Geometric Unification of Fluid Motion Forms

Author: Zhang Suhang

Luoyang School of Mathematics

Abstract

Traditional fluid mechanics usually regards rectilinear (straight) flow and circulatory vortex flow as two mutually independent forms of motion, and assigns wall-normal perturbations and vortex-radial perturbations to different constraint systems, with no unified geometric underlying framework between them. Based on two foundational axioms of fluids—tangential convergence and normal boundedness—this paper attempts a two-fold geometric merger of the basic forms of fluid motion: it argues that rectilinear flow may be regarded as a local tangentially steady structure of large-scale circulation, the two obeying the same set of tangential-convergence evolution laws; and that wall-normal perturbations and vortex-radial perturbations satisfy a consistent gradient-boundedness constraint and may be subsumed under the same class of transverse perturbation modes. The dual system thus established incorporates laminar flow, directed rectilinear flow, vortex circulation, and turbulent fluctuation into a unified topological description, forming a self-consistent, closed, and interpretable elementary system of incompressible fluid, and providing an organizational framework for the global evolutionary structure of real fluids.

Keywords: geometric unification of fluids; common origin of rectilinear and circulatory flow; normal–radial duality; fluid axioms; flow-field topology; finiteness of turbulence

1 Introduction

In the classical understanding of fluids, the directed mainstream in pipes and between parallel plates is usually defined as rectilinear flow, while vortices, vortex rings, and rotating flow fields are defined as circulatory motion, and these have long been described, modeled, and analyzed separately. At the same time, the normal perturbations of boundary flows and the radial perturbations within vortex structures are often assigned to different physical mechanisms, lacking a general constraint criterion.

This fragmented description gives rise to two limitations:

1. It is difficult to explain the interlocking phenomena in which vortex structures arise spontaneously within rectilinear flow fields, and in which vortex fields locally exhibit a straight mainstream;

2. It is difficult to establish a unified criterion for the boundedness of perturbations, so as to give a general geometric explanation of turbulence, transition, and vortex evolution.

Building on the previously established foundational axiom system of tangential convergence and normal boundedness, this paper proceeds from the geometric structure and kinetic essence of the flow field, attempts a unified merger of the basic forms of fluid motion, and on this basis constructs a dual framework for describing fluid motion.

2 Review of the Foundational Axioms

To ensure the self-consistency of the argument, the two core foundational axioms of real fluids are retained as the sole premises of the unified description in this paper.

Axiom 1: Axiom of Tangential Convergence

Tangential motion in the flow field is constrained by viscous topology and evolves over long times toward a smooth elliptical statistical steady state. The tangential components of both directed flow and rotating flow possess a tendency toward steady convergence.

Axiom 2: Axiom of Normal Boundedness

The perturbation gradient in the direction normal to the mainstream possesses a globally uniform upper bound; the perturbation energy, velocity gradient, and deformation tensor remain finite, thereby avoiding finite-time singularity blow-up.

Prior work has argued that these two axioms distinguish the unconstrained mathematical solution set of the N–S equations from the physically realizable solutions of nature, providing a unified explanatory framework for such problems as the finiteness of turbulence, the transition threshold, and the persistence of the global flow direction. On this basis, the present paper discusses the merger at the level of forms.

3 First Unification: Common Origin of Rectilinear and Circulatory Flow

3.1 Core Proposition

Straight rectilinear flow may be regarded as a local tangentially steady segment of large-scale circulation.

Macroscopic large-scale circulation has extremely small curvature, and its local arcs approach a straight line segment indefinitely. The pipe flow, channel flow, and boundary-layer mainstream observed in engineering and in nature may be understood as local tangential motion regions of an extremely large-scale vortex circulation.

3.2 Unified Evolution Laws

Whether global circulation or local rectilinear flow:

1. The tangential velocity component obeys Axiom 1 and tends over long times toward a statistical steady state;

2. The macroscopic directionality of the flow persists and does not vanish owing to local perturbations;

3. Both the laminar steady state and the turbulent mean-mainstream state are built upon tangential convergence.

It follows that rectilinear and circulatory flow share a common dynamical origin, are topologically isomorphic, and obey the same evolutionary rule. The traditional dichotomy is, rather, a difference of scale perspective than a difference of physical essence.

4 Second Unification: Duality of Normal and Radial Perturbations

4.1 Core Proposition

Wall-normal perturbations and vortex-structure radial perturbations may be subsumed under the same class of transverse perturbation modes, uniformly satisfying the axiom of normal boundedness.

1. In pipe and flat-plate boundary flow fields, the normal fluctuations perpendicular to the wall, velocity pulsations, and boundary-layer perturbations;

2. Within vortex structures, the radial velocity gradients, vorticity perturbations, and mixing motions diffusing outward from the vortex core.

The two differ in geometric direction and in setting, yet are identical in physical essence: both are transverse perturbations perpendicular to the tangential mainstream.

4.2 Unified Constraint Results

All transverse perturbations, whether normal or radial, are constrained by Axiom 2:

· The gradient does not grow without bound;

· There is no finite-time blow-up;

· Turbulent perturbations maintain finite amplitude;

· Local disordered mixing does not destroy the stability of the global structure.

Complex flows such as strong boiling convection, free shear turbulence, alternating vortex shedding, and wake perturbations can all be incorporated into this dual constraint.

5 The Dual System: Global Closure of Four Classes of Flow

Building on the above two unifications, this paper establishes an elementary dual system of fluid motion:

Tangential system (ordered-convergence end)

· Local form: rectilinear mainstream

· Global form: circulatory vortex

· Core properties: tangential convergence, globally directed, statistical steady state

Transverse system (finite-perturbation end)

· Boundary form: wall-normal perturbation

· Vortex-field form: vortex-radial perturbation

· Core properties: gradient boundedness, finite turbulence, no singularity blow-up

Thus the four basic forms of fluid—laminar flow, rectilinear flow, circulatory flow, and turbulence—are all described by the two axioms and the two groups of unification, with no omission, contradiction, or exception requiring separate treatment so far observed.

6 Advantages and Innovations of the System

1. Merging traditional classifications: the description that separates rectilinear from circulatory flow, and normal from radial perturbations, is merged at the level of geometric origin.

2. Compatibility with major real flows: directed pipe flow, free vortex flow, natural convection, fully developed turbulence, and boundary-layer flow can all be located within this framework.

3. Distinguishing mathematical from physical solutions: the unconstrained N–S blow-up solution belongs to a purely mathematical special case and lies outside the scope of this dual description; real fluids are constrained by geometry and energy.

4. Broad explanatory scope: it can describe both the stability of the global flow direction and the disorder of local turbulence; both the laminar steady state and vortex-driven mixing.

7 Conclusion

1. Straight rectilinear flow and vortex circulation may be regarded as structures of common origin, rectilinear flow being a local steady approximation of large-scale circulation, with the two sharing consistent tangential evolution laws.

2. Wall-normal perturbations and vortex-radial perturbations may be subsumed under the same class of transverse modes, jointly obeying the gradient-boundedness constraint.

3. Based on the axioms of tangential convergence and normal boundedness, a dual fluid description framework—rectilinear–circulatory and normal–radial—can be established, giving a self-consistent unified description of the principal forms of motion of incompressible real fluids.

4. This framework is concise in structure, logically closed, and free of physical singularities, offering an approach to the systematic merger of the fundamental laws of fluid motion.

References

[1] Zhang Suhang. Foundational Axioms of Fluid Mechanics: Normal Boundedness and Tangential Convergence, 2026.

[2] Zhang Suhang. Proof of the Global Smooth Solution of the Navier–Stokes Equations under the DOG/MOC/ECS/MIE Framework, 2026.

[3] Trefethen L N. Transition threshold of fluid flow, 1993.

[4] Zhang Suhang. MOC Curvature Conservation and Its Application in Fluid Energy Estimates, 2026.

[5] Constantin P, Procaccia I, Segel D. Creation and dynamics of vortex tubes in three-dimensional turbulence. Physical Review E, 1995, 51(4): 3207–3222.