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stretching sheet in porous medium with heat sink or source effect
Accepted Manuscript
MHD stagnation-point flow and heat transfer past a non-isothermal shrinking/stretching sheet in porous medium with heat sink or source effect G.S. Seth , A.K. Singha , M.S. Mandal , Astick Banerjee , Krishnendu Bhattacharyya PII: DOI: Reference:
S0020-7403(17)30877-9 10.1016/j.ijmecsci.2017.09.049 MS 3960
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
6 April 2017 20 September 2017 22 September 2017
Please cite this article as: G.S. Seth , A.K. Singha , M.S. Mandal , Astick Banerjee , Krishnendu Bhattacharyya , MHD stagnation-point flow and heat transfer past a non-isothermal shrinking/stretching sheet in porous medium with heat sink or source effect, International Journal of Mechanical Sciences (2017), doi: 10.1016/j.ijmecsci.2017.09.049
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Highlights MHD stagnation-point flow and heat transfer past a non-isothermal shrinking/stretching sheet in porous medium in presence of heat sink or source is investigated.
The solutions of steady flow are obtained even with higher shrinking rate due to magnetic field and porous medium.
Dual solutions for some cases of shrinking sheet are found, but solution in stretching sheet case is always unique.
For direct variation of wall temperature along the sheet heat absorption occurs and it grows with heat source.
In inverse variation of wall temperature, a smooth zik-zak nature of temperature profile is noticed with normal heat transfer characteristic for shrinking sheet case.
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MHD stagnation-point flow and heat transfer past a non-isothermal shrinking/stretching sheet in porous medium with heat sink or source effect G.S. Setha, A.K. Singhaa, M.S. Mandalb, Astick Banerjeec and Krishnendu Bhattacharyyad* a
Department of Applied Mathematics, Indian School of Mines, Dhanbad–826004, India
Department of Mathematics, Government General Degree College, Kalna–I, Burdwan–713405, West Bengal c
d
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b
Mohulara Jr. High School, Birbhum–731236, West Bengal, India
Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi–221005, Uttar
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Pradesh, India
Abstract
The MHD stagnation-point flow of electrically conducting fluid and heat transfer past a nonisothermal shrinking/stretching sheet in a porous medium in presence of heat sink or source are investigated. The governing equations are transformed by similarity transformation
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technique and the converted equations are solved numerically by shooting method. Computed results are presented in some figures. It is obtained that the boundary layer solutions of steady
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flow exist with higher shrinking rate for the presence of magnetic field. Also, similar fact is observed when the porous parameter increases. The similarity solutions for some cases of shrinking sheet are of dual nature; it means that there exists two solutions for higher
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shrinking rate and the solution is unique for all stretching sheet cases. The impact of variable wall temperature along the sheet is massive on the thermal flow and the temperature field.
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For direct variation of wall temperature along the sheet heat absorption is noted for first solution in shrinking sheet case and it increases with the introduction of heat source; but, the
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presence of heat sink controls the heat absorption and it helps to return back to normal heat transfer process. Whereas, smooth zik-zak nature of temperature profiles is found for the inverse variation of wall temperature for shrinking sheet case with heat transfer from the surface to the ambient fluid layer. On the other hand, for stretching sheet case normally heat transfer occurs, but for higher values of heat source parameter and for larger magnitudes of power-law exponent of inverse variation of wall temperature heat absorption is also noticed.
*
Corresponding author. Email Addresses: [email protected], [email protected], [email protected].
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Key words: MHD; stagnation-point flow; heat transfer; porous medium; non-isothermal shrinking sheet; heat sink or source; dual solutions.
1. Introduction The flow due to a shrinking sheet, i.e., the reverse type of flow to that of the flow over
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a stretching sheet has given less attention though it is quite important in many physical as well as the industrial problems, such as, shrinking balloon problem, shrinking film, packaging of products etc. On the other hand, the analysis of heat transfer in the flow field is quite important in the view of its impact on the final quality of the manufactured product in the industry. The flow due to a linearly stretching sheet was first described by Crane [1] and the
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reverse flow, i.e., the flow developed due to shrinking sheet was discussed by Wang [2]. Whereas, the existence, uniqueness and non-existence of steady flow past porous shrinking sheet with mass suction through the porous sheet was reported by Miklavčič and Wang [3] and they showed that it depends on the applied mass suction. The heat transfer in stretching sheet flow was analysed by Gupta and Gupta [4] and that of shrinking sheet by Fang and
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Zhang [5].
Actually, the flow due to shrinking sheet is much more interesting physically because
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to have the boundary layer flow the diffusion of generated vorticity needs to prevent. This goal of controlling vorticity diffusion to maintain the structure of boundary layer can be achieve by some other external forces. External magnetic field is one of such forces which
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can control the vorticity generated due to shrinking and also can confine it in the boundary layer region. The effect of magnetic field on the flow of electrically conducting Newtonian
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fluid over a shrinking sheet was studied by Fang and Zhang [6] and that of for second grade fluid by Hayat et al. [7]. This type of flow with magnetohydrodynamic (MHD) effects has
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many applications in many engineering problems such as petroleum industries, plasma studies, geothermal energy extractions, the boundary layer control in the field of aerodynamics and many others. Later, Hayat et al. [8,9] discussed the effect of magnetic field for rotating flow of a second grade fluid past a shrinking surface and analysed the mass transfer in MHD flow of a upper-convected Maxwell fluid over a porous shrinking sheet in presence of chemical reaction. Fang et al. [10] showed the slip effects on MHD flow of Newtonian fluid due to a permeable shrinking sheet. The effects of thermal radiation, heat source/sink and suction/injection on unsteady MHD flow and heat transfer past a shrinking 3
ACCEPTED MANUSCRIPT sheet were obtained by Bhattacharyya [11]. The three dimensional MHD flow of Maxwell fluid and heat transfer past a permeable stretching/shrinking surface in presence in presence of nano-particles and convective boundary condition were reported by Jusoh et al. [12]. In addition to the magnetic field, if the medium of flow be a porous medium then also the vorticity generation due to shrinking of sheet can be controlled and a boundary layer flow becomes possible. The flow in porous medium is very realistic case and has many applications in industries, specially, in the petroleum and geothermal industries. Also, this
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type of flows is very useful in solid matrix heat exchanges, groundwater movement, nuclear waste disposal, surface catalysis of chemical reactions, etc. The flow and heat transfer over a shrinking sheet in a porous medium was demonstrated by Mohd and Hashim [13]. Nadeem, and Awais [14] described the effects of variable viscosity, variable thermo-capillarity on the flow and heat transfer in a thin film on a horizontal porous shrinking sheet through a porous
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medium. Vyas and Srivastava [15] discussed the radiative heat transfer in boundary layer flow over an exponentially shrinking permeable sheet in porous medium. The flow and heat transfer past a permeable shrinking sheet in a porous medium with a second-order slip was presented by Yasin et al. [16].
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The boundary layer stagnation-point flow over shrinking sheet is also possible without any other external force and the boundary layer exist upto certain value of the ratio of
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shrinking rate of the sheet to straining rate of the stagnation-point flow. The boundary layer stagnation-point flow of Newtonian fluid over a shrinking sheet was investigated by Wang [17] and he found dual solutions in certain conditions. Some important effects and results on
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the stagnation-point flow towards a shrinking sheet are reported by Ishak et al. [18], Bhattacharyya and Layek [19], Yacob et al. [20], Bhattacharyya et al. [21], Yacob et al. [22]
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and Bhattacharyya [23]. The effect of magnetic field on stagnation-point flow of a electrically conducting fluid and heat transfer over a shrinking sheet was studied by Lok et al. [24] and
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Mahapatra et al. [25]. Abbas et al. [26] investigated the momentum and heat transfer characteristics of stagnation-point flow due to a shrinking sheet in the presence of a transverse magnetic field and partial slip condition at the boundary. Whereas, the stagnationpoint flow due to shrinking sheet embedded in porous medium with heat transfer was illustrated by Rosali et al. [27] and Rosali and Ishak [28]. Chaudhary et al. [29,30] discussed MHD flow stagnation flow with constant wall temperature and unsteady flow with thermal radiation over stretching or shrinking sheet in porous medium. Recently, Majeed et al. [31]
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ACCEPTED MANUSCRIPT studied the stagnation point flow and heat transfer of a ferromagnetic fluid past a stretching/shrinking surface in a porous medium in presence of heat source/sink. So, the study of heat transfer in MHD flow induced by a shrinking sheet in porous medium is important physical mechanism having various applications in industries. In this communication, the simultaneous effects of variable surface temperature and heat sink or source on the heat transfer characteristics for MHD stagnation-point flow towards a shrinking/stretching sheet embedded in a porous medium are investigated. The behaviour of
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this type of flow and heat transfer is not yet reported. Here, the temperature along the surface is taken in power-law variation. After mathematical formulation of this complicated model, numerical solutions are derived and discussed.
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2. Mathematical formulation of the Problem
Let us consider the steady MHD stagnation-point flow of viscous incompressible electrically conducting fluid and heat transfer over a non-isothermal shrinking/stretching sheet in porous medium with heat sink/source effect. The shrinking/stretching velocity of the sheet and the straining velocity are taken linearly variable in the horizontal direction. Using
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boundary layer approximation, the governing equations of motion and the energy equation
u v 0, x y
dU s u u 2u B02 v Us 2 (U s u ) (u U s ) x y dx y k1 T T 2T Q0 v (T T ) , x y c p y 2 c p
(2)
(3)
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and u
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u
(1)
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can be written as:
where u and v are velocity components in x- and y-directions respectively, (=/) is the
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kinematic viscosity of fluid, is the density of fluid, is the coefficient of fluid viscosity, Us is the straining velocity, B0 is the strength uniform magnetic field normal to the sheet, is the electrical conductivity, k1 is the permeability of the porous medium, T is the temperature, is the thermal conductivity, cp is the specific heat and Q0(Js1m3K1) is dimensional heat source or sink coefficient. The corresponding boundary conditions for the velocity components and temperature are
u U w ( x) cx, v 0 at y 0; u U s ( x) ax as y , 5
(4)
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(5)
where c and a(>0) are the shrinking/stretching(00) constant and straining constant respectively, Tw is the variable surface temperature along the sheet where the flow takes place, T is the free stream temperature assumed to be constant, T0 is a constant depended on the thermal properties of the liquid and n is a power-law exponent defining the rate temperature variation along the sheet. A sketch of the physical problem is given in Fig1.
xU s f ( ), T T (Tw T ) ( ) and y
Us x
,
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The following similarity transformations are introduced: (6)
where is the stream function defined in the usual notation as u y & v x and is the similarity variable.
So, the equation continuity (1) satisfies automatically with such construction and using (6)
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the equations (2) and (3) reduce to the following self-similar ordinary differential equations:
f ff f 2 M (1 f ) k ( f 1) 1 0 and Pr[ f nf ] 0 ,
(7) (8)
where primes denote differentiation with respect to , M B02 (a ) is the magnetic
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parameter, k (ak1 ) is the porous parameter and Pr c p is the Prandtl number and
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Q0 ( c p a) is the heat source (<0) or sink (>0) parameter. The boundary conditions (4) and (5) also reduce to f ( ) 0, f ( ) B at 0; f ( ) 1 as
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and ( ) 1 at 0; ( ) 0 as ,
(9) (10)
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where B c a is the velocity ratio parameter. The quantities of physical as well as engineering interest are the local skin friction coefficient and the local Nusselt number which are defined as
w xqw , Nux , 2 U s (Tw T )
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(11)
where w is the surface shear stress and qw is the surface heat flux and which are defined as
u
T
w , qw . y y 0 y y 0
(12)
Hence, the local skin friction coefficient C f and the local Nusselt number Nu x are obtained as: 6
ACCEPTED MANUSCRIPT 1/2 C f Re1/2 (0) , x f (0), Nux Re x
(13)
where Re x xU s is the local Reynolds number.
3. Numerical method for solution The nonlinear self-similar ordinary differential equations (7) and (8) with the boundary conditions (9) and (10) form a boundary value problem (BVP) and it can be solved
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using shooting method [21,23] by converting it to an initial value problem (IVP). In this method, it is necessary to choose a suitable finite value of , say and also the aforesaid higher order equations are replaced by the following first-order system: f p p q
h
h Pr(np fh )
with the boundary conditions, f (0) 0, p(0) B, (0) 1.
(14)
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2 q p fq M (1 p ) k ( p 1) 1
(15)
(16)
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In order to integrate (14) and (15) with (16) as an IVP, the values for q(0) i.e. f(0) and h(0)
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i.e. (0) are required but those are not there in the mathematical formulation of the problem. The initial guess values for f(0) and (0) are selected and then fourth order Runge-Kutta method is adopted to obtain an approximate solution. Then the computed values for f() and
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() at (=15) are compared with the given boundary conditions f()=1 and ()=0, and the values of f(0) and (0) are adjusted using “Secant method” to get better approximation
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for the solution. Here the step-size is taken as =0.01 and the total procedure is repeated
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until the obtained results are converged up to the desired accuracy level i.e. 105.
4. Statements of results and discussion The MHD stagnation-point flow and heat transfer past a shrinking/stretching sheet
embedded in porous medium are discussed considering the effects of heat sink or source and variable temperature of the sheet. The combine effects of magnetic field, porous medium, internal heat generation/absorption in presence of variable wall temperature explores many important results for the heat transfer in stagnation-point flow over shrinking/stretching sheet. Due to occurrence of magnetic field and because of the medium of flow being porous the 7
ACCEPTED MANUSCRIPT steady state solutions of the flow dynamics and heat transfer exist for larger magnitude of B [for B<0, i.e., for shrinking sheet case and for stretching sheet case(B>0) solutions always exist], i.e., for higher shrinking rate compared to the straining rate. Similar to the case M=0 and k=0 [17], here also dual similarity solutions are obtained for B1 and unique for B>1. Similar to Wang [17], for M=0 and k=0 dual solutions obtained for 1.246571.
Further, for M=0.05 and k=0 dual solutions are found for
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1.30832
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problem.
4.1 Validation of obtained results by comparison
A comparison of results obtained using the above described numerical scheme for M=0 and k=0 (without magnetic field and porous medium) with the results by Wang [17], Lok et al.
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[24] (M=0, i.e., with no magnetic field) and Rosali and Ishak [28] are presented in Table1 for stretching sheet (B>0) and in Table2 for shrinking sheet (B<0). From both tables an obvious
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observation can be made that the computed results are in excellent agreement with the published data. Now, we have the confidence to say that the numerical method produces
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correct results.
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4.2 Variations in skin friction coefficient Cf and Nusselt number Nux The local skin friction coefficient Cf and the local Nusselt number Nux are directly related to the quantities f(0) and (0) respectively. So, the values of f(0) and (0) are presented in
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Fig2-Fig10 for various values of physical parameters to explain the effects on Cf and Nux. The graphs Fig2 and Fig4 confirm the aforesaid condition of existence of solution, i.e., for larger values of M and k the range of B, where solution found, increases. It causes due to the Lorentz force (induced due to interaction of electric and magnetic fields) and the resistance of porous medium. Actually, due to the presence of those forces the vorticity generation due to shrinking of sheet reduces and the steady boundary layer flow is maintained. In addition, it is observed that the value of local skin friction coefficient is larger for higher values of M and k for first solution of shrinking sheet and for only solution of stretching sheet (upto B<1) and 8
ACCEPTED MANUSCRIPT for second solution of shrinking sheet it is smaller. The value of (0), i.e., the local Nusselt number increases (decreases) with M and k (Fig3 and Fig5) for first (second) solution of shrinking sheet for direct variation of surface temperature (n>0) and for inverse variation (n<0) the results are exactly reverse. Also, the nature in stretching case is same as that of the first solution of shrinking case. Whereas, the values of (0)(>0) for inverse variation (n=1.4) shows increasing nature with larger Pr for both solutions of shrinking sheet and also
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for stretching sheet, i.e., heat transfers from the sheet to the ambient fluid layer and it is faster for larger Pr (Fig6). The behaviour in stretching sheet case is same for direct variation (n=0.7) also. But, for both solutions of shrinking case (B<1) in direct variation heat transfer rate reduces. From Fig7 it is concluded that when the strength of the heat sink increases or the strength of the heat source decreases the heat transfer rate enhances in all solutions of
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shrinking and stretching cases except some situation of inverse variation of shrinking case. As the value of n increases from negative to positive the rate of heat transfer reduces for both solutions of shrinking sheet cases (B<1) and for stretching case it causes increment of heat transfer rate (Fig8). In Fig9 and Fig10, the values of (0) plotted against n for different values of Pr and for a specific value of the ratio of shrinking rate to the straining rate, i.e.,
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for B=1.3. It is found that as n increases (from 1.5 to 1.5) the effect of Prandtl number on
remains the same.
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the heat transfer changes its character but the effect of heat sink or source parameter almost
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4.2 Variations in dimensionless velocity f() Dual velocity profile for various magnetic field strengths, i.e., for different values of
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magnetic parameter M are depicted in Fig11 and the stream lines for one of such values of M are presented in Fig12 for the case of shrinking sheet, B=1.28. Also for stretching sheet
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case, B=0.5, the variation in velocity profile for various M is plotted in Fig13 and in addition stream lines of the flow for M=0.02. As the strength of the magnetic field enhances the magnitude of Lorentz force increases and it results a reduction in the velocity boundary layer thickness for first solution in shrinking sheet and for the only solution in stretching sheet. Opposite effect is found for second solution of shrinking sheet, which is quite expected. Actually, out of these two solutions in certain cases of shrinking sheet (B1) the first solution is physically stable [16] and found in real situations and it can be confirmed from the fact that this solution is the only solution for B>1, i.e., for some cases of shrinking sheet and all cases of stretching sheet. Effects of change in the permeability of the porous medium, i.e., 9
ACCEPTED MANUSCRIPT the medium of flow, signified by the increment in porous parameter k, on the velocity are similar to that of for magnetic parameter and those variations in velocity and stream lines are presented in Fig14, Fig15 and Fig16. The permeability decreases when porous parameter increases and consequently it causes a reduction in velocity boundary layer thickness. For less permeability, the fluid inside the boundary layer assumes the free stream velocity quicker which narrow down the region where velocity variation is found, i.e., it reduces the boundary layer thickness. For change of the values of B the velocity profiles is affected and those are
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presented in Fig17 for shrinking as well as stretching sheet cases. Due to increase in B the velocity at a point increases (for stretching sheet and stable first solution of shrinking sheet) and it makes the boundary layer thinner. It means that for B<0 when shrinking rate is not much greater than the straining rate then boundary layer becomes smaller and for 0
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be thinner.
4.3 Variations in dimensionless temperature ()
The effects of magnetic field on the dimensionless temperature profiles for shrinking sheet
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and stretching sheet cases are depicted in Fig18 and Fig19, respectively, in presence of power-law variation of surface temperature. Two cases of power-law variation of surface
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temperature are considered; first one, direct variation along the sheet, i.e., n>0 and second one, the inverse variation along the sheet, i.e., n<0. Similar to that velocity boundary layer, the Lorentz force reduces the thermal boundary layer thickness for first temperature solution
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and the action is reverse for the second solution of shrinking sheet case. Though the effect is not so significant in the case of stretching sheet case for small change in magnetic parameter,
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but it shows similar outcome to that of first solution in shrinking sheet case. For the thermal boundary layer, the effect of change of porous parameter is also similar to the consequence of
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variation in magnetic parameter and those can be found in Fig20 and Fig21 for both direct variation and inverse variation along the sheet. Different temperature profiles for several values of velocity ratio parameter B with two values of n (0.7 and –1.4) are presented in Fig22 and Fig23 for shrinking sheet and stretching sheet cases respectively. As B increases the temperature in the boundary layer region and thermal boundary layer thickness reduces for first solution in shrinking sheet case and only solution of stretching sheet case; except for inverse variations, i.e., for n=–1.4 where only initially which means near the sheet i.e., for small values of temperature increases in both cases shrinking (first solution) as well as 10
ACCEPTED MANUSCRIPT stretching, but finally thermal boundary becomes thinner. Next, the influences of the thermal conductivity, i.e., the effects of Prandtl number Pr on the temperature distribution are illustrated in Fig24, Fig25 and Fig26. It is found that the thermal boundary layer thickness becomes larger as Prandtl number decreases. Actually, the Prandtl number Pr is inversely proportional to thermal conductivity of the fluid, so for smaller values of Pr confirms a superior effect of conduction on thermal energy flow and consequently thermal boundary layer thickness increases. In addition it is very much clear from obtained results that the
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temperature near the sheet happens to be greater than the surface temperature for direct variation case (n>0) and this nature can also controlled by making Prandtl number slightly small and also the wavy nature of the of the temperature profile for inverse variation (n<0) can be controlled by small Pr. Finally, the effects of the internal heat generation or absorption, i.e., the heat source (<0) or sink (>0) parameter and the variations of the wall
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temperature, i.e., direct (n>0) and inverse (n<0) variations on the dimensionless temperature are described through graphical structure in Fig27–Fig32. It is clear from the results that the heat sink stabilizes the thermal boundary layer. For heat source, the heat absorption found and it grows with the strength of the heat source for direct variation case (n>0) of wall
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temperature (Fig27) for shrinking sheet flow. Whereas, for inverse variation case with heat source though heat absorption is not found the profile curves show an interesting character
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and the character is that those have a smooth zik-zak nature in shrinking sheet case (Fig28). It initially decreases (with heat transfer from the sheet) then increases (even more than the initial value 1, i.e., temperature slightly away from the sheet is higher than that of wall
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temperature) and again acquires decreasing nature and goes to below zero level (it means that before the temperature get the value which is in the free stream it becomes less than free
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stream value) and finally get increasing character to approach towards zero asymptotically. The wavy character becomes more prominent when the strength of the heat source increases
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(Fig28) and also when the power-law exponent of inverse variation turns out to be larger in magnitude (Fig31). So, the introductions of inverse variation of wall temperature and heat source de-stabilize the thermal flow, whereas the heat sink can control it to make a typical heat flow and for direct variation of surface temperature the heat absorption is found in some situations (Fig30). For stretching sheet case, the heat absorption is always found when the strength of the heat source is large for both types of variations in wall temperature (Fig29).
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ACCEPTED MANUSCRIPT Finally, it is important to note that all the velocity as well as temperature profiles are convergent asymptotically to the zero level with desire accuracy and it ensures the correctness of numerical scheme again.
5. Concluding remarks The main outcomes of the of study of heat transfer in MHD stagnation-point flow past
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a non-isothermal shrinking/stretching sheet in porous medium in presence of heat sink or source effect are summarized as follows: (a)
The steady flow solutions are reported for higher shrinking rate due to the presence of magnetic field and due to the medium of flow being porous.
(b)
Dual solutions for some cases of shrinking sheet with mutually opposite effects of
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physical parameters, like, magnetic parameter, porous parameter and velocity ratio parameter on the two velocity profiles are found. But the effects of all those parameters on the single velocity profile for stretching sheet case are same as that of first solution of shrinking sheet case. (c)
Local skin friction coefficient, i.e., the wall shear stress increases with magnetic field
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strength and the porous parameter for first solution of shrinking sheet and for only solution of stretching sheet and for second solution of shrinking sheet it decreases. Temperature of the fluid layer near the surface is greater than the wall temperature, i.e.,
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(d)
heat absorption occurs for direct variation of wall temperature along the sheet and it flourishes with increasing values of the strength of heat source and power-law exponent
(e)
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of direct variation for both solutions of shrinking sheet case. On the other hand, smooth zik-zak nature of temperature profiles are observed for
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inverse variation of wall temperature for shrinking sheet case with normal heat transfer characteristic and it become more prominent when the magnitude of power-law
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exponent of inverse variation increases and the heat source parameter become large. This nature can be controlled by heat sink.
(f)
For heat source, the temperature inside the boundary layer has an interesting character. In some point it is higher than the wall temperature and some other point it is lower than the free stream temperature.
(g)
For stretching sheet the heat absorption occurs for some cases of inverse variation of wall temperature and higher values of heat source parameter.
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ACCEPTED MANUSCRIPT Acknowledgement The authors want to express their thanks to the reviewers for their valuable comments and suggestions which led to definite improvement of the paper.
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(2011) 16-21.
[23] K. Bhattacharyya, Heat transfer in unsteady boundary layer stagnation-point flow towards a shrinking sheet, Ain Shams Eng. J. 4 (2013) 259-264.
[24] Y.Y. Lok, A. Ishak, I. Pop, MHD stagnation-point flow towards a shrinking sheet, Int. J. Numer. Meth. Heat Fluid Flow 21 (2011) 61-72. [25] T.R. Mahapatra, S.K. Nandy, A.S. Gupta, Momentum and heat transfer in MHD stagnation-point flow over a shrinking sheet, ASME J Appl. Mech. 78 (2011) 021015.
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ACCEPTED MANUSCRIPT [26] Z. Abbas, T. Masood, P.O. Olanrewaju, Dual solutions of MHD stagnation point flow and heat transfer over a stretching/shrinking sheet with generalized slip condition, J. Central South University 22 (2015) 2376-2384. [27] H. Rosali, A. Ishak, I. Pop, Stagnation point flow and heat transfer over a stretching/shrinking sheet in a porous medium, Int. Commun. Heat Mass Transfer 38 (2011) 1029-1032.
medium, AIP Conference Proceedings 1571 (2013) 949.
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[28] H. Rosali, A. Ishak, Stagnation-point flow over a stretching/shrinking sheet in a porous
[29] S. Chaudhary, S. Singh, S. Chaudhary, Numerical solution for magnetohydrodynamic stagnation point flow towards a stretching or shrinking surface in a saturated porous medium, Int. J. Pure Appl. Math. 106 (2016) 141-155.
[30] S. Chaudhary, M.K. Choudhary, R. Sharma, Effects of thermal radiation on
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hydromagnetic flow over an unsteady stretching sheet embedded in a porous medium in the presence of heat source or sink, Meccanica 50 (2015) 1977-1987. [31] A. Majeed, A. Zeeshan, M.M. Rashidi, M.B. Arain, Stagnation point flow of ferromagnetic particle-fluid suspension over a stretching/shrinking surface in a porous
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medium with heat source/sink, Caspian J. Appl. Sci. Res. 5 (2016) 34-44.
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Table1 Comparison values of f(0) for stretching (B>0). B
Present study with M=0, k=0
Wang [17]
Lok et al. [24]
Rosali and Ishak [28]
with M=0
with K=0
1.2325878
1.232588
1.232588
1.232588
0.1
1.1465609
1.14656
1.146561
1.146561
0.2
1.0511300
1.05113
1.051130
1.051130
0.5
0.7132949
0.71330
0.713295
1
0
0
0.00000
2
1.8873070
−1.88731
−1.887307
1.887307
5
10.2647495
−10.26475
−10.264749
10.264749
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0
0. 713295
Table2 Comparison values of f(0) for shrinking sheet (B<0). Present study with M=0, k=0
Wang [17]
Lok et al. [24]
Rosali and Ishak [28]
with M=0
with K=0
1.4022405
1.40224
1.402241
1.402241
0.5
1.4956697
1.49567
1.495670
1.495670
0.75
1.4892981
1.48930
1.489298
1.489298
1.3288169
1.32882
1.328817
1.328817
[0]
[0]
1.0822316
1.08223
1.082233
1.082231
[0.116702]
[0.116702]
0.55430
0.584303
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1.15
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0.25
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B
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1.2
1.2465
1.24657
[0.1167023] 0.9324728
[0.2336491] 0.5842915 [0.5542856]
[0.554295]
0.5745268 [0.5639987]
[ ] Second solution.
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0.554294
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Fig1 Physical sketch of the problem.
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Fig2 The values of f(0) vs. B for various M.
Fig3 The values of (0) vs. B for various M with (a) n>0 and (b) n<0. 17
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Fig4 The values of f(0) vs. B for various k.
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Fig5 The values of (0) vs. B for various k with (a) n>0 and (b) n<0.
Fig6 The values of (0) vs. B for various Pr with (a) n>0 and (b) n<0.
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Fig7 The values of (0) vs. B for various with (a) n>0 and (b) n<0.
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Fig8 The values of (0) vs. B for various n.
Fig9 The values of (0) vs. n for various Pr.
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Fig10 The values of (0) vs. n for various .
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Fig11 Dual velocity profiles for various values of M for shrinking sheet.
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Fig12 Stream function for M=0.02 for (a) Fist solution and (b) Second solution with k=0.05
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and B=1.28 (shrinking sheet).
Fig13 (a) Velocity profiles for various values of M and (b) Stream function for M=0.02 (for
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stretching sheet).
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Fig14 Dual velocity profiles for various values of k for shrinking sheet.
Fig15 Stream function for k=0.05 for (a) Fist solution and (b) Second solution with M=0.05
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and B=1.28 (shrinking sheet).
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Fig16 (a) Velocity profiles for various values of k and (b) Stream function for k=0.05 (for
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stretching sheet).
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Fig17 (a) Dual velocity profiles (for shrinking sheet) and (b) Single velocity profiles (for
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stretching sheet) for various values of B.
Fig18 Dual temperature profiles for various values of M with (a) n>0 and (b) n<0 for shrinking sheet. 23
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Fig19 Temperature profiles for various values of M with (a) n>0 and (b) n<0 for stretching
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sheet.
Fig20 Dual temperature profiles for various values of k with (a) n>0 and (b) n<0 for
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shrinking sheet.
Fig21 Temperature profiles for various values of k with (a) n>0 and (b) n<0 for stretching sheet. 24
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Fig22 Dual temperature profiles for various values of B with (a) n>0 and (b) n<0 for
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shrinking sheet.
Fig23 Temperature profiles for various values of B with (a) n>0 and (b) n<0 for stretching
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sheet.
Fig24 Dual temperature profiles for various values of Pr with n>0 for shrinking sheet.
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Fig25 Dual temperature profiles for various values of Pr with n<0 for shrinking sheet.
sheet.
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Fig26 Temperature profiles for various values of Pr with (a) n>0 and (b) n<0 for stretching
Fig27 Dual temperature profiles for various values of with n>0 for shrinking sheet.
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Fig28 Dual temperature profiles for various values of with n<0 for shrinking sheet.
sheet.
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Fig29 Temperature profiles for various values of with (a) n>0 and (b) n<0 for stretching
Fig30 Dual temperature profiles for various values of n>0 for shrinking sheet.
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Fig31 Dual temperature profiles for various values of n<0 for shrinking sheet.
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Fig32 Temperature profiles for various values of (a) n>0 and (b) n<0 for stretching sheet.
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Graphical abstract
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MHD stagnation-point flow and heat transfer past a non-isothermal shrinking/stretching sheet in porous medium with heat sink or source effect G.S. Seth , A.K. Singha , M.S. Mandal , Astick Banerjee , Krishnendu Bhattacharyya PII: DOI: Reference:
S0020-7403(17)30877-9 10.1016/j.ijmecsci.2017.09.049 MS 3960
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
6 April 2017 20 September 2017 22 September 2017
Please cite this article as: G.S. Seth , A.K. Singha , M.S. Mandal , Astick Banerjee , Krishnendu Bhattacharyya , MHD stagnation-point flow and heat transfer past a non-isothermal shrinking/stretching sheet in porous medium with heat sink or source effect, International Journal of Mechanical Sciences (2017), doi: 10.1016/j.ijmecsci.2017.09.049
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Highlights MHD stagnation-point flow and heat transfer past a non-isothermal shrinking/stretching sheet in porous medium in presence of heat sink or source is investigated.
The solutions of steady flow are obtained even with higher shrinking rate due to magnetic field and porous medium.
Dual solutions for some cases of shrinking sheet are found, but solution in stretching sheet case is always unique.
For direct variation of wall temperature along the sheet heat absorption occurs and it grows with heat source.
In inverse variation of wall temperature, a smooth zik-zak nature of temperature profile is noticed with normal heat transfer characteristic for shrinking sheet case.
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MHD stagnation-point flow and heat transfer past a non-isothermal shrinking/stretching sheet in porous medium with heat sink or source effect G.S. Setha, A.K. Singhaa, M.S. Mandalb, Astick Banerjeec and Krishnendu Bhattacharyyad* a
Department of Applied Mathematics, Indian School of Mines, Dhanbad–826004, India
Department of Mathematics, Government General Degree College, Kalna–I, Burdwan–713405, West Bengal c
d
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b
Mohulara Jr. High School, Birbhum–731236, West Bengal, India
Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi–221005, Uttar
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Pradesh, India
Abstract
The MHD stagnation-point flow of electrically conducting fluid and heat transfer past a nonisothermal shrinking/stretching sheet in a porous medium in presence of heat sink or source are investigated. The governing equations are transformed by similarity transformation
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technique and the converted equations are solved numerically by shooting method. Computed results are presented in some figures. It is obtained that the boundary layer solutions of steady
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flow exist with higher shrinking rate for the presence of magnetic field. Also, similar fact is observed when the porous parameter increases. The similarity solutions for some cases of shrinking sheet are of dual nature; it means that there exists two solutions for higher
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shrinking rate and the solution is unique for all stretching sheet cases. The impact of variable wall temperature along the sheet is massive on the thermal flow and the temperature field.
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For direct variation of wall temperature along the sheet heat absorption is noted for first solution in shrinking sheet case and it increases with the introduction of heat source; but, the
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presence of heat sink controls the heat absorption and it helps to return back to normal heat transfer process. Whereas, smooth zik-zak nature of temperature profiles is found for the inverse variation of wall temperature for shrinking sheet case with heat transfer from the surface to the ambient fluid layer. On the other hand, for stretching sheet case normally heat transfer occurs, but for higher values of heat source parameter and for larger magnitudes of power-law exponent of inverse variation of wall temperature heat absorption is also noticed.
*
Corresponding author. Email Addresses: [email protected], [email protected], [email protected].
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Key words: MHD; stagnation-point flow; heat transfer; porous medium; non-isothermal shrinking sheet; heat sink or source; dual solutions.
1. Introduction The flow due to a shrinking sheet, i.e., the reverse type of flow to that of the flow over
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a stretching sheet has given less attention though it is quite important in many physical as well as the industrial problems, such as, shrinking balloon problem, shrinking film, packaging of products etc. On the other hand, the analysis of heat transfer in the flow field is quite important in the view of its impact on the final quality of the manufactured product in the industry. The flow due to a linearly stretching sheet was first described by Crane [1] and the
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reverse flow, i.e., the flow developed due to shrinking sheet was discussed by Wang [2]. Whereas, the existence, uniqueness and non-existence of steady flow past porous shrinking sheet with mass suction through the porous sheet was reported by Miklavčič and Wang [3] and they showed that it depends on the applied mass suction. The heat transfer in stretching sheet flow was analysed by Gupta and Gupta [4] and that of shrinking sheet by Fang and
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Zhang [5].
Actually, the flow due to shrinking sheet is much more interesting physically because
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to have the boundary layer flow the diffusion of generated vorticity needs to prevent. This goal of controlling vorticity diffusion to maintain the structure of boundary layer can be achieve by some other external forces. External magnetic field is one of such forces which
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can control the vorticity generated due to shrinking and also can confine it in the boundary layer region. The effect of magnetic field on the flow of electrically conducting Newtonian
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fluid over a shrinking sheet was studied by Fang and Zhang [6] and that of for second grade fluid by Hayat et al. [7]. This type of flow with magnetohydrodynamic (MHD) effects has
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many applications in many engineering problems such as petroleum industries, plasma studies, geothermal energy extractions, the boundary layer control in the field of aerodynamics and many others. Later, Hayat et al. [8,9] discussed the effect of magnetic field for rotating flow of a second grade fluid past a shrinking surface and analysed the mass transfer in MHD flow of a upper-convected Maxwell fluid over a porous shrinking sheet in presence of chemical reaction. Fang et al. [10] showed the slip effects on MHD flow of Newtonian fluid due to a permeable shrinking sheet. The effects of thermal radiation, heat source/sink and suction/injection on unsteady MHD flow and heat transfer past a shrinking 3
ACCEPTED MANUSCRIPT sheet were obtained by Bhattacharyya [11]. The three dimensional MHD flow of Maxwell fluid and heat transfer past a permeable stretching/shrinking surface in presence in presence of nano-particles and convective boundary condition were reported by Jusoh et al. [12]. In addition to the magnetic field, if the medium of flow be a porous medium then also the vorticity generation due to shrinking of sheet can be controlled and a boundary layer flow becomes possible. The flow in porous medium is very realistic case and has many applications in industries, specially, in the petroleum and geothermal industries. Also, this
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type of flows is very useful in solid matrix heat exchanges, groundwater movement, nuclear waste disposal, surface catalysis of chemical reactions, etc. The flow and heat transfer over a shrinking sheet in a porous medium was demonstrated by Mohd and Hashim [13]. Nadeem, and Awais [14] described the effects of variable viscosity, variable thermo-capillarity on the flow and heat transfer in a thin film on a horizontal porous shrinking sheet through a porous
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medium. Vyas and Srivastava [15] discussed the radiative heat transfer in boundary layer flow over an exponentially shrinking permeable sheet in porous medium. The flow and heat transfer past a permeable shrinking sheet in a porous medium with a second-order slip was presented by Yasin et al. [16].
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The boundary layer stagnation-point flow over shrinking sheet is also possible without any other external force and the boundary layer exist upto certain value of the ratio of
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shrinking rate of the sheet to straining rate of the stagnation-point flow. The boundary layer stagnation-point flow of Newtonian fluid over a shrinking sheet was investigated by Wang [17] and he found dual solutions in certain conditions. Some important effects and results on
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the stagnation-point flow towards a shrinking sheet are reported by Ishak et al. [18], Bhattacharyya and Layek [19], Yacob et al. [20], Bhattacharyya et al. [21], Yacob et al. [22]
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and Bhattacharyya [23]. The effect of magnetic field on stagnation-point flow of a electrically conducting fluid and heat transfer over a shrinking sheet was studied by Lok et al. [24] and
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Mahapatra et al. [25]. Abbas et al. [26] investigated the momentum and heat transfer characteristics of stagnation-point flow due to a shrinking sheet in the presence of a transverse magnetic field and partial slip condition at the boundary. Whereas, the stagnationpoint flow due to shrinking sheet embedded in porous medium with heat transfer was illustrated by Rosali et al. [27] and Rosali and Ishak [28]. Chaudhary et al. [29,30] discussed MHD flow stagnation flow with constant wall temperature and unsteady flow with thermal radiation over stretching or shrinking sheet in porous medium. Recently, Majeed et al. [31]
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ACCEPTED MANUSCRIPT studied the stagnation point flow and heat transfer of a ferromagnetic fluid past a stretching/shrinking surface in a porous medium in presence of heat source/sink. So, the study of heat transfer in MHD flow induced by a shrinking sheet in porous medium is important physical mechanism having various applications in industries. In this communication, the simultaneous effects of variable surface temperature and heat sink or source on the heat transfer characteristics for MHD stagnation-point flow towards a shrinking/stretching sheet embedded in a porous medium are investigated. The behaviour of
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this type of flow and heat transfer is not yet reported. Here, the temperature along the surface is taken in power-law variation. After mathematical formulation of this complicated model, numerical solutions are derived and discussed.
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2. Mathematical formulation of the Problem
Let us consider the steady MHD stagnation-point flow of viscous incompressible electrically conducting fluid and heat transfer over a non-isothermal shrinking/stretching sheet in porous medium with heat sink/source effect. The shrinking/stretching velocity of the sheet and the straining velocity are taken linearly variable in the horizontal direction. Using
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boundary layer approximation, the governing equations of motion and the energy equation
u v 0, x y
dU s u u 2u B02 v Us 2 (U s u ) (u U s ) x y dx y k1 T T 2T Q0 v (T T ) , x y c p y 2 c p
(2)
(3)
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and u
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u
(1)
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can be written as:
where u and v are velocity components in x- and y-directions respectively, (=/) is the
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kinematic viscosity of fluid, is the density of fluid, is the coefficient of fluid viscosity, Us is the straining velocity, B0 is the strength uniform magnetic field normal to the sheet, is the electrical conductivity, k1 is the permeability of the porous medium, T is the temperature, is the thermal conductivity, cp is the specific heat and Q0(Js1m3K1) is dimensional heat source or sink coefficient. The corresponding boundary conditions for the velocity components and temperature are
u U w ( x) cx, v 0 at y 0; u U s ( x) ax as y , 5
(4)
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(5)
where c and a(>0) are the shrinking/stretching(00) constant and straining constant respectively, Tw is the variable surface temperature along the sheet where the flow takes place, T is the free stream temperature assumed to be constant, T0 is a constant depended on the thermal properties of the liquid and n is a power-law exponent defining the rate temperature variation along the sheet. A sketch of the physical problem is given in Fig1.
xU s f ( ), T T (Tw T ) ( ) and y
Us x
,
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The following similarity transformations are introduced: (6)
where is the stream function defined in the usual notation as u y & v x and is the similarity variable.
So, the equation continuity (1) satisfies automatically with such construction and using (6)
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the equations (2) and (3) reduce to the following self-similar ordinary differential equations:
f ff f 2 M (1 f ) k ( f 1) 1 0 and Pr[ f nf ] 0 ,
(7) (8)
where primes denote differentiation with respect to , M B02 (a ) is the magnetic
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parameter, k (ak1 ) is the porous parameter and Pr c p is the Prandtl number and
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Q0 ( c p a) is the heat source (<0) or sink (>0) parameter. The boundary conditions (4) and (5) also reduce to f ( ) 0, f ( ) B at 0; f ( ) 1 as
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and ( ) 1 at 0; ( ) 0 as ,
(9) (10)
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where B c a is the velocity ratio parameter. The quantities of physical as well as engineering interest are the local skin friction coefficient and the local Nusselt number which are defined as
w xqw , Nux , 2 U s (Tw T )
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(11)
where w is the surface shear stress and qw is the surface heat flux and which are defined as
u
T
w , qw . y y 0 y y 0
(12)
Hence, the local skin friction coefficient C f and the local Nusselt number Nu x are obtained as: 6
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(13)
where Re x xU s is the local Reynolds number.
3. Numerical method for solution The nonlinear self-similar ordinary differential equations (7) and (8) with the boundary conditions (9) and (10) form a boundary value problem (BVP) and it can be solved
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using shooting method [21,23] by converting it to an initial value problem (IVP). In this method, it is necessary to choose a suitable finite value of , say and also the aforesaid higher order equations are replaced by the following first-order system: f p p q
h
h Pr(np fh )
with the boundary conditions, f (0) 0, p(0) B, (0) 1.
(14)
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2 q p fq M (1 p ) k ( p 1) 1
(15)
(16)
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In order to integrate (14) and (15) with (16) as an IVP, the values for q(0) i.e. f(0) and h(0)
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i.e. (0) are required but those are not there in the mathematical formulation of the problem. The initial guess values for f(0) and (0) are selected and then fourth order Runge-Kutta method is adopted to obtain an approximate solution. Then the computed values for f() and
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() at (=15) are compared with the given boundary conditions f()=1 and ()=0, and the values of f(0) and (0) are adjusted using “Secant method” to get better approximation
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for the solution. Here the step-size is taken as =0.01 and the total procedure is repeated
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until the obtained results are converged up to the desired accuracy level i.e. 105.
4. Statements of results and discussion The MHD stagnation-point flow and heat transfer past a shrinking/stretching sheet
embedded in porous medium are discussed considering the effects of heat sink or source and variable temperature of the sheet. The combine effects of magnetic field, porous medium, internal heat generation/absorption in presence of variable wall temperature explores many important results for the heat transfer in stagnation-point flow over shrinking/stretching sheet. Due to occurrence of magnetic field and because of the medium of flow being porous the 7
ACCEPTED MANUSCRIPT steady state solutions of the flow dynamics and heat transfer exist for larger magnitude of B [for B<0, i.e., for shrinking sheet case and for stretching sheet case(B>0) solutions always exist], i.e., for higher shrinking rate compared to the straining rate. Similar to the case M=0 and k=0 [17], here also dual similarity solutions are obtained for B1 and unique for B>1. Similar to Wang [17], for M=0 and k=0 dual solutions obtained for 1.24657
Further, for M=0.05 and k=0 dual solutions are found for
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problem.
4.1 Validation of obtained results by comparison
A comparison of results obtained using the above described numerical scheme for M=0 and k=0 (without magnetic field and porous medium) with the results by Wang [17], Lok et al.
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[24] (M=0, i.e., with no magnetic field) and Rosali and Ishak [28] are presented in Table1 for stretching sheet (B>0) and in Table2 for shrinking sheet (B<0). From both tables an obvious
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observation can be made that the computed results are in excellent agreement with the published data. Now, we have the confidence to say that the numerical method produces
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correct results.
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4.2 Variations in skin friction coefficient Cf and Nusselt number Nux The local skin friction coefficient Cf and the local Nusselt number Nux are directly related to the quantities f(0) and (0) respectively. So, the values of f(0) and (0) are presented in
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Fig2-Fig10 for various values of physical parameters to explain the effects on Cf and Nux. The graphs Fig2 and Fig4 confirm the aforesaid condition of existence of solution, i.e., for larger values of M and k the range of B, where solution found, increases. It causes due to the Lorentz force (induced due to interaction of electric and magnetic fields) and the resistance of porous medium. Actually, due to the presence of those forces the vorticity generation due to shrinking of sheet reduces and the steady boundary layer flow is maintained. In addition, it is observed that the value of local skin friction coefficient is larger for higher values of M and k for first solution of shrinking sheet and for only solution of stretching sheet (upto B<1) and 8
ACCEPTED MANUSCRIPT for second solution of shrinking sheet it is smaller. The value of (0), i.e., the local Nusselt number increases (decreases) with M and k (Fig3 and Fig5) for first (second) solution of shrinking sheet for direct variation of surface temperature (n>0) and for inverse variation (n<0) the results are exactly reverse. Also, the nature in stretching case is same as that of the first solution of shrinking case. Whereas, the values of (0)(>0) for inverse variation (n=1.4) shows increasing nature with larger Pr for both solutions of shrinking sheet and also
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for stretching sheet, i.e., heat transfers from the sheet to the ambient fluid layer and it is faster for larger Pr (Fig6). The behaviour in stretching sheet case is same for direct variation (n=0.7) also. But, for both solutions of shrinking case (B<1) in direct variation heat transfer rate reduces. From Fig7 it is concluded that when the strength of the heat sink increases or the strength of the heat source decreases the heat transfer rate enhances in all solutions of
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shrinking and stretching cases except some situation of inverse variation of shrinking case. As the value of n increases from negative to positive the rate of heat transfer reduces for both solutions of shrinking sheet cases (B<1) and for stretching case it causes increment of heat transfer rate (Fig8). In Fig9 and Fig10, the values of (0) plotted against n for different values of Pr and for a specific value of the ratio of shrinking rate to the straining rate, i.e.,
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for B=1.3. It is found that as n increases (from 1.5 to 1.5) the effect of Prandtl number on
remains the same.
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the heat transfer changes its character but the effect of heat sink or source parameter almost
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4.2 Variations in dimensionless velocity f() Dual velocity profile for various magnetic field strengths, i.e., for different values of
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magnetic parameter M are depicted in Fig11 and the stream lines for one of such values of M are presented in Fig12 for the case of shrinking sheet, B=1.28. Also for stretching sheet
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case, B=0.5, the variation in velocity profile for various M is plotted in Fig13 and in addition stream lines of the flow for M=0.02. As the strength of the magnetic field enhances the magnitude of Lorentz force increases and it results a reduction in the velocity boundary layer thickness for first solution in shrinking sheet and for the only solution in stretching sheet. Opposite effect is found for second solution of shrinking sheet, which is quite expected. Actually, out of these two solutions in certain cases of shrinking sheet (B1) the first solution is physically stable [16] and found in real situations and it can be confirmed from the fact that this solution is the only solution for B>1, i.e., for some cases of shrinking sheet and all cases of stretching sheet. Effects of change in the permeability of the porous medium, i.e., 9
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presented in Fig17 for shrinking as well as stretching sheet cases. Due to increase in B the velocity at a point increases (for stretching sheet and stable first solution of shrinking sheet) and it makes the boundary layer thinner. It means that for B<0 when shrinking rate is not much greater than the straining rate then boundary layer becomes smaller and for 0
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be thinner.
4.3 Variations in dimensionless temperature ()
The effects of magnetic field on the dimensionless temperature profiles for shrinking sheet
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and stretching sheet cases are depicted in Fig18 and Fig19, respectively, in presence of power-law variation of surface temperature. Two cases of power-law variation of surface
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temperature are considered; first one, direct variation along the sheet, i.e., n>0 and second one, the inverse variation along the sheet, i.e., n<0. Similar to that velocity boundary layer, the Lorentz force reduces the thermal boundary layer thickness for first temperature solution
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and the action is reverse for the second solution of shrinking sheet case. Though the effect is not so significant in the case of stretching sheet case for small change in magnetic parameter,
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but it shows similar outcome to that of first solution in shrinking sheet case. For the thermal boundary layer, the effect of change of porous parameter is also similar to the consequence of
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variation in magnetic parameter and those can be found in Fig20 and Fig21 for both direct variation and inverse variation along the sheet. Different temperature profiles for several values of velocity ratio parameter B with two values of n (0.7 and –1.4) are presented in Fig22 and Fig23 for shrinking sheet and stretching sheet cases respectively. As B increases the temperature in the boundary layer region and thermal boundary layer thickness reduces for first solution in shrinking sheet case and only solution of stretching sheet case; except for inverse variations, i.e., for n=–1.4 where only initially which means near the sheet i.e., for small values of temperature increases in both cases shrinking (first solution) as well as 10
ACCEPTED MANUSCRIPT stretching, but finally thermal boundary becomes thinner. Next, the influences of the thermal conductivity, i.e., the effects of Prandtl number Pr on the temperature distribution are illustrated in Fig24, Fig25 and Fig26. It is found that the thermal boundary layer thickness becomes larger as Prandtl number decreases. Actually, the Prandtl number Pr is inversely proportional to thermal conductivity of the fluid, so for smaller values of Pr confirms a superior effect of conduction on thermal energy flow and consequently thermal boundary layer thickness increases. In addition it is very much clear from obtained results that the
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temperature near the sheet happens to be greater than the surface temperature for direct variation case (n>0) and this nature can also controlled by making Prandtl number slightly small and also the wavy nature of the of the temperature profile for inverse variation (n<0) can be controlled by small Pr. Finally, the effects of the internal heat generation or absorption, i.e., the heat source (<0) or sink (>0) parameter and the variations of the wall
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temperature, i.e., direct (n>0) and inverse (n<0) variations on the dimensionless temperature are described through graphical structure in Fig27–Fig32. It is clear from the results that the heat sink stabilizes the thermal boundary layer. For heat source, the heat absorption found and it grows with the strength of the heat source for direct variation case (n>0) of wall
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temperature (Fig27) for shrinking sheet flow. Whereas, for inverse variation case with heat source though heat absorption is not found the profile curves show an interesting character
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and the character is that those have a smooth zik-zak nature in shrinking sheet case (Fig28). It initially decreases (with heat transfer from the sheet) then increases (even more than the initial value 1, i.e., temperature slightly away from the sheet is higher than that of wall
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temperature) and again acquires decreasing nature and goes to below zero level (it means that before the temperature get the value which is in the free stream it becomes less than free
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stream value) and finally get increasing character to approach towards zero asymptotically. The wavy character becomes more prominent when the strength of the heat source increases
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(Fig28) and also when the power-law exponent of inverse variation turns out to be larger in magnitude (Fig31). So, the introductions of inverse variation of wall temperature and heat source de-stabilize the thermal flow, whereas the heat sink can control it to make a typical heat flow and for direct variation of surface temperature the heat absorption is found in some situations (Fig30). For stretching sheet case, the heat absorption is always found when the strength of the heat source is large for both types of variations in wall temperature (Fig29).
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ACCEPTED MANUSCRIPT Finally, it is important to note that all the velocity as well as temperature profiles are convergent asymptotically to the zero level with desire accuracy and it ensures the correctness of numerical scheme again.
5. Concluding remarks The main outcomes of the of study of heat transfer in MHD stagnation-point flow past
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a non-isothermal shrinking/stretching sheet in porous medium in presence of heat sink or source effect are summarized as follows: (a)
The steady flow solutions are reported for higher shrinking rate due to the presence of magnetic field and due to the medium of flow being porous.
(b)
Dual solutions for some cases of shrinking sheet with mutually opposite effects of
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physical parameters, like, magnetic parameter, porous parameter and velocity ratio parameter on the two velocity profiles are found. But the effects of all those parameters on the single velocity profile for stretching sheet case are same as that of first solution of shrinking sheet case. (c)
Local skin friction coefficient, i.e., the wall shear stress increases with magnetic field
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strength and the porous parameter for first solution of shrinking sheet and for only solution of stretching sheet and for second solution of shrinking sheet it decreases. Temperature of the fluid layer near the surface is greater than the wall temperature, i.e.,
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(d)
heat absorption occurs for direct variation of wall temperature along the sheet and it flourishes with increasing values of the strength of heat source and power-law exponent
(e)
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of direct variation for both solutions of shrinking sheet case. On the other hand, smooth zik-zak nature of temperature profiles are observed for
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inverse variation of wall temperature for shrinking sheet case with normal heat transfer characteristic and it become more prominent when the magnitude of power-law
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exponent of inverse variation increases and the heat source parameter become large. This nature can be controlled by heat sink.
(f)
For heat source, the temperature inside the boundary layer has an interesting character. In some point it is higher than the wall temperature and some other point it is lower than the free stream temperature.
(g)
For stretching sheet the heat absorption occurs for some cases of inverse variation of wall temperature and higher values of heat source parameter.
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ACCEPTED MANUSCRIPT Acknowledgement The authors want to express their thanks to the reviewers for their valuable comments and suggestions which led to definite improvement of the paper.
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[2]
C.Y. Wang, Liquid film on an unsteady stretching sheet, Q. Appl. Math. 48 (1990) 601610.
[3]
M. Miklavčič, C.Y. Wang, Viscous flow due a shrinking sheet, Q. Appl. Math. 64 (2006) 283-290.
[4]
P.S. Gupta, A.S. Gupta, Heat and mass transfer on a stretching sheet with suction and blowing, Can. J. Chem. Eng. 55 (1977) 744-746.
T. Fang, J. Zhang, Thermal boundary layers over a shrinking sheet: an analytical
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[1]
solution, Acta Mech. 209 (2010) 325-343. [6]
T. Fang, J. Zhang, Closed-form exact solution of MHD viscous flow over a shrinking sheet, Commun. Nonlinear Sci. Numer. Simulat. 14 (2009) 2853-2857. T. Hayat, Z. Abbas, M. Sajid, On the analytic solution of magnetohydrodynamic flow
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[7]
of a second grade fluid over a shrinking sheet, ASME J. Appl. Mech. 74 (2007) 1165-
[8]
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1171.
T. Hayat, T. Javed, M. Sajid, Analytic solution for MHD rotating flow of a second grade fluid over a shrinking surface, Phys. Lett. A 372 (2008) 3264-3273. T. Hayat, Z. Abbas, N. Ali, MHD flow and mass transfer of a upper-convected Maxwell
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[9]
fluid past a porous shrinking sheet with chemical reaction species, Phys. Lett. A 372
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(2008) 4698-4704.
[10] T. Fang, J. Zhang, S. Yao, Slip magnetohydrodynamic viscous flow over a permeable
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shrinking sheet, Chin. Phys. Lett. 27 (2010) 124702. [11] K. Bhattacharyya, Effects of radiation and heat source/sink on unsteady MHD boundary layer flow and heat transfer over a shrinking sheet with suction/injection, Front. Chem. Sci. Eng. 5 (2011) 376-384.
[12] R. Jusoh, R. Nazar, I. Pop, Flow and heat transfer of magnetohydrodynamic threedimensional Maxwell nanofluid over a permeable stretching/shrinking surface with convective boundary conditions, Int. J. Mech. Sci. 124-125 (2017) 166-173.
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ACCEPTED MANUSCRIPT [13] N.F. Mohd, I. Hashim, MHD flow and heat transfer adjacent to a permeable shrinking sheet embedded in a porous medium, Sains Malaysiana 38 (2009) 559-565. [14] S. Nadeem, M. Awais, Thin film flow of an unsteady shrinking sheet through porous medium with variable viscosity, Physics Letters A 372 (2008) 4965-4972. [15] P. Vyas, N. Srivastava, Radiative Boundary layer flow in porous medium due to exponentially shrinking permeable sheet, ISRN Thermodynamics 2012 (2012) art.id 214362.
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[16] M.H.M. Yasin, A. Ishak, I. Pop, Boundary layer flow and heat transfer past a permeable shrinking surface embedded in a porous medium with a second-order slip: A stability analysis, Appl. Therm. Eng. 115 (2017) 1407–1411.
[17] C.Y. Wang, Stagnation flow towards a shrinking sheet, Int. J. Nonlinear Mech. 43 (2008) 377-382.
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[18] A. Ishak, Y.Y. Lok, I. Pop, Stagnation-point flow over a shrinking sheet in a micropolar fluid, Chem. Eng. Commun. 197 (2010) 1417-1427.
[19] K. Bhattacharyya, G.C. Layek, Effects of suction/blowing on steady boundary layer stagnation-point flow and heat transfer towards a shrinking sheet with thermal radiation,
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Int. J. Heat Mass Transfer 54 (2011) 302-307.
[20] N.A. Yacob, A. Ishak, I. Pop, Melting heat transfer in boundary layer stagnation-point
(2011) 16-21.
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flow towards a stretching/shrinking sheet in a micropolar fluid, Comput. Fluids 47
[21] K. Bhattacharyya, S. Mukhopadhyay, G.C. Layek, Slip effects on boundary layer
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stagnation-point flow and heat transfer towards a shrinking sheet, Int. J. Heat Mass Transfer 54 (2011) 308-313.
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[22] N.A. Yacob, A. Ishak, I. Pop, Melting heat transfer in boundary layer stagnation-point flow towards a stretching/shrinking sheet in a micropolar fluid, Comput. Fluids 47
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(2011) 16-21.
[23] K. Bhattacharyya, Heat transfer in unsteady boundary layer stagnation-point flow towards a shrinking sheet, Ain Shams Eng. J. 4 (2013) 259-264.
[24] Y.Y. Lok, A. Ishak, I. Pop, MHD stagnation-point flow towards a shrinking sheet, Int. J. Numer. Meth. Heat Fluid Flow 21 (2011) 61-72. [25] T.R. Mahapatra, S.K. Nandy, A.S. Gupta, Momentum and heat transfer in MHD stagnation-point flow over a shrinking sheet, ASME J Appl. Mech. 78 (2011) 021015.
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ACCEPTED MANUSCRIPT [26] Z. Abbas, T. Masood, P.O. Olanrewaju, Dual solutions of MHD stagnation point flow and heat transfer over a stretching/shrinking sheet with generalized slip condition, J. Central South University 22 (2015) 2376-2384. [27] H. Rosali, A. Ishak, I. Pop, Stagnation point flow and heat transfer over a stretching/shrinking sheet in a porous medium, Int. Commun. Heat Mass Transfer 38 (2011) 1029-1032.
medium, AIP Conference Proceedings 1571 (2013) 949.
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[28] H. Rosali, A. Ishak, Stagnation-point flow over a stretching/shrinking sheet in a porous
[29] S. Chaudhary, S. Singh, S. Chaudhary, Numerical solution for magnetohydrodynamic stagnation point flow towards a stretching or shrinking surface in a saturated porous medium, Int. J. Pure Appl. Math. 106 (2016) 141-155.
[30] S. Chaudhary, M.K. Choudhary, R. Sharma, Effects of thermal radiation on
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hydromagnetic flow over an unsteady stretching sheet embedded in a porous medium in the presence of heat source or sink, Meccanica 50 (2015) 1977-1987. [31] A. Majeed, A. Zeeshan, M.M. Rashidi, M.B. Arain, Stagnation point flow of ferromagnetic particle-fluid suspension over a stretching/shrinking surface in a porous
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medium with heat source/sink, Caspian J. Appl. Sci. Res. 5 (2016) 34-44.
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Table1 Comparison values of f(0) for stretching (B>0). B
Present study with M=0, k=0
Wang [17]
Lok et al. [24]
Rosali and Ishak [28]
with M=0
with K=0
1.2325878
1.232588
1.232588
1.232588
0.1
1.1465609
1.14656
1.146561
1.146561
0.2
1.0511300
1.05113
1.051130
1.051130
0.5
0.7132949
0.71330
0.713295
1
0
0
0.00000
2
1.8873070
−1.88731
−1.887307
1.887307
5
10.2647495
−10.26475
−10.264749
10.264749
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0
0. 713295
Table2 Comparison values of f(0) for shrinking sheet (B<0). Present study with M=0, k=0
Wang [17]
Lok et al. [24]
Rosali and Ishak [28]
with M=0
with K=0
1.4022405
1.40224
1.402241
1.402241
0.5
1.4956697
1.49567
1.495670
1.495670
0.75
1.4892981
1.48930
1.489298
1.489298
1.3288169
1.32882
1.328817
1.328817
[0]
[0]
1.0822316
1.08223
1.082233
1.082231
[0.116702]
[0.116702]
0.55430
0.584303
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1.15
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1
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0.25
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B
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1.2
1.2465
1.24657
[0.1167023] 0.9324728
[0.2336491] 0.5842915 [0.5542856]
[0.554295]
0.5745268 [0.5639987]
[ ] Second solution.
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0.554294
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Fig1 Physical sketch of the problem.
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Fig2 The values of f(0) vs. B for various M.
Fig3 The values of (0) vs. B for various M with (a) n>0 and (b) n<0. 17
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Fig4 The values of f(0) vs. B for various k.
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Fig5 The values of (0) vs. B for various k with (a) n>0 and (b) n<0.
Fig6 The values of (0) vs. B for various Pr with (a) n>0 and (b) n<0.
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Fig7 The values of (0) vs. B for various with (a) n>0 and (b) n<0.
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Fig8 The values of (0) vs. B for various n.
Fig9 The values of (0) vs. n for various Pr.
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Fig10 The values of (0) vs. n for various .
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Fig11 Dual velocity profiles for various values of M for shrinking sheet.
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Fig12 Stream function for M=0.02 for (a) Fist solution and (b) Second solution with k=0.05
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and B=1.28 (shrinking sheet).
Fig13 (a) Velocity profiles for various values of M and (b) Stream function for M=0.02 (for
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stretching sheet).
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Fig14 Dual velocity profiles for various values of k for shrinking sheet.
Fig15 Stream function for k=0.05 for (a) Fist solution and (b) Second solution with M=0.05
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and B=1.28 (shrinking sheet).
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Fig16 (a) Velocity profiles for various values of k and (b) Stream function for k=0.05 (for
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stretching sheet).
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Fig17 (a) Dual velocity profiles (for shrinking sheet) and (b) Single velocity profiles (for
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stretching sheet) for various values of B.
Fig18 Dual temperature profiles for various values of M with (a) n>0 and (b) n<0 for shrinking sheet. 23
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Fig19 Temperature profiles for various values of M with (a) n>0 and (b) n<0 for stretching
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sheet.
Fig20 Dual temperature profiles for various values of k with (a) n>0 and (b) n<0 for
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shrinking sheet.
Fig21 Temperature profiles for various values of k with (a) n>0 and (b) n<0 for stretching sheet. 24
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Fig22 Dual temperature profiles for various values of B with (a) n>0 and (b) n<0 for
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shrinking sheet.
Fig23 Temperature profiles for various values of B with (a) n>0 and (b) n<0 for stretching
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sheet.
Fig24 Dual temperature profiles for various values of Pr with n>0 for shrinking sheet.
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Fig25 Dual temperature profiles for various values of Pr with n<0 for shrinking sheet.
sheet.
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Fig26 Temperature profiles for various values of Pr with (a) n>0 and (b) n<0 for stretching
Fig27 Dual temperature profiles for various values of with n>0 for shrinking sheet.
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Fig28 Dual temperature profiles for various values of with n<0 for shrinking sheet.
sheet.
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Fig29 Temperature profiles for various values of with (a) n>0 and (b) n<0 for stretching
Fig30 Dual temperature profiles for various values of n>0 for shrinking sheet.
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Fig31 Dual temperature profiles for various values of n<0 for shrinking sheet.
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Fig32 Temperature profiles for various values of (a) n>0 and (b) n<0 for stretching sheet.
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Graphical abstract
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