Explaining the Origin of Quantum Phenomena via the Taylor Column in Viscous Rotating Fluids

3.1 Viscous electromagnetic wave solution from the momentum theorem

In the infinite small flowing area with continuous parameters, according to the momentum theorem F≡D(mu)/Dt, there always exists the momentum equation in differential form [2] (in the following equation, the subscript j of the stress tensor σ_ji denotes the normal direction of the surface on which the stress acts, and the subscript i denotes the projection direction)

From Eq. (1), if the pressure ∂p_i/∂x_i and body force ρf_i are both relatively small one quantity level than the deviatoric stress tensor(τ_ji) and u_i = Const ≠ 0, the momentum equation with uniform kinematic viscosity v and expansion viscosity coefficient μ′ = 0 can be written as [1]:

In Eq. (2), the mass density ρ is now a variable (non-relativistic) and also a vector quantity. when u_i = Const ≠ 0, ρu_i in v∇2(ρui) becomes ρ_ic, Here in after, ρ is often denoted as:ρˆ. At this time, if both the term ∂(ρu_j)/∂x_j and ∂(∇·ρ)/∂x_i are also relative small quantities, then we have:

The parabolic conduction given by Eq. (3) has the wave solution, which amplitude attenuated with time (among k is wave vector, circular frequency ω = -ivk2 [5] and hereinafter imaginary unit i follows the Roman italic type)。

If taking the solution as a wave, the second-order spatial distribution scale R of ρ_i is the wavelength λ, the second-order time distribution scale δt is defined as the period T, and the scale U of convective velocity u_j is wavespeed c. So kinematic viscosity v=UR=Uλ=UcT=c2δt, Substitute it in Eq. (3) and take the time derivative. If the viscosity is uniform and changing with wave, furthermore make δt→∂t and has constraint δt∇2(∂ρˆ/∂t)≪∇2 ρˆ, available obtain:

This is the wave equation with constant amplitude ρ₀ and the wave velocity c is no longer infinite but a constant (now ρˆ is a vector quantity).

Then we define new variables: E=±iρˆ and B≡−δt∇∧E. Taking time derivative of B, if the condition: δ t∇∧ (∂ρ^/∂t)≪∇∧ρ^ is further satisfied, we can derive two governing equations of Maxwell’s equations for electromagnetic waves [6]. Here, E and B are the electric vector and magnetic vector respectively.

3.2 Analytical solution of the viscous Sullivan vortex

In the cylindrical coordinate system (r, θ, z), the analytical solution of the Sullivan vortex can be obtained by solving the continuity equation for incompressible flow, Dρ/Dt = 0 coupled with the Navier–Stokes equations. The specific expressions for the axial velocity V_z, radial velocity V_r, tangential velocity V_θ, tangential vorticity ω_θ, and axial vorticity ω_z of the vortex solution [3, 4, 7] (with the radial vorticity ω_r ≡ 0) are as follows:

In Eq. (9), ω₀ denotes the axial vorticity ω_z at r = 0 (i.e., the axial vorticity at the vortex center). The constraint conditions for this solution are as follows [4, 7]:

Steady state: ∂/∂t→0; axisymmetry: ∂/∂θ→0; geostrophic effect is negligible; whole-axis rigid shift of the tangential velocity V_θ and radial velocity V_r: ∂/∂z →0 with boundedness at r→0; kinematic viscosity and suction intensity a vary slowly, and the relation 4a² = D²–N² usually holds (here N denotes the static stability satisfying -g-ρ⁻¹ ∂p/∂z =N²z, and D denotes the transverse (tangential and radial) divergence). In other words, the physical quantity a incorporates net buoyancy and convergence/divergence effects.

If we substitute r=xν/|a| in Eq. (9), it is easy to find ω_z has no direct correlation with a and v. Therefore plot the velocity, vorticity, and streamlines of the Sullivan vortex according to Eqs. (5)–(9) with substituting r=xν/|a|, then Figure 1 shows three types of scales plotted when taking a = −1.0 and v =1.0:

Physical quantities in Figure 1 (color, unit): V_θ (green solid line, ω₀ν/|a|), V_r (blue, |a|ν), V_z (black, 2az), ω_z (red, z|a|3/ν), ω_θ (orange, z); horizontal axis r (ν/|a|).

Physical quantities in Figure 1 (right subfigure): vertical axis is ω_z; embedded streamlines correspond to Ψ(zv)=2.0 (green dots) and Ψ(zv)=-2.0 (green dashed line). The right subfigure also embeds the solution values under the mesoscale and the ellipse in the figure indicates its location.

The left subfigure shows that near r = 5.3 and r = 6.0, there exist numerical value of V_z (2az, black) or V_r(|a|ν, blue) on the order of 10⁶ and 10⁸, respectively. In contrast, the right subfigure indicates that ω_z at those locations is much smaller than 10⁻³⁰⁰ω₀ and is strictly zero.

Therefore, this mathematical solution can serve as a classical basis for the irrotational and extremely rapid motion of micro-particles.

3.3 Basic requirements for producing viscosity

The Navier–Stokes equations can be derived from the momentum given by Eq. (1) if mass conservation holds Dm/Dt=0 orDρ/Dt+ρ∇⋅V=0 [1]. Then according to the Taylor–Proudman theorem, if the disturbulence velocity of an incompressible (Dρ/Dt = 0) vortex satisfies the constraints of the theorem, the disturbulence velocity can undergo whole-axis rigid shift within the vortex, as a result the vortex will exhibit viscosity with respect to this disturbulence velocity.

In the Sullivan vortex solution, the tangential vorticity ω_θ contains no angular velocity, and only the axial vorticity ω_z incorporates vorticity. As shown in Figure 1, ω_z = 0 in the extremely high-speed region of the Sullivan vortex if the suction intensity a < 0. Therefore, the constraint ζ≫U/L required by the theorem must be satisfied by the vorticity of its spin. Moreover, the radial radius must be sufficiently small to ensure that the spin velocity is negligible compared to the orbital revolving velocity of the Sullivan vortex, thereby maintaining the momentum equation.

At this time, the disturbance velocity that induces viscosity originates from the environment surrounding the vortex surface, and the second-order scale R of the disturbance velocity corresponds to the wavelength λ of the environmental disturbance. Similarly, when the wave velocity c is extremely fast, the viscosity required for the wave solution is also comparable.

3.4 Main constraints for two viscous solutions: Uniform velocity, ultra-rapid rotation, and infinitesimal scale

As shown in Figure 1, the viscous vortex can exhibit extremely fast V_z or V_r if the suction intensity a < 0; these vortices with irrotational moving velocity are generated by the constraint ζ≫U/L. Therefore, if it is regarded as the disturbance velocity, then ∂V/∂z→0 always holds and the vortex can be easily realize the uniform velocity u_i = Const ≠ 0 required by Eqs. (2) and (3) at this region. Otherwise when the suction intensity a > 0 if If the vorticity ω is extremely large, keeping other terms at a smaller scale than ω∂V/∂z, the vortex propagating as a wave in Eq. (4) can still undergo a rigid shift with ∂m/∂t ≠ 0. In this case, the uniform velocity approximation also holds if the wave speed c is on a much larger scale than the spin velocity of the vortex. Furthermore, when ω is extremely large, the necessary constraint for a Taylor column rigid shift – that is, ω∂V/∂z→0 – often requires the disturbance velocity to also be uniform.

Due to the ultra-rapid vorticity ω and the constraint ∂p_i/∂x_i and ρf_i are smaller scale, the vortex is often required to be sufficiently small to infinitesimal, hence the term vortex particle (without requiring closure). Inside the vortex particle, disturbance originate from adjacent vortices. both its characteristic scale R and length scale L are equivalent to the geometric radius r of the vortex particle. Therefore, we obtainν/|a| =r from the scaling relations v ~UR and a~U/L. Consequently, the tangential velocity V_θ (ω₀ν/|a|) can still be a constant velocity under the ultra-rapid ω₀ required by the constraint ζ≫U/L. In this case, if the wave speed c is very extremely fast, the wave vector and the axial direction of the vortex particle (this axis does not need to be the rotation central axis) do not need to be syntropy, and the uniform velocity approximation can still hold.

3.5 Ultra-rapidly rotating vortex particles tend to radiate steady-state oscillation modes with ultra-high frequencies

In the ultra-rapidly rotating vortex particles (where V_r=-ar generally holds; see subsequent content), a special steady-state disturbance velocity (V_r′, V_θ′, V_z′) exists.

During infinite ultra-rapid rotation, centrifugal stabilization inhibits the radial propagation of local strain and prevents the local accumulation of low-frequency components of the radial perturbed velocity V_r′,consequently, V_r′ tends to be a small perturbed velocity and it results in DV_r′/Dt→∂V_r′/∂t. Meanwhile the steady state constraint for rigid shift requires ∂V_r′/∂t→0, moreover whole-axis rigid shift implies ∂V_z′/∂z→0. Furthermore the continuity equation would enforce ∂V_θ′/∂θ→0 if the small perturbation leads to ∂V_r′/∂ r→0; combining all these factors, the radial viscous term which ρv(∇2Vr′−r−2Vr′−2r−2∂Vθ′/ ∂θ)→0) follows naturally. Additionally, infinite ultra-rapid rotation tends to be adiabatic and leading to ∂p′/∂r→0. Therefore, for the radial Navier–Stokes equation of the perturbed velocity (coordinate system following moving votex) on the infinite ultra-rapid vortex particles, it can be simplified as:

This opposite tangential disturbance velocity, V_θ′ = − 2V_θ, reduces the measured value of the tangential velocity in the inertial coordinate system to zero: V_θ →0.

Consequently, a perturbing entity (V_r, 0, V_z + V_z′) with a steady-state ultra-high-frequency oscillation mode tends to radiate along the z-direction of the ultra-rapidly rotating vortex particle, especially when the disturbance velocity is tangential (or when the radial perturbation velocity V_r′ contains no second-order terms).

In a static vacuum, mutual disturbance between two identical vortex particles can generate this tangential perturbed velocity V_θ′. Notably, the vacuum not only satisfies μ′= 0 and uniform viscosity v but also meets the constraints required for deriving Eq. (3) from Eq. (2): ∂(ρu_j)/∂x_j→0 and ∂(∇·ρ)/∂x_i≪∇²ρ i.e., ∇ · ρ→0 (isotropic or transverse wave).

3.6 Vortex particles tend to be long-lived in a wave group

For the imaginary wave with the solution ρˆ=ρ0ei(ωt−kr), its long-persistence requires a constant amplitude ρ₀ and necessitates constraint δ t∇²(∂ρˆ/∂t)≪∇2ρˆ, such that Eq. (3) can degenerate into Eq. (4).

Given that ∇2(∂^ρ/∂t) =∇2(ωiρ^) and ω ~δ t⁻¹ is a scalar function for the wave solution, the constraint δ t∇²ω = 0 is generally required.

In a vacuum, the angular frequency ω of wave ρˆ is generated by the angular velocity Ω of the vortex particle. Due to the constant rotation, constraint ∇∧[(DΩ/Dt)∧r′]≪ω∂V′/∂z of the vortex particle is generally required, that is DΩ/Dt = 0, and the vorticity dynamics equation is v∇2ω = Dω /Dt for adiabatic, potential and plane incompressible flow, so a vortex particle with constant rotation typically satisfies Dω/Dt = 0, making it easy to achieve ∇²ω = 0.

The constraint δt∇∧(∂ρ^/∂t)≪∇∧ρ^ required for the electromagnetic wave solution generally necessitates the requirement of the constraint δt∇ω∧ρ^=0(other term δtωi∇∧ρ^=i∇∧ρ^ adjusts the phase of∇∧ρ^),it implies the constraint ∇ω⊥ρ^ or ∇ω=0.

Therefore, constant-rotation vortex particles in a wave group – characterized by uniform viscosity that varies with transverse waves (where “uniformity + constancy = constant”) – tend to be long-lived.

3.7 Viscous wave with pseudo-variational mass, viscous vortex particle, imaginary ρˆ wave, and strong penetrability

At extremely high frequencies, the higher-order terms in the Taylor series expansion of D(mu)/Dt can no longer be neglected. When the second-order term becomes explicit, it can exhibit viscosity, forming viscous wave solutions or viscous vortex particle solutions depending on the contextual occasion. Meanwhile, terms of orders higher than the third can manifest as unsteady, evanescent virtual particles.

In a vacuum, the key properties of micro-particles (spin, ultra-high speed, uniform velocity, wave nature, vibration, and lifetime) can be correlated with the 10 conditions starting from the momentum given by Eq. (1). In anisotropic expanding–contracting media, if vibrations can still propagate at a uniform velocity, extremely fast frequencies can induce pseudo-variation of mass – that is, virtual displacement involving shape and volume changes [8] – such that the term δm→0 while the term δ(δm) ≠ 0, forming viscous waves with pseudo-variational mass. However, if extremely fast-frequency vibrations cannot propagate uniformly in expanding–contracting anisotropic media, the term ∂m/∂t, lacking the amplification factor ω from the wave solution e^iωt will be discarded from the momentum equation, resulting only in viscous vortex particles. An extremely large ω tends to render the rate-of-change term of u_i in Eq. (1) always negligible, easily manifesting as bulk strain waves with variable wave vectors (i.e., matter waves) that possess strong penetrability. Since the momentum equation conventionally adopts ρ (density), this pseudo-mass-variation viscous wave (discarded if it contains the term m or the term δm) is also referred to as a density wave.

Since the changing of suction intensity ‘a’ can alter the radial distance of vortex particles to induce bulk volume deformation, this pseudo-variation can further generate another virtual wave iρˆ=ρ0ei(ωt−kr+π/2) (where the phase difference is exactly π/2 [9]) through the mutual deformation between radial distance and axial extension. At the initial moment t = 0, it has no real part at r = 0; thus, the virtual wave i ρˆ propagates between fixed ends. In the steady state, if the two vortex particles exchange iρˆ exactly, Coulomb force can be transmitted.

In summary, vortex particles propagating as pseudo-mass-variation viscous waves (satisfying Dm/Dt = 0 & ∂²m/∂t² ≠ 0) can not only be long-lived but also integrate with vacuum fluctuations ∂/∂t due to δ t→∂t (characteristic time tending to local time derivative). Photons with a wave speed of approximately 3.0 × 10⁸ m s⁻¹ or electrons always accompanied by iρˆ -waves can be generated under normal conditions with a tangential velocity V_θ ~ 10⁰ m s⁻¹, exhibiting wave–particle duality. During propagation, they can undergo variations (one-dimensional extreme fast oscillation with m→0 but ρ ≠ 0), thus allowing changes in shape, axial direction, or size. However, their inherent properties remain invariant across different inertial frames – hence the speed of light ‘c’ is independent of the inertial frame.

4.1 Quantum vortex particles and the numerial value of the reduced Planck constant ℏ

The left of Figure 2 plots the radial velocity V_r = −ar + 6 vr−1(1−e−ar2/2ν) (blue dashed line for a = 1.0, green dashed line for a = − 1.0) and tangential velocity V_θ (blue and green solid lines for ω₀ = 1.0) derived from vortex solutions given by Eqs. (6) and (7) using IEEE754 double-precision floating-point arithmetic. The parameters are set as v=1.0 and a = ± 1.0 such that the unit of r is ν/|a| for plotting.

The left of Figure 2 shows that radial velocity V_r (dashed lines)exhibits discontinuous step changes, which result from the rounding operation following the minor variations in r if the viscous term r⁻¹(1- e−ar2/2ν) becomes a primary term of V_r. According to the Taylor–Proudman theorem, the adjacent abruptly changing V_r tends to violate the viscous constraint ζ≫U/L required for the Sullivan vortex solution, and then,at the first endpoint r₁ = 1.05367, the opposite ± V_r is most susceptible to being truncated into inviscid vortex particles by this viscous velocity.

Numerical analysis indicates that the critical radial position r₁ = 1.05367 remains unchanged when the radial velocity V_r, is multiplied by any arbitrary constant. The specific value and order of magnitude of r₁ are only correlated with ν/|a|. This numerial value is fairly close to the reduced Planck constant ℏ ~1.05457 [10, 11].

4.2 Mathematical and physical correlations of ℏ

For a unit-mass vortex particle that satisfies the Navier–Stokes equations and yields shown, the Sullivan vortex solution, if it undergoes steady, inviscid, and homentropic flow in a potential field while still abiding by the Euler momentum equation, its total energy H (kinetic, internal, and pressure energy) must satisfy theCrocco equation: ∇H=V∧ ω. In the cylindrical coordinate system (r, θ, z), since the radial vorticity component ω_r ≡ 0 and the small scale of z and r render the tangential vorticity component ωθ=−6zra2v−1 e−ar2/2ν →0 an absolute smaller scale relative to the axial vorticity component ω_z, the following relation holds:

In the mathematical operations for taking limits, r∂θ = rδθ ≈ rtanθ = δr = ∂r. Thus, the component e_θr⁻¹∂H/∂θ corresponding to V_r ∧ ω_z is equal to e_θ∂H/∂r. That is, in the gradient ∇H of the scalar H in Eq. (11), both of its two components correspond to ∂H/∂r. We tentatively adopt ∂H/∂r=Vr ω_z in the following discussion.

The total energy H is obtained via floating-point arithmetic as H = ∫drV_rω_z. For integration, the parameters are set as a=1.0,v=1.0＆&ω0=1.0, with the initial value taken as 0 and the integration interval as [0, r].

The calculation results are plotted in the right panel of Figure 2, with same values at another two scales embedded in the figure. Calculations show that the variation δH of the total energy ∑H of vortex particle is independent of ‘a’ and is only linearly correlated with v; that is ∑H contains the component:

The first endpoint of δH is the same as that of V_r, which is also r₁ = 1.05367, and the specific value of r₁ is only correlated with ν/|a|.

At the surface of the vortex particle, the scale U of the disturbance velocity is equal to the wave speed c, and the second-order scale R of the disturbance velocity corresponds to the outer radius r of the vortex particle; thus, we have v =cr. Then, substituting the central vorticity ω₀ (i.e., the angular frequency of oscillation) in Eq. (12), we obtain δH∞v∞vω0=crω0. In addition, when calculating the total quantity, the kinematic viscosity v=μ/ρ should be multiplied back by the mass m (where m is usually obtained via one-dimensional accumulation of density ρ). Therefore, a relation for the reduced Planck constant exists [10]:

Floating-point arithmetic requires integer storage and retrieval [12], such that all operations – including the summation of total energy ∑H – are subject to rounding and integerization. The fact that the same critical radial position r₁ arises in both the calculation of V_r and the integral ∫drV_rω_z is also due to the rounding of the term 1−e−ar2/2ν being correlated only with ν/|a|. Since a constant r₁ is obtained for the calculation of any arbitrary H, hence r₁ may serve as the numerical fundamental unit of H with its order of magnitude to be determined,the scale of 10⁻⁸ in the left panel is the result of IEEE754 double-precision floating-point arithmetic of 2-53.

The following paragraphs provide supplementary explanations.

If the axial vorticity ω_z is extremely large such that the ∂/∂z term and the geostrophic term are negligible, meanwhile ∂/∂θ→0, the steady azimuthal equation simplifies to:

Integrating Eq. (14), we obtain: δH = ∫drV_rω_z = v ω_z (initial value = 0), Thus, δH is linearly correlated with v.

Within the vortex particle where r≦r₁, the axial vorticity satisfies ω_z ≡ 1.0ω₀ (the first endpoint of ω_z being r₁ = 1.46422, which is greater than the first endpoint of V_r at r₁ = 1.05367). That is to say, the condition ∂ω_z/∂r = 0 holds inside the vortex particle, such that Eq. (14) is not applicable here.

In this inviscid case, we may yield the integral ∫drV_rω_z = -aω₀r²/2 with the radial velocity given by V_r = −ar (initial value = 0). Since ω_z ≡ 1.0ω₀, the integral of the other component V_θω_z in Eq. (11), where initial value = 0, is expressed as:

Within a vortex particle, suction intensity ‘a’ is determined solely by its intrinsic angular frequency ω₀, such that |a ~ D/2 = ω₀|. It thus follows that 2∫drV_θω_z = ∫drV_rω_z, that is, the radial velocity inside the vortex particle satisfies |V_r = 2V_θ|.

For a homentropic flow, the flow field is isotropic, and the energy is continuous at the surface of the vortex particle. Combining Eqs. (12) and (13), we obtain: ∫drVθωz=ℏω0/2 [10];Furthermore, since ∂V/∂θ→0 and ∂P/∂θ→0 generally lead to ∂H/∂θ→0, this term exists as an integration constant, namely the zero-point energy.

When the azimuthal velocity V_θ is present, the total energy ∑H of the vortex particle contains the term Vθωz~ℏω0/2, the spin possesses a factor of 1/2, so a vortex particle is corresponding to a Fermion. If the azimuthal velocity V_θ is balanced by the disturbance velocity V_θ′ = − 2V_θ to form a steady extreme high frequency oscillation state with V_θ→0, the spin does not possesse a factor of 1/2, now a vortex particle is corresponding to a Boson.

The mutual disturbance of two co-orbital vortex particles induces oppositely directed azimuthal perturbed velocities ± V_θ′.During the extreme high-frequency rotation, the angular accelerations Ω1∧Ω2 always keep the two particles coaxial under macroscopic perturbation deviations. Since the oppositely directed V_θ′∧ ω_z generate oppositely directed radial perturbed velocities ± V_r′, the particles cannot remain co-orbital. In contrast, when the antiparallel vorticities satisfy ω₁ = -ω₂ = ω_z, the two particles possess identical V_r′and thus can stay co-orbital. This is exactly the manifestation of the Pauli exclusion principle.

The uncertainty relation δ t δ ε≧ℏ/2 can be rewritten as δtω₀ ≧1 from the comparison expression δ ε =ℏω₀/2.

The time scale δt of state variation is much larger than the time scale t′ = L/U of the perturbed velocity [10], so ω₀ ≧1/δ t i.e., ζ≫1/t′ and the constraints ζ≫U/L are equivalent to it. The uncertainty principle δx δ p≧ℏ/2 remains same.

From the comparison expression ℏ =mcr, the relations: ℏ/m ~ cλ ~ UR=ν are obtained. Therefore, the Schrödinger equation is derived from the momentum theorem, and the factor 1/2 indicates that the viscosity originates from V_θ (i.e., the zero-point energy). The imaginary unit i among the Schrödinger equation signifies that a particle is subjected to numerous perturbations that can always act as driving forces, which indicates that its velocity has no unique direction or magnitude and must therefore be expressed as a complex velocity.

However, the solutions to Eqs. (5)–(9) do not involve radial constraints [7]; thus, they are irrelevant to the radial Schrödinger solutions.

If integrating over the interval [0,r], only the radial component of viscous force, that is, the work done by v ∇²V_r contributes; but V_r = −ar inside the vortex particle, so it does not have the second-order term required for viscosity, and δH only contains the perturbed energy at the surface. Thus, similar to the turbulent case, the value of r₁ = 1.05367 is also approximately 1‰ smaller than the measured value of ℏ ~1.05457.

The viscous solution exhibits a characteristic – when observed using high-energy particles, the suction intensity increases with the rise in energy, which, in turn, causes the observed radius of the vortex particle to become infinitesimal.

Formally ℏ =mcr takes the form of angular momentum; however, from a dynamical perspective, it is not the rotational angular momentum of the vortex particle (since H contains the internal energy term, the term including V_z and the pressure energy), but rather the dynamic viscosity constrained by both the momentum theorem and its specific case – Taylor–Proudman theorem.

Different from molecular viscosity, turbulent viscosity, shear viscosity and structural viscosity (for non-Newtonian fluids) under the constraint of mass conservation, it is the mass viscosity that is induced by matter waves derived from vortex viscosity v =UR under the constraint of velocity conservation. Since the viscosity occurs between oscillators with virtual mass variation, if the environment converts energy into the mass m of the vortex particle, the vortex particle must in turn convert this mass m into vibration immediately. Consequently, there exist two correlations of δ ε ~ mc² between the vibration frequency and energy, namely a c⁻⁴ correlation. Therefore, ℏ is on the order of ℏ ~10⁻³⁴ if taking the unit of joule-seconds (J = kgm²s⁻²).

The left part of Figure 2 shows that the azimuthal velocities, V_θ, are identical when a = ± 1.0 (the blue straight line represents V_θ at a = 1.0, and the green straight line represents the V_θ + 1.0 × 10⁻⁸ curve at a = − 1.0).

It is evident that the zero-point energy is constant, and thus antiparticles can exist (the ± V_z terms affect the phases ± cosθ and ± isinθ).

However, the first endpoint, r₁ = 1.490, is larger when the suction intensity, a = −1.0, and its total energy, ∑H, is twice that at a = + 1.0. It can be concluded that vortex particles generated by divergence also tend to converge in a convergent field. Therefore, antiparticles can exist, whereas antimatter cannot.

5.1 Convection term nuclear force, nuclear geometric radius, and theoretical value of ℏ

The left part of Figure 3 plots the curves of −V_r∂V_r/∂r for v =1.0 (the orange curve denotes a = 1.0 and the red curve denotes a = − 1.0; unit |a |a|v |). The transition point between repulsion (with the same sign as r) and attraction(with the opposite sign to r), that is, the radial distance r where V_r∂V_r/∂r = 0 is slightly greater than 1.0 × 10⁻⁸ν/|a|.

Herein, the unit ν/|a| of r is generally determined by the compression mechanism, that is, by disturbance. If the disturbance takes the form of a far-field sinusoidal wave, both the second-order scale R and the first-order scale L are equal to the wavelength λ. From the relations v~URanda~D/2~U/L, we obtain r ~ 10⁻⁸λ.

For a given vacuum permittivity ε₀, its corresponding permeability μ₀ = 2πrFI⁻² represents the distance at which a current I (unit in amperes, A) per unit length generates a magnetic force F (×2) per unit magnitude (here, I² reflects the magnetic flux). In other words, the characteristic distance ν/|a| of vacuum electromagnetic perturbations is on the order of 10⁻⁷, consistent with μ₀ = 4π × 10⁻⁷ (kgms⁻²A⁻²). Therefore, the geometric outer radius r₀ of atomic nuclei in a vacuum is on the order of r₀ ~ 10⁻¹⁵ m [13], while the r₀ of atomic nuclei exposed to visible light is approximately on the order of 10⁻¹⁴m.

However, the value of 10⁻⁸ in the graphical relation r ~ 10⁻⁸ν/|a| is variable, it is associated with the rounding of IEEE754 double-precision floating-point arithmetic (2⁻⁵³ = 1.110 × 10⁻¹⁶; via mantissa multiplication by 2 followed by rounding). In other words, 10⁻⁸ is only valid when the dimensionless term 1−e−ar2/2v is rounded at a precision of 10⁻¹⁶. Mathematically, the rounding scheme for the dimensionless quantity 1−e−ar2/2v is determined by the physical hard-cutoff mechanism of the radial velocity V_r = −ar + 6v r−1(1−e−ar2/2ν) (also together with the axial velocity V_z = Const which makes ω∂V/∂z→0). Owing to viscosity, the radial velocity relation degenerates to V_r ≈ 2ar (in the sense of mean value), while suction intensity ‘a’ itself varies slowly, which is also determined by the hard cutoff of the radial distance r.

Assume that r ~ 10^-nν/|a|, then 1−e−ar2/2v (i.e.,ar2/2v) is of the order of 10⁻²ⁿ, so the maximum admissible order of magnitude which 10⁻²ⁿ equals the order of the maximum cutoff distance δr_m. Physically, the maximum cutoff distance δr_m must be approximately one order of magnitude smaller than the radial distance r, that is, the outer radius r₀ of the vortex particle. Combining with the relation ν/|a|~μ0~10−7, we obtain r₀ ~ 10^-n−7, leading to the following expression:

It follows that the value of 10⁻⁸ in the plot is jointly determined by 1-e−ar2/2v, the rigid perturbation of the Taylor column, and the relation μ₀ ~ 10⁻⁷.

All smaller orders of magnitude are subject to rounding, yielding the theoretical value: ℏ →1.111• =1.05410.

For completeness, the left part of Figure 3 also plots the radial velocity V_r curves at v =1.0 (the blue dashed line for a = 1.0 and the green dashed line for a = − 1.0) and the azimuthal velocity V_θ line (the green solid line; with ω₀ = 1.0; the V_θ values are identical for ± a).

5.2 Viscous nuclear force, domain solution, wave function, nuclear radius constant, nuclear force range, and electric quadrupole moment

The green region shown in the right part of Figure 3 corresponds to the calculation results when the radial viscous term v(∇2Vr−r−2Vr−2r−2∂Vθ/∂θ) is defined as the mathematical identity involving differential operations: v∂[r−1∂(rVr)/∂r]/∂r, the plot shows that it presents a domain solution (v=1.0; a=1.0, with identical values for ± a); If the radial viscous term is rewritten as another differential identity (the blue region):v ∂[r−1∂(−ar2+6v (1−e−ar2/2ν))/∂r]/∂r, the resulting domain width will be different; If it is further expressed as a non-differential identity, − 6a²re−ar2/2ν (the orange line), no domain solution will be obtained.

This is because floating-point operations require rounding to the nearest integer. When differential operations are involved, rounding errors must be recovered, which gives rise to domain solutions. Similarly, the rigid shift in the Taylor–Proudman theorem truncates all relatively smaller scale; when variables are driven by the variation of cutoff values, domain solutions will also emerge. Domain solutions are also constrained by the second-order term of V_r that is actual existence or not, which is required for viscous forces. Domain solutions lead to asymptotic freedom in the inner region. Domain solutions together with the constraint ζ≫U/L allow photons (and electrons) to exhibit waves mˆ with variable morphologies.

From the two correlations of vortex particles – E~ℏ ω0/2 and V_r ~ 2V_θ – it can be inferred that the wave function Ψ of the radial Schrödinger equation [12] corresponds to V_r ~ the radial velocity just with complex velocity, while the negative energy term [10] EΨ corresponds to V_rℏ ω₀ /2 ~ V_θω₀, which is radial coriolis force.

The suction intensity is a < 0 for repulsive forces, thus the red line in the left part of Figure 3 may correspond to the radius constant of nuclear force interaction; when taking ν/|a| =10⁻⁷m, it is slightly less than 1.632 × 10⁻¹⁵ m [13, 14]. The suction intensity is a > 0 for electrostatic attractive forces, hence the orange line corresponds to the charge radius constant 1.154 × 10⁻¹⁵ m [13].

The left part of Figure 3 shows that the maximum of the first endpoint, r₁ (the green dashed line), is r₁ = 1.490 with a < 0. Therefore, we can observe that at the size 2r₁ ~ 2.98 × 10⁻¹⁵ m [13], the convective nuclear force term, −V_r∂V_r/∂r, disappears abruptly due to the vanishing of viscous V_r at that point.

The mosaic adhesion exhibited by the domain solutions on the right part of Figure 3 allows for the linear superposition of the three-dimensional bulk of incompressible flows, such that the nuclear radius is proportional to the cube root of the nucleon number. Furthermore, the shape of the domain solution corresponds to the electric quadrupole moment [13].

6.1 The three-quark model of the proton, the classical origin of positive electric charge, and the exotic nucleus

The attenuation of the perturbed velocity will probably satisfy the rigid-displacement constraint. Thus, to maintain the inviscid property of the point–vortex system, the perturbed velocity must exhibit neutral stability. In a steady state, that is, when a standing wave is formed, the perturbed velocity will be evenly divided into j modes by n uniformly distributed vortex particles (0 < j≦n, n≧2; mathematically, the case of n = j corresponds to j = 0).

From the induction equation of the inviscid two-dimensional point–vortex system, it is required [4] that the amplitude of its inherent disturbance wave satisfies the following condition for neutral stability:

This corresponds to two admissible modes, namely n = j and n = 7 (with j = 3 or j = 4). Among these, the system tends to be chaotic when n≧4 [4]. Mesons with n = j = 2 are prone to transverse splitting under one-dimensional forces, and protons and neutrons with n = j = 3 can remain stable indefinitely as two common composite particles.

Inside the composite particle, the viscous wave ρˆ associated with virtual mass variation can degenerate into a vortex-particle solution due to deceleration when passing through the vortex particles. At this location, three uniformly distributed identical vortex particles (with the same circulation Γ) induce only azimuthal velocity; their radial motion is independent of the three circulation values of Γ. Under the constraints of centroid moment conservation and moment of inertia conservation, the three vortex particles satisfying Γ₁⁻¹ + Γ₂⁻¹ + Γ₃⁻¹ = 0 can collapse [4] toward the center of the composite particle.Owing to neutral stability, these particles (Γ₁, Γ₂, Γ₃) can revert to wave ρˆ states, then propagate along the central axial direction and transform into three-dimensional imaginary waves (iρˆ), endowing the composite particle (∑Γ) with electric charge. Since the circulation Γ affects the induced velocity and the sign of ± a, it can give rise to the special electric charge properties of quarks [11].

After the repulsive force is balanced by the pressure of the newly generated wave inside the composite particle, the major axis of the proton must be slightly larger than 1.632 × 10⁻⁸ ν/|a| (in consistent units), to maintain stability by the attractive component in the convective nuclear force under the divergent condition of a < 0 (refer to the red line in the left part of Figure 3). This renders the proton in a contracted state externally, and the divergent condition a < 0 far from the environment corresponds to positive electric charge.

Under the constraint of angular momentum conservation, the quasi-two-dimensional induced vortices governed by the strong nuclear force tend to precess into a spherical shape. Consequently, the measured positions of quarks are randomly distributed, the total spin of nucleons is determined by their precessional angular velocity, and the mass density inside the nucleus increases from the ordinary value of 10³ kgm⁻³ to the exotic value of 10¹⁷ kgm⁻³ [13].

The radius of the circumference where vortex particles reside is determined primarily by the suction intensity, a. If ‘a’ takes the form of a wave solution along the radial direction, the vortex particles can be distributed in spherical shells, and forbidden bands emerge accordingly.

In this case, the radial velocity, V_r, readily acquires a second-order term; the radial viscous-force arises subsequently, the nuclear force becomes increasingly variable, and exotic nuclei come into existence [13].

6.2 Decay, electron antineutrino, and γ radiation from the perspective of viscous vortex particles

If α-decay is caused by external disturbance of nuclei with r≦10⁻¹⁵ m that induce the rearrangement of protons and neutrons, the constraint ζ≫U/L must hold critically. Correspondingly, at r≦10⁻¹⁷m, since the first-order scale L decreases by two orders of magnitude 10⁻², the constraint ζ≫U/L is no longer satisfied. In the absence of rigid shift, viscosity vanishes, induced effects become explicit perturbations, and the presence of a continuous time derivative ∂mˆ/∂t readily triggers β-decay [11, 13]. The critical state approximately maintains mass conservation, which results in three key phenomena: α-decay occurs exclusively in heavy nuclei, heavy-ion emission is rare [13], and the rest mass of the electrons accompanying β-decay is three orders of magnitude 10⁻³ smaller than that of protons.

From the primitive constraint of condition ③: ∇ ∧ [v ∇2(V′+Ω∧r’)]≪ω∂V′/∂zie.,Ek≪1, it can be inferred [3, 4] that, due to the vorticity adjustment during decay, the angular velocity Ω of nucleons is no longer uniform. In this case, an additional constraint must also be satisfied:

That is, it requires a thin-layer perturbation in which the axial scale is much smaller than the transverse scale. Therefore, the conservation of vorticity and the rigid-shift constraint, required for uniformly rotating nucleons to maintain viscous nuclear forces, result in β-decay being accompanied only by quasi-2-dimensional electron antineutrinos (e.g., during collapse).

Gamma rays with extremely fast spin, such as ω≧1,019 s⁻¹ can be generated via transverse compression and axial stretching. This compression and stretching increase the axial scale relative to the transverse scale, causing the constraint ζ≫U/L to break along the z-direction, thereby leading to radiation.