Skip to main content

Chapter 32
Pascal and Archimedes Pressure Balance

Fluid and Mechanical Media
Author: Mark Anthony Higgins
© Mark Anthony Higgins 11 October 2025
Revised Date: 23 February 2026
Audio Overview
Previous Chapter 31
Next Chapter 33
Flow Algebra 
flow "Pascal and Archimedes — Pressure Balance" v1.0

# ---------- PURPOSE ----------
# Unifies Pascal’s law (pressure transmission) and Archimedes’ principle (buoyancy)
# as one pressure-field balance. Adds a boundary coherence index (IB) to account for
# losses (leaks, compliance, trapped gas, surface roughness) and optional compressibility.

# ---------- PARAMETERS ----------
params:
  units: "SI"
  rho_fluid: positive        # fluid density [kg/m^3]
  g:         positive = 9.80665
  IB:        0.0..1.0        # coherence of pressure transmission (1 = ideal)
  compressible: false        # if true, allow small density changes with p
  beta:      positive?       # bulk compressibility [1/Pa], when compressible = true
  gamma_cap: real?           # optional surface tension / capillary head [Pa] adjustment
  p_atm:     real = 101325   # ambient reference pressure [Pa]

# ---------- STATE ----------
state Fluid:
  p0: real                   # reference pressure at free surface [Pa]
  z0: real = 0               # reference elevation [m]

state PascalCell:
  A_in:  positive            # input piston area [m^2]
  A_out: positive            # output piston area [m^2]
  F_in:  real                # applied input force [N]
  F_out: real                # delivered output force [N]
  p_in:  real
  p_out: real
  loss_pascal: real          # effective pressure loss [Pa]

state Hydrostatic:
  z:     real[]              # depths/elevations [m] (+upwards)
  p:     real[]              # pressures at positions z [Pa]

state Body:                   # object (solid) for buoyancy analysis
  V:       positive          # displaced volume when fully submerged [m^3]
  V_disp:  real              # actual displaced volume [m^3]
  rho_obj: positive          # object density [kg/m^3]
  depth:   real              # immersion depth reference [m]
  F_b:     real              # buoyant force [N]
  F_w:     real              # weight [N]
  F_net:   real              # net vertical force [N]
  IB_b:    0.0..1.0 = 1.0    # buoyancy coherence (surface roughness/entrained bubbles)

# ---------- INVARIANTS ----------
invariants:

  # --- Hydrostatic pressure field (with optional capillary term) ---
  # p(z) = p0 + rho g (z0 - z) + gamma_cap
  for k in range(len(Hydrostatic.z)):
    if not params.compressible:
      Hydrostatic.p[k] = Fluid.p0 + params.rho_fluid * params.g * (Fluid.z0 - Hydrostatic.z[k]) + (params.gamma_cap or 0)
    else:
      # small-strain barotropic correction: rho ≈ rho0 * (1 + beta*(p - p0))
      # Use linearised form for clarity (engine may iterate if desired)
      dp = params.rho_fluid * params.g * (Fluid.z0 - Hydrostatic.z[k])
      Hydrostatic.p[k] = Fluid.p0 + dp + (params.gamma_cap or 0)

  # --- Pascal’s law with coherence (transmission of pressure) ---
  # Ideal: p_in = p_out = F/A. With loss: p_out = IB * p_in.
  PascalCell.p_in  = PascalCell.F_in  / PascalCell.A_in
  PascalCell.loss_pascal = (1 - params.IB) * PascalCell.p_in
  PascalCell.p_out = PascalCell.p_in - PascalCell.loss_pascal
  PascalCell.F_out = PascalCell.p_out * PascalCell.A_out

  # --- Archimedes buoyancy (with coherence) ---
  # F_b = IB_b * rho_fluid * g * V_disp
  Body.F_b = Body.IB_b * params.rho_fluid * params.g * Body.V_disp
  Body.F_w = Body.rho_obj   * params.g * Body.V
  Body.F_net = Body.F_b - Body.F_w

  # --- Float / sink conditions (qualitative guards) ---
  float_condition:
    # At static equilibrium for a floating body: F_b = F_w with 0 <= V_disp <= V
    if 0 <= Body.V_disp <= Body.V:
      assert abs(Body.F_b - Body.F_w) <= 1e-6 or Body.F_net == 0

# ---------- TRANSFORMS ----------
transforms:

  # Compute required input force for a target output force (hydraulic press)
  pascal_reverse:
    PascalCell.F_in = (PascalCell.F_out / PascalCell.A_out) * PascalCell.A_in / max(params.IB, 1e-12)

  # Determine displaced volume for a floating body (solve V_disp from F_b = F_w)
  set_float_equilibrium:
    Body.V_disp = clamp(Body.F_w / (Body.IB_b * params.rho_fluid * params.g), 0, Body.V)

  # Pressure at depth utility
  pressure_at_depth:
    p_depth = Fluid.p0 + params.rho_fluid * params.g * (Fluid.z0 - z_query) + (params.gamma_cap or 0)

# ---------- MONITORS ----------
monitors:
  coherence_bounds:
    assert 0 <= params.IB <= 1
    assert 0 <= Body.IB_b  <= 1

  hydraulic_gain_check:
    # Ideal gain ~ A_out / A_in scaled by IB
    gain = PascalCell.F_out / max(PascalCell.F_in, 1e-12)
    gain_ideal = params.IB * (PascalCell.A_out / PascalCell.A_in)
    assert abs(gain - gain_ideal) <= 1e-9

  buoyancy_limits:
    assert 0 <= Body.V_disp <= Body.V

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

  # 1) Hydraulic press with coherence loss
  hydraulic_press_demo:
    given params:
      rho_fluid: 850        # e.g., oil
      IB: 0.97
    given PascalCell:
      A_in:  0.0005         # 5e-4 m^2 (small plunger)
      A_out: 0.02           # 2e-2 m^2  (large piston)
      F_in:  200            # N
    expect:
      p_in   = F_in / A_in
      p_out  = IB * p_in
      F_out  = p_out * A_out
      note: "Output force close to ideal gain (A_out/A_in) but reduced by IB."

  # 2) Hydrostatic pressure difference
  hydrostatic_column:
    given params:
      rho_fluid: 1000
      g: 9.81
    given Fluid:
      p0: 101325
      z0: 0
    given Hydrostatic:
      z: [0, -2, -5]        # 0 m, 2 m below, 5 m below
    expect:
      p[1] - p[0] ≈ rho_fluid * g * 2
      p[2] - p[0] ≈ rho_fluid * g * 5

  # 3) Floating block (find immersion fraction)
  floating_block:
    given params:
      rho_fluid: 1000
    given Body:
      V: 0.02               # m^3
      rho_obj: 600          # kg/m^3 (wood)
      IB_b: 0.99
    transforms:
      set_float_equilibrium: {}
    expect:
      immersion_fraction = V_disp / V ≈ rho_obj / (IB_b * rho_fluid)   # ~ 0.606
      F_net ≈ 0

  # 4) Sinking object (fully submerged)
  sinking_solid:
    given params:
      rho_fluid: 1000
    given Body:
      V: 0.01
      rho_obj: 3000
      V_disp: 0.01
      IB_b: 1.0
    expect:
      F_b = rho_fluid * g * V
      F_w = rho_obj   * g * V
      F_net = F_b - F_w < 0   # sinks

# ---------- NOTES ----------
notes:
  - Pascal (ideal): pressure is transmitted undiminished → \( p = F/A \) uniform in a connected static fluid.
  - With coherence factor \( I_B \), effective transmitted pressure is \( p_{\text{out}} = I_B\, p_{\text{in}} \).
  - Archimedes: buoyant force equals the weight of displaced fluid:
      \( F_b = \rho_{\text{fluid}} g V_{\text{disp}} \).
    Coherence \( I_{B_b} \) moderates real-world effects (entrained gas, wetting, surface roughness).
  - Hydrostatics: \( p(z) = p_0 + \rho g (z_0 - z) \) (plus optional capillary correction).
  - Floating condition: \( F_b = F_w \Rightarrow V_{\text{disp}}/V = \rho_{\text{obj}} / (\rho_{\text{fluid}} I_{B_b}) \) (clamped to \([0,1]\)).
  - Hydraulic gain: ideal \( F_{\text{out}}/F_{\text{in}} = A_{\text{out}}/A_{\text{in}} \); coherence reduces it by \( I_B \).
Overview

Pascal’s and Archimedes’ principles are two expressions of the same law of pressure coherence.
One governs how pressure transmits within a continuous fluid; the other describes how that pressure acts upon an immersed boundary.
Together, they form the static counterpart of Bernoulli and continuity — mapping not motion, but tension balance within a resting field.

🔹 1. Pascal’s Law — Pressure as Boundary Equilibrium

In an enclosed, static fluid, pressure acts equally in all directions and transmits uniformly through the medium.

$$ p = \frac{F}{A} = \text{constant (for a coherent fluid).} $$

If a force FinF_{\mathrm{in}} is applied to a piston of area AinA_{\mathrm{in}}, the resulting pressure is transmitted to an output piston of area AoutA_{\mathrm{out}}:

$$ p_{\mathrm{in}} = p_{\mathrm{out}} = \frac{F_{\mathrm{in}}}{A_{\mathrm{in}}}. $$

Thus the output force becomes:

$$ F_{\mathrm{out}} = \frac{A_{\mathrm{out}}}{A_{\mathrm{in}}}\, F_{\mathrm{in}}. $$

This geometric amplification is the foundation of hydraulic systems — force multiplied through pressure uniformity.
In practice, however, no system is perfectly coherent.
We introduce a coherence index IBI_B to represent loss due to compliance, leaks, trapped gas, or imperfect sealing:

pout​=IB​pin​,Fout​=IB​Ain​Aout​​Fin​.

  • When IB=1I_B = 1, the fluid behaves as an incompressible continuum.
  • When IB<1I_B < 1, coherence diminishes — the boundary’s internal memory is imperfect, and part of the energy

$$ p(z) = p_0 + \rho g (z_0 - z). $$

Here p0p_0 is the surface pressure at height z0z_0, ρ\rho is the fluid density, and gg is gravitational acceleration.
This simple relation defines the pressure field within the fluid — a vertical coherence gradient where potential energy translates directly into pressure tension.

When small capillary or surface effects are relevant, a correction term may be added:

$$ p(z) = p_0 + \rho g (z_0 - z) + \gamma_{\mathrm{cap}}. $$

representing curvature or surface tension acting at the fluid boundary.

🧩 3. Archimedes’ Principle — Coherence Between Medium and Body

When a body is placed in a fluid, the surrounding pressure field creates an upward resultant known as buoyant force:

$$ F_b = \rho_{\mathrm{fluid}}\, g\, V_{\mathrm{disp}}. $$

It equals the weight of the displaced fluid.
In reality, this upward curvature coupling is not always perfect — surfaces trap bubbles, wetting may be partial, or micro-roughness can break phase contact.
A buoyancy coherence index IBb
I_{B_b} accounts for this:

$$ F_b = I_{B_b}\, \rho_{\mathrm{fluid}}\, g\, V_{\mathrm{disp}}. $$

The balance between buoyancy and weight determines the system’s behaviour:

$$ F_{\mathrm{net}} = F_b - F_w = g\, \big(\rho_{\mathrm{fluid}} I_{B_b} V_{\mathrm{disp}} - \rho_{\mathrm{obj}} V\big). $$

  • If Fb>FwF_b > F_w→ the object rises (undercoherent or light).
  • If Fb=FwF_b = F_w → it floats (balanced coherence).
  • If Fb<FwF_b < F_w→ it sinks (overdense or incoherent coupling).

In HST terms, floating occurs when the boundary curvature of the solid and the fluid are phase-matched — the surface mediates equal pressure flux in both directions.

🔸 4. Unified Coherence Field — Static Energy Gradient

Both Pascal and Archimedes describe static field coherence:

  • Pascal’s law maintains lateral coherence — equal pressure transmission through enclosed geometry.
  • Archimedes’ law maintains vertical coherence — gravitational curvature compensated by pressure rise with depth.

Together they satisfy:

\( \nabla p = \rho \mathbf{g}, \quad \text{and } p = \text{uniform at constant depth.} \)

They express equilibrium not as stillness, but as complete coherence — every part of the field perfectly aware of every other part through static tension.

🧠 Summary

Pascal’s and Archimedes’ laws are two expressions of the same geometric truth:
pressure is coherent curvature within a static field.

  • In Pascal, that coherence spreads laterally, transmitting tension through the medium.
  • In Archimedes, it rises vertically, countering weight through displaced curvature.

Together they reveal that a static fluid is not an absence of motion, but a perfectly resolved balance of forces —
a standing wave of coherence between boundary, weight, and depth.

When IB=1I_B = 1, the field is whole; when IB<1I_B < 1, the geometry leaks memory.
The calm surface of a liquid hides a perfect equilibrium of invisible curvatures —
a static mirror of the same coherence that drives motion, light, and form.

Previous Chapter 31
Next Chapter 33
Related Chapters
Privacy Policy
Site Map
Kireth Research
38 Thorndale Croft, Wetwang, Driffield
East Riding of Yorkshire YO25 9XZ
Telephone:
01377 236594
Designed By
Kireth Web Design