International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) ...
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) ...
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) ...
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) ...
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) ...
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) ...
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) ...
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) ...
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) ...
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) ...
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) ...
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) ...
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) ...
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) ...
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A numerical solution of mhd heat transfer in a laminar liquid film on an unstead

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A numerical solution of mhd heat transfer in a laminar liquid film on an unstead

  1. 1. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 49 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 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 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. Several parameter effects have been shown with the aid of graphs. The important observation in this study is, for high values of unsteadiness parameter S reduces the surface temperature and the temperature-dependent heat absorption is one better suited for effective cooling purpose as temperature-dependent heat generation enhance the temperature in the boundary layer. Key words: Internal heat generation, Liquid film, similarity transformation, unsteady stretching surface, viscous dissipation. 1. INTRODUCTION Boundary layer flow and heat transfer in a laminar liquid film on an unsteady stretching sheet has received a 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 laminar 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 INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING AND TECHNOLOGY (IJMET) ISSN 0976 – 6340 (Print) ISSN 0976 – 6359 (Online) Volume 4, Issue 5, September - October (2013), pp. 49-62 © IAEME: www.iaeme.com/ijmet.asp Journal Impact Factor (2013): 5.7731 (Calculated by GISI) www.jifactor.com IJMET © I A E M E
  2. 2. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 50 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 magnetohydrodynamic (MHD) flow are very important industrially and have applications in different areas of research such as petroleum production and metallurgical processes. Crane [1] 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 [1] are subsequently extended by many authors Refs.[2-3] to explore various aspects of the flow and heat transfer occurring in an infinite domain of the fluid surrounding the stretching sheet. Wang [4, 5], Usha and Shridharan [6], Chen [7], Andersson et al. [8] and Dandapat et al. [9]. Abel et al [10] have discussed about the Heat transfer in a liquid film over an unsteady stretching surface with viscous dissipation in presence of external magnetic field. Aziz et.al [11] 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. 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 combined 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. 2. 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 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
  3. 3. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 51 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≤ < . With the same analogy the expression for the surface temperature ( , )sT x t given by equation (2) represents a situation in which the sheet temperature decreases from 0T at the slit in proportion to 2 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 ( ) ( ) 1 2 0, 1- .B x t B tα − = (3) The particular form of the expressions for ( , )U x t , ( , )sT x t and ( , )B x 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 ( )h t lying on the horizontal stretching sheet (Fig.1). 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 (1) 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 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 02 ( ).s p p T T T k T u u v Q T T t x y C y C y µ ρ ρ  ∂ ∂ ∂ ∂ ∂ + + = + + −  ∂ ∂ ∂ ∂ ∂  (6) 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 1 2 b υ      so that the scale ratio is large enough i.e., ( ) 1 2 1 b x >> υ . 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 , 0, at 0,su U v T T y= = = = (7)
  4. 4. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 52 0 at , u T y h y y ∂ ∂ = = = ∂ ∂ (8) at . dh v y h dt = = (9) 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). Similarity transformations: We now introduce dimensionless variables andf θ and the similarity variable η as ( ) ( ) 1 2 , , , 1 b x y t x f t υ ψ η α   =   −  (10) ( ) ( ) ( ) 2 3 2 0, , 1 , 2 ref bx T x y t T T tα θ η υ −  = − −    (11) ( ) 1 2 . 1 b y t η υ α   =   −  (12) 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 ∂   ′= =   ∂ −  (13) ( ) 1 2 . 1 b v f x t ψ υ η α ∂   = − = −   ∂ −  (14) The mathematical problem defined in equations (4) – (8) 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  ′ ′′ ′ ′′ ′′′ ′+ + − = −    (15) ( ) 2S Pr 3 (2 ) EcPr , 2 f f fθ ηθ γ θ θ θ  ′ ′ ′ ′′ ′′+ + − − = −   (16)
  5. 5. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 53 (0) 1, (0) 0, (0) 1,f f θ′ = = = (17) ( ) 0, ( ) 0,f β θ β′′ ′= = (18) S ( ) . 2 f β β = (19) Where a prime denotes the differentiation with respect toη and S b = α is the dimensionless measure of the unsteadiness. Further, the dimensionless film thickness 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   =   −  (20) 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 (20) with respect to t, in the form ( ) 1 2 . 2 1 dh dt b t α β υ α   = −   −  (21) Thus the kinematic constraint at ( )y h t= given by equation (9) transforms into the free surface condition (21). It is noteworthy that the momentum boundary layer equation defined by equation (16) subject to the relevant boundary conditions (17) – (19) is decoupled from the thermal field; on the other hand the temperature field ( )θ η is coupled with the velocity field ( )f η . The most important characteristics of flow and heat transfer are the shear stress sτ and the heat flux sq on the stretching sheet that are defined as 0 0 (22) (23) s y s y u y T q k y = =  ∂ =   ∂   ∂ = −   ∂  τ µ where µ is the fluid dynamic viscosity. The local skin friction coefficient fC and the local Nusselt number xNu for fluid flow in a thin film can be expressed as ( ) 1 0 2 2 2 2Re 0 y f x u y C f U µ ρ −=  ∂ −  ∂  ′′≡ = − (24) ( ) 1/2 3/2 0 1 1 '(0)Re , 2 x x ref y x T Nu t T y α θ − =  ∂ ≡ − = −  ∂  (25)
  6. 6. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 54 where Rex Ux υ = , the local Reynolds number and refT denotes the same reference temperature (temperature difference) as in equation (2). 3. NUMERICAL APPROACH The non-linear differential equations (15) and (16) with appropriate boundary conditions given in (17) to (19) are solved numerically, by the most efficient numerical shooting technique with fourth order Runge–Kutta algorithm (see references [12] and [13]). 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 β . Eqs. (15) and (16) are integrated numerically by fourth order Runge–Kutta scheme from 0 to= =η η β with (0) 0, (0) 1 and (0) 1f f ′= = =θ and guessed trail values (0), (0) and .f ′′ ′θ β However, the numerical solution thus obtained will not generally satisfy the right-end boundary conditions ( ) 0, (0) 0 ( ) / 2.f and f S′′ ′= = =β θ β β 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 (18) and (19). 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 S β up to a tolerance of 6 10− . For further details on the numerical procedure, the readers are referred to [12,13,14] . 4. RESULTS AND DISCUSSION The exact solution do not seem feasible for a complete set of equations (15)-(16) 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 noteworthy 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 for 0 2S≤ ≤ . Present results are compared with some of the earlier published results in some limiting cases are shown in Table 1 and Table 2. The effects of various parameters influencing the dynamics are shown in Fig.2 – Fig.11. Fig.2 shows the variation of film thickness β with the unsteadiness parameter S. 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 [5]. 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 γ on the surface temperature ( )θ β are respectively have been already discussed by Abel et al [10]. 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.
  7. 7. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 55 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 = 1.2. 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 sT to wall velocity the 0T as observed by Anderson et al [8]. 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.10(a) and 10(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.11(a) and Fin.11(b) presents the effect of temperature-dependent heat generation/absorption γ on the temperature profile for different values of unsteadiness parameter S. For 0<γ reduces the temperature and for 0>γ enhances the temperature in the fluid. The dimensionless wall temperature gradient '(0)θ− takes a higher value at a large Prandtl number Pr. The effect '(0)θ− for 1.2S = only marginally exceeds that for 0.8S = for Pr 1> (see fig.12). The dimensionless wall temperature gradient '(0)θ− takes a uniform value at certain moderate values of Eckert number Ec, while the effect of '(0)θ− decreases with their increasing Ec (see fig. 13). Table 1 and Table 2 give the comparison of present results with that of Wang [5] and Aziz et.al [11]. Without any doubt, from these tables, we can claim that our results are in excellent agreement with that of references [5 & 11] under some limiting cases. Table.3 tabulates the values of surface temperature ( )1θ for various values of Mn, Pr, Ec andγ . This table also reveals that Mn and γ 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 laminar liquid film over a heated stretching surface in the presence of a 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, i) The effect of transverse magnetic field on a viscous incompressible fluid is to suppress the velocity field which in turn causes the enhancement of the temperature field. ii) The viscous dissipation effect is characterized by Eckert number (Ec) in the present analysis. It is observed that the dimensionless temperature will increases when the fluid is being heated ( 0)Ec > but decreases when the fluid is being cooled ( 0)Ec < . This is the effect of viscous dissipation is to enhance the temperature in the boundary layer. iii) For a wide range of Pr, the effect of viscous dissipation is found to increase the dimensionless free surface temperature (1)θ for the fluid cooling case. The impact of viscous dissipation on (1)θ diminishes in the two limiting cases: Pr 0 and Pr→ → ∞ , in which situations (1)θ approaches unity and zero respectively.
  8. 8. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 56 iv) 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. Fig.1. Schematic representation of a liquid film on an elastic sheet TABLE 1: Comparison of values of skin friction coefficient ( )0f ′′ with Mn = 0.0 S Wang [5] Aziz et.al [11] Present work Β ( )0f ′′− ( )0f β ′′− 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 2.151994 2.680943 1.245794 2.151990 1.245805 1.0 1.543620 1.972380 1.277762 1.543616 1.972384 1.277768 1.543617 1.277769 1.2 1.127780 1.442631 1.279177 1.127780 1.442625 1.174986 1.127780 1.279171 1.4 0.821032 1.012784 1.233549 0.821032 1.012784 1.233549 0.821033 1.233545 1.6 0.576173 0.642397 1.114937 0.576173 0.642397 1.114937 0.576176 1.114941 1.8 0.356389 0.309137 0.867414 0.356389 0.309137 0.867414 0.356390 0.867416 Note: Wang [5] and Aziz [11] have used different similarity transformation due to which the value of ( )0f β ′′ in his paper is the same as ( )0f ′′ of our results. For Sli t u T h(t y = 0 y = y x
  9. 9. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 57 TABLE 2: Comparison of values of surface temperature ( )1θ and wall temperature gradient ( )0θ′− with Mn = Ec =γ = 0.0 Pr Wang [5] Aziz et. al[11] Present work ( )1θ ( )0θ′− ( )0θ β ′− ( )1θ ( )0θ′− ( )0θ β ′− ( )1θ ( )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 0.097956 3.591125 1.668746 0.097825 1.671919 2 0.024941 5.244150 2.436884 0.025083 5.074186 2.357904 0.024869 2.443914 3 0.008785 6.514440 3.027170 0.008545 5.926547 2.753984 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 [5] has used different similarity transformation due to which the value of ( )0θ β ′− in his paper is the same as ( )0θ′− of our results.
  10. 10. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 58 TABLE 3: Values of surface temperature ( )1θ for various values of Mn, Pr, Ec, γ and S. Mn Pr Ec γ ( )1θ S = 0.8 S = 1.2 0.0 1.0 0.02 0.1 0.118639 0.296847 1.0 1.0 0.02 0.1 0.250815 0.413568 2.0 1.0 0.02 0.1 0.358547 0.495749 3.0 1.0 0.02 0.1 0.439666 0.557227 4.0 1.0 0.02 0.1 0.506920 0.604382 5.0 1.0 0.02 0.1 0.564159 0.642261 6.0 1.0 0.02 0.1 0.605107 0.673786 7.0 1.0 0.02 0.1 0.644046 0.699351 8.0 1.0 0.02 0.1 0.676447 0.721720 1.0 0.001 0.02 0.1 0.997829 0.998886 1.0 0.01 0.02 0.1 0.978616 0.988952 1.0 0.1 0.02 0.1 0.814440 0.897785 1.0 1.0 0.02 0.1 0.225360 0.421320 1.0 2.0 0.02 0.1 0.085194 0.228930 1.0 5.0 0.02 0.1 0.009701 0.061819 1.0 10.0 0.02 0.1 -0.000264 0.012560 1.0 100.0 0.02 0.1 -0.001574 -0.000572 1.0 1.0 0.01 0.1 0.226444 0.422094 1.0 1.0 0.1 0.1 0.216691 0.415427 1.0 1.0 0.2 0.1 0.205854 0.407387 1.0 1.0 0.5 0.1 0.173345 0.384166 1.0 1.0 1.0 0.1 0.119162 0.345464 1.0 1.0 2.0 0.1 0.010796 0.268060 1.0 1.0 3.0 0.1 -0.097570 0.190646 1.0 1.0 4.0 0.1 -0.205937 0.113252 1.0 1.0 5.0 0.1 -0.314303 0.035849 1.0 1.0 0.02 -0.5 0.190930 0.366775 1.0 1.0 0.02 -0.2 0.223926 0.393744 1.0 1.0 0.02 -0.1 0.236696 0.403400 1.0 1.0 0.02 0.0 0.250515 0.413420 1.0 1.0 0.02 0.1 0.265505 0.423823 1.0 1.0 0.02 0.2 0.281804 0.434630 1.0 1.0 0.02 0.5 0.340312 0.469708
  11. 11. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 59 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 0 1 2 3 4 5 S β Fig.2 Variation of film thickness β with unstudieness 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 0.2 0.4 0.6 0.8 S=1.2 S=0.8θ(β) Mn Fig.4. Variation of surface temperature θ(β) with the Magnetic parameter Mn 1E-3 0.01 0.1 1 10 100 1000 0.0 0.2 0.4 0.6 0.8 1.0 Fig.5. Variation of surface temperature θ(β) for S=0.8(solid line) and S=1.2(broken line) with the Prandtl number Pr Pr θ(β) S = 0.8 S = 1.2 0.01 0.1 1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 S = 1.2 S = 0.8 θ(β) Ec Fig.6. Variation of surface tem perature θ(β) with the Eckert num ber Ec -0.4 -0.2 0.0 0.2 0.4 0.0 0.1 0.2 0.3 0.4 0.5 S=1.2 S=0.8 Q θ(β) Fig.7. Variation of surface temperature θ(β) with the Heat source/sink parameter Q
  12. 12. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 60 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 η Fig. 8(a). Variation in the velocity profiles f '(η) for different values of m agnetic parameter Mn with S = 0.8 S = 0.8 Mn = 0,1,2,3,4 β = 1.067175 β = 1.184197 β = 1.350880 β = 1.616880 β = 2.151992 f '(η) 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 profiles f '(η) for different values of m agnetic param eter Mn with S = 1.2 η S = 1.2 Mn = 0,1,2,3,4 β = 0.627910 β = 0.690238 β = 0.775795 β = 0.903878 β = 1.127780 f '(η) 0.0 0.4 0.8 1.2 1.6 0.0 0.2 0.4 0.6 0.8 1.0 S = 0.8 Pr=0.001 Pr=0.01 Pr=0.1 Pr=1.0 Pr=2.0 Pr=5.0 Pr=10.0 Pr=100 β = 1.616880 Fig.9(a). Variation in the temperature profiles θ(η) for different values of Prandtl number Pr with S = 0.8 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 S = 1.2 θ(η) Fig.9(b). Variation in the temperature profiles θ(η) for different values of Prandtl number Pr with S = 1.2 η Pr=100.0 Pr=10.0 Pr=5.0 Pr=2.0 Pr=1.0 Pr=0.1 Pr=0.001 Pr=0.01 β = 0.903878 0.0 0.4 0.8 1.2 1.6 0.0 0.2 0.4 0.6 0.8 1.0 Fig.10(a). Variation in the temperature profiles θ(η) for different values of Eckert number Ec with S = 0.8 θ(η) η S = 0.8 Ec=0.01,0.1,0.5,1,2 β = 1.616880 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 S = 1.2 β = 0.903878 Fig.10(b). Variation in the temperature profiles θ(η) for different values of Eckert number Ec with S = 1.2 θ(η) η Ec = 0.01,0.1,0.5,1,2
  13. 13. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 61 0.0 0.4 0.8 1.2 1.6 0.0 0.2 0.4 0.6 0.8 1.0 S = 0.8 γ = - 0.5 γ = -0.2 γ = -0.1 γ = 0.0 γ = 0.1 γ = 0.2 γ = 0.5 η θ(η) Fig.11(a). Variation in the temperature profiles θ(η) for different values of Heat source/sink γ with S = 0.8 β = 1.616880 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 S = 1.2 η θ(η) Fig.11(b). Variation in the temperature profiles θ(η) for different values of Heat source/sink γ with S = 1.2 β = 0.903878 γ = 0.0 γ = -0.1 γ= -0.2 γ = -0.5 γ = 0.1 γ = 0.2 γ = 0.5 1E-3 0.01 0.1 1 10 100 1E-3 0.01 0.1 1 10 100 Pr −θ'(0) Fig.12. Dimensionless temperature gradient −θ'(0) at the sheet vs Prandtl number for S=0.8 (solid lines) and S=1.2 (broken lines) S=0.8 S=1.2 0.01 0.1 1 0.01 0.1 1 Ec −θ'(0) Fig.13. Dimensionless temperature gradient −θ'(0) at the sheet vs Eckert number for S=0.8 (solid lines) and S=1.2 (broken lines) S = 1.2 S = 0.8 REFERENCES [1]. L.J. Crane, flow past a stretching plate, Z. Angrew. Math. Phys. 21 (1970) 645-647. [2]. B.K. Dutta, A.S. Gupta, cooling of a stretching sheet in a various flow, Ind. Eng. Chem. Res. 26 (1987) 333-336. [3]. C.K. Chan, M.I. Char, Heat transfer of a Continuous stretching surface with suction or Blo3wing, J. math. Anal. Appl. 135 (1988) 568-580. [4]. C.Y. Wang, Liquid film on an unsteady stretching surface, Quart Appl. Math 48 (1990) 601- 610. [5]. C. Wang, Analytic solutions for a liquid film on an unsteady stretching surface, Heat Mass Transfer 42 (2006) 759–766. [6]. R. Usha, R. Sridharan, on the motion of a liquid film on an unsteady stretching surface, ASME Fluids Eng. 150 (1993) 43-48. [7]. 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. 135(2006) 128-135 [8]. 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 (2000) 69-74.
  14. 14. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 62 [9]. 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 (2007) 991-996. [10]. Heat transfer in a liquid film over an unsteady stretching surface with viscous dissipation in presence of external magnetic field. Published in Applied Mathematical Modeling 33 (2009) 3430-3441. [11]. R.C. Aziz · I. Hashim · A.K. Alomari, Thin film flow and heat transfer on an unsteady stretching sheet with internal heating Meccanica (2011) 46: 349–357 [12]. S.M. Roberts, J.S. Shipman, Two point boundary value problems: Shooting Methods, Elsevier, New York, 1972. [13]. S. D. Conte, C.de Boor, Elementary Numerical Analysis, McGraw-Hill, NewYork, 1972. [14]. T. Cebeci, P. Bradshaw, Physical and computational aspects of convective heat transfer, Springer-Verlag, New York, 1984. [15]. 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. [16]. 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. [17]. 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. [18]. 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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