Viscosity and Fluid Suction/Injection Effects on Free Convection Flow from a Vertical Plate in a Porous Medium Saturated with a Pseudoplastic Fluid

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1 ISSN (print), (online) International Journal of Nonlinear Science Vol.8(4) No.,pp.7-38 Viscosity and Fluid Suction/Injection Effects on Free Convection Flow from a Vertical Plate in a Porous Medium Saturated with a Pseudoplastic Fluid Driss Achemlal,, Mohammed Sriti, Mohamed El Haroui, Mohamed Guedda University of Sidi Mohamed Ben Abdellah, Polydisciplinary Faculty of Taza, LIMAO, BP.3, Taza, Morocco University of Picardie Jules-Verne, Faculty of Mathematics and computer Science, 33, Rue Saint-Leu 839, Amiens, France (Received June 4, accepted October 4) Abstract: This paper investigates the effects of the fluid viscosity, the fluid suction/injection and the power law fluid index on the free convection flow from a heated vertical plate in a porous medium saturated with a pseudoplastic fluid in the presence of an internal heat generation. The problem is studied for power law fluid exponents between and. The governing equations are transformed into an autonomous third-order nonlinear degenerate ordinary differential equation by means of similarity transformations. These equation is then solved numerically by the fifth-order Runge-Kutta scheme associated with the shooting iteration technique. Also, the effect of the governing physical parameters on the velocity and temperature, the local Nusselt number and the local skin-friction profiles have been computed and studied with help of graphs. Keywords: free convection, pseudo-plastic fluid, suction/injection,porous medium Introduction In recent years, the study of convective heat transfer in a saturated porous medium with non-newtonian power law fluid has attracted many investigators because of its wide range of engineering applications, e.g., transport processes in chemical industry, storage of nuclear waste material, discoveries of the flow of oil in petroleum reservoirs and food processing. Recent monographs by [ 5] give an excellent summary of the work on the subject. Considerable attention has been devoted to the problems of non-newtonian fluids such as molten plastics, pulps, polymers, slurries, emulsions and many others. The main reason for this is probably that the fluids, which do not obey the Newtonian postulate that the stress tensor is directly proportional to the deformation tensor. Among the most popular rheological models for non-newtonian fluids, there is the power-law or Ostwald de-waele model : τ = m u n u y y, () n where ν = γ u y (γ = m ρ ) is the kinematic viscosity, m and γ are positive constant. Here, n is called power-law index, that the case n = corresponds to a Newtonian fluid and the case < n < is the power law relation proposed as being descriptive of pseudoplastic non-newtonian fluids and n > describes the dilatant fluid. Mehta and Rao [6] have studied the problem of free convection in non-newtonian fluids for vertical surfaces using different methods. Kumari et al. [7] considered the free-convection boundary-layer fow of a non-newtonian fuid along a vertical wavy surface. Pascal and Pascal [8] have annalyzed the free convection in a non-newtonian fluid saturated porous medium with lateral mass flux. Free convection in non-newtonian fluids along a horizontal plate in a porous medium is disscussed by Gorla and Kumari in [9]. Transient free convection from a vertical plate to a non-newtonian fluid in a porous medium is invetigated by Haq and Mulligan in []. Abdelgaied and Eid [] have studied the natural convection of non-newtonian power-law fluid over axisymmetric and two-dimensional bodies of arbitrary shape in fluid-saturated porous media. Corresponding author. address: driss achemlal@yahoo.fr Copyright c World Academic Press, World Academic Union IJNS.4..5/83

2 8 International Journal of Nonlinear Science, Vol.8(4), No., pp.7-38 Table : Nomenclature A wall temperature parameter x coordinate along the plate a equivalent thermal diffusivity y coordinate normal to the plate B suction/injection velocity parameter β thermal expansion coefficient Cf skin friction coefficient η similarity variable C p specific heat of the fluid θ dimensionless temperature f dimensionless stream function λ temperature exponent suction/injection parameter µ dynamic viscosity g gravitational acceleration ν kinematic viscosity H the plate length ρ fluid density K permeability of the porous medium σ viscosity parameter K t thermal conductivity of the porous medium σ c critical viscosity parameter Nu x local Nusselt number τ shear stress n power law index φ internal heat generation P e x local peclet number ψ stream function Q w local heat flux on the plate surface w wall plate condition Ra x modified local Rayleigh number infinity plate condition RE relative error derivative with respect to η Re x local Reynolds number T fluid temperature u velocity component in x direction V w lateral mass flux v velocity component in y direction It is really known that viscosity of many fluids depends strongly by temperature and this change influence also the flow. Thus, one can make significant errors when such viscosity variations are not considered. The effect of variable viscosity on free convective heat transfer over a Non-isothermal body of arbitrary shape in a non-newtonian fluid saturated porous medium with internal heat generation has been discused by Bagai and Nishad in []. The variable viscosity effects on the onset of convection in porous media have been studied by Kassoy and Zebib in [3]. Variable viscosity effects on free and mixed convection boundary layer flow from a horizontal surface in a saturated porous medium with variable heat flux have been discussed by Kumari in [4]. Thus, the aim of the this paper is to investigate the effects of the variable viscosity, the fluid suction/injection and the power law fluid index on the free convection boundary layer flow adjacent to a heated vertical plate in a porous medium saturated by a pseudoplastic fluid taking into account an internal heat generation. Mathematical models A vertical plate embedded in a saturated porous medium by a pseudoplastic fluid with applied lateral mass flux in the direction normal to the plate proportional to x λ quantity is considered as shown in figure. The temperature distribution of the plate has been assumed as T w = T + A x λ, where x is the distance measured along the vertical plate and λ is the constant temperature exponent. T is the temperature away from the plate assumed constant and A is a positive constant. The Cartesian coordinates x and y are measured, respectively, along and perpendicular to the plate. The flow is assumed two-dimensional, steady and laminar for an incompressible fluid. The convective fluid and the porous medium are in local thermodynamic equilibrium anywhere and no dissipation of energy by viscosity. The fluid and medium properties are assumed to be constant, except for the viscosity and density of the fluid. Radiation heat transfer is considered negligible with respect to other modes of heat transfer. Inertia effects of the porous medium are negligible, which is appropriate when the Reynolds number is small. By considering the assumptions mentioned above, the governing equations for this model based on Boussinesq ap- IJNS for contribution: editor@nonlinearscience.org.uk

3 D. Achemlal, M. Sriti et.al:viscosity and Fluid Suction/Injection Effects 9 Fig. : Vertical heated plate in a saturated porous medium. proximation are given by : u x + v y = u n = ρ g β K (T T ) µ u T x + v T y = a T y + φ ρ C p ρ = ρ [ β (T T ) ] () The boundary conditions associated with the problem are : { y = x, v = Vw (x) T = T w (x) y = x, u = T = T (3) where u and v are, respectively, the velocity components along x and y axes, T is the temperature of the fluid and φ is the internal heat generation. The constants µ, a, g and ρ are, respectively, dynamic viscosity, thermal diffusivity, gravitational acceleration and density. C p and β are, respectively, the specific heat at a constant pressure and the coefficient of thermal expansion, V w = B x λ is the lateral mass flux, where B is a constant. Here in this paper the viscosity of the fluid is assumed to vary linearly with temperature as: µ = µ ( + α T T T w T ) (4) where α is a constant. The non-linearity of the model and the complexity of the phenomena encountered (boundary layer, instability, geometry of the porous medium,...) make difficult its direct resolution. The transformation of the PDE system, describing the problem studied in a single non-linear differential equation becomes indispensable. So, we apply the similarity transformations used by Postelnicu et al in [6]: η(x, y) = y x Ra/ x, ψ(x, y) = a Ra / x f(η) θ(η) = T T ( ρ g β K (Tw T ) x n, Ra x = T w T a n µ φ = ρ C p a (T w T ) Ra x x where, ψ(x, y) is the stream function defined as (u, v) = ( ψ/ y, ψ/ x) which identically satisfies the mass conservation. e η ) n (5) IJNS homepage:

4 3 International Journal of Nonlinear Science, Vol.8(4), No., pp.7-38 Substituting Eqs.(5) into Eqs.() we obtain the following system of ordinary differential equations: (f (η)) n θ = + σ θ (η) λ f (η) θ(η) + λ + n n f(η) θ (η) + e η = η = f() =, f () = ( + σ ) n, θ() = η f ( ) =, θ( ) = By injecting the first equation of the system Eqs.(6) in the second, we get the non-linear differential Eq.(7) coupled with the boundary conditions. f (η) + M f (η) f (η) + R f(η) f (η) N f (η) + P f (η) n e η = η = f() =, f () = ( + σ ) n (7) η f ( ) = where = n B (λ + n) a ( ν g β K A )/n x λ (n ) n parameter. M = nσ n +, R = λ+n n, N = The local heat flux on the plate surface is given by : Q w = K t ( T y is the suction/injection parameter and σ = α θ is called the viscosity λ n(+σ) and P = n(+σ) are constants. ) = K t A n+ n y= S /n x λ(n+) n n θ () (9) where S = g β K a n ν, θ () is the gradient of the temperature at the plate surface and K t is the thermal conductivity. The local heat flux is uniform at the plate surface when λ = n The physical quantities of most interest in such problems are the Nusselt number and the skin friction coefficient. The heat transfer coefficient, in terms of the Nusselt number, is given by : n+. Nu x = θ () Ra / x () Expression for shear stress τ can be developed from the similarity solution : τ = µ ( + σ) an n x n Ran x f (η) f (η) () Thus, the skin friction coefficient, Cf, is defined by : Cf = τ ρ U = ( + σ) Ran x Re x P e x f (η) n (6) (8) f (η) () where Re x ν and P e x = U x a. The particular values of τ and Cf at η = represents, respectively, the n dimensionless shear stress and the dimensionless skin friction coefficient on the plate surface. = U x n 3 Numerical procedure The third-order ODE Eq.(7) governed by the boundary conditions, is non-linear. However, it is still difficult to solve it analytically. For this, we can routinely rewrite it as a system of three first-order ODEs by posing : g = f(η), g = f (η), g 3 = f (η) (3) So the non-linear differential Eq.(7) is transformed to the system of differential equations of the first-order coupled with the boundary conditions as follows : g = g g = g 3 g 3 = M g g3 R g g 3 + N g P g n e η (4) η = g =, g = ( + σ ) n η g = IJNS for contribution: editor@nonlinearscience.org.uk

5 D. Achemlal, M. Sriti et.al:viscosity and Fluid Suction/Injection Effects 3 The system of differential Eqs.(4) subject to the boundary conditions has been solved numerically by the fifth-order Runge-Kutta scheme associated with the shooting iteration technique. Since we have the initial conditions on g and g, it would be natural to seek the condition on g 3 at η = (g 3 ()). For given value of λ the value of g 3 () is estimated and the differential equations of the system (4) were integrated until the boundary condition at infinity, g (η) decay exponentially to. If the boundary condition at infinity is not satisfied, then the numerical routine uses the calculate correction to the estimated value of g 3 (). This process is repeated iteratively until exponentially decaying solution in g is obtained. A step size of η =. was found to be sufficient to give results that converge to within an error of 6 in nearly all cases. The value of η was chosen as large as possible. 4 Results and discussion Similarity solutions are obtained for selected values of the temperature exponent λ, power law index n, viscosity parameter σ and fluid suction/injection parameter. However, only the physical solutions were selected for λ =, /3 and. We note that the suction corresponding to ( > ), injection to ( < ) and impermeable plate to ( = ). The sign of the temperature gradient θ () determines the direction of the heat flow. θ () < corresponds to the heat flows from the wall to the porous medium, θ () > corresponds to the heat flows from the porous medium to the wall. The situation when θ () = corresponds to an adiabatic surface where no heat transfer takes place. Note that the fluid is more viscous in the boundary layer area for σ >, less viscous for σ < and a uniform fluid viscosity for σ =. The comparisons are made with Grosan and Pop [5] results in terms of θ () at σ = and = for selected values of the power law fluid exponent n and the temperature exponent λ, are presented in Table.. We find that all relative errors from Grosan and Pop [5] results are less than.3% for λ = and λ =. The present method yields results which are generally in very good agreement with those of Grosan and Pop [5]. We are therefore confident that our results are correct. Table : Values of θ () for σ = and =. θ () = Nu x Ra / x n λ = λ = Present Results [5] RE / [5] Present Results [5] RE / [5] % % % % % % 4. Velocity profiles Figures, 3 and 4 show for n =.5, the effect of σ on the dimensionless velocity profiles in the boundary layer area for an impermeable plate and for λ = (isothermal plate), /3 and. We see that, in the boundary layer area, the velocity profiles are amplified for less viscous fluids (µ < µ ). Away from the dynamics boundary layer, the evolution of the profiles is reversed and characterized by rapid stabilization for less viscous fluids when compared to more viscous fluids (µ > µ ) where there is a slow tendency the velocity flow to. This evolution is essentially due to the influence of the viscosity on the flow velocity. On the other hand, we also note that there is a reduction of the velocity profiles when the exponent λ increases. Figures 5,6 and 7 depict respectively for σ =.5, σ = and σ =.5 the contours of the vertical flow velocity in the boundary layer area of an isothermal and impermeable plate (T w = 65 C) embedded in a saturated porous medium by a pseudoplastic fluid (n =.5). We can deduce from these figures that the ability of the fluid to flow near the plate increases in the case where the fluid is less viscous near the latter (σ =.5). In Figures 8, 9 and we present for σ =., the dimensionless velocity profiles around an impermeable plate for various values of the power law exponent n and for three values of the exponent parameter λ (, /3 and ). Here, it is clear that the increase in n allows to amplify the velocity profiles for all values of λ. This is justified by the susceptibility of the fluid near a Newtonian fluid behavior to flow easily. It is also remarkable here that when going from λ = to λ =, the profiles are reduced further. IJNS homepage:

6 3 International Journal of Nonlinear Science, Vol.8(4), No., pp n =, = 3 n =.5, = f '( ) f '( ) Fig. : f versus η at λ =, = and n =.5 for selected values of σ. Fig.3 : f versus η at λ = /3, values of σ. = and n =.5 for selected 4.9 x 4 f '( ) 3 n =, = 4 6 x (m) y (m) Fig.4 : f versus η at λ =, = and n =.5 for selected values of σ. Fig.5 : Contour plots of the vertical velocity in saturated porous medium (T = 5 C) near a heated vertical plate (H = m, T w = 65 C) for λ =, =, n =.5 and σ = x x x (m).5.5 x (m) y (m) y (m) Fig.6 : Contour plots of the vertical velocity in saturated porous medium (T = 5 C) near a heated vertical plate (H = m, T w = 65 C) for λ =, =, n =.5 and σ =. Fig.7 : Contour plots of the vertical velocity in saturated porous medium (T = 5 C) near a heated vertical plate (H = m, T w = 65 C) for λ =, =, n =.5 and σ =.5. IJNS for contribution: editor@nonlinearscience.org.uk

7 D. Achemlal, M. Sriti et.al:viscosity and Fluid Suction/Injection Effects 33 f '( ),9,7,5,3 n =.4 n =.5 n =.6 n =.7 f '( ) n =.4 n =.5,7 n =.6 n =.7,5,3,, Fig.8 : f versus η at λ =, = and σ =. for selected values of n. Fig.9 : f versus η at λ = /3, = and σ =. for selected values of n. f '( ) n =.4 n =.5,7 n =.6 n =.7,5,3 = x (m) x , y (m) 4 Fig. : f versus η at λ =, = and σ =. for selected values of n. Fig. : Contour plots of the vertical velocity in saturated porous medium (T = 5 C) near a heated vertical plate (H = m, T w = 65 C) for λ =, =, σ =. and n = x x (m) y (m) Fig. : Contour plots of the vertical velocity in saturated porous medium (T = 5 C) near a heated vertical plate (H = m, T w = 65 C) for λ =, =, σ =. and n =.7. Figures and shows respectively for n =. and n =.7 the contours of the vertical flow velocity in the boundary layer area of an isothermal and impermeable plate (T w = 65 C) embedded in a saturated porous medium by a pseudoplastic fluid (σ =., µ > µ ). Here, it is also clear that the fluids whose rheological behavior close to a IJNS homepage:

8 34 International Journal of Nonlinear Science, Vol.8(4), No., pp.7-38 Newtonian fluid can flow easily when compared to the others pseudoplastic fluids. 4. Temperature profiles The displayed figures 3, 4 and 5, plot for n =.5, the effect of the viscosity parameter σ on the dimensionless temperature distribution around a heated and an impermeable vertical plate for selected values of the temperature exponent λ. It is clearly observed that the more viscous fluids are cooled slowly with η, which implies an enlargement of the thermal boundary layer thickness for all values of λ., c,, c n, n, Fig.3 : θ versus η at λ =, = and n =.5 for selected values of σ. Fig.4 : θ versus η at λ = /3, = and n =.5 for selected values of σ., n=, = Fig.5 : θ versus η at λ =, = and n =.5 for selected values of σ. Fig.6 :Isotherm plots of the temperature in saturated porous medium (T = 5 C) near a heated vertical plate (H = m, T w = 65 C) for λ =, =, n =.5 and σ =.5. It sounds obvious for an isothermal plate λ =, that the surface of the plate becomes adiabatic for σ c =.445. For λ = /3, we observe, in this case, that the critical value where the surface is almost adiabatic has occurred at σ c =.5. For σ < σ c, the surface heat flow is always positive and it is directed from the plate to the porous medium. On the other hand, for σ > σ c, the temperature profiles have maxima where the heat flow is transferred from the porous medium to the plate. In the case of a uniform lateral mass flux to the plate (V w = B) and a temperature that varies linearly with x (λ = ), the surface heat flow is always positive (directed from the plate to the porous medium) for all selected values of the parameter σ. IJNS for contribution: editor@nonlinearscience.org.uk

9 D. Achemlal, M. Sriti et.al:viscosity and Fluid Suction/Injection Effects 35 Fig.7 : Isotherm plots of the temperature in saturated porous medium (T = 5 C) near a heated vertical plate (H = m, T w = 65 C) for λ =, =, n =.5 and σ =. Fig.8 :Isotherm plots of the temperature in saturated porous medium (T = 5 C) near a heated vertical plate (H = m, T w = 65 C) for λ =, =, n =.5 and σ =.5., = -. = -.5, =. =.5 =. n, 4 6 8, = -. = -.5, =. =.5 =. n, Fig.9 : θ versus η at λ =, n =.5 and σ =.3 for selected values of. Fig. : θ versus η at λ = /3, n =.5 and σ =.3 for selected values of. Figures 6, 7 and 8 depict respectively for σ =.5, σ = and σ =.5 the isotherm plots in the boundary layer area of an isothermal and impermeable plate (T w = 65 C) embedded in a saturated porous medium by a pseudoplastic fluid (n =.5). We can say here that, in the boundary layer area, the heat exchange by convection from the plate to the porous medium is faster when the fluid is less viscous (µ < µ ). This can be explained by the effect of viscosity on the flow velocity because the less viscous fluids have the ability to flow quickly and consequently to provide the heat easily. Figures 9, and illustrate for n =.5 and σ =., the effect of the fluid suction/injection at the plate on the dimensionless temperature distributions in the boundary layer area. Here, we note that the suction of the fluid at the plate helps to reduce the temperature profiles in the boundary layer area and consequently helps to reduce the thermal boundary layer thickness and this for all selected values of λ. In figures and 3 we present respectively, in the cases of the fluid injection and suction at the plate, the isotherm plots in the boundary layer area of an isothermal plate (T w = 65 C) embedded in a saturated porous medium by a pseudoplastic fluid (n =.5, σ =.3). From these figures, it is also clear that the fuid suction at the plate can braked the heat transfer to the porous medium and consequently to reduce the thickness of the thermal boundary layer. 4.3 Local nusselt number profiles Figure 4 shows for n =.5 and =, the local Nusselt number profiles as function of σ for some values of the temperature exponent λ. Here, it can be noted that the rate of heat transfer at the plate decreases for more viscous fluids (µ > µ ) and amplified with λ. This is normal since the more viscous fluids tend to dissipate less heat compared to less IJNS homepage:

10 36 International Journal of Nonlinear Science, Vol.8(4), No., pp.7-38, = -. = -.5 =. =.5 =. n, Fig. : θ versus η at λ =, n =.5 and σ =.3 for selected values of. Fig. : Isotherm plots of the temperature in saturated porous medium (T = 5 C) near a heated vertical plate (H = m, T w = 65 C) for λ =, σ =.3, n =.5 and =. Fig.3 : Isotherm plots of the temperature in saturated porous medium (T = 5 C) near a heated vertical plate (H = m, T w = 65 C) for λ =, σ =.3, n =.5 and =. viscous fluids. In other words, the rising of the plate temperature can boost the heat transfer more. Figure 5 describes for σ = and =, the local Nusselt number in terms of n for selected values of λ. We deduce, here, that the power law fluid exponent n presents a slight increase in heat transfer in the boundary layer area, unlike the exponent of the temperature λ. 4.4 Local skin friction coefficient profiles Figure 6 shows for n =.5, the effect of viscosity parameter on local skin friction coefficient profiles in the boundary layer area for an isothermal and impermeable plate. We note here that the profiles which correspond to less viscous fluids (µ < µ ) have significant maxima tend towards the plate surface when σ decreases. Away from the plate, Cf approaches quickly to for the less viscous fluids than for the more viscous fluids. This is mainly due to the effect of the fluid viscosity on the local frictional forces in the boundary layer area. 5 Conclusion In this paper, we have discussed the effects of the variable viscosity of the pseudoplastic fluid, the power law fluid index, the fluid suction/injection parameter and the temperature exponent parameter on the free convection flow over a heated IJNS for contribution: editor@nonlinearscience.org.uk

11 D. Achemlal, M. Sriti et.al:viscosity and Fluid Suction/Injection Effects 37 Nu x Ra -/ x,, n, Nu x Ra -/ x,5,3, - -, - - -,5 - -,3 - -,,,3,5 Fig.4 : Nu versus σ at λ =, = and n =.5 for selected values of λ. -,5,7,9 Fig.5 : Nu versus n at λ =, = and σ = for selected values of λ. n, Cf n, Fig.6 : Cf versus η at λ =, = and n =.5 for selected values of σ. vertical flat plate embedded in a porous medium saturated by a pseudoplastic fluid with variable internal heat source. The set of the equations governing the problem are reduced to ordinary differential equations with appropriate boundary conditions. Furthermore, the similarity equations are solved numerically by using the fifth-order Runge-Kutta scheme associated with the shooting iteration technique. The influence of the parameter σ, n, and λ on temperature and velocity, local Nusselt number and local skin-friction profiles have been examined and discussed in details. From the present numerical study, we conclude that : The results of this study were in good agreement with previous studies in the case of a uniform fluid viscosity. More than the fluid is less viscous than the intensity of the flow velocity increases in the boundary layer area and quicklly stabilizes beyond the latter. On the other hand, the velocity profiles reduced when the exponent λ increases. The velocity profiles are amplified for the behavioral fluids close to Newtonian fluid and are reduced with the high temperature of the plate. For all selected thermal states of the plate, the more viscous fluid in the boundary layer area, are cooled slowly which implies an enlargement of the thermal boundary layer thickness. For λ = and λ = /3 the surface of the plate becomes adiabatic for σ = σ c. In the case where σ < σ c the surface heat flow is always positive and it is directed from the plate to the porous medium and for σ > σ c, the temperature profiles have maxima where the heat flow is transferred from the porous medium to the plate. For λ = the surface heat flow is always directed from the plate to the porous medium for all selected values of the parameter σ. IJNS homepage:

12 38 International Journal of Nonlinear Science, Vol.8(4), No., pp.7-38 The suction of the fluid at the plate helps to reduce the temperature profiles in the boundary layer area and consequently to reduce the thermal boundary layer thickness for all selected values of λ. The rate of heat transfer at the plate surface decreases for more viscous fluids and amplified with λ. In other hand, the power law fluid exponent n presents a slight increase in heat transfer but the rising of the plate temperature promotes the latter more. The frictional forces increase when the viscosity of the fluid decreases and tend to rapidly away from the plate unlike to the more viscous fluids. References [] D. B. Ingham, I. Pop. Transport Phenomena in Porous Media. Elsevier, Oxford. 5. [] D. A. Nield, A. Bejan. Convection in Porous Media. Springer, New York. 6. [3] K. Vafai. Handbook of Porous Media. Taylor & Francis(CRC Press), Boca Raton. 5. [4] I. Pop, D. B. Ingham. Convective Heat Transfer : Mathematical and Computational Viscous Fluids and Porous Media. Pergamon, Oxford.. [5] A. Bejan, A. D. Kraus. Heat Transfer Handbook. John Wiley & Sons, Hoboken. 3. [6] K. N. Mehta, K. N. Rao. Buoyancy-induced flow of non-newtonian fluids in a porous medium past a vertical flat plate with non-uniform surface heat flux. Int. J. Engng. Sci, 3(984):97-3. [7] M. Kumari et al. Free-convection boundary-layer flow of a non-newtonian fluid along a vertical wavy surface. International Journal of Heat and Fluid Flow, 8(997): [8] J. P. Pascal, H. Pascal. Free convection in a non-newtonian fluid saturated porous medium with lateral mass flux. International Journal of Non-Linear Mechanics, 3(997): [9] R. S. R. Gorla, M. Kumari. Free convection in non-newtonian fluids along a horizontal plate in a porous medium. Heat and Mass Transfer, 39(3):-6. [] S. Haq, J. C. Mulligan. Transient free convection from a vertical plate to a non-newtonian fluid in a porous medium. Journal of Non-Newtonian Fluid Mechanics, 36(99): [] S. M. Abdelgaied, M. R. Eid. Natural convection of non-newtonian power-law fluid over axisymmetric and twodimensional bodies of arbitrary shape in fluid-saturated porous media. Appl. Math. Mech.- Engl. Ed, 3(): [] S. Bagai, C. Nishad. Effect of variable viscosity on free convective heat transfer over a Non-isothermal body of arbitrary shape in a Non-Newtonian fluid saturated porous medium with internal heat generation. Transport in Porous Media, 94(): [3] D. R. Kassoy, A. Zebib. Variable viscosity effects on the onset of convection in porous media. Physics of Fluids, 8(975):649. [4] M. Kumari. Variable viscosity effects on free and mixed convection boundary-layer flow from a horizontal surface in a saturated porous medium-variable heat flux. In: Mechanics Research Communications, 8(): [5] T. Grosan, I. Pop. Free convection over a vertical flat plate with a variable wall temperature and internal heat generation in a porous medium saturated with a Non-Newtonian fluid. Technische Mechanik, 4(): [6] A. Postelnicu et al. Free convection boundary-layer over a vertical permeable flat plate in a porous medium with internal heat generation. Int comm Heat Mass Transfer, 7(): IJNS for contribution: editor@nonlinearscience.org.uk