Proposal of "Fluid within Fluid" Models

Proposal of "Fluid within Fluid" Models

Michel Felgenhauer, Berlin 2022

There is no theory of Fluid within Fluid-Tubes (FwF). A workable phenomenology of Fluid within Fluid-Tubes is based on the one hand on the vortex physics of Helmholtz and it respects on the other hand the definition of Lagrangian coherent systems of Haller.

The phenomenology of Fluid within Fluid-Tubes quarrels with both established theories and with the recent doctrine.

The paper brings together aspects of traditional vortex models and the theory of Lagrangian Coherent Systems and those of a phenomenology of “Fluid within Fluid Tubes”.

Fundamental assumptions.

In fluid dynamics, vortices denote a rotating motion of fluid elements. Leonhard Euler¹ introduces the equation of motion of a frictionless incompressible fluid around 1755, thus founding modern fluid dynamics. He names the fundamental concept of the vortex vector, today's rotation (x V ).

Joseph-Louis Lagrange² is the origin of the potential theory, which is important for flow fields and from which the field theories develop around 1800. Lagrange published scientific works on differential equations and calculus of variations. William Thomson (Lord Kelvin)³ finds the circulation theorem and is in correspondence with Hermann von Helmholtz in Berlin in the middle of the 19th century, who recognizes the stability of vortices in space and time in frictionless fluids. Helmholtz's Fundamental Theorem of Kinematics (1858) concerns the general change of location of a deformable body of sufficiently small volume as the sum of translation, rotation, and deformation. It is directly applicable to the motion of filamentary structures (vortices). The three vortex theorems were formulated by Hermann von Helmholtz around 1859:

- Helmholtz's first theorem: The strength of a vortex filament is constant along its length.
- Helmholtz's second theorem: A vortex filament cannot end in a fluid; it must extend to the boundaries of the fluid or form a closed path.
- Helmholtz's third theorem: In the absence of rotational external forces, a fluid that is initially irrotational remains irrotational.

The first vortex theorem means that the circulation along the boundary curve of a surface lying entirely on the mantle of a vortex tube vanishes, and that the circulation of different cross-sections of a vortex tube is the same. The second vortex theorem states that vortex tubes are also current tubes, vortices adhere to matter (fluid) and thirdly, particles that once formed a vortex line continue to do so (coherence). The third vortex theorem requires local and temporal constancy of circulation in and around a vortex tube. Helmholtz's vortex theorems are a foundation of the physics of Lagrangian coherents "Fluids within Fluid-Tubes" discussed here. Also interesting is that the classical vortex models Potential vortex, Solid vortex, Rankine vortex and Hamel-Oseen vortex were formulated much later. These vortex models and the Helmholtz vortex theory hand down the prevailing doctrine and shall be briefly named here, because these vortex models contribute to the understanding of the inner milieu of Helmholtz vortex filaments.

The notion of "vortex filaments" links directly to Helmholtz's vortex theory. Vortex filaments serve Helmholtz's vortex theory and postulate a clear position against possible model extensions: the homogeneity of their internal milieu is non-negotiable. The theory of Helmholtz's vortex filaments insists on the temporal consistency of the circulation implicit in a vortex filament const. From the English usage the symbolization of a tube. "Tube" is taken over and is henceforth referred to as "Fluid within Fluid Tubes (FwF)". The Biot-Savart⁴ law is the special formulation of a general field theory. Maxwell⁵ was the first to summarize the corresponding processes in electrodynamics around 1855.

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¹ Leonhard Euler (lateinisch Leonhardus Eulerus; * 15. April 1707 in Basel; + 7. September in Sankt Petersburg) war ein Schweizer Mathematiker, Physiker, Astronom, Geograph, Logiker und Ingenieur. https://de.wikipedia.org/wiki/Leonhard_Euler

² Joseph-Louis de Lagrange (* 25. Januar 1736 in Turin als Giuseppe Lodovico Lagrangia; + 10. April 1813 in Paris) war ein italienischer Mathematiker und Astronom. https://de.wikipedia.org/wiki/Joseph-Louis_Lagrange

³ William Thomson, 1. Baron Kelvin oder kurz Lord Kelvin, OM, GCVO, PC, FRS, FRSE, (* 26. Juni 1824 in Belfast, Provinz Ulster, Vereinigtes Konigreich GroBbritannien und Irland; + 17. Dezember 1907 in Netherhall bei Largs, Schottland) war ein britischer Physiker auf den Gebieten der Elektrizitatslehre und der Thermodynamik. Die Einheit Kelvin wurde nach William Thomson benannt, der im Alter von 24 Jahren die thermodynamische Temperaturskala einfuhrte. Thomson ist sowohl fur theoretische Arbeiten als auch fur die Entwicklung von Messinstrumenten bekannt. https://de.wikipedia.org/wiki/William_Thomson,_1._Baron_Kelvin

⁴ Das Biot-Savart-Gesetz beschreibt das Magnetfeld bewegter Ladungen. Es stellt einen Zusammenhang zwischen der magnetischen Feldstarke H und der elektrischen Stromdichte her und erlaubt die Berechnung raumlicher magnetischer Feldstarkenverteilungen anhand der Kenntnis der raumlichen Stromverteilungen.

⁵ James Clerk Maxwell (* 13. Juni 1831 in Edinburgh; + 5. November 1879 in Cambridge) war ein schottischer Physiker. Er entwickelte einen Satz von Gleichungen (die Maxwell-Gleichungen), welche die Grundlagen der Elektrodynamik sind; insbesondere sagte er 1864 die Existenz von elektromagnetischen Wellen voraus, die Heinrich Hertz als erster 1886 erzeugte und nachwies. Seine nach ihm benannte Feldtheorie ist eine der