The influence of EMHD on Boundary Layer Nanofluid Stagnation Flow over a Stretching Sheet

 

Ch. Nagalakshmi, V.Nagendramma*A.Leelaratnam

Department of Applied Mathematics, SPMVV, Tirupati-517502, A.P, India

 

ABSTRACT:

 

INTRODUCTION:

Analysis on Stagnation nano energy conversion problem of steady nanofluids flows over a stretching sheet has been of abundant importance since last few years. Qualitative analyses of investigations have notable importance on several industrial applications, especially polymer sheet extrusion from a dye, drawing of plastic films, etc. These processes require cooling of the stretching sheet during the manufacturing stages at very high temperature. There are some techniques of cooling the stretching surface. Present investigation considers nanofluid as necessary for more effective cooling of the stretching surface. 

 

After the pioneering work of Khan and Pop [1] several researchers have studied stretching boundary layer flows with heat and mass transfer. Steady hydrodynamic flow past a stretching surface in a quiescent viscous and incompressible fluid with the Oberbeck-Boussinesq approximation has been analyzed by Partha et al. [2] and Bég et al.[3] analyzed cross-diffusion effects. Bég et al. [4] determined stretching flow of a magnetic polymer with the homotopy analysis method.  Afify [5] illustrated heat and mass transfer of a steady, incompressible and electrical conducting fluid over a stretching surface. Awang Kechil and Hashim [6] investigated series solution of MHD flow over non-linearly stretching sheet with chemical reaction. Kiran Kumar et al. [7] illustrated the influence of dissipation and radiation effects on heat transfer characteristics of MHD nanofluid flow due to an exponentially stretching sheet embedded in a porous medium. Nagendramma et al. [8] studied effects of magnetic field and chemical reaction on MHD heat and mass transfer boundary layer flow past an exponentially stretching surface  with slip effects. Ch. Nagalakshmi et al. [9] analyzed heat and mass transfer effects of non-Newtonian nanofluid flow over an explonentially stretching surface with Joule heating.

 

In the present investigation, a steady, incompressible stagnation nanofluid flow for conjugate conduction-convection and multiple slips has been performed. The term “nanofluid” refers to a liquid containing a suspension of submicron solid particles (nanoparticles). There are various models to handle such kinds of problems. In this study, Buongirno model to nanofluid and its properties used Eqs. (10) and (14) to solve the problem.

 

 

 

 

MATHEMATICAL FORMULATION:

If G is a In this problem, consider a steady incompressible nanofluid passing through a slip boundary stretching sheet. The governing continuity, momentum, fluid energy, fluid concentration and sheet conduction equations are formed on the basis of Ohm’s law and Maxwell’s equation with study of Electrical Magneto Hydrodynamics (EMHD) using Boussinesq approximation. The steady mixed convection EMHD nanfluid flow over an infinite vertical sheet have been assumed.  The flow is considered to be in X-direction and y axis is perpendicular to it.  The fluid motion with in the film is due to a slip boundary layer for stretching sheet. The present study considered the Brownian motion and thermophoresis effects model, so that it is one kind of nano particle mixed with fluid thermal system. The geometry of the problem is shown in Fig.1.

 

 

 

Fig. 1.  The physical model and co-ordinate geometry                   

 

The steady two-dimensional boundary layer equations for this fluid in the usual notation are

                                                                                                                                                               (1)

                                    (2)

 

                                                             (3)

 

                                                                                                                                 (4)

                                                                                                                                                (5)

The corresponding boundary conditions are            

  at                                                                       (6)

    as                                                                                                        (7)

where u and v are the velocity components of the fluid in the x and y directions, U is the free stream fluid velocity, respectively. Where is the density, is the specific heat of a fluid at constant pressure,is the heat capacity of the fluid,is heat capacity of the nano particle, is the thermal diffusivity,   is the Brownian diffusion coefficient,  is the theormophoresis diffusion coefficient, and  is the shear stress.

The similarity variables of the flow problem are

By using above similarity variables the equations (1)-(7) becomes

 

                                                                                              (8)

                                                                                                       (9)

                                                                                                                         (10)

                                                                                                                          (11)

The transformed similarity variables are

                                            (12)

 as                                                                    (13)

Where   is the stagnation parameter,   is the magnetic  parameter,  is the dimensionless electric parameter, is the dimensionless thermal free convection  parameter,    is the dimensionless mass free convection  parameter,     is the dimensionless velocity slip parameter,  is the Prandtl number, is temperature slip parameter,  is the Brownian motion parameter and  is the modified convective coefficient,   is a dimensionless thermophoresis parameter,   is the  conduction-convection number,

The physical quantities of engineering interest are the skin friction coefficient , local nusselt number  and local Sherwood number  are given by    

, ,                                                                      (14)

Where

,  ,                                                                                                   (15)

                                                                                 (16)

is Reynolds number.

 

COMPUTATIONAL FINITE DIFFERENCE SOLUTIONS:

The Computational finite difference method (Keller-Box) is employed to solve the transformed, coupled boundary layer problematic defined by eqns. (6) - (7) under (8). Keller-Box method is the most versatile technique available for engineering analysis. This technique has been described succinctly in Cebeci and Bradshaw [13] and Keller [14]. It has been used recently in polymeric flow dynamics by Subba Rao et al. [15-17], Amanulla et al [18,19] for viscoelastic models and Beg et al. [20] for viscoplastic fluid flows with Convective conditions.

 

RESULTS AND INTERPRETATION:

Figs. 2-5 depict the temperature profiles for various values of stagnation point parameter (), Thermal Grashoff number (Gr), Solutal Grashoff number (Gc), Prandtl number (Pr).  From this figures it is clear that the diminishes for enhancing values of .  This is due to the fact that there would be redundant decreasing nature in the thermal boundary layer thickness with increasing values of stagnation point parameter, Thermal Grashoff number, Solutal

Grashoff number and Prandtl number.

 

Figs. 6-8 elucidate that the effect electric parameter, Thermophoresis parameter and Brownian motion parameter on

temperature.  It is clear that the temperature increases with increasing values of E. From figs. 9 and 10 it is observed that the temperature shows redundant effect with enhancing values of conduction-convection parameter and thermal slip parameter where as it shows opposite behavior with velocity slip parameter  (Fig. 11).  Fig.12 reveals that the concentration decreases with increasing values of Schmidt number.  This is due to the fact that solutal boundary layer thickness decreases with higher values of Sc

 

 

Fig. 2 Temperature profile for various values of

 

Fig.3. Temperature profile for various values of

 

Fig.4. Temperature profile for various values of

 

Fig.5. Temperature profile for various values of

 

Fig.6. Temperature profile for various values of

 

 

 

Fig.7. Temperature profile for various values of

 

 

Fig.8. Temperature profile for various values of

 

 

Fig.9. Temperature profile for various values of

 

 

Fig.10. Temperature profile for various values of

 

 

 

Fig.11. Temperature profile for various values of

 

Fig.12. Concenration profile for various values of

 

CONCLUSIONS:

The Runge-Kutta Fourth order method has been successfully applied to perform a comparative study of nano energy conversion problem about conjugate conduction-convection over slip extrusion stretching sheet. This study is based on a novel nano energy conversion problem about extrusion stretching sheet heat transfer. Therefore, the present study is not only a novel study but also a solution for a nano energy conversion system problem. There important conclusions are as follows:

1.      It seemed that the increasing of electric parameter E results in the increasing of temperature  distribution at a particular point of the flow region.

2.      The effect of free convection parameters Gt, Gc and Prandtlnumber Pr on heat transfer process may show that the increasing of value of free convection parameters Gt, Gc or Pr results in the decreasing of temperature  distribution.

3.      The effect of nanofluid parameters Nt and Nb on heat transfer process may show the increasing of values of Nt or Nb and obtaining lower heat transfer effects.

4.      For heat conduction energy conversion part, the conduction convection parameter Ncc will produce a higher heat conduction effect clearly.

5.      For convection nano energy conversion, the free convection nano energy conversion effect is clearly better than the forced convection.

6.      Stagnation effect is important to this study; when stagnation parameter is larger, its heat transfer effect is larger too.

ACKNOWLEDGEMENT:

The authors are grateful to the authorities of Sri Padmavati Mahila Visvavidyalayam, Tirupati for providing research facilities in the campus.

 

CONFLICT OF INTEREST:

The authors declare no conflict of interest.

 

REFERENCES:

1.     W.A. Khan, I. Pop, Boundary-layer flow of a nanofluid past a stretching sheet,  Int. J. Heat Mass Transf. 53 (2010) 2477-2483.

2.     A.A. Afify, MHD free convective flow and mass transfer over a stretching sheet with chemical reaction, Heat Mass Transf. 40 (6-7) (2003) 495-500.

3.     S. Awang Kechil, I. Hashim, Series solution of flow over nonlinearly stretching sheet with chemical reaction and magnetic field, Phys. Lett. A 372 (13) (2008), 2258-2263.

4.     M. K. Partha, P.V.S.N. Murthy, and G.P. Rajasekhar, “Effect of viscous dissipation on the mixed convection heat transfer from an exponentially stretching surface,” Heat and Mass Transf 41, 360-366 (2005).

5.     O. A. Bég, A. Y. Bakier, and V. R. Prasad, “Numerical study of free convection magnetohydrodynamic heat and mass transfer from a stretching surface to a saturated porous medium with Soret and Dufour effects,” Computational Materials Sci 46, 57-65 (2009).

6.     O. A. Bég, U. S. Mahabaleshwar, M. M. Rashidi, N. Rahimzadeh, J-L Curiel Sosa, Ioannis Sarris, and N. Laraqi, “Homotopy analysis of magneto-hydrodynamic convection flow in manufacture of a viscoelastic fabric for space applications,” Int. J. Appl. Maths. Mech 10, 9-49 (2014).

7.     R.V.M.S.S Kiran Kumar, P.Durga Prasad, V. Nagendramma,A. Leelaratnam, S.V.K. Varma, Radiation and Viscous Dissipation Effects on MHD Heat Transfer Flow of Nanofluid over an Exponentially Stretching sheet in a Porous medium, International Journal of Pure and Applied Mathematics Vol. 113( 7 2017), 155-163.

8.     V. Nagendramma, A. Leela Ratnam, G. Sarojamma, MHD Heat and Mass Transfer Flow over an Exponentially stretching sheet in the Presence of Thermal Radiation,Chemical Reaction with Slip Effects, International Journal of Pure and Applied Mathematics, Vol. 109( 9),  2016, 257-265.

9.     Ch. Nagalakshmi1, V. Nagendramma, A. Leelaratnam, Non-Newtonian Nanofluid Flow over an Exponentially Stretching Sheet with Ohmic Effects, International Journal of Innovative Research in Science, Engineering and Technology, Vol. 6(13), 2017.

10.   Kai-Long Hsiao, Stagnation electrical MHD nanofluid mixed convection with slip boundary on a stretching sheet, Applied Thermal Engineering, 98, 2016, 850-861.

 

 

 

 

 

 

Accepted on 21.09.2017      ©A&V Publications All right reserved