Mechanics, Materials Science & Engineering, May 2016
ISSN 2412-5954
MHD Stagnation Point Flow in a Boundary Layer Of a Nano Fluid Over a Stretching Sheet in the Presence of Viscous Dissipation and Chemical Reaction Ch. Achi Reddy 1,a & B. Shankar 2 1
M.L.R. Institute of Technology, Dundigal, Hyderabad, 500 043, India
2
Professor, Department of Mathematics, Osmania University, Hyderabad, 500 007, India
a
achireddy.ch@gmail.com DOI 10.13140/RG.2.1.2022.4881
Keywords: boundary layer flow, exponentially stretching sheet, chemical reaction and viscous dissipation.
ABSTRACT. The paper shows an attempt of numerical investigation on the effect of viscous dissipation and Chemical reaction on a viscous, steady and incompressible fluid over an exponentially stretching sheet within a specified boundary layer. As a formal approach, the model has been adopted with the governing equations and the simulation is carried out with the Keller Box method. The pattern or the profiles of the skin friction coefficient and the heat and mass transfer rates are achieved in execution of mathematical model have been presented in the paper. The enhancement in magnetic parameter leads to a considerable reduction in velocity and Chemical reaction parameter is predominant in controlling the profile of concentration. An increase in Eckert number is observed to cause the enhancement in the temperature profile whereas it decreases the concentration profile. The results obtained in the simulation of Keller box method are in well agreement with realistic situation of the scientific scenario.
1. Introduction. The importance of stagnation point flow has drawn the attention of many researchers due to its growing application in industry. The fluid is said to have reached its stagnation point when local Velocity of the fluid becomes zero. In some situations, flow is stagnated by a solid wall while in others; there is a line interior to a homogeneous fluid domain or the interface between two immiscible fluids [1-3]. A good amount of research is done drawing the attention of several researchers [4-12]. In 1993, during an investigation of new coolants and cooling technologies at Argonne National Laboratory in U.S. Chai invented a new type of fluid called Nano fluid [13]. Nano fluids are fluids that contain small volumetric quantities of nanometre sized particle, Called nanoparticles. The nanoparticles used in Nano fluids are typically made of metals, oxides, carbides, or carbon nanotubes. Common base fluids include water, ethylene glycol and oil. Nano fluids commonly contain up to a 5% volume fraction of nanoparticles to see effective heat transfer enhancements [25-27]. Nano fluids are studied because of their heat transfer properties: they enhance the thermal conductivity and convective properties over the properties of the base fluid. Typical thermal conductivity enhancements are in the range of 15-40% over the base fluid and heat transfer coefficient enhancements have been found up to 40%.Increasing the thermal conductivity of this magnitude cannot be solely attributed to the higher thermal conductivity of the added nanoparticles, and there must be other mechanisms attributed to the increase in performance [28-29]. Stagnation point flow appears in virtually all fields of science and engineering. A flow can be stagnated by a solid wall or a free stagnation point or a line can exist in the interior of the fluid Domain. The study of stagnation point flow as pioneered by Hiemenzin 1911 [15] who solved the two dimensional stagnation point problem using a similarity transformation.
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Magyari and Keller [16] investigated the steady boundary layers flow on a stretching continuous surface with exponential temperature distribution while Partha et al. [17] analysed the effects of viscous dissipation on the mixed convection heat transfer from an exponentially stretching surface. In the present study we have investigated the viscous dissipation for different values of velocity ratio parameter and observed that the nanoparticle volume decreases with the increase of chemical reaction parameter for = 0.1 and 2.1. 2. Mathematical Formulation. Consider a steady, two-dimensional boundary layer stagnation-point are assumed to be of the forms , and , respectively. Where a > 0 and b are constants, x is the coordinate measure along the stretching surface and is length of the sheet. A nonis imposed parallel to the y-axis, where
o polarization of charges is negligible [22]. Figure (1) shows that the temperature T and the Nanoare obtained when y
The governing boundary layer equations of the conservation Law of mass, momentum, energy and
(1)
(2)
(3)
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(4)
Here u and v are the velocity components in the x and y directions respectively, is the viscosity; is the electrical conductivity.
,
where k is the thermal conductivity; is the heat capacitance of the base fluid.
,
where
is the heat capacitance of the nanoparticle; DB
is the Brownian di usion coe cient;
DT
is the thermophoresis di usion coe cient;
(5)
where
is the Stefan-Boltzmann constant; k
is the mean absorption coe cient.
It is assumed that the temperature di erence between the free stream T and local temperature T is small enough expanding T4 in a Taylor series about T and neglecting higher order terms results.
(6)
After substituting Eqs. (5) and (6) in Eq. (3), it will be reduces to
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Mechanics, Materials Science & Engineering, May 2016
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(7)
The subjected boundary conditions are
(8)
The prescribed temperature and concentration on the surface of the sheet is assumed to be T (x) = T + T0ex/2l and C (x) = C + C0ex/2l where T0, C0 are the reference temperature and concentration respectively, now, the non-linear partial di erential equations for the purpose of a stream function
,
(9)
follows
(10)
As such Eq. (10), Eqs. (2), (4) and (7) reduce to the following system of nonlinear ordinary di erential equations.
(11) (12) (13)
where (14)
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Mechanics, Materials Science & Engineering, May 2016
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, ,
where, prime denote the di erentiation with respect to
,
is the velocity ratio parameter;
v Pr is the Prandtl number; Le is the Lewis number; M is the magnetic parameter; A is the heat source parameter, S
is the suction parameter. ,
where is the radiation parameter, , where Nb is the Brownian motion parameter and Nt is the thermophoresis parameter; Ec Eckert number; is chemical reaction parameter; The corresponding boundary conditions Eq. (8) are transformed into f ( ) 0, f ( ) 1, ( ) 1, ( ) 1 at f ( ) , ( ) 0, ( ) 0 as
0,
(15)
The parameters of practical interest in the formulated problem are velocity, heat and mass transfer respectively, which are presented in terms of Skin friction Cf, Nusselt number Nu and Sherwood numbers Sh. Using the transformed variables (10), the non-dimensional expressions for the Skin friction coe cient , the reduced Nusselt number and the reduced Sherwood number Skin friction Coe cient. The Skin friction coe cient
,
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Mechanics, Materials Science & Engineering, May 2016
where
ISSN 2412-5954
is the local wall Shear stress;
Nusselt Number of convection to conductive heat transfer across the boundary
(16)
Sherwood Number. The Sherwood number (Sh) is also called the mass transfer Nusselt number. It represents the ratio of convective to di usive mass transport
,
where
is the local Reynolds number based on the stretching velocity.
3. Numerical Procedure. The equations (11) - (14) subject to the boundary conditions (15) are solved - di erence scheme known as Keller box method. The method has the following four basic steps. 1. Reduce equations (11) 2. Write the di erence equations using central di erences. 3.
-vector.
4. Use the Block - tridiagonal elimination technique to solve the linear system. 4. Results and Discussion.
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Mechanics, Materials Science & Engineering, May 2016
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Table 1. Comparison of the reduced Nusselt number [19]
[20]
[21]
Present results
0
0.9548
0.9548
0.9548
0.9548
0
0
1.4714
1.4714
1.4714
1.4715
3
0
0
1.8691
1.8691
1.8691
1.8692
1
0
1.0
0.5315
0.5312
0.5312
0.5311
1
1.0
0
---
0.8611
0.8611
0.8611
1
1.0 1.0
---
0.4505
0.4505
0.4505
Pr
M
N
1
0
2
Table 2. Values of the reduced Nusselt number , reduced Sherwood number skin friction coefficient for various values of Eckert number. Nb = Nt = M = Ec
= = 0.1
Le
N
0.1
10
1.0
0.5176
3.6996
1.2856
0.2
10
1.0
0.4976
3.7107
1.2856
0.3
10
1.0
0.4776
3.7219
1.2856
0.4
10
1.0
0.4375
3.7442
1.2856
Table 3. Values of the reduced Nusselt number , reduced Sherwood number skin friction coefficient for various values of Chemical Reaction parameter. Nb = Nt = M = Pr
, and the
, and the
= EC= 0.1
Le
N
0.1
1.00
10
1.0
0.5176
3.6996
1.2856
0.2
1.00
10
1.0
0.5175
3.8407
1.2856
0.3
1.00
10
1.0
0.5174
3.9757
1.2856
0.4
1.00
10
1.0
0.5172
4.1055
1.2856
The system of ordinary differential equations [11-13] has been solved numerically using Keller-box method. From the numerical computation, the main physical quantities of interest namely the local Skin friction coefficient, the local Nusselt number and the local Sherwood number are obtained and the results are presented in Table 2 and Table 3. From Table 2 it is observed that with the increase in Eckert number, there is a decrease in rate of heat transfer and increase in mass transfer. From Table 3 it is observed that with increase in chemical reaction parameter, there is no significant change in rate of heat transfer but there is an increase in the rate of mass transfer. Figure 2 shows the effects of the magnetic Parameter M on the flow field velocity different values of the Velocity ratio parameter , =0.1, 1 and 2.1. MMSE Journal. Open Access www.mmse.xyz
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for three
Mechanics, Materials Science & Engineering, May 2016
ISSN 2412-5954
Fig.2. Velocity profile against for different values of M
When =0.1 the velocity profile and the boundary layer thickness decrease with an increase in M. When =2.1, higher value of the Lorentz force further reduces the velocity and consequently the thickness of boundary layer reduces. When =1, there is no influence of magnetic field and attains a constant value of 1 for any value of indicating that there is no boundary layer of fluid, as shown by a dotted line in Fig 2. This means that in the case when the external stream velocity becomes equal to the stretching velocity. The flow is not influenced by the different values of the incorporated flow parameters. When =2.1 the flow velocity increases indicating the decrease in thickness of boundary layer with an increase in M. As compared to =0.1case, the boundary layer thickness decreases causing an inverted boundary structure.
Fig. 3. Temperature profile against for different values of Ec
Figure 3 shows an increase in Nano fluid Temperature with increases in the viscous dissipation parameter, Eckert number. Which can be attributed to the action of viscous heating. Concentration increases with an increase in the viscous dissipation parameter, Eckert number as shown in Fig. 4
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Mechanics, Materials Science & Engineering, May 2016
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Fig. 4. Concentration profile against for different values of Ec
Fig. 5. Concentration profile against for different values of
The influence of chemical reaction parameter on concentration profile is shown in Fig. 5. Concentration decreases with an increase in the chemical reaction parameter indicating that the nanoparticle volume fraction decreases with the increase of chemical reaction parameter, while effect chemical reaction parameter is not significant on the temperature profile.
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Fig. 6. Temperature profile profile against for different values of N
Summary. A numerical study corresponding to the flow and heat transfer in a steady flow region of Nano fluid over an exponential stretching surface and effects of Chemical reaction parameter and Eckert number are examined and discussed in detail. The main observations of the present study are as follows. (I) An increase in the magnetic parameter is to reduce the velocity profile; (II) An increase in the Eckert number increases the temperature profile while it reduces the concentration profile; (III)With increasing values of Chemical reaction parameter ( ) the concentration profile decreases. References [1] -4. [2] T.R. Mahapatra and A.S. Gupta. Heat transfer in stagnationHeat Mass Tran. 2002; 38, 517-21.
eet.
[3] T.R. Mahapatra and A.S. Gupta. StagnationEng. 2003; 81, 258-63. [4] stretching sheet. Int. J. Nonlinear Mech. 2004; 39, 1227-35. [5] Y.Y. Lok, N. Amin and I. Pop. NonInt. J. Nonlinear Mech. 2006; 41, 622-7. [6] S. Nadeem, A. Hussain and M. Khan. HAM solutions for boundary laye stagnation point towards a stretching sheet. Comm. Nonlinear Sci. Numer. Simulat. 2010; 15, 47581. [7] F. Labropulu, D. Li and I. Pop. Non-orthogonal stagnationin a nontransfer. Int. J. Therm. Sci. 2010; 49, 1042-50. [8] A. Ishak, Y.Y. Lok and I. Pop. StagnationChem. Eng. Comm. 2010; 197, 1417-27. [9] N. Bachok, A. Ishak and I. Pop. Boundary-8.
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[10] stretching surface. Int. J. Appl. Math. Mech. 2011; 7, 31-7. [11] N. Bachok, A. Ishak and I. Pop. On the homogeneous-heterogeneous reactions e ects. Comm. Nonlinear Sci. Numer. Simulat. 2011; 16, 4296-302. [12] S.U.S. Choi, Z-G. Zhang, W. Yu, F.E. Lockwood and E.A. Grullce, Appl. Phys. Lett. 79, 2252 (2001). [13] E. Vol. 42, no. 5, pp. 577-585, 1999. [14]
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[27] Transfer, vol. 20, no. 3, pp. 417-430, 1993. [28] F.H. Hady, E.S. Ibrahim, S.M. Abdel-
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