Available Online Through ISSN 2229 – 5046

International Journal of Mathematical Archive-6(2), 2015, 73-78 Available online through www.ijma.info ISSN 2229 – 5046

MHD BOUNDARY LAYER FLOW AND HEAT TRANSFER OF AN UPPER-CONVECTED MAXWELL FLUID

Dr. Anuj Kumar Jhankal1 and Manoj Kumar*2

1Army cadet College wing, Indian Military Academy, Dehradun, India. 2Arya Institute of Engineering Technology & Mgt., Omaxe City, Jaipur-302026, India.

(Received On: 09-02-15; Revised & Accepted On: 28-02-15)

ABSTRACT An analysis is made to study boundary layer flow and heat transfer of an upper-convected Maxwell fluid in presence of transverse magnetic field. The upper-convected Maxwell model is used to characterize the non-Newtonian fluid behavior. Using similarity transformation, the governing boundary layer equations are transformed into self-similar nonlinear ordinary differential equations, which are then solved numerically using a very efficient RKF45 (Runge- Kutta-Fehlberg forth-fifth) method. The effects of various parameters like magnetic parameter, Deborah number and Prandtl number on the velocity and temperature profiles as well as on the local Skin-friction coefficient and the local Nusselt number are presented and discussed through graphs and tables.

Mathematics Subject Classification [MSC]: 76A05, 76D10, 76M20, 76W05, 80A20.

Keywords: MHD, boundary layer flow, heat transfer, upper-convected Maxwell fluid, numerical study.

NOMENCLATURE

B0 Constant applied magnetic field Cp Specific heat at constant pressure f Dimensionless stream function M Magnetic field parameter Pr Prandtl number T Temperature of the fluid U Free stream velocity u, v Velocity component of the fluid along the x and y directions, respectively x, y Cartesian coordinates along the surface and normal to it, respectively

Greek symbols β Deborah number η Dimensionless similarity variable ρ Density of the fluid μ Viscosity of the fluid ν Kinematic viscosity Ψ Stream function λ Retardation time σe Electrical conductivity Thermal conductivity Dimensionless temperature κ Superscriptθ ′ Derivative with respect to η

Subscripts w Properties at the plane ∞ Free stream condition

Corresponding Author: Manoj Kumar*2 1Army cadet College wing, Indian Military Academy, Dehradun, India. International Journal of Mathematical Archive- 6(2), Feb. – 2015 73 Anuj Kumar Jhankal1 and Manoj Kumar*2/ MHD Boundary Layer Flow and Heat Transfer of an Upper-Convected Maxwell Fluid / IJMA- 6(2), Feb.-2015.

INTRODUCTION

The subject of MHD is largely perceived to have been initiated by Swedish electrical engineer Hannes Alfv n [1] in 1942. If an electrically conducting fluid is placed in a constant magnetic field, the motion of the fluid induces currents which create forces on the fluid. The production of these currents has led to the design of among other devices𝑒𝑒́ the MHD generators for electricity production. The equations which describe MHD flow are a combination of continuity equation and Navier-Stokes equations of fluid dynamics and Maxwell’s equations of electromagnetism. The governing equations are differential equations that have to be solved either analytically or numerically.

In recent years, the studies of boundary layer flows of non-Newtonian fluid have received considerable attention due to their numerous industrial and engineering applications. The non-Newtonian fluids in view of their diverse rheological properties cannot be examined through one constitutive relationship between shear stress and rate of strain. Many models of non-Newtonian fluids exist. Maxwell model has become the most popular as it can predict stress relaxation and also excludes the complicating effects of shear dependent viscosity Mukhopadhyay and Gorla [7] (2012). In view of its simplicity, this fluid model has acquired special status amongst model has acquired special status amongst the recent workers in the field. This kind of fluid has significant application in viscoelastic problems which have small dimensionless relaxation time. Sarpakaya [12] (1961) was the first researcher to study the MHD flow a of non- Newtonian fluid. Prandtl’s boundary layer theory proved to be of great use in Newtonian fluids as Navier-Stokes equations can be converted into much simplified boundary layer equation which is easier to handle. Wang and Tan [14] (2011) discussed the flow of Maxwell fluid in a porous medium. Zierep and Fetecau [16] (2007) studied Rayleigh- Stokes problem using Maxwell fluid and find exact solution. Sadeghy et al.[10] (2006) did a comparative study for Sakiadis flow of an upper-convected Maxwell fluid on a rigid plate. In view of the Maxwell model, several researchers have analyzed the MHD flow of a Maxwell fluid under various aspects of rotation, thermal radiation, heat and mass transfer, chemical reaction, suction/injection, thermophoresis and heat source/sink (Bataller [4] (2011); Zheng et al.[15] (2013); Noor [8] 2012; Vajravelu [13] (2012); Eosboee et al.[5] (2010)).

In this study we aim at investigating boundary layer flow and heat transfer of an upper-convected Maxwell fluid in presence of transverse magnetic field. The governing partial differential equation is transformed into ordinary differential equation by means of similarity transformations. This equation is solved numerically by RKF45 (Runge- Kutta-Fehlberg forth-fifth) method using symbolic algebra software Maple. The effects of various parameters are presented with the aid of graphs & tables and discussed.

FORMULATION OF THE PROBLEM

Let us consider the steady and incompressible MHD boundary layer flow an electrically conducting fluid obeying UCM model. The x- and y-axes are taken along and perpendicular to the flow, respectively. A uniform magnetic field of strength B0 is imposed along the y-axis. It is assumed that external field is zero, the electric field owing to polarization of charges and Hall effect are neglected. Under the usual boundary layer approximations, the governing equation of continuity, momentum and energy [Pai [9] (1956), Schlichting [11] (1964), Bansal [2] (1977)] under the influence of externally imposed transverse magnetic field [Jeffery [7] (1966), Bansal [3] (1994)] are: + = 0 (1) 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 2 2 2 2 2 2 2 0 𝜕𝜕𝜕𝜕 +𝜕𝜕𝜕𝜕 + 2 + 2 + 2 = 2 (2) 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕 𝑢𝑢 2 𝜕𝜕 𝑢𝑢 𝜕𝜕 𝑢𝑢 𝜕𝜕 𝑢𝑢 𝜎𝜎𝑒𝑒 𝐵𝐵 𝑢𝑢 𝜕𝜕𝜕𝜕 𝑣𝑣 𝜕𝜕𝜕𝜕+ 𝜆𝜆 �𝑢𝑢=𝜕𝜕𝑥𝑥 2 𝑢𝑢𝑢𝑢 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝑣𝑣 𝜕𝜕𝑦𝑦 � 𝜈𝜈 𝜕𝜕𝑥𝑥 − 𝜌𝜌 𝑢𝑢 (3) 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕 𝑇𝑇

𝜌𝜌𝐶𝐶𝑝𝑝 �𝑢𝑢 𝜕𝜕𝜕𝜕 𝑣𝑣 𝜕𝜕𝜕𝜕 � 𝜅𝜅 𝜕𝜕𝑦𝑦 Along with the boundary conditions for the problem are given by ( , 0) = , ( , 0) = 0, ( , 0) = ( ), ( , ∞) 0, ( , ∞) (4) ∞ 𝑤𝑤 𝑢𝑢 𝑥𝑥 𝑈𝑈 𝑣𝑣 𝑥𝑥 𝑇𝑇 𝑥𝑥 𝑇𝑇 𝑥𝑥 𝑢𝑢 𝑥𝑥 → 𝑇𝑇 𝑥𝑥 → 𝑇𝑇 Ψ Ψ The continuity equation (1) is satisfied by introducing a stream function Ψ such that u = and v = y x ∂ ∂

∂ − ∂ The momentum equation can be transformed into the corresponding ordinary nonlinear differential equation by the following transformation:

= , = ( ) 𝑈𝑈 𝜂𝜂, 𝑦𝑦� 𝛹𝛹 √𝜈𝜈𝜈𝜈𝜈𝜈𝑓𝑓 𝜂𝜂 𝜈𝜈𝜈𝜈 ( ) = ∞ (5) 𝑇𝑇−𝑇𝑇 ∞

𝜃𝜃 𝜂𝜂 𝑇𝑇𝑤𝑤 −𝑇𝑇

Where η is the independent similarity variable. The transformed nonlinear ordinary equation is 2 ′′′ 2 ′ ′′ + 2 ′′′ + ′2 ′′ + ′′ 2 ′ = 0 (6) (2 ) = 0 (7) 𝑓𝑓′′ − 𝛽𝛽� 𝑓𝑓′𝑓𝑓 𝑓𝑓 ′ 𝑓𝑓 𝑓𝑓 𝜂𝜂𝑓𝑓 𝑓𝑓 � 𝑓𝑓𝑓𝑓 − 𝑀𝑀𝑓𝑓 The𝜃𝜃 − transformed𝑃𝑃𝑃𝑃 𝑓𝑓 𝜃𝜃 − boundary𝜃𝜃 𝑓𝑓 conditions are: = 0: = 0, = 1, = 1 : = 0,′ = 0 (8) 𝜂𝜂 𝑓𝑓 𝑓𝑓 𝜃𝜃 2 Where prime′ denotes differentiation with respect to , = 0 is the magnetic parameter, = is the Deborah 2 𝜂𝜂 → ∞ 𝑓𝑓 𝜃𝜃 𝜎𝜎𝑒𝑒 𝐵𝐵 𝑥𝑥 𝜆𝜆𝑈𝑈 number and = is the Prandtl number. 𝜂𝜂 𝑀𝑀 𝜌𝜌 𝑈𝑈 𝛽𝛽 𝑥𝑥 𝜇𝜇 𝐶𝐶𝑝𝑝

𝜅𝜅 RESULTS AND𝑃𝑃𝑃𝑃 DISCUSSIONS

The system of governing equations (6)-(7) together with the boundary condition (8) is non-linear ordinary differential equations depending on the various values of magnetic parameter M, Deborah number β and Prandtl number Pr. The non-linear differential equations (6) and (7) subject to the boundary conditions (8) are solved numerically using Runge- Kutta-Fehlberg Forth-Fifth order method. To solve this equation we adopted symbolic algebra software Maple. Maple uses the well known Runge-Kutta-Feulberg Forth-Fifth (RKF45) order method to generate the numerical solution of boundary value problem.

In order to verify the accuracy of our present method, we have compared our results with HPM and BVP Eosboee et al. [5]. Table 1 compares the values of velocity component ( ) for M=1.0 and β=0.4. The comparisons are found to be in excellent agreement. 𝑓𝑓 𝜂𝜂 Table-1: Numerical values of f( ) compared with the result obtained by Eosboee et al. [5], when M=1.0 and β=0.4. Eosboee et al. [5] ( ) η Present paper results ( ) η HPM BVP 𝐟𝐟 𝛈𝛈 0 0 0 0 𝐟𝐟 𝛈𝛈 0.5 0.38662 0.386325 0.386128 1.0 0.60613 0.605299 0.604469 1.5 0.72485 0.724257 0.722237 2.0 0.78562 0.786745 0.782737 2.5 0.81461 0.818662 0.811468 3.0 0.82698 0.834332 0.822121 3.5 0.83119 0.841253 0.832975 4.0 0.83192 0.843124 0.843124

Table 2 represents the Skin friction parameter against magnetic parameter M for various values of Deborah number β. It is noted that the effect of increasing M is to decrease the skin friction at the surface. The positive values of f ′′(0) imply that fluid exerts the drag force on the surface, and the negative values f ′′ (0) imply the opposite meaning. Thus as M becomes very large, there will be a decrease in skin friction f ′′ (0), and then a decrease in the drag force at the surface. Results also shows that an increase in the Deborah number β leads to increases the skin friction f ′′(0) and the drag force at the surface.

Table-2: Numerical values of Skin friction coefficient f''(0) M β=0 β=0.1 β=0.2 β=0.3 β=0.4 β=0.5 β=0.6 β=0.7 0 -0.49664 -0.48124 -0.46522 -0.44854 -0.43117 -0.41316 -0.39465 -0.37598 0.5 -0.83234 -0.82348 -0.81449 -0.80536 -0.79608 -0.78665 -0.77706 -0.76731 1 -1.08534 -1.07869 -1.07198 -1.06521 -1.05836 -1.05144 -1.04445 -1.03738 1.5 -1.29338 -1.28783 -1.28223 -1.27660 -1.27092 -1.26520 -1.2594 -1.25362 2 -1.4733 -1.46844 -1.46354 -1.45860 -1.45364 -1.44864 -1.44361 -1.43855

Table 3 represents the Nusselt number against magnetic parameter M for various values of Deborah number β. It is noted that the rate of heat transfer increases with magnetic parameter M. While, the rate of heat transfer decreases with Deborah number β.

Table-3: Numerical values of Nusselt number when Pr = 1.0 ( ) M β=0 β=0.1 β=0.2 β=0.3 ′ β=0.4 β=0.5 β=0.6 β=0.7 −𝛉𝛉 𝟎𝟎 0 -1.45012 -1.45406 -1.45818 -1.46249 -1.46699 -1.47166 -1.47642 -1.48113 0.5 -1.36997 -1.37189 -1.37384 -1.37582 -1.37784 -1.37989 -1.38198 -1.38411

1 -1.30857 -1.30973 -1.31089 -1.31206 -1.31325 -1.31445 -1.31567 -1.3169

1.5 -1.25892 -1.2597 -1.26048 -1.26127 -1.26207 -1.26287 -1.26368 -1.26449 2 -1.21729 -1.21786 -1.21842 -1.21899 -1.21956 -1.26287 -1.22072 -1.2213

Figures 1 and 2 show the effect of magnetic parameter M for the given values of Deborah number β. An increase in the M leads in decrease of both the velocity components f(η) and f ′(η). Accordingly, the thickness of the momentum boundary layer decreases. This happen due to the Lorentz force arising from the interaction of magnetic and electric fields during the motion of the electrically conducting fluid. To reduce momentum boundary layer thickness the generated Lorentz force enhances the fluid motion in the boundary layer region.

Figure 1: Effects of magnetic parameter M on Figure 2: Effects of magnetic parameter M on f ( ), f( ), when β=0.5. when β=0.5 ′ η Figuresη 3 and 4 show the effect of Deborah number β on velocity profiles for the given values of magnetic parameter M. An increase in Deborah number β is seen to decrease both velocity components f(η) and f ′(η) at a given point and thereby decreases the thickness of the boundary layer with an increase in β.

Figure 3: Effects of Deborah number β on f( ), when Figure 4: Effects of Deborah number β on f ( ), M=2.5. when M=2.5. ′ η η Figures 5 and 6 illustrate the temperature profile for various values of magnetic parameter M and Deborah number β respectively. We infer from this figure that the temperature increase with an increase in the magnetic parameter M and Deborah number β.

Figure 5: Effect of magnetic parameter M on the Figure 6: Effects of Deborah number β on the temperature profiles, when β=0.5 and Pr=1.0 temperature profiles, when M=2.5 and Pr=1.0

Figure 7 is plotted for the temperature profiles for different values of Prandtl number (Pr). An increase in the Prandtl number is seen to decrease the fluid temperature θ(η) above the surface. That is not surprising realizing the fact that the thermal boundary becomes thinner for larger the Prandtl number. Therefore, with an increase in Prandtl number the rate of thermal diffusion drops. The reason is that smaller values of Prandtl number are equivalent to increasing the thermal conductivities, and therefore heat is able to diffuse away from the heated surface more rapidly than for higher values of Prandtl number.

Figure 7: Temperature profiles for different values of Prandtl number (Pr), when M=0 and β=0.

CONCLUSION

In the present study, we have theoretically studied the two-dimensional boundary layer flow and heat transfer of an upper-convected Maxwell fluid in the presence of transverse magnetic field. The governing partial differential equations are transformed into ordinary differential equations by means of similarity transformations. The resulting non-linear ordinary differential equations are solved numerically by RKF45 (Runge-Kutta-Fehlberg forth-fifth) method. The velocity and temperature profiles are discussed numerically and presented through graphs. The numerical values of Skin-friction coefficient and Nusselt number are derived, discussed numerically and presented through tables. From the study, following conclusions can be drawn: 1. The study reveals that due to increase of magnetic parameter M, the momentum boundary layer thickness reduces and the temperature at a point increases. Thus we conclude that we can control the velocity field and temperature by introducing magnetic field. 2. The effect of Deborah number β is to decreases the momentum boundary layer thickness and the thermal boundary layer thickness.

3. The temperature as well as the thermal boundary layer thickness decrease with increasing values of Prandtl number. 4. The skin friction parameter decrease with magnetic parameter M but increases with the increasing values of Deborah number β. 5. The heat transfer is enhanced for increase of magnetic parameter M, this heat transfer is very important in production engineering to improve the quality of the final product.

REFERENCES

1. Alfven, H.: Existence of electromagnetic-hydrodynamic waves”, Nature, 150, (1942), 3805, 405. 2. Bansal, J. L.: Viscous Fluid Dynamics, Oxford & IBH Pub. Co., New Delhi, India. (1977) 3. Bansal,́ J. L.): Magnetofluiddynamics of Viscous Fluids, Jaipur Publishing House, Jaipur, India. (1994 4. Bataller, R. C.: Magnetohydrodynamic flow and heat transfer of an upper-convected Maxwell fluid due to a stretching sheet, FDMP, vol.7, no.2, (2011) pp.153–173. 5. Eosboee, M. E., Pourmahmoud, N., Mirzaie, I., Khameneh, P. M., Majidyfar, S. and Ganji, D. D.: Analytical and Numerical analysis of MHD Boundary Layer Flow of an Incompressible Upper-Convected Maxwell Fluid, International Journal of Engineering Science and Technology, Vol. 2(12), (2010), pp. 6909-6917. 6. Jeffery, A.: Magnetohydrodynamics, Oliver and Boyed, New York, USA, (1966). 7. Mukhopadhyay S. and Gorla, R. S. R.: Unsteady MHD boundary layer flow of an upper convected Maxwell fluid past a stretching sheet with first order constructive/destructive chemical reaction, Journal of Naval Architecture and Marine Engineering, vol. 9, (2012),pp. 123–133. 8. Noor, N. F. M.: Analysis for MHD flow of a Maxwell fluid past a vertical stretching sheet in the presence of thermophoresis and chemical reaction, World Academy of Science, Engineering and Technology, vol. 64, (2012), pp. 1019–1023. 9. Pai, S. I.: Viscous Flow Theory: I – Laminar Flow, D.Van Nostrand Co., New York, USA, (1956). 10. Sadeghy, K., Hajibeygi, H. and Taghavi, S. M.: Stagnation-Point Flow of Upper-Convected Maxwell Fluids. International Journal of Non-Linear Mechanics, Vol.41, (2006),pp. 1242-1247. 11. Schichting, H.: Boundary Layer Theory, sixth ed., McGraw-Hill, New York, (1964). 12. Sarpakaya, T.: Flow of non-Newtonian fluids in a magnetic field,” AIChE Journal, vol. 7, (1961),pp. 324–328. 13. Vajravelu, K., Prasad, K. V., Sujatha, A., and Ng, C. O.: MHD flow and mass transfer of chemically reactive upper convected Maxwell fluid past porous surface, Appl. Math. Mech. -Engl. Ed., vol. 33, no. 7, (2012)pp. 899–910. 14. Wang, S. and Tan, W. C.: Stability Analysis of Soret-Driven Double-Diffusive Convection of Maxwell Fluid in a Porous Medium. International Journal of Heat and Fluid Flow, Vol.32, (2011),pp. 88-94. 15. Zheng, L., Liu, L., and Zhang, X.: Maxwell fluids unsteady mixed flow and radiation heat transfer over a stretching permeable plate with boundary slip and nonuniform heat source/sink, Journal of Heat Transfer, vol. 135, (2013), pp. 031705. 16. Zierep, J. and Fetecau, C.: Energetic Balance for the Rayleigh-Stokes Problem of a Maxwell Fluid. International Journal of Engineering Science, Vol.45, (2007), 617-627.

Source of support: Nil, Conflict of interest: None Declared

[Copy right © 2014. This is an Open Access article distributed under the terms of the International Journal of Mathematical Archive (IJMA), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.]