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Commun Nonlinear Sci Numer Simulat 15 (2010) 1553–1564
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Effect of variable viscosity on MHD non-Darcy mixed convective heat transfer over a stretching sheet embedded in a porous medium with non-uniform heat source/sink Dulal Pal a,*, Hiranmoy Mondal b a b
Department of Mathematics, Visva-Bharati University, Santiniketan, West Bengal 731235, India Department of Mathematics, Bengal Institute of Technology and Management, Santiniketan, West Bengal 731236, India
a r t i c l e
i n f o
Article history: Received 25 June 2009 Accepted 4 July 2009 Available online 10 July 2009 PACS: 44.20+b 44.30.+v 44.25.+f 47.15.Cb 83.60.Np 87.10.Ed Keywords: Magnetohydrodynamics Heat transfer Stretching surface Ohmic dissipation Porous medium
a b s t r a c t An analysis has been presented to investigate the effect of temperature-dependent viscosity on non-Darcy MHD mixed convective heat transfer past a porous medium by taking into account of Ohmic dissipation and non-uniform heat source/sink. Thermal boundary layer equation takes into account of viscous dissipation and Ohmic dissipation due to transverse magnetic field and electric field. The governing fundamental equations are first transformed into system of ordinary differential equations using self-similarity transformation and are solved numerically by using the fifth-order Runge–Kutta–Fehlberg method with shooting technique for various values of the physical parameters. The effects of variable viscosity, porosity, Eckert number, Prandtl number, magnetic field, electric field and non-uniform heat source/sink parameters on velocity and temperature profiles are analyzed and discussed. Favorable comparisons with previously published work on various special cases of the problem are obtained. Numerical results on the development of the local skin-friction co-efficient and local Nusselt number with non-uniform heat source/sink are tabulated for various physical parameters to show the interesting aspects of the solution. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Recently, studies on the boundary layer flow and heat transfer due to a stretching surface have become more and more important in number of engineering applications that includes both metal and polymer sheets. For instance, it occurs in the extrusion of a polymer sheet from a die or in the drawing of plastic films, which are then cooled in a cooling bath and during cooling reduction of both thickness and width takes place. The quality of the final product depends on the rate of heat transfer at the stretching surface. Crane [1] gave an exact similarity solution in closed analytical form for steady boundary layer flow of an incompressible viscous fluid caused due to stretching of an elastic sheet which moves in its own plane with a velocity varying linearly with distance from a fixed point. Ishak et al. [2] studied mixed convection boundary layers in the stagnation-point flow of an incompressible viscous fluid over a stretching vertical sheet. In certain porous media applications such as those involving heat removal from nuclear fuel debris, underground disposal of radiative waste material, storage of food stuffs, the study of heat transfer is of much importance. Comprehensive reviews * Corresponding author. Tel./fax: +91 3463 261029. E-mail addresses: [email protected] (D. Pal), [email protected] (H. Mondal). 1007-5704/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2009.07.002
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Nomenclature a, b A0 Cb Cf Cp f g A ; B q000 Ec F Ha ! J E1 F Cf Rex k1 Pr Nux x; y u; v ! E ! B0 Cp T Tw T1
constant parameters of temperature distribution on the stretching surface quadratic drag co-efficient skin friction specific heat at constant pressure dimensionless velocity variable acceleration due to gravity co-efficients of space and temperature-dependent heat source/sink non-uniform heat source/sink Eckert number emperical constant (Forchheimer number) Hartmann number Joule current local electromagnetic parameter local inertia co-efficient local skin-friction co-efficient local Reynolds number porous parameter Prandtl number Nusselt number horizontal distance and vertical distance velocity component in the x-direction and y-direction uniform electric field uniform transverse magnetic fields Specific heat at constant pressure temperature variable stands for stretching sheet temperature temperature far away from the stretching sheet
Greek symbols h non-dimensional temperature parameter co-efficient of thermal expansion bT g similarity variable w stream function m kinematic viscosity q1 density of the fluid porosity of the porous medium constant hr r electrical conductivity j permeability of the porous medium k buoyancy or mixed convection parameter
of the convection through porous media have been reported by Nield and Bejan [3] and by Ingham and Pop [4]. Considering the effect of temperature difference between the surface and ambient fluid some research work have been carried out by Vajravelu and Rollins [5] and Vajravelu and Nayfeh [6] on flow and heat transfer considering temperature-dependent heat source/sink. Ali [7] analyzed the effect of lateral mass flux on the natural convection boundary layer induced by a heated vertical plate embedded in a saturated porous medium with an exponential decaying heat generation. The use of magnetic field that influences heat generation/absorption process in electrically conducting fluid flows has important engineering applications. This interest stems from the fact that hydromagnetic flows and heat transfer have been applied in many industries. For example, in many metallurgical processes such as drawing of continuous filaments through quiescent fluids, and annealing and tinning of copper wires, the properties of the end product depend greatly on the rate of cooling involved in these processes. Numerous attempts have been made to analysis the effect of transverse magnetic field on boundary layer flow characteristics (Andresson [8], Char [9] and Lawrence and Rao [10]). Pavlor [11] gave an exact similarity solution to the MHD boundary layer equations for the steady two-dimensional flow caused solely by the stretching of a plane elastic surface in the presence of a uniform magnetic field. Vajravelu and Rollins [12] studied heat transfer in an electrically conducting fluid over a stretching surface taking into account the magnetic field only. However, all these works have neglected electric field which is also one of the important parameters to alter the momentum and heat transfer characteristic in Newtonian boundary layer flow. Recently, Abel et al. [13] studied viscoelastic MHD flow and heat transfer over a stretching sheet with viscous with Ohmic dissipations.
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Most of the existing analytical studies for this problem are based on the constant physical properties of the ambient fluid. However, it is known that these properties may change with temperature, especially for fluid viscosity. To accurately predict the flow and heat transfer rates, it is necessary to take into account this variation of viscosity. Seddeek [14] analyzed nonDarcian effect on forced convection heat transfer over a flat plate in a porous medium with temperature-dependent viscosity. Salem [15] studied the problem of flow and heat transfer of electrically conducting viscoelastic fluid having temperaturedependent viscosity as well as thermal conductivity fluid over a continuously stretching sheet in the presence of a uniform magnetic field for the case of power-law variation in the sheet temperature. It is worth mentioning that non-Darcian forced flow boundary layers from a very important group of flows, the solution of which is of great importance in many practical applications such as in biomechanical problems, in filtration transpiration cooling and geothermal. Singh and Tewani [16] studied the effect of thermal stratification on non-Darcian free convection flow by using the Ergun model [17] to include the inertia effect. It is well known that there exists non-Darcian flow phenomena bodies inertia effect and solid-boundary viscous resistance. These non-Darcian effects include non-uniform porosity distribution and thermal dispersion. Tien and Hunt [18] analyzed these non-Darcian effects applicable for a boundary layer flow in porous beds. The study of heat source/sink effects on heat transfer is very important in view of several physical problems. Aforementioned studies include only the effect of uniform heat source/sink (i.e. temperature-dependent heat source/sink) on heat transfer. Abo-Eldahab and El-Aziz [19] have included the effect of non-uniform heat source with suction/blowing, but confined to the case of viscous fluid only. Abel et al. [20] investigated on non-Newtonian boundary layer flow past a stretching sheet taking into account of non-uniform heat source and frictional heating. Abel and Mahesha [21] studied the magnetohydrodynamic boundary layer flow and heat transfer characteristic of a non-Newtonian viscoelastic fluid over a flat sheet with variable thermal conductivity in the presence of thermal radiation and non-uniform heat source. They have reported that the combined effect of variable thermal conductivity, radiation and non-uniform heat source have significant impact in controlling the rate of heat transfer in the boundary layer region. The previous studies are based on the constant physical parameters of the fluid. For most realistic fluids, the viscosity shows a rather pronounced variation with temperature. It is known that the fluid viscosity changes with temperature [22]. Thus it is necessary to take into account the variation of viscosity with temperature in order to accurately predict the heat transfer rates. Ali [23] investigated the effect of variable viscosity on mixed convection heat transfer along a moving surface. Lai and Kulacki [24] analyzed the effects of variable viscosity on convective heat transfer along a vertical surface in a saturated porous medium. Pantokratoras [25,26] studied laminar free-convection over a vertical isothermal plate with uniform blowing or suction in water with variable physical properties. Later, Kafoussian and Williams [27] investigated on free forced convective boundary layer flow past a vertical isothermal flat plate considering temperature-dependent viscosity of the fluid. Recently, Pantokratoras [28] made a theoretical study to investigate the effect of variable viscosity on the classical Falkner–Skan flow with constant wall temperature and obtained results for wall shear stress and the wall heat transfer for various values of ambient Prandtl numbers varying from 1 to 10,000. In view of the above discussions, authors envisage to investigate the effect of variable viscosity on MHD non-Darcy flow and heat transfer over a continuous stretching sheet with electric field in presence of Ohmic dissipation and non-uniform heat source/sink of heat. The flow is subjected to a transverse magnetic field normal to the plate. The Forchheimer’s extension is used to describe the fluid flow in the porous medium. Highly non-linear momentum and heat transfer equations are solved numerically using fifth-order Runge–Kutta–Fehlberg method with shooting technique (Na, [29]). The effects of various parameters on the velocity and temperature profiles as well as on local skin-friction co-efficient and local Nusselt number are presented in graphical and in tabular form. It is hoped that the results obtained from the present investigation will provide useful information for application and also serve as a complement to the previous studies. 2. Mathematical formulations 2.1. Flow analysis Consider two-dimensional study of incompressible electrically conducting fluid flow over a continuous stretching sheet embedded in a porous medium. The fluid properties are assumed to be isotropic and constant, except for the fluid viscosity l which is assumed to vary as be an inverse linear function of temperature T, in the form (see Lai and Kulacki [24])
1
l where
¼
1
l1
½1 þ cðT T 1 Þ
ð1Þ
l1 is the ambient fluid dynamic viscosity. Eq. (1) can be written as follows 1
l
¼ aðT T r Þ
ð2Þ
where
a¼
c 1 and T r ¼ T 1 l1 c
ð3Þ
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Both a and T r are constant and their values depend on the reference state and the thermal property of the fluid, i.e. c. In general, a > 0 for liquids and a < 0 for gases. Consider the uniform flow of velocity U 1 and temperature T 1 through a highly porous medium bounded by a semi-infinite flat plate parallel to the flow. Also, hr is a constant which is defined by
hr ¼
Tr T1 1 ¼ Tw T1 cðT w T 1 Þ
ð4Þ
and primes denote differentiation with respect to g. It is worth mentioning here that for c ! 0 i.e. l ¼ l1 (constant) then hr ! 1. It is also important to note that hr is negative for liquids and positive for gases. The flow model is based on the following assumption that the flow is steady, incompressible, laminar and the fluid viscosity which is assumed to be an inverse linear function of temperature. ! The flow region is exposed under uniform transverse magnetic fields B0 ¼ ð0; B0 ; 0Þ and uniform electric field ! E ¼ ð0; 0; E0 Þ (see Fig. 1). Since such imposition of electric and magnetic fields stabilizes the boundary layer flow (Dandapat and Mukhopadhyay [30]). It is assumed that the flow is generated by stretching of an elastic boundary sheet from a slit by imposing two equal and opposite forces in such a way that velocity of the boundary sheet is of linear order of the flow direc! ! tion. We know from Maxwell’s equation that r: B ¼ 0 and r E ¼ 0. When magnetic field is not so strong then electric ! ! ! ! ! field and magnetic field obey Ohm’s law J ¼ rð E þ q B Þ, where J is the Joule current. The viscous dissipation and velocity of the fluid far away from the plate are assumed to be negligible. We assumed that magnetic Reynolds number of the fluid is small so that induced magnetic field and Hall effect may be neglected. We take into account of magnetic field effect as well as electric field in the momentum equation. Under the above stated physical situation, the governing boundary layer equations for momentum and energy for mixed convection under Boussinesq’s approximation are
@u @ v þ ¼0 @x @y 1 @u @u 1 @ @u r m C 2 pbffiffiffi u2 þ gbT ðT T 1 Þ ¼ þ u v l ðE B B uÞ þ u 0 0 0 @y q1 @y @y q1 k 2 @x k
ð5Þ ð6Þ
where u and v are the velocity components in the x and y directions, respectively; m is the kinematic viscosity; g is the acceleration due to gravity; q1 is the density of the fluid; bT is the co-efficient of thermal expansion; T, T w and T 1 are the temperature of the fluid inside the thermal boundary layer, the plate temperature and the fluid temperature in the free stream, respectively; k is the permeability of the porous medium; is the porosity of the porous medium; C b is the form of drag coefficient which is independent of viscosity and other physical properties of the fluid but is dependent on the geometry of the medium. The third and fourth terms on the right hand side of Eq. (6) stand for the first-order (Darcy) resistance and secondorder porous inertia resistance, respectively. It is assumed that the normal stress is of the same order of magnitude as that of the shear stress in addition to usual boundary layer approximations for deriving the momentum boundary layer Eq. (6). The following appropriate boundary conditions on velocity are appropriate in order to employ the effect of stretching of the boundary surface causing flow in x-direction as
u ¼ U w ðxÞ ¼ bx;
v¼0
at y ¼ 0
u ¼ 0 as y ! 1
ð7Þ
To solve the governing boundary layer Eq. (6), the following similarity transformations are introduced: 0
u ¼ bxf ðgÞ;
v
pffiffiffiffiffiffiffiffiffi ¼ bm1 f ðgÞ;
sffiffiffiffiffiffi b y g¼
m1
Fig. 1. Schematic diagram of the problem under consideration.
ð8Þ
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Here f ðgÞ is the dimensionless stream function and g is the similarity variable. Substitution of Eq. (8) in the Eq. (6) results in a third-order non-linear ordinary differential equation of the following form.
f 000
000 02 h ff 1 h0 f 00 h h f þ 1 þ þ 1 þ F f 02 kh þ k1 f 0 Ha2 ðE1 f 0 Þ ¼ 1 2 2 hr hr h hr hr
where k1 ¼ mkb1
ð9Þ
qffiffiffiffi is the porous parameter, Ha ¼ qrbB0 is Hartmann number, E1 ¼ BE0 0bx is the local electric parameter, F ¼ pC bffiffik x is
the local inertia co-efficient, k ¼ gbT ðTbw2 xT 1 Þ is the buoyancy or mixed convection parameter. In view of the transformations, the boundary conditions (7) take the following non-dimensional form on stream function f as
f ð0Þ ¼ 0;
f 0 ð0Þ ¼ 1;
f 0 ð1Þ ¼ 0
ð10Þ
The physical quantities of interest are the skin-friction co-efficient C f , which is defined as
Cf ¼
sw q1 U 2 =2
ð11Þ
where wall sharing stress
sw
sw is given by
@u ¼l @y y¼0
ð12Þ
Using the non-dimensional variables (8), we get from Eqs. (11) and (12) as
1 hr ¼ C f Re1=2 f 00 ð0Þ x 2 hr h
ð13Þ
where Rex ¼ mUx is the local Reynolds number. 1 We now consider the heat transfer in the flow using appropriate boundary conditions in the next section. 2.2. Similarity solution of the heat transfer equation The governing boundary layer heat transfer equation with viscous and Ohmic dissipations, and non-uniform heat source/ sink is given by [20]:
u
@T @T j @2T l @u 2 r 1 þ þ ðuB E Þ2 þ q000 þv ¼ @x @y q1 C p @y2 q1 C p @y q1 C p 0 0 q1 C p
ð14Þ
where C p is the specific heat at constant pressure and j is the thermal conductivity. The non-uniform heat source/sink, q000 , is modeled as
q000 ¼
kuw ðxÞ ½A ðT w T 1 Þf 0 þ ðT T 1 ÞB xm 1
ð15Þ
where A and B are the co-efficient of space and temperature-dependent heat source/sink, respectively. Here we make a note that the case A > 0; B > 0 corresponds to internal heat generation and that A < 0; B < 0 corresponds to internal heat absorption. We consider non-isothermal temperature boundary condition as follows:
x2 l y!1
T ¼ T w ¼ T 1 þ A0 T ! T1
as
at y ¼ 0 ð16Þ
where A0 is the parameters of temperature distribution on the stretching surface, T w stands for stretching sheet temperature and T 1 is the temperature far away from the stretching sheet. We introduce a dimensionless temperature variable hðgÞ of the form:
h¼
T T1 Tw T1
ð17Þ
where expression for T w T 1 is given by Eq. (16). Making use of the Eq. (17) in Eq. (14) we obtain the non-dimensional thermal boundary layer equation as
h i h h00 1 Pr ð2f 0 h f h0 Þ Ha2 Ec ðE1 f 0 Þ2 ¼ Ec Prðf 00 Þ2 ðA f 0 þ B hÞ hr
ð18Þ
1 q1m C 2 2 where Pr ¼ 1 hhr Pr1 is the Prandtl number and Pr1 ¼ j1 p is the ambient Prandtl number, and Ec ¼ Ab0 cl p is the Eckert number. Temperature boundary conditions (16) take the following non-dimensional form
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hð0Þ ¼ 1;
hð1Þ ¼ 0
ð19Þ
The local Nusselt number which are defined as
Nux ¼
xqw
ð20Þ
jðT w T 1 Þ
where qw is the heat transfer from the sheet is given by
@T qw ¼ j @y y¼0
ð21Þ
Using the non-dimensional variables (17), we get from Eqs. (20) and (21) as
Nux =Re1=2 ¼ h0 ð0Þ x
ð22Þ
The set of transformed governing Eqs. (9) and (18) are solved numerically as described in the next section. 3. Numerical solutions The Eqs. (9) and (18) are highly non-linear ordinary differential equations which are solved numerically by most efficient fifth-order Runge–Kutta–Fehlberg integration scheme with shooting method. The most important factor of this method is to choose the appropriate finite values of g ! 1. In order to determine g1 for the boundary value problem stated by Eqs. (9) and (18), we start with some initial guess value for some particular set of physical parameters to obtain f 00 ð0Þ. The solution procedure is repeated with another large value of g1 until two successive values of f 00 ð0Þ differ only by the specified significant digit. The last value of g1 is finally chosen to be the most appropriate value of the limit g ! 1 for that particular set of parameters. The value of g may change for another set of physical parameters. Once the finite value of g is determined then the coupled boundary value problem given by Eqs. (9), (10), (18) and (19) are solved numerically using the method of superposition (Na, [29]). In this method the third-order non-linear Eq. (9) and second-order Eq. (18) have been reduced to five simultaneously ordinary differential equations for which only three initial conditions are given. Thus we employ shooting technique with Runge–Kutta–Fehlberg scheme to determine two more unknowns in order to convert the boundary value problem to initial value problem. Once all the five initial conditions are determined then we solve this system of simultaneous equation using fifth-order Runge–Kutta–Fehlberg integration scheme. The results are presented for major parameters including the magnetic field parameter, the porosity parameter, electric field parameter, variable viscosity parameter, Prandtl number and buoyancy number. A systematic study on the effect of the various parameters on flow and heat transfer characteristic is carried out. 4. Results and discussion Numerical solutions for effects of Non-Darcy mixed convection heat transfer over a stretching sheet in presence of magnetic field are reported. The results are presented graphically from Figs. 2–10 and conclusions are drawn that the flow field and other quantities of physical interest have significant effects. Comparisons with previously published works are
Fig. 2. Influence of the Hartmann number Ha, on the dimensionless velocity profile f 0 ðgÞ.
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Fig. 3. Variation of velocity profile for different values of Electric parameter E1 .
Fig. 4. Variation of velocity profile for different values of porosity
of the porous medium.
performed and excellent agreement between the results are obtained. Non-linear ordinary differential equations are integrated by Runge–Kutta–Fehlberg method with shooting technique. Comparison of our results of h0 ð0Þ with those obtained by Chen [31], Grubka and Bobba [32] and Ishak et al. [33] (see Table 1) in absence of buoyancy force and magnetic field show a very good agreement. The values of skin-friction co-efficient f 00 ð0Þ for different values of Hartmann number Ha, Eckert number Ec , local electric field E1 and Prandtl number Pr are presented in Table 2. Physically, positive sign of f 00 ð0Þ implies that the fluid exerts a drag force on the sheet and negative sign implies the opposite. It is seen from this table that the skin-friction co-efficient f 00 ð0Þ decreases with an increase in the electric parameters E1 and Prandtl number Pr in absence of Hartmann number Ha. It is also observed that the skin-friction coefficient increases with increasing the value of Hartmann number Ha and Eckert number Ec . Table 3 gives the values of wall temperature gradient h00 ð0Þ for different values of Hartmann number (Ha), Eckert number ðEc Þ, local electric parameter ðE1 Þ and Prandtl number (Pr). Analysis of the tabular data shows that magnetic field enhances the rate of heat transfer across the stretching sheet to the fluid. However, the effect of Prandtl number (Pr), in absence of local electric field parameter ðE1 Þ, is to reduce the rate of heat transfer from boundary stretching sheet to the fluid whereas in the presence of local electric parameter ðE1 Þ, the effect of Prandtl number (Pr) is to increase the rate of heat transfer. Thus application of electric field may change the limitation of heat transfer.
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Fig. 5. Variation of temperature profile for different values of Prandtl number Pr.
Fig. 6. Effects of buoyancy or mixed convection parameter k on the velocity profile in the boundary layer.
Fig. 2 shows the effect of Hartmann number (Ha) on velocity profiles by keeping other physical parameter fixed. Fig. 2 depicts that the effect of Hartmann number is to reduce the velocity distribution in the boundary layer which results in thinning of the boundary layer thickness. The decrease in the velocity profile is due to the fact that the transverse magnetic field has a tendency to retard the motion of the fluid as Hartmann number increases the Lorentz force. Fig. 3 is the plot of velocity profile for various values of electric field parameter E1 . It is clearly observed from this figure that the effect of electric parameter E1 is to increase velocity throughout the boundary layer but more significantly far away from the stretching sheet. Analysis of the graph reveals that the effect of local electric field parameter E1 is to shift the streamlines away from the stretching boundary. This shifting of streamlines is seen little away from the stretching sheet. This is because Lorentz force arising due to electric field acts as an accelerating force in reducing the frictional resistance. Fig. 4 is the plot of velocity profile for various values of porosity . It is clearly observed from this figure that the effect of porosity on velocity is to increase its value throughout the boundary layer but more significantly little away from the stretching sheet. This is due to fact that the obstruction in the motion of the fluid reduces as the porosity increases (pore size increases) hence the velocity increases is the boundary layer. Fig. 5 represents the graph of temperature profile for different values of Prandtl number Pr. It is seen that the effect of Prandtl number Pr is to decrease, temperature throughout the boundary layer, which results in decrease of the thermal
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λ θ ε
θη
θ Fig. 7. Effect of non-uniform heat source/sink parameter A on temperature profiles.
Fig. 8. Effect of temperature-dependent heat source/sink parameter B on temperature profiles.
boundary layer thickness with the increase of values of Prandtl number Pr. The increase of Prandtl number means slow rate of thermal diffusion. Fig. 6 represents the variations of velocity distribution in the boundary layer profiles for various values of mixed convection parameter or buoyancy parameter k. It is observed from this figure that the velocity distribution increases with increasing the buoyancy parameter k, this is due to the fact the boundary layer thickness increases with k. Fig. 7 depicts the effect of space-dependent heat source/sink parameter A . It is observed that the boundary layer generates the energy, which causes the temperature profiles to increase with increasing the values of A > 0 (heat source) where as in the case of A < 0 (absorbtion) boundary layer absorbs energy resulting in the temperature to fall considerably with decreasing in the value of A < 0. The effect of temperature-dependent heat source/sink parameter B on heat transfer is demonstrated in Fig. 8. This graph illustrates that energy is released when B > 0 which causes the temperature to increase, whereas energy is absorbed by decreasing the values of B < 0 resulting in the temperature to drop significantly near the boundary layer. Figs. 9 and 10 display results for the velocity and temperature distribution, respectively, for different values of fluid viscosity parameter, hr . The figures indicate that as hr ! 0, the boundary layer thickness decreases and the velocity distribution becomes shallower whereas the temperature distribution approaches a linear shape. Fig. 10 shows the variation of the dimensionless temperature parameter hðgÞ for various values of hr for both air and liquid. It is seen that for air, the
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Fig. 9. Effect of variable viscosity parameter hr on the velocity profile f 0 ðgÞ.
Fig. 10. Effect of variable viscosity parameter hr on the temperature profile hðgÞ.
Table 1 Comparison of local Nusselt number h0 ð0Þ for Ha ¼ 0; k ¼ 0 and various values of Pr with Chen [31], Grubka and Bobba [32] and Ishak et al.[33]. Pr
Chen [31]
Grubka and Bobba [32]
Ishak et al. [33]
h0 ð0Þ present result
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0
1.33334 – 2.50972 – – – – – – 4.79686
1.3333 – 2.5097 – – – – – – 4.7969
1.3333 – 2.5097 – – – – – – 4.7969
1.333333 2.000000 2.509725 2.938785 3.316482 3.657772 3.971512 4.263460 4.537612 4.796873
temperature decreases very rapidly with g and its value decreases with increase in hr whereas in the case of liquid, the temperature decreases very steadily with g. Further it is observed that the decrease in the temperature with hr is not very remarkable near the boundary in this case. This effect is much noticeable little away from the stretching sheet.
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D. Pal, H. Mondal / Commun Nonlinear Sci Numer Simulat 15 (2010) 1553–1564 Table 2 The skin-friction co-efficient f 00 ð0Þ for different values of Hartmann number Ha, Eckert number Ec , local electric parameter E1 and Prandtl number Pr. Ha
Ec
E1
Pr
f 00 ð0Þ ðk1 ¼ 0; k ¼ 0; F ¼ 0Þ
0.0 0.0 1.0 1.0
0.0 1.0 1.0 1.0
0.0 1.0 0.0 1.0
3.0 5.0 3.0 3.0
1.000000 1.000000 1.414214 0.6561953
Table 3 The wall temperature gradient h0 ð0Þ for different values of Hartmann number Ha, Eckert number Ec , local electric parameter E1 and Prandtl number Pr with Abel et al. [13]. Ha
Ec
E1
Pr
h0 ð0Þ ðk1 ¼ 0; k ¼ 0; F ¼ 0Þ
0.0 0.0 0.0 0.0 0.0 1.0 1.0 1.0
0.0 0.0 0.0 1.0 1.0 1.0 1.0 1.0
0.0 0.0 1.0 1.0 1.0 0.0 1.0 0.0
3.0 5.0 3.0 3.0 5.0 3.0 3.0 5.0
2.509725 3.316482 2.509725 1.745137 2.219414 0.459952 2.288068 0.366367
5. Conclusions Mathematical analysis has been carried out to study the effect of variable viscosity on MHD non-Darcy boundary layer flow and heat transfer characteristics in an incompressible electrically conducting fluid over a linear stretching sheet in the presence of Ohmic dissipation and non-uniform heat source/sink. Highly non-linear third-order momentum boundary layer equation is converted into a ordinary differential equation using similarity transformations. Fifth-order Runge–Kutta–Fehlberg method with shooting is used to solve momentum and heat transfer equations numerically. The effects of various physical parameters like Prandtl number, Hartmann number and local electric field parameter on velocity and temperature profiles are obtained. The following main conclusions can be drawn from the present study: (i) Boundary layer flow attain minimum velocity for higher values of Hartmann number (Ha). (ii) The effect of the local electric field is to increase velocity distribution in the boundary layer more significant little away from the stretching sheet. (iii) The effect of increasing the values of Prandtl number (Pr) is to decrease temperature largely near the stretching sheet and the thermal boundary layer thickness decreases with Prandtl number. (iv) The effect of porosity parameter is to increase velocity profile throughout the boundary layer. (v) Buoyancy parameter is to increase the velocity distribution in the momentum boundary layer. (vi) The effect of non-uniform surface and temperature-dependent heat source/sink parameters is to generate temperature for heat source and absorb temperature for heat sink values. Hence non-uniform heat sink is better for cooling purposes.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
Crane LJ. Flow past a stretching plate. Z Angew Math Phys 1970;21:645–7. Ishak A, Nazar R, Pop I. Mixed convection boundary layer in the stagnation-point flow toward a stretching vertical sheet. Meccanica 2006;41:509–18. Nield DA, Bejan A. Convection in porous media. New York: Springer; 1992. Ingham DB, Pop I. Transfer phenomena in porous media. Oxford: Pergamon; 1998. Vajravelu K, Rollins D. Hydromagnetic flow of a second grade fluid over a stretching sheet. Appl Math Comput 2004;148:783–91. Vajravelu K, Nayfeh J. Convective heat transfer at a stretching sheet. Acta Mech 1993;96:47–54. Ali ME. The effect of lateral mass flux on the natural convection boundary layers induced by a heated vertical plate embedded in a saturated porous medium with internal heat generation. Int J Thermal Sci 2007;46:157–63. Andersson HI. MHD flow of a viscoelastic fluid past a stretching surface. Acta Mech 1992;95:227–30. Char MI. Heat and mass transfer in a hydromagnetic flow of viscoelastic fluid over a stretching sheet. J Math Anal Appl 1994;186:674–89. Lawrence PS, Rao BN. The non-uniqueness of the MHD flow of a viscoelastic fluid over a stretching sheet. Acta Mech 1995;112:223–8. Pavlor KB. Magnetohydrodynamic flow of an incompressible viscous fluid caused by the deformation of a plane surface. Magn Gidrodin 1974;4:146–7. Vajravelu K, Rollins D. Heat transfer in electrically conducting fluid over a stretching sheet. Int J Nonlinear Mech 1992;27(2):265–77. Abel MS, Sanjayanand E, Nandeppanavar MM. Viscoelastic MHD flow and heat transfer over a stretching sheet with viscous and Ohmic dissipation. Commun Nonlinear Sci Numer Simul 2008;13:1808–21. Seddeek MA. Effect of non-Darcian on forced convection heat transfer over a flat plate in a porous medium with temperature dependent viscosity. Int Commun Heat Mass Transfer 2005;32:258–65.
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[15] Salem AM. Variable viscosity and thermal conductivity effects on MHD flow and heat transfer in viscoelastic fluid over a stretching sheet. Phys Lett A 2007;369:315–22. [16] Singh P, Tewani K. Non-Darcy free convection from vertical surfaces in thermally stratified porous medium. Int J Eng Sci 1993;31:1233–42. [17] Ergun S. Fluid flow through packed columns. Chem Eng Prog 1952;48:89–94. [18] Tien CL, Hunt ML. Boundary-layer flow and heat transfer in porous beds. Chem Eng Prog 1987;21:53–63. [19] Abo-Eldahab EM, El-Aziz MA. Blowing/suction effect on hydromagnetic heat transfer by mixed convection from an inclined continuously stretching surface with internal heat generation/absorption. Int J Thermal Sci 2004;43:709–19. [20] Abel MS, Siddheshwar PG, Nandeppanawar MM. Heat transfer in a viscoelastic boundary layer flow over a stretching sheet with viscous dissipation and non-uniform heat source. Int J Heat Mass Transfer 2007;50:960–6. [21] Abel MS, Mahesha N. Heat transfer in MHD viscoelastic fluid flow over a stretching sheet with variable thermal conductivity, non-uniform heat source and radiation. Appl Math Model 2008;32:1965–83. [22] Das UN, Deka R, Soundalgekar VM. Effects of mass transfer on flow past an impulsively started infinite vertical plate with constant heat flux and chemical reaction. Forsch Ingenieurwes 1994;60:284–7. [23] Ali ME. The effect of variable viscosity on mixed convection heat transfer along a vertical moving surface. Int J Thermal Sci 2006;45:60–9. [24] Lai EC, Kulacki FA. Effects of variable viscosity on connective heat transfer along a vertical surface in a saturated porous medium. Int J Heat Mass Transfer 1990;33:1028–31. [25] Pantokratoras A. Laminar free-convection over a vertical isothermal plate with uniform blowing or suction in water with variable physical properties. Int J Heat Mass Transfer 2002;45:963977. [26] Pantokratoras A. Further results on the variable viscosity on flow and heat transfer to a continuous moving flat plate. Int J Eng Sci 2004;42:18911896. [27] Kafoussias NG, Williams EW. Effects of temperature-dependent viscosity on free-forced convective laminar boundary layer flow past a vertical isothermal flat plate. Acta Mech 1995;110:123137. [28] Pantokratoras A. The Falkner–Skan with constant wall temperature and variable viscosity. Int J Thermal Sciences 2006;45:378–89. [29] Na TY. Computational method in engineering boundary value problems. New York: Academic Press; 1979. [30] Dandapat BS, Mukhopadhyay A. Finite amplitude long wave instability of a film of conducting fluid flowing down an inclined plane in presence of electromagnetic field. Int J Appl Mech Eng 2003;8:379–93. [31] Chen CH. Laminar mixed convection adjacent to vertical continuously stretching sheets. Heat Mass Transfer 1998;33:471–6. [32] Grubka LG, Bobba KM. Heat transfer characteristics of a continuous stretching surface with variable temperature. ASME J Heat Transfer 1985;107:248–50. [33] Ishak A, Nazar R, Pop I. Hydromagnetic flow and heat transfer adjacent to a stretching vertical sheet. Heat Mass Transfer 2007. doi:10.1007/s00231007-0322.
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Effect of variable viscosity on MHD non-Darcy mixed convective heat transfer over a stretching sheet embedded in a porous medium with non-uniform heat source/sink Dulal Pal a,*, Hiranmoy Mondal b a b
Department of Mathematics, Visva-Bharati University, Santiniketan, West Bengal 731235, India Department of Mathematics, Bengal Institute of Technology and Management, Santiniketan, West Bengal 731236, India
a r t i c l e
i n f o
Article history: Received 25 June 2009 Accepted 4 July 2009 Available online 10 July 2009 PACS: 44.20+b 44.30.+v 44.25.+f 47.15.Cb 83.60.Np 87.10.Ed Keywords: Magnetohydrodynamics Heat transfer Stretching surface Ohmic dissipation Porous medium
a b s t r a c t An analysis has been presented to investigate the effect of temperature-dependent viscosity on non-Darcy MHD mixed convective heat transfer past a porous medium by taking into account of Ohmic dissipation and non-uniform heat source/sink. Thermal boundary layer equation takes into account of viscous dissipation and Ohmic dissipation due to transverse magnetic field and electric field. The governing fundamental equations are first transformed into system of ordinary differential equations using self-similarity transformation and are solved numerically by using the fifth-order Runge–Kutta–Fehlberg method with shooting technique for various values of the physical parameters. The effects of variable viscosity, porosity, Eckert number, Prandtl number, magnetic field, electric field and non-uniform heat source/sink parameters on velocity and temperature profiles are analyzed and discussed. Favorable comparisons with previously published work on various special cases of the problem are obtained. Numerical results on the development of the local skin-friction co-efficient and local Nusselt number with non-uniform heat source/sink are tabulated for various physical parameters to show the interesting aspects of the solution. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Recently, studies on the boundary layer flow and heat transfer due to a stretching surface have become more and more important in number of engineering applications that includes both metal and polymer sheets. For instance, it occurs in the extrusion of a polymer sheet from a die or in the drawing of plastic films, which are then cooled in a cooling bath and during cooling reduction of both thickness and width takes place. The quality of the final product depends on the rate of heat transfer at the stretching surface. Crane [1] gave an exact similarity solution in closed analytical form for steady boundary layer flow of an incompressible viscous fluid caused due to stretching of an elastic sheet which moves in its own plane with a velocity varying linearly with distance from a fixed point. Ishak et al. [2] studied mixed convection boundary layers in the stagnation-point flow of an incompressible viscous fluid over a stretching vertical sheet. In certain porous media applications such as those involving heat removal from nuclear fuel debris, underground disposal of radiative waste material, storage of food stuffs, the study of heat transfer is of much importance. Comprehensive reviews * Corresponding author. Tel./fax: +91 3463 261029. E-mail addresses: [email protected] (D. Pal), [email protected] (H. Mondal). 1007-5704/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2009.07.002
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Nomenclature a, b A0 Cb Cf Cp f g A ; B q000 Ec F Ha ! J E1 F Cf Rex k1 Pr Nux x; y u; v ! E ! B0 Cp T Tw T1
constant parameters of temperature distribution on the stretching surface quadratic drag co-efficient skin friction specific heat at constant pressure dimensionless velocity variable acceleration due to gravity co-efficients of space and temperature-dependent heat source/sink non-uniform heat source/sink Eckert number emperical constant (Forchheimer number) Hartmann number Joule current local electromagnetic parameter local inertia co-efficient local skin-friction co-efficient local Reynolds number porous parameter Prandtl number Nusselt number horizontal distance and vertical distance velocity component in the x-direction and y-direction uniform electric field uniform transverse magnetic fields Specific heat at constant pressure temperature variable stands for stretching sheet temperature temperature far away from the stretching sheet
Greek symbols h non-dimensional temperature parameter co-efficient of thermal expansion bT g similarity variable w stream function m kinematic viscosity q1 density of the fluid porosity of the porous medium constant hr r electrical conductivity j permeability of the porous medium k buoyancy or mixed convection parameter
of the convection through porous media have been reported by Nield and Bejan [3] and by Ingham and Pop [4]. Considering the effect of temperature difference between the surface and ambient fluid some research work have been carried out by Vajravelu and Rollins [5] and Vajravelu and Nayfeh [6] on flow and heat transfer considering temperature-dependent heat source/sink. Ali [7] analyzed the effect of lateral mass flux on the natural convection boundary layer induced by a heated vertical plate embedded in a saturated porous medium with an exponential decaying heat generation. The use of magnetic field that influences heat generation/absorption process in electrically conducting fluid flows has important engineering applications. This interest stems from the fact that hydromagnetic flows and heat transfer have been applied in many industries. For example, in many metallurgical processes such as drawing of continuous filaments through quiescent fluids, and annealing and tinning of copper wires, the properties of the end product depend greatly on the rate of cooling involved in these processes. Numerous attempts have been made to analysis the effect of transverse magnetic field on boundary layer flow characteristics (Andresson [8], Char [9] and Lawrence and Rao [10]). Pavlor [11] gave an exact similarity solution to the MHD boundary layer equations for the steady two-dimensional flow caused solely by the stretching of a plane elastic surface in the presence of a uniform magnetic field. Vajravelu and Rollins [12] studied heat transfer in an electrically conducting fluid over a stretching surface taking into account the magnetic field only. However, all these works have neglected electric field which is also one of the important parameters to alter the momentum and heat transfer characteristic in Newtonian boundary layer flow. Recently, Abel et al. [13] studied viscoelastic MHD flow and heat transfer over a stretching sheet with viscous with Ohmic dissipations.
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Most of the existing analytical studies for this problem are based on the constant physical properties of the ambient fluid. However, it is known that these properties may change with temperature, especially for fluid viscosity. To accurately predict the flow and heat transfer rates, it is necessary to take into account this variation of viscosity. Seddeek [14] analyzed nonDarcian effect on forced convection heat transfer over a flat plate in a porous medium with temperature-dependent viscosity. Salem [15] studied the problem of flow and heat transfer of electrically conducting viscoelastic fluid having temperaturedependent viscosity as well as thermal conductivity fluid over a continuously stretching sheet in the presence of a uniform magnetic field for the case of power-law variation in the sheet temperature. It is worth mentioning that non-Darcian forced flow boundary layers from a very important group of flows, the solution of which is of great importance in many practical applications such as in biomechanical problems, in filtration transpiration cooling and geothermal. Singh and Tewani [16] studied the effect of thermal stratification on non-Darcian free convection flow by using the Ergun model [17] to include the inertia effect. It is well known that there exists non-Darcian flow phenomena bodies inertia effect and solid-boundary viscous resistance. These non-Darcian effects include non-uniform porosity distribution and thermal dispersion. Tien and Hunt [18] analyzed these non-Darcian effects applicable for a boundary layer flow in porous beds. The study of heat source/sink effects on heat transfer is very important in view of several physical problems. Aforementioned studies include only the effect of uniform heat source/sink (i.e. temperature-dependent heat source/sink) on heat transfer. Abo-Eldahab and El-Aziz [19] have included the effect of non-uniform heat source with suction/blowing, but confined to the case of viscous fluid only. Abel et al. [20] investigated on non-Newtonian boundary layer flow past a stretching sheet taking into account of non-uniform heat source and frictional heating. Abel and Mahesha [21] studied the magnetohydrodynamic boundary layer flow and heat transfer characteristic of a non-Newtonian viscoelastic fluid over a flat sheet with variable thermal conductivity in the presence of thermal radiation and non-uniform heat source. They have reported that the combined effect of variable thermal conductivity, radiation and non-uniform heat source have significant impact in controlling the rate of heat transfer in the boundary layer region. The previous studies are based on the constant physical parameters of the fluid. For most realistic fluids, the viscosity shows a rather pronounced variation with temperature. It is known that the fluid viscosity changes with temperature [22]. Thus it is necessary to take into account the variation of viscosity with temperature in order to accurately predict the heat transfer rates. Ali [23] investigated the effect of variable viscosity on mixed convection heat transfer along a moving surface. Lai and Kulacki [24] analyzed the effects of variable viscosity on convective heat transfer along a vertical surface in a saturated porous medium. Pantokratoras [25,26] studied laminar free-convection over a vertical isothermal plate with uniform blowing or suction in water with variable physical properties. Later, Kafoussian and Williams [27] investigated on free forced convective boundary layer flow past a vertical isothermal flat plate considering temperature-dependent viscosity of the fluid. Recently, Pantokratoras [28] made a theoretical study to investigate the effect of variable viscosity on the classical Falkner–Skan flow with constant wall temperature and obtained results for wall shear stress and the wall heat transfer for various values of ambient Prandtl numbers varying from 1 to 10,000. In view of the above discussions, authors envisage to investigate the effect of variable viscosity on MHD non-Darcy flow and heat transfer over a continuous stretching sheet with electric field in presence of Ohmic dissipation and non-uniform heat source/sink of heat. The flow is subjected to a transverse magnetic field normal to the plate. The Forchheimer’s extension is used to describe the fluid flow in the porous medium. Highly non-linear momentum and heat transfer equations are solved numerically using fifth-order Runge–Kutta–Fehlberg method with shooting technique (Na, [29]). The effects of various parameters on the velocity and temperature profiles as well as on local skin-friction co-efficient and local Nusselt number are presented in graphical and in tabular form. It is hoped that the results obtained from the present investigation will provide useful information for application and also serve as a complement to the previous studies. 2. Mathematical formulations 2.1. Flow analysis Consider two-dimensional study of incompressible electrically conducting fluid flow over a continuous stretching sheet embedded in a porous medium. The fluid properties are assumed to be isotropic and constant, except for the fluid viscosity l which is assumed to vary as be an inverse linear function of temperature T, in the form (see Lai and Kulacki [24])
1
l where
¼
1
l1
½1 þ cðT T 1 Þ
ð1Þ
l1 is the ambient fluid dynamic viscosity. Eq. (1) can be written as follows 1
l
¼ aðT T r Þ
ð2Þ
where
a¼
c 1 and T r ¼ T 1 l1 c
ð3Þ
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Both a and T r are constant and their values depend on the reference state and the thermal property of the fluid, i.e. c. In general, a > 0 for liquids and a < 0 for gases. Consider the uniform flow of velocity U 1 and temperature T 1 through a highly porous medium bounded by a semi-infinite flat plate parallel to the flow. Also, hr is a constant which is defined by
hr ¼
Tr T1 1 ¼ Tw T1 cðT w T 1 Þ
ð4Þ
and primes denote differentiation with respect to g. It is worth mentioning here that for c ! 0 i.e. l ¼ l1 (constant) then hr ! 1. It is also important to note that hr is negative for liquids and positive for gases. The flow model is based on the following assumption that the flow is steady, incompressible, laminar and the fluid viscosity which is assumed to be an inverse linear function of temperature. ! The flow region is exposed under uniform transverse magnetic fields B0 ¼ ð0; B0 ; 0Þ and uniform electric field ! E ¼ ð0; 0; E0 Þ (see Fig. 1). Since such imposition of electric and magnetic fields stabilizes the boundary layer flow (Dandapat and Mukhopadhyay [30]). It is assumed that the flow is generated by stretching of an elastic boundary sheet from a slit by imposing two equal and opposite forces in such a way that velocity of the boundary sheet is of linear order of the flow direc! ! tion. We know from Maxwell’s equation that r: B ¼ 0 and r E ¼ 0. When magnetic field is not so strong then electric ! ! ! ! ! field and magnetic field obey Ohm’s law J ¼ rð E þ q B Þ, where J is the Joule current. The viscous dissipation and velocity of the fluid far away from the plate are assumed to be negligible. We assumed that magnetic Reynolds number of the fluid is small so that induced magnetic field and Hall effect may be neglected. We take into account of magnetic field effect as well as electric field in the momentum equation. Under the above stated physical situation, the governing boundary layer equations for momentum and energy for mixed convection under Boussinesq’s approximation are
@u @ v þ ¼0 @x @y 1 @u @u 1 @ @u r m C 2 pbffiffiffi u2 þ gbT ðT T 1 Þ ¼ þ u v l ðE B B uÞ þ u 0 0 0 @y q1 @y @y q1 k 2 @x k
ð5Þ ð6Þ
where u and v are the velocity components in the x and y directions, respectively; m is the kinematic viscosity; g is the acceleration due to gravity; q1 is the density of the fluid; bT is the co-efficient of thermal expansion; T, T w and T 1 are the temperature of the fluid inside the thermal boundary layer, the plate temperature and the fluid temperature in the free stream, respectively; k is the permeability of the porous medium; is the porosity of the porous medium; C b is the form of drag coefficient which is independent of viscosity and other physical properties of the fluid but is dependent on the geometry of the medium. The third and fourth terms on the right hand side of Eq. (6) stand for the first-order (Darcy) resistance and secondorder porous inertia resistance, respectively. It is assumed that the normal stress is of the same order of magnitude as that of the shear stress in addition to usual boundary layer approximations for deriving the momentum boundary layer Eq. (6). The following appropriate boundary conditions on velocity are appropriate in order to employ the effect of stretching of the boundary surface causing flow in x-direction as
u ¼ U w ðxÞ ¼ bx;
v¼0
at y ¼ 0
u ¼ 0 as y ! 1
ð7Þ
To solve the governing boundary layer Eq. (6), the following similarity transformations are introduced: 0
u ¼ bxf ðgÞ;
v
pffiffiffiffiffiffiffiffiffi ¼ bm1 f ðgÞ;
sffiffiffiffiffiffi b y g¼
m1
Fig. 1. Schematic diagram of the problem under consideration.
ð8Þ
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Here f ðgÞ is the dimensionless stream function and g is the similarity variable. Substitution of Eq. (8) in the Eq. (6) results in a third-order non-linear ordinary differential equation of the following form.
f 000
000 02 h ff 1 h0 f 00 h h f þ 1 þ þ 1 þ F f 02 kh þ k1 f 0 Ha2 ðE1 f 0 Þ ¼ 1 2 2 hr hr h hr hr
where k1 ¼ mkb1
ð9Þ
qffiffiffiffi is the porous parameter, Ha ¼ qrbB0 is Hartmann number, E1 ¼ BE0 0bx is the local electric parameter, F ¼ pC bffiffik x is
the local inertia co-efficient, k ¼ gbT ðTbw2 xT 1 Þ is the buoyancy or mixed convection parameter. In view of the transformations, the boundary conditions (7) take the following non-dimensional form on stream function f as
f ð0Þ ¼ 0;
f 0 ð0Þ ¼ 1;
f 0 ð1Þ ¼ 0
ð10Þ
The physical quantities of interest are the skin-friction co-efficient C f , which is defined as
Cf ¼
sw q1 U 2 =2
ð11Þ
where wall sharing stress
sw
sw is given by
@u ¼l @y y¼0
ð12Þ
Using the non-dimensional variables (8), we get from Eqs. (11) and (12) as
1 hr ¼ C f Re1=2 f 00 ð0Þ x 2 hr h
ð13Þ
where Rex ¼ mUx is the local Reynolds number. 1 We now consider the heat transfer in the flow using appropriate boundary conditions in the next section. 2.2. Similarity solution of the heat transfer equation The governing boundary layer heat transfer equation with viscous and Ohmic dissipations, and non-uniform heat source/ sink is given by [20]:
u
@T @T j @2T l @u 2 r 1 þ þ ðuB E Þ2 þ q000 þv ¼ @x @y q1 C p @y2 q1 C p @y q1 C p 0 0 q1 C p
ð14Þ
where C p is the specific heat at constant pressure and j is the thermal conductivity. The non-uniform heat source/sink, q000 , is modeled as
q000 ¼
kuw ðxÞ ½A ðT w T 1 Þf 0 þ ðT T 1 ÞB xm 1
ð15Þ
where A and B are the co-efficient of space and temperature-dependent heat source/sink, respectively. Here we make a note that the case A > 0; B > 0 corresponds to internal heat generation and that A < 0; B < 0 corresponds to internal heat absorption. We consider non-isothermal temperature boundary condition as follows:
x2 l y!1
T ¼ T w ¼ T 1 þ A0 T ! T1
as
at y ¼ 0 ð16Þ
where A0 is the parameters of temperature distribution on the stretching surface, T w stands for stretching sheet temperature and T 1 is the temperature far away from the stretching sheet. We introduce a dimensionless temperature variable hðgÞ of the form:
h¼
T T1 Tw T1
ð17Þ
where expression for T w T 1 is given by Eq. (16). Making use of the Eq. (17) in Eq. (14) we obtain the non-dimensional thermal boundary layer equation as
h i h h00 1 Pr ð2f 0 h f h0 Þ Ha2 Ec ðE1 f 0 Þ2 ¼ Ec Prðf 00 Þ2 ðA f 0 þ B hÞ hr
ð18Þ
1 q1m C 2 2 where Pr ¼ 1 hhr Pr1 is the Prandtl number and Pr1 ¼ j1 p is the ambient Prandtl number, and Ec ¼ Ab0 cl p is the Eckert number. Temperature boundary conditions (16) take the following non-dimensional form
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hð0Þ ¼ 1;
hð1Þ ¼ 0
ð19Þ
The local Nusselt number which are defined as
Nux ¼
xqw
ð20Þ
jðT w T 1 Þ
where qw is the heat transfer from the sheet is given by
@T qw ¼ j @y y¼0
ð21Þ
Using the non-dimensional variables (17), we get from Eqs. (20) and (21) as
Nux =Re1=2 ¼ h0 ð0Þ x
ð22Þ
The set of transformed governing Eqs. (9) and (18) are solved numerically as described in the next section. 3. Numerical solutions The Eqs. (9) and (18) are highly non-linear ordinary differential equations which are solved numerically by most efficient fifth-order Runge–Kutta–Fehlberg integration scheme with shooting method. The most important factor of this method is to choose the appropriate finite values of g ! 1. In order to determine g1 for the boundary value problem stated by Eqs. (9) and (18), we start with some initial guess value for some particular set of physical parameters to obtain f 00 ð0Þ. The solution procedure is repeated with another large value of g1 until two successive values of f 00 ð0Þ differ only by the specified significant digit. The last value of g1 is finally chosen to be the most appropriate value of the limit g ! 1 for that particular set of parameters. The value of g may change for another set of physical parameters. Once the finite value of g is determined then the coupled boundary value problem given by Eqs. (9), (10), (18) and (19) are solved numerically using the method of superposition (Na, [29]). In this method the third-order non-linear Eq. (9) and second-order Eq. (18) have been reduced to five simultaneously ordinary differential equations for which only three initial conditions are given. Thus we employ shooting technique with Runge–Kutta–Fehlberg scheme to determine two more unknowns in order to convert the boundary value problem to initial value problem. Once all the five initial conditions are determined then we solve this system of simultaneous equation using fifth-order Runge–Kutta–Fehlberg integration scheme. The results are presented for major parameters including the magnetic field parameter, the porosity parameter, electric field parameter, variable viscosity parameter, Prandtl number and buoyancy number. A systematic study on the effect of the various parameters on flow and heat transfer characteristic is carried out. 4. Results and discussion Numerical solutions for effects of Non-Darcy mixed convection heat transfer over a stretching sheet in presence of magnetic field are reported. The results are presented graphically from Figs. 2–10 and conclusions are drawn that the flow field and other quantities of physical interest have significant effects. Comparisons with previously published works are
Fig. 2. Influence of the Hartmann number Ha, on the dimensionless velocity profile f 0 ðgÞ.
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Fig. 3. Variation of velocity profile for different values of Electric parameter E1 .
Fig. 4. Variation of velocity profile for different values of porosity
of the porous medium.
performed and excellent agreement between the results are obtained. Non-linear ordinary differential equations are integrated by Runge–Kutta–Fehlberg method with shooting technique. Comparison of our results of h0 ð0Þ with those obtained by Chen [31], Grubka and Bobba [32] and Ishak et al. [33] (see Table 1) in absence of buoyancy force and magnetic field show a very good agreement. The values of skin-friction co-efficient f 00 ð0Þ for different values of Hartmann number Ha, Eckert number Ec , local electric field E1 and Prandtl number Pr are presented in Table 2. Physically, positive sign of f 00 ð0Þ implies that the fluid exerts a drag force on the sheet and negative sign implies the opposite. It is seen from this table that the skin-friction co-efficient f 00 ð0Þ decreases with an increase in the electric parameters E1 and Prandtl number Pr in absence of Hartmann number Ha. It is also observed that the skin-friction coefficient increases with increasing the value of Hartmann number Ha and Eckert number Ec . Table 3 gives the values of wall temperature gradient h00 ð0Þ for different values of Hartmann number (Ha), Eckert number ðEc Þ, local electric parameter ðE1 Þ and Prandtl number (Pr). Analysis of the tabular data shows that magnetic field enhances the rate of heat transfer across the stretching sheet to the fluid. However, the effect of Prandtl number (Pr), in absence of local electric field parameter ðE1 Þ, is to reduce the rate of heat transfer from boundary stretching sheet to the fluid whereas in the presence of local electric parameter ðE1 Þ, the effect of Prandtl number (Pr) is to increase the rate of heat transfer. Thus application of electric field may change the limitation of heat transfer.
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Fig. 5. Variation of temperature profile for different values of Prandtl number Pr.
Fig. 6. Effects of buoyancy or mixed convection parameter k on the velocity profile in the boundary layer.
Fig. 2 shows the effect of Hartmann number (Ha) on velocity profiles by keeping other physical parameter fixed. Fig. 2 depicts that the effect of Hartmann number is to reduce the velocity distribution in the boundary layer which results in thinning of the boundary layer thickness. The decrease in the velocity profile is due to the fact that the transverse magnetic field has a tendency to retard the motion of the fluid as Hartmann number increases the Lorentz force. Fig. 3 is the plot of velocity profile for various values of electric field parameter E1 . It is clearly observed from this figure that the effect of electric parameter E1 is to increase velocity throughout the boundary layer but more significantly far away from the stretching sheet. Analysis of the graph reveals that the effect of local electric field parameter E1 is to shift the streamlines away from the stretching boundary. This shifting of streamlines is seen little away from the stretching sheet. This is because Lorentz force arising due to electric field acts as an accelerating force in reducing the frictional resistance. Fig. 4 is the plot of velocity profile for various values of porosity . It is clearly observed from this figure that the effect of porosity on velocity is to increase its value throughout the boundary layer but more significantly little away from the stretching sheet. This is due to fact that the obstruction in the motion of the fluid reduces as the porosity increases (pore size increases) hence the velocity increases is the boundary layer. Fig. 5 represents the graph of temperature profile for different values of Prandtl number Pr. It is seen that the effect of Prandtl number Pr is to decrease, temperature throughout the boundary layer, which results in decrease of the thermal
D. Pal, H. Mondal / Commun Nonlinear Sci Numer Simulat 15 (2010) 1553–1564
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λ θ ε
θη
θ Fig. 7. Effect of non-uniform heat source/sink parameter A on temperature profiles.
Fig. 8. Effect of temperature-dependent heat source/sink parameter B on temperature profiles.
boundary layer thickness with the increase of values of Prandtl number Pr. The increase of Prandtl number means slow rate of thermal diffusion. Fig. 6 represents the variations of velocity distribution in the boundary layer profiles for various values of mixed convection parameter or buoyancy parameter k. It is observed from this figure that the velocity distribution increases with increasing the buoyancy parameter k, this is due to the fact the boundary layer thickness increases with k. Fig. 7 depicts the effect of space-dependent heat source/sink parameter A . It is observed that the boundary layer generates the energy, which causes the temperature profiles to increase with increasing the values of A > 0 (heat source) where as in the case of A < 0 (absorbtion) boundary layer absorbs energy resulting in the temperature to fall considerably with decreasing in the value of A < 0. The effect of temperature-dependent heat source/sink parameter B on heat transfer is demonstrated in Fig. 8. This graph illustrates that energy is released when B > 0 which causes the temperature to increase, whereas energy is absorbed by decreasing the values of B < 0 resulting in the temperature to drop significantly near the boundary layer. Figs. 9 and 10 display results for the velocity and temperature distribution, respectively, for different values of fluid viscosity parameter, hr . The figures indicate that as hr ! 0, the boundary layer thickness decreases and the velocity distribution becomes shallower whereas the temperature distribution approaches a linear shape. Fig. 10 shows the variation of the dimensionless temperature parameter hðgÞ for various values of hr for both air and liquid. It is seen that for air, the
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Fig. 9. Effect of variable viscosity parameter hr on the velocity profile f 0 ðgÞ.
Fig. 10. Effect of variable viscosity parameter hr on the temperature profile hðgÞ.
Table 1 Comparison of local Nusselt number h0 ð0Þ for Ha ¼ 0; k ¼ 0 and various values of Pr with Chen [31], Grubka and Bobba [32] and Ishak et al.[33]. Pr
Chen [31]
Grubka and Bobba [32]
Ishak et al. [33]
h0 ð0Þ present result
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0
1.33334 – 2.50972 – – – – – – 4.79686
1.3333 – 2.5097 – – – – – – 4.7969
1.3333 – 2.5097 – – – – – – 4.7969
1.333333 2.000000 2.509725 2.938785 3.316482 3.657772 3.971512 4.263460 4.537612 4.796873
temperature decreases very rapidly with g and its value decreases with increase in hr whereas in the case of liquid, the temperature decreases very steadily with g. Further it is observed that the decrease in the temperature with hr is not very remarkable near the boundary in this case. This effect is much noticeable little away from the stretching sheet.
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D. Pal, H. Mondal / Commun Nonlinear Sci Numer Simulat 15 (2010) 1553–1564 Table 2 The skin-friction co-efficient f 00 ð0Þ for different values of Hartmann number Ha, Eckert number Ec , local electric parameter E1 and Prandtl number Pr. Ha
Ec
E1
Pr
f 00 ð0Þ ðk1 ¼ 0; k ¼ 0; F ¼ 0Þ
0.0 0.0 1.0 1.0
0.0 1.0 1.0 1.0
0.0 1.0 0.0 1.0
3.0 5.0 3.0 3.0
1.000000 1.000000 1.414214 0.6561953
Table 3 The wall temperature gradient h0 ð0Þ for different values of Hartmann number Ha, Eckert number Ec , local electric parameter E1 and Prandtl number Pr with Abel et al. [13]. Ha
Ec
E1
Pr
h0 ð0Þ ðk1 ¼ 0; k ¼ 0; F ¼ 0Þ
0.0 0.0 0.0 0.0 0.0 1.0 1.0 1.0
0.0 0.0 0.0 1.0 1.0 1.0 1.0 1.0
0.0 0.0 1.0 1.0 1.0 0.0 1.0 0.0
3.0 5.0 3.0 3.0 5.0 3.0 3.0 5.0
2.509725 3.316482 2.509725 1.745137 2.219414 0.459952 2.288068 0.366367
5. Conclusions Mathematical analysis has been carried out to study the effect of variable viscosity on MHD non-Darcy boundary layer flow and heat transfer characteristics in an incompressible electrically conducting fluid over a linear stretching sheet in the presence of Ohmic dissipation and non-uniform heat source/sink. Highly non-linear third-order momentum boundary layer equation is converted into a ordinary differential equation using similarity transformations. Fifth-order Runge–Kutta–Fehlberg method with shooting is used to solve momentum and heat transfer equations numerically. The effects of various physical parameters like Prandtl number, Hartmann number and local electric field parameter on velocity and temperature profiles are obtained. The following main conclusions can be drawn from the present study: (i) Boundary layer flow attain minimum velocity for higher values of Hartmann number (Ha). (ii) The effect of the local electric field is to increase velocity distribution in the boundary layer more significant little away from the stretching sheet. (iii) The effect of increasing the values of Prandtl number (Pr) is to decrease temperature largely near the stretching sheet and the thermal boundary layer thickness decreases with Prandtl number. (iv) The effect of porosity parameter is to increase velocity profile throughout the boundary layer. (v) Buoyancy parameter is to increase the velocity distribution in the momentum boundary layer. (vi) The effect of non-uniform surface and temperature-dependent heat source/sink parameters is to generate temperature for heat source and absorb temperature for heat sink values. Hence non-uniform heat sink is better for cooling purposes.
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