛♣t❡rs ♣r❡s❡♥ts t❤❡ ❛❜s♦❧✉t❡❧② ❡ss❡♥t✐❛❧s ♦❢ t✇♦ ♣r❡r❡q✉✐s✐t❡ s✉❜❥❡❝ts ✭❢♦r ❛ ❝♦✉rs❡ ♦♥ ❈❋❉✮✿ ✢✉✐❞✲❞②♥❛♠✐❝s ❛♥❞ ❤❡❛t✲tr❛♥s❢❡r ✐♥ ❈❤❛♣t❡r ✸✱ ❛♥❞ ✷✵ ✶✳ ■♥tr♦❞✉❝t✐♦♥ ♥✉♠❡r✐❝❛❧✲♠❡t❤♦❞s ✐♥ ❈❤❛♣t❡r ✹✳ ❆❢t❡r ❜✉✐❧❞✐♥❣ t❤❡ ❢♦✉♥❞❛t✐♦♥ ♦❢ t❤❡ ❝♦✉rs❡ ✐♥ P❛rt ■✱ ❋✐❣✳ ✶✳✺ s❤♦✇s t❤❛t t❤❡ ❜♦♦❦ st❛rts ✇✐t❤ t❤❡ P❛rt ■■ ♦♥ ❈❋❉ ❢♦r ❛ ❈❛rt❡s✐❛♥ ●❡♦♠❡tr②✱ ❛♥❞ ❡♥❞s ✇✐t❤ P❛rt ■■■ ♦♥ ❈❋❉ ❢♦r ❛ ❈♦♠♣❧❡① ●❡♦♠❡tr②✳ ❋♦r t❤❡ P❛rt ■■✱ ✐t ✐s ❢❡❧t ❜❡tt❡r t♦ ✐♥tr♦❞✉❝❡ t❤❡ ❈❋❉ ❞❡✈❡❧♦♣♠❡♥t ✇✐t❤ t❤❡ ✢✉① ❜❛s❡❞ s♦❧✉t✐♦♥✲♠❡t❤♦❞♦❧♦❣② ✭❢♦r ❛ ❜❡tt❡r ❛♣♣r❡❝✐❛t✐♦♥ ♦❢ t❤❡ ♣❤②s✐❝❛❧ ❧❛✇ ❜❛s❡❞ ❋❱▼✮✱ ❢♦❧❧♦✇❡❞ ❜② t❤❡ ❝♦❡✣❝✐❡♥t ♦❢ ▲❆❊s ❜❛s❡❞ ♠❡t❤♦❞♦❧♦❣② ✭❋✐❣✳ ✶✳✹✮✳ ❚❤❡ P❛rt ■■ ♣r❡s❡♥ts ❈❋❉ ❞❡✈❡❧♦♣♠❡♥t ❢♦r ❡①♣❧✐❝✐t ❛s ✇❡❧❧ ❛s ✐♠♣❧✐❝✐t ♠❡t❤♦❞✱ ✉♥✐❢♦r♠ ❛s ✇❡❧❧ ❛s ♥♦♥✲✉♥✐❢♦r♠ ❣r✐❞✱ ❛♥❞ st❛❣❣❡r❡❞ ❛s ✇❡❧❧ ❛s ❝♦✲❧♦❝❛t❡❞ ❣r✐❞✳ ❆♥② ❞❡✈❡❧♦♣♠❡♥t ❛❝t✐✈✐t② ✐♥✈♦❧✈❡s ❧❡❛r♥✐♥❣ t♦ ❞❡✈❡❧♦♣ ❛s ✇❡❧❧ ❛s t❡st t❤❡ ❞✐✛❡r❡♥t ❝♦♠♣♦♥❡♥ts✱ ❜❡❢♦r❡ t❤❡✐r ❛ss❡♠❜❧② t♦ ♦❜t❛✐♥ t❤❡ ✜♥❛❧ ♣r♦❞✉❝t✳ ❆s t❤❡ ♣r♦❞✉❝t ❡♥✈✐s❛❣❡❞ ❤❡r❡ ✐s ❛ ❈❋❉ s♦❢t✇❛r❡✱ t❤✐s ❜♦♦❦ ✐s str✉❝t✉r❡❞ ✇✐t❤ ❛ ♠♦❞✉❧❡✲❜②✲♠♦❞✉❧❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ t❤❡ ♥✉♠❡r✐✲ ❝❛❧ ❢♦r♠✉❧❛t✐♦♥✳ ■t ♣r❡s❡♥ts t❤❡ ♠❡t❤♦❞♦❧♦❣② ♦❢ ❞❡✈❡❧♦♣✐♥❣ ❛s ✇❡❧❧ ❛s t❡st✐♥❣ t❤❡ ❞✐✛❡r❡♥t ❝♦♠♣♦♥❡♥ts✴♠♦❞✉❧❡s✴s♦❧✈❡rs ♦❢ t❤❡ s♦❢t✇❛r❡✿ ❞✐❢✲ ❢✉s✐♦♥ ❛♥❞ ❛❞✈❡❝t✐♦♥✱ ❛♥❞ t❤❡✐r ❝♦♠❜✐♥❛t✐♦♥ ❛s ❝♦♥✈❡❝t✐♦♥❀ ♣r❡s❡♥t❡❞ ❛s s❡♣❛r❛t❡ ❝❤❛♣t❡rs ♦❢ P❛rt ■■✳ ❚❤✉s✱ ❋✐❣✳ ✶✳✺ s❤♦✇s t❤❛t P❛rt ■■ ♦❢ t❤✐s ❜♦♦❦ ❝♦♥s✐sts ♦❢ ❝❤❛♣t❡rs ✇✐t❤ t❤❡ t✐t❧❡ ❛s ❝♦♠♣✉t❛t✐♦♥❛❧ ❤❡❛t ❝♦♥❞✉❝t✐♦♥ ✭❢♦r ❞✐✛✉s✐♦♥✮✱ ❝♦♠♣✉t❛t✐♦♥❛❧ ❤❡❛t ❛❞✈❡❝t✐♦♥✱ ❝♦♠♣✉t❛✲ t✐♦♥❛❧ ❤❡❛t ❝♦♥✈❡❝t✐♦♥✱ ❛♥❞ ❝♦♠♣✉t❛t✐♦♥❛❧ ✢✉✐❞✲❞②♥❛♠✐❝s✳ ❚❤❡ ❝♦♠✲ ♣✉t❛t✐♦♥❛❧ ✢✉✐❞✲❞②♥❛♠✐❝s ❝♦rr❡s♣♦♥❞s t♦ ◆❛✈✐❡r✲❙t♦❦❡s s♦❧✈❡r ❛s t❤❡ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ ❞✐✛✉s✐♦♥ ❛♥❞ ❛❞✈❡❝t✐♦♥ s♦❧✈❡r✱ ❛❧♦♥❣ ✇✐t❤ ❛ s♦✉r❝❡ t❡r♠✳ ❚❤❡ ♣❤②s✐❝❛❧ ❡①❛♠♣❧❡ ♦❢ t❤❡ ✐♥❞✐✈✐❞✉❛❧ s♦❧✈❡r ✐s ♠♦r❡ ❝♦♠✲ ♠♦♥❧② ❡♥❝♦✉♥t❡r❡❞ ✐♥ ❤❡❛t✲tr❛♥s❢❡r ❜✉t ♥♦t ✐♥ ✢✉✐❞✲❞②♥❛♠✐❝s✳ ❚❤✉s✱ t❤❡ ❞✐✛✉s✐♦♥✱ ❛❞✈❡❝t✐♦♥ ❛♥❞ ❝♦♥✈❡❝t✐♦♥ s♦❧✈❡r ✭♣r❡s❡♥t❡❞ ✐♥ ❈❤❛♣t❡r ✺✲✼ ♦❢ P❛rt ■■✮ ❛r❡ ❝♦♥s✐❞❡r❡❞ ❤❡r❡ ❢♦r ❤❡❛t✲tr❛♥s❢❡r❀ ❛❧t❤♦✉❣❤ t❤❡② ❛r❡ ❛❧s♦ ❡♥❝♦✉♥t❡r❡❞ ✭✐♥ ❝♦♠❜✐♥❛t✐♦♥✮ ✐♥ ✢✉✐❞✲❞②♥❛♠✐❝s✳ P❛rt ■■ ✐s ♦♥ ✉♥✐❢♦r♠ ❛s ✇❡❧❧ ❛s ♥♦♥✲✉♥✐❢♦r♠ ❈❛rt❡s✐❛♥ ❣r✐❞✱ ❛♥❞ P❛rt ■■■ ✐s ♦♥ t❤❡ ❝✉r✈✐❧✐♥❡❛r ❣r✐❞❀ ✇✐t❤ t❤❡ ❛♣♣r♦♣r✐❛t❡ ❣r✐❞ ❣❡♥❡r❛t✐♦♥ ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ✜rst ❝❤❛♣t❡r ♦❢ P❛rt ■■ ❛♥❞ ■■■✳ P❛rt ■■■ ♦♥ ❈❋❉ ❢♦r ❝♦♠♣❧❡① ❣❡♦♠❡tr② st❛rts ✇✐t❤ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❤❡❛t ❝♦♥❞✉❝t✐♦♥ ❛♥❞ ❡♥❞s ✇✐t❤ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ✢✉✐❞✲❞②♥❛♠✐❝s✳ ❋♦r ❈❋❉ ❞❡✈❡❧♦♣♠❡♥t✱ t❤❡r❡ ❛r❡ ✜✈❡ ♠❛✐♥ st❡♣s✿ ✜rst✱ ❣r✐❞✲ ❣❡♥❡r❛t✐♦♥❀ s❡❝♦♥❞✱ ✜♥✐t❡ ✈♦❧✉♠❡ ♠❡t❤♦❞❀ t❤✐r❞✱ s♦❧✉t✐♦♥ ♠❡t❤♦❞✲ ♦❧♦❣②❀ ❢♦✉rt❤✱ ❝♦❞❡ ❞❡✈❡❧♦♣♠❡♥t❀ ❛♥❞ ✜❢t❤✱ t❡st✐♥❣✳ ❚❤❡ ✜rst ❝❤❛♣t❡r ♦❢ P❛rt ■■ ❛s ✇❡❧❧ ❛s P❛rt ■■■ ♦❢ t❤✐s ❜♦♦❦ ♣r❡s❡♥ts t❤❡ ✜rst st❡♣✳ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈❋❉ ✷✶ ❚❤❡ r❡♠❛✐♥✐♥❣ st❡♣s ❛r❡ ❞✐s❝✉ss❡❞ ✐♥ t❤❡ s❡♣❛r❛t❡ s❡❝t✐♦♥s ❛♥❞ t❤❡ ❡①❛♠♣❧❡ ♣r♦❜❧❡♠s ♦❢ t❤❡ ✈❛r✐♦✉s ❝❤❛♣t❡rs ♦❢ P❛rt ■■ ❛♥❞ ■■■✳ ❚❤❡ ❝♦♠♣✉t❡r✲♣r♦❣r❛♠s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❢♦✉rt❤ st❡♣ ✐s ♣r❡s❡♥t❡❞ ❤❡r❡ ❢♦r t❤❡ ❜❛s✐❝ ♠♦❞✉❧❡s ♦♥ t❤❡ ❤❡❛t ❝♦♥❞✉❝t✐♦♥✱ ❛❞✈❡❝t✐♦♥✱ ❛♥❞ ❝♦♥✲ ✈❡❝t✐♦♥ ✐♥ ❈❤❛♣t❡r ✺✲✼❀ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ t❤❡ ✢✉✐❞✲❞②♥❛♠✐❝s✳ ❚❤❡ ♣r♦❣r❛♠s ❛r❡ ✇r✐tt❡♥ ✉s✐♥❣ ❙❝✐❧❛❜ − ❛♥ ♦♣❡♥✲s♦✉r❝❡ s♦❢t✇❛r❡ ❢♦r ♥✉✲ ♠❡r✐❝❛❧ ❝♦♠♣✉t❛t✐♦♥s✱ ✇✐t❤ ❛♥ ✐♥✲❜✉✐❧t ❣r❛♣❤✐❝❛❧ ✉s❡r ✐♥t❡r❢❛❝❡✳ ❈❋❉ ❛♣♣❧✐❝❛t✐♦♥ ❛♥❞ ❛♥❛❧②s✐s ❛r❡ ♣r❡s❡♥t❡❞ ❜② ❝❡rt❛✐♥ ❝❛r❡❢✉❧❧② ❞❡s✐❣♥❡❞ ❡①❛♠♣❧❡ ❛♥❞ ❡①❡r❝✐s❡ ♣r♦❜❧❡♠s✳ ❲✐t❤ t❤✐s t❤r❡❡✲♣r♦♥❣❡❞ ❛♣♣r♦❛❝❤ ♦❢ ❞❡✈❡❧♦♣♠❡♥t✱ ❛♣♣❧✐❝❛t✐♦♥✱ ❛♥❞ ❛♥❛❧②s✐s✱ t❤❡ ♣✉r♣♦s❡ ♦❢ t❤✐s ❜♦♦❦ ✐s t❤❛t t❤❡ r❡❛❞❡r ✇✐❧❧ ❧❡❛r♥ t♦ ❛♣♣r❡❝✐❛t❡ t❤❡ t❤❡♦r② ❛s ✇❡❧❧ ❛s ✉t✐❧✐③❡ ✐t ❢♦r t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❛ ♥❡✇ ❈❋❉ s♦❢t✇❛r❡✱ ♦r ❛♥ ❡①✐st✐♥❣ ♦♥❡ ❝❛♥ ❜❡ ✉s❡❞ ✐♥t❡❧❧✐❣❡♥t❧② ❢♦r t❤❡ ❈❋❉ ❛♣♣❧✐❝❛t✐♦♥ ❛♥❞ ❛♥❛❧②s✐s✳ ❚❤✉s✱ ❛ r❡❛❞❡r ✇✐❧❧ ❜❡ ✜r♠❧② s❡t ♦♥ t❤❡ ♣❛t❤ ♦❢ ❜❡❝♦♠✐♥❣ ❛ ❈❋❉ ❊①♣❡rt✳ ✷ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈❋❉✿ ❉❡✈❡❧♦♣♠❡♥t✱ ❆♣♣❧✐❝❛t✐♦♥✱ ❛♥❞ ❆♥❛❧②s✐s ❈♦♠♣✉t❛t✐♦♥❛❧ ✢✉✐❞ ❞②♥❛♠✐❝s ✐♥✈♦❧✈❡s ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❛ s♦❢t✇❛r❡✱ ✐ts ❛♣♣❧✐❝❛t✐♦♥ ❢♦r ❛ ✢✉✐❞✲❞②♥❛♠✐❝s ♣r♦❜❧❡♠ ✭t♦ ♦❜t❛✐♥ t❤❡ s❝✐❡♥t✐✜❝❛❧❧②✲ ❡①❝✐t✐♥❣ ❛s ✇❡❧❧ ❛s ❡♥❣✐♥❡❡r✐♥❣✲r❡❧❡✈❛♥t r❡s✉❧ts✮✱ ❛♥❞ ❛♥❛❧②s✐s ♦❢ t❤❡ r❡s✉❧ts ✭❢♦r ❛ ✉♥✐✜❡❞ ❝❛✉s❡✲❛♥❞✲❡✛❡❝t st✉❞②✮✳ ❆ ❞❡t❛✐❧❡❞ ♦✈❡r✈✐❡✇ ♦♥ t❤❡ ❞❡✈❡❧♦♣♠❡♥t✱ ❛♣♣❧✐❝❛t✐♦♥✱ ❛♥❞ ❛♥❛❧②s✐s ❛s♣❡❝ts ♦❢ ❈❋❉ ✐s ♣r❡✲ s❡♥t❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r✱ t♦ ❛♥s✇❡r ❤♦✇ ❈❋❉ ✇♦r❦s❄ ✷✳✶ ❈❋❉ ❉❡✈❡❧♦♣♠❡♥t ❈❋❉ ❞❡✈❡❧♦♣♠❡♥t ✐♥✈♦❧✈❡s ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❝♦♠♣✉t❡r✲♣r♦❣r❛♠s ✭❛❧s♦ ❝❛❧❧❡❞ ❛s ❝♦❞❡s ✇❤✐❝❤ ❧❡❛❞ t♦ ❛ s♦❢t✇❛r❡✮ ❛s ✇❡❧❧ ❛s t❡st✐♥❣✳ ❚❤❡ ❞❡✲ ✈❡❧♦♣♠❡♥t ❢♦❧❧♦✇s ❛ ♥✉♠❡r✐❝❛❧ ♠❡t❤♦❞♦❧♦❣② ✇❤✐❝❤ ❝♦♥s✐sts ♦❢ ✜✈❡ ♠❛✐♥ st❡♣s✿ ✜rst✱ ❣r✐❞✲❣❡♥❡r❛t✐♦♥❀ t❤✐r❞✱ s♦❧✉t✐♦♥✲♠❡t❤♦❞♦❧♦❣②❀ s❡❝♦♥❞✱ ❞✐s❝r❡t✐③❛t✐♦♥✲♠❡t❤♦❞❀ ❢♦✉rt❤✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❡♥❣✐♥❡❡r✐♥❣✲ ♣❛r❛♠❡t❡rs❀ ❛♥❞ ✜❢t❤✱ t❡st✐♥❣✳ ❚❤❡② ❛r❡ s❤♦✇♥ s❝❤❡♠❛t✐❝❛❧❧② ✐♥ ❋✐❣✳ ✷✳✶ ❛♥❞ ♣r❡s❡♥t❡❞ ✐♥ s❡♣❛r❛t❡ s✉❜s❡❝t✐♦♥s ❜❡❧♦✇✳ ❈❋❉ ❞❡✈❡❧♦♣♠❡♥t ❤❛s ❜❡❡♥ ♠❛✐♥❧② ❝♦♥tr✐❜✉t❡❞ ❜② t✇♦ ❞✐✛❡r❡♥t t②♣❡s ♦❢ ♣❡♦♣❧❡✿ ✢✉✐❞✲❞②♥❛♠✐❝✐sts ❛♥❞ ❝♦♠♣✉t❡r✲s❝✐❡♥t✐sts✳ ❋❧✉✐❞✲ ❞②♥❛♠✐❝✐sts ❞❡✈❡❧♦♣❡❞ ❛ ♥❡✇ ♦r ❛♥ ✐♠♣r♦✈❡❞ ♠❛t❤❡♠❛t✐❝❛❧✲♠♦❞❡❧ ❛♥❞✴♦r ❛ ♥✉♠❡r✐❝❛❧✲♠❡t❤♦❞♦❧♦❣② ✇❤✐❝❤ ❝❛♥ ❣✐✈❡ ❛❝❝✉r❛t❡ r❡s✉❧ts ✐♥ ❛ ❧❡❛st ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦st✳ ❲❤❡r❡❛s✱ ❝♦♠♣✉t❡r✲s❝✐❡♥t✐sts ❞❡✈❡❧✲ ♦♣❡❞ ❛ ♥❡✇ ✭♦r ❛♥ ✐♠♣r♦✈❡❞✮ ❝♦♠♣✉t❡r✲s②st❡♠ ❛♥❞✴♦r ♣r♦❣r❛♠♠✐♥❣✲ ♣❛r❛❞✐❣♠ t♦ r❡❞✉❝❡ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦st✳ ❚❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ✷✸ Introduction to Computational Fluid Dynamics: Development, Application and Analysis, First Edition.

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