Kelvin's circulation theorem

Kelvin's circulation theorem In fluid mechanics, Kelvin's circulation theorem (named after William Thomson, 1st Baron Kelvin who published it in 1869) states In a barotropic ideal fluid with conservative body forces, the circulation around a closed curve (which encloses the same fluid elements) moving with the fluid remains constant with time.[1][2] Stated mathematically: {stile di visualizzazione {frac {matematica {D} Gamma }{matematica {D} t}}=0} dove {stile di visualizzazione Gamma } is the circulation around a material contour {stile di visualizzazione C(t)} . Stated more simply this theorem says that if one observes a closed contour at one instant, and follows the contour over time (by following the motion of all of its fluid elements), the circulation over the two locations of this contour are equal.

This theorem does not hold in cases with viscous stresses, nonconservative body forces (for example a coriolis force) or non-barotropic pressure-density relations.

Contenuti 1 Mathematical proof 2 Poincaré–Bjerknes circulation theorem 3 Guarda anche 4 Notes Mathematical proof See also: Euler equations (fluid dynamics) The circulation {stile di visualizzazione Gamma } around a closed material contour {stile di visualizzazione C(t)} è definito da: {stile di visualizzazione Gamma (t)=unto _{C}{simbolo audace {tu}}cdot mathrm {d} {simbolo audace {S}}} where u is the velocity vector, and ds is an element along the closed contour.

The governing equation for an inviscid fluid with a conservative body force is {stile di visualizzazione {frac {matematica {D} {simbolo audace {tu}}}{matematica {D} t}}=-{frac {1}{rho }}{simbolo audace {nabla }}p+{simbolo audace {nabla }}Phi } where D/Dt is the convective derivative, ρ is the fluid density, p is the pressure and Φ is the potential for the body force. These are the Euler equations with a body force.

The condition of barotropicity implies that the density is a function only of the pressure, cioè. {displaystyle rho =rho (p)} .

Taking the convective derivative of circulation gives {stile di visualizzazione {frac {matematica {D} Gamma }{matematica {D} t}}=unto _{C}{frac {matematica {D} {simbolo audace {tu}}}{matematica {D} t}}cdot mathrm {d} {simbolo audace {S}}+unguento _{C}{simbolo audace {tu}}cdot {frac {matematica {D} matematica {d} {simbolo audace {S}}}{matematica {D} t}}.} For the first term, we substitute from the governing equation, and then apply Stokes' theorem, così: {displaystyle unto_{C}{frac {matematica {D} {simbolo audace {tu}}}{matematica {D} t}}cdot mathrm {d} {simbolo audace {S}}=int _{UN}{simbolo audace {nabla }}times left(-{frac {1}{rho }}{simbolo audace {nabla }}p+{simbolo audace {nabla }}Phi right)cdot {simbolo audace {n}},matematica {d} S=int _{UN}{frac {1}{ro ^{2}}}sinistra({simbolo audace {nabla }}rho times {simbolo audace {nabla }}pright)cdot {simbolo audace {n}},matematica {d} S=0.} The final equality arises since {stile di visualizzazione {simbolo audace {nabla }}rho times {simbolo audace {nabla }}p=0} owing to barotropicity. We have also made use of the fact that the curl of any gradient is necessarily 0, o {stile di visualizzazione {simbolo audace {nabla }}volte {simbolo audace {nabla }}f=0} for any function {stile di visualizzazione f} .

For the second term, we note that evolution of the material line element is given by {stile di visualizzazione {frac {matematica {D} matematica {d} {simbolo audace {S}}}{matematica {D} t}}= sinistra(matematica {d} {simbolo audace {S}}cdot {simbolo audace {nabla }}Giusto){simbolo audace {tu}}.} Quindi {displaystyle unto_{C}{simbolo audace {tu}}cdot {frac {matematica {D} matematica {d} {simbolo audace {S}}}{matematica {D} t}}=unto _{C}{simbolo audace {tu}}cdot a sinistra(matematica {d} {simbolo audace {S}}cdot {simbolo audace {nabla }}Giusto){simbolo audace {tu}}={frac {1}{2}}unguento _{C}{simbolo audace {nabla }}sinistra(|{simbolo audace {tu}}|^{2}Giusto)cdot mathrm {d} {simbolo audace {S}}=0.} The last equality is obtained by applying gradient theorem.

Since both terms are zero, we obtain the result {stile di visualizzazione {frac {matematica {D} Gamma }{matematica {D} t}}=0.} Poincaré–Bjerknes circulation theorem A similar principle which conserves a quantity can be obtained for the rotating frame also, known as the Poincaré–Bjerknes theorem, named after Henri Poincaré and Vilhelm Bjerknes, who derived the invariant in 1893[3][4] e 1898.[5][6] The theorem can be applied to a rotating frame which is rotating at a constant angular velocity given by the vector {stile di visualizzazione {simbolo audace {Omega }}} , for the modified circulation {stile di visualizzazione Gamma (t)=unto _{C}({simbolo audace {tu}}+{simbolo audace {Omega }}volte {simbolo audace {r}})cdot mathrm {d} {simbolo audace {S}}} Qui {stile di visualizzazione {simbolo audace {r}}} is the position of the area of fluid. From Stokes' theorem, this is: {stile di visualizzazione Gamma (t)=int _{UN}{simbolo audace {nabla }}volte ({simbolo audace {tu}}+{simbolo audace {Omega }}volte {simbolo audace {r}})cdot {simbolo audace {n}},matematica {d} S=int _{UN}({simbolo audace {nabla }}volte {simbolo audace {tu}}+2{simbolo audace {Omega }})cdot {simbolo audace {n}},matematica {d} S} The Vorticity of a velocity field in fluid dynamics is defined by: {stile di visualizzazione {simbolo audace {omega }}={simbolo audace {nabla }}volte {simbolo audace {tu}}} Quindi: {stile di visualizzazione Gamma (t)=int _{UN}({simbolo audace {omega }}+2{simbolo audace {Omega }})cdot {simbolo audace {n}},matematica {d} S} See also Bernoulli's principle Euler equations (fluid dynamics) Helmholtz's theorems Thermomagnetic convection Notes ^ Katz, Plotkin: Low-Speed Aerodynamics ^ Kundu, P and Cohen, io: Fluid Mechanics, pagina 130. Stampa accademica 2002 ^ Poincaré, H. (1893). Théorie des tourbillons: Leçons professées pendant le deuxième semestre 1891-92 (vol. 11). Gauthier-Villars. Article 158 ^ Truesdell, C. (2018). The kinematics of vorticity. Pubblicazioni Courier Dover. ^ Bjerknes, V., Rubenson, R., & Lindstedt, UN. (1898). Ueber einen Hydrodynamischen Fundamentalsatz und seine Anwendung: besonders auf die Mechanik der Atmosphäre und des Weltmeeres. Kungl. Boktryckeriet. PA Norstedt & Söner. ^ Chandrasekhar, S. (2013). Hydrodynamic and hydromagnetic stability. Courier Corporation. Categorie: Equations of fluid dynamicsFluid dynamicsEquationsWilliam Thomson, 1st Baron Kelvin