MHD convection flow of viscous incompressible fluid over a stretched vertical flat plate in the presence of thermal radiation, viscous dissipation and Hall current effect using similarity solutions and HPM technique
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The effect of thermal radiation, viscous dissipation and hall current of the MHD convection flow of the viscous incompressible fluid over a stretched vertical flat plate has been discussed by using regular perturbation and homotophy perturbation technique with similarity solutions. The influence of various physical parameters on velocity, cross flow velocity and temperature of fluid has been obtained numerically and through graphs.
![Vivek Kumar Sharma. Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 9( Version 5), September 2014, pp.20-27
www.ijera.com 21 | P a g e
II. Homotopy Perturbation Method:
The Homotopy Perturbation Method is a combination of classical Perturbation Technique and Homotopy Theory, which has eliminated the limitations of the traditional perturbation methods. A brief introduction of Homotopy Perturbation Method is given below: ํฟ ํข +ํ ํข โํ ํ =0,ํโฮฉ โฆ (a) with boundary conditions ํต ํข, ํํข ํํ =0,ํโฮ โฆ (b) here ํฟ is the linear operator, ํ is Nonlinear operator, ํต is boundary operator and ํ ํ is known analytic function and ฮ is the boundary of the domain ฮฉ . A Homotopy ํฃ ํ,ํ :ฮฉ x [0,1]โR for the problem mentioned in equation (a) is ํป ํฃ,ํ = 1โํ ํฟ ํฃ โํฟ ํฃ0 +ํ ํฟ ํฃ +ํ ํฃ โํ ํ =0 โฆ (c) or ํป ํฃ,ํ =ํฟ ํฃ โํฟ ํฃ0 +ํ ํฟ ํฃ0 +ํ ํฃ โํ ํ =0 โฆ (d) where ํโ[0,1] is an embedding parameter and ํฃ0 is an initial approximation of equation (a) which satisfies boundary conditions. It follows from equation (c) and equation (d) that ํป ํฃ,0 =ํฟ ํฃ โํฟ ํฃ0 ํํํ ํป ํฃ,1 =ํฟ ํฃ +ํ ํฃ โํ ํ โฆ (e) The changing process of p from zero to unity is just that of ํฃ ํ,ํ from ํฃ0(ํ) to ํฃ ํ . In topology, this is called deformation and ํฟ ํฃ โํฟ ํฃ0 and ํฟ ํฃ +ํ ํฃ โํ ํ are called homotopic in topology. Let ํฃ=ํฃ0+ํํฃ1+ํ2ํฃ2+โฏ โฆ (f) And setting ํ=1 result in an approximate solution of equation (a) ํข=limํโ1ํฃ=ํฃ0+ํฃ1+ํฃ2+โฏ โฆ (g) The series of equation (g) is convergent for most of the cases. However, the convergent rate is depends upon the nonlinear operator ํ ํฃ , the following options are already suggested by He (1999):
1. The second derivative of ํ ํฃ with respect to ํฃ must be small because the parameter may be relatively large i.e. ํโ1.
2. The norm of ํฟโ1 ํํ ํํข must be smaller than one so that the series is convergent.
III. Mathematical Analysis:
Consider the steady incompressible mixed convection flow of a viscous electrically-conducting fluid past a three dimensional uniformly stretched flat plate in the vertical direction. The stationary frame of reference (x, y, z) is chosen such that velocity is proportional to the distance from the fixed origin O also the x-axis is along the direction of motion of the surface, the y-axis is normal to the surface and the z-axis is transverse to the xy-plane. The external magnetic field is assumed to be constant H0 and is applied in the positive y-direction also the sheet has a variable temperature ํํค(ํฅ) at the surface while ํโ is the free stream temperature yield that ํํค(ํฅ)>ํโ corresponds to a heated plate and ํํค(ํฅ)<ํโ corresponds to a cooled plate. It is assumed that the electron pressure gradient, the ion slip and the thermo-electric effects are neglected and in influence of Hall effects the generalized Ohmโs law can be written as ํ=ํ(ํธ+ํํํโํปโ ํํ ํํํ ํโํป) โฆ (1) Where ํํ and ํ stand for the electron number density and the electric charge, respectively and the electrical conductivity, ํ is given by](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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Similar to MHD convection flow of viscous incompressible fluid over a stretched vertical flat plate in the presence of thermal radiation, viscous dissipation and Hall current effect using similarity solutions and HPM technique
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MHD convection flow of viscous incompressible fluid over a stretched vertical flat plate in the presence of thermal radiation, viscous dissipation and Hall current effect using similarity solutions and HPM technique
- 1. Vivek Kumar Sharma. Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 9( Version 5), September 2014, pp.20-27 www.ijera.com 20 | P a g e MHD convection flow of viscous incompressible fluid over a stretched vertical flat plate in the presence of thermal radiation, viscous dissipation and Hall current effect using similarity solutions and HPM technique Vivek Kumar Sharma, Divya Saxena Jagan Nath University, Jaipur, Rajasthan, India Abstract: The effect of thermal radiation, viscous dissipation and hall current of the MHD convection flow of the viscous incompressible fluid over a stretched vertical flat plate has been discussed by using regular perturbation and homotophy perturbation technique with similarity solutions. The influence of various physical parameters on velocity, cross flow velocity and temperature of fluid has been obtained numerically and through graphs. Key words: Thermal Radiation; Viscous Dissipation; Hall Current effect; MHD flow; Homotophy Perturbation Technique; Stretched flat plate I. Introduction: The influence of stretching sheet and the various combinations of additional effects of boundary layer flow problem has many industrial applications such as polymer sheet or filament extrusion from a dye or long thread between feed roll or wind-up roll, glass fiber and paper production, drawing of plastic films, liquid films in condensation process. These highly applicable phenomena in practical problems attract many researchers. The pioneering studies have been performed by Sakiadis (1961). The study of stretching surfaces and the several combinations of additional effects on the stretching problems are important in many practical applications because the production of sheeting material arises in a number of industrial manufacturing processes and includes both metal and polymer sheets. In the manufacture of the latter, the material is in a molten phase when thrust through an extrusion die and then cools and solidifies some distance away from the die before arriving at the collecting stage. The quality of the resulting sheeting material, as well as the cost of production, is affected by the speed of collection and the heat transfer rate, and a knowledge of the flow properties of the ambient fluid is clearly desirable in Banks and Zaturska (1986) Sparrow and Abraham (2005) pointed the very important practical problem of the thermal processing of sheet-like materials which is a necessary operation in the production of paper, linoleum, polymeric sheets, roofing shingles, insulating materials, fine-fiber mattes. The effects of Hall current and chemical reaction on the hydro magnetic flow of a stretching vertical surface is studied by Salem and Abd El-Aziz (2008). Ghosh (2009) have studied the Hall effects in a parallel plate channel, while Abd El-Aziz (2010) has analyzed the effects of Hall currents on the flow and heat transfer of an electrically conducting fluid over an unsteady stretching surface in the presence of a strong magnetic field. Gnaneswara Reddy and Bhaskar Reddy (2011) investigated mass transfer and heat generation effects on MHD free convection flow past an inclined vertical surfacein a porous medium. MHD boundary- layer flow over a stretching surface with internal heat generation or absorption was studied by Basiri Parsa (2013). Gnaneswara Reddy (2012) analyzed thermophoresis, viscous dissipation and joule heating effects on steady MHD heat and mass transfer flow over an inclined radiative isothermal permeable surface with variable thermal conductivity.Ali, Nazar and Arifin were consider the effect of Hall current on MHD mixed convection boundary layer flow over a stretched vertical flat plate.Reasently Gnaneswara Reddy (2014) study the Influence of thermal radiation, viscous dissipation and Hall current on MHD convection flow over a stretched vertical flat plate. The aim of the present paper is to investigate the influence of thermal radiation, viscous dissipation and hall current on steady MHD mixed convection boundary layer flow over a stretched vertical flat plate. The non linear coupled partial differential equations are reduced using similarity solutions and further these are solved by Homotophy Perturbation Method. The effects of various governing parameters on the velocity, cross flow velocity, temperature, skin-friction coefficient and Nusselt number are own in figures and tables and discussed further in detail. RESEARCH ARTICLE OPEN ACCESS
- 2. Vivek Kumar Sharma. Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 9( Version 5), September 2014, pp.20-27 www.ijera.com 21 | P a g e II. Homotopy Perturbation Method: The Homotopy Perturbation Method is a combination of classical Perturbation Technique and Homotopy Theory, which has eliminated the limitations of the traditional perturbation methods. A brief introduction of Homotopy Perturbation Method is given below: ํฟ ํข +ํ ํข โํ ํ =0,ํโฮฉ โฆ (a) with boundary conditions ํต ํข, ํํข ํํ =0,ํโฮ โฆ (b) here ํฟ is the linear operator, ํ is Nonlinear operator, ํต is boundary operator and ํ ํ is known analytic function and ฮ is the boundary of the domain ฮฉ . A Homotopy ํฃ ํ,ํ :ฮฉ x [0,1]โR for the problem mentioned in equation (a) is ํป ํฃ,ํ = 1โํ ํฟ ํฃ โํฟ ํฃ0 +ํ ํฟ ํฃ +ํ ํฃ โํ ํ =0 โฆ (c) or ํป ํฃ,ํ =ํฟ ํฃ โํฟ ํฃ0 +ํ ํฟ ํฃ0 +ํ ํฃ โํ ํ =0 โฆ (d) where ํโ[0,1] is an embedding parameter and ํฃ0 is an initial approximation of equation (a) which satisfies boundary conditions. It follows from equation (c) and equation (d) that ํป ํฃ,0 =ํฟ ํฃ โํฟ ํฃ0 ํํํ ํป ํฃ,1 =ํฟ ํฃ +ํ ํฃ โํ ํ โฆ (e) The changing process of p from zero to unity is just that of ํฃ ํ,ํ from ํฃ0(ํ) to ํฃ ํ . In topology, this is called deformation and ํฟ ํฃ โํฟ ํฃ0 and ํฟ ํฃ +ํ ํฃ โํ ํ are called homotopic in topology. Let ํฃ=ํฃ0+ํํฃ1+ํ2ํฃ2+โฏ โฆ (f) And setting ํ=1 result in an approximate solution of equation (a) ํข=limํโ1ํฃ=ํฃ0+ํฃ1+ํฃ2+โฏ โฆ (g) The series of equation (g) is convergent for most of the cases. However, the convergent rate is depends upon the nonlinear operator ํ ํฃ , the following options are already suggested by He (1999): 1. The second derivative of ํ ํฃ with respect to ํฃ must be small because the parameter may be relatively large i.e. ํโ1. 2. The norm of ํฟโ1 ํํ ํํข must be smaller than one so that the series is convergent. III. Mathematical Analysis: Consider the steady incompressible mixed convection flow of a viscous electrically-conducting fluid past a three dimensional uniformly stretched flat plate in the vertical direction. The stationary frame of reference (x, y, z) is chosen such that velocity is proportional to the distance from the fixed origin O also the x-axis is along the direction of motion of the surface, the y-axis is normal to the surface and the z-axis is transverse to the xy-plane. The external magnetic field is assumed to be constant H0 and is applied in the positive y-direction also the sheet has a variable temperature ํํค(ํฅ) at the surface while ํโ is the free stream temperature yield that ํํค(ํฅ)>ํโ corresponds to a heated plate and ํํค(ํฅ)<ํโ corresponds to a cooled plate. It is assumed that the electron pressure gradient, the ion slip and the thermo-electric effects are neglected and in influence of Hall effects the generalized Ohmโs law can be written as ํ=ํ(ํธ+ํํํโํปโ ํํ ํํํ ํโํป) โฆ (1) Where ํํ and ํ stand for the electron number density and the electric charge, respectively and the electrical conductivity, ํ is given by
- 3. Vivek Kumar Sharma. Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 9( Version 5), September 2014, pp.20-27 www.ijera.com 22 | P a g e ํ= ํ2ํํํํ ํํ โฆ (2) Where ํํ and ํํ are the electron collision time and the mass of an electron, respectively. The effect of Hall current gives rise to a force in the z-direction resulting in a cross-flow in this direction and thus the flow becomes three-dimensional. Using the boundary layer variables it is observed that the physical variable does not depends on the z -coordinate. Under the above mentioned assumptions and usual Boussinesque approximation, the governing equations for the relevant fluid flow are given as: Continuity equation: ํํข ํํฅ + ํํฃ ํํฆ =0 โฆ (3) Momentum equations: ํข ํํข ํํฅ +ํฃ ํํข ํํฆ =ํฃ ํ2ํข ํํฆ2+ํํฝ ํโํโ โ ํํต02 ํ 1+ํ2 (ํข+ํํค) โฆ (4) ํข ํํค ํํฅ +ํฃ ํํค ํํฆ =ํฃ ํ2ํค ํํฆ2+ ํํต02 ํ 1+ํ2 (ํํขโํค) โฆ (5) Energy Equation: ํข ํํ ํํฅ +ํฃ ํํ ํํฆ = ํ ํํํ ํ2ํ ํํฆ2+ ํ ํํํ ํํข ํํฆ 2+ ํํค ํํฆ 2 โ 1 ํํํ ํํํ ํํฆ โฆ (6) With the boundary conditions ํข=ํขํค ํฅ ,ํฃ=0,ํค=0,ํ=ํํค ํฅ ํํก ํฆ=0 ํขโ0,ํคโ0,ํโํํค ํํ ํฆโโ โฆ (7) Using the Rosseland approximation, the radiative heat flux ํํ is given by ํํ=โ 4ํํ 3ํํ ํํ4 ํํฆ โฆ (8) Where ํํ is the StefanโBoltzmann constant and ํํ is the mean absorption coefficient. Considering Rosseland approximation, the present analysis is limited to optically thick fluids. If the temperature difference within the flow is sufficiently small, then Eq (8) can be linearized by expanding ํ4 using the Taylor series about ํโ, neglecting higher order terms as ํ4โ 4ํโ 3ํโ3ํโ 4 โฆ (9) Using Eq (8) and Eq (9) the Eq (6) reduces to ํข ํํ ํํฅ +ํฃ ํํ ํํฆ =ํผ(1+ํ ) ํ2ํ ํํฆ2+ ํ ํํํ ํํข ํํฆ 2+ ํํค ํํฆ 2 โฆ (10) Where ํผ= ํ ํํํ is the thermal diffusivity and ํ =16ํโํโ 33ํํโ is the radiation perameter. It is assumed that ํขํค ํฅ and ํํค(ํฅ) are varies linearly with variable ํฅ as ํขํค ํฅ =ํํฅ ,(ํ>0) ํํํ ํํค ํฅ =ํโ+ํํฅ โฆ (11) where c and a are constants. Noted that For ํ>0,(ํํค(ํฅ)>ํโ), the plate is heated and for ํ<0,(ํํค(ํฅ)<ํโ), the plate is cooled. IV. Method of solution: We introduce the similarity solutions for Eq(3), Eq(4), Eq(5) and Eq(10) in the form of ํข=ํํฅํโฒ ํ ,ํฃ=โ ํํ 12 ํ ํ ,ํค=ํํฅํ ํ ,ํ ํ = ํโํโ ํํคโํโ , ํ= ํ ํ 12 ํฆ,ํ= ํํต02 ํํ ,ํบํํฅ= ํํฝ(ํํคโํโ)ํฅ3 ํ2,ํ ํํฅ= ํขํค(ํฅ) ํ , ํ= ํบํํฅ ํ ํํฅ 2,ํํ= ํ ํผ ,ํธํ= ํ2 ํํ(ํํคโํโ) โฆ (12) Where Pr is the Prandtl Number, M is the megnetic parameter ,m is the hall perameter,ฮป is the constant of buoyancy or mixed convection perameter, ํบํํฅis the local Grashof Number , ํ ํํฅ is the local Reynold Number and Ec is the Eckert Number. Substituting these assumptions in Eq (4), Eq(5) and Eq(10), we get ํโฒโฒโฒ+ํํโฒโฒโํโฒ2โ ํ 1+ํ2 ํโฒ+ํํ +ํํ=0 โฆ (13) ํโฒโฒ+ํํโฒโํโฒํ+ ํ 1+ํ2 ํํโฒโํ =0 โฆ (14) 1+ํ ํโฒโฒ+Pr ํํโฒโํโฒํ+ํธํ ํโฒโฒ2+ํโฒ2 =0 โฆ (15) The corresponding boundary conditions are
- 4. Vivek Kumar Sharma. Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 9( Version 5), September 2014, pp.20-27 www.ijera.com 23 | P a g e ํํก ํ=0โถ ํ=0,ํโฒ=1,ํ=0,ํ=1 ํํ ํโโโถํโฒโ0,ํโ0,ํโ0 โฆ (16) The Homotophy for the above three equations are following: ํป ํ,ํ = 1โํ ํโฒโฒโฒโ ํ 1+ํ2ํโฒโ 1โ ํ 1+ํ2 ํโํ +ํ ํโฒโฒโฒ+ํํโฒโฒโํโฒ2โ ํ 1+ํ2 ํโฒ+ํํ +ํํ =0 โฆ (17) ํป ํ,ํ = 1โํ ํโฒโฒโ ํ 1+ํ2ํโ 1โ ํ 1+ํ2 ํโํ+ 4โ ํ 1+ํ2 ํโ2ํ +ํ ํโฒโฒ+ํํโฒโํโฒํ+ ํ 1+ํ2 ํํโฒโ ํ=0 โฆ (18) ํป ํ,ํ = 1โํ 1+ํ ํโฒโฒโ 1+ํ ํโํ +ํ 1+ํ ํโฒโฒ+Pr ํํโฒโํโฒํ+ํธํ ํโฒโฒ2+ํโฒ2 =0 โฆ (19) Let ํ=ํ0+ํํ1+ํ2ํ2+โฏ ํ=ํ0+ํํ1+ํ2ํ2+โฏ ํ=ํ0+ํํ1+ํ2ํ2+โฏ โฆ (20) Substituting these assumptions in Eq(17), Eq(18) and Eq(19) and comparing the coefficient of like powers of p, we get ํ0โถ ํ0โฒโฒโฒโ ํ 1+ํ2ํ0โฒ โ 1โ ํ 1+ํ2 ํโํ=0 โฆ (21) ํ1โถ ํ1โฒโฒโฒโ ํ 1+ํ2ํ1โฒ+ 1โ ํ 1+ํ2 ํโํ+ํ0ํ0โฒโฒโํ0โฒ 2โ ํ 1+ํ2ํํ0+ํํ0=0 โฆ (22) ํ0โถ ํ0โฒโฒโ ํ 1+ํ2ํ0โ 1โ ํ 1+ํ2 ํโํ+ 4โ ํ 1+ํ2 ํโ2ํ=0 โฆ (23) ํ1โถ ํ1โฒโฒโ ํ 1+ํ2ํ1+ 1โ ํ 1+ํ2 ํโํโ 4โ ํ 1+ํ2 ํโ2ํ+ํ0ํ0โฒ โํ0โฒ ํ0+ ํ 1+ํ2ํํ0โฒ =0โฆ (24) ํ0โถ 1+ํ ํ0โฒโฒโ 1+ํ ํโํ=0 โฆ (25) ํ1โถ 1+ํ ํ1โฒโฒ+ 1+ํ ํโํ+Pr ํ0 ํ0โฒ โํ0โฒ ํ0+ํธํ ํ0โฒโฒ 2+ํ0โฒ 2 =0 โฆ (26) Now the corresponding boundary conditions are ํํก ํ=0โถํ0=0,ํ1=0 ,โฆ, ํ0โฒ =1, ํ1โฒ=0 ,โฆ,ํ0=0,ํ1=0,โฆโฆ,ํ0=1,ํ1=0,โฆโฆ ํํ ํโโ: ํ0โฒ โ0,ํ1โฒโ0,โฆ,ํ0โ0,ํ1โ0,โฆ, ํ0โ0,ํ1โ0,โฆโฆ โฆ (27) The solutions of the Eq(21) to Eq(26) under the corresponding boundary conditions Eq(27) are ํ0=1โํโํ โฆ (28) ํ0=ํโํโํโ2ํ โฆ (29) ํ0=ํโํ โฆ (30) ํ1= โ ํ 1+ํ2+ ํํ 1+ํ2โํ ํ 1+ํ2 ํ 1+ํ2+1 + ํํ/2 ํ 1+ํ2 2+ ํ 1+ํ2 + ํ 1+ํ2+ ํํ 1+ํ2โํ ํ 1+ํ2 1โ ํ1+ํ2โํํ/2ํ1+ํ24โํ1+ํ2ํโํ1+ํ2ํโํ1+ํ2+ํํ1+ํ2โํ1โํ1+ํ2ํโํ+ํํ1+ํ224โํ 1+ํ2ํโ2ํ โฆ (31) ํ1= ํ(1+ํ) 1+ํ2 1โ ํ 1+ํ2 ํโํ โ 6 + 2ํํ 1+ํ2 โ ํ 1+ํ2 4 โ ํ 1+ํ2 ํโ2ํ + โ ํ(1+ํ) 1+ํ2 1โ ํ 1+ํ2 + 6+2ํํ1+ํ2โํ1+ํ24โํ1+ํ2ํโํ1+ํ2ํ โฆ (32) ํ1=โ 1+ํ ํโํ+ํํํโํโPrํธํ 12 ํโ2ํโ 49 ํโ3ํ+ 14 ํโ4ํ โฆ (33) Where 1+ํ =ํํ 1+ํธํ1136 โก The valuesํ,ํ and ฮธ are obtained as given below ํ=ํํํํโ1ํ=ํ0+ํ1+ํ2+โฏ ํ=limํโ1ํ=ํ0+ํ1+ํ2+โฏ ํ=limํโ1ํ=ํ0+ํ1+ํ2+โฏ Now
- 5. Vivek Kumar Sharma. Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 9( Version 5), September 2014, pp.20-27 www.ijera.com 24 | P a g e ํ= 1โํโํโ ํ 1+ํ2+ ํํ 1+ํ2โํ ํ 1+ํ2 ํ 1+ํ2+1 + ํํ 2 ํ 1+ํ2 2+ ํ 1+ํ2 + ํ 1+ํ2+ ํํ 1+ํ2โํ ํ 1+ํ2 1โ ํ 1+ํ2 โ ํํ 2 ํ 1+ํ2 4โ ํ 1+ํ2 ํ โ ํ 1+ํ2ํ โ ํ 1+ํ2+ ํํ 1+ํ2โํ 1โ ํ 1+ํ2 ํโํ+ ํํ 1+ํ22 4โ ํ 1+ํ2 ํโ2ํ+ โฆ โฆ (34) ํ= ํโํโํโ2ํ+ โ ํ(1+ํ) 1+ํ2 1โ ํ 1+ํ2 + 6+ 2ํํ 1+ํ2โ ํ 1+ํ2 4โ ํ 1+ํ2 ํ โ ํ 1+ํ2ํ + ํ(1+ํ) 1+ํ2 1โ ํ 1+ํ2 ํโํโ 6+ 2ํํ 1+ํ2โ ํ 1+ํ2 4โ ํ 1+ํ2 ํโ2ํ+โฏ โฆ (35) ํ=ํโํโ 1+ํ ํโํ+ํํํโํโPrํธํ 12 ํโ2ํโ 49 ํโ3ํ+ 14 ํโ4ํ +โฏ โฆ (36) Hence the velocities for the flow given as ํข= ํํฅ ํโํโ ํ 1+ํ2+ ํํ 1+ํ2โํ 1โ ํ 1+ํ2 โํํ/ 4โ ํ 1+ํ2 ํ โ ํ 1+ํ2ํ + ํ1+ํ2+ํํ1+ํ2โํ1โํ1+ํ2ํโํโํํ/4โํ1+ํ2ํโ2ํ+โฆ โฆ (37) ํฃ= โ ํํ 12 1โํโํโ ํ 1+ํ2+ ํํ 1+ํ2โํ ํ 1+ํ2 ํ 1+ํ2+1 + ํํ 2 ํ 1+ํ2 2+ ํ 1+ํ2 + ํ1+ํ2+ํํ1+ํ2โํํ1+ํ21โํ1+ํ2โํํ2ํ1+ํ24โํ1+ํ2ํโํ1+ํ2ํโํ1+ํ2+ํํ1+ํ2โ ํ1โํ1+ํ2ํโํ+ํํ1+ํ224โํ1+ํ2ํโ2ํ+ โฆ โฆ (38) ํค= ํํฅ ํโํโํโ2ํ+ โ ํ(1+ํ) 1+ํ2 1โ ํ 1+ํ2 + 6+ 2ํํ 1+ํ2โ ํ 1+ํ2 4โ ํ 1+ํ2 ํ โ ํ 1+ํ2ํ + ํ(1+ํ)1+ํ21โํ1+ํ2ํโํโ6+2ํํ1+ํ2โํ1+ํ24โํ1+ํ2ํโ2ํ+โฆ ... (39) And the temperature is given as ํ= ํํคโํโ ํโํโ 1+ํ ํโํ+ํํํโํโPrํธํ 12 ํโ2ํโ 49 ํโ3ํ+ 14 ํโ4ํ +โฏ +ํโ โฆ (40) Here ฮป > 0 corresponds to the assisting flow (heated plate), also ฮป < 0 corresponds to the opposing flow (cooled plate) and ฮป = 0 corresponds to the forced convection flow. V. Skin Friction Coefficient and Heat Transfer Coefficient: The quantities of physical interest are the coefficient of skin friction ํถํฅ and ํถํง as well as heat transfer coefficient (Nusselt Number) Nu are defined as ํถํฅ= ํํคํฅ 2 ํํขํค , ํถํง= ํํคํง 2 ํํขํค ,ํํข= ํฅํํค ํ ํํคโํโ โฆ โฆ (41)
- 6. Vivek Kumar Sharma. Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 9( Version 5), September 2014, pp.20-27 www.ijera.com 25 | P a g e Where k being thermal conductivity of the fluid ํํคํฅ and ํํคํง are the shear stresses in the directions of x and z respectively also ํํค is the heat flux from the surface of the flat plate are given by ํํคํฅ=ํ ํํข ํํฆ ํฆ=0,ํํคํง=ํ ํํค ํํฆ ํฆ=0,ํํค=โํ ํํ ํํฆ ํฆ=0 Using eq.(12) and eq.(37) we get ํถํฅํ ํํฅ 12 =ํโฒโฒ 0 , ํถํงํ ํํฅ 12 =ํโฒ 0 ,ํํขํ ํํฅ 12 =โํโฒ(0) โฆ (42) Where ํโฒโฒ 0 =โ1+ ํ 1+ํ2+ ํํ 1+ํ2โํ 1+ ํ 1+ํ2 + ํํ 1+ํ2 2+ ํ 1+ํ2 โฆ (43) ํโฒ 0 =1โ ํ(1+ํ) 1+ํ2 1โ ํ 1+ํ2 + 6โ ํ 1+ํ2+ 2ํํ 1+ํ2 2โ ํ 1+ํ2 โฆ (44) โํโฒ 0 =โํ +ํํโ 23 ํํํธํ โฆ (45) VI. Results and discussion: The Table 1 elucidates the effect of increasing values of hall parameter m and mixed convection parameter ฮป, the values of ํโฒโฒ 0 ,ํโฒ(0) also increases and ํโฒโฒ 0 ,ํโฒ(0) increases due to decrement of magnetic parameter M. Table 2 shows that the value of ํโฒ(0) increase due to increase in radiation parameter R and Eckert Number Ec but this decreases due to increase in Prandtal Number Pr. It has been observed from the fig. 1 that the velocity increase due to increase in hall parameter m and mixed convection parameter ฮป also velocity decreases due to increase in magnetic field parameter M. It is observed from the fig.2 that the cross flow velocity increases due to increase in hall parameter m and decrease due to increase in magnetic field parameter M. It has been observed that the temperature increases due to increase in radiation parameter R and Eckert Number Ec also increases due to decrease in Prandtal Number Pr. m M ฮป f''(0) h'(0) 1 1 1 -0.815301 2.613270 1 0.8 1 -0.725536 3.503314 1 1 2 -0.229515 2.613270 2 1 1 -0.560156 3.026459 Table 1: Numerical Values of the skin-friction coefficient R Pr Ec (-ฮป'(0)) 1 0.1 0.01 0.900667 1.1 0.1 0.01 1.000667 1 0.2 0.01 0.801333 1 0.1 0.05 0.903333 Table 2: Numerical values of the heat transfer rate -ฮธ'(0)
- 7. Vivek Kumar Sharma. Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 9( Version 5), September 2014, pp.20-27 www.ijera.com 26 | P a g e Figure 1: velocity profile for different values of M, m and ฮป Figure 2: Cross flow velocity profile for different values of M and m 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 M=0.001,m=3, M=0.001,m=3, M=0.001,m=1, M=0.01,m=1 ๏จ f' ๏ฌ๏ฝ๏ฐ ๏ฌ๏ฝ๏ฐ๏ฎ๏ฐ๏ต ๏ฌ๏ฝ๏ฐ๏ฎ๏ฐ๏ต ๏ฌ๏ฝ๏ฐ๏ฎ๏ฐ๏ต 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 M=1,m=1 M=1,m=1.5 M=1.4,m=1 ๏จ h
- 8. Vivek Kumar Sharma. Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 9( Version 5), September 2014, pp.20-27 www.ijera.com 27 | P a g e Figure 3: Temperature profiles for different values of R, Pr and Ec Refrences: [1.] Sakiadis BC. Boundary layer behavior on continuous solid surfaces. AlChe J 1961; 7:26โ8. [2.] Banks WHH, Zaturska MB. Eigen solutions in boundary-layer flow adjacent to a stretching wall. IMA J Appl Math 1986; 36:263โ73. [3.] Sparrow EM, Abraham JP. Universal solutions for the stream wise variation of the temperature of a moving sheet in the presence of a moving fluid. Int J Heat Mass Transfer 2005; 48:3047โ56. [4.] Abraham JP, Sparrow EM. Friction drags resulting from the simultaneous imposed motions of a free stream and its bounding surface. Int J Heat Fluid Flow 2005; 26:289โ95. [5.] He J.H. (2005): โApplications of Homotophy Perturbation Method to Nonlinear wave equationsโ.Chaos Solutions Fractals 26, pp. 295-300. [6.] Hemeda A.A.(2012): โHomotophy Perturbation Method for solving System of Nonlinear Coupled EquationsโVol.6,2012 no.96,4787-4780. [7.] Salem AM, Abd El-Aziz M. Effect of Hall currents and chemical reaction on a hydromagnetic flow of a stretching vertical surface with internal heat generation/absorption. Appl Math Model 2008; 32(7):1236โ 54. [8.] Ghosh SK, Anwar Beยดg O, Narahari M. Hall effects on MHD flow in a rotating system with heat transfer characteristics. Meccanica 2009; 44:741โ65. [9.] Abd El-Aziz M. Flow and heat transfer over an unsteady stretching surface with Hall effect. Meccanica 2010; 45:97โ109. [10.] Gnaneswara Reddy M, Bhaskar Reddy N. Mass transfer and heat generation effects on MHD free convection flow past an inclined vertical surface in a porous medium. J Appl Fluid Mech 2011; 4(3):7โ 11. [11.] Basiri Parsa A, Rashidi MM, Hayat T. MHD boundary-layer flow over a stretching surface with internal heat generation or absorption. Heat Trans โ Asian Res 2013; 42(6):500โ14. [12.] Gnaneswara Reddy M. Effects of thermophoresis, viscous dissipation and joule heating on steady MHD heat and mass transfer flow over an inclined radiative isothermal permeable surface with variable thermal conductivity. Int J Heat Technol 2012; 30(1):99โ110. [13.] Ali FM, Nazar R, Arifin NM, Pop I. Effect of Hall current on MHD mixed convection boundary layer flow over a stretched vertical flat plate. Meccanica 2011; 46:1103โ12. [14.] Gnaneswara Reddy M. Influence of thermal radiation, viscous dissipation and Hall current on MHD convection flow over a stretched vertical flat plate.Ain Shams Engg. Journal (2014)5. 169-175 00.511.522.533.544.5500.10.20.30.40.50.60.70.80.91 R=0.01,Pr=1,Ec=0.01R=0.05,Pr=1,Ec=0.01R=0.01,Pr=0.9,Ec=0.01R=0.01,Pr=1,Ec=0.5 ๏จ ๏ฑ