[#8] Navier-Stokes Simplification and the Rotational Pressure Gradient
Part II/Anatomy of a Particle: Cavitation Vortex & the Birth of Mass/Ch. 1 §1.1–1.2
▶ Part II begins — The Birth of a Particle
✦ Post #8 · Part II, Ch. 1 §1.1–1.2
Part II — Anatomy of a Particle: Cavitation Vortex & the Birth of Mass
Navier-Stokes Simplification
and the Rotational Pressure Gradient
Where did particles come from? — The mathematical inevitability of Space Fluid tearing itself apart
Author Kim Donghak (HeeRim)
Section Part II · Ch. 1 · §1.1–1.2
Type Equation Derivation
Series progress — 8 / 26 posts
§ 1.1 — Introduction
Part II's Question: How Was Matter Born?
In Part I, we established the existence of the Space Fluid (Axiom 1) and measured its properties: viscosity η_s ≈ 10⁹ Pa·s, sound speed c. Now Part II confronts the deeper question.
Electrons, quarks, protons — these fundamental particles are not solid grains that appeared from nowhere in empty space.
They are structures formed when the Space Fluid,
unable to sustain extreme rotational stress,
tears itself open.
The mathematical tool that shows this happening as a necessity — not a coincidence — is the Navier-Stokes (N-S) governing equation: the fluid-mechanics version of Newton's second law (F = ma), which describes the motion of any fluid with complete generality.
§ 1.1 — Starting Point
The Navier-Stokes Equation (Full Form)
With Space Fluid density ρ_s, viscosity μ_s, velocity vector u_s, and static pressure P, the governing equation is:
Navier-Stokes equation — applied to Space Fluid
ρ_s [ ∂u_s/∂t + (u_s · ∇)u_s ] = −∇P + μ_s ∇²u_s
ρ_s : Space Fluid density
u_s : Space Fluid velocity vector
P : static pressure
μ_s : Space Fluid dynamic viscosity (≈ η_s)
To apply this to "the instant a fundamental particle is born — the extreme-rotation limit," three physical assumptions are introduced to strip the equation down to its essential form.
§ 1.1 — Simplification
Three Assumptions Reduce the Equation
1
Steady Flow
∂u_s/∂t = 0
Once a particle settles into its orbit, its velocity is no longer changing in time. We capture the moment of stabilised rotation. The time-derivative term vanishes from the left side.
2
No Body Forces
f_body = 0
At the microscopic scale, with a single spontaneously forming vortex, there are no external volume forces such as gravity. We describe the isolated, self-generating case.
3
Axisymmetric Pure Rotation (Swirl)
u_s = u_θ(r) · ê_θ
A vortex revolves around its central axis in circles. In cylindrical coordinates (r, θ, z), only the azimuthal velocity u_θ is dominant. Radial and axial velocities are negligible and drop out.
§ 1.2 — Result of Simplification
The Equation That Emerges — Intuitive and Clean
Applying all three assumptions and retaining only the radial (r-direction) component yields a remarkably direct equation:
Simplified radial N-S equation
− ρ_s · u_θ² / r = −∂P/∂r
Left side
Centrifugal force — the tendency of rotating fluid to fly outward. Grows explosively as rotational velocity u_θ increases or radius r shrinks.
Right side
Pressure gradient — the rate of pressure change with distance. An inward pressure difference must exactly balance the centrifugal force to hold the vortex together.
Physical meaning
"The faster the fluid rotates, the more steeply pressure must drop toward the centre."
The equality ρ_s · u_θ²/r = ∂P/∂r means that for a vortex to maintain its shape, a low-pressure zone must form at its core. The faster the rotation, the deeper the pressure drop. This is the same result Bernoulli's equation predicts — and it sets up the catastrophic threshold that produces a particle.
§ 1.2 — Critical Condition
When Core Pressure Hits Zero — the Cavitation Threshold
Integrating this equation over r gives the actual pressure at the vortex core. With P_∞ as the ambient background pressure of the Space Fluid:
As rotational speed u_θ increases, the subtracted term grows and P_center falls. At a critical rotational speed, P_center approaches zero. But fluid pressure cannot go below zero — the absolute vacuum. When that threshold is crossed, the Space Fluid cannot simply continue rotating. Something else must happen.
Post #9 covers exactly that moment — when the Space Fluid physically tears, forming a void bubble (Cavitation). This is the mechanism by which a particle is born.
§1.1–1.2 Core Insight
N-S equation + three assumptions → "The faster the rotation, the lower the core pressure." (Centrifugal force = pressure gradient)
This simple equation is the mathematical prelude to Cavitation — the birth of matter. No external energy was injected, no parameters invented. Only the N-S equation and Axiom 1 (space is fluid). The conditions for particle creation emerge automatically from the governing equation of fluid mechanics. Nature follows this equation and tears itself open.
▶ Next post — #9 · Part II §1.3
Space Tears — Cavitation and the Birth of a Particle
When P_center reaches the critical threshold, the Space Fluid tears and forms a Void Bubble — a Cavitation Vortex Core with a physical boundary. This is a particle. The same mechanism that destroys ship propellers at high speed is happening at the microscopic scale of the cosmos.