international journal of modern mathematical sciences issn

Copyright © 2014 by Modern Scientific Press Company, Florida, USA

International Journal of Modern Mathematical Sciences, 2014, 11(1): 1-12

International Journal of Modern Mathematical Sciences

Journal homepage:www.ModernScientificPress.com/Journals/ijmms.aspx

ISSN:2166-286X

Florida, USA

Article

Generating Exact Solutions of Two-Dimensional Incompressible

Navier-Stokes Equations

Muhammad Abul Kalam Azad (Ph.D student), Professor Dr. Laek Sazzad Andallah*

Department of Mathematics, Jahangirnagar University, Dhaka, Bangladesh

*Author to whom correspondence should be addressed; Email:[email protected]

Article history: Received 1 June 2014, Received in revised form 10 July 2014, Accepted 14 July 2014,

Published 20 July 2014.

Abstract: This paper investigates exact solutions of two dimensional incompressible Navier-

Stokes equations (2D NSEs) for a time dependent pressure gradient term using Orlowski and

Sobczyk transformation (OST) and Cole-Hopf transformation (CHT).Separation of variables

method is applied to find the solutions of 2D heat equation.

Keywords: Burgers equation, Cole-Hopf transformation (CHT), Orlowski and Sobczyk

transformation (OST).

Mathematical Subject Classification (2000): 35Q35, 65N06, 34K28

1. Introduction

The Navier–Stokes equations are important governing equations in the fluid dynamics which

describe the motion of fluid. These equations arise from applying Newton’s second law to fluid motion,

together with the assumption that the fluid feels forces due to pressure, viscosity and perhaps an external

force. They are useful because they describe the physics of many things of academic and economic

interest.

Int. J. Modern Math. Sci. 2014, 11(1): 1-12


2

However, NSEs are nonlinear in nature and it is difficult to solve these equations analytically. Despite

the concentrated research on Navier Stokes equations, their universal solutions are not achieved. The full

solutions of the three-dimensional NSEs remain one of the open problems in mathematical physics.

The exact solutions for the NSEs can be obtained are of particular cases. Exact solutions on the other

hand are very important for many reasons. They provide reference solutions to verify the accuracies of

many approximate methods. In order to understand the non-linear phenomenon of NSE, one needs to

study 2D NSEs. Since these equations posses’ diffusion, advection and pressure gradient terms of the full

3D NSEs, these incorporate all the main mathematical features of the full 3D NSEs.

Applying OST we have reduced 2D NSEs to 2D viscous Burgers’ equations .These equations are

mathematical model which are widely used for various physical applications, such as modeling of gas

dynamics and traffic flow ,shock waves in examining the chemical reaction-diffusion model of

Brusselator etc. So a number of analytical and numerical studies on NSEs as well as Burgers’ equations

have been conducted to find the exact solutions of the governing equations.

M.A.K. Azad, L.S. Andallah [1] investigated on analytical solutions of 2D incompressible

Navier-Stokes equations for a particular exponentially decreasing time dependent function. Prof. Sanjay

Mittal [2] studied on analytical solutions of simplified form of Navier-Stokes equation. Mohammad

Mehedi Rashidi [3] presented new analytical solutions of the 3D NSE. Maria Carmela Lombardo [4]

studied analytical solutions of the time dependent incompressible NSE on an half plane. Hikmat G.

Hasanov [5] found a method of 2D nonlinear Laplace transformation for solving the NSE. Hui, W. H.[6]

studied on the 2D incompressible viscous flow in which the local vorticity is proportional to the stream

function perturbed by a uniform stream. Muhammad R. Mohyuddinet. el.[7] presented analytical

solutions of 2D NSE governing the unsteady incompressible flow. The solutions have been obtained

using hondograph-Legendre transformation method. M.A.K. Azad, L.S. Andallah [8] investigated

analytical solutions of 1D Navier-Stokes equations. Nursalawati Rusli et al.[9] presented numerical

computation of a two dimensional Navier- Stokes equations using an improved finite difference method.

M. A. K. Azad, L.S. Andallah [10] studied on an explicit finite difference scheme for 1D Navier-Stokes

Equation.

Vineet Kumar Srivastava, Ashutosh, Mohammad Tamsir [11] derived a general analytical

solution of the three dimensional homogeneous coupled unsteady nonlinear generalized viscous Burgers’

equation via CHT and separation of variables method. Vineet Kumar Srivastava, Mohammad Tamsir,

Utkarsh Bhardwaj, YVSS Sanyasiraju [12] used Crank-Nikolson scheme for numerical solutions of two-

dimensional coupled Burgers’ equations. C. A. J. Fletcher [13] generated exact solution of the two-

dimensional Burgers’ equation. M.C. Kweyu, W. A. Manyonge, A. Koross, V. Ssemaganda [14]

generated varied sets of exact initial and Diritchlet boundary conditions for the 2D Burgers’ equations



3

from general analytical solutions via CHT. Lei Zhang, Lisha Wang, Xiaohua Ding [15] developed exact

finite-difference schemes for the 2D nonlinear coupled viscous Burgers’ equations using the analytical

solutions. Mohammad Tamsir, Vineet Kumar Srivastava [16] used a semi-impicit finite difference

approach to find the numerical solutions of the two- dimensional coupled Burgers’ equations. T.X. Yan,

L.S. Yue [17] presented variable separable solutions for the (2+1) –dimensional equation.

In this paper, we reduce 2D NSEs into 2D Burgers equations by applying OST. Then after

applying CHT 2D Burgers’ equations will be reduced to 2D diffusion equation. By using separation of

variables method we will solve diffusion equation. Then applying CHT and inverse OST we get the exact

solutions of 2D NSEs.

2.Mathematical Formulation

The dimensionalised governing equations of the fluid flow are given respectively by the

continuity equation

0*

*

*

*

y

v

x

u

x-momentum equation

2*

*2

2*

*2

*

*

*

**

*

**

*

* 1

y

u

x

u

x

p

y

uv

x

uu

t

u

y-momentum equation

2*

*2

2*

*2

*

*

*

**

*

**

*

* 1

y

v

x

v

y

p

y

vv

x

vu

t

v

where u and v are the velocity components in the x and y directions respectively, p is the pressure,

is

the constant density and

is the kinematic viscosity.

Using the dimensionless definitions in [ 9],

2

******

,,,,,U

pp

U

vv

U

uu

h

yy

h

xx

h

Utt

The dimensionalised governing equations are then converted into the non-dimensional form 2D

Navier-Stokes equations (2D NSEs) as

yyxxxyxt uupvuuuu Re

1 (1)

yyxxyyxt vvpvvuvv Re

1 (2)

0 yx vu (3)



4

where ;Re

1

Uh

- ).();( tgptfp yx

As a result equation (1) can be rewritten as

yyxxyxt uutfvuuuu Re

1)( (4)

“ (2)” can be written as

yyxxyxt uutgvvuvv Re

1)( (5)

3. Generating Exact Solutions of 2D NSEs

We apply OST [18] as

.,,),(),,()',,'(',',),(' tWtyxvvtWtyxutyxutttyytxx (6)

With t t

dWtdftW0 0

)()(,)()( (7)

And t t

dWtdftW0 0

)()(,)()( (8)

Here dftWt

0

= tF 0

= 0FtF .

Again dWtt

0

= dFFt

0

0

= tFG 00

= 00 GtFtG

Now dftWt

0

= tF 0

= 0FtF .

Again dWtt

0

= dFFt

0

0

= tFG 00



5

00 GFttG

Thus we get 00' GtFtGxx

9

00' GFttGyy

10

tt ' 11

.0),,(',,'' FtFtyxutyxu

12

.0),,(',, FtFtyxvtyxv

13

Now 0FtFutt

u

tft

u

tft

t

t

u

t

y

y

u

t

x

x

u

'

'

''

'

''

'

'

tft

t

t

uGFttGy

ty

uGtFtGx

tx

u

.0000

'

'

.'

'0

'

'0

'

'tf

t

uFtF

y

uFtF

x

u

0' FtFuxx

u

x

u

'

x

t

t

u

x

y

y

u

x

x

x

u

'

'

''

'

'

x

t

t

uGFttGy

xx

uGtFtGx

xx

u

'

'00

'

'00

'

'

.'

'

x

u

0' FtFuyy

u

y

u

'

y

t

t

u

y

y

y

u

y

x

x

u

'

'

''

'

'

y

t

t

uGFttGy

yy

uGtFtGx

yx

u

'

'00

'

'00

'

'



6

.'

'

y

u

Also

x

u

xx

u2

2

'

'

x

u

x

x

t

x

u

tx

y

x

u

yx

x

x

u

x

'

'

'

'

'

'

'

'

'

'

'

'

x

t

xt

uGFttGy

xxy

uGtFtGx

xx

u.

'00.

'00.

'

22

2

2

.'

'2

2

x

u

y

u

yy

u2

2

'

'

y

u

y

y

t

y

u

ty

y

y

u

yy

x

y

u

x

'

'

'

'

'

'

'

'

'

'

'

'

y

t

y

u

tGFttGy

yy

uGtFtGx

yyx

u

''0000

'

' '

2

22

0'

'

'1

'

'0

2

22

y

u

ty

u

yx

u

.'

'2

2

y

u

0FtFvtt

v

tft

v

tft

t

t

v

t

y

y

v

t

x

x

v

'

'

''

'

''

'

'

tft

t

t

vGFttGy

ty

vGtFtGx

tx

v

'

'0000

'

'

tft

vFtF

y

vFtF

x

v

'

'0

'

'0

'

'



7

0' FtFvxx

v

x

v

'

x

t

t

v

x

y

y

v

x

x

x

v

'

'

'

'

'

x

t

t

vGFttGy

xy

vGtFtGx

xx

v

'

'00

'

'00

'

'

.'

'

x

v

0)(' FtFvyy

v

y

v

'

y

t

t

v

y

y

y

v

y

x

x

v

'

'

''

'

'

y

t

t

vGFttGy

yy

vGtFtGx

yx

v

.

'

'00.

'

'00.

'

'

.'

'

y

v

Also

x

v

xx

v2

2

'

'

x

v

x

x

t

x

v

tx

y

x

v

yx

x

x

v

x

'

'

'

'

'

'

'

'

'

'

'

'

x

t

x

v

tGFttGy

xxy

vGtFtGx

xx

v

''0000

'

' '2

2

2

0'

'

'01

'

' 2

2

2

x

u

txy

v

x

v

.'

'2

2

x

v

y

v

yy

v2

2



8

'

'

y

v

y

y

t

y

v

ty

y

y

v

yy

x

y

v

x

'

'

'

'

'.

'

'

'

'

'

'

'

y

t

yt

vGFttGy

yy

vGtFtGx

yyx

v

'0000

'

' 2

2

22

010'

' 2

2

2

2

2

yt

v

y

v

yx

v

.'

'2

2

y

v

Substituting the transformed derivatives in “(2)” , we get

yyxxyxtyx uutfuFtFvuFtFutfuFtFuFtFu

Re

10000

yyxxyxt uuuvuuu

Re

1

14

Similarly, substituting the transformed derivatives in “(3)” , we get

yyxxyxt vvvvvuv

Re

1

15

From “ (4)” , we obtain

0 yx vu

16

Again, we know that the non-dimensional form of 2D Burgers equations are

.Re

12*

*2

2*

*2

*

**

*

**

*

*

y

u

x

u

y

uv

x

uu

t

u

(17)

.Re

12*

*2

2*

*2

*

**

*

**

*

*

y

v

x

v

y

vv

x

vu

t

v

(18)

So, “ (14)”, “ (15)”are the transformed 2D Navier-Stokes equations after applying OST and these are

analogous to non-dimensional form of 2D Burgers equation “ (17)”, “ (18)”.

Now we need to solve “(14)” , “(15)” with initial conditions

.,,0,,,,0,, 00 yxwhereyxvyxvyxuyxu

“(14)” and “(15)” can be linearized by the Cole-Hopf transformation [11],[13],[14],[19]



9

xtyxu

Re

2),,('

(19)

And

ytyxv

Re

2),,('

(20)

By using these transformations 2D coupled Burgers’ equations can be reduced to 2D diffusion equation

.Re

1''''' yyxxt

(21)

which is the well-known second order PDE called heat or diffusion equation [11], [13],[14],[19].

Consider a general solution of “(21)” of the form similar analogy to [11],[14].

tTyYxXyxayaxaatyx 4321,,' 22

which is the sum of the solution yxayaxaatyx

43211 ,,' and the separable solution

tTyYxXtyx ,,'2 where a1,a2,a3,a4 are arbitrary constants and X,Y,T are functions of

x´,y´,t´ respectively. Then “(21)” becomes

TYXYTXTXY Re

1

Dividing by XYT/Re on both sides, we get

Y

Y

X

X

T

T

Re

Since the left side is a function of t´ alone, while the right side is a function of x´ and y´, we see that each

side must be a constant, say 2(which is needed for boundedness). Thus

0Re

2

T

T

23

2

Y

Y

X

X

24

“(24)” can be written as

2

Y

Y

X

X

And since the left side depends only on x´ while the right side depends only on y´ each side must be a

constant, say -µ2 . Thus



10

02 XX

25

022 YY

26

Solutions to “(25)”, “(26)” and “(23)” are given by

Re3

22

2

22

211

2

,sincos,sincos

t

ebTycybYxcxbX

27

It follows that a solution to “(21)” is given by

Re3

22

2

22

2112

2

sincossincos,,

t

ebycybxcxbtyx

28

Thus the general solution of 2D diffusion equation is given by

Re3

22

2

22

2114321

2

sincossincos,,

t

ebycybxcxbyxayaxaatyx

29

By using CHT we obtain the general solution of 2D viscous Burgers equation as

Re3

22

2

22

2114321

Re3

22

2

22

21142

2

2

sincossincos

sincoscossin

Re

2,,

t

t

ebycybxcxbyxayaxaa

ebycybxcxbyaa

tyxu

30

Re3

22

2

22

2114321

Re3

2222

2

22

21143

2

2

cossincos

cossinsincos

Re

2,,

t

t

ebycybxcxbxyayaxaa

ebycybxcxbxaa

tyxv

31

Now applying inverse OST we get the exact solutions of 2D NSEs as

tyxuFtFtyxu ,,0,, 32

tyxvFtFtyxv ,,0,, 33

where x´,y´,t´,u´,v´ will be determined from equations (9)-(13).

4. Conclusions

In this study, we have shown how to generate exact solutions of 2D NSEs by using OST and CHT

with the help of separation of variables method. By using same method one can easily find out exact

solutions 3D and 1D NSE . The method is simple and can be used to find exact solutions of the governing

equations when we set initial and boundary conditions.



11

References

[1] M. A. K. Azad, L.S. Andallah, Analytical solutions of 2D incompressible Navier-Stokes equations

for time dependent pressure gradient., International Journal of Scientific & Engineering Research,

5(6)(2014): 878-886.

[2] Prof. Sanjay Mittal, Analytical solutions of simplified form of Navier-Stokes equation. 8th Indo-

German Winter Academy, MahimMistra Indian institute of technology, Kanpur, India, December

13-19, 2009, p 1-31.

[3] Mohammad Mehedi Rashdi, New analytical solution of the three-dimensional Navier-Stokes

equation. Modern Physics Letters B, World Scientific, 23(6)(2009): 3147-3155.

[4] Maria Carmela Lombardo, Analytic solutions of the Navier-Stokes equations. Rendiconti Del

Circolo Mathematico Di Palermo, 50(2)(2001): 299-311.

[5] Hikmat G. Hasanov, Method of two-dimensional nonlinear Laplace Transformation for solving the

Navier-Stokes equation. GSTF Journal on computing, 3(1)(2013):153.

[6] Hui, W.H., Exact solutions of the unsteady two dimensional Navier-Stokes equations. Research

Gate, 38(5)(1987): 689-702.

[7] Muhammad R. Mohyuddin et. el.,Exact solutions of the time dependent Navier-Stokes equations

by Hodograph-Legendre transformation method., Tamsui Oxford Journal of Mathematical

Sciences, 24(3)(2007): 257-268.

[8] M. A. K. Azad, L.S. Andallah, An analytical solution of 1D Navier-Stokes equation. International

Journal of Scientific & Engineering Research, 5( 2) (2014): 1342-1348.

[9] Nursalawati Rusli et al., Numerical computation of a two dimensional Navier- Stokes equations

using an improved finite difference method.,Mathematika, 27(1)(2011): 1-9.

[10] M. A. K. Azad, L.S. Andallah, An Explicit Finite Difference Scheme for 1D Navier-Stokes

Equation. International Journal of Scientific & Engineering Research, 5(3)(2014): 873-881.

[11] Vineet Kumar Srivastava, Ashutosh, Mohammad Tamsir, Generating exact solution of three

dimensional coupled unsteady nonlinear generalized viscous Burgers’ equation, International

Journal of Mathematical Sciences, 5(1)(2013): 1-13.

[12] Vineet Kumar Srivastava, Mohammad Tamsir, Utkarsh Bhardwaj, YVSS Sanyasiraju, Crank-

Nikolson Scheme for Numerical Solutions of two-dimensional Coupled Burgers’ Equations.

International Journal of Scientific & Engineering Research, 2(5)(2011):1-7.

[13] C. A. J. Fletcher, Generating exact solution of the two-dimensional Burgers’ equation, Int. J.

Numer. Math. Fluids, 3(1983): 213-216.



12

[14] M.C. Kweyu, W. A. Manyonge, A. Koross, V. Ssemaganda, Numerical solutions of the Burgers’

system in two dimensional under varied initial and boundary conditions. Applied Mathematical

Sciences, 6(113)(2012): 5603-5615.

[15] Lei Zhang, Lisha Wang, Xiaohua Ding, Exact finite-difference scheme and nonstandard finite

difference scheme for coupled Burgers’ equation. Advances in Difference Equations,

2014(2014):122.

[16] Mohammad Tamsir, Vineet Kumar Srivastava, A semi-impicit finite difference approach for two-

dimensional coupled Burgers’ equations .International Journal of Scientific & Engineering

Research, 2(6)(2011): 1-6.

[17] T. X. Yan, L.S. Yue, Variable separable solutions for the (2+1)-dimensional Burgers equation,

Chin. Phys. Lett., 20(3)(2003): 335-337.

[18] A. Orlowski and K. Sobczyk, Rep Math. Phys., 27(1989): 59.

[19] E. Hopf, Commun. Pure Appl. Math. 3(1950): 201-212.

[20] J.D. Cole, Q. Appl. Math. 9(1951): 225-232.

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