משפט הסירקולציה של קלווין

משפט הסירקולציה של קלווין הוא משפט חשוב במכניקת הזורמים הקובע שבזורם ברוטרופי מושלם הסירקולציה מסביב ללולאה הנעה עם הזורם קבועה בזמן. המשפט הוכח בשנת 1869 על ידי לורד קלווין וקרוי על שמו.

ניסוח מתמטי והוכחה

תהי $C$ לולאה, התלויה בזמן וזזה עם הזורם $C(t)$ . הסירקולציה מסביב ללולאה מוגדרת על ידי $\Gamma (t)=\oint _{C(t)}\mathbf {u} (t)\cdot d\ell$ כאשר $\mathbf {u} (t)$ הוא שדה המהירות. משפט הסירקולציה של קלווין קובע כי עבור זורם ברוטרופי מושלם ${\frac {D\Gamma }{Dt}}=0$ , כאשר הנגזרת היא נגזרת חומרית.

ההוכחה מתבססת על חישוב ישיר של הנגזרת:

${\frac {D\Gamma (t)}{Dt}}={\frac {D\oint _{C(t)}\mathbf {u} (t)\cdot d\ell }{Dt}}=\oint _{C}{\frac {D\mathbf {u} }{Dt}}\cdot d\ell +\oint _{C}\mathbf {u} \cdot {\frac {Dd\ell }{Dt}}$

כעת לפי משואת אוילר מתקיים:

${\frac {D\mathbf {u} }{Dt}}=-{\frac {1}{\rho }}\nabla p+\nabla \Phi$

כאשר $p$ הוא הלחץ, $\rho$ הצפיפות ו- $\Phi$ הפוטנציאל של הכוחות החיצוניים (כדוגמת כבידה) ולכן עבור האיבר השמאלי מתקיים:

$\oint _{C}{\frac {D\mathbf {u} }{Dt}}\cdot d\ell =\oint _{C}\left[-{\frac {1}{\rho }}\nabla p+\nabla \Phi \right]\cdot d\ell =\iint _{A}\left[\nabla \times \left(-{\frac {1}{\rho }}\nabla p+\nabla \Phi \right)\right]{\hat {n}}dA=\iint _{A}{\frac {1}{\rho ^{2}}}\left(\nabla \rho \times \nabla p\right){\hat {n}}dA=0$

כאשר המעבר הראשון התבצע לפי משוואת אוילר, השני לפי חוק סטוקס, השלישי נובע מכך שהרוטור של גרדיאנט של שדה מתאפס, והרביעי נובע מברוטרופיות הנוזל (נוזל ברוטרופי הוא נוזל בו $\nabla \rho \times \nabla p=0$ ).

עבור האיבר הימני מתקיים:

$\oint _{C}\mathbf {u} \cdot {\frac {Dd\ell }{Dt}}=\oint _{C}\mathbf {u} \cdot \left(d\ell \cdot \nabla \right)\mathbf {u} ={\frac {1}{2}}\oint _{C}\nabla \left(\mathbf {u} ^{2}\right)d\ell =0$

כאשר המעבר האחרון מתבסס על כך שסירקולציה של שדה משמר מתאפסת על כל לולאה. מאחר ששני האיברים מתאפסים ניתן להסיק כי ${\frac {D\Gamma }{Dt}}=0$ .

משפט פואנקרה-ביירקנס

הרחבה למשפט הסירקולציה עבור מערכות מסתובבות ניתנה על ידי אנרי פואנקרה וווילהלם ביירקנס בסוף המאה ה-19. על-פי משפט פואנקרה-ביירקנס, במערכת המסתובבת במהירות זוויתית קבועה הניתנת על ידי הווקטור $\mathbf {\Omega }$ ניתן להגדיר סירקולציה מוכללת לפי $\Gamma (t)=\oint _{C}\left(\mathbf {u} +\Omega \times \mathbf {r} \right)\cdot d\ell$ . הסירקולציה המוכללת הזו היא גודל משתמר.

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