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Chapter 31
Bernoulli and Continuity Flows

Fluid and Mechanical Media
Author: Mark Anthony Higgins
© Mark Anthony Higgins 11 October 2025
Revised Date: 23 February 2026
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Flow Algebra 
flow "Bernoulli and Continuity Flows" v1.0

# ---------- PURPOSE ----------
# Expresses fluid motion through energy conservation (Bernoulli) and mass conservation (Continuity)
# as coherent boundary flows. Unifies pressure, velocity, and potential energy as manifestations of
# field coherence in motion. Includes compressibility, coherence factor (IB), and loss terms for
# viscous or incoherent flow.

# ---------- PARAMETERS ----------
params:
  units: "SI"
  rho:   positive           # fluid density [kg/m³]
  IB:    0.0..1.0           # boundary coherence index (1 = ideal flow, 0 = fully dissipative)
  g:     positive = 9.80665 # gravitational acceleration [m/s²]
  mu:    real?              # dynamic viscosity [Pa·s], optional
  include_viscous_loss: true
  include_compressibility: false

# ---------- STATE ----------
state Streamline:
  p:  real        # pressure [Pa]
  v:  real        # flow velocity [m/s]
  h:  real        # height [m]
  phi: real       # potential energy per unit mass [J/kg]
  zeta: real      # total head [J/kg]
  loss: real      # energy loss term [J/kg]

state FlowField:
  A[]:  positive  # cross-sectional areas [m²]
  v[]:  real      # local velocities [m/s]
  rho[]: positive # local densities [kg/m³]
  Q:    real      # volumetric flow rate [m³/s]

# ---------- INVARIANTS (CLASSICAL FORM) ----------
invariants:
  # --- Bernoulli’s principle (ideal flow, along a streamline) ---
  # p + ½ρv² + ρgh = constant
  if params.IB == 1 and not params.include_viscous_loss:
    Streamline.zeta = Streamline.p/params.rho + 0.5*Streamline.v**2 + params.g*Streamline.h
    d/ds(Streamline.zeta) = 0
    note: "Perfect coherence (IB=1) → total head constant."

  # --- Generalised Bernoulli (with coherence & loss terms) ---
  if params.IB < 1 or params.include_viscous_loss:
    Streamline.loss = (1 - params.IB)*(0.5*Streamline.v**2) + viscous_term
    Streamline.zeta = Streamline.p/params.rho + 0.5*Streamline.v**2 + params.g*Streamline.h - Streamline.loss
    note: "Loss term represents incoherence or viscous dissipation."

  # --- Optional viscous term (Poiseuille-like correction) ---
  if params.include_viscous_loss:
    viscous_term = 32 * params.mu * FlowField.Q / (pi * FlowField.A**2)
  else:
    viscous_term = 0

  # --- Continuity equation ---
  # A₁v₁ = A₂v₂ = Q (for incompressible steady flow)
  if not params.include_compressibility:
    FlowField.Q = FlowField.A[0]*FlowField.v[0]
    for i in range(len(FlowField.A)):
      assert abs(FlowField.A[i]*FlowField.v[i] - FlowField.Q) < 1e-9
  else:
    # compressible form: ρ₁A₁v₁ = ρ₂A₂v₂
    FlowField.Q = FlowField.rho[0]*FlowField.A[0]*FlowField.v[0]
    for i in range(len(FlowField.A)):
      assert abs(FlowField.rho[i]*FlowField.A[i]*FlowField.v[i] - FlowField.Q) < 1e-9

# ---------- TRANSFORMS ----------
transforms:
  # Compute velocity from Bernoulli between two points (1, 2)
  velocity_from_pressure_drop:
    Δp = p1 - p2
    v2 = sqrt((2/params.rho) * (Δp + params.rho*params.g*(h1 - h2)))
    note: "Derived from p₁ + ½ρv₁² + ρgh₁ = p₂ + ½ρv₂² + ρgh₂"

  # Compute flow rate through an orifice
  orifice_flow:
    FlowField.Q = Cd * FlowField.A[0] * sqrt(2*(p_up - p_down)/params.rho)
    Cd = params.IB             # discharge coefficient proportional to coherence
    note: "IB acts as effective discharge coefficient; lower IB = more loss."

  # Compute head loss as deviation from ideal Bernoulli
  head_loss:
    h_loss = (1 - params.IB) * (Streamline.v**2 / (2*params.g))

# ---------- MONITORS ----------
monitors:
  coherence_limit:
    assert 0 <= params.IB <= 1
  incompressible_flow_check:
    if not params.include_compressibility:
      assert all(abs(FlowField.A[i]*FlowField.v[i] - FlowField.Q) < 1e-9)
  energy_consistency:
    # Check approximate energy conservation when IB ≈ 1
    delta_zeta = abs((Streamline.p/params.rho + 0.5*Streamline.v**2 + params.g*Streamline.h) - Streamline.zeta)
    assert delta_zeta < 1e-3 when params.IB > 0.9

# ---------- SCENES ----------
scenes:

  ideal_pipe_flow:
    given params:
      rho: 1000
      g: 9.81
      IB: 1.0
    given FlowField:
      A: [0.002, 0.001]
      v: [2.0, 4.0]
    expect:
      FlowField.Q = A[0]*v[0] = 0.004
      note: "Ideal continuity satisfied — doubling velocity halves area."

  bernoulli_height_difference:
    given params:
      rho: 1000
      IB: 1.0
      g: 9.81
    given Streamline:
      p: 101325
      v: 0
      h: 10
    transforms:
      velocity_from_pressure_drop:
        p2 = 101325 - 1000*9.81*10
    expect:
      v2 ≈ sqrt(2*g*h) ≈ 14 m/s
      comment: "Potential energy converts entirely into kinetic energy."

  viscous_loss_demo:
    given params:
      rho: 1000
      mu: 0.001
      IB: 0.85
      include_viscous_loss: true
    given FlowField:
      A: [0.002]
      v: [3.0]
      Q: 0.006
    expect:
      Streamline.loss > 0
      Streamline.zeta reduced by ~15%
      comment: "Lower coherence (IB<1) simulates viscous and turbulent losses."

# ---------- NOTES ----------
notes:
  - Bernoulli’s equation describes energy coherence along a streamline:
    \( p + \tfrac{1}{2}\rho v^2 + \rho g h = \text{constant} \).
  - The Continuity equation enforces conservation of mass:
    \( A v = \text{constant} \) (incompressible) or \( \rho A v = \text{constant} \) (compressible).
  - The boundary coherence index \( I_B \) maps how much total energy is preserved
    along the streamline. IB = 1 → perfectly coherent (inviscid, laminar); IB < 1 → dissipative or turbulent.
  - Viscous or incoherent flow introduces entropy and head loss, reducing total head (zeta).
  - In HST terms, Bernoulli’s constant is the scalar of **boundary tension coherence** — the balance between pressure curvature, velocity phase, and gravitational potential.
  - Continuity represents **field conservation** — the preservation of flow topology across boundary cross-sections.
  - The coherence-limited form generalises both classical and quantum fluid flows: turbulence and superfluidity become opposite ends of the same coherence spectrum.
Overview

At the heart of fluid motion lies coherence — the ability of pressure, velocity, and potential to remain phase-linked as energy travels through space.
Bernoulli’s equation and the continuity law express this harmony.
One tracks the energy coherence along a streamline; the other ensures spatial continuity across boundaries.
Together they form the foundation of coherent flow: the conservation of both tension and form within a moving field.

🔹 1. Bernoulli’s Law — Energy Coherence Along a Streamline
In its classical form:

$$ p + \frac{1}{2}\rho v^2 + \rho g h = \text{constant}. $$

Each term represents a different curvature mode of the same boundary field:

  • p/ρp / \rho: pressure potential — stored compression energy,
  • 12v2 \tfrac{1}{2}v^2: kinetic curvature — flow alignment in space,
  • ghg h: gravitational potential — geometric elevation term.

When coherence is perfect (IB=1I_B = 1), the sum remains constant along a streamline: energy redistributes but total curvature is preserved.
This expresses the fundamental law of coherent exchange — energy neither created nor lost, only shifted between its spatial forms.

🌀 2. Coherence Loss and Viscous Dissipation

Real flows are rarely ideal.
Viscous friction, turbulence, and microscopic scattering introduce boundary incoherence, breaking perfect curvature symmetry.
The generalised Bernoulli equation becomes:

$$ p + \frac{1}{2}\rho v^2 + \rho g h = \text{constant} - \text{loss}. $$

with loss parameterised as:

$$ \text{loss} = (1 - I_B)\,\frac{1}{2}\rho v^2. $$

Here, IBI_B (the boundary coherence index) measures how faithfully energy phase is conserved along the streamline.

  • IB=1I_B = 1: inviscid, laminar, perfectly coherent.
  • IB<1I_B < 1: turbulent, dissipative, coherence degraded.

In HST terms, these losses correspond to boundary decoherence — energy escaping the geometric loop of the flow and dispersing as entropy or micro-vorticity.

⚖️ 3. Continuity Equation — Conservation of Flow Topology
For steady, incompressible flow:

$$ A_1 v_1 = A_2 v_2 = Q. $$

where AA is cross-sectional area, vv is mean velocity, and QQ is volumetric flow rate.
In the compressible form:

$$ \rho_1 A_1 v_1 = \rho_2 A_2 v_2. $$

This expresses not just mass conservation, but field continuity:
each streamtube preserves the topological product of density, area, and velocity — a measure of coherent flux geometry.

In HST interpretation, continuity is a boundary consistency law.
The fluid’s holographic surface retains constant curvature flux across its cross-sections, ensuring information (mass-energy distribution) remains self-consistent as the field moves.

🔸 4. Pressure–Velocity Coupling as Curvature Feedback
Pressure and velocity are not opposites but complementary curvature responses:
  • Increased velocity reduces static pressure because curvature shifts from compression to motion.
  • Deceleration reverses the process: kinetic curvature folds back into static pressure.

This exchange forms a coherence loop within the flow — one that continually balances compression and translation.
It is the same logic that governs optical wave propagation, electromagnetic tension, and even quantum amplitude exchange: coherence moves, but it does not vanish.

🌐 5. From Fluids to Fields

Bernoulli’s and continuity equations together describe flow coherence — the local conservation of energy and topology.
When written through HST logic, they reveal the general rule:

$$ \nabla \!\cdot (\rho \mathbf{v}) = 0, \qquad \frac{D}{Dt}\!\left(\frac{p}{\rho} + \frac{1}{2}v^2 + g h\right) = -\,\frac{d\,\text{loss}}{dt}. $$

These are the boundary-symmetric forms of motion — local energy curvature and global continuity evolving as one equation of coherence.
Turbulence, then, is not random chaos but coherence fragmentation — a breakdown of shared curvature across scales.

🧠 Summary

Bernoulli’s law and continuity are not isolated principles; they are two faces of coherence conservation.
Bernoulli describes how energy tension stays balanced along a streamline.

Continuity ensures that the geometry of that tension remains unbroken.

Together they express the same deeper symmetry:
that motion, structure, and energy are unified through boundary coherence —
the seamless flow of curvature through space.

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