Delusions in Theoretical Hydrodynamics

Theoretical hydrodynamics may lead one into serious delusions. This article is focused on three of them. First, using flowing around a sphere as an example it is shown that the known potential solutions of the flow-around problems are not unique and there exist nonpotential solutions. A nonpotential solution has been obtained for flowing around a sphere. A general solution of the problem of flowing around an arbitrary surface has been obtained in the quadrature form. To single out a physically realisable solution among a great number of others, it is necessary to add supplementary conditions to the known boundary ones, in particular, to find a solution with the minimum total energy. The hypothesis explaining the reason for stalled flows by viscosity is erroneous. When considering a flow-around problem one should use stalled and broken solutions of the continuity equation along with the continuous ones. If the minimum total energy is achieved by the continuous solution, it is a continuous flow that will be implemented. If it is achieved by the broken solution, a stalled flow will be realised. Second, the hydrodynamics of a flow is considered exclusively at each point of it. Differential equations are used to describe the flows that are written for a randomly small volume of a flow, i.e., for a point. The integral characteristics of a flow and its inertial properties are neglected in the consideration, which results in the misunderstanding of the mechanism of the formation of a vortex. The reason for the formation of vortices is related to viscosity, which is a mistake. The formation of vortices is the result of the inhomogeneity of the acceleration field and the inertial properties of a flow. Third, the fictitious values of viscous stresses are used in hydrodynamics. As a matter of fact, viscosity is the momentum diffusion and it should be described by the diffusion equation included into the Euler system of equations for a viscous fluid. The momentum diffusion leads to the necessity of including the volume momentum sources produced by diffusion into the continuity equation and excluding the viscosity forces from the equation of motion. The problem of a viscous fluid flowing around a thin plate has been solved analytically, the velocity profiles satisfying the experiment have been obtained. The superfluidity of helium is not its property. It is How to cite this paper: Ivanchin, A. (2018) Delusions in Theoretical Hydrodynamics. World Journal of Mechanics, 8, 387-415. https://doi.org/10.4236/wjm.2018.89029 Received: August 27, 2018 Accepted: September 26, 2018 Published: September 29, 2018 Copyright © 2018 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access


Introduction
In theoretical hydrodynamics there exist serious delusions, both mathematical and physical ones, which lead to erroneous conclusions and misunderstanding in the physics of the flow.
The Euler system of the differential equations of the mechanics of fluid consists of the continuity equation, the momentum-conservation equation or the so-called equation of motion, the energy equation and that of thermodynamic relations [1]. At present the continuity equation is the starting one for steady-state flows. When solving it one derives the field of velocities to define the pressure from the equation of motion.
It is considered that if the potential solution of the continuity equation is found, which is thought to be unique without proof, then the problem has been solved. This is a delusion. In addition to the potential solution, the continuity equation has some nonpotential solutions, which leads to a revision of the knowledge about the physics of the flow of fluids. To obtain a physically realizable solution, it is necessary to add supplementary conditions, e.g. the minimum energy condition, to the boundary and initial ones that are used at present. Other variants are also possible.
The second delusion is that only differential characteristics of a flow are taken into account, the conservation laws are written for a point, whereas the integral characteristics, such as the moment of inertia, are ignored. As a result, the mechanism of the formation of vortices is ignored in the consideration.
The third delusion is the mechanism of viscosity. It is believed that under a viscous flow there appear shear stresses. This theory, in spite of its rather long

Flowing around a Sphere
The convolution of two functions [2] ( ) 1 , , f x y z and ( ) 2 , , f x y z is denoted by * and determined in this way Continuity equation. The continuity equation for a steady-state flow is written as [1] ( ) Here I is the vector of the momentum density at a point in the flow, V is the velocity at this point, ρ is the fluid density, ( ) , , x y z is the differential gradient operator. The vector is denoted by the bold type, its modulus by the usual one.
If the medium is noncompressible, const ρ = , then the continuity Equation (3) is written as [1] ( ) The Equations ( (3) and (6)), in fact, coincide, with only the right side differing by the presence of the constant coefficient, which for linear equations is not essential.
For a compressible gas it is necessary to use thermodynamic relations. The dependence between the density ρ and the pressure p for the adiabatic process is as follows [3] 0 0 The enthalpy for the adiabatic process is written as The Bernoulli potential is Having derived the momentum by (3) it is possible to define the velocity field from (9).

Equation of motion. The equation of motion of a fluid when no volume
forces or viscosity exist is the following [1] Summation is made with respect to the recurrent indices from 1 to 3.
Otherwise (10) is written as Here the symbol ∇ × stands for the differential rotor operation, is the angular velocity. The value of is the density of the Coriolis force. It is worth noting that the Coriolis force is a force rather than the moment of force, it cannot induce the rotation of a fluid.
In the general case, according to (11), the velocity is a nonpotential vector. The pressure is a potential, since the force produced by it is the gradient of the pressure.
For an incompressible fluid (11) is written as For the potential field V = ∇Ψ , where Ψ is the velocity potential, then The lower index 0 denotes the parameters whose value is known at some point of the flow. It should be noted that (16) is true only for the potential fields of the momentum and an incompressible fluid.
There are two ways of mathematical description of the motion of a fluid. In the Langrange method an elementary particle of the mass is taken and the equation of motion is written for it. Its coordinates are varied in time.
In the Euler method, it is not a material particle that is taken but a fixed volume with a fluid flowing through its surface, and the balance of the forces is considered in this volume. Using the Langrange method it is very simple to show the difference between the potential and nonpotential flow. For the arbitrary element of the mass, according to (12), the angular velocity (15) of its motion in the potential flow is zero, it does not rotate and keeps its orientation even when following a curved trajectory as shown in Figure 1. In the nonpotential flow the angular velocity (12) is not zero and the element of the mass changes its orientation during motion as shown in Figure 2.

A Point Source
The point source is assigned as Here µ is the power of the momentum source. The Equation (3) is written as ( ) As shown in [4] this equation has two solutions. The first is the well-known potential one In the present article the Cartesian components are written in the braces. The broken brackets denote that the vector is written in the spherical coordinates with the following sequence of the components: the radial r, the zenith ϑ , the azimuth ϕ . Substituting ∇Φ from (19) into (18) one derives the Poisson differential equation for the potential Φ : The solution (20) called a potential is ( )  1 r at r → ∞ , its first derivative as 3 1 r and so on. The derivative δ-function of order n is a multipole of order n. Inclusion into the consideration of the nth derivatives in (28) means the addition of the multipoles of the nth order to the solution (25).
The potential source dipole. The homogeneous flow with a constant momentum density oriented parallel to the z-axis and directed from +∞ to −∞ is written as The field of the dipole source with the z-axis as the axis of symmetry is obtained by differentiating (19) with respect to z and the replacement of µ by the dipole moment µ  { } w I 1 cos , 1 sin , 0 2 The neutron vector dipole. The neutron vector dipole (24) with the z-axis as the axis of symmetry is written in the spherical coordinates as The field (39) decreases at infinity as 2 1 r due to the neutron component (24), whereas the potential field (31) decreases as 3 1 r . The energetics of the flow around a sphere. The value of ( ) is the inherent field of a streamlined sphere. The kinetic energy of the field momentum (40) K is Integration is performed with respect to the external part of the surface of the sphere. The values of 1 K and 2 K are the self-energies of the potential and the neutron components, respectively. The value of µ  is arbitrary.
is the interaction energy of the potential and the neutron components.
The potential energy should be added to the kinetic energy (41) to get the total energy. The pressure p acts as the potential energy density for an incompressible fluid, whereas enthalpy for a compressible one. The pressure p can be derived from the equation of motion (14), substituting there the velocity (40) one obtains a complex differential equation. There is no point in solving it, since it is impossible to produce continuous flowing around a sphere.

Solution of the Problem of Flowing through the Sources on the Surface
Let us illustrate the solution of the problem of flowing using the example of flowing around a sphere with the radius R with the centre at the origin of the The modulus of the normal of the external flow to its surface, according to . It means that the surface of the sphere S is the carrier of the delta-function of the simple layer [2], and in (3) the function of the source in the spherical coordinates is In the spherical coordinates only the radial component has the source on the sphere, the direction of the normal to the surface coincides with the coordinate line, therefore, the source (47) is a scalar. In the general case of an arbitrary surface it will be a vector. The flow of the vector n w through the area element dS will be cos d w S ϑ , its contribution to the potential at the point r will be The lower index 1 means that the given value refers to the sphere S, the radial component of the area element is The problem is axially symmetric relative to the z-axis, therefore, it is possible to assume 0 ϕ = without a loss of generality. The velocity potential is Here it is designated ν is an arbitrary parameter. The expansion in terms of the Legendre polynomials is used The Legendre polynomial is expressed in this way [5]: and (58) will be written as The function (61) is the solution of the Equation (3)

Inertial Effects
Vortices in a flow occur under stalled flow when a low-density zone is formed behind a streamlined body, which is clearly seen in flowing around an orthogonal plate ( Figure 3).
The contribution of viscosity into the flow energetics is small because viscosity is essential only at the sides of the plate A and B with a small length. This contribution is much less than the kinetic energy of the vortices formed behind a streamlined plate. The vortex diameter is comparable with its length rather than with its thickness which is much less than the vortex diameter. Therefore, it is  The principal question is which moments of force produce vortices. Pressure cannot produce them in principle, since its force is the pressure gradient, which is a potential value and its work along a closed contour is always zero. If there are no external forces, only the forces produced by the pressure act in a fluid.
Let us consider a fluid ring in a flow with the radius r and the thickness dr.
The element of the ring mass in the cylindrical coordinates is Its moment of inertia is The origin of the coordinates is in the centre of the domain mass.
The moment of the force acting on the ring with the inner radius r and the external radius

Momentum Diffusion
At present it is considered that in the flow of a gas or a fluid there exist tangent or the so-called viscous stresses, their value, according to the Newton law is Here ν is the viscosity coefficient, ij p is the stress tensor and the nondiagonal components in it are the viscous stresses [1]. The equation of motion (10) has the form The physical reasons for the pressure in a gas and a condensed media are principally different. In the case of a gas, the pressure on the surface is produced only due to the collision of the molecules moving chaotically with the surface. So the source of the pressure in a gas is the kinetic energy of the molecules. The potential energy of the intermolecular interaction does not make contribution to the pressure, since the average distance between the molecules is much larger than the radius of the action of the interatomic forces. This property of a gas is the reason for the implementation of the thermodynamic relations, in particular, the Clapeyron-Mendelyeev law. For the tangential stresses to exist, it is necessary that the molecules be located near the equilibrium position and if they move there would be a force making them return to the previous position.
In condensed media the situation is different. The molecules are at distances at which the potential energy of the interatomic interaction is great, it has to be overcome in compression or tension and due to it the pressure is produced.
Here η is the coefficient of the momentum diffusion.  The diffusion momentum J is derived from the diffusion equation If a fluid moves along the abscissa at the velocity w − , then the derivative with respect to time in (78) should be considered total and then (78) is written as If a fluid is immobile 0 w = , then (79) is written as Let us introduce the scales of the distance L, the time m t , the momentum The value J 0 . The average distance between the molecules λ from the Clapeyron-Mendelyeev law is written as is the gas constant, k is the Boltzmann constant, A N is the Avogadro number, T is the absolute temperature, p is the gas pressure. For the air λ is by an order of magnitude larger than the interatomic distance in condensed substances.
The average velocity of the progressive motion of a molecule along the normal to the strip is Here µ is the molecule mass. This relation is valid for monoatomic gases. which a molecule will cover the length λ will be equal to The average length of the free path of the molecules in a gas is much larger than λ , it is possible to neglect the colliding ones and consider that half the molecules located at the distance λ from the strip reach its surface since the other half of them move back from it.
If 0 N t t < < , then in the sum in (99) summing should be interrupted when N n t t − > and the upper limit in the integral will be t rather than 0. Thus, (100) is the solution of the problem when Note that as a result of the summation, the momentary character in time of the point source disappears, it becomes constant in time during its movement due to the summation. However, the momentum character of the spatial coordinates remains but changes to the form ( ) ( ) x wt y Here it is designated     In front of the plate there also exists a diffusion momentum shown in Figure   9. However, it disappears quickly with increasing distance from the plate up the flow. The Here P p l = ∆ is the difference of pressure per unit of the tube length.
For the stationary flow of a viscous fluid through a round tube of the radius R due to the viscosity there is a radial dependence of the velocity ( ) u u r = that is smaller near the tube wall than on the axis. As the boundary condition let us take the adherence condition The equation of the momentum diffusion for an incompressible fluid in the cylindrical coordinates is written as The velocity u does not depend on z, then The relations (116) and (124) coincide, which means that in the experiments on the measurement of the viscosity coefficient it is the coefficient of the momentum diffusion which is determined rather than that of viscosity.
Coefficients of the momentum diffusion and thermal conductivity. The relation of the viscosity coefficient to that of thermal conductivity is called the Prandtl number [1]. The mechanism of the momentum diffusion and the heat transfer is ensured in the same way-by the chaotic motion of the gas molecules and their collision. On the average for a monoatomic gas a kinetic energy of kT/2 falls at one degree of freedom and 3kT/2 at three degrees of freedom, with the energy, i.e. the heat transferred by all three degrees of freedom. Unlike the heat transfer, the momentum has a definite direction, which is that of the flow of a fluid. The chaotic motion in this direction does not affect the transfer of the momentum that is transferred in the deterministic process. So only two directions remain for the diffusion. Therefore, the Prandtl number for monoatomic gases should be 2/3, which is really observed [7].
Superfluidity of helium. Fluid helium at a temperature below 2.6 K possesses superfluidity [8], i.e. the ability to flow at a large velocity through capillars, which means that the viscosity coefficient is either by orders of magnitude less than the known values or zero.
The diffusion mechanism of viscosity is responsible for superfluidity. At low temperatures the thermal energy of 3kT/2 falling at a helium atom is small and by the laws of quantum mechanics the energy transfer is possible only by quanta.
If the energy transferred is less than a quantum, then the momentum cannot be transferred from a streamlined body to helium atoms, which rules out its diffusion, which in turn means that the coefficient of cohesion in (95) 0 α = .
Superfluidity is not the property of helium nor a phase transition leading to the disappearance of viscous stresses that do not exist in nature. It is the feature of the interaction of helium atoms with the atomic lattice of the surface at low temperatures.
One In hydrodynamics d'Alembert's paradox is well-known: from the potential solution of the problem of the flow around a sphere by an incompressible ideal fluid it follows that the resistance of the flow is zero. However, helium does not manifest superfluidity in flowing around and the hydrodynamic resistance does exist. Superfluidity does not rule out the formation of stalled flows of helium, which produces resistance, according to the laws of hydrodynamics.
To explain superfluidity, a hypothesis [8] was suggested according to which at a temperature of 2.6 K in fluid helium there occurs a phase transition separating into two phases: a superfluid one with zero viscosity and a usual one, viscous. As it follows from the present article this is a mistake because there is no need to involve phase transitions to consider superfluidity.

Discussion
In theoretical hydrodynamics there exist serious delusions preventing one from theoretical understanding of flows. 2) The neglect of the integral characteristics of the flow and the attention fixed only on the differential ones. As a result, the inertial properties of the flow are ignored in the consideration, and for this reason the mechanism of the formation of vortices by stalled flows is not clear. The formation of vortices is erroneously explained by the nonexisting viscous stresses. As a result, there is no mathematical theory of stalled flows even for the simplest problems, such as flowing around an orthogonal plate or a sphere. The consequence is also the absence of a consistent theory of turbulent flows.
3) The use of the viscous stresses that are fictious forces. They enter the equation of motion but do not exist in reality.
The momentum diffusion leads to the necessity of including into the continuity Then the fact that the Prandtl number for monoatomic gases is 2/3 finds the explanation.
The mathematical description of the flow of a viscous medium should include • the equation of the momentum diffusion; • the continuity equation that includes the diffusion momentum as the volume momentum source; • the equation of motion without viscous stresses; • the energy equation.
One should also take into account the integral characteristics of the flow and find the regions forming vortices. One should take into account nonuniqueness of the solution of a hydrodynamic problem and the existence of nonpotential solutions. It is necessary to include some additional conditions along with the initial and boundary ones to find a physically realisable solution, e.g. the condition of the minimum total energy of a system.

Conclusion
The analysis made here shows that at present in hydromechanics there exist serious gaps and erroneous notions both in the physical and mathematical aspects. These drawbacks block the application of theoretical hydrodynamics for the solution of practical problems and understanding of the processes related to

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.