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Kelvin circulation: the circulation around a $C^{1}$ material loop is constant in the barotropic inviscid model

Statement

Let $O \subseteq R^{3}$ be open, let $I \subseteq R$ be a nonempty open interval, let $u : I \times O \to R^{3}$ and $ρ : I \times O \to (0, \infty)$ be $C^{1}$ , let $p : (0, \infty) \to R$ be $C^{1}$ on the positive density interval $(0, \infty)$ , with enthalpy $h (r) : = \int_{1}^{r} p^{'} (s) / s d s$ for $r > 0$ as in Incompressible Newtonian and barotropic inviscid fluid constitutive models, and let $V : I \times O \to R$ be $C^{1}$ , with the barotropic inviscid stress $σ = - p (ρ) I_{3}$ ; assume the local equations (M) of Barotropic inviscid momentum in gradient form, Bernoulli along streamlines and the irrotational constant on $I \times O$ :

$\partial_{t} ρ + div (ρ u) = 0, ρ D_{t} u = ρ (- \nabla V) + g, g_{i} = \sum_{j} \partial_{j} σ_{i j},$

so that, by part (i) of that theorem, $D_{t} u = - \nabla (h (ρ) + V)$ on $I \times O$ .

Let $t_{0} < t_{1}$ with $[t_{0}, t_{1}] \subseteq I$ and let $γ : [t_{0}, t_{1}] \times [0, 1] \to O$ be $C^{1}$ with

$γ (t, 0) = γ (t, 1) and \partial_{t} γ (t, s) = u (t, γ (t, s)) ((t, s) \in [t_{0}, t_{1}] \times [0, 1]) .$

Here $C^{1}$ on the closed parameter rectangle means that $γ$ is continuous and its first partial derivatives exist and are continuous there, with one-sided derivatives on the edges.

Then the circulation

$Γ (t) : = \int_{0}^{1} \sum_{i = 1}^{3} u_{i} (t, γ (t, s)) \partial_{s} γ_{i} (t, s) d s$

is constant on $[t_{0}, t_{1}]$ . The integral is an ordinary Riemann integral of a continuous real function; a singleton time interval has a trivially constant circulation and requires no time differentiation.

Facts & Assumptions

Given: The fields $u, ρ, p, V$ and the local equation; the material loop $γ$ .

[F1]

By Barotropic inviscid momentum in gradient form, Bernoulli along streamlines and the irrotational constant(i), $D_{t} u = - \nabla (h (ρ) + V)$ on $I \times O$ ; the enthalpy satisfies $h \in C^{1}$ .

[F2]

Chain rule for the composite $s \mapsto φ (t, γ (t, s))$ and for $t \mapsto u (t, γ (t, s))$ : $\partial_{s} (u (t, γ)) = (D u) (t, γ) \partial_{s} γ$ , and $\partial_{t} (u (t, γ)) = \partial_{t} u (t, γ) + (D u) (t, γ) \partial_{t} γ$ (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , A total derivative computes every directional derivative, and its matrix is the Jacobian, Euclidean spatial calculus: gradient, divergence and material derivative).

[F3]

The identity $u \cdot (D u v) = \frac{1}{2} \nabla (∣ u ∣_{2}^{2}) \cdot v$ for $C^{1}$ fields: componentwise, $\sum_{i} u_{i} \partial_{j} u_{i} v_{j} = \frac{1}{2} \partial_{j} (\sum_{i} u_{i}^{2}) v_{j}$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , A total derivative computes every directional derivative, and its matrix is the Jacobian, Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

[F4]

If $g$ and $\partial_{t} g$ are continuous on a compact rectangle, FTC gives $h^{- 1} (g (t + h, s) - g (t, s)) = h^{- 1} \int_{t}^{t + h} \partial_{t} g (τ, s) d τ$ . Uniform continuity makes these quotients converge uniformly in $s$ to $\partial_{t} g (t, s)$ ; the uniform integral-error bound therefore gives $\frac{d}{d t} \int g (t, s) d s = \int \partial_{t} g (t, s) d s$ . The same argument with the coordinate roles exchanged permits differentiation in $s$ of $\int_{t_{0}}^{t} U (τ, s) d τ$ when $U$ and $\partial_{s} U$ are continuous; endpoints use one-sided derivatives (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ , A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous, Uniformly close integrable functions have integrals differing by at most the interval length times their uniform error).

[F5]

Fundamental theorem of calculus: $\int_{0}^{1} \partial_{s} F (s) d s = F (1) - F (0)$ for $C^{1}$ $F$ ; and $\int_{0}^{1} \partial_{s} F = 0$ when $F (0) = F (1)$ (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ , The first fundamental theorem: if $f$ is integrable on $[a, b]$ and continuous at $c$ , then $F^{'} (c) = f (c)$ ; in particular a continuous $f$ has $F$ as a primitive, The integral with oriented limits: $\int_{a}^{a} f : = 0$ and $\int_{b}^{a} f : = - \int_{a}^{b} f$ , For $a < c < b$ : $f$ is integrable on $[a, b]$ if and only if it is integrable on $[a, c]$ and on $[c, b]$ , and then $\int_{a}^{b} f = \int_{a}^{c} f + \int_{c}^{b} f$ ; with the oriented form for arbitrary $a, b, c$ ).

Proof

technique · direct; differentiate the parameterized circulation and recognize the integrand as a total $s$-derivative

1.1givenF2F4algebra

Put $U (t, s) : = u (t, γ (t, s))$ . The chain rule makes $U$ a $C^{1}$ map, with $\partial_{t} U = \partial_{t} u + (D u) \partial_{t} γ$ and $\partial_{s} U = (D u) \partial_{s} γ$ . Although $γ$ is only assumed $C^{1}$ , its material equation supplies the needed mixed derivative: from $γ (t, s) = γ (t_{0}, s) + \int_{t_{0}}^{t} U (τ, s) d τ$ , differentiating in $s$ gives $\partial_{s} γ (t, s) = \partial_{s} γ (t_{0}, s) + \int_{t_{0}}^{t} \partial_{s} U (τ, s) d τ$ by [F4]. Therefore $\partial_{t} \partial_{s} γ = \partial_{s} U = (D u) \partial_{s} γ$ , continuously. Thus the circulation integrand $U \cdot \partial_{s} γ$ has a continuous time derivative, and [F4] gives $Γ^{'} (t) = \int_{0}^{1} [(\partial_{t} u + (D u) \partial_{t} γ) \cdot \partial_{s} γ + u \cdot (D u) \partial_{s} γ] d s$ for $t$ in the time interior.

2.1step 1.1F2F3algebra

Since $\partial_{t} γ = u (t, γ)$ , the first term in step 1.1 is $(\partial_{t} u + (D u) u) \cdot \partial_{s} γ = (D_{t} u) \cdot \partial_{s} γ$ . The second term is exactly $u \cdot (D u) \partial_{s} γ = \partial_{s} (\frac{1}{2} ∣ u (t, γ) ∣_{2}^{2})$ by the chain rule and [F3]. Hence $Γ^{'} (t) = \int_{0}^{1} [(D_{t} u) \cdot \partial_{s} γ + \partial_{s} (\frac{1}{2} ∣ u ∣_{2}^{2})] d s .$ No equality between $(D u) u$ and $(D u)^{T} u$ , and no symmetry of $D u$ , is used.

3.1step 2.1F1F2F5algebra∎

Substituting [F1], $Γ^{'} (t) = \int_{0}^{1} [- \nabla (h (ρ) + V) \cdot \partial_{s} γ + \partial_{s} \frac{1}{2} ∣ u ∣_{2}^{2}] d s = - \int_{0}^{1} \partial_{s} [(h (ρ) + V) (t, γ (t, s))] d s + [\frac{1}{2} ∣ u ∣_{2}^{2}]_{s = 0}^{s = 1}$ by the chain rule [F2] and the fundamental theorem [F5]. Since $γ (t, 1) = γ (t, 0)$ , both boundary terms vanish, so $Γ^{'} (t) = 0$ for every $t \in (t_{0}, t_{1})$ ; [F4] also gives continuous one-sided endpoint derivatives, so The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ yields $Γ (t) - Γ (t_{0}) = \int_{t_{0}}^{t} Γ^{'} (τ) d τ = 0$ on $[t_{0}, t_{1}]$ .

Remarks

The loop is material: its points move with the fluid velocity, which is the hypothesis $\partial_{t} γ = u (γ)$ . The theorem is conditional on the barotropic inviscid local equation and uses the gradient form $D_{t} u = - \nabla (h + V)$ ; viscosity, non-barotropic pressure or a non-conservative body force invalidate the argument.
The circulation is defined as an oriented line integral with the parameterization $(t, s) \mapsto γ (t, s)$ ; a different parameterization of the same oriented loop gives the same value by Substitution: if $φ$ is differentiable on $[c, d]$ with $φ^{'}$ integrable and $f$ is continuous on an interval containing $φ ([c, d])$ , then $\int_{φ (c)}^{φ (d)} f = \int_{c}^{d} (f \circ φ) φ^{'}$ for a C1 orientation-preserving change of parameter with endpoints fixed. No claim is made about the existence of loops with prescribed circulation, nor about vortex dynamics.

Kelvin circulation: the circulation around a $C^{1}$ material loop is constant in the barotropic inviscid model

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

Kelvin circulation: the circulation around a C1C1 material loop is constant in the barotropic inviscid model

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

Kelvin circulation: the circulation around a $C^{1}$ material loop is constant in the barotropic inviscid model