CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pendingverified 2026-10-23 (gpt-6.1-sol)

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Unsteady pathlines differ from streamlines

Physical scope

Smooth kinematic velocity field on Newtonian spacetime, with pathlines and streamlines defined on the same spatial region. No momentum equation, constitutive law, or empirical flow is asserted.

Statement refuted

Whenever both curves exist on a common interval, for every $C^{1}$ velocity field $u : I \times Ω \to R^{3}$ , every initial point $x_{0} \in Ω$ , and every frozen time $t_{0} \in I$ , the pathline $X$ satisfying $X (t_{0}) = x_{0}$ equals the streamline through $x_{0}$ at time $t_{0}$ when the streamline parameter is identified with $t - t_{0}$ .

Statement references

David Tong, Fluid Mechanics, Part I, §1.1.1 (path lines and streamlines)

Facts & Assumptions

Given: An open time interval $I$ , a spatial region $Ω$ , and the definitions of pathline and frozen-time streamline from the stated velocity field.

[F1]

Classical frames, primitive quantities and SI units: a pathline solves $d X / d t = u (t, X (t))$ ; a streamline at fixed $t_{0}$ solves $d Y / d s = u (t_{0}, Y (s))$ with $s$ measured in seconds.

Counterexample

Given: Let $I = R$ , $Ω = R^{3}$ , and $u (t, x, y, z) = (α y t, β, 0)$ , where $α \neq 0$ has units s $^{- 2}$ and $β \neq 0$ has units m s $^{- 1}$ . Choose frozen time $t_{0} = 0$ and initial point $(x_{0}, 0, z_{0})$ .

1.1givenF1algebra

Solve the pathline equations $\dot{y} = β$ , $\dot{z} = 0$ , and $\dot{x} = α t y (t)$ with $y (0) = 0$ , $z (0) = z_{0}$ , $x (0) = x_{0}$ . They give $y (t) = β t$ , $z (t) = z_{0}$ , and $x (t) = x_{0} + α β t^{3} / 3$ .

2.1F1givenstep 1.1algebra

At frozen time $t_{0} = 0$ , the velocity is $u (0, x, y, z) = (0, β, 0)$ , so by [F1] the streamline through the same initial point is $Y (s) = (x_{0}, β s, z_{0})$ . Under $s = t - t_{0} = t$ , its $x$ -coordinate is constant, whereas step 1.1 has $x (t) = x_{0} + α β t^{3} / 3$ , which differs for every $t \neq 0$ . Thus the pathline and streamline are distinct.

3.1givenstep 2.1algebra∎

Since $\partial_{t} u = (α y, 0, 0)$ is not identically zero, this is an unsteady field. The explicit distinct curves in step 2.1 refute the universal assertion for an unsteady flow.

Unsteady pathlines differ from streamlines

Statement refuted

Statement references

Facts & Assumptions

Counterexample

Depends on

Dependency tree · 2 levels

Sources