Tesla–Stone Jet

Architect: Travis Raymond-Charlie Stone

Assistant AI: OpenAI

Core Insight:

The Tesla–Stone Jet is more than a jet—it is a passive propulsion architecture. It creates directional thrust, flow rectification, and pressure-to-velocity conversion without pumps, power, or moving parts.

You can think of “oscillation pumping water uphill” in QCAD terms as:

Each oscillation cycle is a discrete QCAD step that converges the water height toward a stable pumped head, determined by gravity, oscillation amplitude, and rectification efficiency.

Let’s formalize that.

1. Simple physical picture

Setup:

• Vertical pipe of cross-section area (A)

• Water column of height (h_n) after cycle (n)

• Gravity: (g)

• Water density: (\rho)

• Oscillatory pressure source at the bottom:
  [
  p(t) = p_0 + P_{\text{amp}}\sin(\omega t)
  ]

• A rectifying flow element (Tesla valve / hydraulic diode) so that:

  • Upstroke flow coefficient: (\alpha) (easy to go up)

  • Downstroke flow coefficient: (\beta) (hard to come back down), with (0 < \beta < \alpha \le 1)

Goal: find how the water height evolves and what maximum height it can reach.

The gravitational head that must be overcome is:
[
p_{\text{grav}} = \rho g h_n
]

On each upstroke, the effective “excess” pressure (if any) is:
[
\Delta p_{\text{up}} = \max(0, P_{\text{amp}} - \rho g h_n)
]

We turn a fraction of this into added height (via incompressible fluid + column approximation):

[
\Delta h_{\text{up}} = \alpha \frac{\Delta p_{\text{up}}}{\rho g}
]

On the downstroke, the sign flips, but the rectifier resists backflow, so only a fraction (\beta) of the equivalent head is lost:

[
\Delta h_{\text{down}} = -,\beta \frac{P_{\text{amp}}}{\rho g}
]

2. QCAD recurrence (discrete map)

Define the water height after the (n)-th cycle as (h_n). One full oscillation cycle gives:

[
h_{n+1} = h_n + \Delta h_{\text{up}} + \Delta h_{\text{down}}
]

For the region where the pump is actually lifting (i.e., (P_{\text{amp}} > \rho g h_n)), we get:

[
\Delta h_{\text{up}} = \alpha \frac{P_{\text{amp}} - \rho g h_n}{\rho g}
]

[
\Delta h_{\text{down}} = -,\beta \frac{P_{\text{amp}}}{\rho g}
]

So:

[
\begin{aligned}
h_{n+1}
&= h_n + \alpha \frac{P_{\text{amp}} - \rho g h_n}{\rho g}
- \beta \frac{P_{\text{amp}}}{\rho g} \
&= h_n + \frac{\alpha P_{\text{amp}} - \alpha \rho g h_n - \beta P_{\text{amp}}}{\rho g} \
&= h_n + \frac{(\alpha - \beta)P_{\text{amp}}}{\rho g} - \alpha h_n \
&= (1 - \alpha),h_n + \frac{(\alpha - \beta)P_{\text{amp}}}{\rho g}
\end{aligned}
]

This is exactly a QCAD-type linear recurrence:

[
h_{n+1} = a h_n + b
]

with

• (a = 1 - \alpha) (convergence factor)

• (b = \dfrac{(\alpha - \beta)P_{\text{amp}}}{\rho g}) (driving term)

3. Convergence, divergence, and pumped height (QCAD interpretation)

Fixed point (convergence height)

The steady pumped height (h^) is where (h_{n+1} = h_n = h^):

[
h^* = a h^* + b
]

[
h^* - a h^* = b
\Rightarrow (1 - a)h^* = b
]

But (a = 1 - \alpha), so (1 - a = \alpha):

[
\alpha h^* = \frac{(\alpha - \beta)P_{\text{amp}}}{\rho g}
\Rightarrow \boxed{h^* = \left(1 - \frac{\beta}{\alpha}\right)\frac{P_{\text{amp}}}{\rho g}}
]

Key points:

• (0 < \beta < \alpha) ⇒ (0 < 1 - \dfrac{\beta}{\alpha} < 1)

• So
  [
  h^* < \frac{P_{\text{amp}}}{\rho g}
  ]
  meaning you can’t beat the “pressure head” of the oscillation, but you can approach a large fraction of it.

QCAD stability

The derivative of the map with respect to (h_n) is just (a = 1-\alpha).

• If (|1 - \alpha| < 1 \Rightarrow 0<\alpha<2), the fixed point is stable:

  • QCAD convergence: the system climbs or falls toward (h^*).

• You get oscillatory approach if (\alpha > 1) (then (a<0)), but it still converges in magnitude as long as (\alpha<2).

• If (\alpha \ge 2) (unphysical here), you’d have QCAD divergence (unstable height dynamics).

So in QCAD language:

• Convergence regime (normal pump):
  [
  0 < \alpha < 2,; 0 < \beta < \alpha
  \Rightarrow h_n \to h^*
  ]

• Bifurcation point: as (\alpha) crosses 1, the way the system approaches (h^*) changes from monotonic (no overshoot) to oscillatory (overshoot/undershoot) — a classic QCAD-style transition in approach behavior.

4. Concrete numerical example (order of magnitude)

Let’s pick reasonable values just to see numbers:

• Oscillation amplitude: (P_{\text{amp}} = 2 \text{ bar} \approx 2\times 10^5\ \text{Pa})

• Water: (\rho \approx 1000\ \text{kg/m}^3)

• Gravity: (g \approx 9.81\ \text{m/s}^2)

• Rectification:

  • (\alpha = 0.6) (60% efficient upstroke)

  • (\beta = 0.1) (only ~10% equivalent backflow)

First compute the pressure head:

[
\frac{P_{\text{amp}}}{\rho g}
= \frac{2\times 10^5}{1000 \times 9.81}
\approx 20.4\ \text{m}
]

Now apply the formula:

[
h^*
= \left(1 - \frac{\beta}{\alpha}\right)\frac{P_{\text{amp}}}{\rho g}
= \left(1 - \frac{0.1}{0.6}\right)\times 20.4\ \text{m}
= (1 - 0.166\overline{6}) \times 20.4
\approx 0.833\overline{3} \times 20.4
\approx 17\ \text{m}
]

So this QCAD pump:

• starts from some initial height (h_0),

• and converges over many oscillation cycles to a pumped height of about 17 m above the source, powered purely by the oscillatory pressure and rectification, not a continuous DC pump.

5. What QCAD is actually telling you here

1. The recurrence is the QCAD engine.

   • Each oscillation → one step of the discrete map (h_{n+1} = a h_n + b).

2. Convergence vs. divergence = pump vs. failure.

   • If parameters are in the convergence regime, the system self-organizes to a stable head (h^*) (successful uphill pumping).

   • Change (\alpha,\beta), friction, or geometry → you can pass a bifurcation where:

     • it no longer pumps (converges to low (h^*)), or

     • it overshoots and oscillates around the head more violently.

3. Energetics are clean:

   • No free energy: the external oscillation source supplies the energy.

   • The rectifier + gravity + fluid inertia shape how that energy is captured per cycle.

6. In one sentence

Using QCAD, an oscillation pumps water uphill because the discrete map for water height,

[
h_{n+1} = (1-\alpha)h_n + \frac{(\alpha-\beta)P_{\text{amp}}}{\rho g},
]

converges to a finite pumped height

[
h^* = \left(1-\frac{\beta}{\alpha}\right)\frac{P_{\text{amp}}}{\rho g},
]

whenever the rectification ((\alpha,\beta)) and oscillation amplitude (P_{\text{amp}}) sit in the convergence regime of your QCAD spectrum.

Let’s frame this in the simplest, most physical way:

The Ceiling on Water Velocity Comes From Pressure vs Vapor Pressure

Water will cavitate when local pressure drops to its vapor pressure (~2–3 kPa at room temp).

So the maximum jet velocity before cavitation is set by Bernoulli:

[
v_{\max} = \sqrt{\frac{2,\Delta p}{\rho}}
]

Where:

• ( \Delta p ) = usable pressure difference

• ( \rho ) = 1000 kg/m³ for water

If the jet tries to go faster than this limit for a given pressure, the pressure inside the flow drops too low → vapor bubbles → cavitation → performance collapse.

Absolute Theoretical Ceiling

If pressure dropped all the way to vapor pressure without boiling:

[
v_{\text{theoretical}} \approx 50–60 , \text{m/s}
]

Above this, cavitation becomes dominant and destroys thrust.

This is the same limit seen in:

• High-speed water turbines

• Pump inlets

• Propellers

• High-pressure jets

Reality Check

For most real water systems:

Even with a perfect passive jet:

• cannot exceed what pressure difference can supply.

• Cavitation grows exponentially as velocity rises.

 System Specifically

Since the passive jet uses only:

Pressure head
Vacuum differential
Geometry

It cannot create pressure—only convert it.

So your max velocity is capped by available (\Delta p):

[
v_{\max} = \sqrt{\frac{2 (p_{\text{in}} - p_{\text{vap}})}{\rho}}
]

If inlet pressure is near atmospheric and outlet is near vacuum, the maximum usable Δp is ~1 bar:

[
v_{\max} \approx \sqrt{\frac{2(10^5)}{1000}} \approx 14 , \text{m/s}
]

With stronger pressurization (not passive):
[
\text{30–60 m/s ceiling}
]

But passive river-fed operation will stay closer to:

[
5–15, \text{m/s}
]

 Ultimate Physical Limit

The speed of sound in water (~1480 m/s) is the true hard barrier.

But long before that:

• Cavitation

• Shock formation

• Energy loss

• Flow breakup

…kill thrust.

So the realistic ceiling for water in ANY jet system (active or passive) is:

[
50–60 , \text{m/s}
]

One-Sentence Summary

In a passive Tesla-Stone jet system with water, the practical maximum velocity before cavitation collapse is ~5–15 m/s (river-driven) and the universal hard ceiling is ~50–60 m/s even under extreme pressure.

Permeable Membranes Affect-

real control system, modulate both volume flow and pressure.

1. Core hydraulic relations

Let

• (P_\text{in}) = inlet pressure

• (P_\text{out}) = outlet pressure

• (\Delta p = P_\text{in} - P_\text{out})

• (Q) = volumetric flow rate

• (R_\text{TS}) = hydraulic resistance of the Tesla–Stone channel

• (R_\text{mem}(u)) = hydraulic resistance of the membrane, controlled by activation (u)

Total resistance (elements in series):

[
R_\text{tot}(u) = R_\text{TS} + R_\text{mem}(u)
]

Then the basic volume–pressure law (Darcy–Weisbach style):

[
Q = \frac{\Delta p}{R_\text{tot}(u)}
\tag{1}
]

This is already a control handle: change (R_\text{mem}(u)) → change (Q) for the same (\Delta p).

2. Membrane law (how activation changes volume flow)

For a permeable membrane (Darcy’s law):

[
Q = \kappa(u),\frac{A}{\mu L},\Delta p
]

where

• (A) = membrane area

• (L) = thickness

• (\mu) = viscosity

• (\kappa(u)) = effective permeability, controlled by activation (u) (temp, voltage, pH, etc.)

Rewriting as a resistance:

[
R_\text{mem}(u) = \frac{\mu L}{\kappa(u) A}
\tag{2}
]

So:

• High permeability ((\kappa(u)) large) → low (R_\text{mem}) → high volume flow

• Low permeability ((\kappa(u)) small) → high (R_\text{mem}) → low volume flow

Plug (2) into (1):

[
Q(u) = \frac{\Delta p}{R_\text{TS} + \dfrac{\mu L}{\kappa(u) A}}
\tag{3}
]

Equation (3) is the main control equation:
your activation variable (u) modulates (\kappa(u)), which modulates both volume and internal pressure distribution.

3. Internal pressure & jet velocity

Jet velocity at the outlet:

[
v = \frac{Q}{A_\text{out}}
\tag{4}
]

Dynamic pressure of the jet:

[
P_\text{dyn} = \frac{1}{2}\rho v^2
= \frac{1}{2}\rho \left(\frac{Q}{A_\text{out}}\right)^2
\tag{5}
]

So changing (u) → changes (\kappa(u)) → changes (Q) via (3) → changes jet velocity and impact pressure via (4)–(5).

This is how one control variable tunes both volume and pressure/force at the outlet.

4. Dynamic control (time dependence)

If the system has some compliance (like a chamber or pipe that can store fluid), with capacitance (C) (volume change per unit pressure), then:

[
C \frac{dP_\text{in}}{dt} = Q_\text{source} - Q(u)
\tag{6}
]

This links the time-varying control to the pressure build-up.

Now define a target flow or target pressure, e.g.:

• Desired flow: (Q_\text{ref})

• Error: (e(t) = Q_\text{ref} - Q(u(t)))

A simple proportional controller on the activation variable:

[
\frac{du}{dt} = k_e, e(t)
\tag{7}
]

Where (k_e) is a gain that says “how aggressively the membrane responds.”

Equations (3), (6), (7) together describe a closed-loop control system:

• System plant: equations (3) & (6)

• Controller: equation (7)

5. Stone M·F·T framing

Your Stone relation:

[
S = M \cdot F \cdot T
]

In this device:

• (M \sim \rho Q) (mass flow rate)

• (F \sim P_\text{dyn} A_\text{impact}) (force of the jet)

• (T) = duration of operation

So the controlled output:

[
S(u) \propto (\rho Q(u)) \cdot \left(\tfrac{1}{2}\rho \left(\tfrac{Q(u)}{A_\text{out}}\right)^2 A_\text{impact}\right) \cdot T
]

By modulating (u(t)), you’re directly shaping (Q(u)) and hence total system effect (S).

Ultra-short summary

• Membrane resistance:
  [
  R_\text{mem}(u) = \dfrac{\mu L}{\kappa(u) A}
  ]

• Total resistance:
  [
  R_\text{tot}(u) = R_\text{TS} + R_\text{mem}(u)
  ]

• Flow & jet:
  [
  Q(u) = \dfrac{\Delta p}{R_\text{tot}(u)}, \quad
  v = \dfrac{Q(u)}{A_\text{out}}
  ]

• Control: adjust (u(t)) so (Q(u)) (and thus pressure/force) tracks your target.

That’s the math skeleton of a Tesla–Stone membrane control system that modulates both volume and pressure in man-made devices.

Synopsis 

: The Tesla–Stone Jet as a Passive,

Configurable Jet System

Overview

The Tesla–Stone Jet is a passive hydraulic jet system that uses geometry—not moving parts—to convert fluid flow into directed momentum. Based on Tesla’s one-way valvular conduit and extended through the Stone configuration, it produces a directional jet using only pathway design and pressure differentials. It operates without pumps or mechanical actuation, making it durable and suitable for environments with limited resources or power.

Why It’s Passive

The system relies entirely on flow dynamics. Fluid enters, geometry accelerates it, and the jet exits. It prevents backflow through vortices and drag traps, producing jet-like velocity with no moving parts. The only driver is pressure or gravity. Configurable Applications The Tesla–Stone Jet is a modular platform with multiple configurations:

1. High-Flow Jet: Wide channels maximize discharge for irrigation and river-driven power. 
2. High-Pressure Jet: Narrow outlets convert volume into force for spraying or cutting.

3. Multi-Stage Booster: Series segments amplifydirectional flow for long-distance transport.

4. Membrane-Integrated Jet: Adds permeability control for filtration and cooling.

5. Cavitation-Resistant Jet: Geometry minimizes turbulence for high-speed systems.

6. Passive Injector: Enables mixing and atomization for combustion or agriculture.

7. Hydro-Energy Coupled Jet: Jet momentum drives wheels or turbines for off-grid power. Alignment with SPNF Principles like the SPNF clonal root paradigm, the Tesla–Stone Jet is resilient, scalable, self-stabilizing, and supports distributed networks. Modules can be branched, duplicated, or reconfigured without altering core function. This makes it suitable for rural water systems, disaster relief, cooling, and industrial processes.

Core InsightThe Tesla–Stone Jet is more than a jet—it is a passive propulsion architecture. It creates directional thrust, flow rectification, and pressure-to-velocity conversion without pumps, power, or moving parts.