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1. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 4, July - August (2013) © IAEME 387 MHD AND HEAT TRANSFER IN A THIN FILM OVER AN UNSTEADY STRETCHING SURFACE WITH COMBINED EFFECT OF VISCOUS DISSIPATION AND NON-UNIFORM HEAT SOURCE Anand H. Agadi1* , M . Subhas Abel2 and Jagadish V. Tawade3 1* Department of Mathematics, Basaveshwar Engineering College, Bagalkot-587102, INDIA 2 Department of Mathematics, Gulbarga University, Gulbarga- 585 106, INDIA 3 Department of Mathematics, Bheemanna Khandre Institute of Technology, Bhalki-585328 ABSTRACT We have studied two-dimensional flow of a thin film over a horizontal stretching surface. The flow of a thin fluid film and subsequent heat transfer from the stretching surface is investigated with the aid of similarity transformation. The transformation enables to reduce the unsteady boundary layer equations to a system of non-linear ordinary differential equations. Numerical computation for the resulting nonlinear differential equations is obtained by Runge-Kutta fourth order method with efficient shooting technique, which agrees well with the analytic solution. It is shown that the heat fluxes from the liquid to the elastic sheet decreases with S forPr 0.1≤ and increases with S forPr 1≥ . Some important findings reported in this work reveals that the effect of non-uniform heat source have significant impact in controlling rate heat transfer in the boundary layer region. Key words: Eckert number, MHD, Prandtl number, thin film, unsteady stretching surface. 1. INTRODUCTION The study of the flow resulting from a stretching boundary is important in process industry such as the extrusion of sheet material into the coolant environment. The tangential velocity imparted by the stretching sheet induces motion in the extruding fluid which ultimately solidified and formed a sheet. In fact, stretching imparts a unidirectional orientation to the extrudate, thereby improving its mechanical properties and the quality of the final product. Crane [1] first modeled this flow configuration as a steady two-dimensional boundary layer flow caused by the stretching of a sheet which moves in its own plane with velocity varying with distance from the slit and obtained an exact INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING AND TECHNOLOGY (IJMET) ISSN 0976 – 6340 (Print) ISSN 0976 – 6359 (Online) Volume 4, Issue 4, July - August (2013), pp. 387-400 © IAEME: www.iaeme.com/ijmet.asp Journal Impact Factor (2013): 5.7731 (Calculated by GISI) www.jifactor.com IJMET © I A E M E
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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 4, July - August (2013) © IAEME 388 solution analytically. This simple configuration has attracted several researchers [2 - 6] for the last four decades and is extensively studied. The hydrodynamics of the finite fluid domain ( thin liquid film), over a stretching sheet was first considered by Wang [8] who reduced the unsteady Nervier–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 [9]). Wang [10] himself has used homotopy analysis method to reinvestigate the thin film flow over a stretching sheet. Of late the works of Wang [8] to the case of finite fluid domain are extended by several authors [11-15] for fluids of both Newtonian and non-Newtonian kinds using various velocity and thermal boundary conditions. Motivated by all these works we contemplate to study the effects of non-uniform heat source, and viscous dissipation in presence of thermal radiation on the flow and heat transfer in a thin liquid film over an unsteady stretching sheet, which is subjected to an external magnetic field. 2. MATHEMATICAL FORMULATION 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 1. The continuous sheet at 0y = is parallel with the x-axis and moves in its own plane with the velocity ( ), (1 ) bx U x t tα = − (1) where b and α are both positive constants with dimension per time. The surface temperature sT of the stretching sheet is assumed to vary with the distance x from the slit as ( ) 32 2 0, (1 ) 2 s ref bx T x t T T tα υ − = − − (2) Where 0T is the temperature at the slit and refT can be taken as a constant reference temperature such that 00 refT T≤ ≤ . The term 2 (1 ) bx tυ α− can be recognized as the Local Reynolds number based on the surface velocityU . The expression (1) for the velocity of the sheet ( , )U x t reflects that the elastic sheet which is fixed at the origin is stretched by applying a force in the positive x-direction and the effective stretching rate (1 ) b tα− increase with time as 0 1α≤ < . The applied transverse magnetic field is assumed to be of variable kind and is chosen as ( ) ( ) 1 2 0, 1- .B x t B tα − = (3) The sheet is assumed to have velocity U as defined in equation (1) and the flow field is exposed to the influence of an external transverse magnetic field of strength B as defined in equation (3). The velocity and temperature fields of the liquid film obey the following boundary layer equations
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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 4, July - August (2013) © IAEME 389 0, u v x y ∂ ∂ + = ∂ ∂ (4) 2 2 2 , u u u u B u v u t x y y σ υ ρ ∂ ∂ ∂ ∂ + + = − ∂ ∂ ∂ ∂ (5) 22 2 p p p T T T k T u q u v t x y C y C y C µ ρ ρ ρ ′′′ ∂ ∂ ∂ ∂ ∂ + + = + + ∂ ∂ ∂ ∂ ∂ (6) The non-uniform heat source/sink (see [30]) is modeled as 0 0 ( ) [ *( ) ( ) *],w s ku x q A T T f T T B xυ ′′′ ′= − + − (7) Where A* and B* are the coefficients of space and temperature dependent heat source/sink respectively. Here we make a note that the case 0*,0* >> BA corresponds to internal heat generation and that 0*,0* << BA corresponds to internal heat absorption. The associated boundary conditions are given by , 0, at 0,su U v T T y= = = = (8) 0 at , u T y h y y ∂ ∂ = = = ∂ ∂ (9) at . dh v y h dt = = (10) At this juncture we make a note that the mathematical problem is implicitly formulated only for 0x ≥ . 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 assumed to be negligible. The viscous shear stress u y τ µ ∂ = ∂ and the heat flux T q k y ∂ = − ∂ vanish at the adiabatic free surface (at y = h). We now introduce dimensionless variables andf θ and the similarity variable η as ( ) ( ) 1 2 , , , 1 x y t f b x t ψ η υ α = − (11) ( ) 1 2 . 1 b y t η υ α = − (12)
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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 4, July - August (2013) © IAEME 390 ( ) ( ) ( ) 0 2 3 2 , , , 2 1 ref T T x y t b x T t θ η υ α − − = − (13) The physical stream function ( ), ,x y tψ automatically assures mass conversion given in equation (4). The velocity components are readily obtained as: ( ), 1 bx u f y t ψ η α ∂ ′= = ∂ − (14) ( ) 1 2 . 1 b v f x t ψ υ η α ∂ = − = − ∂ − (15) The mathematical problem defined in equations (4) – (6) and (8) – (9) transforms exactly into a set of ordinary differential equations and their associated boundary conditions: ( ) 2 Mn , 2 S f f f ff f f η ′ ′′ ′ ′′ ′′′ ′+ + − = − (16) ( ) 2S Pr 3 2 EcPr ( * * ), 2 f f f A f Bθ ηθ θ θ θ θ ′ ′ ′ ′′ ′′ ′+ + − = − − + (17) (0) 1, (0) 0, (0) 1,f f θ′ = = = (18) ( ) 0, ( ) 0,f β θ β′′ ′= = (19) S ( ) . 2 f β β = (20) Here S b α ≡ is the dimensionless measure of the unsteadiness and the prime indicates differentiation with respect toη . Further, β denotes the value of the similarity variable η at the free surface so that equation (12) gives ( ) 1 2 . 1 b h t β υ α = − (21) Yet β 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 (21) with respect to t, in the form
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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 4, July - August (2013) © IAEME 391 ( ) 1 2 . 2 1 dh dt b t αβ υ α = − − (22) Thus the kinematic constraint at ( )y h t= given by equation (10) transforms into the free surface condition (22). It is noteworthy that the momentum boundary layer equation defined by equation (16) subject to the relevant boundary conditions (18) – (20) is decoupled from the thermal field; on the other hand the temperature field ( )θ η is coupled with the velocity field ( )f η . Since the sheet is stretched horizontally the convection least affects the flow and hence there is a one-way coupling of velocity and thermal fields. 3. NUMERICAL SOLUTION The non-linear differential equations (16) and (17) with appropriate boundary conditions given in (18) to (20) are solved numerically, by the most efficient numerical shooting technique with fourth order Runge–Kutta algorithm (see references [16]). The non-linear differential equations (16) and (17) are first decomposed to a system of first order differential equations in the form ( ) ( ) 20 1 2 1 2 1 2 1 0 2 1 20 1 1 0 1 1 0 1 0 2 , , +Mn , 2 S , Pr 3 2 Ec Pr ( * * ) . 2 df df df f f S f f f f f f d d d d d f f f A f B d d η η η η θ θ θ θ ηθ θ θ θ η η = = = + + − ′= = + + − − + + (25) Corresponding boundary conditions take the form, 1 0 0(0) 1, (0) 0, (0) 1,f f θ= = = (26) 2 1( ) 0, ( ) 0,f β θ β= = (27) 0 S ( ) . 2 f β β = (28) Here 0 0( ) ( ) and ( ) ( ).f fη η θ η θ η= = The above boundary value problem is first converted into an initial value problem by appropriately guessing the missing slopes 2 1(0) and (0)f θ . The resulting IVP is solved by shooting method for a set of parameters appearing in the governing equations and a known value of S. The value of β is so adjusted that condition (28) holds. This is done on the trial and error basis. The value for which condition (28) holds is taken as the appropriate film thickness and the IVP is finally solved using this value of β. The step length of h = 0.01 is employed for the computation purpose. 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 previous iterative values of ( )f β matches with the value of 2 Sβ up to a tolerance of 6 10− . Once the convergence in achieved we integrate the resultant ordinary differential equations using standard fourth order Runge–Kutta method with the given set of parameters to obtain the required solution.
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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 4, July - August (2013) © IAEME 392 4. RESULTS AND DISCUSSION The exact solution do not seem feasible for a complete set of equations (16)-(17) because of the non linear 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 note worthy to mention that the solution exists only for small value of unsteadiness parameter0 2S≤ ≤ . Moreover, when 0S → the solution approaches to the analytical solution obtained by Crane [1] with infinitely thick layer of fluid ( β → ∞ ). The other limiting solution corresponding to 2S → represents a liquid film of infinitesimal thickness ( 0β → ). The numerical results are obtained for0 2S≤ ≤ . Present results are compared with some of the earlier published results in some limiting cases which are tabulated in Table 1. The effects of magnetic parameter on various fluid dynamic quantities are shown in Fig.2 – Fig.13 for different unsteadiness parameter. Fig.2 shows the variation of film thickness β with the unsteadiness parameter. It is evident from this plot that the film thickness β decreases monotonically when S is increased from 0 to 2. This result concurs with that observed by Wang [10]. The variation of film thickness β with respect to the magnetic parameter Mn is projected in Fig.3 for different values of unsteadiness parameter. It is clear from this plot that the increasing values of magnetic parameter decreases the film thickness. The result holds for different values of unsteadiness parameter S. The variation of free-surface velocity ( )f β′ with respect to Mn is shown in Fig.4. The free surface velocity behaves almost as a constant function of Mn as can be seen from Fig.4. The effect of Mn on the wall shear stress parameter ( )0f ′′− is illustrated in Fig.5. Clearly, increasing values Mn results in increasing the wall shear stress. Fig.6 demonstrates the effect of Mn on the free- surface temperature ( )θ β . From this plot it is evident that the free surface temperature increases monotonically with Mn. Fig.7 highlights the effect of Mn on the dimensionless wall heat flux ( )0θ′− . It is found from this plot that the dimensionless wall heat flux ( )0θ′− decreases with the increasing values of Mn. The effect of Mn on ( )f β′ , ( )0f ′′− , ( )θ β and ( )0θ′− is observed to be same for different values of unsteadiness parameter S. The effect of Mn on the axial velocity is depicted in Figs.8(a) and (b) for two different values of S. From these plots it is clear that the increasing values of magnetic parameter decreases the axial 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 both the values of S = 0.8 and S = 1.2. Figs.9 (a) and (b) depicts the effect of Mn on temperature profiles for two different values of S. The results show that the thermal boundary layer thickness increases with the increasing values of Mn. The increasing frictional drag due to the Lorentz force is responsible for increasing the thermal boundary layer thickness. Fig.10 (a) and 10(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 sT to wall velocity the 0T .
7.
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 4, July - August (2013) © IAEME 393 Fig.11 (a) and 11(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. Fig.12 (a) and Fig.12 (b) presents the effect of space dependent heat source/sink parameter A* and Fig.13 (a) and Fig.13 (b) presents the effect of temperature dependent heat source/sink parameter B* on the temperature profile for different values of unsteadiness parameter S .The effect of sink parameter ( * 0, * 0)A B< < reduces the temperature in the fluid as the effect of source parameter ( * 0, * 0)A B> > enhances the temperature. For effective cooling of the sheet, heat sink is preferred. Table 1 and Table 2 give the comparison of present results with that of Wang [10]. Without any doubt, from these tables, we can claim that our results are in excellent agreement with that of Wang [10] under some limiting cases. Table 3 tabulates the values of surface temperature ( )θ β for various values of Mn, Pr, Ec, A* and B*. This table also reveals that Mn, Ec, A* and B* proportionately increase the surface temperature whereas opposite effect is seen in case of Pr. 5. CONCLUSIONS A theoretical study of the boundary layer behavior in a liquid film over an unsteady stretching sheet is carried out including the effects of a variable transverse magnetic field including viscous dissipation and non-uniform heat source/sink. The effect of several parameters controlling the velocity and temperature profiles are shown graphically and discussed briefly. Some of the important findings of our analysis obtained by the graphical representation are listed below. 1. 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. 2. For a wide range of Pr, the effect viscous dissipation is found to increase the dimensionless free-surface temperature ( )θ β for the fluid cooling case. The impact of viscous dissipation on ( )θ β diminishes in the two limiting cases: Pr 0 and Pr→ → ∞ , in which situations ( )θ β approaches unity and zero respectively. 3. 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 increase 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. 4. The effect of non-uniform heat source/sink parameter is to generate temperature for increasing positive values and absorb temperature for decreasing negative values. Hence non- uniform heat sinks are better suited for cooling purpose.
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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 4, July - August (2013) © IAEME 394 TABLE 1: Comparison of values of skin friction coefficient ( )0f ′′ with Mn = 0.0 S Wang [10] Present Results β ( )0f ′′ ( )0f β ′′ β ( )0f ′′ 0.4 5.122490 -6.699120 -1.307785 4.981455 -1.134098 0.6 3.131250 -3.742330 -1.195155 3.131710 -1.195128 0.8 2.151990 -2.680940 -1.245795 2.151990 -1.245805 1.0 1.543620 -1.972380 -1.277762 1.543617 -1.277769 1.2 1.127780 -1.442631 -1.279177 1.127780 -1.279171 1.4 0.821032 -1.012784 -1.233549 0.821033 -1.233545 1.6 0.576173 -0.642397 -1.114937 0.576176 -1.114941 1.8 0.356389 -0.309137 -0.867414 0.356390 -0.867416 Note: Wang [10] has used different similarity transformation due to which the value of ( )0f β ′′ in his paper is the same as ( )0f ′′ of our results. TABLE 2: Comparison of values of surface temperature ( )θ β and wall temperature gradient ( )0θ′− with Mn = Ec = A* = B* = 0.0 Pr Wang [10] Present Results ( )θ β ( )0θ′− ( )0θ β ′− ( )θ β ( )0θ′− S = 0.8 and β = 2.15199 0.01 0.960480 0.090474 0.042042 0.960438 0.042120 0.1 0.692533 0.756162 0.351378 0.692296 0.351920 1 0.097884 3.595790 1.670913 0.097825 1.671919 2 0.024941 5.244150 2.436884 0.024869 2.443914 3 0.008785 6.514440 3.027170 0.008324 3.034915 S = 1.2 and β = 1.127780 0.01 0.982331 0.037734 0.033458 0.982312 0.033515 0.1 0.843622 0.343931 0.304962 0.843485 0.305409 1 0.286717 1.999590 1.773032 0.286634 1.773772 2 0.128124 2.975450 2.638324 0.128174 2.638431 3 0.067658 3.698830 3.279744 0.067737 3.280329 Note: Wang [10] has used different similarity transformation due to which the value of ( )0θ β ′− in his paper is the same as ( )0θ′− of our results.
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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 4, July - August (2013) © IAEME 395 TABLE 3: Values of surface temperature ( )θ β for various values of Mn, Pr, Ec, A*, B* and S Mn Pr Ec A* B* ( )θ β S = 0.8 S = 1.2 0.0 1.0 0.02 0.05 0.05 0.257696 0.496022 1.0 1.0 0.02 0.05 0.05 0.420739 0.618190 2.0 1.0 0.02 0.05 0.05 0.526782 0.692995 5.0 1.0 0.02 0.05 0.05 0.695757 0.806962 8.0 1.0 0.02 0.05 0.05 0.776253 0.913333 1.0 0.01 0.02 0.05 0.05 1.030899 1.009712 1.0 0.1 0.02 0.05 0.05 0.931433 0.959465 1.0 1 0.02 0.05 0.05 0.420739 0.618190 1.0 10 0.02 0.05 0.05 0.011137 0.061941 1.0 100 0.02 0.05 0.05 0.000095 0.000238 1.0 1.0 0.01 0.05 0.05 0.420304 0.617857 1.0 1.0 1.0 0.05 0.05 0.463423 0.650865 1.0 1.0 2.0 0.05 0.05 0.506978 0.684207 1.0 1.0 5.0 0.05 0.05 0.637642 0.784232 1.0 1.0 0.02 0.05 0.05 0.227566 0.423871 1.0 1.0 0.02 0.05 0.05 0.420739 0.618190 1.0 1.0 0.02 0.05 0.05 0.715871 0.838324 1.0 1.0 0.02 0.05 0.05 0.826899 0.906104 1.0 1.0 0.02 -0.4 0.05 0.379098 0.586395 1.0 1.0 0.02 0.0 0.05 0.416112 0.614657 1.0 1.0 0.02 0.4 0.05 0.453127 0.642920 1.0 1.0 0.02 0.05 -0.4 0.353675 0.578066 1.0 1.0 0.02 0.05 0.0 0.412373 0.613518 1.0 1.0 0.02 0.05 0.4 0.487372 0.652540
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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 4, July - August (2013) © IAEME 396 Fig.1 Schematic representation of a liquid film on an elastic sheet 0.0 0.5 1.0 1.5 2.0 0 10 20 30 40 50 S β Fig.2 Variation of film thickness β with unsteadiness parameter S with Mn = 0.0 0 2 4 6 8 0.0 0.4 0.8 1.2 1.6 2.0 Fig.3 Variation of film thickeness β with magnetic parameter Mn β S = 1.2 S = 0.8 Mn 0 2 4 6 8 0.0 0.1 0.2 0.3 0.4 0.5 f'(β) Mn Fig.4 Variation of free-surface velocity f ' (β) with magnetic parameter Mn S = 0.8 S = 1.2 0 2 4 6 8 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 -f''(0) Mn Fig.5 Variation of wall shear stress parameter -f '' (0) with Magnetic parameter Mn S = 1.2 S = 0.8 Forc Slit u Ts h(t) y = 0 y = h y x
11.
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 4, July - August (2013) © IAEME 397 0 2 4 6 8 0.0 0.2 0.4 0.6 0.8 1.0 Fig.6 Variation of free-surface temperature θ(β) with the Magnetic parameter Mn S = 1.2 S = 0.8 θ(η) η 0 2 4 6 8 0.4 0.6 0.8 1.0 1.2 Fig. 7 Dimensionless emperature gradien -θ '(η) at the sheet vs Magnetic parameter Mn for S = 0.8 and S = 1.2 S = 1.2 S = 0.8 -θ'(0) Mn 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.2 0.4 0.6 0.8 1.0 f '(η) η Fig. 8(a) Variation in the velocity profile f ' (η) for different values of magnetic parameer Mn with S = 0.8 Mn = 0, 1, 2, 5, 8 β = 0.806512 β = 0.979193 β = 1.350880 β = 1.616880 β = 2.151990 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 Fig. 8(b) Variation in the velocity profile f ' (η) for different values of magnetic parameer Mn with S = 1.2 f ' (η) η β = 0.483049 β = 0.579900 β = 0.775795 β = 0.903878 β = 1.127780 M = 0, 1, 2, 5, 8 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.2 0.4 0.6 0.8 1.0 Fig.9(a) Variation oin the temperature profile θ (η) for different values of magnetic parameter Mn with S = 0.8 η θ (η) β = 0.806512 β = 0.979193 β = 1.350880 β = 1.616880 β = 2.151990 M = 0, 1, 2, 5, 8 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 M = 0, 1, 2, 5, 8 η θ (η) Fig.9(b) Variation on the temperature profile θ (η) for different values of magnetic parameter Mn with S = 1.2 β = 0.483049 β = 0.579900 β = 0.775795 β = 0.903878 β = 1.127780
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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 4, July - August (2013) © IAEME 398 0.0 0.4 0.8 1.2 1.6 0.0 0.2 0.4 0.6 0.8 1.0 β = 1.61688 Mn = 1, Ec = 0.02, A* = B* = 0.05 Fig.10(a) Variation of the temperature profile θ (η) for different values of Prandtl number Pr with S = 0.8 Pr=100 Pr=10 Pr=5 Pr=1 Pr=0.1 θ(η) η 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Mn=1, Pr = 1, Ec = 0.02, A* = B* = 0.05 Fig.10(b) Variation in he temperaure profile θ (η) for different values of Prandtl number Pr with S = 1.2 Pr = 100 Pr = 10 Pr = 5 Pr = 1 Pr = 0.1 β = 0.903878 θ (η) η 0.0 0.4 0.8 1.2 1.6 0.0 0.2 0.4 0.6 0.8 1.0 Fig.11(a) Variation in the temperature profile θ (η) for different values of Eckert number Ec with S = 0.8 Mn=1, Pr = 1, A* = B* = 0.05 Ec = 0.01, 1, 2, 5 β = 1.61688 θ(η) η 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Fig.11(b) Variation in the temperature profile θ (η) for different values of Eckert number Ec with S = 1.2 Mn = 1, Pr = 1, A* = B* = 0.05 β = 0.903878 Ec = 0.01, 1, 2, 5 θ (η) η 0.0 0.4 0.8 1.2 1.6 0.0 0.2 0.4 0.6 0.8 1.0 Fig 12(a). Variation of temperature profile θ (η) for different values of space dependent heat source/sink A * with S = 0.8 Mn = 1, Pr = 1.0, Ec = 0.02, B* = 0.5 A* = -0.4, -0.2, 0, 0.2, 0.4 β 1.61688 θ (η) η 0.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 Fig 12(b). Variation of temperature profile θ (η) for different values of space dependent heat source/sink A * with S = 1.2 A* = -0.4, -0.2, 0, 0.2, 0.4 Mn=1, Pr = 1, Ec = 0.02, B* = 0.05 β = 0.903878 θ (η) η
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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 4, July - August (2013) © IAEME 399 0.0 0.4 0.8 1.2 1.6 0.0 0.2 0.4 0.6 0.8 1.0 Fig 13 (a). Variation of the temperature profile θ(η) for different values of temperature dependent heat source/sink B * with S = 0.8 Mn = 1, Pr = 1, Ec = 0.02, A* = 0.05 B* = -0.4, -0.2, 0, 0.2, 0.4 β = 1.61688 θ (η) η 0.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 Fig 13 (b). Variation of the temperature profile θ(η) for different values of temperature dependent heat source/sink B * with S = 1.2 B* = -0.4, -0.2, 0, 0.2, 0.4 Mn = 1, Pr = 1, Ec = 0.02, A* = 0.05 β = 0.903878 θ (η) η REFERENCES [1] L.J. Crane, flow past a stretching plate, Z. Angrew. Math. Phys. 21 (1970) 645-647. [2] P.S. Gupta, A.S. Gupta, Heat and Mass transfer on a stretching sheet with suction or blowing, Can. J. Chem. Eng. 55 (1977) 744-746. [3] L.G. Grubka, K.M, Bobba, Heat Transfer characteristics of a continuous stretching surface with variable temperature, J. Heat Transfer 107 (1985) 248-250. [4] B.K. Dutta, A.S. Gupta, cooling of a stretching sheet in a various flow, Ind. Eng. Chem. Res. 26 (1987) 333-336. [5] A. Chakrabarti, A.S. Gupta, Hydromagnetic flow and heat transfer over a stretching sheet, Q. Appl. Math. 37 (1979) 73-78. [6] M.S. Abel, P.G. Siddheshwar, Mahantesh M. Nandeppanavar, Heat transfer in a viscoelastic boundary layer flow over a stretching sheet with viscous dissipation and non-uniform heat source. Int. J. Heat Mass Transfer, 50(2007), 960-966. [7] M.S. Abel, Mahantesh M. Nandeppanavar, Jagadish V. Tawade, Heat transfer in a Walter’s liquid B fluid over an impermeable stretching sheet with non-uniform heat source/ sink and elastic deformation. Commun Nonlinear Sci Numer Simulat 15 (2010) 1791–1802. [8] C.Y. Wang, Liquid film on an unsteady stretching surface, Quart Appl. Math 48 (1990) 601- 610. [9] S.M. Roberts, J.S. Shipman, Two point boundary value problems: Shooting Methods, Elsevier, New York, 1972. [10] C. Wang, Analytic solutions for a liquid film on an unsteady stretching surface, Heat Mass Transfer 42 (2006) 759–766. [11] R. Usha, R. Sridharan, On the motion of a liquid film on an unsteady stretching surface, ASME Fluids Eng. 150 (1993) 43-48. [12] E.M. Sparrow, J.L. Gregg, A boundary-layer treatment of laminar film condensation, ASME J. Heat Transfer 81 (1959) 13-18. [13] C.-H. Chen, Effect of viscous dissipation on heat transfer in a non-Newtonian liquid film over an unsteady stretching sheet, J. Non-Newtonian Fluid Mech. 135 (2006) 128. [14] B. S. Dandapat, S. Maity, A. Kitamura, Liquid film flow due to an unsteady stretching sheet, Int. J. non-Linear Mech. 43 (2008) 880-886.
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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 4, July - August (2013) © IAEME 400 [15] Z. Abbas, T. Hayat, M. Sajid, S. Asghar, Unsteady flow of a second grade fluid film over an unsteady stretching sheet, Math. Comp. Modelling 48 (2008) 518-526. [16] S.D. Conte, C. de Boor, Elementary Numerical Analysis, McGraw-Hill, New York, 1972 [17] T. Cebeci, P. Bradshaw, Physical and computational aspects of convective heat transfer, Springer-Verlag, New York, 1984. [18] Dr P.Ravinder Reddy, Dr K.Srihari and Dr S. Raji Reddy, “Combined Heat and Mass Transfer in MHD Three-Dimensional Porous Flow with Periodic Permeability & Heat Absorption”, International Journal of Mechanical Engineering & Technology (IJMET), Volume 3, Issue 2, 2012, pp. 573 - 593, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359. [19] M N Raja Shekar and Shaik Magbul Hussain, “Effect of Viscous Dissipation on MHD Flow and Heat Transfer of a Non-Newtonian Power-Law Fluid Past a Stretching Sheet with Suction/Injection”, International Journal of Advanced Research in Engineering & Technology (IJARET), Volume 4, Issue 3, 2013, pp. 296 - 301, ISSN Print: 0976-6480, ISSN Online: 0976-6499. [20] M N Raja Shekar and Shaik Magbul Hussain, “Effect of Viscous Dissipation on MHD Flow of a Free Convection Power-Law Fluid with a Pressure Gradient”, International Journal of Advanced Research in Engineering & Technology (IJARET), Volume 4, Issue 3, 2013, pp. 302 - 307, ISSN Print: 0976-6480, ISSN Online: 0976-6499. [21] Dr. Sundarammal Kesavan, M. Vidhya and Dr. A. Govindarajan, “Unsteady MHD Free Convective Flow in a Rotating Porous Medium with Mass Transfer”, International Journal of Mechanical Engineering & Technology (IJMET), Volume 2, Issue 2, 2011, pp. 99 - 110, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359.
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