Generalized sedeonic equations of hydrodynamics
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Generalized sedeonic equations of hydrodynamics V. L. Mironov a
, S. V. Mironov
Institute for Physics of Microstructures, Russian Academy of Sciences, GSP-105, Nizhny Novgorod 603950, Russia Received: 8 June 2020 / Accepted: 18 August 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We present a generalization of the equations of hydrodynamics based on the noncommutative algebra of space-time sedeons. It is shown that for vortex-less flow the system of Euler and continuity equations is represented as a single nonlinear sedeonic second-order wave equation for scalar and vector potentials, which is naturally generalized on viscous and vortex flows. As a result we obtained the closed system of four equations describing the diffusion damping of translational and vortex motions. The main peculiarities of the obtained equations are illustrated on the basis of the plane wave solutions describing the propagation of sound waves.
1 Introduction The analogy between the equations of hydrodynamics and electrodynamics has been actively discussed for a long time. Apparently first, some similarity between vortex dynamics of fluid and electromagnetic phenomena induction was pointed out by Helmholtz in [1]. Subsequently, several attempts were made to describe the fluid dynamics by vector fields (similar to electric and magnetic fields) satisfying some Maxwell-like equations [2–10]. However, a common drawback of the approach used in these works is that the equation for the vortex component of the fluid motion is obtained simply by taking the “curl” operator from the Euler equation for velocity and therefore it is not independent. In particular, in [4] the linearized equations for a free isentropic compressible fluid reduce to the following form: ∂E = J, ∂t ∂H = 0, [∇ × E] + ∂t (∇ · E) = g, c2 [∇ × H] −
(∇ · H) = 0,
(1)
where vector fields E and H are defined by the following expressions: ∂v − ∇h, ∂t H = [∇ × v], E=−
(2)
a e-mail: [email protected] (corresponding author)
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708
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Eur. Phys. J. Plus
(2020) 135:708
and the field sources ∂ (∇ · v) − h, ∂t ∂ ∂ 2v (3) J = 2 + (∇h) + c2 [∇ × [∇ × v]]. ∂t ∂t Here v is the velocity of the fluid, h is the enthalpy per unit mass, c is the speed of sound [4]. In form, the system (1) coincides with Maxwell’s equations. However, these equations do not have any predictive power, since the field sources are determined through the quantities v and h, which themselves must be found from the equations. In addition, by substituting the definition of fields (2) and sources (3) into Eq. (1), we obtain the identity. A similar situation is observed in the works of other authors. During the past decades the essential progress is observed in the reformulation of the equations for electromagnetic field and fluid motion based on the different algebras of hypercomplex numbers such as quaternions [11–14] and octonions [15–18], which take into account the symmetry of physical values with respect to operatio
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