Thermal radiation effect on MHD stagnation point flow of a Carreau fluid with convective boundary condition

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1 Proceedings of ICFM International Conference on Frontiers in Mathematics March 6-8,, Gauhati University, Guahati, Assam, India Available online at Thermal radiation effect on MHD stagnation point flo of a Carreau fluid ith convective boundary condition S.SUNEETHA, K. GANGADHAR and N. BHASKAR REDDY* Applied Mathematics Department, Yogi Vemana University, Kadapa- 6 suneethayvu@gmail.com Mathematics Department, Acharya Nagarjuna University Ongole Campus, Ongole kgangadharmaths@gmail.com *Mathematics Department, Sri Venkatesara University, Tirupati - 7 nbrsvu@gmail.com Abstract: This paper analyzes the effect of thermal radiation on a to-dimensional stagnation-point flo of an in-compressible magneto-hydrodynamic Carreau fluid toard a shrinking surface in the presence of heat transfer and convective boundary condition. Using the similarity transformations, the governing equations have been transformed into a system of ordinary differential equations. The resultant differential equations are solved numerically by using bvp4c MATLAB solver. The behavior of the velocity and temperature as ell as skin-friction coefficient and the local Nusselt number for different values of the governing parameters, namely, magnetic parameter, poer la index parameter, suction/bloing parameter, radiation parameter and convective parameter are discussed in detail. Keyords: Carreau fluid, convective surface boundary condition, radiation, Stagnation point flo. I. INTRODUCTION Prandtl s boundary layer theory demonstrated to be enormous utilize in Netonian fluids as the Navier Stokes equations can be transformed into a great deal basic equations hich are easier to handle. The study of boundary layer flo over a stretching sheet is a topic of great attention due to a variety of applications in designing cooling system hich included liquid metals, MHD generators, accelerators, pumps and flo meters. The stretching sheet idea in Crane s (97) boundary layer flo of a Netonian fluid past a stretching sheet has been examined and extended by numerous authors. Laminar mixed convection adjacent to vertical, continuously stretching surface studied by Chen (998). A stagnation point occurs henever a flo impinges on a solid object Pioneering ork on -dimensional stagnation point flo problem as first studied by Hiemenz ho used the similarity transformations approach to reduce the Navier-Stokes equations to non-linear ordinary differential equations.. In this contest Nadeem and Hussain () discussed HAM solutions for boundary layer flo in the region of stagnation point toards a stretching sheet. Makinde and Charles () conducted a computational dynamics on the hydrodynamic stagnation point flo toards a stretching sheet. But little attention has been paid to the four-parameter Carreau inelastic model ith the stress formulation n u yx y u y is frequently used in chemical engineering. It fits reasonably ell ith the suspensions of polymers behavior in many flo situations. This model describes the behavior of a purely viscous fluid hose viscosity changes ith increasing rate of deformation. Unlike the poer-la or Ostald-De Waele model, it predicts a viscosity that remains finite as the shear rate approaches zero. For that reason, the Carreau constitutive equation suits ell for free surface flos. Among the recent studies in the theory of Carreau fluid include peristaltic flo and heat transfer. Ali and Hayat (7) presented the analytic solution of the mathematical modelling for the flo of incompressible Carreau fluid in an asymmetric channel ith sinusoidal all variations. Hayat et al. () examined the MHD peristaltic flo of a Carreau fluid in a channel ith different aveforms. Olajuon () investigated the convection heat and mass transfer in a hydromagnetic Carreau fluid past a vertical porous plate in presence of thermal radiation and thermal diffusion. Mohamed Y. Abou-zeid (9) studied the system of non linear partial differential equations, hich describe the unsteady flo of MHD non-netonian fluid ith heat and mass transfer past a porous plate through a non-darcy porous medium. Akbar et al. (4) studied the to-dimensional stagnation-point flo of an incompressible Carreau fluid toard a shrinking surface. Shakhaoath et al. () investigated the effect of thermal radiation on non-netonian magnetohydrodynamic mixed convective poer-la fluid flo past a vertical stretching sheet ith heat generation and chemical reaction effects. Gangadhar () investigated the temperature increases ith an increasing the convective parameter. The effects of thermal radiation on MHD stagnation-point flo past a stretching sheet ith heat generation as studied by Zhu et al.,(). El-Arabay () analyzed the effect of suction and injection on the flo and heat transfer characteristics for a continuous moving plate in a micropolar fluid ith the effect of radiation. The present study investigates the to-dimensional stagnation-point flo of an in-compressible magnetohydrodynamic Carreau fluid toard a shrinking surface in the presence of heat transfer and convective boundary condition. Using the similarity transformations, the governing equations have been transformed into a set of ordinary differential equations, hich are nonlinear and cannot be solved analytically, therefore, bvp4c MATLAB solver has been used for solving it. The results for velocity and temperature functions are carried out for the ide range of important parameters namely; magnetic parameter, stretching/shrinking parameter, convective parameter, radiation parameter and Prandtl number. The skin friction and the rate of heat transfer have also been computed. II. MATHEMATICAL FORMULATION Consider a steady laminar to dimensional stagnation point flo of incompressible and electrically conducting Carreau fluid on a stretching surface in the presence of radiation and magnetic field of strength, B, applied in the positive y direction as shon in the figure. The induced magnetic field due to the ISBN:

2 Proceedings of ICFM International Conference on Frontiers in Mathematics March 6-8,, Gauhati University, Guahati, Assam, India Available online at motion of the electrically conducting fluid and the pressure gradient are neglected. over a all coinciding ith a plane y=, the flo is being confined to y>, the flo is generated due to the linear stretching. Extra stress tensor for carreau fluid is ( n ) () ij ( ) ij In hich is the extra stress tensor, is the zero shear rate ij viscosity, Γ is the time constant, n is the poer la index and is defined as ij ji i j Here is the second invariant strain tensor. Flo equations for carreau fluid model and energy equation after applying the boundary layer approximations can be defined as follos. Continuity equation u v x y Linear momentum equation u u x + v u y =u u e e x +n u y + s B u e - u r ( )G ( ) + n n - é uù ê ú u ë y û y Energy equation u T v T T () q r x y y y The boundary conditions are T u u( x) ax, v ( x), ht T at y=o y u u ( ), as (6) x bx T T y here u and v are the velocity components in the x - and y - directions, respectively, T is the fluid temperature in the boundary layer, ν is kinematic viscosity, σ is the electrical conductivity, ρ is the density, and ( is the thermal k / c ) p diffusivity ith k is the fluid thermal conductivity, is the heat capacity pressure, it is noticed that for poer la index (n=) our c p () () (4) problem reduced to the case of Netonian fluid hile for n > phenomena remains for non-netonian fluid respectively. The boundary conditions are T u u( x) ax, v ( x), ht T y at y= u u ( ), x bx T T y as (7) In hich b > is constant, e assume that u (x)=ax and u e (x)=bx are the velocities near and aay from the all respectively, h is the all heat transfer coefficient. The radiative heat flux q r is described by Roseland approximation such that q r 4 * T 4 * y (8) here * and k * are the Stefan-Boltzmann constant and the mean absorption coefficient, respectively. We assume that the temperature differences ithin the flo are sufficiently small so 4 that they T can be expressed as a linear function after using 4 Taylor series to expand T about the free stream temperature and neglecting higher-order terms. This result is the folloing approximation: 4 4 T 4T T T (9) In vie of equations (8) and (9), equation () reduces to T T 6 * T T u v () x y kk* y The continuity equation () is satisfied by the Cauchy Riemann equations x u () and v y here ( xy, ) is the stream function. In order to transform equations (4), (6) and () into a set of ordinary differential equations, the folloing similarity transformations and dimensionless variables are introduced. b T T a B y, b xf,,, M,Pr T T b b b x 6 * T We, R kk * () here f ( ) is the dimensionless stream function, θ is the dimensionless temperature, η is the similarity variable, λ is the stretching/shrinking parameter, We is the Weissenberg number, R is the radiation parameter, M is the magnetic parameter, Pr is the Prandtl number. In vie of equations () and (), the equations (4) and () transform into ( n ) We f ff '' ' M () (4) R " f ' Pr The corresponding boundary conditions are f () S, (),, '() Bi () at η=, as T () ISBN:

3 Proceedings of ICFM International Conference on Frontiers in Mathematics March 6-8,, Gauhati University, Guahati, Assam, India Available online at here the primes denote differentiation ith respect to and v the S, s > (i.e v <) corresponds to suction, s< a (i.e v >) corresponds to bloing, convective parameter h Bi b is the layer thickness decreases, these results are similar to the findings by Akbar et al. [8]. The all shear stress is given by u n u x y The coefficient of skin friction is defined as C f u The dimensionless form of skin friction is defined as, (6) n=.,,. We=., =-, M=, S=, Pr=R=Bi= n We Re xc f ' ' (7) The dimensionless form of Nusselt number Nu x is defined as x T Nux Re x ' T T y y (8) Our main aim is to investigate ho the values of f () and '() vary in terms of the various parameters. III. SOLUTION OF THE PROBLEM The set of equations () to (4) ere reduced to a system of first-order differential equations and solved using a MATLAB boundary value problem solver called bvp4c. This program solves boundary value problems for ordinary differential equations of the form y ' f x, y, p, a x b, by implementing a collocation method subject to general nonlinear, to-point boundary conditions g y( a), y( b), p. Here p is a vector of unknon parameters. Boundary value problems (BVPs) arise in most diverse forms. Just about any BVP can be formulated for solution ith bvp4c. The first step is to rite the ODEs as a system of first order ordinary differential equations. The details of the solution method are presented in Shampine and Kierzenka[]. IV. RESULTS AND DISCUSSION The governing equations () - (4) subject to the boundary conditions () are integrated as described in section. In order to get a clear insight of the physical problem, the velocity and temperature have been discussed by assigning numerical values to the parameters encountered in the problem. Figure shos the effect of the poer la index parameter on the non-dimensional velocity profiles. We observe that the velocity increases ith the influence of n in shrinking sheet. These findings are similar to the results reported by Akbar et al., (4). The results are quite different in the case of stretching sheet. Figures illustrate the effect of suction/bloing parameter on the velocity for both cases hen shrinking case (λ= ) and in the absence of shrinking parameter (λ=). We observed that the velocity increases ith increasing S. Moreover, the boundary - 4 Fig. (a) Velocity for different values of n n=.,,. We=., =, M=, S=, Pr=R=Bi= 4 Fig. (b) Velocity for different values of n S =, 6, 7 We=., =, M=, n=, Pr=R=Bi= 4 Fig. (a). Velocity for different values of S ISBN:

4 Proceedings of ICFM International Conference on Frontiers in Mathematics March 6-8,, Gauhati University, Guahati, Assam, India Available online at The variation of the velocity profiles ith the magnetic parameter in both cases of shrinking sheet and stretching sheet is shon in Figure 4. It is observed that the velocity increases ith an increasing magnetic parameter in case of shrinking sheet and the results are quite different hen in the case of stretching sheet. Figures, 6 & 7 illustrates the effects of the suction/bloing parameter, convective parameter and radiation parameter on the temperature profiles in both cases of shrinking case (λ = -) and stretching case (λ = ). It is observed that temperature of the fluid reduces ith a rising the suction/bloing parameter and also, it is observed that temperature of the fluid increases ith a rising the convective parameter and radiation parameter hen both cases of shrinking sheet and stretching sheet M=.,,, n=, =, We=., S=, Pr=.7, R=, Bi =..8 S =, 6, Fig.4(b) Velocity for different values of M n=, =, We=., M=, Pr=.7, R=, Bi =.. We=., =, M=, n=, Pr=R=Bi= 4 Fig.(b). Velocity for different values of S M=.,,, n=, = -, We=., S=, Pr=.7, R=, Bi = S =, 6, 7 4 Fig. (a) Temperature for different values of S Variations of the Skin friction and Nusselt number for the different values of S, n, M and λ are presented in Table. It is noticed that the skin friction and Nusselt number increases ith an increasing the suction/bloing parameter. It is observed that the skin friction and Nusselt number decreases ith an increase in the poer la index parameter. Also, it is observed that the skin friction reduces here as Nusselt number increases ith an increase in the Magnetic parameter or stretching/shrinking parameter. Table shos that the Nusselt number increases ith an increase in the Prandtl number or convective parameter and also observed that Nusselt number reduces ith an increasing the radiation parameter. Fig.4(a) Velocity for different values of M ISBN:

5 Proceedings of ICFM International Conference on Frontiers in Mathematics March 6-8,, Gauhati University, Guahati, Assam, India Available online at n=, =, We=., M=, Pr=.7, R=, Bi =..6 S=, n=, = -, We=., M=, Pr=.7, Bi= S =, 6, 7.. R =.,.6,... 4 Fig. (b) Temperature for different values of S 4 Fig. 7(a) Temperature for different values of R.4.6. S=, n=, = -, We=., M=, Pr=.7, R=. S=, n=, =, We=., M=, Pr=.7, Bi= Bi =.,,.,... R =.,.6,.. 4 Fig. 6(a) Temperature for different values of Bi 4 Fig.7(b) Temperature for different values of R Table. shos that the present results perfect agreement to the previously published data... S=, n=, =, We=., M=, Pr=.7, R=.... Bi =.,,., Fig. 6(b) Temperature for different values of Bi Table. Computations for the values of '(), '() (ith M=Pr=Bi=We=R=) for various values of λ S n m λ '() '() ISBN:

6 Proceedings of ICFM International Conference on Frontiers in Mathematics March 6-8,, Gauhati University, Guahati, Assam, India Available online at λ Table. Computations for the values of '() (ith M=Pr=Bi=We=R=) for various values of λ Pr R Bi '() Table. Comparison for the values of skin friction (ith M=Pr=Bi=We=R=) for various values of λ.. Present results Akbar et al. (4) Mahapatra and Nandy () Wang (8) Lok et al. (6) III. CONCLUSIONS In the present paper, a to-dimensional stagnation-point flo of an in-compressible magneto-hydrodynamic Carreau fluid toard a shrinking surface in the presence of heat transfer and convective boundary condition has been studied. The governing equations are approximated to a system of non-linear ordinary differential equations by similarity transformation. Numerical calculations are carried out for various values of the dimensionless parameters of the problem. It has been found that,. The velocity decreases in case of Shrinking and increases in the case of stretching ith an increase in the poer la index parameter.. The temperature of the fluid reduces ith a rising the suction/bloing parameter and also, increases ith a rising the convective parameter and radiation parameter hen both cases of shrinking sheet and stretching sheet.. The skin friction reduces here as Nusselt number increases ith an increase in the Magnetic parameter or stretching/shrinking parameter. REFERENCES [] Akbar, N.S., Nadeem, S., Rizan U Haq, Shiei Ye, MHD stagnation point flo of Carreau fluid toard a permeable shrinking sheet: Dual solutions, Ain Shams Engineering Journal, 9 (4). [] Ali, N., Hayat, T., Peristaltic Motion of a Carreau Fluid in an Asymmetric Channel, Appl. Math. Comput.,9,, -(7). [] Chen, C.H., Laminar mixed convection adjacent to vertical, continuously stretching sheets, Heat Mass Transfer,, (998). [4] Crane, L.J., Flo past a stretching sheet, Z Ange Math Phys (ZAMP),, (97). [] El-Arabay, H.A.M., Effect of suction/injection on the flo of a micropolar fluid past a continuously moving plate in the presence of radiation, Int. J. Heat Mass Transfer, 46, (). [6] Gangadhar, K., Soret and Dufour Effects on Hydromagnetic Heat and Mass Transfer over a Vertical Plate ith a Convective Surface Boundary Condition and Chemical Reaction, Journal of Applied Fluid Mechanics, 6,, 9-(). [7] Hayat, T., Saleem, N., Ali, N., Effect of Induced Magnetic Field on Peristaltic Transport of acarreau Fluid, Commun Nonlinear Sci. Numer. Simulat,, 9, 47-4(). [8] Hiemenz,K, Die grenzschicht an einem in dengleich formingen Flussigkeitsstrmeinge-tauchtengradenKrei szylinder, Dinglers polytech.j., 6, 9, 4 [9] Lok, Y.Y., Amin, N., and Pop. I, Non-orthogonal stagnation point toards a stretching sheet, Int J Non- Linear Mech., 4, 4, 6 67(6). [] Mahapatra, T.R., Nandy, Stability of dual solutions in Stagnation point flo and hea transfer over a porous shrinking sheet ith thermal radiation, Meccanica, 48, (). [] Makinde O.D. and Charles W.M., Computational dynamics of hydromagnetic stagnation flo toards a stretching sheet, Appl. Comput.Math., 9(),4- (). [] Mohamed Y. Abou-zeid, Numerical Treatment of Heat and Mass Transfer of MHD Flo of Carreau Fluid ith Diffusion and Chemical Reaction through a Non Darcy Porous Medium, The Open Mathematics Journal,, (9). [] Nadeem S. Hussain A, HAM solutions for boundary layer flo in the region of stagnation point toards a stretching sheet, Commun Nonlin Sci Number Sim,, 47-8(). [4] Olajuon, B.I., Convection heat and mass transfer in a hydromagnetic Carreau fluid past a vertical porous plate in presence of thermal radiation and thermal diffusion, Thermal Science,,, S4-S (). [] Shakhaoath, K Md., Ifsana, K., Haider Ali, B Md., Non-Netonian MHD mixed convective poer-la fluid flo over a vertical stretching sheet ith thermal radiation, heat generation and chemical reaction effects Acad. Res. Int., (), 8 9 (). [6] Shampine, L. F. and Kierzenka J., Solving boundary value problems for ordinary differential equations in MATLAB ith bvp4c, Tutorial Notes(). [7] Wang, C.Y., Stagnation flo toards a shrinking sheet, Int J Non- Linear Mech., 4,77 8(8). [8] Zhu J, Zheng L.C., and Zhang X.X., The influence of thermal radiation on MHD stagnation point flo past a stretching sheet ith heat generation, Acta Mech.Sin.,7(4),-9(). ISBN: