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Numerical Solution of Uniform Suction/ Blowing Effect on

Newtonian Fluid Flow Due to a Stretching Cylinder

Dr. M. Shafique1, Dr. Farooq Ahmad 2, Sajjad Hussain3, Sifat Hussain 4 1 Department of Mathematics, Gomal University, D. I. Khan, Pakistan

2 Mathematics Department, Government Degree College Darya Khan (Bhakkar) 30000, Punjab, Pakistan 3, 4 Centre for Advanced Studies in Pure and Applied Mathematics, B. Z. Uni., Multan, Pakistan

Abstract -- The numerical solution of uniform suction /blowing effect on flow and heat transfer due to a stretching cylinder is investigated for Newtonian fluids. The partial differential equations are converted in to ordinary differential equations by using similarity transformations. The resulting equations are solved by using successive over relaxation method and Simpson’s (1/3) rule. The results have been obtained for various values of the parameters involved in the equations of motion, namely Reynolds number R, suction/ injection parameter γ and Prandtl number Pr, to study their effect on velocity and temperature fields as well as on the Nusselt number and the skin friction coefficient. The results are given both in tabular and graphical forms.

Key Words – SOR Method, Uniform Suction / Blowing, Stretching Cylinder.

I. INTRODUCTION

Fluid dynamics due to a stretching surface is important in extrusion processes. Crane [1] discussed a closed form exact solution of the Navier-Stokes equations subject to two dimensional stretching of a flat surface. Brady and Acrivos [2] examined the exact similarity in solutions of a flow inside a stretching channel and inside a stretching cylinder. Chiam [3] investigated steady two dimensional stagnation point flow of an incompressible fluid towards a stretching surface. Chen [4] analyzed mixed convection cooling of heated stretching surface. Ali and Al-Yousef [5] considered laminar mixed convection boundary layers induced by a linearly stretching permeable surface. Ali [6] studied the effect of variable viscosity on mixed convection heat transfer along a vertical moving surface. Mahaputra and Gupta [7] analyzed steady, two dimensional stagnation point flow of viscoelastic fluid towards a stretching surface.

A solid surface induces effects on the flow field around an immersed cylinder. Several researchers including Bosh and Rodi [8], Choi and Lee [9,10], Cigada et al.[11], Malavasi and Guadagnini[12] Malavasi and Zappa[13] have considered this phenomenon under various situations. Also, Ishak et al. [14] studied the uniform suction /blowing effects on fluid flow and heat transfer due to a stretching cylinder.

In the present work, we extended the work of Ishak et al. [14] for rang

0.1

≤ ≤

R

20

and obtained numerical solutions of the fluid flow problem by using SOR method and Simpson’s (1/3) rule with the formula of Adams-Moulton. Our numerical scheme is more stable for the extended range of R

than that of previous work. The results have been computed for several values of the parameters namely, R the Reynolds number, Pr the Prandtl numbers and

γ

the suction/ injection parameter.

Ishak [14] solved this problem by using Keller Box method for the range

0.5

≤ ≤

R

10

. The Keller Box method is more complicated than our simplest numerical scheme. Also, the previous method is not efficient because it works for smaller range of R than that of our numerical techniques. The accuracy of the results is checked very carefully by performing calculations on three different grid sizes.

II. BASIC ANALYSIS

Navier-Stokes equation and continuity equation for steady and incompressible fluid flow in the absence of the body force are given by

2

1

(

)

V

V

V

π

υ

ρ

− ∇ = − ∇

+

⋅∇

,

0

=

V

, and the energy equation is

2 2

2 2

1

( ) ( )

p

T T T T T

c u w k

r z r r r z

ρ

∂ + ∂ = ∂ + ∂ +∂

∂ ∂

Where

ρ

, υ,

π

and V are density, kinematics viscosity, pressure and velocity of the fluid respectively. It is assumed that heat dissipation function

Φ

=

0

A stretching tube of radius a in the axial direction, lying in a fluid at rest causes the fluid flow. The axis of tube is taken in the z-direction and r-axis is measured in the radial direction.

( , )

V V u w

=

is the velocity field and the velocity

(2)

48 

denotes the temperature of ambient fluid where

T

w

>

T

. The body force and the viscous dissipation are neglected.

Under these assumptions, the set of equations become:

0,

u u

w

r

r

z

+ +

=

(1)

2 2

2 2 2

1

,

u u u u u u

u w

r z r r r r r z

π

ρ⎛ ∂ + ∂ ⎞= −∂ +μ⎛∂ + ∂ − +∂ ⎞

∂ ∂ ∂ ∂ ∂ ∂ ⎝ ⎠ ⎝ ⎠ (2) 2 2 2 2 1 ,

w w w w w

u w

r z z r r r z

π

ρ⎛ ∂ + ∂ ⎞= −∂ +μ⎛∂ + ∂ +∂

∂ ∂ ∂ ∂ ∂ ∂

⎝ ⎠ ⎝ ⎠ (3)

2 2 2 2 1 , p T

T T T T

C u w K

r z r r r z

ρ

⎛ ∂ + ∂ ⎞= ⎛⎜∂ + ∂ +∂ ⎟⎞

∂ ∂ ∂ ∂

⎝ ⎠ (4)

The equations (1) to (4) are solved subject to the following boundary conditions:

, , , when ,

0, , when ,

w w w

u u w w T T r a

w T T r

= = = =

⎨ = = → ∞

⎩ (5)

where uw=−ca

γ

,ww=2cz with c is a positive constant of dimension [1/Time] and

γ

is a constant and it corresponds to mass injection for

γ

<

0

and mass suction forγ >0.

The following similarity transformations are used to obtain the governing equations in their ordinary differential form

( )

, 2 ( ) and ( ) ,

w

f T T

u ca w cf z

T T

η

η θ η

η ∞ ∞ ⎧ = − = = ⎨ − ⎪⎩ (6)

where r 2 a

η= ⎜ ⎟⎛ ⎞

⎝ ⎠

is dimensionless variable. The prime denotes

the differentiation with respect to

η

.

The mass conservation equation (1) is satisfied. The equations (2) to (4) yield.

2

(

),

f

′′

+

η

f

′′′

=

R f

ff

′′

(7)

(1

R

Pr )

f

0,

ηθ

′′

+ +

θ

=

(8) where 2

2 ca R

ν

= , Pr Cp

K

ρν

= and ν μ ρ

= is the coefficient of kinematics viscosity.

The boundary conditions (5) become:

, 1, 1, 1,

0, 0, .

f f

f

γ

θ

η

θ

η

= = = =

⎨ ′ = = → ∞

(9)

The pressure π can be found from equation (2) as:

2 2

2( ) 2 ( ).

2

c a f c f

ρ

π π η ρ ν η

η

∞ ′

= − −

In order to obtain numerical solution of the equations (7) and (8) with faster convergence, the variable

η

is compressed as in [6]

, so that 1 as 0.

x

e x

η

=

η

→ → (10) and we obtain

(

2

)

2

,

xxx x xx x xx x

f

+

f

f

=

R f

ff

+

ff

(11)

Pr

0.

xx

R

f

x

θ

+

θ

=

(12) The associated boundary conditions become:

,

1,

1,

0,

0,

0,

,

x x

x

f

f

x

e f

x

γ

θ

θ

=

=

=

=

⎧⎪

=

=

→ ∞

⎪⎩

(13)

where the suffixes denote the differentiation with respect to x. III. FINITE DIFFERENCE EQUATIONS In order to solve the equation (11) numerically, we put

.

x

f

=

P

(14) The equations (11) and (12) become

(

2

)

2 ,

Pr 0.

xx x x

xx x

P P P R P fP Pf

R f

θ

θ

+ − = + ⎪ ⎨ + = ⎪⎩ (15)

The boundary conditions (3.13) become

, 1, 1, 0,

0, 0, .

x

f P x

e P x

γ

θ

θ

− = = = = ⎧ ⎨ = = → ∞

⎩ (16)

We approximate the equations in (15) by using central difference approximation at a typical point

n

x

x

=

of the interval [0,

), we obtain

1 1 1 1 1 1

2 2

2( ) 2 ( ) ( )

2 ( ) 4 2 ,

n n n n n n n

n n n n n

P P h P P Rhf P P

h P f RP P h P

+ + − − + − − + + − −

⎧⎪

= + +

⎪⎩ (17)

1 1 1 1

2(

θ

n+

+

θ

n

)

+

R

Pr

f h

n

(

θ

n+

θ

n

) 4 ,

=

θ

n (18) where

h

denotes the grid size and the symbols used denote

( )

n n

f = f x ,Pn =P x( )n , and θ θn= ( )xn . For computational purposes, we replace the interval [0, ∞) by [0, β) where β is sufficiently large.

IV. COMPUTATIONAL PROCEDURE

(3)

49 

The order of the sequence of iterations is as follows: 1. The equations (17) and (18) are solved to calculate the

values of P and

θ

subject to the boundary conditions:

1, 1, when 0,

0, 0, when .

P x

P x

θ

θ

= = =

= = → ∞

2. The computed solutions of P are then employed into the equations (14) for the calculation of f with the condition:

when 0.

f =

γ

x=

3. In order to accelerate the speed of convergence of the SOR method, the optimum value of the relaxation parameter

ω

optis estimated by the formula

2

2 .

1 1

opt

ω

ρ

= + −

where the range of

ω

opt is 1<

ω

opt <2

and ρ denotes the spectral radius of the associated Jacobi iteration matrix.

4. The above procedure is repeated until convergence is obtained according to the criterion

1 6

maxUn+ Un <10− where n denotes the number of

iterations and U stands for each of the functional value. For higher order accuracy, the above steps 1 to 4 are repeated for step sizes

2 hand

4

h.

V. RESULTS AND DISCUSSION

The effects of the flow parameters namely R, γ and Pr

have been examined for the velocity and temperature profiles. The results have been presented in tabular as well graphical forms for several values of these parameters.

The comparison of the present results with the previous results by Ishak et al. [14] is given in Table 1 to Table 3. The Table 2 shows that all the values of f′′(1)are negative that means, the fluid is under the action of a dragging force due to stretching surface. Figure 1 to Figure 3

demonstratef(η)for γ = −0.5,0,0.5 respectively for various

values of R. Figure 4 to Figure 6 show f′(η) for

0.5,0,0.5

γ

= − respectively for different values of R. Figure 7 depicts f′(η)for R=10 and for different values of

1.2, 1,1 and 1.2

γ

= − − . It is noted that

f

(

η

)

decreases for increasing values of R irrespective of γ is positive, negative or zero as shown in Figure 1 to Figure 3. The velocity gradient also increases for increasing values of R as can be seen in Figure 4 to Figure 6. Figure 7 depicts that the velocity gradient at the surface increases for increasing values of γ and it implies that the wall shear stress also increases. It is worth mentioning that the velocity field is not affected by Prandtle number Pr.

The Figure 8 and Figure 9 demonstrate the temperature distributions for Pr 0.7= (such as air) and Pr 7= (such as water) for a fixed value of R and different values of γ. Also, the Figure 10 and Figure 11 demonstrate the temperature distributions for Pr 0.7= (such as air) and Pr 7= (such as water) for a fixed value of γ and different values of R. It is found that θ decreases for increasing values of γ. Also, θ

[image:3.612.98.511.573.691.2]

decreases for increasing the distance from the surface and then becomes zero at some larger distance from the surface. For increasing values of γ, the thermal boundary layer thickness decreases for some fixed value of γ. The behavior of temperature profiles for injection is different from that of suction. It is noted that −θ′(1)decreases to zero for larger values of Pr in case of injection. It means that there is no heat transfer at the surface. Thus the wall shear stress and heat transfer rate at the surface can be reduced by injection.

Table 1: Comparison of

f

(

)

for possible values of Reynolds number R.

(4)

50 

Table 2: Comparison of − f ′′(1) for possible values of Reynolds number R.

γ

Present R=2.0 Ishak et al Present R=5.0 Ishak et al Present R=10.0 Ishak et al -1.2 0.798736 --- 0.838972 --- 0.843940 --- -1.0 0.890127 --- 0.971818 --- 0.995908 --- -0.5 1.184350 1.1810 1.485243 1.4811 1.682201 1.6776

0.0 1.595297 1.5941 2.410499 2.4175 3.318044 3.3445 0.5 2.141963 2.1468 3.882652 3.9308 6.435548 6.6222 1.0 2.818490 --- 5.774575 --- 10.371458 --- 1.2 3.118564 --- 6.591984 --- 11.996690 ---

Table 3: Compression of −θ ′(1) for possible values of

γ

and R.=10.

Pr

Present

γ

=

0

.

Ishak et al

0

Present

γ

=

0

Ishak et al

.

5

Present

γ

=

0

.

5

Ishak et al

0.1 1.050543 --- 0.873923 --- 1.546597 --- 0.7 1.702547 1.5683 0.406289 0.2573 4.276681 4.1961 2.0 3.026962 3.0360 0.067486 0.0600 11.012555 11.1517 7.0 6.155753 6.1592 0.000476 0.0000 35.076481 36.6120 10.0 7.462454 7.4668 0.000000 0.0000 48.662162 51.7048 15.0 9.258509 --- 0.000000 --- 70.205160 ---

-0.5 0 0.5 1 1.5

1 2 η 3 4

[image:4.612.62.523.135.649.2]

f

Figure 1: The similarity profile f(

η

) for γ=-0.5 and

R=0.1, 2, 5, 10 and 20 from top to bottom.

0 0.4 0.8 1.2 1.6

1 1.4 1.8 2.2 2.6

f

[image:4.612.338.526.419.622.2]

η

Figure 2: The similarity profile f(η) for γ=0.0 and

(5)

51 

0 0.5 1 1.5 2

1 1.4 1.8 2.2 2.6

f

[image:5.612.355.553.153.359.2]

η

Figure 3: The similarity profile f(η)for γ=0.5 and

R=0.1, 2, 5, 10 and 20 from top to bottom.

0 0.2 0.4 0.6 0.8 1

1 2 3 4

η

5 6 7 8

[image:5.612.79.278.153.354.2]

f'

Figure 5: The velocity profile f′( )η for γ=-0.5 and R=0.1, 2, 5 and 20 from top to bottom.

0 0.2 0.4 0.6 0.8 1

1 2 3 4

η

5 6 7 8

[image:5.612.79.279.433.632.2]

f'

Figure 4: The velocity profile f′( )η for γ=0.0 and R=0.1, 2, 5 and 20 from top to bottom.

0 0.2 0.4 0.6 0.8 1

1 2 3 4 η 5 6 7 8

f'

Figure 6: The velocity profile f′( )η for γ=0.5 and

[image:5.612.355.553.440.638.2]
(6)

52  0

0.2 0.4 0.6 0.8 1

1 2 3 4 η 5 6 7 8

[image:6.612.81.287.160.356.2]

f'

Figure 7: The velocity profile ( )f′η for R=10 and

γ=-1.2,-1, 0 and 1.2 from top to bottom.

0 0.2 0.4 0.6 0.8 1 1.2

1 2 η 3 4

[image:6.612.353.557.163.362.2]

θ

Figure 8: Temperature profile

θ

(

η

)

for R=10, Pr=0.7 and γ=-0.5, 0, 0.5 from top to bottom

0 0.2 0.4 0.6 0.8 1 1.2

1 2

η

3 4

[image:6.612.353.558.446.658.2]

θ

Figure 9: Temperature profile

θ

(

η

)

for R=10, Pr=7 and γ=-0.5, 0, 0.5 from top to bottom.

0 0.2 0.4 0.6 0.8 1

1 2

η

3 4

θ

[image:6.612.79.289.449.657.2]
(7)

53 

0 0.2 0.4 0.6 0.8 1

1 2

η

3 4

[image:7.612.58.576.128.713.2]

θ

Figure 11 Temperature profile θ(η) for γ=0.5, Pr=7

and R=1, 6, 12 from top to bottom.

VI. REFERENCES

[1] I. J. Crane, Flow past a stretching plate Zeit. Angew. Math. Phys., 21(1970) 645-647.

[2] J. F. Brady and A. Acrivos, Steady flow in a channel or tube with an accelerating surface velocity. An exact solution to the Navier-Stokes equations with reverse flow. J. Fluid Mech., 112 (1982) 127-150.

[3] T. C. Chiam, Stagnation point flow towards a stretching plate, J. Phys. Soc. Japan, 63 (1994) 2443-2444.

[4] C. H. Chen, Mixed convection cooling of heated continuously stretching surface, Heat Mass Transfer, 36 (2000) 79-86.

[5] M. E. Ali and F. Al-Yousef, Laminar mixed convection boundary layers induced by a linearly stretching permeable surface, Int. J. Heat Mass Transfer, 45(2002) 4221-4250.

[6] M. E. Ali, The effect of variable viscosity on mixed convection heat transfer along a vertical moving surface, Int. J. Thermal Sci., 45 (2006) 60-69.

[7] T. R. Mahaputra and A. S. Gupta, Stagnation-point flow of viscoelastic fluid towards a stretching surface, Int. J. Nonlinear Mech., 39, (2004) 811-820.

[8] G. Bosh and W. Rodi, Simulation of vortex shedding past squar cylinder near a wall, Int. J. Heat Fluid Flow, 17 (1996) 267-275.

[9] J. H. Choi and S. J. Lee, Ground effect of flow around an elliptic cylinder in a turbulent boundary layer, J. Fluids Struct., 14 (2000) 679-709.

[10] J. H. Choi and S. J. Lee Flow characteristics around an inclined elliptic cylinder in a turbuleat boundary layer, J. Fluids Struct., 15 (2001)1123-1135.

[11] A. Cigada, S. Malavasi and M. Vanali, Effects of an asymmetrical confined flow on a rectangular cylinder, J. Fluid Struct., 22 (2006) 213-227.

[12] S. Malavasi and A. Guadagnini, Interaction between a rectangular cylinder near a solid wall, J. Fluid Struct., 23(8) (2007) 1137-1148.

[13] S. Malavasi and E. Zappa, Fluid-dynamic forces and wake frequencies on a tilted rectangular cylinder near a solid wall, Meccanica, 44 (1)(2009) 91-101.

[14] Ishak, A., Nazar, R. and Pop, I. (2008), Uniform suction / blowing effect on flow and heat transfer due to a stretching cylinder. Applied Mathematical Modelling, 32, 2059 – 2066

[15] C. F. Gerald, Applied Numerical Analysis, Addison-Wesley Pub. Co. N. Y. (1989).

[16] W. E. Milne, Numerical Solution of Differential Equations, Dover publications, Inc., New York (1970).

[17] G. D. Smith, Numerical Solution of Partial Differential Equations, Clarendon Press, Oxford (1979).

Figure

Table 1: Comparison of f(∞) for possible values of Reynolds number R.
Figure 2: The similarity profile f(η) for γ=0.0 and R=0.1, 2, 5, 10 and 20 from top to bottom
Figure 4: The velocity profile f η′( )for γ=0.0 and R=0.1, 2, 5 and 20 from top to bottom
Figure 10: Temperature profile  θ(η) for  γ=-0.5, Pr=7
+2

References

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