a class of exact solutions of navier-stokes equations for ...
A CLASS OF EXACT SOLUTIONS OF NAVIER-STOKES EQUATIONS FOR INCOMPRESSIBLE FLUID OF VARIABLE VISCOSITY FOR DEFINED VORTICITY DISTRIBUTION R. K. Naeem Department of Mathematics, University of Karachi, Karachi, Pakistan. Email: [email protected] Received 13 December 2010; accepted 14 January 2011
ABSTRACT A class of exact solutions is obtained of Navier-Stokes equations for incompressible fluid of variable viscosity. This class consist of flows for which the vorticity distribution is defined by equation (11). The Navier-Stokes equations are transformed into simple ordinary differential equations by introducing a variable defined by equation (23) and a class of exact solutions is determined. The effect of parameters n, m, a, b and U on the velocity components are plotted and discussed. Keywords: Exact Solutions, Navier-Stoke Equations, Solutions to the flow equations of an incompressible fluid of variable viscosity, A class of exact solutions to Navier Stokes equations.
1 INTRODUCTION Exact solutions of Navier-Stokes equations are very important, because they serve a dual purpose. First, they provide a solution to be a flow that has some physical meaning. Second, such solutions can be used as accuracy checks for experimental, numerical, and asymptotic methods. Some exact solutions of Navier-Stokes equations are determined in cases where these equations can be linearized and in cases where these equations can be reduced to ordinary differential equations for which the solution is possible. Some exact solutions of Navier-Stokes equations are also reported in which the vorticity distribution is prescribed such that the governing equations written in terms of the stream function become linear. The reader interested in details of these exact solutions may refer to Taylor(Taylor 1923), Kamp de Feriet( Kamp de Feriet 1930), Kovasznay (Kovasznay 1948), Wang (Wang 1966), Lin and Tobak (Lin and Tobak 1986), Hui (Hui 1987), Wang (Wang 1989) ,Wang (Wang 1990), Wang (Wang 1991), Chandna and Oku-UKpong (Chandna and Oku-UKpong 1994), Naeem and Jamil (Naeem and Jamil 2005), Naeem and Jamil (Naeem and Jamil 2006) and Naeem and Younus (Naeem and Younus 2010) and references therein. The objective of this paper is to indicate a class of exact solutions of the equations governing the steady plane flows of incompressible fluid of variable viscosity for which the vorticity distribution is defined by equation (11).We point out that exact solutions to Navier-Stokes Int. J. of Appl. Math and Mech. 7 (4): 97-118, 2011.
R. K. Naeem
98
equations, to be best of our knowledge, are not yet presented for the vorticity distribution characterized by equation (11). The outline of the paper is as follows. In section 2, we present the basic flow equations and express them in terms of the function and a new variable defined by equations (12) and (23), respectively. In section 3, we determined the exact solutions. In section 4, we discuss the influence of the pertinent parameters involved in the solutions on the fluid motion. The conclusion is given in section 5.
2 BASIC EQUATIONS The basic non-dimensional equations governing the steady plane flow of incompressible fluid of variable viscosity, in the absence of external force and with no heat additions from Naeem and Jamil (2006) are: (1) ,
(2)
,
(3) ,
(4)
The various symbols used here have their usual meanings. Equation (1) implies the existence of the streamfunction such that .
(5)
The system of equations (1-4), employing equation (5), transforms into the following system of equations: ,
(6) ,
(7) ,
where the vorticity function
and the generalized energy function
,
(8) are defined by (9)
.
Int. J. of Appl. Math and Mech. 7 (4): 97-118, 2011.
(10)
A Class Of Exact Solutions Of Navier-Stokes Equations
Once a solution of system of equations (6-8) is determined, the pressure equation (10).
99
is obtained from
We shall investigate fluid motion for which the vorticity distribution is proportional to the streamfunction perturbed by a stream periodic in x and y, a uniform stream and an exponential stream. Therefore we set (11) where
and
are real constants.
On substituting ,
(12)
in equation (11), we get (13) Equation (9), employing equations (11) and (12), becomes (14) Inserting equations (12) and (14) in equation (6), we get
(15) On introducing the functions H and M defined by (16) and (17) the equation (15) becomes (18)
Equation (7), utilizing equations (12) and (14), becomes
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(19) Inserting equations (16) and (17) in equation (19), we get
,
(20)
Equation (8), employing equation (12), becomes
, Equations (18) and (20), on using the integrability condition
(21)
, yields
.
(22)
Equation (22) is the equation that must be satisfied by the viscosity and the function for the motion of a steady incompressible fluid of variable viscosity in which the vorticity distribution is given by equation (11). On introducing the new variable
defined by
,
(23)
the equations(13),(21) and (22) become , ,
(24) (25) (26)
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where .
(27)
3 EXACT SOLUTIONS In this section we determine the solutions of the system of equations (24-26) for the following three cases: Case I :
.
Case II:
.
Case III: We consider these three cases separately. Case I For this case solution of equation (24) in the physical plane is given by
where
and
are real constants.
Equation (26), employing equations (28) and (29), yields
where
and
are real constants.
The solution of equation (25) for
is
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and for
is
(33) In equations (32) and (33),
and
are real constants.
In order to determine the pressure p we require to solve equations (18) and (20). These equations for and are solved as follows: Equations (18) and (20), for
, becomes ,
(34) (35)
On substituting (36) the equations (34) and (35) becomes
where
,
(37)
,
(38)
is given by equation(28).
The solution of equations (37) and (38) is
+ where
is a real constant.
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(39)
A Class Of Exact Solutions Of Navier-Stokes Equations
103
Inserting equation (39) in equation (36), we obtain
(40) Equation (39) is the solution of equations (18) and (20). The pressure p can easily be obtained on inserting equations (40) and (16) in equation (10). Proceeding in the same manner as above, the solution of equations (18) and (20) for
, where
is
(41)
is a real constant.
In this case the streamfunction
and velocity components for n ≠ 1 are
,
(42) , ,
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(43) (44)
R. K. Naeem
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and for n = 1 are ,
(45) ,
(46)
.
(47)
Case II For this case the solution of the system of equations (24-26) is
,
(50)
(51)
32 2 4 +6 where
and
are all real constants.
The solution of equations (18) and (20) for
Int. J. of Appl. Math and Mech. 7 (4): 97-118, 2011.
is
,
=1
(52)
A Class Of Exact Solutions Of Navier-Stokes Equations
105
(53) and for
is
, where
and
(54)
are real constants.
In this case the stream function
and velocity components u and v for ,
are (55)
, ,
(56) (57)
and for m = 1 are ,
(58) , .
(59) (60)
Case III For this case we have ,
(61) ,
(62)
,
(63)
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,
(64)
,
(65)
,
(66) where
and
are all real constants.
4 RESULTS AND DISCUSSION A class of exact solutions of the equations governing the steady plane flows of incompressible fluid of variable viscosity are determined for which the vorticity distribution is defined by equation (11), and the effect of parameters n, m, a, b and U are depicted in figures(1-22). Figures (1-12) are for Case-I and figures (13-22) are for Case-II. Figures (1)and (2) present the effect of parameter n on the velocity components u and v in the direction of y and x, respectively. It is obvious form these figures that velocity components u and v have oscillating behavior and both are increasing function of n in absolute value. The amplitude of oscillation increase with the increase of the parameter n. Figures (3) and (4) show effect of parameters b and a on velocity components u and v in the y and x directions, respectively. The effects are similar to effects of n in figures (1) and (2). Figures (5-8) illustrate the influence of parameter U on the velocity components u and v in y and x directions, respectively. These figures indicate that oscillations decrease with increase in U and the magnitude of the velocity components u and v increase with increase in U. Figures(9-12) present the effect of parameters U, a and b on the velocity components u and v. The figures show that the magnitude of velocity components increase with increase in U, a and b. Figures (13-22) present influence of the parameters m, a, b and U on the velocity components. These figures show that velocity components are increasing function of parameters in absolute value.
5 CONCLUSION A class of exact solutions of equations governing the steady plane motion of incompressible fluid of variable viscosity are obtained for which the vorticity distribution is given by equation (11). The effect of parameters n, m, b and U of interest on the velocity components are plotted and discussed. Int. J. of Appl. Math and Mech. 7 (4): 97-118, 2011.
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APPENDIX
30 25 20
n =10 n =15 n= 20
15 10 u(y) 5 0 -5 -10 -15 -20 0
0.5
1 y
1.5
2
Figure 1: The effect of n on velocity component u for a = 0 A1=A2=A3=B1=B2=U=b= 1, in the direction of y. 20 15 10 5 0 v(x)
-5
-10 -15 -20 -25 -30 0
n =10 n =15 n=20 0.5
1 x
1.5
2
Figure 2: The effect of n on velocity component v for A 1= A 2 =A 3= B 1= B 2 =U= a =1, b = 0 in the direction of x.
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400 350 300
b = 1.5 b=2 b = 2.5
250 u(y)200 150 100 50 0 -50 0
0.5
1 y
1.5
2
Figure 3: The effect of b on velocity component u for A 1 =A2= A3= B1= B2 =U =1, a = 0, n = 10 in the direction of y.
50 0 -50 -100 v(x)-150 -200 -250 -300 -350 -400 0
a = 1.5 a=2 a = 2.5 0.5
1 x
1.5
2
Figure 4: The effect of a on velocity component v for A 1= A 2 =A 3= B 1= B 2 =U=1, n = 10, b = 0 in the direction of x.
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1200 U = 10 U = 30 U = 35
1000 800 u(y)
600 400 200 0 0
0.5
1 y
1.5
2
Figure 5: The effect of U on velocity component u for A 1=A 2=A 3=B 1=B 2= 1, a = 0, n = 10, b=1.5 in the direction of y.
6000 5000
U = 130 U = 160 U = 180
4000 u(y) 3000 2000 1000 0 0
0.5
1 y
1.5
2
Figure 6: The effect of U on velocity component u for A1=A2=A3=B1=B2= 1, a = 0, n = 10, b=1.5 in the direction of y. A1=A2=A3=B1=B2= 1, a = 0, n = 10, b=1.5 in the direction of y.
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0 -200 -400 v(x)
-600 -800
-1000 -1200 -1400 0
U = 10 U = 30 U = 40 0.5
1 1.5 2 x Figure 7: The effect of U on velocity component v for A 1= A 2 =A 3= B 1= B2 =1, n = 10, b = 0, a = 1.5 in the direction of x.
0 -500 -1000 -1500 v(x)-2000 -2500 -3000 -3500 -4000 -4500 0
U = 90 U = 120 U = 150 0.5
1 1.5 2 x Figure 8: The effect of U on velocity component v for A 1= A 2 =A 3= B 1= B 2 =1, n = 10, b = 0, a = 1.5 in the direction of x.
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60 b = 1.5 b = 1.7 b=2
50 40 u(y) 30 20 10 0 0
0.5
1 1.5 2 y Figure 9: The effect of b on velocity component u for A 1=A 2=A 3=B 1=B 2=U=1, a = 0, n = 1 in the direction of y.
50 45 40
U=2 U = 2.5 U=3
35 30 u(y) 25 20 15 10 5 0 0
0.5
1 y
1.5
2
Figure 10: The effect of U on velocity component u for A 1=A 2=A 3=B 1=B 2=1, a = 0, n = 1, b=1.5 in the direction of y.
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0 -5 -10
U=2 U = 2.5 U=3
-15 -20 v(x) -25 -30 -35 -40 -45 -50 0
0.5
1 1.5 2 x Figure 11: The effect of U on velocity component v for A 1= A 2 =A 3= B 1= B 2 = n = 1, b = 0, a = 1.5 in the direction of x.
0 -10
a = 1.5 a = 1.7 a=2
-20 v(x) -30 -40 -50 -60 0
0.5
1 1.5 2 x Figure 12: The effect of a on velocity component v for A 1= A 2 =A 3= B 1= B 2 =U= n = 1, b = 0 in the direction of x.
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1400 m=2 m = 2.5 m=3
1200 1000 u(y)
800 600 400 200 0 0
0.5
1 1.5 2 y Figure 13: The effect of m on velocity component u for A 1=A 2=A 3=C1=C2=U=1, a = 0, b = 1 in the direction of y.
4
12
x 10
10
b = 2.1 b = 2.3 b = 2.5
8 u(y)
6 4 2 0 0
0.5
1 y
1.5
2
Figure 14: The effect of b on velocity component u for A 1=A 2=A 3=C1=C2=U=1 , a = 0, m =2 in the direction of y.
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2000 U=5 U = 10 U =15
1800 1600 1400 1200 u(y) 1000 800 600 400 200 0 0
0. 1 1. 2 y 5 5 Figure15: The effect of U on velocity component u for A1= A2= A3=C1= C2=1, b = 1.5, m = 2, a = 0 in the direction of y.
0 -100 -200 -300 v(x) -400 -500 -600 -700 0
m = 2.3 m = 2.5 m = 2.7 0.5
1 1.5 2 x Figure 16: The effect of m on velocity component v for A 1=A 2=A 3=C1=C2=U=a=1, b= 0 in the direction of x .
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0 -200 -400 v(x)
-600 -800
-1000 -1200
a = 1.3 a = 1.4 a = 1.5
-1400 0
0.5
1 1.5 2 x Figure 17: The effect of a on velocity component v for A 1=A 2=A 3=C1=C2=U=1, m= 2, b= 0 in the direction of x.
0 -200 -400 -600 v(x) -800 -1000 -1200 -1400 -1600 -1800 0
U=2 U=7 U = 10 0.5
1 x
1.5
2
Figure 18: The effect of U on velocity component v for A 1=A 2=A 3=C1=C2=1, m= 2, a = 1.5, b= 0 in the direction of x.
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200 0 -200 u(y)
-400 -600 -800
-1000 -1200 0
b = 1.5 b = 1.7 b = 1.9 0.5
1 1.5 2 y Figure 19: The effect of b on velocity component u for A 1=A 2=A 3=C1=C2=1, m= 1, a=0, U=10 in the direction of y.
100 0 -100 -200 -300 u(y) -400 -500 -600 -700 -800 -900 0
U = 10 U = 20 U = 30 0.5
1 y
1.5
2
Figure 20: The effect of U on velocity component u for A 1=A 2=A 3=C1=C2= m=1, b= 1.5, a =0 in the direction of y.
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1200 1000
a = 1.5 a = 1.7 a = 1.9
800 v(x)
600 400 200 0 -200 0
0.5
1 1.5 2 x Figure 21: The effect of a on velocity component v for A 1=A 2=A 3=C1=C2= m=1, b= 0, U=10 in the direction of x.
900 800 700
U =10 U =20 U =30
600 500
v(x)40 0 300 200 100 0
-1000
0.5
1
1.5
2
x Figure 22: The effect of U on velocity component v for A1=A2=A3=C1=C2= m=1, b= 0, a=1.5 in the direction of x.
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REFERENCES Hui WH (1987), Exact solutions of the unsteady two-dimensional Navier-Stokes equations, Journal of Applied Mathematics and Physics, 38, pp. 689-702 Kampe de JF (1930). Sur quelques cas d’integration des equations du movement plan d’un fluid visqueux incompressible. Proc. Int Congr. Appl. Mech., 3rd. Stockholm, 1, pp. 334-338. Kovasznay LIG (1948). Laminar flow behind a two-dimensional grid. Proc. Cambridge Phil. Soc., 44, pp. 58-62. Lin SP and Tobak M (1986). Reversed flow above a plate with suction. American Institute of Aeronautics and Astronautics Journal, 24(2), pp.334-335 Naeem RK and Jamil M (2005). A class of exact solutions of an incompressible fluid of variable viscosity, Quaid-e-Awam University Research Journal of Engineering, Science and Technology, 6(1,2), pp.11-18. Naeem RK and Jamil M (2006). On plane steady flows of an incompressible fluid with variable viscosity. Inernational. Journal of Applied Mathematics and Mechanics, 2(3), pp.3251. Naeem RK and Younus S. (2010), Exact solutions of the Navier-Stokes equations for incompressible fluid of variable viscosity for prescribed vorticity distributions, International Journal of Applied Mathematics and Mechanics, 6(5), pp. 18-38. Taylor GI (1923). On the decay of vortices in a viscous fluid, Phil.Mag Series, 6(46). Wang CY (1966). On a class of exact solutions of the Navier-Stocks equations. Journal of Applied. Mechanics, 33, pp.696-698 Wang CY (1989), Exact solutions of the unsteady Navier-Stokes Equations, Applied Mechanics Review ,42, 269-282, 1989. Wang CY (1990). Exact solutions of the Navier- Stokes equations-the generalized Beltrami flows, review and extension, Acta Mechanica, 81, 69-74. Wang CY (1991). Exact solutions of the steady-state Navier-Stocks equations, Annual Review: Fluid Mechanics, 23, pp.159-177
Int. J. of Appl. Math and Mech. 7 (4): 97-118, 2011.
ABSTRACT A class of exact solutions is obtained of Navier-Stokes equations for incompressible fluid of variable viscosity. This class consist of flows for which the vorticity distribution is defined by equation (11). The Navier-Stokes equations are transformed into simple ordinary differential equations by introducing a variable defined by equation (23) and a class of exact solutions is determined. The effect of parameters n, m, a, b and U on the velocity components are plotted and discussed. Keywords: Exact Solutions, Navier-Stoke Equations, Solutions to the flow equations of an incompressible fluid of variable viscosity, A class of exact solutions to Navier Stokes equations.
1 INTRODUCTION Exact solutions of Navier-Stokes equations are very important, because they serve a dual purpose. First, they provide a solution to be a flow that has some physical meaning. Second, such solutions can be used as accuracy checks for experimental, numerical, and asymptotic methods. Some exact solutions of Navier-Stokes equations are determined in cases where these equations can be linearized and in cases where these equations can be reduced to ordinary differential equations for which the solution is possible. Some exact solutions of Navier-Stokes equations are also reported in which the vorticity distribution is prescribed such that the governing equations written in terms of the stream function become linear. The reader interested in details of these exact solutions may refer to Taylor(Taylor 1923), Kamp de Feriet( Kamp de Feriet 1930), Kovasznay (Kovasznay 1948), Wang (Wang 1966), Lin and Tobak (Lin and Tobak 1986), Hui (Hui 1987), Wang (Wang 1989) ,Wang (Wang 1990), Wang (Wang 1991), Chandna and Oku-UKpong (Chandna and Oku-UKpong 1994), Naeem and Jamil (Naeem and Jamil 2005), Naeem and Jamil (Naeem and Jamil 2006) and Naeem and Younus (Naeem and Younus 2010) and references therein. The objective of this paper is to indicate a class of exact solutions of the equations governing the steady plane flows of incompressible fluid of variable viscosity for which the vorticity distribution is defined by equation (11).We point out that exact solutions to Navier-Stokes Int. J. of Appl. Math and Mech. 7 (4): 97-118, 2011.
R. K. Naeem
98
equations, to be best of our knowledge, are not yet presented for the vorticity distribution characterized by equation (11). The outline of the paper is as follows. In section 2, we present the basic flow equations and express them in terms of the function and a new variable defined by equations (12) and (23), respectively. In section 3, we determined the exact solutions. In section 4, we discuss the influence of the pertinent parameters involved in the solutions on the fluid motion. The conclusion is given in section 5.
2 BASIC EQUATIONS The basic non-dimensional equations governing the steady plane flow of incompressible fluid of variable viscosity, in the absence of external force and with no heat additions from Naeem and Jamil (2006) are: (1) ,
(2)
,
(3) ,
(4)
The various symbols used here have their usual meanings. Equation (1) implies the existence of the streamfunction such that .
(5)
The system of equations (1-4), employing equation (5), transforms into the following system of equations: ,
(6) ,
(7) ,
where the vorticity function
and the generalized energy function
,
(8) are defined by (9)
.
Int. J. of Appl. Math and Mech. 7 (4): 97-118, 2011.
(10)
A Class Of Exact Solutions Of Navier-Stokes Equations
Once a solution of system of equations (6-8) is determined, the pressure equation (10).
99
is obtained from
We shall investigate fluid motion for which the vorticity distribution is proportional to the streamfunction perturbed by a stream periodic in x and y, a uniform stream and an exponential stream. Therefore we set (11) where
and
are real constants.
On substituting ,
(12)
in equation (11), we get (13) Equation (9), employing equations (11) and (12), becomes (14) Inserting equations (12) and (14) in equation (6), we get
(15) On introducing the functions H and M defined by (16) and (17) the equation (15) becomes (18)
Equation (7), utilizing equations (12) and (14), becomes
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(19) Inserting equations (16) and (17) in equation (19), we get
,
(20)
Equation (8), employing equation (12), becomes
, Equations (18) and (20), on using the integrability condition
(21)
, yields
.
(22)
Equation (22) is the equation that must be satisfied by the viscosity and the function for the motion of a steady incompressible fluid of variable viscosity in which the vorticity distribution is given by equation (11). On introducing the new variable
defined by
,
(23)
the equations(13),(21) and (22) become , ,
(24) (25) (26)
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101
where .
(27)
3 EXACT SOLUTIONS In this section we determine the solutions of the system of equations (24-26) for the following three cases: Case I :
.
Case II:
.
Case III: We consider these three cases separately. Case I For this case solution of equation (24) in the physical plane is given by
where
and
are real constants.
Equation (26), employing equations (28) and (29), yields
where
and
are real constants.
The solution of equation (25) for
is
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and for
is
(33) In equations (32) and (33),
and
are real constants.
In order to determine the pressure p we require to solve equations (18) and (20). These equations for and are solved as follows: Equations (18) and (20), for
, becomes ,
(34) (35)
On substituting (36) the equations (34) and (35) becomes
where
,
(37)
,
(38)
is given by equation(28).
The solution of equations (37) and (38) is
+ where
is a real constant.
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(39)
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103
Inserting equation (39) in equation (36), we obtain
(40) Equation (39) is the solution of equations (18) and (20). The pressure p can easily be obtained on inserting equations (40) and (16) in equation (10). Proceeding in the same manner as above, the solution of equations (18) and (20) for
, where
is
(41)
is a real constant.
In this case the streamfunction
and velocity components for n ≠ 1 are
,
(42) , ,
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(43) (44)
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and for n = 1 are ,
(45) ,
(46)
.
(47)
Case II For this case the solution of the system of equations (24-26) is
,
(50)
(51)
32 2 4 +6 where
and
are all real constants.
The solution of equations (18) and (20) for
Int. J. of Appl. Math and Mech. 7 (4): 97-118, 2011.
is
,
=1
(52)
A Class Of Exact Solutions Of Navier-Stokes Equations
105
(53) and for
is
, where
and
(54)
are real constants.
In this case the stream function
and velocity components u and v for ,
are (55)
, ,
(56) (57)
and for m = 1 are ,
(58) , .
(59) (60)
Case III For this case we have ,
(61) ,
(62)
,
(63)
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,
(64)
,
(65)
,
(66) where
and
are all real constants.
4 RESULTS AND DISCUSSION A class of exact solutions of the equations governing the steady plane flows of incompressible fluid of variable viscosity are determined for which the vorticity distribution is defined by equation (11), and the effect of parameters n, m, a, b and U are depicted in figures(1-22). Figures (1-12) are for Case-I and figures (13-22) are for Case-II. Figures (1)and (2) present the effect of parameter n on the velocity components u and v in the direction of y and x, respectively. It is obvious form these figures that velocity components u and v have oscillating behavior and both are increasing function of n in absolute value. The amplitude of oscillation increase with the increase of the parameter n. Figures (3) and (4) show effect of parameters b and a on velocity components u and v in the y and x directions, respectively. The effects are similar to effects of n in figures (1) and (2). Figures (5-8) illustrate the influence of parameter U on the velocity components u and v in y and x directions, respectively. These figures indicate that oscillations decrease with increase in U and the magnitude of the velocity components u and v increase with increase in U. Figures(9-12) present the effect of parameters U, a and b on the velocity components u and v. The figures show that the magnitude of velocity components increase with increase in U, a and b. Figures (13-22) present influence of the parameters m, a, b and U on the velocity components. These figures show that velocity components are increasing function of parameters in absolute value.
5 CONCLUSION A class of exact solutions of equations governing the steady plane motion of incompressible fluid of variable viscosity are obtained for which the vorticity distribution is given by equation (11). The effect of parameters n, m, b and U of interest on the velocity components are plotted and discussed. Int. J. of Appl. Math and Mech. 7 (4): 97-118, 2011.
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APPENDIX
30 25 20
n =10 n =15 n= 20
15 10 u(y) 5 0 -5 -10 -15 -20 0
0.5
1 y
1.5
2
Figure 1: The effect of n on velocity component u for a = 0 A1=A2=A3=B1=B2=U=b= 1, in the direction of y. 20 15 10 5 0 v(x)
-5
-10 -15 -20 -25 -30 0
n =10 n =15 n=20 0.5
1 x
1.5
2
Figure 2: The effect of n on velocity component v for A 1= A 2 =A 3= B 1= B 2 =U= a =1, b = 0 in the direction of x.
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108
400 350 300
b = 1.5 b=2 b = 2.5
250 u(y)200 150 100 50 0 -50 0
0.5
1 y
1.5
2
Figure 3: The effect of b on velocity component u for A 1 =A2= A3= B1= B2 =U =1, a = 0, n = 10 in the direction of y.
50 0 -50 -100 v(x)-150 -200 -250 -300 -350 -400 0
a = 1.5 a=2 a = 2.5 0.5
1 x
1.5
2
Figure 4: The effect of a on velocity component v for A 1= A 2 =A 3= B 1= B 2 =U=1, n = 10, b = 0 in the direction of x.
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1200 U = 10 U = 30 U = 35
1000 800 u(y)
600 400 200 0 0
0.5
1 y
1.5
2
Figure 5: The effect of U on velocity component u for A 1=A 2=A 3=B 1=B 2= 1, a = 0, n = 10, b=1.5 in the direction of y.
6000 5000
U = 130 U = 160 U = 180
4000 u(y) 3000 2000 1000 0 0
0.5
1 y
1.5
2
Figure 6: The effect of U on velocity component u for A1=A2=A3=B1=B2= 1, a = 0, n = 10, b=1.5 in the direction of y. A1=A2=A3=B1=B2= 1, a = 0, n = 10, b=1.5 in the direction of y.
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110
0 -200 -400 v(x)
-600 -800
-1000 -1200 -1400 0
U = 10 U = 30 U = 40 0.5
1 1.5 2 x Figure 7: The effect of U on velocity component v for A 1= A 2 =A 3= B 1= B2 =1, n = 10, b = 0, a = 1.5 in the direction of x.
0 -500 -1000 -1500 v(x)-2000 -2500 -3000 -3500 -4000 -4500 0
U = 90 U = 120 U = 150 0.5
1 1.5 2 x Figure 8: The effect of U on velocity component v for A 1= A 2 =A 3= B 1= B 2 =1, n = 10, b = 0, a = 1.5 in the direction of x.
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60 b = 1.5 b = 1.7 b=2
50 40 u(y) 30 20 10 0 0
0.5
1 1.5 2 y Figure 9: The effect of b on velocity component u for A 1=A 2=A 3=B 1=B 2=U=1, a = 0, n = 1 in the direction of y.
50 45 40
U=2 U = 2.5 U=3
35 30 u(y) 25 20 15 10 5 0 0
0.5
1 y
1.5
2
Figure 10: The effect of U on velocity component u for A 1=A 2=A 3=B 1=B 2=1, a = 0, n = 1, b=1.5 in the direction of y.
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112
0 -5 -10
U=2 U = 2.5 U=3
-15 -20 v(x) -25 -30 -35 -40 -45 -50 0
0.5
1 1.5 2 x Figure 11: The effect of U on velocity component v for A 1= A 2 =A 3= B 1= B 2 = n = 1, b = 0, a = 1.5 in the direction of x.
0 -10
a = 1.5 a = 1.7 a=2
-20 v(x) -30 -40 -50 -60 0
0.5
1 1.5 2 x Figure 12: The effect of a on velocity component v for A 1= A 2 =A 3= B 1= B 2 =U= n = 1, b = 0 in the direction of x.
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1400 m=2 m = 2.5 m=3
1200 1000 u(y)
800 600 400 200 0 0
0.5
1 1.5 2 y Figure 13: The effect of m on velocity component u for A 1=A 2=A 3=C1=C2=U=1, a = 0, b = 1 in the direction of y.
4
12
x 10
10
b = 2.1 b = 2.3 b = 2.5
8 u(y)
6 4 2 0 0
0.5
1 y
1.5
2
Figure 14: The effect of b on velocity component u for A 1=A 2=A 3=C1=C2=U=1 , a = 0, m =2 in the direction of y.
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2000 U=5 U = 10 U =15
1800 1600 1400 1200 u(y) 1000 800 600 400 200 0 0
0. 1 1. 2 y 5 5 Figure15: The effect of U on velocity component u for A1= A2= A3=C1= C2=1, b = 1.5, m = 2, a = 0 in the direction of y.
0 -100 -200 -300 v(x) -400 -500 -600 -700 0
m = 2.3 m = 2.5 m = 2.7 0.5
1 1.5 2 x Figure 16: The effect of m on velocity component v for A 1=A 2=A 3=C1=C2=U=a=1, b= 0 in the direction of x .
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115
0 -200 -400 v(x)
-600 -800
-1000 -1200
a = 1.3 a = 1.4 a = 1.5
-1400 0
0.5
1 1.5 2 x Figure 17: The effect of a on velocity component v for A 1=A 2=A 3=C1=C2=U=1, m= 2, b= 0 in the direction of x.
0 -200 -400 -600 v(x) -800 -1000 -1200 -1400 -1600 -1800 0
U=2 U=7 U = 10 0.5
1 x
1.5
2
Figure 18: The effect of U on velocity component v for A 1=A 2=A 3=C1=C2=1, m= 2, a = 1.5, b= 0 in the direction of x.
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116
200 0 -200 u(y)
-400 -600 -800
-1000 -1200 0
b = 1.5 b = 1.7 b = 1.9 0.5
1 1.5 2 y Figure 19: The effect of b on velocity component u for A 1=A 2=A 3=C1=C2=1, m= 1, a=0, U=10 in the direction of y.
100 0 -100 -200 -300 u(y) -400 -500 -600 -700 -800 -900 0
U = 10 U = 20 U = 30 0.5
1 y
1.5
2
Figure 20: The effect of U on velocity component u for A 1=A 2=A 3=C1=C2= m=1, b= 1.5, a =0 in the direction of y.
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117
1200 1000
a = 1.5 a = 1.7 a = 1.9
800 v(x)
600 400 200 0 -200 0
0.5
1 1.5 2 x Figure 21: The effect of a on velocity component v for A 1=A 2=A 3=C1=C2= m=1, b= 0, U=10 in the direction of x.
900 800 700
U =10 U =20 U =30
600 500
v(x)40 0 300 200 100 0
-1000
0.5
1
1.5
2
x Figure 22: The effect of U on velocity component v for A1=A2=A3=C1=C2= m=1, b= 0, a=1.5 in the direction of x.
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