In other words, what phenomenon or what arising unaccounted forces reduce homogeneous fluid into inhomogeneous one is an open problem. In this connection, the goal of our studies is to reveal the mechanism of relation between the variability of the fluid density on the stress function of the electronic filed changing in thickness. Within the linear variability of the density and electronic field stress in thickness , we suggest this dependence in the form fig.
Variability of the density of fluid in the nanotube in thickness. And the stress of the electronic field decreases from the boundary of the contact inside.
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Taking into account the variability character of the density of fluid in low dimensional systems depending on variability of the parietal electronic field of the form 1 , the determining equations of ideal fluid hydromechanics with regard to parietal physical field electronic field effect will be as follows:. Thus, the determining equations of hydromechanics in low dimensional systems, in particular in nanotubes, with regard to parietal physical field electronic field effect phenomenon are reduced to 7 equations with respect to 3 velocity components , initial density , the function of variability of density in thickness, the pressure and the influence function of the physical field stress.
Below, based on the solutions of equations 2 - 6 we will show quality and quantity effect of the influence function of the physical field stress on hydrodynamics of fluid in low dimensional systems. Generalization of the Bernoulli equation in low dimensional systems with regard to parietal physical field effect. In this case, differential equations of motion of ideal fluid with regard to parietal physical field effect 2 have a solution in the form of integrals under the conditions: if the condition is a total differential of ; secondly, the mass forces have the potential , i.
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Under these conditions, we get a so called generalized Bernoulli equation for compressible barotropic fluid with regard to parietal electronic field effect of the form:. This equation holds for all lines of the stream tube passing through the points of the line in thickness situated in the interval. In the case of incompressible fluid the generalized Bernoulli equation 7 takes the form:. In the case of fluid being under the action of gravity force of potential energy , the Bernoulli integral for the parietal stream line with regard to electron field effect takes the form:.
Each summand of the left side of formula 9 has a special name and certain sense: the quantity is called geometrical height and determines the distance of the considered particle of fluid from some horizontal plane accepted as a coordinate plane; the quantity is said to be speed height and expresses the height to which would rise the material point if we throw it in a vacuum vertically upwards with initial speed ; we call the quantity piezometric height with regard to parietal electronic field effect that shows what height should have fluid column of stream line passing through the point , for the pressure at its bottom be equal to.
Thus, equation 9 shows that the sum of three mentioned heights has one and the same value at all the points of the given stream line. The equation shows that the greater the velocity of the fluid pressure in the given stream line tube the less pressure is observed at this point, and vice versa, the less the velocity, the greater value has the pressure. Secondly, equation 10 shows that hydrodynamic pressure is always less than hydrostatic pressure that we get if. Thirdly, the dependence of pressure on velocity when passing from one stream line tube to another one with regard to electronic field stress effect changing in thickness, is shown.
We give formula 10 in the compact form:. Here for. It is seen from 11 that pressure and fluid velocity distribution in the cross section will be in the following form. In the domain it will be in the form:. This means that the pressure on the stream line near the interface is twice less than the quantity of pressure given by the Bernoulli formula fig.
The velocity of fluid flow near the boundary will be greater than in the domain fig.
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Fig 5. The plot of pressure distribution in depth of fluid with regard to effect of parietal electronic field effect. Fig 6. The plot of velocity distribution in depth of fluid with regard to effect of parietal electronic field effect.
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In the paper the mechanism of the parietal physical phenomenon holding between a solid and fluid in low dimensional systems is revealed. The parietal physical field is represented in the form of stress of the electronic field ; the linear dependence between the fluid density function and the function of variability of the electronic field in depth in the form is suggested; the determining equations of ideal fluid hydrodynamics with regard to parietal physical field effect in low dimensional systems are constructed in the form 2 - 6 ; the generalized Bernouilli equation in low dimensional systems with regard to parietal physical field effect was offered in the form 7 ; the essential quality and quantity effect of parietal physical field phenomenon on ideal fluid hydrodynamics in low dimensional systems was shown.
Darrigol O. Bitween hydrodynamics and elasticity theory: the first five births of the Navier-Stokes equations. Archive for History of Exact Sciences. Kalra A. Hammer G. Osmotic water transport through carbon nanotube arrays. Size and temperature effects on viscosity of water inside carbon nanotubes. Nanoscale Research Letters. Holt J. Science, v. Kotsalis E.
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Multiphase water flow inside carbon nanotubes. Internation Journal of Multiphase Flow, 30, p. Abdullayeva S. Ministry of communication and information technologies of the Republic of Azerbaijan. Research Center of Higher Technologies. ISBN , Baku, p. Punyamurtula, Ling Liu, Patricia J. Szoszkiwicz R. B, John A. Thomas, Alan J. Ottoleo Kuter-Arnebeck. Pressure-driven water flow throgh carbon nanotubes: Insights from molecular dinamics simulation. International Journal of Thermal Sciences, 49, p. Technical Aesthetics. The monography offers a detailed analysis of the methods for constructing mathematical models of transient non-isothermal flows of gas mixtures, multicomponent fluids, and gas--liquid fluids through systems of long branched pipelines including annular sections.
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