Baroclinic Multiphase Vortex System · FunnelB.com
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Sustained fluid circulation in a thermally isolated cavity — driven solely by the gravitational field · Genesis@FunnelB.com · 2026
01 — System
A sealed rigid cavity with a fixed tilt angle. Inside: air and water, separated by a tilted interface. The interface is not a free surface — it is a dynamic annular seal sliding along the cavity wall. It cannot rotate to horizontal. The density field is locked by geometry.
As long as the gravitational field exists, the pressure gradient exists. As long as the cavity is tilted, the baroclinic term is non-zero. The driving condition does not decay:
Shallow Segment
Gravity drives descent
Air above. Short water column. Low pressure. Interface objects: positive apparent weight. Water moves down under direct gravitational body force.
f = ρ_w · g ↓
Deep Segment
Pressure drives ascent
Air below. Communicating-vessel pressure exceeds air pressure. Interface objects: negative apparent weight. Water moves up under gravity-sustained pressure gradient.
∇P = ρ_w · g ↑
The system collects from both directions simultaneously. Gravity acts on both sides through the full pressure field. Neither driving condition decays. The same gravitational field — through a tilted density interface — creates path asymmetry between the two segments.
02 — First Principles
Gravitational body force
Gravity acts on every fluid element simultaneously, maintaining the pressure field on both sides continuously.
f = ρg · ∇P = ρg (sustained by gravity)
Density discontinuity at interface
Water and air are distinct phases. The density field is a delta function at the interface — not a diffusible concentration gradient. Mixing arguments do not apply.
∇ρ = (ρ_w − ρ_a) · δ(φ) · ∇φ
Baroclinic condition enforced by geometry
When the cavity tilt θ > 0°, density and pressure gradients are non-collinear. The baroclinic cross product is non-zero — by geometric necessity, not assumption.
∇ρ × ∇P ≠ 0 (for all θ > 0°)
Bjerknes circulation theorem
Whenever the baroclinic term is non-zero, circulation must be continuously generated. This is a mathematical consequence of the theorem, not a hypothesis.
dΓ/dt = ∬ (1/ρ²)(∇ρ × ∇P) · dA ≠ 0
The open question
Circulation exists. The baroclinic source is sustained. Whether the full-cycle net work integral is zero or non-zero has not been analytically resolved for this two-phase configuration.
W_net = ∮ ( ∫_V1 + ∫_V2 + ∫_Σ(t) [P]_Σ n·u_Σ dA ) dt = ?
03 — The Challenge
The standard proof that "net work is zero in a conservative field" relies on the pressure field being globally single-valued. In a two-phase system with a dynamic interface Σ(t), this assumption fails. The pressure is discontinuous at the interface:
Full-Cycle Net Work Integral
The third term — the interface jump integral — cannot be dismissed by single-phase conservative field arguments. A complete answer must evaluate or prove the vanishing of this term under the given geometric constraints.
We do not claim this device works. We claim: until the interface jump integral is fully evaluated for this two-phase configuration, the assertion "net work is zero" is mathematically incomplete.
Public Record · Physics Stack Exchange #869919 · 25 days · No answer
In two-phase flow with dynamic interface Σ(t), how is the full-cycle work integral evaluated when [P]_Σ = γκ + f is non-zero?
The question has been publicly posted and is open for response from the fluid mechanics community. We welcome rigorous mathematical engagement.
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We ask whether the field, through structure,
delivers sustained output.
The mathematics will decide."
FunnelB Physical Research Group · [email protected]
04 — Contact
Whether you are a physicist, engineer, or institution — if you have a complete analytical response to the interface jump integral, or wish to discuss collaboration, we want to hear from you.