Generator of solutions for 2D Navier-Stokes equations Текст научной статьи по специальности «Математика»

УДК 532.516:517.958

Generator of Solutions for 2D Navier-Stokes Equations

Alexander V. Koptev*

Makarov State University of Maritime and Inland Shipping, Dvinskaya, 5/7, Saint-Petersburg, 198035,

Russia

Received 05.03.2014, received in revised form 15.04.2014, accepted 25.05.2014 On the paper under consideration the investigation of Navier-Stokes equations for 2D viscous incompressible fluid flow is present. An analysis is based on the first integral of these equations. It is revealed that all ratios are reduced to one governing equation which can be considered as a generator of solutions.

Keywords: viscous incompressible fluid, differential equation, partial derivative, nonlinearity, integral, generator of solutions.

Introduction

The Navier-Stokes equations describe a motion of fluids and gases the presence of viscosity. These equations are used in areas where the effects of viscous friction play a significant role. Hydrology, meteorology, shipbuilding, tribology, oil production, pipeline transport, cardiology, are just some of areas where traditionally used the Navier-Stokes equations [1, 2]. However, despite the great practical significance, many issues associated with the Navier-Stokes equations are studied not enough and need further research [3, 4]. One of the main problems is the lack of a constructive method of solution. How to solve the equations of Navier-Stokes equations with complete preservation of nonlinear terms — this is an unsolved question which is still relevant for today.

In this paper the Navier-Stokes equations for case of 2D unsteady motion of a viscous incompressible fluid are under consideration and attempts to develop a comprehensive approach to solution construction. If external forces have the potential initial system of equations in dimensionless variables has the form

du du du d (p + Ф) dt dx dy dx Re \dx2 dy2

_1_ fcfu d^u

(1)

dv dv dv d (p + Ф) 1 f d2v d2v

dt dx dy dy Re \dx2 dy2

du dv о dx dy

(2)

(3)

The major unknowns in the equations (1-3) are components of velocity u, v and pressure p; $ — the potential of external forces, which is a given function; Re is the Reynolds number, representing a positive parameter,

Re= LU0, v

where L — the scape of length, U0 — the velocity scape, v — the coefficient of kinematic viscosity.

* Alex.Koptev@mail.ru © Siberian Federal University. All rights reserved

1. First integral

The suggested procedure of solution construction for equations (1-3) is based on the first integrals of these equations. General description and conclusion ratios, representing first integral for case of the 3D equations, given in works [5, 6]. In the particular case of 2D motion conclusion is presented in the work [7]. For the most simple case like this of the first integral is reduced to five equations as the next

^ U2 1 d fdy 1 dy2

p + + d +2 dt[lx + -W

ai + pi,

2 f du dv\

+ Re V dx + dy)

d2y3 d2y3 d fdy1

3 + ^r + d

dy2

Re

dv du\ dx dy J

1 d 2 dy

dx2 dxdy

3^2

dy2 dt \ dx dy

+ 2(ai - pi),

dx

1 d ( d y2

2dx \ dx

1 d id^2 dV1 + 2 ~dt\dx + ~dy

+ dy(a2 +S)

d y A d ,o ^ ) + dx (P2 - S) •

d

- dt (a2 + p2),

(4)

(5)

(6)

(7)

(8)

In ratios (4-8) together with major unknowns u, v, p appear three new associate unknown ^i, y2, ^3. These values were not in the original equations (1-3), but they occur as a result of the first integration. Name as stream pseudo-functions were entered for these unknowns [5, 6, 7]. The meaning of other designations of the following: U is the velocity modulus, U = Vu2 + v2, d is dissipative term of the calculated by formula

2 V dx2 + dy2

(9)

aj, pj, (j = 1, 2), S — an arbitrary functions in two variables arising over integration. Each of these five functions is not dependent on any of the three possible arguments, respectively, x, y, or t

daj dx

dj dy

dj dt

0.

(10)

Therefore we face the system of five partial differential equations (4-8) with respect to six unknown u, v, p, y^ y2, y3. These equations have some advantages before the original ones. First advantage is that the order of the derivative on major unknown u, v, p is one less than the same for Navier-Stokes equations (1-3). Another advantage is that the equations (4-8) also allow conversion and consistent simplification.

2

u

1

uv

2. Generator of solutions

Let's analyze the values appearing in equations (4-8) and do some conversion. An unknown p is present only in equation (4) with additively. This equation should be used to determine p at the final stage when the rest of the unknown u, v, y2, y3 have already been found.

To determine the latter have a system of four equations (5-8). In these equations appear three associated unknown y^ y2, y3. Despite the similar signs, these unknown have different connotation. According to (9) unknown y3 determines the dissipative term d. For this reason, y3 aptly called the dissipative stream pseudo-function. While the y1 and y2, according to (7-8), determine the velocities u, v. These unknown logical to name as velocity stream pseudo-functions.

Unknowns ^i, ^2, appear in the equations (5-8) differently. Unknown is present in two equations from four, in only equations (5-6). These equations can be used to determine when ^1, ^2, u, v have been found. If to unite the left parts of all the terms, not containing

these two equations can be represented as

f1 = _ ^ + , (11)

dx2 dy2 ' dxdy'

where f1, f2 are some expressions containing u, v, aj, fy.

Two equations (11-12) can be solved for only in the case when the left parts meet the the condition of consistency. This condition can be obtained as follows. Calculate the derivative with respect to x from (11) and the derivative with respect to y from (12). Folding the results, we arrive at the equation

f + f = _ d^ll (13) dx dy dx3

Next we calculate the derivative with respect to y from (13) and the second derivative with respect to x from (12). Subtracting the results we arrive at the equation

f = f (14)

dxdy dx2 dy2

Equation (14) is necessary ("almost sufficient") condition of consistency. If this equation is satisfied, then an unknown can be found. To do this, equation (12) twice in succession to integrate in x and y respectively. The result for the solution it is enough to allow two remaining equation of system (7-8) together with equation (14).

It is vital that the functions f1 and f2, appearing in equation (14), are determined only by the unknown u, v, Moreover, u, v are expressed through the according to (7-8).

So, the unknown u, v you can exclude. Enough instead of u, v substitute in (14) right parts of (7-8). Therefore, we come to one equation with respect to The result of transform of

this equation can be represented as

1 ( dAu dAv\ 1 d ( dA^2\ , N

- v + vAu + R tly- + ftTj + 2at (-V + = ° (15)

Here instead of u, v means of their expression through the according to (7-8). And A

denotes the Laplace operator in the variables x and y.

So, to determine we have one equation (15). At this stage there is a whole range of

possibilities. We can set arbitrarily and by equation (15) allow Can, on the contrary, set arbitrarily and the unknown to find out of (15). We can also impose some conditions on

and and subject to the conditions both of the unknown to find from equation (15). In any case, both unknown and determined as solution of (15).

After that, we can find all the other interesting values. On the equations (7-8), there exist u, v. Then using (6) is defined And finally, equation (4) with account of (9) is p.

As a result of one equation with two unknowns generates solutions of the 2D Navier-Stokes equations. This equation is (15) and it can be called a generator of solutions. Note that with this approach, all of the nonlinear terms of the equations is completely stored and we obtain the exact solution of the Navier-Stokes equations.

3. Implementation

We give a concrete example of implementation and consider how the work described above generating ratio.

We construct a class of solutions of the 2D Navier-Stokes equations, which holds equalities as the next = A(t) exp (kix ± liy), and y2 = B(t) exp (k2x ± l2y), where A(t), B(t) are some functions in time, kj, lj — some constants.

Suppose, for simplicity, aj = 0, [3j = 0, (j = 1,2), S = 0. Calculating velocities, according to (7-8), we arrive at the expressions

u = 1 (-A(t)l2 exp (kix + liy) + B(t)k2l2 exp (k2x + l2y)) , (16)

v = -2 (-A(t)liki exp (kix + liy) + B(t)k2 exp (k2x + l2y)) . (17)

Further changes are the following. Substitute the expressions for yi, y2, u, v in (15). We obtain the equality to zero of a linear combination of five different functions: exp (2kix ± 2liy), exp (2k2x ± 2l2y), exp ((ki + k2)x ± (li ± l2)y), exp (kix ± liy), exp (k2x ± l2y).

To ensure the feasibility of equations (15), to the coefficients on each of these functions put equal to zero. The result is five ratios. First two of them are identical. Three other ones are as follows

k2li (k2li - kil2)(k2 + l2 - k2 - l2) = 0, (18)

f=^^ <19> "B = B . kl±l. (20)

dt Re v '

Expressions (19-20) are ordinal differential equations with respect to A(t) and B(t). Solutions for these equations are defined by expressions

A(t) = A(0)exp(cit), (21)

B(t) = B(0) exp(c2t), (22)

where A(0), B(0) are some nonzero constants, while decay constants over time defined as

k2 ± l2

The ratio of (18) is the characteristic equation for the exponent indicators. Its solution leads to two main non-trivial possibilities

F = T, (25)

ki k2

l2 ± k2 = l2 ± k2. (26)

For each specific task selection of roots is carried out differently. Let us consider in more detail the second possibility. We assume that kj, lj are non-zero and satisfy (26). Then decay constants are the same ci = c2 = c. Let's find for this case all the remaining unknown. For velocities receive expression

u = i (-A(0)l2 exp(kix ± liy ± c ■ t) ± B(0)k2l2 exp(k2x ± hy ± ct)) . (27)

v = -1 (-A(0)ki/i exp(kix + liy + c • t) + 5(0)^ exp(k2x + l2y + ct)) . (28)

For unknown we have equation (6), which in light of the previous equation is transformed

to

= 1(B(0)2k3l2 exp (2k2x + 2l2y + 2c • t) + A(0)2/3k1 exp (2k1x + 2l1y + 2c • t) -dx dy 4

- A(0)B(0)k2li(ki/2 + k2li) exp ((ki + k2)x + (li + l2)y + 2c • t))+

+ ^ (A(0)lik2 exp(kix + liy + ct) + B(0)k2l2 exp(k2x + hy + ct)) . (29) Re

The consistent integration of this equation in x and y with zero additive functions leads to result

1 k2 l2 ^3 = (0)^-42 exp (2k2x + 2l2y + 2c • t) + A(0)24 exp (2kix + 2liy + 2c • t) -

- A(0)B(0)k^krW-+4 exp ((ki + k2)x + (li + l2)y + 2c • t))+

(ki + k2)(li + l2)

+ -^(A(0)ki exp(kix + liy + ct) + B(0)l2 exp(k2x + l2y + ct)). (30) Re

Note that (30) satisfies equation (5). Direct substituting (30) into (5) leads to relation

2(ki + k2)(li + l2)(lil2 - kik2) = (kil2 + lik2)((li + l2)2 - (ki + k2)2), (31)

which is obviously valid, due to (26).

Therefore, unknown is defined. Now you should contact to (4). From this equation, taking into account (9), we find p + $. Calculations lead to a result

p + $ = A(0)B(0) k2li • (k2li - kil2) exp ((ki + k2)x + (li + l2)y + 2ct). (32)

4 /i + ¿2

So, the major unknowns are defined. Constructed solution correspond to the wave motion of a viscous fluid. The most interesting case is when the solution fade over time. For this case decay constant c should be negative. According to formulas (23-24), this can realize if

k2 + j < 0. (33)

This inequality can only be performed on complex values kj, j.

Let's consider, for example, the simplest case, when kj, j purely imaginary. Suppose kj = inj, j = imj, where i is the imaginary unit, nj, mj are some real numbers. Then k2 + j = -n2 - m2 and inequality (33) is obviously valid. Substituting kj = inj, j = imj on (27-28), (32) and separating real and imaginary parts, we arrive at expressions as the next

1 /_n2_mm2

u = — (A(0)mi cos(nix + miy) - B(0)n2m2 cos(n2x + m2y)) exp ( -^-i • t ) , (34)

2 i Re

1 ( ) n2 m2

v = -- (A(0)nimi cos(nix + miy) - B(0)n2 cos(n2x + m2y)) exp ( -^-2 • t) , (35)

2 2 Re

^ A(0)B(0) (ni — n2)

p + $ = ----,-r • n2mi(n2mi - nim2)x

4 (mi + m2 )

.. / —2ni — 2mi N x cos((ni + n2)x + (mi + m2)y) • exp I -r-• t J . (36)

Solution of the 2D Navier-Stokes equations constructed above corresponds to standing waves in deep water [2].

If in expressions (27-28),(32) values kj, j believe common species complex numbers, kj = rj + isj, lj = £j + iZj, then we obtain the solution corresponding to progressive waves in deep water. For this case decay constant is a complex number with negative real part c = -A2 + iw. On addition, should be performed ratios as the next

r 2 - s2 + e2 - z2 = r2 - s2 + e2 - z2, (37)

r 1S 1 + £iCi = r2«2 + 6C2, (38)

A2 = s2 - r2 + - £2 A = Re

2(r is 1 + £ 1Ç1 )

w = —Re— •

As a result of velocities are obtained in the form u = 1 B(0) ((r2£2 - S2£2) cos(s2X + Z2y + wt)-

(39)

(40)

y + wt)) exp(r 2X + £2y - a2j

22

coB(S1X + Z1 y + wt) + +2£1Z1 sin(s1 x + Z1y + wt) exp(r1x + £1y - A2t)) , (41)

-(r2Z2 + S2£2) sin(s2X + Z2y + wt)) exp(r2x + £2y - A2t)-- 1 A(0) ((£2 - C2) cos(s1X + Z1 y + wt) +

v = -2B(0) ((r2 - s2) cos(s2x ± Z2y ± wt)-

-2r2S2 sin(s2x ± Z2y ± wt)) exp(r2x ± Z2y - A2t)± ± 1 A(0) ((ri£i - siZi) cos(six ± Ziy ± wt)-

(riZi ± si£i) sin(six ± Ziy ± wt)) exp(rix ± £iy - A2t). (42)

Here rj, sj, £j, Zj are some real numbers satisfying two equations (37-38) and inequality r2 - s2 ± £2 - Z2 < 0. An exact solutions of 2D Navier-Stokes equations constructed above are new.

Conclusion

Thus, for 2D Navier-Stokes equations the generator of solutions is constructed. It is equation (15). It represent the equation of the fifth order with respect to two unknowns. This equation opens way to construction of exact solutions. Allowing this equation we can consistently find all unknown.

This procedure leads to some conclusions about structure of solutions for the 2D Navier-Stokes equations. For anyone solution u, v, p all the relationships representing the first integral and all subsequent ratio discussed above are hold. So, the velocities must be of the type described by equations (7-8). Similarly, pressure p must be defined by equation (4). That is, without consideration of additive functions, p must be represented as the sum of four different in nature components.

Thus, each of the major unknown u, v, p must be presented with sum of components of a specific type and it general structure is clear.

References

[1] L.G.Loitsynskiy, Mechanics of Fluid and Gas, Nauka, Moscow, 1987 (in Russian).

[2] N.E.Kochin, I.A.Kibel, N.V.Rose, Theoretical Hydromechanics, Part 2, Nauka, Moscow, 1967 (in Russian).

[3] O.A.Ladijzenskaia, The Mathematical Theory of Viscous Incompressible Fluid, Gordon and Breach, New York, 1969.

[4] Charles L. Fefferman. Existence and smoothness of the Navier-Stokes equation, Preprint, Princeton Univ., Math. Dept., Princeton, NJ, USA, (2000), 1-5.

[5] A.V.Koptev, Integrals of Navier-Stokes equations, Trudy Sredne-volzhskogo Matematich-eskogo Obshchestva,Saransk, 6(2004), no. 1, 215-225 (in Russian).

[6] A.V.Koptev, First integral and ways of further integration of Navier-Stokes equations, Izvestia Rossiyskogo gosudarstvennogo pedagogicheskogo universiteta im. Ge'rtsena, Saint-Petersburg, 147(2012), 7-17 (in Russian).

[7] A.V.Koptev, How integrate the Navier-Stokes equations, Physical Mechanics, Saint-Petersburg state university, Saint-Petersburg, 8(2004), 218-226 (in Russian).

Генератор решений 2В-уравнений Навье-Стокса

Александр В. Коптев

Рассматриваются уравнения Навье-Стокса для ËD-движений вязкой несжимаемой жидкости. Исследование основывается на первом интеграле этих уравнений. Показано, что существует одно определяющее соотношение, которое можно рассматривать как генератор решений.

Ключевые слова: вязкая несжимаемая жидкость, дифференциальное уравнение, частная производная, нелинейность, интеграл, генератор решений.