From the Newton’s laws to motions of the fluid and superfluid vacuum: vortex tubes, rings, and others Valeriy I. Sbitnev∗ B. P. Konstantinov St. Petersburg Nuclear Physics Institute, NRC Kurchatov Institute, Gatchina, Leningrad district, 188350, Russia; Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, Berkeley, CA 94720, USA (Dated: August 14, 2018) Owing to three conditions (namely: (a) the velocity is represented by sum of irrotational and solenoidal components; (b) the fluid is barotropic; (c) a bath with the fluid undergoes vertical vibrations) the Navier-Stokes equation admits reduction to the modified Hamilton-Jacobi equation. The modification term is the Bohmian (quantum) potential. This reduction opens possibility to define a complex-valued function, named the wave function, which is a solution of the Schr¨odinger equation. The solenoidal component being added to the momentum operator poses itself as a vector potential by analogy with the magnetic vector potential. The vector potential is represented by the solenoidal velocity multiplied by mass of the fluid element. Vortex tubes, rings, and balls along with the wave function guiding these objects are solutions of this equation. Motion of the vortex balls along the Bohmian trajectories gives a model of droplets moving on the fluid surface. The physical vacuum represents a peculiar superfluid medium. It consists of Bose particle-antiparticle pairs having nonzero masses and the angular momenta because of rotating about the center of the masses. Bundles of the vortex lines can transmit a torque from one rotating classical disk to other. Keywords: Navier-Stokes; Schr¨odinger; Faraday waves; wave function; probability density; inter- ference; droplet; vortex tube; vortex ring; superfluid vacuum I. INTRODUCTION the particle from its source up to a detector along a most optimal path. In this case the wave func- tion plays a more active role. Quantum mechanics beginning with its birth up to present time shows triumphal contribu- It is amazing, confirmation of the de Broglie- tion to many technological areas of human activ- Bohm theory came from the area faraway from ities. As for ontological basis, as ironic as it may quantum mechanics. This is an experiment with sound, different interpretations of quantum me- droplets which bounce on a silicon oil surface chanics, contradicting one another in some sub- at moving through an obstacle containing two tle points, are applied for describing one and the slits [8]. The authors have shown that in the far same physical experiment. The interpretations field an interference pattern arises after passing compete each with other for the sake of clearness of many droplets. It was found that the Faraday of ontological understanding this discipline [14]. waves play a role of the guiding wave for the Two well known interpretations of quantum me- droplets bouncing on the silicon oil surface, a arXiv:1403.3900v2 [physics.flu-dyn] 20 Jun 2014 chanics are the Copenhagen interpretation and bath with which is subjected to small vertical the interpretation based on the de Broglie-Bohm vibrations. theory. Both interpretations have in their foun- Where is truth? In order to answer on this dation the concept of the wave-particle dualism. question, it makes sense to begin from funda- The Copenhagen interpretation gives priority to mentals. Let us begin from the most fundamen- the wave function and its collapse is conditioned tal laws of classical physics. They are three New- by detecting a particle. The wave function de- ton’s laws first published in Mathematical Prin- scribes only a probabilistic position of the par- ciples of Natural Philosophy in 1687 [37]. The ticle in the physical space. In turn, De Broglie- Newton’s laws read: (a) the first law postulates Bohm theory says that the wave function guides existence of inertial reference frames: an object that is at rest will stay at rest unless an external force acts upon it; an object that is in motion ∗Electronic address: [email protected] will not change its velocity unless an external 2 force acts upon it. The inertia is a property of curve lies. This vector characterizes a quantita- the bodies to resist to changing their velocity; tive measure of the vortex motion and it is called (b) the second law states: the net force applied vorticity. Vector product [~v ~ω] is perpendicu- to a body with a mass M is equal to the rate lar to the both vectors ~v and× ~ω. It represents of change of its linear momentum in an inertial the Coriolis acceleration of the body under ro- reference frame tating it around the vector ~ω. In turn, the term v d~v ρM [~ ~ω] represents the Coriolis force acting on F~ = M~a = M ; (1) the fluid× element in the volume ∆V . d t The term (5) entering in the Navier-Stokes (c) the third law states: for every action there is equation is responsible for emergence of vortex an equal and opposite reaction. structures. The vortex tubes, rings, and balls Leonard Euler had generalized Newton’s laws are considered briefly in Sec. 2 The latter objects of motion on deformable bodies that are assumed are good models of the droplets moving on the as a continuum [16, 21]. We rewrite the second fluid surface. In Sec. 3 we transform the Navier- Newton’s law for the case of deformed medium. Stokes to the Schr¨odinger-like equation. The Let us imagine that a volume ∆V contains a fluid transformation is achieved due to introduction medium of the mass M. We divide Eq. (1) by ∆V of a material dependent parameter which substi- and determine the time-dependent mass density tutes the Plank constant. Solutions of the equa- ~ ρM = M/∆V . In this case F is understood as tion disclose interference patterns arising from the force per volume. Then the second law in the motion of the vortex balls along optimal this case takes a form: paths, the Bohmian trajectories, through an ob- v v stacle containing slits. The Schr¨odinger equation ~ dρM ~ d~ dρM F = = ρM + ~v . (2) describing flow of a superfluid medium known as d t d t d t the physical vacuum is considered in Sec. 4. Here The total derivatives in the right side can be we deal with vortex lines which arise due to rota- written down through partial derivatives: tion of the virtual electron-positron pairs. The lines can self-organize into twisted vortex bun- dρM ∂ρM = +(~v )ρM , (3) dles in a rotating cylindrical container. Kinetic d t ∂ t ∇ energy of these vortex bundles is sufficient to be- d~v ∂~v = + (~v )~v. (4) gin to rotate a top disk covering the container. d t ∂ t ∇ Concluding remarks are given in Sec. 5. Eq. (3) equated to zero is seen to be the conti- nuity equation. As for Eq. (4) we may rewrite II. THE NAVIER-STOKES EQUATION AND the rightmost term in detail MOTION OF VORTICES v 2 (~v )~v = [~v [ ~v]]. (5) Taking into account the continuity equation ∇ ∇ 2 − × ∇ × let us rewrite Eq. (2) by omitting the rightmost As follows from this formula the first term, mul- term and specify forces which should be in this tiplied by the mass, is gradient of the kinetic equation: energy. It is a force applied to the fluid element ~ ∂~v 2 F for its shifting on the unit of length, δs. The ρM +(~v )~v = P +µ ~v + . (6) ∂ t ∆V second term is acceleration of the fluid element ·∇ ! −∇ ∇ directed perpendicularly to the velocity. Let the The following set of forces is presented in this fluid element move along some curve in 3D space. equation: (i) the first term is the pressure gra- Tangent to the curve in each point refers to ori- dient. It takes place when there is a differ- entation of the body motion. In turn, the vector ence in pressure across a surface; (ii) the sec- ~ω = [ ~v] is orientated perpendicularly to the ond term represents the viscosity of a Newtonian plane,∇ where × an arbitrarily small segment of the fluid (here µ is the dynamic viscosity, its units 3 are N s/m2 = kg/(m s)); (iii) F/~ ∆V is a body 0.3 0.15 force per· unit volume· acting on the fluid. So, we (a) wrote down the Navier-Stokes equation for the 0.2 0.1 m/s viscous incompressible Newtonian fluid [29]. The rad/s term (~v )~v in Eq. (6) can be rewritten in de- 0.1 0.05 , , tail as shown·∇ in Eq. (5). The vector ~ω = [ ~v] ∇ × in Eq. (5), named vorticity, underlies the vortex 0 0 formation. Let us consider the simplest exam- 0 0.5r 1m 1.5 ples. y A. Helmholtz vortices max Let the force F~ be conservative. Then by ap- plying to Eq. (6) the curl we obtain right away the equation for the vorticity: ∂ ~ω +(~ω )~v = ν 2~ω. (7) ∂ t ·∇ ∇ Here ν = µ/ρM is the kinematic viscosity. Its di- mension is [m2/s]. It corresponds to the diffusion coefficient. For that reason the energy stored in min the vortex will dissipate in thermal energy. As a result the vortex, with the lapse of time, will dis- x appear. For supporting the vortex activity, the energy has to be supplied permanently. Trivial FIG. 1: Vortex tube oriented along the axis z: (a) Vorticity solution of Eq. (7) is seen to be ~ω = 0. ω and velocity u of a flow around the center as functions of remoteness from this center. The velocity u vanishes in the We suppose that the fluid is ideal, barotropic, center and tends to zero on infinity. (b) Cross-section of the and the mass forces are conservative [28].