From the jacobian technique to the variational formulation of rotational flows

Gargalas, Dimitrios; Γαργάλας Δημήτριος

dc.contributor.author	Gargalas, Dimitrios	en
dc.contributor.author	Γαργάλας Δημήτριος	el
dc.date.accessioned	2026-03-11T07:56:53Z
dc.date.available	2026-03-11T07:56:53Z
dc.identifier.uri	https://dspace.lib.ntua.gr/xmlui/handle/123456789/63923
dc.identifier.uri	http://dx.doi.org/10.26240/heal.ntua.31618
dc.rights	Αναφορά Δημιουργού-Μη Εμπορική Χρήση-Όχι Παράγωγα Έργα 3.0 Ελλάδα	*
dc.rights.uri	http://creativecommons.org/licenses/by-nc-nd/3.0/gr/	*
dc.subject	Jacobian determinants	en
dc.subject	Determinant Derivatives	en
dc.subject	4-momentum	en
dc.subject	Hamilton’s Principle	en
dc.subject	Variational Principles	en
dc.subject	Constrained Action Functional	en
dc.subject	Clebsch Representation	en
dc.subject	Conservation of Parcels’ Labels	en
dc.subject	Euler’s equations	en
dc.subject	Gauge Symmetry	en
dc.title	From the jacobian technique to the variational formulation of rotational flows	en
heal.type	masterThesis
heal.classification	Mathematical Modeling of Rotational Flows	en
heal.language	en
heal.access	free
heal.recordProvider	ntua	el
heal.publicationDate	2025-10
heal.abstract	The understanding of the fluid motion, especially the rotational flows, is of fundamental importance in the understanding of a large number of interesting phenomena. A pivotal point in the formulation of any flow phenomenon is the underlying description of the fluid kinematics. Historically, the movement of the fluid is described either through the “positioning” of the fluid in space and time, inside the flow field, or in terms of infinitesimal masses of fluid (fluid parcels) identified uniquely by an identifier (label) at some labeling time. Nevertheless, these two descriptions of the flow must both be considered in order to completely describe the flow of fluid. The derivation of equations, governing the flow of the fluid, through the variational principles, highlights these considerations since, for the appropriate constraints to be applied on an action functional in the Eulerian description, the Lagrangian description of the flow must also be considered. Through the Herivel-Lin approach, an almost universally acceptable approach was established, considering as constraints, applied on the Eulerian action functional, the conservation of mass, as well as the conservation of the parcels’ labels, which directly stems from the Lagrangian description. Following Rylov’s perspective, we approach the formulation of these constraints in accordance with the Herivel-Lin approach but making an effort to state the exact nature of the constraints taking into account only the necessary considerations for their validity. Since the Herivel-Lin approach assumes a bijective and sufficiently smooth point transformation exists between the Lagrangian and the Eulerian description of the flow, we conducted a thorough study of the mathematical structure of the Jacobian determinant generated by such mappings that allow for the aforementioned descriptions of the flow to exist compatibly with respect to each other. We, further, established identities originating from the application of derivative operators (partial and total) on the structure of the determinant expressing such mappings. We then established a Lagrangian time-like variable, accounting for the passage of time through the successive states of the individual parcels of the fluid, in order to ensure that, throughout their motion, the fluid parcels are uniquely identified by a specific label, conserved throughout the course of the flow. Under these considerations and through the use of 4-vectors, we proceeded to construct an expression for the 4-momentum vector in terms of the derivatives of the determinants generated by mappings between the Eulerian and the Lagrangian spatio-temporal variables. We then proceeded to demonstrate that, since the aforementioned expression of the 4-momentum is derived from the consideration of mappings that allow for both descriptions of the fluid to be stated compatibly, the 4-momentum in terms of the Jacobian determinant identically satisfies the conservation of mass as well as the conservation of the parcels’ labels (both of which constitute the constraints applied to the action functional in the Herivel-Lin approach). We then conduct a variational study, over a constrained action functional using the aforementioned expressions for the 4-momentum as a constraint, by means of Lagrange multipliers. The constraints on the 4-momentum are expressed through the Jacobian, generated by the transformations from the Eulerian to the Lagrangian spatio-temporal variables. A first set of dynamic equations is, thus, derived from the variational study. Making use of the aforementioned set of dynamic equations, the Lagrange multipliers concerning the constraint over the 4-momentum field are then integrated out, providing explicit forms for the Lagrange multipliers. Using these expressions a modified action functional is constructed. We then proceeded to conduct an additional variational study over the second form of the constrained action functional resulting in a set of dynamic equations that includes a generalized Clebsch representation of the velocity field, the continuity equation as well as an equation, governing the evolution of the field of the parcels’ labels. A survey is finally carried out, with the aim of interpreting the physical meaning of the dynamic equations derived from the variations of the action functional, leading to results, indicating that, in rotational flows, the phenomena taking place inside the flow field are more complex, as an additional undefined function arises from the dynamic equation describing the evolution of the field of the parcels’ labels throughout the course of the flow. Nevertheless, we demonstrate that, regardless of the values of the undefined function, Euler’s equations of fluid motion can be derived from the set of dynamic equations produced from the variational study of the constrained action functional and as such, the undefined function expresses a gauge symmetry, rendering the evolution of the field of the parcel labels dynamically indeterminate.	en
heal.advisorName	Athanassoulis, Gerassimos	en
heal.advisorName	Αθανασούλης, Α. Γεράσιμος	el
heal.committeeMemberName	Belibassakis, Kostas	en
heal.committeeMemberName	Flokas, Helena	en
heal.academicPublisher	Εθνικό Μετσόβιο Πολυτεχνείο. Σχολή Ναυπηγών Μηχανολόγων Μηχανικών	el
heal.academicPublisherID	ntua
heal.numberOfPages	215 σ.	el
heal.fullTextAvailability	false