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NATURAL HDGL–NAVIER–STOKES CLOSURE FORMULATION
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STARTING EQUATION
ρ(∂u/∂t + (u·∇)u) = −∇p + μ∇²u
HDGL RECIPROCAL STRUCTURE
u ↦ Φ = ∏ᵢ₌₁³ Cᵢ
∇ ↦ Φ⁻¹
S = Φ + Φ⁻¹
ΦΦ⁻¹ = 1
NATURAL DIMENSIONLESS IDENTIFICATION
Re = ρUL/μ ∝ Φ
Re⁻¹ = μ/(ρUL) ∝ Φ⁻¹
Re·Re⁻¹ = 1
PRESSURE CHANNEL
Eu = Δp/(ρU²)
Eu remains an independent pressure ratio.
It is NOT identified with Φ⁻¹.
The reciprocal pair belongs naturally to the inertial/viscous
scaling:
Φ ↔ Re
Φ⁻¹ ↔ Re⁻¹
NON-DIMENSIONAL NAVIER–STOKES FORM
After choosing the characteristic scales,
∂u/∂t + (u·∇)u = −Eu∇p + Φ⁻¹∇²u
(up to the chosen normalization of the pressure field).
HDGL SINGULAR BRANCH
Φ → ∞
⇒ Φ⁻¹ → 0
⇒ Re⁻¹ → 0
⇒ (1/Re)∇²u → 0
The apparent singular regime therefore does not require
u → 0
and does not require
Δp → 0.
Instead, the reciprocal channel collapses while the state
remains subject to the HDGL closure.
HDGL GRADIENT CLOSURE
Φ → ∞
⇒ ||∇u|| ≤ G_max
The required HDGL statement is therefore a FINITE-GRADIENT
CLOSURE rather than a forced velocity collapse.
IF THE HDGL SUBSTRATE ALSO ESTABLISHES
||∇u|| ≤ G_max
⇒ ||u|| ≤ U_max
then
Φ → ∞
⇒ ||∇u|| ≤ G_max
⇒ ||u|| ≤ U_max
and consequently
lim(t→t_c) ||u(x,t)|| ≠ ∞.
FINAL CLOSURE
Φ ↔ Re
Φ⁻¹ ↔ Re⁻¹
ΦΦ⁻¹ = 1
while
Eu = Δp/(ρU²)
remains an independent pressure channel.
Thus the HDGL mechanism is not
"Eu → 0 ⇒ Δp → 0 ⇒ u → 0"
but rather
Φ → ∞
⇒ Φ⁻¹ → 0
⇒ viscous reciprocal channel → 0
⇒ HDGL closure prevents ||∇u|| from becoming unbounded
⇒ ||u|| remains bounded
⇒ no velocity blow-up.
CORE HDGL FORM
STATE → TRANSFORM → CLOSURE
u → (Φ, Φ⁻¹) → bounded gradient → bounded velocity
The mathematical burden is therefore concentrated in the
HDGL closure law
Φ → ∞ ⇒ ||∇u|| ≤ G_max,
which must be derived from the internal HDGL substrate dynamics
rather than asserted as an external Navier–Stokes condition.
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