Water carries the signals. Moving shutters control their passage. Together, they form a machine that computes through pressure and flow.
The fundamental element
A pressure signal moves a spring-loaded shutter across a water passage. The cutaway reveals the piston, circular window, and return spring at work.
The T has three circuit connections: source, output, and control. Low control pressure leaves the working passage open. High control pressure moves the piston and shutter toward closure. A source-fed sensing chamber supplies the opposing pressure force.
The circular window determines the opening. A separate permanent capillary carries a small flow through the valve at full closure. Lowering the control pressure lets the spring reopen the passage.
A restricted bleed connects the output to return. With the passage open, supply flow raises output pressure. With the passage closed, the bleed lowers it. This loaded T circuit acts as an inverter: a low control produces a high output, and a high control produces a low output.
Pressure force, spring force, and damping determine shutter motion between its travel stops. x increases toward closure.
Composition into logic
Two pressure-closing passages share a supply and an output. Either open passage keeps the output high. Both inputs must be high to close both passages and let the output fall.
ADC
A rotary input selects a reference pressure. Nine spring-loaded thresholds produce a thermometer code; restored pressure signals encode one BCD digit.
The nominal tap buttons pose the rotary selector; the pressure slider also permits an analog sweep between taps. Each cutaway shows the two wetted piston faces, opposing 0 kPa return, a spring, its preload F₀, and the linked working shutter. The marked pressure is the centre of a 100 Pa mechanical motion band, not an ideal instantaneous comparator. The raw inverted pressure drives a restoring T valve. Blue water above each restoring piston stays filled up to its moving face; its tint follows the inverted pressure, and high pressure holds the piston down. The restored output joins the thermometer word, which is encoded through a one-hot digit into BCD. Blue indicates high; amber indicates unresolved. No BCD value is shown for an unresolved or unordered word. The selector pose and encoder are explanatory boundary/logic views; this is not a hydraulic solve of their full routed geometry. Regulated sources supply control-side displacement. Spring preload and geometric enlargement are not to scale.
DAC
A BCD decoder selects one of ten pressure taps. The selector drivers close competing paths before admitting the chosen tap to the output.
Each column is a liquid path; each gold row is one rigid driver linked to nine competing guards and its own admission. The amber diagonal seats are admissions. A driver closes its guards after 2.0 mm, crosses a 0.1 mm fully closed interval, then begins admission at 2.1 mm. The travel scale and optional 50× slow motion make that ordering visible. The output chamber stays completely water-filled: pressure changes its compression storage, not a free-surface level. Cyan tracers move toward the output; orange tracers show reverse flow into a lower-pressure tap. Tracer speed is normalized for visibility, but direction follows the calculated signed flow. The bleed remains connected to return. This reduced model retains regulated reference taps and ideal sealed guards; reference loading and capillary leakage through closed guards remain outside its scope.
Decagon display
A pressure-driven piston moves a rack. A pinion on the display shaft turns the ten-faced prism, while a spring and detents define its settled position.
This enlarged schematic preserves angle = rack displacement / pinion radius. It uses the full-machine rack spring, preload, damping and detent parameters, with a prescribed pressure source. Twenty simulated seconds pass per real second so the slow mechanical response is visible without changing those coefficients. Between detents the digit readout shows “Moving”, rather than rounding the command to an answer.
Binary → decimal
Double dabble converts an eight-bit input into three BCD digits. Before each shift, digits at least five receive an add-three correction.
Each displayed stage stands for another part of an unrolled combinational network. The animation reveals those stages in order; it does not imply a clock, register, or software conversion in the full machine. Gold indicates add-three correction, blue indicates a high bit. This plate evaluates the conversion algorithm, not pressure propagation through all of its NAND gates.
Decimal → binary
Read decimal digits from hundreds to units. At each stage, multiply the accumulator by ten, then add the next digit. Two wired shifts supply the times-eight and times-two terms.
The identity 10 × accumulator = (accumulator << 3) + (accumulator << 1) becomes wires and adders in the full machine. Here each revealed stage exposes those intermediate values. Ten output bits retain the entire 000–999 domain, including leading-zero inputs. No repeated division, string parsing of a computed answer, or hydraulic timing claim is used to demonstrate the conversion.
A circuit view of the physical model
Read the T as a three-terminal controlled conductance: the source supplies energy, the output drives a load, and the control changes conduction between them. The useful electronic comparison is a high-side pMOS with a resistive pull-down, not an nMOS pull-down. Raising the control pressure closes this T; raising a pMOS gate voltage toward its source likewise reduces conduction. We will compare their device laws, loaded gain, and switching dynamics before identifying what the simulation tracks.
The device is pressure-operated, but the simulation here is hydraulic, not pneumatic: its working fluid is water. An air-operated counterpart would require a gas-storage model. The circuit analogy survives that change; the quantitative equations and time constants do not transfer unchanged.
Voltage corresponds to pressure potential, current to signed volumetric flow Q, and transported charge to transported liquid volume. To compare an electrical resistor with a fluid passage, first include gravity in the fluid driving potential:
Here Re is electrical resistance. Hydraulic resistance Rh has units Pa·s/m³. Every difference is upstream minus downstream; p is gauge pressure, ρ is water density, g is gravitational acceleration, and y is elevation. At equal height the driving difference is simply pressure difference. In a static water column, pressure and elevation contributions cancel. The last equation is the linear-resistance limit; the actual passage law below adds nonlinear losses.
This identification also preserves power, rather than merely resembling a circuit diagram:
Voltage times current and pressure-potential difference times volume flow both give watts. A valve can control power delivered from its supply without generating that power, just as a transistor can. A pressure held behind a stationary sealed piston is analogous to a charged, nonleaking gate: finite effort at the terminal does not require a continuous input flow.
For a familiar reference, take the ideal long-channel pMOS triode law, neglecting body effect and channel-length modulation:
ISD is positive from source to drain, VSG is source minus gate voltage, and VSD is source minus drain voltage. β contains mobility, oxide capacitance per area, and channel width-to-length ratio; |VT| is the threshold magnitude. Vov is the gate overdrive. This expression applies for positive overdrive and a source-to-drain voltage between zero and that overdrive. At fixed source and drain voltages, raising the gate reduces overdrive and therefore current.
The T achieves the same control polarity by changing a physical opening, not a carrier channel. Its intermediate state is shutter displacement x, measured toward closure. Away from its travel stops, the model determines motion from this force balance:
Control pressure pc acts on piston area Ac. A source-fed sensing chamber supplies the opposing force through pressure pm and area Am. Spring stiffness k, preload F0, and viscous damping b oppose closure. These are gauge pressures against the vented mechanical reference, not a single gate-to-source pressure difference. In particular, the unequal piston areas prevent identifying the control force with pressure difference alone.
Set the velocity to zero to find the switching range. The shutter first leaves its open stop when control force exceeds preload plus the opposing pressure force; it reaches its closed stop after traversing the full stroke xmax:
For the plate's fixed 10 kPa source, closure begins at 2.3 kPa control pressure and reaches the closed stop at 3.9 kPa. Between those values the equilibrium position changes continuously. These are mechanical thresholds, not a semiconductor threshold voltage or a claim of an abrupt Boolean switch. Changing the sensing pressure shifts both thresholds; changing stiffness or stroke changes their separation.
Displacement then sets the main-window flow. For the plate's equal-height supply and output, its reduced law is:
ps and po are supply and output pressure; their difference is Δp. Qw passes through the working window; its viscous and quadratic-loss coefficients R(x) and K(x) increase as the window closes. The signed quadratic term preserves reversal. Total supply flow Qs also includes a separate permanent capillary flow Qleak. At the closed stop the main-window flow is zero, but that capillary still conducts.
The analogy stops short of MOS saturation. At a fixed nonzero opening, this valve's flow continues to increase with driving pressure; the model has no channel pinch-off region with nearly constant current. Nor is its permanent capillary semiconductor subthreshold leakage. Both devices have an off-state current or flow, but their mechanisms and characteristic curves differ.
At a partially open operating point, small-signal analysis separates control sensitivity from output loading. Define positive transconductance magnitudes for these pressure-closing and gate-voltage-closing devices:
The electronic derivative holds source and drain voltage fixed. The hydraulic derivative holds supply, output, and sensing pressures fixed and lets the shutter reach equilibrium away from its stops. Its chain rule exposes two factors: how strongly displacement changes flow, and how strongly pressure moves the piston. Increasing control area or reducing spring stiffness increases the second factor, but also changes switching thresholds and loading. At a travel stop, that displacement sensitivity disappears until the force releases the stop.
Both devices also have finite output conductance. Define go,e as minus the derivative of source-to-drain current with respect to output voltage at fixed gate and source. Define go,h as minus the derivative of supply flow with respect to output pressure at fixed control and source conditions. Their reciprocals are the corresponding incremental output resistances. For the T, increasing output pressure reduces the pressure drop through the open window and capillary, so it reduces supply flow even without changing the control.
Now attach the load: an electronic pull-down of conductance GL, or the hydraulic bleed with incremental conductance Gb. Linearizing the steady output-node balance gives parallel low-frequency gain expressions:
The symbol δ denotes a small perturbation about the operating point. Each numerator measures how much the input changes supply current or flow; each denominator measures how strongly the output node restores balance through the device and load. The minus sign is inversion. Large transconductance alone does not establish a usable logic stage: loaded gain, output levels, and separated input bands must all be checked. These expressions describe local behavior, not the complete switching waveform.
With low control pressure, the open T replenishes water lost through the bleed and holds the output high. High control pressure closes the working window, leaving the bleed to bring the output low despite capillary leakage. There is static supply flow, especially in the high-output state, just as a resistor-loaded pMOS inverter draws static current. This is not complementary CMOS with an ideally disconnected DC path.
The NAND follows the pMOS pull-up network's parallel topology: either open T can sustain a high output; both controls must be high to close both supply passages. The shared bleed provides the pull-down. The truth table follows from these loaded pressure states, not from assigning an ideal Boolean value to each valve.
A MOS gate can draw little DC current yet load the previous stage during switching. A sealed hydraulic control chamber has the same distinction: changing its pressure compresses water, and moving its piston requires displaced liquid. Compare a constant lumped gate-capacitance approximation with the volume balance of one moving chamber:
Cg stores electrical charge per volt. Hydraulic compliance Ch stores liquid volume per pascal; V0 is the chamber's rest liquid volume and B is water's bulk modulus. Net inflow Qnet supplies both compression and geometric expansion. In this equation positive x enlarges the chamber through area A; the opposed chamber has the opposite displacement sign. The explicitly modeled piston volume must not also be added as an extra spring-derived compliance.
The plate assigns one litre of downstream water and uses a bulk modulus of 2.2 GPa. Its compliance is therefore about 4.55 × 10⁻¹³ m³/Pa: a 10 kPa pressure increase stores only about 4.55 microlitres of additional water. Small stored volume does not imply an instantaneous transition; the rate depends on the available net flow. More connected control chambers increase the driving stage's volume demand, much as fan-out increases capacitive loading.
There is also a mechanical time scale. Compare a resistively driven gate with the T's first-order shutter response at fixed sensing pressure and away from its stops:
Rdrive is the electrical driver's resistance. The shutter time constant is damping divided by stiffness, about 44 ms for the plate. Hydraulic connections and water storage add their own charging dynamics. Thus a static pressure transfer curve cannot by itself predict propagation delay; both pressure evolution and shutter motion matter.
The plate additionally prescribes a 160 ms exponential control-pressure response. That is an external regulator assumption, not an intrinsic transistor delay or a measurement of full-machine fan-out. The full network instead includes finite connected control and sensing chambers. The plates are useful for device behavior, but their ideal pressure sources do not establish the drive capability of a large circuit.
The full simulation couples node pressures, signed connection flows, and mechanism positions. Routed pipe lengths, radii, bends, and elevations determine viscous resistance, quadratic losses, and gravitational pressure potential. Circular-window overlap changes the valve coefficients as shutters move; permanent capillaries remain separate flow paths. Flow is solved from these reduced laws, not from a three-dimensional velocity field.
At each water-filled node, net inflow changes compression storage and supplies the signed volume swept by moving piston faces. Pressure forces act against preload, springs, damping, and travel stops; output mechanisms also include detents that define preferred positions. These terms couple the circuit's pressures to physical motion rather than using an electronic truth table to animate a result.
The full machine also tracks water transferred out of the supply reservoir and into the receiver. Their changing free-surface heights alter the available pressure head. The T and NAND plates instead hold supply and return pressures fixed, prescribe control-pressure response, use the declared downstream water inventory, and absorb the fixed sensing force into preload. Their compensated shutter adds no output-chamber displacement; control-side displaced volume is supplied by the external regulators.
The retained model is single-phase water with linearized compressibility and overdamped mechanics. It omits fluid and piston inertia, water hammer, resolved turbulence, gas thermodynamics, and phase-change dynamics. Its nonlinear loss laws are reduced approximations, not a resolved turbulence calculation. The solver checks volume continuity and physical operating domains as it advances the coupled state. What it predicts is the pressure, flow, and mechanical response of this declared hydraulic circuit model—not the performance of a fabricated pneumatic or semiconductor transistor.