POC Bounded Classical – Law L07: Bernoulli’s Equation – SSM

A bounded two-lane view of Bernoulli’s equation so the same downstream pressure `P2` can be tagged calm, borderline, or stressed.

Domain. Fluids / Pipe Flow / Lab Rig
Classical law.

For steady, incompressible, inviscid flow along a streamline (take a mostly horizontal pipe so heights are nearly equal):

P1 + 0.5 * rho * v1^2 ≈ P2 + 0.5 * rho * v2^2

Rearranged to solve for downstream pressure:

P2 = P1 + 0.5 * rho * (v1**2 - v2**2)

What this shows

In this law POC, we take Bernoulli’s Equation in a simple horizontal pipe and show how:

The Classical calculation gives the scalar downstream pressure P2, and
The Shunyaya Symbolic Mathematics (SSM) version keeps that pressure magnitude intact but adds a bounded alignment lane a in (-1,+1), indicating whether this particular Bernoulli snapshot is calm, borderline, or stressed.

When pressures and flow velocities are measured cleanly, Shunyaya Symbolic Mathematics (SSM) collapses to the classical result and you effectively get the same answer.

When gauge readings jump a bit or velocity estimates are noisy (pulsating pump, bubbles, sensor jitter), the alignment lane a and its band surface posture that the classical scalar P2 alone does not show.

POC display policy (simple).

For this POC, we use:

a_semantics = "drift-positive"

so larger printed a means more drift / more risk. This is a display choice only; you can flip semantics without changing the math or phi((m,a)) = m.

1) Setup (inputs)

Imagine a water flow rig in a lab:

A pump drives water through a pipe that narrows from a wider section (1) to a narrower section (2).
You measure pressure at section (1) with a gauge, and estimate pressure at section (2) from Bernoulli.
You measure flow rates or velocities with some jitter (due to pump ripple, small bubbles, etc.).

We model:

Fluid density (water, approx.):
- rho_m = 1000 kg/m^3, rho_a = +0.02 — nearly constant, small uncertainty.
Upstream pressure (gauge):
- P1_m = 200000 Pa (≈ 2.0 bar), P1_a = +0.10 — some gauge noise.
Velocities from flow measurement:
- v1_m = 1.5 m/s, v1_a = +0.30 — upstream velocity (slight jitter)
- v2_m = 3.0 m/s, v2_a = +0.20 — higher velocity in narrower section, still somewhat noisy

We treat the pipe as roughly horizontal so rho * g * h terms are approximately equal and cancel out in the comparison.

In SSM:

Each physical quantity is represented as (m,a) with a in (-1,+1).
Collapse is phi((m,a)) = m.

Our target:

Use P2 = P1 + 0.5 * rho * (v1**2 - v2**2)
Compare classical pressure P2 vs SSM pressure (m_P2, a_P2).

2) Classical calculation

Ignoring alignment, we compute P2 from Bernoulli between section 1 and 2:

# classical illustration (no external packages required)

rho = 1000.0    # kg/m^3
P1  = 200000.0  # Pa  (section 1)
v1  = 1.5       # m/s (section 1)
v2  = 3.0       # m/s (section 2)

P2 = P1 + 0.5 * rho * (v1**2 - v2**2)

print(P2)       # 196625.0 Pa

Compute step by step:

v1**2 = 2.25
v2**2 = 9.00
(v1**2 - v2**2) = -6.75
0.5 * rho * (v1**2 - v2**2) = 0.5 * 1000 * (-6.75) = -3375 Pa
P2 = 200000 - 3375 = 196625 Pa

(Depending on rounding you’ll see values around 1.97e5 Pa.)

Classical result.

P2 ≈ 1.97 × 10^5 Pa (slightly lower pressure in the faster region, as expected).

A typical write-up would just report this scalar P2 and maybe a percentage drop, without clearly surfacing how noisy the flow measurements were.

3) SSM calculation (same magnitude + bounded alignment lane)

In Shunyaya Symbolic Mathematics (SSM), we:

Treat each ingredient as (m,a) in (-1,+1).
Combine the dynamic pressure term alignments for section 1 and 2. We conceptually have: P2 = P1 + (0.5 * rho * v1**2) - (0.5 * rho * v2**2) So we first build alignments for 0.5 * rho * v1**2 and 0.5 * rho * v2**2. For each section’s dynamic pressure term, we combine density and velocity lanes by adding rapidities: a_dyn1 := tanh(atanh(a_rho_c) + atanh(a_v1_c)) a_dyn2 := tanh(atanh(a_rho_c) + atanh(a_v2_c)) (Here we keep it simple and fold the v**2 effect into the velocity-lane semantics, not into the SSM math structure itself.)
Then we combine the pressure and dynamic terms via a “signed sum” view:
- P2 is P1 plus a contribution from dyn1 and minus a contribution from dyn2.
- For this POC, we approximate the net alignment as a weighted sum over the three contributing terms:
a_c := clamp(a, -1+eps, +1-eps) u := atanh(a_c) U += w * u with w := |contribution_magnitude|^gamma W += w a_P2 := tanh( U / max(W, eps) ) where the contributions are roughly:
- +P1_m with lane P1_a
- +0.5 * rho_m * v1_m**2 with lane a_dyn1
- -0.5 * rho_m * v2_m**2 with lane a_dyn2
We use their absolute magnitudes for weights, so each sizeable term’s posture flows into a_P2.
Keep the pressure magnitude as the classical P2_m: P2_m = P1_m + 0.5 * rho_m * (v1_m**2 - v2_m**2) phi((P2_m, a_P2)) = P2_m

So:

The pressure magnitude stays about 1.97 × 10^5 Pa.
The bounded alignment lane a_P2 tells you how “reliable” this Bernoulli-based pressure estimate is for this particular measurement snapshot.

4) Tiny script (copy-paste)

# scenario_L07_bernoulli.py  (ASCII-only, top-level prints)

import math

def clamp(a, e=1e-6):
    return max(-1 + e, min(1 - e, float(a)))

def ssm_align_sum(a_list, eps=1e-6):
    """
    Sum of hyperbolic rapidities:
    a_out := tanh(atanh(a1_c) + atanh(a2_c) + ...)
    """
    U = 0.0
    for a_raw in a_list:
        a = clamp(a_raw, eps)
        U += 0.5 * math.log((1.0 + a) / (1.0 - a))
    return math.tanh(U)

def ssm_align_weighted(pairs, gamma=1.0, eps=1e-12):
    """
    pairs: iterable of (a_raw, m_term)
    where m_term is the magnitude of that contribution.
    """
    U = 0.0
    W = 0.0
    for a_raw, m_term in pairs:
        a = clamp(a_raw)
        # atanh(a) = 0.5 * ln((1+a)/(1-a))
        u = 0.5 * math.log((1.0 + a) / (1.0 - a))
        w = abs(float(m_term)) ** gamma
        U += w * u
        W += w
    return math.tanh(U / max(W, eps))

# 1) law-specific inputs: Bernoulli between section 1 and section 2
# P2 = P1 + 0.5 * rho * (v1**2 - v2**2)

# density (m, a)
rho_m, rho_a = 1000.0, +0.02    # kg/m^3

# upstream pressure (m, a)
P1_m, P1_a = 200000.0, +0.10    # Pa

# velocities (m, a)
v1_m, v1_a = 1.5, +0.30         # m/s
v2_m, v2_a = 3.0, +0.20         # m/s

# 2) classical magnitude for P2

P2_m = P1_m + 0.5 * rho_m * (v1_m**2 - v2_m**2)

# 3) SSM alignments

# dynamic pressure alignments for sections 1 and 2
a_dyn1 = ssm_align_sum([rho_a, v1_a])
a_dyn2 = ssm_align_sum([rho_a, v2_a])

# contribution magnitudes
dyn1_m = 0.5 * rho_m * (v1_m**2)
dyn2_m = 0.5 * rho_m * (v2_m**2)

# we treat P2 as composed of: +P1, +dyn1, -dyn2
# for posture, we use absolute magnitudes as weights
a_P2 = ssm_align_weighted(
    [
        (P1_a,   P1_m),
        (a_dyn1, dyn1_m),
        (a_dyn2, dyn2_m),
    ],
    gamma=1.0,
    eps=1e-12,
)

print("Classical:")
print("  P1 =", f"{P1_m:.0f}", "Pa")
print("  P2 =", f"{P2_m:.0f}", "Pa")

print("SSM (downstream pressure lane):")
print("  P2 =", f"m={P2_m:.0f}, a={a_P2:+.4f}")

You can attach bands in your shared runner, for example:

|a| < 0.20 → A+ (calm Bernoulli snapshot)
0.20 <= |a| < 0.50 → A0 (borderline)
|a| >= 0.50 → A- (stressed / noisy flow state)

5) What to expect

Numerically:

Classical: P2 only slightly lower than P1 (around 1.96–1.97 × 10^5 Pa).
SSM:
- m = P2_m (same magnitude as classical)
- a_P2 ≈ +0.1039 under drift-positive semantics

Under the sample band policy:

|a| < 0.20 → A+ (calm Bernoulli snapshot)
0.20 <= |a| < 0.50 → A0 (borderline)
|a| >= 0.50 → A− (stressed / noisy flow state)

we get:

a_P2 ≈ +0.1039 → comfortably in A+ (calm).

Interpretation:

The downstream pressure P2 is numerically reasonable and posturally calm for this snapshot.
Even though the pump and velocities carry some drift (v1_a, v2_a, P1_a non-zero), the overall posture remains low-risk for this particular configuration.

If the pump were smoother and velocities even more stable (smaller v1_a, v2_a, P1_a), then for the same geometry and flow regime you would see:

a_P2 move even closer to 0, and
SSM ≡ Classical not just in magnitude, but also in posture, marking that Bernoulli snapshot as a very calm, high-confidence pressure estimate.

6) Why this helps in the real world

Fluid labs can distinguish between “a nice Bernoulli example” and “a numerically nice but noisy run”, using the alignment lane a as a compact, quantitative posture signal.
Process engineers can see which P2 estimates are safe to rely on for tuning pumps, valves, or orifices, and which ones might have come from jittery operating conditions.
Dashboards in industrial systems can color or band derived pressures by a, revealing where derived values look fine but are fed by unstable signals.

7) License and scope

License. CC BY-NC 4.0 (non-commercial, attribution required).
Scope. Observation-only; not for critical use.

This POC is for thinking, experimentation, and education around bounded classical laws. It is not a safety case, design guarantee, or regulatory tool.

Navigation

Previous: POC Bounded Classical – Law L06: Conservation of Momentum – SSM
Next: POC Bounded Classical – Law L08: Snell’s Law – SSM

Disclaimer (summary):
All Shunyaya Symbolic Mathematics (SSM) POCs for popular laws are observation-only examples and must not be used for design, certification, or safety-critical decisions.

A bounded two-lane view of Bernoulli’s equation so the same downstream pressure P2 can be tagged calm, borderline, or stressed.

A bounded two-lane view of Bernoulli’s equation so the same downstream pressure `P2` can be tagged calm, borderline, or stressed.