Investigating the Mathematical Foundations of the Euler and Navier-Stokes Equations  

作　　者：Zaki Harari Zaki Harari(Novelant Scientific Research Inc., Toronto, Canada)

机构地区：[1]Novelant Scientific Research Inc., Toronto, Canada

出　　处：《Open Journal of Fluid Dynamics》2024年第4期242-258,共17页流体动力学(英文)

摘　　要：This study examines the mathematical foundations of the Euler and Navier-Stokes equations of fluid dynamics, identifying some inconsistencies in the mathematical definitions of flow velocity and the material derivative. We show that the flow velocity of a fluid parcel, which in the Lagrangian description is traditionally modeled as a bivariate function of the presumed independent variables of initial parcel position and time, is more accurately defined as a parametric function of time, with the initial parcel position treated as a time-dependent parameter. This finding leads to the result that the standard form of the material derivative in the Lagrangian description is mathematically inconsistent. We also show that if the fluid flow is non-unidirectional, then the map from parcel position to flow velocity becomes a one-to-many map, leading to the conclusion that the flow velocity is not a valid mathematical function of position in both the Lagrangian and Eulerian descriptions under such conditions. Therefore, if flow velocity is not a valid mathematical function of position, we conclude that the inability to integrate the Euler and Navier-Stokes differential equations in the spatial domain implies the nonexistence of a mathematical solution of these equations under these conditions. Additionally, through mathematical and theoretical analysis, supported by experimental and numerical simulations, we uncover challenges in the material consistency of the definition of the material derivative in the Eulerian description. This inconsistency leads to a decoupling between the Lagrangian and Eulerian descriptions, especially under complex non-unidirectional flow conditions and multi-directional flows with intersecting pathlines. We also show that the Eulerian description is a quasi-continuum mechanics model that, when applied to certain fluids, especially gases and low-viscosity liquids where intermolecular forces are weak or intermediate, limits the ability to accurately model the bi-directional transmission of deformatThis study examines the mathematical foundations of the Euler and Navier-Stokes equations of fluid dynamics, identifying some inconsistencies in the mathematical definitions of flow velocity and the material derivative. We show that the flow velocity of a fluid parcel, which in the Lagrangian description is traditionally modeled as a bivariate function of the presumed independent variables of initial parcel position and time, is more accurately defined as a parametric function of time, with the initial parcel position treated as a time-dependent parameter. This finding leads to the result that the standard form of the material derivative in the Lagrangian description is mathematically inconsistent. We also show that if the fluid flow is non-unidirectional, then the map from parcel position to flow velocity becomes a one-to-many map, leading to the conclusion that the flow velocity is not a valid mathematical function of position in both the Lagrangian and Eulerian descriptions under such conditions. Therefore, if flow velocity is not a valid mathematical function of position, we conclude that the inability to integrate the Euler and Navier-Stokes differential equations in the spatial domain implies the nonexistence of a mathematical solution of these equations under these conditions. Additionally, through mathematical and theoretical analysis, supported by experimental and numerical simulations, we uncover challenges in the material consistency of the definition of the material derivative in the Eulerian description. This inconsistency leads to a decoupling between the Lagrangian and Eulerian descriptions, especially under complex non-unidirectional flow conditions and multi-directional flows with intersecting pathlines. We also show that the Eulerian description is a quasi-continuum mechanics model that, when applied to certain fluids, especially gases and low-viscosity liquids where intermolecular forces are weak or intermediate, limits the ability to accurately model the bi-directional transmission of deformat

关 键 词：Euler’s Equation Navier-Stokes Equations Fluid Dynamics Fluid Mechanics 

分 类 号：O17[理学—数学]