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Barotropic Kelvin and Bernoulli with exact hypotheses

Statement

Let $I \subseteq R$ be an open time interval, let $Ω \subseteq R^{3}$ be open, and let $P : (0, \infty) \to R$ be $C^{2}$ . Suppose $u : I \times Ω \to R^{3}$ has continuous first time, first spatial, and second spatial derivatives; $ρ : I \times Ω \to (0, \infty)$ has continuous first time and first spatial derivatives; and $V : Ω \to R$ is time-independent and $C^{2}$ . Assume pointwise

$\partial_{t} ρ + div (ρ u) = 0, \partial_{t} u + (u \cdot \nabla) u = - ρ^{- 1} \nabla (P (ρ)) - \nabla V .$

For any reference density $ρ_{r e f} > 0$ , define $h (q) = \int_{ρ_{r e f}}^{q} P^{'} (r) / r d r$ for $q > 0$ . For a steady velocity field, a streamline is a $C^{1}$ curve $x : J \to Ω$ , where $J \subseteq R$ is an interval, satisfying $x^{'} (s) = u (x (s))$ with $u (x)$ denoting its time-independent spatial value. If $u$ and $ρ$ are independent of time, every streamline has $∣ u ∣^{2} / 2 + h (ρ) + V$ constant along it. If in addition $curl u = 0$ on a connected component of $Ω$ , this quantity is constant throughout that component.

For any closed $C^{1}$ material loop $X : J \times [0, 1] \to Ω$ satisfying $X (t, 0) = X (t, 1)$ , $X_{t} (t, s) = u (t, X (t, s))$ , with $X_{s}$ and $X_{t s}$ continuous, the circulation $Γ (t) = \int_{0}^{1} u (t, X (t, s)) \cdot X_{s} (t, s) d s$ is constant on $J$ . Independently of the Euler assumptions, every $C^{2}$ vector field $v$ on an open subset of $R^{3}$ obeys the identity

$(v \cdot \nabla) v = \nabla (∣ v ∣^{2} / 2) - v \times curl v .$

No claim is made for shocks, nonbarotropic pressure, nonconservative body forces, or viscous circulation. Under SI interpretation, $u$ is in metres per second, $ρ$ in kilograms per cubic metre, $P$ in pascals, and $h, V$ in square metres per second squared.

Statement references

Iain W. Stewart, MIT 8.09/8.309, Chapter 6, §6.4

David Tong, Fluid Mechanics lecture notes, Chapter 1, §§2.2.1–2.2.2

Formal barotropic system, Cartesian derivative conventions, and streamline definition: Formal parameterized fluid systems and observables.

Facts & Assumptions

Given: Use the positive density, $C^{2}$ equation of state, conservative time-independent potential, classical pointwise equations, steady-flow hypothesis where stated, and material-loop regularity in the Statement. The result is a conditional deduction from the stated smooth model.

[F1]

Formal parameterized fluid systems and observables: the barotropic Euler equations are $ρ_{t} + div (ρ u) = 0$ and $u_{t} + (u \cdot \nabla) u = - ρ^{- 1} \nabla (P (ρ)) - \nabla V$ , with positive density, $P \in C^{2}$ , and time-independent $V \in C^{2}$ ; divergence, gradient, curl and convective derivative use Cartesian coordinates.

Proof

1.1givenalgebra

In coordinates, $((v \cdot \nabla) v)_{i} = \sum_{j} v_{j} \partial_{j} v_{i}$ and $\partial_{i} (∣ v ∣^{2} / 2) = \sum_{j} v_{j} \partial_{i} v_{j}$ . Let $ϵ_{i j k}$ be the alternating symbol with $ϵ_{123} = 1$ . Using $(v \times curl v)_{i} = \sum_{j, k, l, m} ϵ_{i j k} ϵ_{k l m} v_{j} \partial_{l} v_{m} = \sum_{j} v_{j} \partial_{i} v_{j} - \sum_{j} v_{j} \partial_{j} v_{i}$ gives the asserted identity component by component.

2.1givenF1step 1.1algebra

The fundamental theorem of calculus and the chain rule give $\nabla h (ρ) = (P^{'} (ρ) / ρ) \nabla ρ = ρ^{- 1} \nabla (P (ρ))$ . For a steady solution, $u_{t} = 0$ , so dotting the momentum equation with $u$ and applying step 1.1 yields $u \cdot \nabla (∣ u ∣^{2} / 2 + h (ρ) + V) = 0$ , because $u \cdot (u \times curl u) = 0$ . Along a streamline $x^{'} (s) = u (x (s))$ , the chain rule makes the derivative of this quantity zero, proving streamline constancy. If $curl u = 0$ , step 1.1 and the momentum equation give $\nabla (∣ u ∣^{2} / 2 + h (ρ) + V) = 0$ . A differentiable function with zero gradient is constant along every polygonal path; in an open connected subset of $R^{3}$ , polygonally reachable points form a nonempty relatively open and relatively closed set (each point has a ball contained in the open set). Connectedness therefore gives constancy on that component.

3.1givenF1step 2.1algebra∎

By the assumed continuous mixed derivatives, differentiate under the integral sign and commute $X_{t s} = \partial_{s} X_{t}$ . The chain rule and $X_{t} = u (t, X)$ give $\partial_{t} (u (t, X) \cdot X_{s}) = (\partial_{t} u + (u \cdot \nabla) u) (t, X) \cdot X_{s} + \partial_{s} (∣ u (t, X) ∣^{2} / 2)$ . The momentum equation and step 2.1's enthalpy-gradient identity turn this into $- \partial_{s} (h (ρ (t, X)) + V (X)) + \partial_{s} (∣ u (t, X) ∣^{2} / 2)$ . Integrating in $s$ gives the endpoint difference of $∣ u ∣^{2} / 2 - h (ρ) - V$ , which is zero because $X (t, 0) = X (t, 1)$ and all fields are single-valued. Hence $d Γ / d t = 0$ on $J$ .

Depends on

Formal parameterized fluid systems and observables