Kelvin's Circulation Theorem
- Fluid Mechanics 1.6
Vorticity and Circulation
In fluid dynamics, vorticity is a measure of the local rotation of fluid elements. It is defined mathematically as the curl of the velocity field:
\[ \bs{\omega} = \curl \b{v} \]
Circulation, on the other hand, is a measure of the total “twisting” or “rotational” motion of the fluid around a closed loop. It is defined as the line integral of the velocity field around a closed curve $C$:
\[ \Gamma = \oint_C \b{v} \cdot \dd{\b{l}} \]
By the Stokes’ theorem, circulation can also be related to vorticity through the surface integral of vorticity over a surface $S$ bounded by the curve $C$:
\[ \Gamma = \iint_S \bs{\omega} \cdot \dd{\b{S}} \]
Kelvin’s Circulation Theorem
In fluid mechanics, Kelvin’s circulation theorem 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.
Mathematically, this can be expressed as:
\[ \mdv{\Gamma} = 0 \]
Proof
\[ \mdv{\Gamma} = \oint_C \mdv{\b{v}} \cdot \dd{\b{l}} + \oint_C \b{v} \cdot \mdv{\dd{\b{l}}} \]
From the Euler equation for an ideal fluid:
\[ \mdv{\b{v}} = -\frac{1}{\rho} \grad p + \grad \Phi \]
Substituting this,
\[ \begin{align*} \oint_C \mdv{\b{v}} \cdot \dd{\b{l}} &= \oint_C \left( -\frac{1}{\rho} \grad p + \grad \Phi \right) \cdot \dd{\b{l}} \nl &= \iint_S \curl \left( -\frac{1}{\rho} \grad p + \grad \Phi \right) \cdot \dd{\b{S}} \nl &= \iint_S \frac{1}{\rho^2} (\grad \rho \times \grad p) \cdot \dd{\b{S}} \end{align*} \]
since $\curl(\grad f) = 0$ for any scalar function $f$. For a barotropic fluid, $p = p(\rho)$, so $\grad p=\pdv{p}{\rho}\grad \rho$, and thus $\grad \rho \times \grad p = 0$. Therefore, the first term vanishes:
\[ \oint_C \mdv{\b{v}} \cdot \dd{\b{l}} = 0 \]
Now, consider the second term. Let’s parametrize the curve $C$ by a parameter $a$ such that $\b{l} = \b{l}(a, t)$, where $a$ is constant for a given fluid element. Then,
\[ \dd{\b{l}(a,t)} = \b{l}(a + \dd{a}, t) - \b{l}(a, t) = \pdv{\b{l}(a,t)}{a} \dd{a} \]
Taking the material derivative,
\[ \begin{align*} \mdv{\dd{\b{l}}} &= \pdv{\b{v}(\b{l}(a,t), t)}{a} \dd{a} = \pdv{\b{v}}{\b{l}} \cdot \pdv{\b{l}}{a} \dd{a} \nl &= (\dd{\b{l}} \cdot \grad) \b{v} \end{align*} \]
Thus, the second term becomes:
\[ \begin{align*} \oint_C \b{v} \cdot \mdv{\dd{\b{l}}} &= \oint_C \b{v} \cdot (\dd{\b{l}} \cdot \grad) \b{v} \nl &= \frac{1}{2} \oint_C \grad(v^2) \cdot \dd{\b{l}} \nl &= 0 \end{align*} \]
by the gradient theorem. Since both terms vanish, we have:
\[ \mdv{\Gamma} = 0 \]
It, of course, holds for ideal fluids; because the barotropic condition is more general than the isentropic (adiabatic) condition. The conservation of circulation can be intuitively interpreted as meaning that the vorticity moves with the fluid elements.
