Viscous Fluid Flowing around the Plate: Turbulence

Diffusion of momentum gives rise to viscosity. This article presents a solution in the explicit form of the equation of the momentum diffusion for a viscous fluid flowing around a plate taking into account deceleration. Three characteristic regions of a viscous flow have been described: the mantle, the body of the boundary layer, the viscous sublayer. In the mantle, the effect of viscosity is significant at a considerable distance from the plate. The momentum diffusion is focused in the body of the boundary layer. The diffusion force that produces the momentum of force giving rise to eddies is localized in the viscous sublayer. At the beginning of the plate, a moment of force twists the liquid along the flow, creating eddies that roll along the plate. For this reason, they are pressed against the surface of the plate. But at some distance from the beginning of the plate, the moment of force changes its orientation to the opposite and twists the vortices in the opposite direction, causing the vortices to roll along the plate against the flow. This causes the liquid to detach from the surface of the plate. This is the beginning of turbulence. The diameter of the vortex produced in the viscous sublayer is small being of the order of the thickness of the viscous sublayer. The vortex possesses a large angular velocity. Due to the momentum diffusion and the effect of the eddies combined in passing along the plate, its diameter increases up to the size of the thickness of the boundary layer and even more, whereas its angular velocity decreases down to the values really observed. The value of the critical Reynolds number of the transition from the laminar flow to the turbulent one has been found, and it agrees with the experimental data. The value of the shear stress produced by the viscous fluid on the plate surface has also been obtained. The way of measurement of the friction coefficient characterizing the effect of the plate on the flow has been proposed. It has been shown that the boundary condition of adhesion to the surface of a body flown around, that is applied in the estimation of viscous flows, contradicts the real processes of the flow. How to cite this paper: Ivanchin, A. (2020) Viscous Fluid Flowing around the Plate: Turbulence. Open Journal of Fluid Dynamics, 10, 291-316. https://doi.org/10.4236/ojfd.2020.104018 Received: September 16, 2020 Accepted: November 23, 2020 Published: November 26, 2020 Copyright © 2020 by author(s) 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
At present, the theory of the flow of a viscous fluid has been insufficiently developed. Even the simplest problem of a flow around a plate parallel to the stream cannot be solved using the proposed solutions that are not suitable for the description of the flow near the front edge of the plate. Nor there is an adequate description of the transition of the laminar flow to the turbulent one or the theory of the generation of eddies, etc. The main reason for this situation is the introduction of "viscous stresses" into the above theory, that do not exist in nature [1]. At the same time, it is surprising that when considering a viscous flow, they are absolutely right in saying that the reason for the viscosity is the momentum diffusion. However, instead of studying diffusion, they say that it produces the shear stresses τ .
The Newton law is postulated

Equation of Momentum Diffusion
The right-hand side of (17) does not contain time; it is the solutions of the stationary equation of the momentum diffusion produced by the immobile elementary plate in the moving fluid Thus, the solution of the stationary Equation (18) is derived from the solution of the nonstationary Equation (9).

Interaction of the Flow with the Plate
Until now, the review did not specify the liquid or gas, since this was not necessary. Below we will consider only the flow of gas. This is due to the fact that in a gas, the molecules between collisions move independently of each other. In a liquid, there is no such independence, the molecules are constantly interacting with each other and move in clusters. These differences affect their interaction with the surface of the plate. If gas need to consider the interaction of individual molecules with a surface for fluid interaction of the cluster with it.
When a gas molecule of the mass µ runs into the plate moving at the velocity e v , it acquires a momentum which, on the average, is and directed along the abscissa. The value of α specifies an average fraction of the tangential momentum imparted when the gas molecule collides with the plate surface From the Clapeyron-Mendeleev law it follows that the number of the gas molecules per unit of volume is Hence, the average distance between the molecules is as follows Here k is the Boltzmann constant, T is the absolute temperature, p is the gas pressure. For the air 0 r is by an order of magnitude larger than the interatomic distance in condensed substances.
The molecule free path is [8] Here σ is the square of the section of the molecule scattering. For the air The root-mean square velocity of the molecule thermal movement is The average time during which a molecule covers the length Λ will be Let us consider a parallelepiped adjacent to the plate with the sides dl along the abscissa, Λ along the ordinate and 1 along the z-axis ( Figure 2) and call it an elementary volume. Its lower side, the plate dl , has the square d 1 l × . It contains the molecule number The sixth part of these molecules 6 n Λ move to the plate and over the time t (25) they will reach its surface. As a result of their interaction with the surface atoms of the plate, each gas molecule will get the momentum of the value (19).
The total value of the momentum imparted to the molecules during the time t will be written as Figure 2. Shown is the elementary volume, a parallelepiped with the sides dl along the abscissa x and Λ along the ordinate y adjacent to the plate. The length of the side along the z-axis is 1.
The force acting on the plate of the length l, is written as Since the plate has two surfaces, then the coefficient 2 is added to (34). For the formula (34) to be used correctly, it is necessary that the stress g should be the same across the whole surface of the plate, i.e. the plate length L should be sufficiently small. The value F can be measured experimentally, which makes it possible to estimate the value g, and using the latter, according to (33) one can estimate the friction coefficient α Open Journal of Fluid Dynamics In metal fluids heat conduction is caused by free electrons moving at a high velocity, whereas the momentum diffusion is produced by the molecules moving slowly. As a result, the Prandtl number for metal fluids is much less than unity.

Plate of a Finite Size
Let a plate be located on a segment of the axis of abscissa ( )  The value 1 β  , the value 1 α  and their product (39) 1 γ  . From mechanics, it is known that when two bodies with a different mass collide, the transferred kinetic energy is in inverse proportion to the relation of their masses [9]. When a gas molecule collides with a solid, the above effect takes place.
However, since the solid mass is by orders of magnitude larger than the mass of an air molecule, probably, one should take only the mass of the volume with a diameter of the order of the phonon length rather than the whole mass of the solid. Besides, the part of the kinetic energy of the translation transferred by viscosity is small compared to the kinetic energy of the thermal movement. For instance, for the above values of the parameters the relation of the kinetic energies . The collision occurs tangentially to the solid surface, which further decreases the part of the transferred translation. As a result, when a gas molecule collides with the plate surface, the latter is given a small part of the momentum. Therefore, the value β is so small that The diffusion profile of the zero sector is described by the Equation (18), its solution is (17).
The flow momentum of sector 1 is given by the relation (40), the diffusion equation for it is written as Let us consider that d 0 λ > , the plate is located on the abscissa χ at the point dλ − . The right-hand side (43) describes the source of the diffusion momentum located at that point. The solution of (43) is ( ) Adding (17) and (44) Summing up (46) with respect to n and replacing summation by integration one obtains the diffusion momentum for a plate with a finite length , the sign of minus is taken because the plate is located on the negative section of abscissa on the interval ( ) (47) and (48) (47) and (48) can be neglected in comparison with unity.
Let us introduce the following symbols. If differentiation is performed with respect to the dimensional variables x and y, then grad and div are written with small letters. If differentiation is performed with respect to the dimensionless χ and ζ , then the capital letters are written -Grad, Div.
The diffusion momentum is the vector Note the following feature of the diffusion momentum. With potential flows a change in the velocity results in a change of the pressure, and vice versa. This is not true in the case of the diffusion momentum: it is transferred by diffusion without a change of pressure or density.

Approximation of the Quadrature Formula
Let us consider a case when 2 s c s Since 0 s > , then for 1.7 c > one can approximately assume [10] ( ) erf sign 2 s s s The larger c the higher the accuracy of approximation (52)  To find the domains where (50) and (51) The Equation (54) and Equation (55) have two similar roots The sign of minus is taken from square root for indices 1 and 3 and that of plus for 2 and 4, then with the real roots 1 the Equation (54) has no real roots and (50) is satisfied for all values 0 s ≤ < ∞ .
In this case, In (47) and (67) the substitution is used Similarly, for (55) at There are no real roots ( ) In (47) and (67) the substitution is used If (62) is satisfied, then the same holds true for (58), which means that 0 ϒ = .
Therefore, at a distance of 2 c in front of the plate the diffusion has no effect on the flow.
The interval for (52), where one cannot change erf for sign, is found in the same way. Under the conditions of (50), (51) it is possible to use approximation (61), Equation (67)  , with ϒ further decreasing at a distance from the plate. When a distance of 100 ζ The larger ϒ the slower the flow moves and the weaker its effect on the plate, which, in turn, produces less diffusion momentum. This is clearly seen in Figure   3. At first the deceleration value on the plate surface increases quickly (the curves 1, 2, 3 for the interval ( ) After that the slope of the ϒ curve becomes more flattened, which suggests that the diffusion rate decreases, as it is proportional to the slope. It is the decrease in the slope that indicates stabilization in the ϒ value on the plate surface, since the less diffusion momentum is 1 γ is the characteristic length of the plate, with the plate length 1 λ γ  the profiles of ϒ being similar. Figure 4 shows the ϒ plots at 0.001 γ = for 1000 λ = . These plots are similar to those in Figure 3.

Induced Field
The momentum diffusion field (47)    Substitution of (67) into (49) shows that 0 Γ ≠ , which means that the momentum diffusion produces volume momentum sources without a mass influx.
The total momentum density in dimensionless variables is The continuity equation for T is The potential solution for (69) is The potential  is The symbol * denotes convolution. The potential velocity field is Figure 6 shows the plots of the potential abscissa and ordinate components U for 50 χ = , 100 χ = , 2500 χ = , 5000 χ = . Their value is by an order of magnitude less than that of ϒ (Figure 3). The potential momentum has the largest value near the front edge of the plate.  The general solution is written as follows Here 1 2 , c c are the arbitrary constants. Substituting T into (69) one obtains 1 2 Div Div The ϒ maximum is considerably larger than the maximum of the induced components , U V as can be seen from the plots in Figure 3, Figure 6 and   The radial components of the induced vectors on the circle are written as Here it is designated ( ) Integrating (84) and (85)  The above relation contains one arbitrary constant 1 c , its value can be defined from the condition of the minimum of energy flow. This problem calls for a separate consideration and it is beyond the scope of the present article.

How Eddies Are Generated
The emergence of eddies when a viscous fluid flows around a plate has not been thoroughly studied in theory. That they emerge in a fluid or a gas is explained by the action of the tangential viscous stresses which do not exist in nature. The problem is not trivial. The formation of an eddy requires the momentum of force. However, the question about how viscous stresses produce the momentum of force has not been answered yet. Besides, the simplest experiments reject this hypothesis. If one takes a glass with water and begins to rotate it, one can make sure that to make the water rotate in the glass due to viscosity, one has to rotate it for a long time. However, it should be noted that eddies form without the rotation of a plate or any external flow. In a fluid or a gas the surface forces can have only components normal to the surface, as it happens with pressure, i.e.
there are no shear stresses. This is how they are different from a solid.
If the work of the force on a closed contour is not zero, then it produces the momentum of force. There can be no rotation without it. Therefore, the main problem of the formation of an eddy is the nature of the momentum of force.

Diffusion Force
The total derivative of the diffusion momentum density, i.e. the diffusion force is  Figure 8. The coefficients are negative, so taking into account the minus with the azimuth component, in (100), one obtains the positive azimuth component, and, hence, the moment of force produced by it is positive and directed counter clockwise. The eddy seems to roll along the surface of the plate in the direction of the flow. The above direction of rotation agrees with the observation data on the eddies produced when the plate is flown around by a stream. Besides, the above direction of the moment produces the pressing effect of the flow to the plate. Since the point of the maximum deceleration is located on the plate, the direction of the rotation along the flow as if rolls the eddy in the direction of the movement of the flow. If the rotation is in the opposite direction, it obstructs the movement of the flow along the plate so the eddy "rolls" against the flow breaking it off the surface of plate. In the curve 7 of Figure 8 there is a section near the plate with the positive value of the angle of the slope, that is the place where the negative moment of force starting to rotate the fluid clockwise. From that moment eddies break off the surface of the plate and the laminar flow turns into the turbulent one.
At  For small values of γ the critical dimension of * λ is larger, at present γ is not measured.
The inertia moment of the ring is written as   The inertia moment of the cylinder of the radius r relative to its axis per unit length along the z-coordinate is [9] 4 2 r ρ π (107) Therefore, the increase in the cylinder radius r ten times with the same momentum results in the decrease of the angular velocity 10 4 times, i.e. the angular velocity ω will decrease from 10 5 s −1 down to 10 s −1 . The time necessary for the distribution of the momentum moment of the cylinder with the radius 30 m l onto that with the radius 300 m l is ( ) Attraction of eddies. Theoretically, the rotation of two cylinders in a viscous liquid is discussed [11] [12], but we are not interested in their interaction with a viscous liquid medium, but they are used exclusively to illustrate the joining (summation) of vortices.
In addition to diffusion, the distribution of the momentum can follow the mechanism of "summation". Let there be two rotating cylinders with the radius R and the angular velocity ω (Figure 9). Stability of the circular motion of a fluid. During the rotation of a circular ring the pressure inside it is the same, which holds true for its external part, so the pressure on its external boundary will be the same at all points of the external contour. The difference in the pressures on the internal and external contours gives rise to the centripetal force. The circular motion of a fluid will be stable. If a rotating fluid ring changes into an ellipse whose curvature is not stable, the centripetal force will be unstable either. Where the curvature is larger the difference in the pressure between the internal and the external contours of the ring is larger, and so is the centripetal force. As a result, the pressure inside the elliptical ring will be unstable and it will be smaller with the arcs of a larger curvature and vice versa. It will make the fluid moving inside the ring equilibrate the pressure, which finally leads to a change of the elliptical ring into the circular one. Therefore, the elliptical rotation is unstable and it changes over to the circular one. If due to the action of the volume force, there arises the motion of a fluid rotating along arbitrary trajectories, it evolves into circular eddies.

Summary
In the dynamics of viscous liquids and gases, the concept of viscous stresses is The "sticking" condition is used as a boundary condition in the equation of motion when the plate flows around, that is, the velocity on the surface of the plate is considered to be zero. This is not an experimental fact, it is a postulate accepted without serious justification. The sticking condition for certain types of flow is absurd. For example, the flow of a ceramic surface with a noble gas, argon.
The surface of the argon atoms is not captured, there is no sticking. An incorrect boundary condition leads to solutions that contradict the observed effects. The analysis carried out here uses the kinetic interaction of a gas with a streamlined surface, and the dependence of the surface interaction parameters on the thermodynamic parameters of the flowing gas is found out. This is necessary for creating experimental methods for measuring the parameters of interaction between a gas and a streamlined surface.

Conclusions
Viscosity is the result of the momentum diffusion. Viscous stresses do not exist in nature. In [1], the author considers the momentum diffusion without taking The body of the boundary layer is the region where the effect of the diffusion viscous momentum is significant. The viscous sublayer is the region where the diffusion force producing the moment of force and generating the eddies is essential. The moment of force at the front edge of the plate is directed so that the eddy caused by it would press the flow to the plate surface. Moving along the plate the diffusion moment of force changes its orientation for the opposite one, which leads to the fact that the flow breaks off the plate surface, thus resulting in turbulence.
The eddy produced in the viscous sublayer has a small diameter and a high rotational velocity. Due to the diffusion momentum, the eddy increases in diameter and its angular velocity decreases. The eddy diameter also grows as the eddies combine. As a result, during the time, the flow passes the plate the eddy diameter increases up to the value of the order of the thickness of the boundary layer, with its angular velocity decreasing.
The momentum of the mantle has been obtained in the quadrature form, and it consists of the potential and nonpotential parts.
The consideration of the interaction of the gas with the surface of the plate that is flown around shows that the boundary condition of the adhesion of the flow to the plate surface is rough and it does not allow estimating the diffusion momentum near the front edge of the plate. The interaction of the flow with the plate is characterized by the friction coefficient. It determines the plate lengths for qualitatively similar flows as well as the value of the Reynolds number for the transition from the laminar flow to the turbulent one.
The momentum of the mantle has been obtained in the quadrature form, and it consists of the potential and nonpotential parts.
Here we obtain an analytical solution to the problem of viscous fluid flow around the plate. To understand a physical phenomenon, the analytical solution is more convenient, makes it easier to understand the essence of the phenomenon, and to identify the essential parameters that determine it. It greatly simplifies calculations for engineering applications in the development of structures that use fluid and gas flows, simplifies finding their optimal action, and so on.
Here, the theory of generalized functions [6] [7] was used to obtain an analytical solution. It allows you to get an analytical solution to a wide range of problems in mathematical physics that have real practical significance.

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

Appendix. Curl and Rotation
Since this article is a continuation of [1], one more delusion in theoretical hydrodynamics should be noted. There exists a concept [2] that the value rot 2 v is the angular velocity of the rotation of the elementary volume. However, this is not so.
The concept of rotation implies a body of a finite size that cannot be determined for a point, since the rotation of a point is a meaningless concept. A curl is the differential operator meaningful for a point rather than for an extended object. Therefore, in a general case, it is wrong to say that a curl determines rotation.
It makes sense to determine rotation through a curl for solids whose curl of all points is the same. The linear velocity of an arbitrary point of a rotating solid in the cylindrical coordinates , , r z ϕ with the z-coordinate as the axis of rotation is written as The broken brackets indicate that the vector is written in the cylindrical coordinate system. Hence, one derives