Abstract
From the principle of least action the equation of motion for viscous compressible and charged fluid is derived. The viscosity effect is described by the 4-potential of the energy dissipation field, dissipation tensor and dissipation stress-energy tensor. In the weak field limit it is shown that the obtained equation is equivalent to the Navier-Stokes equation. The equation for the power of the kinetic energy loss is provided, the equation of motion is integrated, and the dependence of the velocity magnitude is determined. A complete set of equations is presented, which suffices to solve the problem of motion of viscous compressible and charged fluid in the gravitational and electromagnetic fields.
Highlights
Since Navier-Stokes equations appeared in 1827 [1], [2], constant attempts have been made to derive these equations by various methods
In book [5] it is considered that Navier-Stokes equations are the extremum conditions of some functional, and a method of finding a solution of these equations is described, which consists in the gradient motion to the extremum of this functional
P0 0 c2 u is the 4-potential of the pressure field, consisting of the scalar potential and the vector potential Π, p0 is the pressure in the reference frame associated with the particle, the ratio p0 0 c2 defines the equation of state of the fluid, f is the pressure field tensor
Summary
Since Navier-Stokes equations appeared in 1827 [1], [2], constant attempts have been made to derive these equations by various methods. Our goal is to provide in general form the four-dimensional stress-energy tensor of energy dissipation, which describes in the curved spacetime the energy density and the stress and energy flux, arising due to viscous stresses This tensor will be derived with the help of the principle of least action on the basis of a covariant 4-potential of the dissipation field. And are some functions of coordinates and time, p0 0 c2 u is the 4-potential of the pressure field, consisting of the scalar potential and the vector potential Π , p0 is the pressure in the reference frame associated with the particle, the ratio p0 0 c2 defines the equation of state of the fluid, f is the pressure field tensor. We will not provide here the intermediate results from [7], and will right away write the equations of motion of the fluid and field, obtained as a result of the variation of the action function (1)
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