Meccanica (2012) 47:863–876 DOI 10.1007/s11012-011-9457-6

Effects of variable fluid viscosity on flow past a heated stretching sheet embedded in a porous medium in presence of heat source/sink Swati Mukhopadhyay · G.C. Layek

Received: 26 April 2010 / Accepted: 15 June 2011 / Published online: 19 July 2011 © Springer Science+Business Media B.V. 2011

Abstract The boundary layer flow and heat transfer of a fluid through a porous medium towards a stretching sheet in presence of heat generation or absorption is considered in this analysis. Fluid viscosity is assumed to vary as a linear function of temperature. The symmetry groups admitted by the corresponding boundary value problem are obtained by using a special form of Lie group transformations viz. scaling group of transformations. These transformations are used to convert the partial differential equations corresponding to the momentum and the energy equations into highly non-linear ordinary differential equations. Numerical solutions of these equations are obtained by shooting method. It is found that the horizontal velocity decreases with increasing temperature-dependent fluid viscosity parameter up to the crossing-over point but increases after that point and the temperature decreases in this case. With the increase of permeability parameter of the porous medium the fluid velocity decreases but the temperature increases at a particular point of the sheet. Effects of Prandtl number on the velocity boundary layer and on the thermal boundary layer are studied and plotted.

S. Mukhopadhyay () · G.C. Layek Department of Mathematics, The University of Burdwan, Burdwan 713104, WB, India e-mail: [email protected] G.C. Layek e-mail: [email protected]

Keywords Scaling group of transformations · Temperature-dependent fluid viscosity · Porous medium · Stretching sheet · Heat generation/absorption Nomenclature F non-dimensional stream function. variable. F∗ first order derivative with respect to η. F second order derivative with respect to η. F  third order derivative with respect to η. F  k permeability of the porous medium. permeability parameter. k1 Pr Prandtl number. dimensional heat generation/absorption Q0 coefficient. p, q variables. T temperature of the fluid. temperature of the wall of the surface. Tw free-stream temperature. T∞ u, v components of velocity in x and y directions. z variable. Greek symbols α1 , α2 , α3 , α4 , α5 , α6 , α  , α  transformation parameters. β  , β  transformation parameters. η similarity variable.  Lie-group transformations. κ the coefficient of thermal diffusivity. λ heat source/sink parameter.

864

μ μ∗ ν∗ ψ ψ∗ ρ θ θ ∗ , θ¯ θ θ 

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dynamic viscosity. reference viscosity. reference kinematic viscosity. stream function. variable. density of the fluid. non-dimensional temperature. variables. first order derivative with respect to η. second order derivative with respect to η.

1 Introduction The study of hydrodynamic flow and heat transfer over a stretching sheet has gained considerable attention due to its applications in industries and important bearings on several technological processes. It has also applications in certain geothermal areas where the shallow surface layers are being stretched though in these cases the velocities are very small. Moreover, the study of a stretching sheet or a moving wall is relevant to several important engineering applications in the field of metallurgy and chemical engineering processes (such as drawing, annealing and tinning of copper wire, etc.). These applications involve the cooling of continuous strips or filaments by drawing them through a quiescent fluid. Crane [1] investigated the flow caused by the stretching of a sheet. Many researchers such as Gupta and Gupta [2], Chen and Char [3], Dutta et al. [4] extended the work of Crane [1] by including the effect of heat and mass transfer analysis under different physical situations. Several authors have considered various aspects of this problem and obtained similarity solutions (Ishak et al. [5–8], Boutros et al. [9], Mahapatra et al. [10], Pal [11, 12] and Aziz et al. [13]). All the above mentioned studies continued their discussions by assuming the uniform fluid viscosity. However, it is known that the physical properties of fluid may change significantly with temperature (Herwing and Gresten [14], Lai and Kulacki [15] and Pop et al. [16], Chaim [17], Abel et al. [18]). The variations of properties with temperature has several practical applications in the field of metallurgy and chemical engineering; in the extrusion process, the heat-treated materials traveling between a feed roll and wind-up roll or on conveyor belt possess the feature of a moving continuous surface.The increase of temperature leads

to a local increase in the transport phenomena by reducing the viscosity across the momentum boundary layer and so rate of heat transfer at the wall is also affected. Therefore, to predict the flow behaviour accurately it is necessary to take into account the viscosity variation for incompressible fluids. Gary et al. [19] and Mehta and Sood [20] showed that, when this effect is included the flow characteristics may change substantially compared to constant viscosity assumption. For lubricating fluids heat generated by internal friction and the corresponding rise in the temperature affects the viscosity of the fluid and so that the fluid viscosity no longer be assumed constant. Mukhopadhyay et al. [21] investigated the MHD boundary layer flow with variable fluid viscosity over a heated stretching sheet. The effects of temperature dependent viscosity and thermal conductivity on flow and heat transfer over a stretching surface in different flow situations and for different fluids were considered by El-Aziz [22], Dandapat et al. [23], Salem [24], Mukhopadhyay and Layek [25], Prasad et al. [26] etc. The increase of temperature leads to the increase in the transport phenomena by reducing the viscosity across the momentum boundary layer and due to which the heat transfer rate at the wall is also affected. A new dimension is added to the above mentioned study of Mukhopadhyay et al. [21] by considering the effects of porous media. Flows through porous media are of principal interest because these are quite prevalent in nature. Such type of flow finds its applications in a broad spectrum of disciplines covering chemical engineering to geophysics. Flow through fluid-saturated porous medium is important in many technological applications, and it has increasing importance with the growth of geothermal energy usage and in astrophysical problems. Several other applications may also benefit from a better understanding of the fundamentals of mass, energy, and momentum transport in porous media, namely, cooling of nuclear reactors, underground disposal of nuclear waste, petroleum reservoir operations, building insulation, food processing, and casting and welding in manufacturing processes. In certain porous media applications, working fluid heat generation (source) or absorption (sink) effects are important. Representative studies dealing with these effects have been reported by authors such as Gupta and Sridhar [27], Abel and Veena [28]. Abel et al. [18] studied the flow and heat transfer of a viscoelastic fluid immersed in a porous medium over a

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Fig. 1 Sketch of the physical problem

non-isothermal stretching sheet, and the fluid viscosity was assumed to vary as a function of temperature. The present work deals with fluid flow and heat transfer over a stretching sheet immersed in a porous media in presence of heat source/sink. Fluid viscosity is assumed to vary as a linear function of temperature. Most of the researchers try to obtain the similarity solutions in such cases using the similarity variables. But in this paper, a special form of Lie group transformations, known as scaling group of transformations is used to find out the full set of symmetries of the problem and then to study which of them are appropriate to provide group-invariant or more specifically similarity solutions. Because group-theoretic method is the only rigorous mathematical method to find all the symmetries of a given differential equation and no ad hoc assumptions or a prior knowledge of the equation under investigation is needed. Moreover, this method unifies almost all known exact integration techniques for both ordinary and partial differential equations. This method can be used as a tool for finding the similarity solutions for those problems for which the similarity solutions can not be found easily by usual method. The system remains invariant due to some relations among the parameters of the scaling group of transformations. Using this transformation, a third order and a second order ordinary differential equations corresponding to momentum and energy equations

are derived. These equations are solved numerically using shooting method. The effects of temperaturedependent fluid viscosity parameter, permeability parameter of the porous media, heat source/sink parameter and the influence of Prandtl number on velocity and temperature fields are investigated and analysed with the help of their graphical representations.

2 Equations of motion We consider steady two-dimensional forced convection flow of a viscous incompressible fluid past a heated stretching sheet immersed in a porous medium in the region y > 0. The flow is generated as a consequence of linear stretching of the sheet, caused by simultaneous application of equal and opposite forces along x-axis keeping the origin fixed [Fig. 1]. The temperature of the sheet is different from that of the ambient medium. The fluid viscosity is assumed to vary with temperature while the other fluid properties are assumed constants. In order to get the effect of temperature difference between the sheet and the ambient fluid we consider temperature dependent heat source/sink in the flow region. In this study we consider that the volumetric rate of heat generation, denoted by q m [w.m−3 ] should be q m = Q0 (T − T∞ ) for T ≥ T∞ and equal to zero for

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T ≤ T∞ where Q0 (> 0) is the heat generation and Q0 (< 0) is the heat absorption constant (Vajravelu and Hadjinicolaou [29]). In flow through porous medium, the pressure drop caused by the frictional drag is directly proportional to the velocity for low speed flow. This is the familiar Darcy’s law which relates the pressure drop and velocity in an unbounded porous medium. At higher velocities, inertial effects become appreciable, causing an increase in the form drag. Experimental observations do indicate that the pressure drop in the bulk of a porous medium is proportional to a linear combination of flow velocity and square of a flow velocity. The square term is caused by the inertial effects offered through the solid boundary. The effects of a solid boundary on flow and heat transfer in a porous medium originate from vorticity diffusion caused by the boundary frictional resistance. This resistance is additional to the bulk frictional drag induced by the solid matrix as characterized by Darcy’s law (Vafai and Tien [30]). The present paper is concerned primarily with mathematical formulation based on Darcy’s law, where the effects of a solid boundary and the inertial effects are neglected. These effects become more significant near boundary and in a media with high porosity (Vafai and Tien [30]). In this paper, we shall limit our consideration to flows where the non-linear Forchheimer term is neglected but the linear Darcy term describing the distributed body force exerted by the porous medium is retained. Here, we have assumed that Reynolds number is very small (typically < 10) (Takhar et al. [31]) i.e. for very slow motion. In this regime viscous forces dominate and inertial forces are therefore negligible. The continuity, momentum and energy equations governing such type of flow are written as (with the application of Darcy’s law) ∂u ∂v + = 0, ∂x ∂y u

  ∂u ∂u 1 ∂ ∂u μ μ − +v = u ∂x ∂y ρ ∂y ∂y ρk =

u

(1)

μ 1 ∂μ ∂T ∂u μ ∂ 2 u + u, − ρ ∂T ∂y ∂y ρ ∂y 2 ρk

∂T ∂T ∂ 2T Q0 +v =κ 2 + (T − T∞ ), ∂x ∂y ρcp ∂y

(2) (3)

where u and v are the components of velocity respectively in x and y directions, T is the tempera-

ture, κ is the coefficient of thermal diffusivity, Q0 (J s−1 m−3 K−1 ) is the dimensional heat generation (Q0 > 0) or absorption (Q0 < 0) coefficient, cp is the specific heat, ρ is the fluid density (assumed constant), μ is the coefficient of fluid viscosity (dependent on temperature), k is the permeability of the porous medium. 2.1 Boundary conditions The appropriate boundary conditions for the problem are given by u = cx,

v = 0,

u → 0,

T → T∞

T = Tw

at y = 0,

as y → ∞.

(4) (5)

Here c(> 0) is constant, Tw is the uniform wall temperature, T∞ is the temperature far away from the sheet. 2.2 Method of solution We now introduce the following relations for u, v and θ as u=

∂ψ , ∂y

v=−

∂ψ ∂x

and θ =

T − T∞ T w − T∞

(6)

where ψ is the stream function. The temperature dependent fluid viscosity is given by (Batchelor [32]), μ = μ∗ [a + b(Tw − T )]

(7)

where μ∗ is the constant value of coefficient of viscosity far away from the sheet and a, b are constants and b(> 0). We have used viscosity-temperature relation μ = a − bT (b > 0) which agrees quite well with the relation μ = 1/(b1 + b2 T ) (Saikrishnan and Roy [33]) when second and higher order terms are neglected. The viscosity-temperature relation used by Saikr1 which gives on ishnan and Roy [33] is μ = (b1 +b 2T ) expansion  −1 b2 1 1+ T μ= b1 b1   b22 2 b2 1 1 − T + 2T − ··· = b1 b1 b1 ≈

b2 1 − 2T b1 b1

(8)

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(provided | bb21 T | < 1) (neglecting second and higher b2 b1 .

order terms) = a − bT where a = b= They took in their study, b1 = 53.41, b2 = 2.43 and so | bb21 T | < 1 gives 0° ≤ T ≤ 23°. Our viscosity-temperature relation also agrees quite well with the relation μ = e−aT (Bird et al. [34]) when second and higher order terms are neglected in the expansions. Range of temperature i.e. (Tw − T∞ ) studied here is (0°–23°C). Coefficient of viscosity μ of a large number of liquids agree very closely with the empirical formula given by μ = c/(a + bT )n where a, b, c, n are constants depending on the nature of liquid. This agrees well with n = 1 for pure water with our formulation for fluid viscosity. Using the relations (6) in the boundary layer equation (2) and in the energy equation (3) we get the following equations 1 b1 ,

∂ψ ∂ 2 ψ ∂ψ ∂ 2 ψ − ∂y ∂x∂y ∂x ∂y 2

Equation (13) may be considered as a point-transformation which transforms co-ordinates (x, y, ψ, u, v, θ ) to the co-ordinates (x ∗ , y ∗ , ψ ∗ , u∗ , v ∗ , θ ∗ ). Substituting (13) in (9) and (10) we get,  ∗ 2 ∗  ∂ ψ ∂ψ ∗ ∂ 2 ψ ε(α1 +2α2 −2α3 ) ∂ψ − e ∂y ∗ ∂x ∗ ∂y ∗ ∂x ∗ ∂y ∗2 = −Aν ∗ eε(3α2 −α3 −α6 )

+ (a + A)ν ∗ eε(3α2 −α3 ) − Aν ∗ θ ∗ e(3α2 −α3 −α6 ) −

∂ψ ∂θ ∂ψ ∂θ ∂ 2θ Q0 − =κ 2 + θ, ∂y ∂x ∂x ∂y ρcp ∂y

∂ψ ∗ ν∗ (a + A)eε(α2 −α3 ) ∗ k ∂y

= κeε(2α2 −α6 )

(9) (10)

∂ψ = 0, ∂x

∂ψ = cx, ∂y ∂ψ → 0, ∂y

θ = 1 at y = 0,

(11) (12)

2.3 Scaling group of transformations We now introduce simplified form of Lie-group transformations namely, the scaling group of transformations (Mukhopadhyay et al. [21]), ∗

 : x = xe

εα1

ψ ∗ = ψeεα3 , ∗

v = ve

εα5

,

∗

y = ye

,

εα2

u∗ = ueεα4 , ∗

θ = θe

εα6

.

,

∂ 2θ ∗ Q0 −εα6 ∗ + e θ . ∂y ∗2 ρcp

= α2 − α3 = α2 − α3 − α6

and

α1 + α2 − α3 − α6 = 2α2 − α6 = −α6 . These relations give and α2 = 0 = α6 .

The boundary conditions yield α1 = α4 , α5 = 0. Thus the set  reduces to a one parameter group of transformations: x ∗ = xeεα1 ,

y ∗ = y,

ψ ∗ = ψeεα1 ,

u∗ = ueεα1 ,

v ∗ = v,

θ = θ ∗.

(16)

Expanding by Taylor’s series we get, x ∗ − x = xεα1 ,

(13)

(15)

α1 + 2α2 − 2α3 = 3α2 − α3 − α6 = 3α2 − α3

α1 = α3

θ → 0 as y → ∞.

(14)

The system will remain invariant under the group of transformations , we would have the following relations among the parameters, namely

∗

where A = b(Tw − T∞ ), ν ∗ = μρ . The boundary conditions (4) and (5) then become

∂ 3ψ ∗ ∂y ∗3

ν ∗ ε(α2 −α3 −α6 ) ∗ ∂ψ ∗ Ae θ , k ∂y ∗  ∗ ∗  ∂ψ ∂θ ∂ψ ∗ ∂θ ∗ eε(α1 +α2 −α3 −α6 ) − ∂y ∗ ∂x ∗ ∂x ∗ ∂y ∗

∂ 2ψ ∂ 3ψ ∗ = −Aν + ν [a + A(1 − θ )] ∂y ∂y 2 ∂y 3 ∂ψ ν∗ [a + A(1 − θ )] , k ∂y

∂ 3ψ ∗ ∂y ∗3

+

∗ ∂θ

−

∂θ ∗ ∂ 2 ψ ∗ ∂y ∗ ∂y ∗2

y ∗ − y = 0,

ψ ∗ − ψ = ψεα1 , ∗

v − v = 0,

u∗ − u = uεα1 , ∗

θ − θ = 0.

(17)

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In terms of differentials, we get,

and

dy dψ du dv dθ dx = = = = = . α1 x 0 α1 ψ α1 u 0 0 From the subsidiary equations dy = 0 which on integration gives y = η (constant)

we get,

(18a)

dx α1 x

=

dθ 0

we get dθ = 0 which on

(18b)

(say). dx α1 x

=

dψ α1 ψ

we get

= constant i.e.

ψ = xF (η)

(18c)

(say)

is the Prandtl number and λ = ν∗

ψ = xF (η),

Q0 ρccp

is

the heat source or sink parameter, k1 = ck is the permeability parameter (Cortell [35]). Taking F ∗ = f and θ¯ = θ the equations (24) and (25) finally take the following form: [a + A(1 − θ )]f  + ff  − Af  θ  − f 2 (26)

and 1  θ + f θ  + λθ = 0. Pr

(27)

The boundary conditions take the following forms f  = 1,

where F is an arbitrary function of η. Thus from (18a)–(18c) we obtain, y = η,

ν∗ κ

(25)

− k1 [a + A(1 − θ )]f  = 0,

Also integrating the equations ψ x

dy 0

¯ =0 θ¯  + Pr(F ∗ θ¯  + λθ) where Pr =

(say).

From the equations integration gives θ = θ (η)

=

dx α1 x

(18)

f = 0,

θ =1

at η∗ = 0,

(28)

and

θ = θ (η).

(19)

f  → 0,

θ → 0 as η∗ → ∞.

(29)

Equations (14) and (15) become F 2 − F F  = −Aν ∗ θ  F  + ν ∗ [a + A(1 − θ )]F  ν∗ − [a + A(1 − θ )]F  , k Q 0 κθ  + F θ  + θ = 0. ρcp

3 Numerical method for solution (20) (21)

The boundary conditions become F  = c,

F = 0,

F  → 0,

θ →0

θ =1

at η = 0.

(22)

as η → ∞.

(23) 



Introducing η = ν ∗α cβ η∗ , F = ν ∗α cβ F ∗ ,   θ = ν ∗α cβ θ¯ in equations (20) and (21) we get 1 α = α = , 2 1 β  = −β = , 2

α  = 0,

with the boundary conditions f (0) = 0,

β  = 0.

The equations (20) and (21) are transformed to F ∗2 − F ∗ F ∗ = −AF ∗ θ¯  + [a + A(1 − θ¯ )]F ∗ ∗ ¯ − k1 [a + A(1 − θ)]F ,

The above equations (26) and (27) along with boundary conditions are solved by converting them to an initial value problem. The above mentioned third order and second order equations are written in terms of first order differential equations as follows: ⎧  ⎪ ⎨ f = z,  z = p, (30) ⎪ ⎩ p  = [z2 −fp+Apq+k1 {a+A(1−θ)}z] , [a+A(1−θ)]   θ = q, (31) q  = − Pr(f q + λθ ),

(24)

f  (0) = 1,

θ (0) = 1.

(32)

In order to integrate (30) and (31) as an initial value problem we require a value for p(0) i.e. f  (0) and q(0) i.e. θ  (0) but no such values are given in the boundary. The suitable guess values for f  (0) and θ  (0) are chosen and then integration is carried out. We

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compare the calculated values for f  and θ at η = 10 (say) with the given boundary condition f  (10) = 0 and θ (10) = 0 and adjust the estimated values, f  (0) and θ  (0), to give a better approximation for the solution. Different values of η (such as η = 6, 8, 9, 10 etc.) are taken in our numerical computations so that numerical values obtained are independent of η chosen. We take the series of values for f  (0) and θ  (0), and apply the fourth order classical Runge-Kutta method with different step-sizes (h = 0.01, 0.001 etc.) so that the numerical results obtained are independent of h. The above procedure is repeated until we get the results up to the desired degree of accuracy, 10−7 .

4 Results and discussions In order to analyse the results, numerical computations have been carried out using method described in the previous section for various values of temperature dependent fluid viscosity parameter (A), permeability parameter (k1 ), heat source or sink parameter λ and the Prandtl number Pr. For illustrations of the results, numerical values are plotted in Figs. 2 to 6. In all figures we take a = 1. In order to assess the accuracy of the method, the results (in case of uniform viscosity) are compared with those (in the absence of suction/blowing and in

869

case of constant surface temperature) of Cortell [35]. The results (presented through Table 1) are found to agree well. At first, we concentrate on the effects of temperature-dependent fluid viscosity on velocity distribution and heat transfer in case of non-porous media in absence of any heat source/sink. In Fig. 2(a), velocity profiles are shown for different values of A (A = 0, 4, 10) with Pr = 0.5 in absence of any heat source or sink and in case of non-porous media. The velocity curves show that the rate of transport decreases with increasing distance (η) of the sheet. In all cases, velocity vanishes at some large distance from the sheet (at η = 10). An anamolus behaviour is noticed here. With increasing A, fluid viscosity decreases resulting the increment of velocity boundary layer thickness. Fluid velocity is found to decrease up to the crossingTable 1 Values of skin-friction [−f  (0)] and wall temperature gradient [−θ  (0)] for two values of k1 with a = 1, A = 0, λ = 0 and Pr = 1 k1

[−f  (0)]

[−θ  (0)]

Cortell [35] (with fw = 0)

Present study

Cortell [35] (CST case)

Present study

1.0

1.414213

1.414213

0.500000

0.500001

2.0

1.732051

1.732051

0.447552

0.447553

Fig. 2a Velocity profiles for several values of viscosity variation parameter A in case of non-porous medium and in absence of heat source/sink

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Fig. 2b Velocity profiles for several values of viscosity variation parameter A in case of porous medium and in presence of heat source/sink

Fig. 3 Temperature distribution for several values of viscosity variation parameter A in case of non-porous medium and in the absence of heat source/sink

over point with the increase in A but after the crossingover point it increases with increasing A i.e. with decreasing viscosity for a particular non-zero value of η which conforms to the real situation. In Fig. 2(b), velocity field is found to decay with increasing value of η for all values of A considered. The combined effects of heat source and permeability parameter are considered in this case. The anamolus behaviour nearly

vanishes in this case. Figure 3 exhibits the temperature profiles for various values of A (A = 0, 4, 10) in case of non-porous media. In each case, temperature is found to decrease with the increase of η until it vanishes at η = 10. But temperature is found to decrease for any non-zero fixed value of η with the increase of A. The increase of temperature-dependent fluid viscosity parameter A makes decrease of thermal

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871

(a)

(b) Fig. 4 (a) Velocity profiles for several values of permeability parameter k1 in case of uniform viscosity and in absence of heat source/sink. (b) Velocity profiles for several values of permeability parameter k1 in case of variable viscosity and in absence of heat source/sink

boundary layer thickness, which results in decrease of temperature profile θ (η). Increase in the values of viscosity parameter has the tendency to increase the thermal boundary layer thickness. This causes to increase the values of θ as shown in Fig. 3. It is also noticed, from Fig. 2, that the velocity decreases as viscosity parameter increases (up to the crossing over point). It is of some interest to note that the effect of fluid viscosity parameter, is to increase the skin fric-

tion parameter and decrease the wall temperature gradient. Now we concentrate on velocity and temperature distribution for the variation of permeability parameter k1 of the porous medium without heat generation or absorption and with Pr = 0.5. Figure 4(a) and Fig. 4(b) demonstrate the effects of permeability parameter (k1 = 0, 0.5, 1) on velocity field in the absence (A = 0) and presence (A = 1) of temperature-

872

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(a)

(b) Fig. 5 (a) Temperature distribution for several values of permeability parameter k1 in case of uniform viscosity and in absence of heat source/sink. (b) Temperature distribution for several values of permeability parameter k1 in case of variable viscosity and in absence of heat source/sink

dependent fluid viscosity parameter A respectively. With increasing k1 , fluid velocity is found to decrease [Figs. 4(a) and 4(b)]. As the porosity of the medium increases, the value of k1 decreases. For large porosity of the medium i.e. for decreasing k1 fluid gets more space to flow as a consequence its velocity increases. Thus, an increase of permeability parameter k1 leads to a decrease of the horizontal velocity

profile, which leads to the enhanced deceleration of the flow and, hence, the velocity decreases. The increase of permeability parameter k1 leads to increase the skin-friction. The permeability parameter k1 introduces additional shear stress on the boundary. As the value of the viscosity variation parameter A is very small (A = 1), the effect of permeability parameter (k1 ) dominates over that of the former (i.e. viscosity

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Fig. 6 Temperature distribution for several values of heat source/sink parameter λ in case of porous medium and variable viscosity

variation parameter A) causing a decrease in fluid velocity [Fig. 4(b)]. Figures 5(a) and 5(b) exhibit that temperature θ (η) in boundary layer increases with increasing permeability parameter k1 in both the cases i.e. in absence (A = 0) and presence (A = 1) of temperaturedependent fluid viscosity parameter A respectively. The thermal boundary layer thickness becomes thinner with the decreasing permeability parameter k1 . The effect of increasing values of k1 opposes the flow in the boundary layer region, which results in more heat transfer from the sheet to the fluid. This is because the presence of the porous medium is to increase the resistance to the fluid motion, this causes the fluid velocity to decrease (see Figs. 4(a) and 4(b)) and due to which there is rise in the temperature in the boundary layer (Figs. 5(a) and 5(b)). In Fig. 6, effects of heat source/sink on the temperature field is shown, taking fixed values for the parameters A = 1, k1 = 0.1, Pr = 0.5 and various values for internal heat generation/absorption parameter λ. In this case, temperature field increases with the increase of heat source parameter λ. This feature prevails up to certain heights and then the process is slowed down and at a far distance from the wall, temperature vanishes. On the other hand, temperature field increases with the decrease in the amount of heat absorption. Again, far away from the wall, such feature is smeared out. It is observed that the thermal boundary layer generates energy, which causes the temperature profiles

to increase with the increasing values of λ > 0 where as in the case of the λ < 0 boundary layer absorbs energy resulting the temperature to fall considerably with the decreasing values of λ. Actually, the internal heat generation/absorption enhances or damps the heat transport. The heat generation source leads to a larger thermal diffusion layer that may increase thermal boundary layer thickness, on the contrary, the thermal boundary layer thickness decreases for heat absorption sink. Figures 7 and 8 depict the velocity and temperature profiles for the effects of Prandtl number Pr on momentum and heat transfer. Fluid velocity decreases with increasing Prandtl number. An increase in Prandtl number reduces the thermal boundary layer thickness. Prandtl number signifies the ratio of momentum diffusivity to thermal diffusivity. It can be noticed that as Pr decreases, the thickness of the thermal boundary layer becomes greater than the thickness of the velocity boundary layer according to the well-known 1 relation δδT ≈ Pr where δT is the thickness of the thermal boundary layer and δ is the thickness of the velocity boundary layer, so the thickness of the thermal boundary layer increases as Prandtl number Pr decreases, and hence temperature profile decreases with increase of Prandtl number Pr (Abel et al. [36]). Figure 8 implies that an increase of Prandtl number Pr results in a decrease of temperature distribution at a particular point. This is due to the fact that there would be a decrease of thermal boundary layer thickness

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Fig. 7 Velocity profiles for several values of Prandtl number Pr in presence of porous medium, variable viscosity and heat source/sink

Fig. 8 Temperature distribution for several values Prandtl number Pr in presence of porous medium, variable viscosity and heat source/sink

with increasing values of Prandtl number Pr. Temperature distribution asymptotically approaches to zero in the free stream region. In heat transfer problems, the Prandtl number Pr controls the relative thickening of momentum and thermal boundary layers. When Prandtl number Pr is small, it means that heat diffuses quickly compared to the velocity (momentum), which means that for liquid metals, the thickness of the thermal boundary layer is much bigger than the momen-

tum boundary layer. Fluids with lower Prandtl number will possess higher thermal conductivities (and thicker thermal boundary layer structures) so that heat can diffuse from the sheet faster than for higher Pr fluids (thinner boundary layers). Hence Prandtl number can be used to increase the rate of cooling in conducting flows. Finally, we compute the values of [−f  (0)] which will determine the local skin friction for two values of

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k1 (with A = 0, a = 1) and the wall temperature gradient for two values of k1 (with Pr = 1 and λ = 0). The values of [−f  (0)] and [−θ  (0)] are given in the Table 1. Our computed results (in case of uniform viscosity i.e. a = 1, A = 0) agree well with the results of Cortell [35] in the absence of suction/blowing (fw = 0) and in case of constant surface temperature (CST case). It is of some interest to note that with increasing k1 , the skin friction parameter [−f  (0)] increases which is the result of thickening of the boundary layer. From this table, it is also noticed that the effect of porosity parameter (k1 ) is to decrease the wall temperature gradient. This result has significance in industrial applications where the power expenditure can be reduced in stretching sheet by increasing the porosity parameter.

5 Conclusion The present study gives the solutions for steady boundary layer flow and heat transfer over a stretching surface embedded in a saturated porous medium with variable fluid viscosity in presence of heat source or sink. Efficient method of Lie group analysis is used to solve the governing equations of motion. This procedure helps in removing the difficulties faced in solving the equations arising from the non-linear character of the partial differential equations. The scaling symmetry group is very essential procedure to comprehend the mathematical model and to find the similarity solutions for such type of flow which have wider applications in the engineering disciplines related to fluid mechanics. The effect of porosity parameter on a viscous incompressible fluid is to suppress the velocity field which in turn causes the enhancement of the temperature field. The results pertaining to the present study indicate that due to internal heat generation thermal boundary layer increases. The boundary-layer edge is reached faster as Pr increases. The increasing Prandtl number has a suppressive effect on temperature. It is hoped that, the physics of flow over the stretching sheet can be utilized as the basis for many engineering and scientific applications with the help of our present model. The results pertaining to the present study may be useful for the different model investigations. The findings of the present problem are also of great interest in geophysics particularly in certain

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geothermal areas where the shallow surface layers are being stretched. Acknowledgement The authors would like to acknowledge the input from the honourable reviewers that has improved the quality of this paper.

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