J.Basic.Appl.Sci.Res., 8(5)1-7, 2018 ISSN 2090-4304
Journal of Basic and Applied Scientific Research
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Zanariah Mohd Yusof, Noraini Ahmad, Nur Idalisa Norddin,Nur Huzainiee Mat Yuzut, Ruhana Jaafar
FacultyofComputer andMathematicalSciences, Universiti TeknologiMARA,Terengganu,Malaysia
Received: March21, 2018 Accepted:June13, 2018
ABSTRACT
The study of boundary layer theory over an exponentially stretching sheet has become important in the recent year due to development in industrial applications that includes both metal and polymer sheets. The additions of parameters such as magnetic, thermal radiation and porous medium have proved to be significantly affecting the momentum and energyproduced by the fluid motion. By considering the boundarylayer flow over an exponentially stretching sheet, the presence of magneto hydrodynamic, thermal radiation and mixed convection are studied. The governing partial differential equations are reduced to ordinary partial equations by applying the similarities transformation. MATLAB software is used to solve the large matrix of Keller-Box method because of the compatibility to run the large data decomposition for the parallel implementation. It is found that the mixed convection increases both fluid movement andheat transfer rate from the exponentiallystretching sheet to the fluid. KEYWORDS: BoundaryLayer, MHD, Keller-Box,Permeability.
INTRODUCTION
The boundary layer flow and heat due to an exponentially stretching surface is important in industrial manufacturing processes that includes both metal and polymer sheets. Study of the flow and heat transfer can be of significant importance since the quality of the final product depends to a large extent on the skin friction and the heat transfer rate at the surface [1]. The magnetic field and radiation have been found significantlyinfluenced bythe thickness of the velocity and thermal boundary layers due to their effects on the viscous drag, temperature distribution and the parameters in the equations of boundary layer problems [2]. In order to achieve the desired characteristics of the final product, it is important to study the influence of the magnetic field and radiation effect in controlling the problem occurs in industrial manufacturing processes. Therefore, it is necessary to study the MHD boundarylayer problemandheat transfer over an exponentiallystretching sheetwith magnetic and radiation effect. It is supported by numerous researchers whose have investigated the heat transfer over stretching surface by considering the effect of magnetic field and radiation where the radiative effects have important applications in physics and engineering particularly in space technology and high temperature processes [3]. The effect of thermal radiation on the forced and free convection flow also important in the process at high temperature [4]. The MHD non-Darcy flow and heat transfer characteristics over a stretching sheet with the presence of thermal radiation and ohmic dissipation was studied [5]. It is found that the thermal boundary layer thickness increase due to the effect of thermal radiation. The steady hydromagnetic two dimensional flow and heat transfer in a stationary electrically conducting and heat-generating fluid driven bya continuously moving porous surface immersed in a fluid-saturated porous medium was analyzed [6]. It was shown the heat transfer characteristics can be enhanced by the porous medium. An exact solution of thermal radiation on magneto hydrodynamics flow over a stretching porous sheet was obtained by [7]. The study about MHD mixed convective flow and heat transfer in viscoelastic fluid over a stretching sheet considered in porous medium have been carried out [8-9]. It is observed that the effect of increasing the porous parameter in the boundarylayer result in decreasesof temperature distribution.
Corresponding Author: Zanariah MohdYusof, Faculty of Computer andMathematicalSciences, Universiti TeknologiMARA, Terengganu,Malaysia.E-mail: zanariah297@tganu.uitm.edu.my
Citation: Zanariah Mohd Yusof, Noraini Ahmad, Nur Idalisa Norddin, Nur Huzainiee Mat Yuzut, Ruhana Jaafar, 2018, Steady Magnetohydrodynamics (MHDs) Boundary Layer Flows over an Exponentially Stretching Surface with the Effect of Radiation and Porous Medium; Journal of Basic and Applied Scientific Research, 8(5)1-7.
METHODOLOGY
Introducing Mathematical Formulation
Thegoverning equations are:
i. Conservative equation:
u v
0
ii. Momentum equation:
u u 2 u B2
u v v u u
x y y 2 K
iii. Energyequation:
T Tk 2T 1 qr
u
x y cp y2 Cp y
Theboundaryconditions for theproblemare:
u 0,T T as y
where
x/L x/ 2L
U Ue and T T Te
w 0 w 0
which are u and v are velocity components along the x and y direction respectively, is the fluid
density, T is the fluid temperature, B is the uniform magnetic field, is the fluid electrical conductivity,
0
is thedynamic viscosity, c is the specific heat, q is the radiative heat flux, U is the reference velocity, T is
pr 00
thereference temperatureand L is the reference length.
Transforming Partial Derivatives (PDE’s) into Ordinary Differential Equations (ODE’s)
All the governing equations must be in ordinary differential equation (ODEs) in order to solve the equations numerically. The governing partial differential equations (PDEs) were transformed into the system of ordinary differential equation (ODEs) by using the similarity transformation method. The similarity variables were introduced and obtained[10]. Thus, the variablesof u,v,T and areshown asbelow:
1 2
Lx vU0 x 2L
f f
u Ue f ( ) v e
0 2L 1
x/2L U0 x/2L
T Te , 2 ey
T
0 2L
J.Basic.Appl. Sci.Res.,8(5)1-7, 2018
After applying the similarityvariables, the governing partial differential equations (PDEs) have been reduced to the systemof nonlinear ordinarydifferential equations (ODEs) as written below [10]:
i. Momentum equation f ff 2f2 Mf 0
ii. Energyequation 4
1 K Pr f f 0
2 x/ 2L
2B0 LgT0 Le
where M is the magnetic parameter, is the buoyancy parameter,
22x/L
U0 U0 e 2L 4T 3 vcp
D is the permeabilityparameter, K is the radiation parameter and Pr is the
x/ L
Prandtl number. The boundaryconditions transformed as follows.
f0 0,f0 1, 0 1 at 0
f 0, 0 as
Numerical Solution
The system of ordinary differential equation has been solved numerically by using finite difference scheme known as Keller-box method in MATLAB.
RESULTS AND DISCUSSION
The validations of results are shown in the Table 1 and it shows a good agreement between the existing results with theresult obtained.The comparison considered the valueof zero for thermal and magnetic parameter, while the
permeabilityparameter is considered the value one on the heat transfer rate at the surface, 0 . Different values of Prandtl number, Pr is also considered. The main purpose of Table 1 is to verify the validity and accuracy of the present analysis.
Table 1: The local heat transfer for the parameters
K | M | Pr | D | | Bidin and Nazar (2009) | Ishak (2011) | Present |
---|---|---|---|---|---|---|---|
0 | 0 | 1 | 0 | 0 | -0.9548 | -0.9548 | -0.9548 |
2 | -1.4714 | -1.4715 | -1.4715 | ||||
3 | -1.8691 | -1.8691 | -1.8691 | ||||
5 | -2.5001 | -2.5002 | |||||
0 | 1 | 1 | 0 | -0.8611 | -0.8596 | ||
1 | -0.7916 | ||||||
1 | 0 | 1 | 0 | -0.5312 | -0.5330 | ||
1 | -0.4570 | ||||||
1 | 1 | 1 | 0 | -0.4505 | -0.4570 | ||
1 | -0.4094 |
Citation: Zanariah Mohd Yusof, Noraini Ahmad, Nur Idalisa Norddin, Nur Huzainiee Mat Yuzut, Ruhana Jaafar, 2018, Steady Magnetohydrodynamics (MHDs) Boundary Layer Flows over an ExponentiallyStretching Surface with the Effect of Radiation and Porous Medium; Journal of Basic and Applied Scientific Research, 8(5)1-7.
Table 2: Values of skin friction coefficient and heat transfer for different values of permeability parameter
Skin Friction Coefficient Heat Transfer Rate D | ||
---|---|---|
0.1 | 1.9643 | 0.4019 |
0.5 | 2.1587 | 0.3761 |
1.0 | 2.3793 | 0.3511 |
2.0 | 2.7682 | 0.3156 |
Table 2 shows the values of skin friction coefficient and heat transfer for different values of permeability parameter,D.From the Table4 above, it shows that the value of skin friction coefficient is increasing while the rate ofheat transfer is decreasing for increasing value ofpermeabilityparameter.
Figure 1: Velocity profiles for permeability parameter
From Figure 1, it shows that the momentum boundarylayer thickness decreases and thus the velocity gradient increase at the surface as the values of D increases. Then, the velocityof the fluid flow is found to decrease. This is related with Table 2 that shows the increasing values in skin friction coefficient and thus the fluid flow tends to decrease. It is shows that the presence of porous medium causes higher restriction to the fluid, which is reduces the velocity of the fluid. The flow of the fluid becomes slower because of the resistance to the flow that presents in the porous medium.
Figure 2: Temperature profiles for permeability parameter
J.Basic.Appl. Sci.Res.,8(5)1-7, 2018
The temperature profiles for the various values of D are shown in the Figure 2. Based on Figure 2, the thermalboundarylayer thickness increases as the values of D increases. In this context, the porous medium effect reduced the temperature distribution where the rate ofheat transfer at the surface as can beseen in Table2 decreases with an increase ofD.
Figure 3: Velocity profiles for magnetic parameter
Figure 4: Temperature profiles for magnetic parameter
Figure 3 and 4 show the velocity profiles and the temperature profiles for different values of magnetic parameter, M while the other parameters are fixed to one. FromFigure 3,it can be clearlyseen that the momentum boundary layer thickness decrease with the increasing values of M . This is because the transverse magnetic field opposesthe transportphenomena.
Figure 4 shows the thermal boundary layer thickness increase as the values of M increase. This situation related to the fact that thevariation of magnetic parameter leads to the variation of the Lorentz force due to magnetic field.Thus, the Lorentz force producesmoreresistance to thetransportphenomena.
Citation: Zanariah Mohd Yusof, Noraini Ahmad, Nur Idalisa Norddin, Nur Huzainiee Mat Yuzut, Ruhana Jaafar, 2018, Steady Magnetohydrodynamics (MHDs) Boundary Layer Flows over an ExponentiallyStretching Surface with the Effect of Radiation and Porous Medium; Journal of Basic and Applied Scientific Research, 8(5)1-7.
Figure 5: Velocity profiles for radiation parameter
The velocity profiles for different values of K is presented in Figure 5. The increasing values of K shows that does not affect the momentum boundary layer thickness same goes with the value of skin friction coefficient. Theunique value of skin friction coefficient thatobtainedis 2.3793.
Figure 6: Temperature profiles for radiation parameter
By referring to Figure 6, it shows that the temperature profiles for various values of K. The thermal boundary layer thickness increase with the increasing values ofK. Meanwhile, the temperature gradient decrease as can be seen in the Figure 6. The heat transfer rate obtained decreasesrelated in decreasing of temperaturegradient.
CONCLUSION
Asteadyboundarylayer flow over an exponentially stretching surface placed in porousmedium in the presence of radiation and magnetic field has been investigated. The effects of radiation, magnetic parameter, permeability parameterandPrandtlnumberonthecharacteristicsofheattransferwereanalyzed.Itcanbeconcluded thattherateof theheattransfer atthesurfacedecreasesasthemagneticparameter increases.Thevelocitydecreaseswith theincrease of magnetic and permeability parameter. Magnetic field and permeability present resistance and influence the momentum transfer where thevelocityof the flow reduces.
J.Basic.Appl. Sci.Res.,8(5)1-7, 2018
REFERENCES