Int. J Latest Trend Math Vol 3 No. 1 December 2014

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1 Int. J Latest Trend Math Vol 3 No. December 04 A Numerical Solution of MHD Heat Transfer in a Laminar Liquid Film on an Unsteady Flat Incompressible Stretching Surface with Viscous Dissipation and Internal Heating M. Subhas Abel*, Jagadish V. Tawade, and 3 Anand Agadi Department of Mathematics, Walchand Institute of Technology, Solapur-6, Maharashtra, INDIA Department of Mathematics, Gulbarga University, Gulbarga , Karnataka, INDIA 3 Department of Mathematics, Basveshawar Engineering College, Bagalkot-5870, Karnataka, INDIA msabel00@yahoo.co.uk 37 Abstract: This study deals with the numerical solution of MHD flow and heat transfer to a laminar liquid film from a horizontal stretching surface. Similarity transformations are used to convert unsteady boundary layer equations to a system of non-linear ordinary differential equations. The resulting non-linear differential equations are solved numerically by using efficient numerical shooting technique with fourth order Runge Kutta algorithm (see references [6] and [7]). The effect of Prandtl number Pr, Eckert number Ec, Magnetic parameter Mn and temperature-dependent parameter on various flow parameters are shown with the aid of graphs. The important observation in this study is, for high values of unsteadiness parameter S reduces the surface temperature which is well in agreement with the earlier published works, under some limiting cases, and temperature-dependent heat absorption is one better suited for effective cooling purpose as temperaturedependent heat generation enhance the temperature in the boundary layer. Key words: Liquid film, unsteady stretching surface, similarity transformation, viscous dissipation, internal heat generation.. Introduction Boundary layer flow and heat transfer in a thin liquid film on an unsteady stretching sheet has received considerable attention from researchers because of their numerous practical applications in many branches of science and technology. The knowledge of flow and heat transfer within a thin liquid film is crucial in understanding the coating process and design of various heat exchangers and chemical processing equipments. Other applications include wire and fiber coating, food stuff processing reactor fluidization, transpiration cooling and so on. The prime aim in almost every extrusion applications is to maintain the surface quality of the extrudate. All coating processes demand a smooth glossy surface to meet the requirements for best appearance and optimum service properties such as low friction, transparency and strength. The problem of extrusion of thin surface layers needs special attention to gain some knowledge for controlling the coating product efficiently. The studies of boundary layer flows of Newtonian and non-newtonian fluids on stretching surfaces have become important, not only because of their technological importance but also in view of the interesting mathematical features presented by the equations governing the flow. Such studies have considerable practical relevance, for example in the manufacture of plastic film, in the extrusion of a polymer sheet from a die and in fibre industries, etc. During the manufacture of these films, the melt issues from a slit and is subsequently stretched to achieve the desired thickness. Such investigations of magneto hydrodynamic (MHD) flow are very important industrially and have applications in different areas of research such as petroleum production and metallurgical International Journal of Latest Trends in Mathematics IJLTCM, E ISSN: Copyright ExcelingTech, Pub, UK (

2 Int. J Latest Trend Math Vol 3 No. December 04 processes. The magnetic field has been used in the process of purification of molten metals from non-metallic inclusions. The study of flow and heat transfer caused by a stretching surface is of great importance in many manufacturing processes such as extrusion process, glass blowing, hot rolling, manufacturing of plastic and rubber sheets, crystal growing, continuous cooling and fibers spinning (Tadmor and Klein 970; Fisher, 976). In all these cases, a study of flow field and heat transfer can be of significant importance because the quality of the final product depends to a large extent on the skin friction coefficient and the surface heat transfer rate. Sakiadis [, ] investigated the flow due to a sheet issuing with constant speed from a slit into a fluid at rest. This flow was of Blasius type, in which the boundary layer thickness increased with the distance from the slit. Sarpakaya [3] was the first researcher to study the MHD flow of a 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. McCormack and Crane [4] gave a similar solution in a closed analytic form for the two dimensional stretching of a flat surface with a velocity proportional to the distance from the slit. Crane [5] was the first among others to consider the steady two-dimensional flow of a Newtonian fluid driven by a stretching elastic flat sheet which moves in its own plane with a velocity varying linearly with the distance from a fixed point. The pioneering works of Crane [5] are subsequently extended by many authors to explore various aspects of the flow and heat transfer occurring in an infinite domain of the fluid surrounding the stretching sheet see [6-]. Wang [, 3], Usha and Shridharan [4], Chen [5, 6],Kumari and Nath [7], Andersson et al. [8, 9] and Dandapat et al. [0,, ]. In the pioneering work of Wang [], the flow of a Newtonian fluid in a thin liquid film past an unsteady stretching sheet was investigated. In his work he reduced the unsteady Navier Stokes equations to a nonlinear ordinary differential equations by means of similarity transformation and solved the same using a kind of multiple shooting method (see Robert and Shipman [5]). Wang [3] himself has used homotopy analysis method to reinvestigate the thin film flow over a stretching sheet. Of late the works of Wang [] to the case of finite fluid domain are extended by several authors [8-4] for fluids of both Newtonian and non-newtonian kinds using various velocity and thermal boundary conditions. Aziz et.al [4] have neglected the magnetic field effect and also used the homotopy analysis method (HAM) for thin film flow and heat transfer on an unsteady stretching sheet with internal heating. There are extensive works in literature concerning the production of thin liquid film either on a vertical wall achieved through the action of gravity or that over a rotating disc achieved through the action of centrifugal forces. If the fluid is very viscous, considerable heat can be produced even though at relatively low speeds, e.g. in the extrusion of plastic, and hence the heat transfer results may alter appreciably due to viscous dissipation. To the author s knowledge, the influence of viscous dissipation on heat transfer in a finite liquid film over a continuously moving surface has not yet been discussed in the literature. Aforementioned studies have neglected the viscous dissipation effect on the heat transfer which is important in view point of desired properties of the outcome. It is the purpose of this present work to investigate the effect of viscous dissipation and internal heat generation along with an external uniform magnetic field for flow and heat transfer analysis in a thin liquid film on an unsteady stretching sheet.. Mathematical modeling Let us consider a thin elastic sheet which emerges from a narrow slit at the origin of a Cartesian co-ordinate system for investigations as shown schematically in Fig. The continuous sheet at y 0 is parallel with the x-axis and moves in its own plane with the velocity bx Ux, t () ( t) 38

3 Int. J Latest Trend Math Vol 3 No. December where b and are both positive constants with dimension per time. The surface temperature T s of the stretching sheet is assumed to vary with the distance x from the slit as bx Tsx, tt0 Tref ( t) where T 0 is the temperature at the slit and 0 Tref T. The term 0 bx ( t) 3 Tref can be taken as a constant reference temperature such that can be recognized as the Local Reynolds number based on the surface velocityu. The expression () for the velocity of the sheet U( x, t ) reflects that the elastic sheet which is fixed at b the origin is stretched by applying a force in the positive x-direction and the effective stretching rate ( t ) increase with time as 0. With the same analogy the expression for the surface temperature Ts ( x, t ) given by equation () represents a situation in which the sheet temperature decreases from T 0 at the slit in proportion to x and such that the amount of temperature reduction along the sheet increases with time. The applied transverse magnetic field is assumed to be of variable kind and is chosen in its special form as B x, t B0 - t. (3) The particular form of the expressions for U( x, t ), T ( x, t ) and Bx (, t) are chosen so as to facilitate the construction of a new similarity transformation which enables in transforming the governing partial differential equations of momentum and heat transport into a set of non-linear ordinary differential equations. Consider a thin elastic liquid film of uniform thickness ht () lying on the horizontal stretching sheet (Fig.). The x-axis is chosen in the direction along which the sheet is set to motion and the y-axis is taken perpendicular to it. The fluid motion within the film is primarily caused solely by stretching of the sheet. The sheet is stretched by the action of two equal and opposite forces along the x-axis. The sheet is assumed to have velocity U as defined in equation () and the flow field is exposed to the influence of an external transverse magnetic field of strength B as defined in equation (3). We have neglected the effect of latent heat due to evaporation by assuming the liquid to be nonvolatile. Further the buoyancy is neglected due to the relatively thin liquid film, but it is not so thin that intermolecular forces come into play. The velocity and temperature fields of the liquid film obey the following boundary layer equations u v 0, (4) x y u u u u B u v u t x y y, T T T k T u u v Q( T s T0 ). t x y Cp y Cp y The pressure in the surrounding gas phase is assumed to be uniform and the gravity force gives rise to a hydrostatic pressure variation in the liquid film. In order to justify the boundary layer approximation, the length scale in the primary flow direction must be significantly larger than the length scale in the cross stream direction. We choose the representative measure of the film thickness to be b s () (5) (6) so that the scale ratio is large enough i.e., x b.

4 Int. J Latest Trend Math Vol 3 No. December 04 This choice of length scale enables us to employ the boundary layer approximations. Further it is assumed that the induced magnetic field is negligibly small. The associated boundary conditions are given by u U, v 0, T Ts at y 0, (7) u T 0 at y h, (8) y y dh v at y h. (9) dt At this juncture we make a note that the mathematical problem is implicitly formulated only for x 0. Further it is assumed that the surface of the planar liquid film is smooth so as to avoid the complications due to surface waves. The influence of interfacial shear due to the quiescent atmosphere, in other words the effect of surface tension is u assumed to be negligible. The viscous shear stress T and the heat flux q k vanish at the y y adiabatic free surface (at y = h). 40 Similarity transformations: We now introduce dimensionless variables b,,, t xyt xf 3 bx Tx, y, tt 0 Tref t, b t y. f and and the similarity variable as (0) () () automatically assures mass conversion given in equation (4). The velocity components are readily obtained as: The physical stream function x, yt, bx u f, y t b v f. (4) x t The mathematical problem defined in equations (4) (8) transforms exactly into a set of ordinary differential equations and their associated boundary conditions: S f f f ff f f Mn, S Pr 3 ( f ) f Ec Pr f, (6) f(0), f(0) 0, (0), (7) f ( ) 0, ( ) 0, (8) (3) (5)

5 Int. J Latest Trend Math Vol 3 No. December 04 S f ( ). (9) Where a prime denotes the differentiation with respect to and S is the dimensionless measure of the b unsteadiness. Further, the dimensionless film thickness denotes the value of the similarity variable at the free surface so that equation () gives b t h. Yet is an unknown constant, which should be determined as an integral part of the boundary value problem. The rate at which film thickness varies can be obtained differentiating equation (0) with respect to t, in the form dh dt bt y. Thus the kinematic constraint at h() t given by equation (9) transforms into the free surface condition (). It is noteworthy that the momentum boundary layer equation defined by equation (6) subject to the relevant boundary conditions (7) (9) is decoupled from the thermal field; on the other hand the temperature field ( ) is coupled with the velocity field f ( ). The most important characteristics of flow and heat transfer are the shear stress s and the heat flux the stretching sheet that are defined as u s y y0 T qs k y y0 where is the fluid dynamic viscosity. The local skin friction coefficient number Nu x for fluid flow in a thin film can be expressed as (0) () () (3) 4 qs on C f and the local Nusselt u y y0 Cf Re x f 0 (4) U x T / '(0)Re 3/ Nux t x, (5) Tref y y0 Ux where Re x, the local Reynolds number and Tref denotes the same reference temperature (temperature difference) as in equation (). 3. Numerical approach

6 Int. J Latest Trend Math Vol 3 No. December 04 The non-linear differential equations (5) and (6) with appropriate boundary conditions given in (7) to (9) are solved numerically, by the most efficient numerical shooting technique with fourth order Runge Kutta algorithm (see references [6] and [7]). The BVP is equivalent to a system of five first order differential equations with six boundary conditions. The crucial part of the numerical solution is to determine the dimensionless film thickness. Eqs. (5) and (6) are integrated numerically by fourth order Runge Kutta scheme from 0to with f(0) 0, f(0) and (0) and guessed trail values f (0), (0) and. However, the numerical solution thus obtained will not generally satisfy the right-end boundary conditions f ( ) 0, (0) 0 and f ( ) S /. At this end Newton Raphson scheme is employed to correct the three arbitrary guess values such that the numerical solution will eventually satisfy the required boundary conditions (8) and (9). The convergence criterion largely depends on fairly good guesses of the initial conditions in the shooting technique. The iterative process is terminated until the relative difference between the current and the 6 previous iterative values of f ( ) matches with the value of S / up to a tolerance of0. For further details on the numerical procedure, the readers are referred to [6, 7, 8] 4. Results and discussion The exact solution do not seem feasible for a complete set of equations (5)-(6) because of the nonlinear form of the momentum and thermal boundary layer equations. This fact forces one to obtain the solution of the problem numerically. Appropriate similarity transformation is adopted to transform the governing partial differential equations of flow and heat transfer into a system of non-linear ordinary differential equations. The resultant boundary value problem is solved by the efficient shooting method. It is noteworthy to mention that the solution exists only for small value of unsteadiness parameter 0 S. Moreover, when S 0 the solution approaches to the analytical solution obtained by Crane [5] with infinitely thick layer of fluid ( ). The other limiting solution corresponding to S represents a liquid film of infinitesimal thickness ( 0 ). The numerical results are obtained for 0 S. Present results are compared with some of the earlier published results in some limiting cases are shown in Table and. The effects of various parameters influencing the dynamics are shown in Fig. Fig.. Fig. shows the variation of film thickness β with the unsteadiness parameter S. It is evident from this plot that the film thickness decreases monotonically when S is increased from 0 to. This result concurs with that observed by Wang [3]. The variation of film thickness β with respect to the magnetic parameter Mn is projected in Fig.3 for different values of unsteadiness parameter. The effect of magnetic parameter Mn, Prandtl number Pr, Eckert number Ec and temperature-dependent parameter on the surface temperature are respectively illustrated from Fig.4-Fig.7. Clearly, increasing values of magnetic parameter Mn causes the surface temperature to blow-up monotonically. Small values of Eckert number Ec almost keep the surface temperature a constant but enhance the surface temperature for higher values. The opposite effect is exhibited in case of Pr i.e., increasing values of Pr decreases the surface temperature. For Prandtl numbers of order unity and below the surface temperature ( ) attains a finite value below and the temperature gradients extend all the way to the free surface. In the limiting case Pr 0, however, the dimensionless surface temperature tends to unity i.e. the temperature T becomes uniform in the vertical direction and equals T s. This is consistent with the trivial solution ( ) obtained from the thermal energy equation (5) when Pr = 0. At sufficiently high Prandtl number, i.e. low thermal diffusivity, the surface temperature remained practically equal to zero. The temperature-dependent heat absorption ( 0) is to reduce the temperature 4

7 Int. J Latest Trend Math Vol 3 No. December 04 distribution significantly throughout the region as the ( 0) brings about the temperature increase throughout the entire region. The observed results hold good for different values of unsteadiness parameter S. The effect of magnetic parameter Mn on the horizontal velocity profiles are depicted in Fig.8 (a) and 8(b) for two different values of unsteadiness parameter S. From both these plots one can make out that the increasing values of magnetic parameter decreases the horizontal velocity. This is due to the fact that applied transverse magnetic field produces a drag in the form of Lorentz force thereby decreasing the magnitude of velocity. The drop in horizontal velocity as a consequence of increase in the strength of magnetic field is observed for S = 0.8 as well as S =.. Fig.9(a) and 9(b) demonstrate the effect of Prandtl number Pr on the temperature profiles for two different values of unsteadiness parameter S. These plots reveals the fact that for a particular value of Pr the temperature increases monotonically from the free surface temperature T s to wall velocity the T 0 as observed by Anderson et al [9]. The thermal boundary layer thickness decreases drastically for high values of Pr i.e., low thermal diffusivity. From these figure we observe that Prandtl number Pr will speed up the cooling of the thin film flow. Fig.0(a) and 0(b) project the effect of Eckert number Ec on the temperature profiles for two different values of unsteadiness parameter S. The effect of viscous dissipation is to enhance the temperature in the fluid film. i.e., increasing values of Ec contributes in thickening of thermal boundary layer. For effective cooling of the sheet a fluid of low viscosity is preferable. Fig.(a) and Fin.(b) presents the effect of temperature-dependent heat generation/absorption on the temperature profile for different values of unsteadiness parameter S. For 0 reduces the temperature and for 0 enhances the temperature in the fluid. The dimensionless wall temperature gradient The effect The dimensionless wall temperature gradient Eckert number Ec, while the effect of 43 '(0) takes a higher value at a large Prandtl number Pr. '(0) for S. only marginally exceeds that for S 0.8 for Pr (see fig.). '(0) takes a uniform value at certain moderate values of '(0) decreases with their increasing Ec (see fig. 3). Table and Table give the comparison of present results with that of Wang [3] and Aziz et.al [4]. Without any doubt, from these tables, we can claim that our results are in excellent agreement with that of references [3 & 4] under some limiting cases. Table.3 tabulates the values of surface temperature for various values of Mn, Pr, Ec and. This table also reveals that Mn and proportionately increase the surface temperature whereas Pr and Ec decreases the surface temperature. 5. Conclusions The present method gives solutions for steady incompressible boundary layer flow of a fluid film over a heated stretching surface in the presence of a variable transverse magnetic field including the viscous dissipation and internal heating effect. Present results reveal that Magnetic field and viscous dissipative effects play significant role on controlling the heat transfer from stretching sheet to the liquid film. The important findings pertaining to the present analysis are. The effect of transverse magnetic field on a viscous incompressible electrically conducting fluid is to suppress the velocity field which in turn causes the enhancement of the temperature field.. The viscous dissipation effect is characterized by Eckert number (Ec) in the present analysis. Comparing to the results without viscous dissipation, one can see that the dimensionless temperature will increases when the fluid is being heated ( Ec 0) but decreases when the fluid is being cooled ( Ec 0). This reveals that effect of viscous dissipation is to enhance the temperature in the thermal boundary layer.

8 Int. J Latest Trend Math Vol 3 No. December For a wide range of Pr, the effect of viscous dissipation is found to increase the dimensionless free surface temperature () for the fluid cooling case. The impact of viscous dissipation on () diminishes in the two limiting cases: Pr 0 and Pr, in which situations () approaches unity and zero respectively. 4. The effect of internal heat generation/absorption is to generate temperature for increasing positive values and absorb temperature for decreasing negative values. However negative value of temperature dependent parameter is better suited for cooling purpose. References:. B.C.Sakiadis, Boundary layer behavior on continuous solid surface: I Boundary layer on a continuous flat surface. AIChE J 96:7():3-5.. B.C.Sakiadis, Boundary layer behavior on continuous solid surface: I Boundary layer equation for two dimensional and axisymmetric flows. AIChE J 96:7: T. Sarpakaya, Flow of non-newtonian fluids in a magnetic field, AIChE. J. 7 (96) McCormack, P. D., P.D. Crane, L.: Physical fluid dynamics. New York: Academic Press L.J. Crane, flow past a stretching plate, Z. Angrew. Math. Phys. (970) P.S. Gupta, A.S. Gupta, Heat and Mass transfer on a stretching sheet with suction or blowing, Can. J. Chem. Eng. 55 (977) B.K. Dutta, A.S. Gupta, cooling of a stretching sheet in a various flow, Ind. Eng. Chem. Res. 6 (987) C.K. Chan, M.I. Char, Heat transfer of a Continuous stretching surface with suction or blowing, J. math. Anal. Appl. 35 (988) A. Chakrabarti, A.S. Gupta, Hydromagnetic flow and heat transfer over a stretching sheet, Q. Appl. Math. 37 (979) N. Afzal, Heat transfer from a stretching surface, Int. J. Heat and Mass Transfer 36 (993) M. Massoudi, I. Christie, Effects of variable viscosity and viscous dissipation on the flow of a third grade fluid in a pipe, Int. J. Non-Linear Mech. 30 (995) C.Y. Wang, Liquid film on an unsteady stretching surface, Quart Appl. Math 48 (990) C. Wang, Analytic solutions for a liquid film on an unsteady stretching surface, Heat Mass Transfer 4 (006) R. Usha, R. Sridharan, On the motion of a liquid film on an unsteady stretching surface, ASME Fluids Eng. 50 (993) Chien-Hsin Chen, Effect of viscous dissipation on heat transfer in a non-newtonian liquid film over an unsteady stretching sheet, J. Non-Newtonian Fluid Mech. 35(006) Chien-Hsin Chen, Marangoni effects on forced convection of power-law liquids in a thin film over a stretching surface, Physics letters A 370 (007) M. Kumari, G. Nath, Unsteady MHD film flow over a rotating infinite disk, Int. J. Engng. Sci. 4 (004) H.I. Anderson, J.B. Aarseth, N. Braud, B.S. Dandapat, Flow of a power-law fluid film on an unsteady stretching surface, J. Non-Newtonian Fluid Mech 6 (996) H.I. Anderson, J.B. Aarseth, B.S. Dandapat, Heat transfer in a liquid film on an unsteady stretching surface, Int. J. Heat Mass Transfer 43 (000) B.S. Dandapat, B. Santra, H.I. Anderson, Thermocapillary in a liquid film on an unsteady stretching surface, Int. J. Heat Mass Transfer 49 (003) B.S. Dandapat, B. Santra, K. Vajravelu, The effects of variable fluid properties and thermocapillarity on the flow of a thin film on an unsteady stretching sheet, Int. J. Heat Mass Transfer 50 (007) B.S. Dandapat, P.C. Ray, The effect of thermocapillarity on the flow of a thin liquid film on a rotating disk, J Phys D Appl Phys 7 (994) T. Hayat, S. Saif, Z. Abbas, The influence of heat transfer in an MHD second grade fluid film over an unsteady stretching sheet, Physics Letters A 7 (008)-9 44

9 Int. J Latest Trend Math Vol 3 No. December R.C. Aziz I. Hashim A.K. Alomari, Thin film flow and heat transfer on an unsteady stretching sheet with internal heating Meccanica (0) 46: S.M. Roberts, J.S. Shipman, Two point boundary value problems: Shooting Methods, Elsevier, New York, S.D. Conte, C. de Boor, Elementary Numerical Analysis, McGraw-Hill, New York, T. Cebeci, P. Bradshaw, Physical and computational aspects of convective heat transfer, Springer-Verlag, New York, F.M. White, Viscous fluid flow, McGraw Hill International Edition, McGraw Hill, New York, Nomenclature: b stretching rate U x y u v T t h S C p sheet velocity - [s ] [m s ] horizontal coordinate [m] vertical coordinate [m] horizontal velocity component vertical velocity component temperature [K] time [s] film thickness [m] Unsteadiness parameter, b specific heat - - [J kg K ] [m s ] [m s ] f dimensionless stream function, Eq. (0) Pr Ec Mn q Re x Prandtl number, k Eckert number, U C T T p s 0 B 0 Magnetic parameter, b T heat flux, k [J s m ] y local Reynolds number, Ux C f skin friction coefficient, Eq. (4) Nu x local Nusselt number, Eq. (5) Greek symbols constant [s ] dimensionless film thickness

10 Int. J Latest Trend Math Vol 3 No. December 04 Q Temperature-dependent parameter, cb similarity variable, Eq. () dimensionless temperature, Eq. () k s thermal diffusivity [m s ] dynamic viscosity [kg m s ] kinematic viscosity density 3 [kg m ] shear stress, u stream function [m s ] / y [kg m s ] [m s ] p 46 Table : Comparison of values of skin friction coefficient f 0 with Mn = 0.0 Wang [3] Aziz et.al [4] Present work S β f 0 f 0 β f 0 f 0 β f Note: Wang [3] and Aziz [4] have used different similarity transformation due to which the value of his paper is the same as f 0 of our results. f 0 in Table : Comparison of values of surface temperature = = 0.0 and wall temperature gradient 0 Pr Wang [3] Aziz et. al[4] Present work with Mn = Ec

11 Int. J Latest Trend Math Vol 3 No. December S = 0.8 and β = S =. and β = Note: Wang [3] has used different similarity transformation due to which the value of same as 0of our results. 0 in his paper is the

12 Int. J Latest Trend Math Vol 3 No. December 04 for various values of Mn, Pr, Ec, and S. Mn Pr Ec Table 3: Values of surface temperature S = 0.8 S =

13 Int. J Latest Trend Math Vol 3 No. December Sli x y T u y = y = h(t For Fig.. Schematic representation of a liquid film on an elastic sheet. Fig : Variation of film thickness β with unstudieness parameter S with Mn=0.0 Fig 3: Variation of film thickness β with magnetic parameter with Mn Fig 4: Variation of surface temperature θ (β) with the magnetic parameter Mn Fig 5: Variation of surface temperature θ (β) with S=0.8 (solid line) and S=.(broken line) with the prand tl number Pr

14 Int. J Latest Trend Math Vol 3 No. December Fig 6: Variation of surface temperature θ (β) with the Eckert number Ec Fig 7: Variation of surface temperature θ (β) with the Heat source/sink parameter Q Fig 8(a): Variation in the velocity profiles f (η) for different values of magnetic parameter Mn with S=0.8 Fig 8(b): Variation in the velocity profiles f (η) for different values of magnetic parameter Mn with S=.

15 Int. J Latest Trend Math Vol 3 No. December 04 5 Fig 9(a): Variation in the temperature profiles θ(η) for different values of Prandtl number Pr with S=0.8 Fig 9(b): Variation in the temperature profiles θ(η) for different values of Prandtl number Pr with S=. Fig 0(a): Variation in the temperature profiles θ(η) for different values of Eckert number Ec with S=0.8 Fig 0(b): Variation in the temperature profiles θ(η) for different values of Eckert number Ec with S=.

16 Int. J Latest Trend Math Vol 3 No. December 04 5 Fig (a): Variation in the temperature profiles θ(η) for different values of Heat source/sink ϒ with S=0.8 Fig (b): Variation in the temperature profiles θ(η) for different values of Heat source/sink ϒ with S=. Fig : Dimensionless temperature gradient - θ(0) at the sheet vs Prandtl number for S=0.8 (solid lines) and S=. (broken lines) Fig 3: Dimensionless temperature gradient - θ(0) at the sheet vs Eckert number for S=0.8 (solid lines) and S=. (broken lines)