Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

To unsubscribe from our mailing list, please click on the following link: unsubscribe

I am looking forward to hearing from you.

Kind regards,

David Rain
Email:join@sciepub.com
Science and Education Publishing (http://www.sciepub.com)

On the Non-Smooth Solutions of 3D Navier-Stokes Equations for the Incompressible Fluid Flows

Khatiashvili Nino

I. Vekua Institute of Applied Mathematics, Iv. Javakhishvili Tbilisi State University, Tbilisi, Georgia

Abstract: In the paper non-stationary 3D incompressible viscous fluid flow over the point, the infinite line, the plane, the rectangular prism and the octahedron are studied. The corresponding Navier-Stokes equations (NSE) with the appropriate initial-boundary conditions are considered. NSE is a very important equation and has various applications in Plasma Physics, Astrophysics, magma physics, geophysical fluids, biophysics, nanofluids, etc. NSE describes significant characteristics of different fluids. The exact solutions are obtained in a very few cases and especially in 2D. In the paper the novel exact non-smooth solutions blow-up in time are obtained for the specific pressure and initial conditions by means of the methods of mathematical physics (the main result). Besides, the solutions for the turbulent flows are given. Those solutions are new and are applied to solving of the problem of some substance transportation in the space by means of the turbulent flow. The profiles of the velocity and substance distribution are constructed by means of “Maple” for the different parameters. The results have applications to the description of atmospheric and ocean currents, nanosciences.

Keywords: incompressible fluid flows, Navier-Stokes Equations

Received April 17, 2021; Revised May 22, 2021; Accepted May 30, 2021

Citation: Khatiashvili Nino, On the Non-Smooth Solutions of 3D Navier-Stokes Equations for the Incompressible Fluid Flows. International Journal of Physics, 9 (3). 178-185. January 2021. doi:10.12691/ijp-9-3-5

1. Introduction

Navier-Stokes equations were formulated by famous scientists Claude-Louis Navier and George Gabriel Stokes in the XIX century and became the subject of inspiration for the generations of scientists [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21].

NSE is a very general system of equations and describes a wide-range physical phenomena such as hydrodynamics, atmospheric and ocean currents, plasma physics, air flow around of aircrafts, etc. With the Maxwell equation they form the system of magnetohydrodynamics equations (MHD) [1,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26].

Many substances at the nanoscale level reveal the fluidic properties (Gold, Silver, etc.) and it is very important to study NSE with MHD for those fluids, as they are part of microfluidic devices [4,5,27,28,29]. This problem was investigated in [29,30,31,32,33,34,35,36,37,38].

There exist a lot of modifications of NSE. The first simplification is a Stokes linear system, which was studied in [1,2,3,4,5,6,7,8,9,10,12,13,14,15,16,17,18,19,21,39,40,41,42,43,44,45,46,47,48,49,50,51,52].

In some spatial cases there were obtained the exact solutions of NSE [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,21,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,80,81].

Analytically and numerically NSE was studied in [1,3,4,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,26,39,40,41,42,52,56,57,60,61,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,82,85].

Many open problems are connected with NSE. For example, the existence of smooth solutions in 3D (The Sixth Millennium Problem) [71].

We are sure that various phenomena of nature are non-smooth and the smoothness is a very rare event. As NSE is the reflection of real physical substances, we seek for the non-smooth solutions without any simplifications. This is the main goal of our research.

We consider an unsteady incompressible 3D viscous fluid flow over the point, straight line, plane, rectangle, rectangular prism and octahedron in the 3D space.

We use the methods of mathematical physics and obtain novel 3D solutions of NSE. Those solutions we apply to the investigation of the turbulence-diffusion equation which describes the substance propagation in the space by means of the turbulent flow.

We consider NSE in the non-dimensional variables and all quantities given in the article are dimensionless.

2. Statement of the Problem

Let us consider NSE for the incompressible viscous fluid in some infinite area In the area (t is the time) where is the infinite domain of space with the boundary we consider the following Navier-Stokes system for the incompressible flow with the equation of continuity [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21].

[picture] | (1)

[picture] | (2)

[picture] | (3)

[picture] | (4)

where is the fluid velocity (to be determined), is the body force, is the pressure, is the constant density, is the constant viscosity of the fluid, and are given. The system (1), (2), (3), (4) is studied in the area with the initial-boundary conditions

[picture] | (5)

[picture] | (6)

We admit that the function has continuous derivatives of the first order except for some lines and the body force has some potential i.e. there exists the function for which and the equations (1), (2), (3) can be rewritten in the form

[picture] | (7)

[picture] | (8)

[picture] | (9)

where is the dynamical pressure, is the certain constant.

3. Non-Smooth Particular Solutions of NSE

Let us consider the following functions

1). In the area ( is the octahedron with the surface given by the formula is some given constant)

[picture] | (10)

[picture] | (11)

[picture] | (12)

are some real constants, . In case of , .

2) In the area (is the infinite prism with the surface is some given constant)

[picture] | (13)

[picture] | (14)

[picture] | (15)

are some real constants, . In case of , is the axis .

3) In the area

[picture] | (16)

[picture] | (17)

[picture] | (18)

are some real constants, .

By direct verification we obtain the following theorem:

Theorem.

1) If and the dynamical pressure is given by the formula

then the functions given by the formulas (10), (11), (12) are solutions in of system (1), (2), (3), (4), satisfying the initial condition (5) and the boundary condition (6)

is the boundary of . The functions (10), (11), (12) are the velocity components of the fluid flow in in the area and in case of over the point with the velocity modulus

[picture] | (19)

2) If and the dynamical pressure is given by the formula

then the functions given by the formulas (13), (14), (15) are solutions in of the system (1), (2), (3), (4), satisfying the initial condition (5) and the boundary condition (6)

is the boundary of .

The functions (13), (14), (15) are the velocity components of the fluid flow in in the area or in case of over the axis with the velocity modulus

[picture] | (20)

3) If and the dynamical pressure is given by the formula

then the functions given by the formulas (16), (17), (18) are solutions in of the system (1), (2), (3), (4) satisfying the initial condition (5) and the boundary condition (6)

The functions (16), (17), (18) are the velocity components of the fluid flow in over the planewith the velocity modulus

[picture] | (21)

Below the profiles of the velocities given by the formulas (19), (20), (21) are constructed for the different parameters by means of “Maple” (Figure 1 - Figure 7).

Figure 1. The profile of the velocity given by (19) in case of

Figure 2. The profile of the velocity given by (19) in case of

Figure 3. The profile of the velocity given by (19) in case of

Figure 4. The profile of the velocity given by (20) in case of

Figure 5. The profile of the velocity given by (20) in case of

Figure 6. The profile of the velocity given by (20) in case of

Figure 7. The profile of the velocity given by (21) in case of

Remark 1. The vortex in the flow will be defined by the formula [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]

1) In case of (10), (11), (12)

2) In case of (13), (14), (15)

3) In case of (16), (17), (18)

Remark 2.

If the dynamical pressure is given by the formula

the functions

where are some real constants, will be solutions of the system (7), (8), (9) , and describe the fluid flow in over the point with the velocity modulus

[picture] | (22)

and the initial-boundary conditions

4. The Stationary Case

We now consider the stationary case for the system (7), (8), (9), (4) in the area

[picture] | (23)

[picture] | (24)

[picture] | (25)

[picture] | (26)

If the dynamical pressure is given by the formula

then the set of functions

is the solution of the stationary system (23), (24), (25), (26) in the area with the boundary condition and the vortex

5. The Solution of 2D Helmholtz Equation for the Turbulent Flow

Here we consider the 2D Helmholtz equation for the turbulent flow in [1,2,3,4,5,6,7,9,10,11,12,13,14,15,16,19,20,21,23,52,72,79]

[picture] | (27)

where is the scalar vorticity, is the eddy viscosity,

[picture] | .(28)

If , for the velocity components given by (13), (14), the solution of the equation (27) is

The solutions (13), (14), represent the fluid flow in plane over the rectangle or in case of over the point with the vortex (28).

6. The Solution of the Turbulence-Diffusion Equation

Let us consider the free turbulence-diffusion equation

[picture] | (29)

where is the substrate distribution in , is the molecular diffusion coefficient, is the turbulence diffusion coefficient is the eddy viscosity, which is function of time [10,11,20,21,23,72,79,86].

The equation (29) represents propagation of some substance with the velocity in the space . Here we consider the case, when the eddy viscosity depends on time linearly , where are certain constants. If the velocity of the substance is given by the formulas (13), (14), (15), then the solution of the equation (29) is given by

where are some constants. The initial distribution of the substance in will be given by the formula

Figure 8. The initial distribution of the substance for

Figure 9. The distribution of the substance with the velocity (19)

Figure 10. The distribution of the substance for with the velocity (19)

At the surface the function satisfies the condition Below (Figure 8, Figure 9, Figure 10) the distribution of the substance is shown for the different parameters. The graphs are constructed by means of “Maple”.

Remark 3.

The function

where are some parameters, represents some class of solitary waves and are the solutions of the cubic nonlinear Schrödinger equation [83,84].

Remark 4.

At the planes and the velocity components change the sign which reflects the rotation of the velocity vector .

7. Discussions

From the mathematics viewpoint the system (1), (2), (3), (4) has the following solution

where is a complex number In this case and those solutions vanish in time, but their physical meaning is unclear.

8. Conclusion

Hence, we obtain the new type of exact non-smooth blow-up in time solutions of 3D NSE which describe the incompressible fluid flow in over the point, the infinite line, the plane, the rectangle, rectangular prism and the octahedron. Also we obtain the exact solution of the turbulent-diffusion equation with the specific initial conditions.

We applied our solutions to the turbulence-diffusion equation and described the substance propagation in the space with the given velocity.

The investigation is theoretical and needs the experimental justifications.

In the future we plan to expand our results to the MHD fluids.

Acknowledgements

I am grateful to Mrs. Tsitsino Gabeskiria for the useful remarks.

My thanks goes to the unknown referee for the significant advices.

References

[1]   | Bachelor, G.K, An Introduction to Fluid Dynamics, Cambridge Univ. Press, Cambridge, 2000.
[2]   | Chorin, A.J., Marsden, J.E., A Mathematical Introduction to Fluid Mechanics, 2-nd ed., Springer, NY, 1984.
[3]   | Constantin, P., Foias, C., Navier-Stokes equations, Chicago Lectures in Mathematics. University of Chicago Press, 1988.
[4]   | Kim, S., Karrila, S., Microhydrodynamics: Principles and Selected Applications, Dover, 2005.
[5]   | Kirby, B.J., Micro and Nanoscale Fluid Mechanics in Microfluidic Devices, Cambridge University Press, 2010.
[6]   | Ladyzhenskaya, O. A., The Mathematical Theory of Viscous Incompressible Flow, Mathematics and its Applications. 2nd Ed., Gordon and Breach, 1969.
[7]   | Lamb Horace, Hydrodynamics (Sixth ed.)., Dover, NY, 1945.
[8]   | Landau, L. D., “New exact solution of the Navier-Stokes equations”, Doklady Akademii Nauk SSSR , 44. 311-314. 1944.
[9]   | Landay, L.D., Lifshitz, E.M., Fluid Mechanics, Course of Theoretical Physics. 6, Pergamon Press, 1987.
[10]   | Lautrup, B., Physics of Continuous Matter, Second Edition: Exotic and Everyday Phenomena in the Macroscopic World, Second Ed., CRC Press, 2011.
[11]   | Lavrentiev, M.A., Shabat, B.V., Problems of hydrodynamics and their mathematical Models (Russian), Nauka, Moskow, 1977.
[12]   | Lions, P.L., Mathematical Topics in Fluid Mechanics, The Clarendon Press, Vols. 1,3 Oxford University Press, NY, 1996.
[13]   | Majda, A.J., Bertozzi, A.L., Vorticity and Incompressible Flow, Vol.27, Cambridge University Press, 2002.
[14]   | Milne-Thompson L.M., Theoretical Hydrodynamics, 5-th ed, Macmillan, 1968.
[15]   | Ockendon, H and Ockendon J., Viscous Flow, Cambridge University Press, 1995.
[16]   | Prandtl, L., Essentials of Fluid Mechanics, third ed.in: Applied Mathematical Sciences, Vol. 158, Springer, NY, 2010.
[17]   | Rosenhead, L., (ed.), Laminar boundary layers, Clarendon Press, 1963.
[18]   | Stokes, G. G., “On the steady motion of incompressible fluids”. Transactions of the Cambridge Philosophical Society 7:, Mathematical and Physical Papers, Cambridge University Press, 1880.
[19]   | Temam, R., Navier-Stokes Equations, Theory and numerical Analysis, AMS Chelsea, 2001.
[20]   | Tychonov, A.N., Samarski, A.A., Partial Differential Equations of Mathematical Physics, vol. I. Translated by S. Radding, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1964.
[21]   | Whitham, G. B., Linear and Nonlinear Waves. Reprint of the 1974 original. Pure and Applied Mathematics , John Wiley & Sons, Inc., New York, 1999.
[22]   | Cermak, J.E., Cochran, L.S., “Physical modeling of the atmospheric surface layer”, J. Wind Eng. Ind. Aerodyn., 41. 935-946. 1992.
[23]   | Chacon-Rebollo, T., Lewandowski R., Mathematical and numerical foundations of turbulence models and applications, in: Modeling and Simulation in Science, Engineering and Technology, Springer, New-York, 2014.
[24]   | Dedner, A., Kröner, D., Rohde, C., Wesenberg, M., Radiation magnetohydrodynamics: analysis for model problems and efficient 3d-simulations for the full system. Analysis and numerics for conservation laws, Springer, Berlin, 2005.
[25]   | Li, J., Recent advances concerning certain class of geophysical flows, in Handbook of Mathematical Analyses in Mechanics of viscous fluids , Springer, 2018, 933-71.
[26]   | Lions, J.L., Temam, R., Wang, S., “New formulations of the primitive equations of atmosphere and applications”, Nonlinearity, 5. 237-88. 1992.
[27]   | Brushnan, B., (Ed.), Springer Handbook of Nanotechnology. Springer, 2010.
[28]   | Nabok, A., Organic and Inorganic Nanostructures, Artech House, Boston/London, 2005.
[29]   | Sarit, K.Das, Stephen, U. Choi, Wenhua Yu., Pradeep, T. Nanoﬂuids Science and Technology, Wiley-Interscience, NY, 2007.
[30]   | Khan, J.A., Mustafa, M., Hayet, A., Alsaedi, A., “Three-dimensional ﬂow of nanoﬂuid over a non-linearly stretching sheet: An application to solar energy”, Int. J. Heat Mass Transf., 86. 158-164. 2015.
[31]   | Rasool, G., Shafiq, A., Khalique, M., Zhang, T., “Magnetohydrodynamic Darcy-Forchheimer nanofluid flow over a nonlinear stretching sheet”, Physica Scripta, 94 (10). 105221. 2019.
[32]   | Rasool, G., Shafiq, A., Khan, I., Baleany, D., Nisar, K., Shahzadi, G., “Entropy Generation and Consequences of MHD in Darcy-Forchheimer Nanofluid Flow Bounded by Non-Linearly Stretching Surface”, Symmetry, 12(4), 652. 2020.
[33]   | Rasool, G., Shafiq, A., “Thermally Enhanced Chemically Reactive Radiative Powell-Eyring Nanofluid Flow via Darcy Medium over Non-linearly Stretching Surface Affected by a Transverse Magnetic Field and Convective Boundary Conditions”, Alexandria Engineering Journal, 59 (4). 2533-2541. 2020.
[34]   | Sadiq, M.A., “MHD Stagnation Point Flow of Nanoﬂuid on a Plate with Anisotropic Slip”, Symmetry, 11(2).132. 2019.
[35]   | Shafiq, A., Sindhu, N., Khalique, M., “Numerical investigation and Sensitivity analysis on bioconvective tangent hyperbolic nanofluid flow towards stretching surface by response surface methodology”, Alexandria Engineering Journal, 59 (6). 4533-4548. 2020.
[36]   | Shafiq, A., Hammouch, Z., Sindhu T., Baleanu, D., “Statistical approach of mixed convective flow of third-grade fluid towards an exponentially stretching surface with convective boundary condition”, In Special Functions and Analysis of Differential Equations, Chapman and Hall/CRC, 307-319. 2020.
[37]   | Shafiq, A., Sindhu T., Al-Mdallal, Q., “A sensitivity study on carbon nanotubes significance in Darcy-Forchheimer flow towards a rotating disk by response surface methodology”, Scientific Reports, 11(1). 1-26, 2021.
[38]   | Shah, Z., Khan, A.S., Dawar, A., Nie, Y., Shah, Z., Khan, W., Islam, S., “Three-Dimensional Nanoﬂuid Flow with Heat and Mass Transfer Analysis over a Linear Stretching Surface with Convective Boundary Conditions”, Appl. Sci., 8(11). 22-44. 2018.
[39]   | Chen, Jeng-Tzong, Hsiao, Chia-Chun, Leu, Shyue-Yuh, “A new method for Stokes problems with circular boundaries using degenerate kernel and Fourier series”, Internat. J. Numer. Methods Engrg. 74 (13). 1955-1987. 2008.
[40]   | Chwang, A., and Wu, T., “Hydromechanics of low-Reynolds-number flow. Singularity method for Stokes flows”, J. Fluid Mech., 62. 65-70. 1974.
[41]   | Giga, Y., “Bounded H-calculus for the hydrostatic Stokes operator on L_p spaces and applications”, Proc.Am.Math. Soc., 145. 3865-3876. 2017.
[42]   | Girault, V., Riviere, B., Wheeler, M., “A discontinuous Galerkin Method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems”, Math. Comp., 74. 53-84. 2005.
[43]   | Khatiashvili, N., Pirumova, K., Janjgava, D., “On the Stokes flow over ellipsoidal type bodies”, Lecture Notes in Engineering and Computer Science, Proceedings of The World Congress on Engineering, 1.IAENG, London,148-151. 2013.
[44]   | Khatiashvili, N., K. Pirumova, K., Janjgava, D., „On some Effective Solutions of Stokes Axisymmetric Equation for a Viscous Fluid”, Proceedings of World Academy of Science, Engineering and Technology, 79, London, 690-694. .2013.
[45]   | Khatiashvili, N., Pirumova, K., Khatiashvili, I., Akhobadze, V., “On the Influence of the Cancer Proteins on the Blood Flow”, Rep. Enlarged Sess. Sem. I. Vekua Inst. Appl. Math., 29. 60-63.2015.
[46]   | Khatiashvili, N., “On the Stokes Flow in a Pipes”, J. Way Science, 355-358. 2020.
[47]   | Khatiashvili, N., “On 2D Free Boundary Problem for the Stokes Flow”, Proceedings of I.Vekua Institute of Applied Mathematics, 70. 61-71.2020.
[48]   | Mitrea, M., Monniaux, S., “The regularity of the Stokes operator and the Fujita-Cato Approach to the Navier-Stokes initial value problem in Lipschitz domains”, J. Funct. Anal. 254. 1522-1574. 2008.
[49]   | Ogata, H., Amano, K., “Fundamental solution method for two-dimensional Stokes flow problems with one-dimensional periodicity”, Jpn. J. Ind. Appl. Math., 27 (2). 191-215. 2010.
[50]   | Papkovich, P.F., “Solution Generale des equations differentials fondamentales delasticite exprimee par trois fonctions harmoniques”, Compt. Rend. Acad. Sci. Paris , 195,513-515. 1932.
[51]   | Zabarankin, M., Ulitko, A. F., “Hilbert formulas for $r$-analytic functions and the Stokes flow about a biconvex lens”, Quart. Appl. Math. , 64 (4). 663-693.2006.
[52]   | Drazin, P.G., Norman, R., The Navier-Stokes equations: a classification of flows and exact solutions, 334. Cambridge University Press, 2006.
[53]   | Ethier, C. R., Steinman, D. A., “Exact fully 3D Navier-Stokes solutions for benchmarking”, International Journal for Numerical Methods in Fluids, 19 (5). 369-375. 1994.
[54]   | Jeffery, G. B., “The two-dimensional steady motion of a viscous fluid”, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 29 (172). 455-465. 1915.
[55]   | Fraenkel, L. E., “Laminar flow in symmetrical channels with slightly curved walls, I. On the Jeffery-Hamel solutions for flow between plane walls”, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 267 (1328). 119-138. 1962.
[56]   | Glushak, A. V., Malaj, N. V., Mironova, N. N., “Solution of the Boundary Value Problem for the Navier-Stokes Equation for the flow of a Gaseous Medium Past a Heated Spheroid”, Differ. Eq., 48 (6). 886-890. 2012.
[57]   | Karman, T., “Über laminare und turbulente Reibung”, Zeitschrift für Angewandte Mathematik und Mechanik, 1 (4). 233-252. 1921.
[58]   | Kerr, O. S., Dold, J. W., “Periodic steady vortices in a stagnation-point flow”, Journal of Fluid Mechanics, 276. 307-325. 1994.
[59]   | Rosenhead, L., “The steady two-dimensional radial flow of viscous fluid between two inclined plane walls,” Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, The Royal Society, 963. 175. 1940.
[60]   | Squire, H. B. , “The round laminar jet”, The Quarterly J. of Mechanics and Applied Mathematics, 4(3). 321-329. 1951.
[61]   | Siddique, I., Iftikhar, S., ‘Unsteady flow of a viscous fluid between two oscillating cylinders’, Appl. Math. Inf. Sci. 6 (3). 483-489. 2012.
[62]   | Yatseyev, V. I., “About a class of exact solutions of viscous fluid equations of motion”, Zhurnal Tekhnicheskoj Fiziki, 20 (11). 1031-1034. 1950.
[63]   | Boffi D., Brezzi F., Fortin M., Mixed Finite Element Methods and Applications, Springer Series in Computatinal Mathematics, Vol. 44, Springer, 2013.
[64]   | Bradshaw, Z., Grujić, Z., “Blow-up scenarios for the 3D Navier-Stokes equations exhibiting sub-criticality with respect to the scaling of one-dimensional local sparseness”, J. Math. Fluid Mech., 16 (2). 321-334. 2014.
[65]   | Burman E. and Fernandez M.A., “Continuous interior penalty finite element method for the time-dependent Navier-Stokes equations: Space discretization and convergence”, Numer. Math., 107. 39-77. 2007.
[66]   | Chen, C., Strain, R.M., Tsai, T., Yau, H.T., “Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations”, Comm. Partial Differ. Eq., 34 (1-3). 203-232. 2009.
[67]   | Girault, V., Raviart, P.A., Fnite element methods for Navier-Stokes equations:Theory and Algorithms, Vol. 5, Springer-Science & Business Media, 1986.
[68]   | Grujic, Z., “A geometric measure-type regularity criterion for solutions to the 3D Navier-Stokes equations”, Nonlinearity, 26. 289-296. 2013.
[69]   | Holdeman, J. T., “A Hermite finite element method for incompressible fluid flow”, Int.J. Numer. Meth. Fluids, 64 (4). 376-408. 2010.
[70]   | Koch, G., Nadirashvili, N., Seregin, G.A., Šverák, V., “Liouville theorems for the Navier-Stokes equations and applications”, Acta Math., 203 (1). 83-105. 2009.
[71]   | Ladyzhenskaya, O. A. “The Sixth Millennium Problem: Navier-Stokes Equations, Existence and Smoothness”, Russ. Math. Surv., 58 (2). 251-286. 2003.
[72]   | Lewandowski, R., “Navier-Stokes equations in the whole space with an eddy viscosity”, J. Math. Anal. Appl., 478 (2). 698-742. 2019.
[73]   | Liu, Q., Hou, Y., Liu, Q., “A two-level method in time and space for solving the Navier-Stokes equations based on Newton iteration”, Comput. Math. Appl., 64 (12). 3569-3579.2012.
[74]   | Masmoudi, N., Rousset, R., “Uniform regularity for the Navier-Stokes equation with Navier boundary condition”, Arch. Ration. Mech. Anal. , 203 (2). 529-575.2012.
[75]   | Schroeder, P.W., Lehrenfeld, C., Linke, A., Lube, G., “Towards computable flows and robust estimates for inf-sup stable FEM applied to the time-dependent incompressible Navier-Stokes equations”, SEMA J., 75. 629-653.2018.
[76]   | Seregin, G., Šverák, V., “On type I singularities of the local axi-symmetric solutions of the Navier-Stokes equations”, Commun. Partial Differ. Eq. 34 (1-3). 171-201. 2009.
[77]   | Tani, A., “Small- time existence for the three dimensional Navier-Stokes equations for an incompressible fluid with a free surface”, Arch. Ration. Mech. Anal. 133 (4). 299-331.1996.
[78]   | Tao, T., “Vanishing vertical viscosity limit of anisotropic Navier-Stokes equation with no-slip boundary condition”, J. Differential Equations 265 (9). 4283-4310. 2018.
[79]   | Taylor, G.I. and Green, A.E., “Mechanism of the Production of Small Eddies from Large Ones”, Proc. R. Soc. Lond. A, 158. 499-521. 1937.
[80]   | Waleed S. K., “Classical Fundamental Unique Solution for the Incompressible Navier-Stokes Equation in RN”, J. Appl. Math. and Physics, 5(4). 30-40. 2017.
[81]   | Wang, C. Y., “Exact solutions of the steady-state Navier-Stokes equations”, Annual Review of Fluid Mechanics, 23. 159-177. 1991.
[82]   | Yang, Z., Yong, W., “Asymptotic analysis of the lattice Boltzmann method for generalized Newtonian fluid flows”, Multiscale Model. Simul. 12 (3). 1028-1045.2014
[83]   | Khatiashvili, N., Shanidze, R., Janjgava, D., “On the effective solutions of the non-linear Schrödinger equation”, Physics and Mathematics of the nonlinear Phenomena, IOP Journal of Physics, 482. 012020. 2014.
[84]   | Khatiashvili N., “On one Class of Elliptic Equations Connected with the Nonlinear Waves”, Transactions of A. Razmadze Math. Inst,, 173. 61-70.2019.
[85]   | Caffarelli, L., Kohn, R., Nirenberg, L., “Partial regularity of suitable weak solutions of the Navier-Stokes equations”, Commun. Pure Appl. Math., 35 (6). 771-780. 1982.
[86]   | Khatiashvili, N., Komurjishvili, O., Kutchava, Z., “On the Numerical Treatment of the Axi-symmetric Turbulent-Diffusion Equation”, Appl. Math. Inform. Mech.,18 (2). 55-61. 2013.