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Barotropic Kelvin and Bernoulli with exact hypotheses — conditional physical prediction

Physical scope

C² smooth barotropic Euler, positive density and conservative body force; C¹ advected loop with continuous mixed derivatives; steady Bernoulli; connected component global constant only if irrotational. Worked example is the actual explicit field/parameter selection in that section. B leaf cannot supply any later item.

Statement

Let $I \subseteq R$ be a nonempty open time interval measured in seconds, $Ω \subseteq R^{3}$ nonempty and open in metres, and $P : (0, \infty) \to R$ be $C^{2}$ . Use SI units: $u$ in $m s^{- 1}$ , $ρ$ in $k g m^{- 3}$ , $P$ in pascals, and $V, h, B$ in $m^{2} s^{- 2}$ . Let $ρ : I \times Ω \to (0, \infty)$ have continuous first time and spatial derivatives, $u : I \times Ω \to R^{3}$ have continuous first time, first spatial and second spatial derivatives, and $V : Ω \to R$ be a time-independent $C^{2}$ potential per unit mass. Adopt the barotropic Euler equations with conservative body acceleration $- \nabla V$ : $\partial_{t} ρ + div (ρ u) = 0, \partial_{t} u + (u \cdot \nabla) u = - ρ^{- 1} \nabla (P (ρ)) - \nabla V .$ For a fixed $ρ_{r e f} > 0$ , define $h (q) = \int_{ρ_{r e f}}^{q} P^{'} (r) / r d r$ . A steady streamline is a $C^{1}$ curve $x : J \to Ω$ satisfying $x^{'} (s) = u (x (s))$ . If $u, ρ$ are steady, then $B = \frac{∣ u ∣^{2}}{2} + h (ρ) + V$ is constant along each streamline. If additionally $curl u = 0$ on a connected component, then $B$ is constant throughout that component; here $curl$ is the Cartesian curl. For every closed $C^{1}$ material loop $X : J \times [0, 1] \to Ω$ with $X (t, 0) = X (t, 1)$ , $X_{t} = u (t, X)$ , and continuous $X_{s}, X_{t s}$ , the circulation $Γ (t) = \int_{0}^{1} u (t, X (t, s)) \cdot X_{s} (t, s) d s$ is constant. The conservative body acceleration and local barotropic branch are model inputs; no statement concerns shocks, nonbarotropic pressure, nonconservative forcing, or viscous circulation.

In particular, take $P (ρ) = c^{2} ρ$ with $c > 0$ in $m s^{- 1}$ , $V = 0$ , angular speed $Ω_{0} \neq 0$ in $s^{- 1}$ , density $ρ_{c}, ρ_{r e f} > 0$ in $k g m^{- 3}$ , angular and reference lengths $R, H$ in metres, and the open cylinder $D = {(x, y, z) : x^{2} + y^{2} < R^{2}, ∣ z ∣ < H}$ for $R, H > 0$ measured in metres. Set $r = (x^{2} + y^{2})^{1 / 2}$ , $e_{θ} = (- y / r, x / r, 0)$ for $r > 0$ , $u = Ω_{0} (- y, x, 0)$ , and $ρ = ρ_{c} \exp (Ω_{0}^{2} r^{2} / (2 c^{2}))$ for $ρ_{c} > 0$ . These smooth fields satisfy both equations. For any $ℓ \in (0, R)$ and $z_{0} \in (- H, H)$ , the closed material loop $X : R \times [0, 2 π] \to D$ is $X (t, θ) = (ℓ \cos (θ + Ω_{0} t), ℓ \sin (θ + Ω_{0} t), z_{0}), t \in R, 0 \leq θ \leq 2 π .$

It satisfies $X (t, 0) = X (t, 2 π)$ and $\partial_{t} X (t, θ) = u (t, X (t, θ))$ . Its circulation is $Γ (t) = \int_{0}^{2 π} u (t, X (t, θ)) \cdot \partial_{θ} X (t, θ) d θ = 2 π Ω_{0} ℓ^{2} .$

Its Bernoulli function is $B = c^{2} \log (ρ_{c} / ρ_{r e f}) + Ω_{0}^{2} r^{2} .$ It is constant along each circular streamline but varies between radii, since $curl u = (0, 0, 2 Ω_{0}) \neq 0$ . This is a conditional prediction of the stated smooth barotropic model, not an empirical claim about viscous flow.

Statement references

Mandre, Fluid Dynamics Lecture Notes, Chapter 5, §§5.2.1–5.2.2, incompressible Bernoulli context only; the chapter does not treat the barotropic/compressible result proved here

Stewart, MIT 8.09, Fluid Mechanics, §6.4, ideal Bernoulli context, printed pp. 120–123

Facts & Assumptions

Given: The smooth positive-density barotropic model, $P \in C^{2}$ , time-independent potential $V$ , material-loop regularity, and the explicit solid-rotation specialization stated above.

[A1]

Single-component nonpolar continuum and material balances: the continuum model includes a specified body-acceleration input. In the general theorem statement the conservative acceleration $- \nabla V$ is an explicit model hypothesis; it is set to zero in the solid-rotation example.

[A2]

Specified wall, thermal, periodic and initial data: boundary data are separate; no wall data are required for this local steady solution.

[F1]

Smooth positive-density barotropic Euler model: this earlier Definition is the unforced positive-density barotropic branch with a specified $C^{1}$ pressure law; its equations match the $V = 0$ specialization. The general time-independent potential force $- \nabla V$ used here is separately and explicitly assumed in the Statement, not inferred from that unforced Definition. The theorem uses the stronger regularity $P \in C^{2}$ .

[F2]

Barotropic Kelvin and Bernoulli with exact hypotheses: for the stated smooth barotropic Euler system with conservative potential, steady Bernoulli is constant along streamlines and globally on connected curl-free components, and circulation is conserved for closed material loops with the stated regularity.

[F3]

Formal parameterized fluid systems and observables: for $u = (u_{x}, u_{y}, u_{z})$ in Cartesian coordinates, $curl u = (\partial_{y} u_{z} - \partial_{z} u_{y}, \partial_{z} u_{x} - \partial_{x} u_{z}, \partial_{x} u_{y} - \partial_{y} u_{x})$ .

Proof

1.1givenA1F1F2

The hypotheses of [F2] hold: $ρ > 0$ , the stated derivatives exist, $P \in C^{2}$ , and $V$ is time-independent and $C^{2}$ . The body force is the conservative input $- \nabla V$ , so [F2] supplies the streamline Bernoulli, curl-free global-constant and material-loop circulation conclusions conditionally on this adopted model.

1.2givenF1F3algebra

In the example, $div u = 0$ and $u \cdot \nabla (r^{2}) = Ω_{0} (- y, x, 0) \cdot (2 x, 2 y, 0) = 0$ throughout the cylinder, including its axis. Since $ρ = ρ_{c} \exp (Ω_{0}^{2} r^{2} / (2 c^{2}))$ , the Cartesian chain rule gives $u \cdot \nabla ρ = 0$ ; together with $\partial_{t} ρ = 0$ , this yields $\partial_{t} ρ + div (ρ u) = 0$ . Also $(u \cdot \nabla) u = - Ω_{0}^{2} (x, y, 0)$ and $\nabla P (ρ) = c^{2} \nabla ρ = ρ Ω_{0}^{2} (x, y, 0)$ , proving momentum balance for $V = 0$ . The Cartesian curl convention in [F3] gives $curl u = (0, 0, 2 Ω_{0})$ for this rotation field.

2.1givenF2step 1.2algebraalgebra

On the radius- $ℓ$ material circle, $u = Ω_{0} ℓ e_{θ}$ and $d X = ℓ e_{θ} d θ$ , hence $Γ = 2 π Ω_{0} ℓ^{2}$ . Since $h (ρ) = c^{2} \log (ρ / ρ_{r e f})$ , substitution gives $B = c^{2} \log (ρ_{c} / ρ_{r e f}) + Ω_{0}^{2} r^{2}$ . The radius is fixed on each circle, but the value varies across radii because $Ω_{0} \neq 0$ .

3.1A1A2F1F2step 1.1step 2.1∎

These are deductions from the stated barotropic Euler model and do not establish its empirical validity or apply to viscous circulation. No wall data are used, under [A2].

Barotropic Kelvin and Bernoulli with exact hypotheses — conditional physical prediction

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources