Applied Mathematics
Vol.06 No.03(2015), Article ID:54629,20 pages
10.4236/am.2015.63050

Viscous Dissipation and Radiation Effects on MHD Boundary Layer Flow of a Nanofluid Past a Rotating Stretching Sheet

M. Wahiduzzaman1*, Md. Shakhaoath Khan2, P. Biswas1, Ifsana Karim2, M. S. Uddin1

1Mathematics Discipline, Khulna University, Khulna, Bangladesh

2Discipline of Chemical Engineering, The University of Newcastle, Callaghan, Australia

Email: *wahidmathku@gmail.com   Received 12 February 2015; accepted 12 March 2015; published 13 March 2015

ABSTRACT

Viscous dissipation and radiative heat transfer in nanofluid with the influence of magnetic field over a rotating stretching surface has been investigated numerically. The steady laminar boun- dary layer flow is considered in this study. The governing boundary-layer equations are formu- lated and transformed into nonlinear ordinary coupled differential equations by using similarity variables. The governing equations are solved numerically using the Nactsheim-Swigert Shooting iteration technique together with the Runge-Kutta six order iteration schemes with the help of a computer programming language Compaq Visual Fortran 6.6a. The simulation results are presented graphically to illustrate influence of well-known parameters on the velocity, temperature and concentration distributions as well as skin-friction coefficient, Nusselt and Sherwood number at the sheet.

Keywords:

Nanofluid, Viscous Dissipation, Thermal Radiation, Stretching Sheet, Nactsheim-Swigert Shooting Iteration Technique 1. Introduction

Crane  first obtained an elegant analytical solution to the boundary layer equations for the problem of steady two-dimensional flow due to a stretching surface in a quiescent incompressible fluid. Carragher and Crane  studied the heat transfer in the flow over a stretching surface in the case when the temperature difference between the surface and the ambient fluid is proportional to the power of distance from fixed point. Na and Pop  studied an unsteady flow past a stretching sheet. The thermal radiation effect is a new dimension added to the study of stretching surface has important applications in physics and engineering particularly in space technology and high temperature process. It plays an important role in controlling the heat transfer process in polymer processing industry. The study of radiative heat transfer within the influence of magnetic field over a stretching surface has newly arrived. Pop et al.  investigated the flow over stretching sheet near a stagnation point taking the effect of thermal radiation. The research of nanofluids has gained huge attention in recent years. Jang and Choi  investigated the analysis of convective instability and heat transfer characteristics of the nanofluids. Jang and Choi  studied the effect of various parameters on nanofluid thermal conductivity. Kumar  inves- tigated the radiative heat transfer with the viscous dissipation effect in the presence of transverse magnetic field. Khan and Pop  have investigated boundary layer flow of a nanofluid past a stretching sheet. Very recently a number of studies of MHD boundary layer fluid (e.g., nanofluid) flow with the effects of thermal radiation, vis- cous dissipation and Nanofluid were reported in the literature  -  . But the effect of rotation is still not get- ting promising attraction to the researchers.

Our prime objective is to extend analysis of Khan and Pop  . This study finds the effect of rotation, thermal radiation and viscous dissipation on boundary layer flow of nanofluid past a stretching sheet of a rotating sys- tem.

2. Methods: Mathematical Model

The steady two dimensional boundary layer flow of a nanofluid past a stretching surface of a rotating system is considered with the linear stretching velocity , where constant and x is coordinate measured along the stretching surface. The flow takes place at where y is the coordinate measured normal to the stretch- ing surface. A steady uniform stress leading to equal and opposite forces is applied along the x-axis, so that the sheet is stretched keeping the origin fixed. The sketch of the physical configuration and coordinate system are shown in Figure 1 by following Khan and Pop  .

It is assumed that at the stretching surface, the temperature and the concentration takes constant value and respectively. The ambient values attained as tends to infinity of and are denoted by and respectively. A uniform magnetic field B is imposed to the plate. The magnetic vector can be taken as and is the radiative heat flux in the y-direction. The velocity of the plate (uniform velocity) considered as . Also a and b are the linear constant parameter, l is the characteristics length and, is the constant whose values depends on the properties of the fluid. Under this assumption and usual boundary layer approximation MHD nanofluid flow is governed by the following equations  :

(1)

(2)

Figure 1. Physical configuration of the flow.

(3)

(4)

(5)

and the boundary condition for the model is

(6)

where, is the thermal diffusivity, k is the thermal conductivity, is the Brownian diffusion coefficient, is the thermophoresis diffusion coefficient, x is the coordinate measured along stretching surface, is the stretching velocity, is the uniform velocity, a and b the linear constant parameter, l is the characteristics length and is the constant whose values depends on the properties of the fluid.

In order to find a similarity solution to the Equations (2) to (5) with boundary conditions (6), the following similarity transformations, dimensionless variables are used in the rest of analysis:

(7)

From the above transformations the non-dimensional, nonlinear, coupled ordinary differential equations are obtained as:

(8)

(9)

(10)

(11)

The transformed boundary conditions are as follows:

(12)

where the notation primes denote differentiation with respect to and the parameters are defined as:

is the Magnetic parameter,

is the Prandtl number,

is the Eckert number,

is the Rotational parameter,

is the Lewis number,

is the Brownian motion parameter,

is the Thermophoresis parameter and

is the Stretching parameter and is the Constant parameter.

The physical quantities of importance in stretching sheet transport modeling   are the skin friction coefficient, the reduced Nusselt number, and reduced Sherwood number, which are calculated respectively by the following equations:

(13)

(14)

(15)

(16)

where is the Reynolds number. The reduced skin friction coefficient, the reduced Nusselt number and reduced Sherwood number has an inordinate influence in the model as these possessions are the function of local Reynolds number. This dimensionless parameter used to predict analogous flow patterns in the boun- dary layer nanofluid flow past a rotating stretching sheet. Also the governing mathematical models describes for the laminar flow which occurs at low Reynolds numbers. Where the viscous forces are dominant, and are cate- gorized by smooth, continuous motion of nanofluid.

The governing equations are transformed into nonlinear ordinary differential equations which depends on the Magnetic parameter, the Radiation parameter, the Prandtl number, the Eckert number, the Rotational parameter, the Lewis number, the Brownian motion parameter, the Thermophoresis parameter, Stretching parameter and constant parameter respectively.

3. Numerical Technique

The obtained non-linear coupled ordinary differential Equations (8)-(11) with boundary condition (12) are solved numerically using Nactsheim-Swigert shooting technique together with Runge-Kutta six order iteration schemes  . For the purpose of this method, the Nactsheim-Swigert shooting iteration technique  together with Runge-Kutta six order iteration scheme is taken and determines the temperature and concentration as a function of the coordinate.

4. Results and Discussion

The heat and mass transfer problem associated with laminar flow of the nanofluids past a stretching surface of a rotating system has been studied. In order to investigated the physical representation of the problem, the numeri- cal values of primary velocity, secondary velocity, temperature and species concentration with the boundary layer have been computed for different parameters as the Magnetic parameter, the Radiation parameter R, the Prandtl number, the Eckert number, the Rotational parameter, the Lewis number, the Brownian motion parameter, the Thermophoresis parameter, the Stretching parameter, and the Con- stant parameter respectively. In Figure 2, dimensionless primary velocity distribution is plotted for different values of Magnetic parameter M. Here consider the values of Stretching parameter. It is observed that as the Magnetic parameter increases the primary velocity decreases gradually.

In Figures 3-6, dimensionless secondary velocity distribution is plotted for different values of Magnetic pa- rameter M, Here consider the values of Stretching parameter, 0.5, 1, 1.5. It is observed that as the

Figure 2. Effect of M on primary velocity profiles.

Figure 3. Effect of M on secondary velocity profiles.

Figure 4. Effect of M on secondary velocity profiles when b/a = 0.5.

Figure 5. Effect of M on secondary velocity profiles when b/a = 1.0.

Figure 6.Effect of M on secondary velocity profiles when b/a = 1.5.

Magnetic parameter increases the secondary velocity increases gradually when and the secondary velocity decreases gradually when, 1, 1.5. In Figure 7, dimensionless secondary velocity distribution is plotted for different values of Rotational parameter. It is observed that as the Rotational parameter increases the secondary velocity decreases gradually. In Figure 8, the dimensionless temperature distribution is plotted for different values of Stretching parameter. It is observed that as the Stretching parameter increases the temperature decreases gradually. In Figure 9, the dimensionless temperature distribution is plotted for different values of Brownian parameter. It is observed that as the Brownian parameter increases the temperature increases gradually. In Figure 10, the dimensionless temperature distribution is plotted for different values of constant parameter m. It is observed that as the constant parameter increases the temperature increases gradually. In Figure 11, the dimensionless temperature distribution is plotted for different values of Eckert number. It is observed that as the Eckert number increases the temperature increases gradually. In Figure 12, the dimensionless temperature distribution is plotted for different values of Lewis number. It is observed that as the Lewis number increases the temperature decreases gradually. In Figure 13, the dimensionless temperature distribution is plotted for different values of Lewis number. It is observed that as

Figure 7. Effect of R’ on secondary velocity profiles.

Figure 8. Effect of b/a on temperature profiles.

Figure 9. Effect of Nb on temperature profiles.

Figure 10. Effect of m on temperature profiles.

Figure 11. Effect of Ec on temperature profiles

Figure 12. Effect of Le on temperature profiles.

Figure 13. Effect of Leon temperature profiles.

For the physical interest of the problem, the dimensionless skin-friction coefficient and, the dimensionless heat transfer rate at the sheet and the dimensionless mass transfer rate at the sheet are plotted against Thermophoresis parameter and illustrated in Figures 26-38.

Figure 26 represents the primary shear stress plotted against the Thermophoresis parameter for different values of Magnetic parameter. It is observed that as the Magnetic parameter increases the primary shear stress decreases gradually.

Figure 27 represents the primary shear stress plotted against the Thermophoresis parameter for different values of Magnetic parameter. It is observed that as the Magnetic parameter increases the primary shear stress increases gradually. Figure 28 represents the secondary shear stress plotted against the Thermophoresis parameter for different values of Magnetic parameter. It is observed that as the

Figure 14. Effect of M on temperature profiles.

Figure 15. Effect of Pr on temperature profiles.

Figure 16. Effect of R on temperature profiles.

Figure 17. Effect of R’ on temperature profiles.

Figure 18. Effect of b/a on concentration profiles.

Figure 19. Effect of m on concentration profiles.

Figure 20. Effect of Ec on concentration profiles.

Figure 21. Effect of M on concentration profiles.

Figure 22. Effect of R on concentration profiles.

Figure 23. Effect of Le on concentration profiles.

Figure 24. Effect of Nt on concentration profiles.

Figure 25. Effect of Nb on concentration profiles.

Figure 26. Effect of M on primary shear stress.

Figure 27. Effect of M on primary shear stress.

Figure 28. Effect of M on secondary shear stress.

Figure 29. Effect of M on secondary shear stress.

Figure 30. Effect of M on secondary shear stress.

Figure 31. Effect of R’ on secondary shear stress.

Figure 32. Effect of m on heat transfer rate.

Figure 33. Effect of Ec on heat transfer rate.

Figure 34. Effect of Nb on heat transfer rate.

Figure 35. Effect of Pr on heat transfer rate.

Figure 36. Effect of R on heat transfer rate.

Figure 37. Effect of Le on mass transfer rate.

Figure 38. Effect of Nb on mass transfer rate.

Figure 38 represents the dimensionless mass transfer rate plotted against the Thermophoresis para- meter for different values of Brownian parameter. It is observed that as the Brownian parameter increases the mass transfer rate decreases gradually.

The performances of the dimensionless skin-friction coefficient, dimensionless heat transfer rate and dimen- sionless mass transfer rate with the function of Thermophoresis parameter are concluded in Table 1.

5. Conclusions

Laminar boundary layer flow of a nanofluid has been investigated for steady flow past a stretching surface of rotating system with the influence of magnetic field and thermal radiation. The results are presented for various parameters. The velocity, temperature and concentration distributions for different parameters are shown graphically. The important findings of the investigation from graphical representation are listed below:

Table 1. Different behavior of the skin-friction coefficient, heat transfer rate, dimensionless mass transfer rate.

a) The temperature and concentration boundary layer thickness increases due to increase of magnetic parameter for nanofluids.

b) The temperature boundary layer thickness increases due to increase of Brownian motion parameter. Also the heat transfer rate at the sheet increases for increasing value of Brownian motion parameter and Thermophoresis parameter. The concentration boundary layer thickness increases due to increase of Thermophoresis parameter and concentration boundary layer thickness decreases due to increase of Brownian motion parameter. Also the mass transfer rate at the sheet decreases due to decrease of Brownian motion parameter.

c) The temperature and concentration boundary layer thickness decreases due to increase of stretching parameter.

d) The temperature boundary layer thickness increases due to increase of constant parameter and concentration boundary layer thickness decreases due to increase of constant parameter. Also the surface heat transfer rate is decreased.

e) The temperature boundary layer thickness increases due to increase of Radiation parameter whereas concentration boundary layer thickness decreases due to increase of Radiation parameter. Also the surface heat transfer rate is increased.

f) The temperature boundary layer thickness increases due to increase of Eckert number where as concentration boundary layer thickness decreases due to increase of Eckert number. Also the surface heat transfer rate is increased.

g) The temperature boundary layer thickness increases due to increase of Rotational parameter for nanofluids.

h) The temperature boundary layer thickness as well as the concentration boundary layer thickness decreases due to increase of Lewis number. Also the mass transfer rate at the sheet decreases.

i) The primary velocity profiles decreases for increasing Magnetic parameter when the stretching parameter takes the value zero but when the Stretching parameter takes value 0.1 then the primary velocity profiles increases. The secondary velocity profiles increase for increasing Magnetic parameter when the stretching parameter takes the value zero, but when the Stretching parameter takes the value (0.5 to 1.5 in present study) then the secondary velocity profiles decreases.

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NOTES

*Corresponding author.