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.

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}.

(1)

(2)

(3)

(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

(5)

(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

(7)

(8)

(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)

(10)

(11)

(12)

are some real constants, . In case of , .

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

(13)

(14)

(15)

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

3) In the area

(16)

(17)

(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

(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

(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

(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).

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

(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

(23)

(24)

(25)

(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}

(27)

where is the scalar vorticity, is the eddy viscosity,

.(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

(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

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