A potential‐based formulation of the classical and relativistic Navier‐Stokes equations

Scholle, Markus; Marner, Florian; Gaskell, Philip

doi:10.1002/pamm.202000231

Cited by 3 publications

(3 citation statements)

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“…According to [4,5], a decisive condition for self-adjointness is the appropriate gauging of the potentials. Different from the gauging used in [1,2] in order to eliminate the last term in the field equations (3), we assume that by gauging the following form of the fourth rank potential:…”

Section: Modified Gauging Of the Tensor Potentialmentioning

confidence: 99%

“…In [1,2] a potential-based formulation of the relativistic Navier-Stokes (NS) equations has been developed, an essential underpinning being analogies drawn with the methodical reduction of Maxwell's equations. We assume a flat space-time and utilise the usual relativistic notation [3] with metric signature (+, −, −, −), that is, (η µν ) = (η µν ) = diag(1, −1, −1, −1), the coordinate vector (x µ ) = (ct, x 1 , x 2 , x 3 ), together with the respective 4-gradient in covariant form given by…”

Section: Introductionmentioning

confidence: 99%

“…allowing to reformulate the relativistic energy-momentum balance in the potential-based first integral form as [1,2]:…”

Section: Introductionmentioning

confidence: 99%

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In search of a variational formulation of the relativistic Navier‐Stokes equations

Scholle

Mellmann

2021

Proc Appl Math & Mech

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Drawing an analogy with Maxwell theory a new Lagrangian is proposed for a variational formulation of the relativistic Navier‐Stokes equations which to‐date has remained elusive. A key feature is the use of tensor potentials, whose degrees of gauge freedom allow for the reformulation of the energy‐momentum equations in a self‐adjoint form. An already existing potential‐based representation of the relativistic field equations is a suitable starting point for the present considerations, which in turn are guided by the already successfully solved case of non‐relativistic, stationary and incompressible flow.

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Section: Modified Gauging Of the Tensor Potentialmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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In search of a variational formulation of the relativistic Navier‐Stokes equations

Scholle

Mellmann

2021

Proc Appl Math & Mech

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Extensions to the Navier–Stokes equations

Wang

2022

Physics of Fluids

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Historically, the mass conservation and the classical Navier-Stokes equations were derived in co-moving reference frame. It is shown that the mass conservation and Navier-Stokes equations are Galilean invariant - they are valid in any arbitrary inertial reference frame. From the mass conservation and Navier-Stokes equations, we can derive a wave equation, which contains the speed of pressure wave as its parameter. This parameter is independent of the speed of the source - the fluid velocity. The speed of pressure wave is reference frame independent. It is well known that Lorentz transformation ensures wave speed invariant in all inertial frames, and Lorentz invariance holds for different inertial observers. Based on these arguments, the Navier-Stokes equations (conservation law for the energy-momentum) can be written in any inertial reference frame, they are transformed from one reference frame into another with the help of the Lorentz transformation. The key issue is that the Lorentz factor is parametrized by the local Mach number. In the instantaneous co-moving reference frame, these equations will degrade to the classical Navier-Stokes equations - the limit of the non-relativistic ones. For subsonic flow, the classical Navier-Stokes equations are only valid as an approximation at very low Mach number. For supersonic flow, namely when the Mach number is greater than one, the Lorentz factor becomes an imaginary number, a faster-than-wave-speed reference frame (complex valued Lorentz transformation) should be applied to solve the supersonic flow problems. The general Navier-Stokes equations embody an intrinsic singularity when local Mach number is equal to one.

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Extension of Galilean invariance to uniform motions for a relativistic equation of fluid flows

Caltagirone

2023

Physics of Fluids

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Galilean invariance and the Lorentz transformation are two pillars of mechanics that an equation of motion must respect. The objective is to extend Galilean invariance to uniform rotational motion and to expansion motion of which motion at constant translational velocity is only a special case. The second goal is to show that the discrete equation of motion is naturally relativistic without using the Lorentz transformation. The realization of these two aims leads to different concepts of special relativity, (i) space is homogeneous and isotropic and (ii) time, which flows regularly, is invariant by translation. These invariances, symmetric for translation and rotation, observed by the discrete equation of motion give it fundamental conservation properties to extend the scope of the equations of fluid mechanics to other fields of physics.

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A potential‐based formulation of the classical and relativistic Navier‐Stokes equations

Cited by 3 publications

References 4 publications

In search of a variational formulation of the relativistic Navier‐Stokes equations

In search of a variational formulation of the relativistic Navier‐Stokes equations

Extensions to the Navier–Stokes equations

Extension of Galilean invariance to uniform motions for a relativistic equation of fluid flows

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