Physical Sciences

Abstract: The description of time in modern physics has been built almost exclusively around translational motion, leaving rotational dynamics in an asymmetric position within the standard single-time framework. This work presents an operational definition of angular time τ as an independent temporal dimension within the angular displacement spacetime framework φ(τ) [1], and provides two complementary lines of investigation.First, a ‌zero-parameter‌ turbulent scaling law, derived solely from the intrinsic 2π periodicity of τ, reproduces all high-precision, high–Reynolds-number DNS data up to order p=9 with absolute errors below 0.0107. This is one of the first scaling law for turbulent intermittency that requires no adjustable parameters [2,3]. The agreement provides an indirect consistency check for the φ(τ) hypothesis, rather than definitive experimental proof for the physical reality of τ.Second, a binary crucial experiment under the zero-area limit (r=0) is proposed: at the rotation center, where both special and general relativity predict zero frequency shift, the φ(τ) framework predicts a non-zero shift proportional to ω². This geometric test draws upon existing clock‑comparison technology for its feasibility, though it has not yet been implemented in laboratory measurements, and provides an unambiguous falsification pathway.This work does not contradict established relativity; rather, it introduces a complementary geometric framework for rotational dynamics that yields falsifiable experimental predictions, and may provide a conceptual bridge between the continuum spacetime of relativity and the discrete nature of quantum mechanics.

Abstract: Plasma-activated water (PAW) selectively kills glioblastoma cells while sparing normal brain cells, but whether this selectivity originates in membrane biophysics, channel-mediated transport, or intracellular clearance is unresolved. We dissect the question for U87-MG glioblastoma cells versus normal astrocytes by combining all-atom molecular dynamics of the two unoxidized plasma membranes with an experiment-anchored transfer-function model of H₂O₂ influx. Simulations of unoxidized bilayers demonstrate that physiological cholesterol enrichment in glioblastoma membranes marginally increases lipid packing and acyl-chain order, conferring no baseline passive permeability advantage on the malignant cell. While this excludes pristine bilayer ordering as an intrinsic source of differential uptake, it establishes that any membrane-level selectivity must arise downstream from regulated protein conduits or differential susceptibility to oxidative pore formation, rather than pre-existing lipid biophysical disparities. We show that intracellular dose is governed by a single dimensionless transport number Φ ("influx"÷"clearance" ) whose regime switch defines selectivity. Global sensitivity analysis reveals that while AQP8 abundance exhibits a modest local variance index () once influx is saturated, channel over-expression serves as a mandatory, non-linear permissive gate: it transitions the malignant cell across the critical transport boundary (Φ>1). Beyond this threshold, the dominant driver dictating the magnitude of cytotoxic divergence is the tumor-specific collapse of catalase-based clearance, with background lipid permeability setting a ceiling. An AlphaFold conformational ensemble (240 models) leaves the AQP8 selectivity filter robustly closed, confirming that resting peroxiporin transport is tightly regulated and rendering the experiment-anchored transfer function the rigorous biophysical choice. Selectivity thus emerges from a regulated transport system—a permissive AQP8 threshold switch paired with an asymmetric clearance collapse—yielding a calculable, platform-testable therapeutic window.

Abstract: We review what is known about finite-time singularity formation ("blow-up") in the unsteady Prandtl boundary-layer equations, from the numerical discovery of the van Dommelen--Shen singularity in 1980 to the rigorous description of its self-similar structure completed in 2018--2022. The survey is organized around five questions: (i) what the singularity is physically (unsteady separation and eruption of the boundary layer); (ii) where it is known to occur (the impulsively started cylinder, vortex-induced separation, colliding wall streams); (iii) what has been proved (the E--Engquist theorem, the Kukavica--Vicol--Wang theorem, the Collot--Ghoul--Ibrahim--Masmoudi description of the stable blow-up pattern, the classification of inviscid singularities); (iv) how blow-up relates to the ill-posedness of the Prandtl equations without monotonicity and to the inviscid limit of the Navier--Stokes equations; and (v) how the reduced one-dimensional system on the symmetry axis, which carries the rigorous theory, sits inside the recent similarity transformation of Sun [Phys.\ Fluids \textbf{36}, 083616 (2024)] as its \(m=1\) member. We also record the explicit affine blow-up solution \(u=-x/(t_*-t)\), \(v=y/(t_*-t)\), which is exact for both Prandtl and Navier--Stokes, and explain why Prandtl blow-up bears no relation to the finite-energy Navier--Stokes blow-up announced in September 2026. A list of open problems closes the survey.

Abstract: A recent similarity solution of the two-dimensional unsteady laminar boundary-layer equations [B. H. Sun, Phys. Fluids 36, 083616 (2024)], expressed in terms of Kummer functions, was proposed as the solution for a semi-infinite flat plate impulsively set into motion, but its boundary conditions cannot be satisfied at any finite time. We show that the solution is, in fact, a Burgers–Townsend strained shear layer superposed on the decaying, zero-pressure-gradient straining flow u = x/t, v = −y/t. For this class the nonlinear terms cancel identically, and the shear component obeys a linear equation that a Lundgren-type change of variables reduces to the heat equation; the Kummer functions are the fractional repeated-integral error functions i^2/3 erfc. The two forms of the published solution are copies of a single flat-wall solution related by Prandtl’s transposition theorem, and the flat-wall representative is an exact solution of the full Navier–Stokes equations, not merely of the boundary-layer equations. The correct physical setting is a sheet stretching at the decaying rate 1/(t + t₀), the S = −1 point of the Wang–Andersson family of unsteady stretching-sheet flows. Within this setting we solve three well-posed initial–boundary-value problems in closed form: (1) impulsive translation of the stretching sheet, whose wall shear stress crosses over from the Rayleigh law to −1.370 ρU_s√ν/t; (2) the decay of a pre-existing shear layer, governed by the exact invariant \( s^3\!\int_0^\infty Y u'\ \)dY and (3) arbitrary wall histories, including power-law ramps with wall-shear factor √3 Γ(p + 1)/Γ(p + 1/2) and an oscillating sheet that relaxes to the classical Stokes layer. All three problems are two-dimensional – both velocity components are nonzero and the field depends on x – in contrast to Stokes’ first and second problems, which are one-dimensional and are recovered only as limits. The published Kummer solution is identified as the self-similar late-time attractor of problem (1). The unsteady Blasius problem for a finite plate remains open.

Abstract: Lithium filaments penetrate solid electrolytes by wedging open pre-existing flaws, and the field has consequently treated the problem as one of fracture mechanics: raise the toughness, close the crack, and penetration is delayed. Here we show that mechanical stress acts on filament growth through a second channel that has been overlooked, and that this channel is exponential rather than linear. Because ion migration in a solid requires a local dilation of the lattice, the ionic conductivity of every solid electrolyte depends on stress through an activation volume ΔV∗, a quantity measured to span 0.6–25 cm³ mol⁻¹ across sulfide and polymer conductors, and never measured at all for the oxide garnets. The stress field of a pressurised filament therefore reshapes the conductivity of the electrolyte that feeds it. Using coupled finite-element simulations of a sharp, Li-filled edge flaw we find that the sign of this feedback is set by the near-tip stress state, and that in an unstressed electrolyte it is adverse: the crack tip sits in tension, transport there is enhanced, and the supply of lithium to the tip rises by 12–22% at the activation volumes measured for sulfide argyrodites (0.58–1.08 cm³ mol⁻¹) and by a factor of 6.6 at the value measured for PEO–LiTFSI (25 cm³ mol⁻¹), even as the supply to the crack flanks falls. We derive a two-constant scaling law, σ ^tip_h = κ (p − λσr) with κ =3.00 and λ = 1.195, which locates a narrow but real operating window, σ_r < p < λσ_r, in which the flaw is open yet its tip remains compressed. Inside that window the feedback reverses sign, giving a predicted 3.6-fold gain in critical current density of which about half is fracture-mechanical in origin. Because the exponent scales with ΔV∗, the effect is a 10–35% correction for stiff inorganic conductors but the dominant term for polymer and polymer-in-ceramic electrolytes, where Γ ≈ 7at crack-tip stresses. The framework predicts that critical current density should correlate negatively with ΔV∗ in unstressed cells, and identifies a concrete architecture—a compressively prestressed gating skin—that converts a known linear benefit into an exponential one.

Abstract: The Kraichnan model provides one of the clearest solvable settings for passive-scalar transport in turbulence, but its standard formulations often presuppose familiarity with stochastic-process and field-theoretic methods. Here we develop the model from the passive-scalar advection--diffusion equation upward, emphasizing its physical content while introducing the required formalism as it becomes necessary. Instead of using the full Navier--Stokes velocity, with nonlinear dynamics and multi-scale statistics not known in closed form, the model prescribes the advecting velocity as a Gaussian random field that is white in time, correlated in space, and allowed to have a rough spatial increment covariance. It is therefore not a literal model of Navier--Stokes turbulence, but a controlled idealization designed to isolate scalar transport by a rough random flow. The central simplification is that Gaussianity and temporal delta-correlation allow mixed velocity--scalar averages to be eliminated exactly, producing a closed hierarchy for equal-time scalar correlation functions. The two-point equation becomes a diffusion equation in separation space, with molecular diffusion supplemented by a scale-dependent turbulent diffusivity determined by the velocity-increment covariance. This gives a direct route to Richardson-type dispersion, the scalar variance cascade, the dissipative anomaly, and the inertial-range scaling of the second-order scalar structure function, F²(r) = ⟨[c(x + r) − c(x)]²⟩ ∼ r^2−ξ . Higher-order equations show how a Gaussian advecting velocity can nevertheless generate non-Gaussian scalar statistics, intermittency, and anomalous scaling through zero modes and statistical conservation laws of multi-particle configurations. We also discuss the Lagrangian particle interpretation, smooth versus rough stochastic flows, slow modes and particle-cluster geometry, compressible and anisotropic extensions, and the relation to renormalization-group and operator-product-expansion descriptions. The emphasis throughout is on the physical meaning of closure, roughness, stochastic particle motion, scalar diffusion, and the connection between Kraichnan-type advection and high-Schmidt-number liquid diffusion.

Abstract: The incompressible Navier–Stokes equations have recently been shown to admit finite-time singularities from smooth data under smooth bounded forcing. Since unbounded velocity gradients are physically inadmissible, the only element of the equations open to revision is the constitutive relation \(\sigma=-pI+F(\nabla u)\). From the Cauchy momentum equation we derive the most general \(F\) compatible with frame indifference, isotropy and the Cayley–Hamilton theorem, \(F=\alpha_1D+\alpha_2D\cdot D\) with invariant-dependent coefficients, and prove that the constant-coefficient second-order model \(\sigma=-pI+2\mu D+4\lambda D\cdot D\) violates the second law in three dimensions, is an exact gradient in two, and cannot bound velocity gradients. The improvement must therefore enter the linear coefficient, and the lowest-order admissible law is \(\sigma=-pI+2\mu(1+\tau^2 D: D) D+4\lambda\ D\cdot\ D\) with a material time \(\tau\). We prove that the second law requires \(\tau\ge|\lambda|/(\sqrt6\mu)\) and, by an exact block decomposition of the tangent operator, that monotonicity requires the sharp bound \(\tau\ge\sqrt{2/3}\,|\lambda|/\mu\), twice the thermodynamic one. Under it the momentum operator is strongly monotone with quartic growth, the equations lie in the Ladyzhenskaya–Lions class \(p=4\), solutions are global, unique and bounded, and the forced blow-up cannot occur. Exact solutions in simple shear and gravity-driven pipe flow give a two-measurement protocol for \(\tau\) and \(\lambda\) with an internal consistency test, and a strained vortex exhibits the arrest of collapse at strain rate \(1/\tau\). A boundary-layer theory consistent to second order shows that \(\lambda\) drops out exactly, that the flat-plate layer has a leading-edge regime \(\delta\propto x^{1/4}\), \(\tau_w\propto x^{-3/4}\) crossing over to Blasius scaling at \(\ell_\tau=\tau^2U^3/(2\nu)\), and that only the wedge flow \(U\propto x^{1/3}\) remains self-similar. Numerical solutions and experimental protocols are given.

Abstract: Discharge schemes in particle models of thunderclouds remove charge from a stochastically grown channel, which does not conserve charge and imposes the channel geometry rather than deriving it. This paper replaces the scheme with a plasma process: when the field in the gap between two oppositely signed charge layers reaches the ionization level the gap is filled with neutral air at the simulation’s own initial density, molecules above threshold separate into a positive ion and an electron, and everything after that follows from the force law that moves every other particle. Charge is conserved at every reaction. Two quantities such schemes leave free are derived rather than chosen: the separation at which a new pair is unbound, from the ambient field, and the timestep, from the electron drift velocity at the reduced field the model itself produces, E/N = 27 Td. In a column whose charge layers are prescribed rather than generated, the same charge spread smoothly through each layer gives 19 kV m⁻¹ and never fires; gathered into local concentrations it gives up to 196 kV m⁻¹ and does. The requirement is a layered and locally concentrated charge structure. Rotation is how this model produces one, but the discharge acts identically on prescribed layers that no vortex has shaped. Against matched controls the field in the firing gap falls 55.8 ± 6.5 per cent over eleven seeds, with relief of neighbouring gaps attenuating to zero by 2.16 km. Without an electron attachment loss the discharge never terminates; with attachment on oxygen all ten discharges self-terminate, at a median 190 ms and 1.44 × 10⁴ m s⁻¹ against 192 ms and 1.69 × 10⁴ m s⁻¹ for flashes measured in a mapped storm using the model’s own definitions. The reach is a factor of four short, which bounds the result to initiation and local field relief rather than to flash extent. Two predictions follow that observation can test: initiation should track the concentration of charge within a layer rather than the layer-averaged field, and flash duration should scale with the oxygen number density that sets the attachment length.

Abstract: At or near atmospheric pressures, most electrical discharges develop as an ionization front known as streamers. In early phases of streamer, the plasma formed is non-thermal. However, the streamers develop into filamentary arc-like plasma if not terminated by limiting the pulse width. We report on the numerical results of screamer development and propagation in pin-plane electrode geometry. The effect of pin diameter on the formation of reactive species in hydrogen/air plasma, which is important for plasma-assisted combustion, is discussed. The Laplacian field has significant enhancement at the electrode tip as the pin radius is decreased. The results show that the initial development is dependent on the tip radius but once away from the tip, the pin radius has very little impact on streamer propagation as the electrode gap is increased.

Abstract: Red blood cell (RBC) aggregation is a sensitive indicator of hemorheological alterations, but conventional methods generally rely on intermittent or end-point measurements. Here, we present a flow-stabilized closed-loop microfluidic platform for continuous measurement of the RBC aggregation index (AI) during controlled circulation and thermal exposure. The system integrates a peristaltic pump, hydraulic flow stabilizer (FS), microfluidic aggregation sensor, and thermal mixer to suppress flow pulsation and RBC sedimentation. The FS increases the characteristic flow-response time by approximately 17.6-fold, markedly attenuating pump-induced fluctuations. Under stabilized conditions, flow rates in the main channel and test chamber remain highly linear with the imposed flow rate, while stable AI measurements are maintained for at least 30 min. The platform also resolves dextran-dependent changes in RBC aggregation during continuous circulation. During thermal exposure from 25 to 47 °C, main-channel image intensity (I_mc) and AI remain nearly constant up to 43 °C, whereas exposure at ≥ 45 °C causes a progressive increase in Imc and a marked decrease in AI. These results demonstrate continuous detection of progressive RBC alterations without interrupting circulation and provide a versatile framework for investigating dynamic RBC damage over extended periods under controlled and reproducible in vitro conditions.

Abstract: Unsteady forced convection is almost always computed by inserting a steady velocity field into a time-dependent energy equation, because exact unsteady solutions of the coupled problem are scarce. We remove the inconsistency by transforming the full Navier–Stokes and energy equations—retaining streamwise diffusion, streamwise conduction and the complete dissipation function—under the diffusion-time-scale variables η = y/δ(x), τ = νt/δ(x)². The energy equation transforms in closed form with the same stretching operator D = η∂_η + 2τ∂_τ and the same parameters α, β, γ as the momentum equation; the terms in ηf_ηθ_η cancel identically, as they do in the momentum balance, and a wall temperature T_w − T_∞ ∝ x^2m keeps the Eckert number constant and contributes a single extra term 2Prβf_ηθ. The coupled system closes in (η, τ) alone if and only if m = −1 when dissipation is retained, and for m = ±1 when it is not: the streamwise strain 4u²_x carries a coefficient that no choice of scaling removes. Because D annihilates ζ = η/√τ = y/√νt, temperature fields θ = Θ(ζ) reduce the energy equation exactly to Θ′′ + 1/2Prζ Θ′ + π⁻¹PrEc e^−ζ2/2 = 0, whose solution is Θ = erfc(√Pr ζ/2)without dissipation and elementary at Pr = 2. Unlike previous unsteady thermal solutions, which satisfy their boundary conditions only as τ → ∞, this one satisfies them exactly at every finite time. It yields Nu_x = 0.3989(1 − 12Ec r)√PrRe_x/τ, a heat transfer coefficient independent of x, a recovery factor r = 2√πPrQ(Pr) equal to unity at Pr = 1, and a thickness ratio δ_T/δ_u = Pr^−1/2 exact at all Pr, with none of the Pr^−1/3 crossover of the steady Pohlhausen layer. The stagnation-point member m = 1 is solved numerically and relaxes onto its steady state within one diffusion time. For the fully closing case m = −1, a diverging wedge, the steady thermal problem is shown to coincide exactly with the vorticity equation at Pr = 1, so that the temperature follows by quadrature, and to be of Helmholtz type in general: it loses uniqueness at a discrete set of critical Prandtl numbers, the first at Pr_c = 1.82 and almost independent of Reynolds number.

Abstract: Similarity reductions of unsteady two-dimensional flow are almost always constructed from the boundary-layer equations, in which streamwise diffusion and the transverse momentum balance are discarded at the outset. We ask when such a reduction is also a reduction of the full Navier–Stokes equations, using the diffusion-time-scale variables \(\eta=y/\delta(x)\), \(\tau=\nu t/[\delta(x)]^2\) of Sun [32]. All chain rules collapse onto the stretching operator \(\mathcal{D}=\eta\partial_\eta+2\tau\partial_\tau\), both momentum equations transform in closed form, and the discarded terms enter through a single coefficient \(\varepsilon=\delta'^2\). For the family \(U=Cx^m\) the reduction closes in \( (\eta,\tau) \) alone if and only if \(m=1\) or \(m=-1\). The second case is a diverging channel, and there the reduced problem is elliptic in \( (\eta,\tau) \), with principal symbol \( (\xi_\eta^2+\varepsilon s^2)^2 \) degenerating only on \(\tau=0\): the similarity time is not an evolution variable, data must be posed at both ends, and marching in \(\tau\) is inconsistent. Solving the resulting boundary-value problem, we find that the influence of the terminal condition decays as \(\exp[-(T-\tau)/L]\), and that \(L\) collapses on the wedge half-angle, \(L\simeq2.2\,\theta_w\), to within 14% over a factor of 16 in Reynolds number, the prefactor depending on the sense of the flow but the scaling not. The upstream influence is therefore geometric rather than viscous: it vanishes with the divergence angle and is essentially independent of Re, so that marching is safe in slender geometries at any Reynolds number and unsafe in wide ones however viscous. Three exact solutions in elementary functions, obtained from the invariance of \(\zeta=\eta/\sqrt{\tau}=y/\sqrt{\nu t}\) under \( D \), and the Jeffery–Hamel steady limit, serve to verify the reduction throughout.

Abstract: Reijtenbagh, Tummers and Westerweel [Phys. Rev. Lett. 130, 174001 (2023)] measured the instantaneous drag on plates accelerated normal to their own plane, reported \(\hat F_a=C_a\rho A\sqrt{\nu aV_a}\) with \(C_a\simeq291\) independent of acceleration and velocity, and rationalized it with a history force built on the one-dimensional Stokes first problem. We solve the correct model problem---an unsteady Hiemenz layer whose strain rate grows linearly in time---using the similarity transformation of Sun [Phys. Fluids 36, 083616 (2024)]. The two-dimensional boundary-layer equation collapses to a single equation in $(\eta,\tau)$ whose coefficients are independent of the streamwise coordinate; its leading-order solution is exact and is a Kummer function of the second kind. The transformation identifies the experimental dimensionless time with the diffusion time, \(\tau=2t^{*}\), and an exact geometric identity returns the empirical law with the same coefficient \(2/\sqrt\pi\), generalized to arbitrary velocity programs as a Basset memory integral. The resulting force is smaller than the measurement by a factor that we show factorizes exactly into the ratio \(h/\delta=\sqrt{a^{*}\mathrm{Re}_h}\) of plate size to diffusion length and a residual factor of two: the boundary layer accelerates fluid over \(\sqrt{\nu t}\) where the plate accelerates it over \(h\). We then test the empirical law against the published data. Digitizing the reported coefficient for all runs shows that \(C_a\) is not constant but rises as \(V_a^{0.36\,[0.28,0.45]}\), excluding the assumed \(\nu^{1/2}\) scaling at about seven standard deviations. Frame-by-frame analysis of the supplemental video shows that \( F_D \approx \rho \, l_a \, \frac{d(\Gamma b_v)}{dt} \) to within $10\%$ throughout the acceleration, with no fitted parameter. The unsteady force on a starting plate is therefore vortical, and a single experiment---repeating a few runs at elevated viscosity---would settle the exponent.

Abstract: We construct the similarity structure of unsteady compressible two-fluid (Hall) plasma boundary layers with separate ion and electron temperatures. Combining the unsteady Howarth–Dorodnitsyn reduction of Stewartson \( [Q.~J.~Mech.~Appl.~Math.~\textbf{4}, 182 (1951)] \) with the diffusion-time similarity ansatz of Sun \( [Phys.~Fluids ~ \textbf{36}, 083616 (2024)] \), we obtain a closed system in \( (\eta,\tau) \) for the reduced stream function, flux function, out-of-plane field and velocity, and two temperatures. Three structural results organise the problem. First, the Hall transport operator is invariant under the Dorodnitsyn map: because the Hall coefficient \( \lambda=1/\mu_0 ne \) scales as the local temperature while the in-plane magnetic derivative \( B_\bot\!\cdot\!\nabla \) transforms with the inverse weight, the two cancel exactly, and the Hall term—together with the magnetic force per unit mass—takes precisely its incompressible form. The pressure gradient does not; it acquires the familiar factor \( \omega=T/T_r \). Second, variable density restores the electron-pressure (Biermann) term, which vanishes identically at constant density. We show it can never enter the in-plane flux equation, and that within a similarity solution at uniform pressure it still vanishes because all thermodynamic gradients are parallel; it survives only through the streamwise pressure gradient, where it reduces to the source \( \theta_{e,\eta}/2M^2 \) and generates a second, independent quadrupole. This Biermann quadrupole dominates the Hall quadrupole below \( M\simeq0.73 \) and exceeds it twelvefold at \( M^2=0.05 \). Third, exact Hall similarity requires a layer of constant thickness (\( m=1 \)) while exact compressible similarity requires constant edge Mach number (\( m=0 \)); the two are disjoint, so no exactly similar compressible two-fluid layer exists. The admissible families are a strongly heated stagnation-type layer, exact to all orders in \( \varepsilon_H \), and a compressible flat plate, exact in the thermodynamics and self-similar through \( O(\varepsilon_H) \), whose Hall subsystem remains a first-order pair with \( f\mapsto\omega f \). Further results: the change of type of the unsteady system is purely kinematic, both equations carrying \( 1-2\tau f_\eta, so \tau_{\rm crit}=1/2 \) independently of field strength, Mach number and wall condition; the Alfvénic degeneracy is not displaced by compressibility, because \( \omega\to1 \) at the layer edge where the transition is decided; an aligned frozen-in field destroys the Mach independence of the transformed flat-plate skin friction, by \( 38\% \) at \( M^2=0.7, M_e^2=16 \); a hot wall expels tangential flux, the wall field falling by a factor of \( 4.6 \) between \( M_e^2=0 \) and \( 16 \); and expansion cooling drives the electron temperature below its free-stream value when equipartition is weak. Numerical solutions reproduce Hiemenz, Blasius–Dorodnitsyn and the compressible recovery factor to six figures, and confirm the predicted \( O(\varepsilon_H^2) \) back-reaction over a sixteenfold range in \( \varepsilon_H \).

Abstract: Artificial intelligence (AI) continues to face fundamental mathematical challenges such as optimization in high-dimensional nonconvex landscapes, generalization under uncertainty, lack of interpretability, and sharp phase transitions in learning dynamics. Similar unresolved problems appear in physics and engineering — for example in turbulence, nuclear fusion, neural information processing, and extreme events. We propose that the universality of the Riemann zeta function provides a unified mathematical foundation for these phenomena. In particular, we introduce the zeta-derived potential S (Re, Im) \( = |\zeta(\Re + i\,\Im)| - \ln|\zeta(\Re + i\,\Im)| - 1, \) which generates a family of self-consistent measures reproducing canonical physical distributions such as Boltzmann, Planck and Kolmogorov spectra. By incorporating the zeros of the zeta function, we develop a zero-aware reparameterization framework that improves optimization, accelerates convergence and provides a principled turbulence closure mechanism. This approach creates a bridge between data, dynamics and statistical measures while preserving analytical properties of \( \zeta(s) \) and basic conservation laws. As a result, it offers a single coherent structure for understanding AI optimization, turbulence modelling and critical transitions in complex systems.

Abstract: The notion of a vortex is fundamental in fluid dynamics, where it broadly refers to rotary fluid mo-tions of various forms. Yet a precise, universal definition has remained elusive. Two fundamental challenges persist in the existing definitions of a vortex. The first is the gap between the analytic perspective that adopts a framework of motion decomposition and the synthetic perspective that em-phasizes the geometric patterns of the composite motion. The second is the gap between precisely defined local measures of rotation and intuitive large-scale descriptions of vortices. The author devel-ops a geometric theory of smooth tangent vector fields on oriented closed surfaces that bridges the an-alytic and synthetic perspectives, and provides a nonlocal definition of a vortex core. Working within the frameworks of the irreducible symmetric-antisymmetric decomposition (iSAD), eigenvalue de-composition (EVD) and Helmholtz-Hodge decomposition (HHD), the author proves two principal results concerning these vortex cores. First, all streamlines therein wind in the same direction, indi-cating a nonlocal rotary motion in the entire vortex core. Second, each vortex core can have at most one axis (center or focus). The theory is illustrated with examples on spheres and tori of various cur-vature, demonstrating how geometry and topology influence the shape of vortex cores. The results are purely mathematical and extend naturally to open surfaces, offering a rigorous foundation for vortex identification across disciplines.

Abstract:

We construct from first principles the complete similarity structure of unsteady incompressible two-fluid (Hall) plasma boundary layers with separate ion and electron temperatures, and analyse the resulting states in full. We first establish four exact degeneracies of the planar problem that constrain any admissible formulation: the Hall force cancels identically from the total momentum equation; the electron-pressure (Biermann) term vanishes identically from the induction equation at constant density; if the out-of-plane field \(B_z\) is set to zero the Hall term drops out of the in-plane flux equation altogether, so that a strictly planar "Hall boundary layer" is indistinguishable from resistive magnetohydrodynamics; and the \(B_z\) magnetic pressure cancels exactly from streamwise momentum. The minimal consistent description is therefore a four-field system in (\(\psi, B_Z, \mu_\perp, \mu_Z\)) coupled to two temperatures. Reducing this system under the two-parameter similarity map \(\eta=y/\delta(x)\), \(\tau=\nu t/\delta^{2}(x)\), we prove an exhaustive classification theorem: exact similarity of the full two-fluid problem requires \(\delta'(x)=0\), which admits precisely two nondegenerate families—a stagnation-type layer with \(U_e\propto x\), exactly self-similar to all orders in the Hall parameter \(\varepsilon_H=Md_i/\delta\), and a degenerate Rayleigh layer—while the Blasius-type growing layer is self-similar exactly through first order in \(\varepsilon_H\) and breaks similarity only at \(O(\varepsilon_H^{2})\). The obstruction is intrinsic: \(d_i\) is an absolute length while \(\delta\) grows, so Hall physics in a growing layer is a leading-edge phenomenon confined to \(x\lesssim x_H\) with \(x_H/d_i=\tfrac12M^{2}\mathrm{Re}_{d_i}\). We derive the reduced systems, their boundary conditions, and a set of von K\'arm\'an integral relations in full detail; obtain the asymptotic hierarchy in \(\varepsilon_H\); and identify the dispersive whistler scale \(d_i/\delta=\varepsilon_H/M\). Numerical solutions of the steady boundary-value problem and of the unsteady initial-value problem—the latter requiring implicit integration because the Hall coupling is a stiff fourth-order whistler operator—confirm every structural prediction, including an exact identity between the viscous and magnetic displacement thicknesses. A linear stability analysis shows the similarity states to be stable, with a leading eigenvalue that reproduces the measured relaxation rate to 0.6%, equals -1 exactly in the field-free limit for all \(P_m\) (a mode we identify analytically), vanishes linearly at the Alfvénic point as -1.36(1-\(M^{2}\)), and becomes complex near \(\varepsilon_H\simeq0.4\), so that relaxation turns oscillatory under the whistler coupling. We close by mapping the accessible regime onto laboratory and space plasmas using Braginskii transport, and by stating plainly where the collisional description fails.

Abstract: Steady compressible magnetohydrodynamic (MHD) boundary layers have been studied since the work of Rossow [32], Bleviss [3] and Bush [5]. What has not been developed is their unsteady similarity structure. This paper supplies it, by combining the Howarth--Dorodnitsyn transformation with the diffusion-time similarity ansatz of Sun [37], and in doing so uncovers a control mechanism that the steady literature did not isolate. Taking the applied field normal to the wall, B \( = B_0(x)\hat{e}_y \), makes the current solenoidal identically in two dimensions, so no auxiliary electrostatic problem arises. The external circuit is characterised by the classical load factor \( k = -E_z/(UB_0) \) [29,38]. The reduction yields a coupled pair of equations in \( (\eta,\tau) \) for the reduced stream function f and reduced total enthalpy g, governed by \( (a,b,c;\Theta,\mathrm{Pr},N,k) \) with \( N=\sigma B_0^2\delta^2/(C\mu_r) \) a squared Hartmann number. Our central structural result is that the load factor cancels identically from the momentum equation once the outer pressure field is treated consistently, and survives only in the enthalpy equation, where the combined Lorentz work and Joule dissipation reduce exactly to J\( \cdot\boldsymbol{E} \) \( =\sigma B_{0}^{2}kU\left(kU-u\right) \). The momentum source is \( N\omega(1-f_\eta) \) for every k; the enthalpy source is \( NE\,\omega\,k(k-f_\eta) \). Three consequences follow. (i) The linear Crocco integral survives the field only at k=0, where the classical \( g\equiv1 \) holds exactly despite an arbitrarily strong field, and at k=1, where it holds only for the cooled wall \( T_w/T_e = 2-\Theta \); for all other k no linear enthalpy--velocity relation exists. (ii) Because \( k(k-f_\eta) \) changes sign at \( f_\eta = k \), a boundary layer operating in generator mode extracts enthalpy near its outer edge. (iii) Consequently the load factor is a thermal-control parameter: sweeping k moves the adiabatic recovery factor from r=0.336 (a 44%reduction in adiabatic wall temperature relative to the field-free value at M_e=4, N=4) through r=1 near \( k_*\simeq0.61 \) and on to thermal runaway, with an optimum cooling load factor \( k_{\rm opt}\simeq0.26 \) that is almost independent of N and is predicted analytically by \( k=f_\eta/2 \). The unsteady analysis gives three further results. The change of type of the reduced system is purely kinematic: both equations carry the same factor \( 1-c\tau f_\eta \), so \( \tau_{\rm crit}=1/c \) is independent of N, k, Mach number, Prandtl number and wall condition, while the thermal mode diverges beyond it a factor 1/Pr faster — confirmed by a \( 10^{44} \) disparity in computed divergences. Exact constancy of the compressibility parameter \( \Theta \) requires \( m=0 \) whereas unconditional well-posedness requires \( m\ge1 \), so the two are mutually exclusive. And for Pr=1 we derive and verify to seven digits the identity \( r = 2f_{\eta\eta}(0)Q-1 \) with \( Q=\int_{0}^{\infty}\exp\left[-\int_{0}^{\eta}f\right]d\eta \). Global solutions in the Williams--Rhyne chart connect the impulsive-start Rayleigh state to the steady state through the region forward marching cannot enter, and confirm that neither field nor compressibility is felt at the impulsive start.

Abstract: The diffusion-time similarity transformation introduced by Sun [Phys. Fluids 36, 083616 (2024)] reduces the two-dimensional unsteady boundary layer equations to a single partial differential equation in the two variables η = y/δ(x) and τ = νt/δ²(x). We extend the reduction to an electrically conducting fluid at low magnetic Reynolds number and show that the reduced problem changes type. Grouping the time-derivative terms reveals that the coefficient of the highest mixed derivative is 1 − Λ with Λ = cτf_η, so that a frozen-coefficient analysis gives the growth rate λ = −k²/(1 − Λ): marching forwards in the diffusion time is parabolic only where Λ < 1 and backward-parabolic beyond. For a power-law outer stream U = Cx^m we find Λ = (1 − m)ut/x, so the threshold coincides with the Stewartson starting-front criterion t = x/u for a flat plate. Integrating the constancy conditions rather than postulating a profile shows that the admissible flows comprise exactly three families—power-law, exponential and uniform—the exponential one being missed by the usual ansatz. Constancy of the magnetic interaction parameter is a separate condition, B₀δ = const, so a uniform applied field is admissible only where the layer does not grow—that is, only at a stagnation point. The reduction is unconditionally well posed for m ≥ 1 and, for m < 1, must be integrated backwards in τ beyond τ = 1/c, which corresponds to marching downstream. A second-order implicit scheme confirms the threshold sharply: for c = 2 the wall shear is grid-converged to six figures up to τ = 0.45 on four successive meshes and diverges non-monotonically in the mesh beyond τ = 0.5. We further derive the two-term strong-field expansion f_ηη(0) ≃ M^1/2 + (a/12 + 2b/3)M^−1/2 for the magnetic interaction parameter M, which reproduces published spectral benchmarks to eight significant figures and fixes the prefactor of the √M friction law at unity.

Abstract: The thermal boundary layer on a flat plate is almost always computed by inserting the steady Blasius velocity field into a time-dependent energy equation, so that the temperature evolves along trajectories that do not. The inconsistency has been unavoidable: no exact solution of the two-dimensional unsteady laminar boundary layer was available until Sun [16] obtained one in terms of Kummer functions, using the diffusion time \(\tau=\nu t/\delta^{2}(x)\) as a similarity variable. We solve the corresponding thermal problem. The similarity transformation applied to the energy equation yields \[ \theta_{\tau}-\alpha f\theta_{\eta}+\gamma\tau(f_{\tau}\theta_{\eta} -f_{\eta}\theta_{\tau})=Pr^{-1}\theta_{\eta\eta}+Ec\,f_{\eta\eta}^{2}, \] the terms in \(\eta f_{\eta}\theta_{\eta}\) cancelling identically; the system closes only for a flat plate. Sun's solution, given through six auxiliary functions, collapses to three Kummer-\(U\) expressions and is invariant under the group generated by \(X=(\eta\tau+2)\partial_{\eta}+2\tau^{2}\partial_{\tau}\), with \(Xf=\tau f-\eta\). On group-invariant temperature fields the convective operator loses all dependence on the velocity field, and the energy equation reduces exactly to \[ \Theta''+2Pr\,\omega\Theta'+Pr\,Ec\,[F'(\omega)]^{2}=0 \] with \(\omega=(3\eta\tau+2)/2\sqrt{3}\tau^{3/2}\). Hence \[ \theta=\operatorname{erfc}(\sqrt{Pr}\,\omega)+Pr\,Ec\,\Phi(\omega), \] together with closed forms for the Nusselt number, a recovery factor \(r=2Pr\,P_{\infty}\), a Reynolds analogy factor \(0.7132Pr^{-1/2}\) and a thickness ratio \(1.0910Pr^{-1/2}\) exact at all $Pr$. Both fields satisfy the governing equations to below \(10^{-17}\). Boundary conditions are met to \(O(\tau^{-1})\), inherited from the parent solution.