The influence of mixed convection boundary layer flow of a viscoelastic fluid over an isothermal horizontal circular cylinder has been analyzed. The b...

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DOI: 10.1134/S0869864317010127

The influence of heat radiation on mixed convection boundary layer flow of a viscoelastic fluid over a circular cylinder with constant surface temperature H. Ahmad , T. Javed, and A. Ghaffari* International Islamic University, Islamabad, Pakistan E-mail: [email protected]* (Received August 31, 2015) The influence of mixed convection boundary layer flow of a viscoelastic fluid over an isothermal horizontal circular cylinder has been analyzed. The boundary layer equations governing the problem are reduced to dimensionless nonlinear partial differential equations and then solved numerically using Keller-box method. Skin friction coefficient and Nusselt number are emphasized specifically. These quantities are displayed against curvature parameter. Effects of mixed convection parameter and radiation-conduction parameter on skin friction coefficient and Nusselt number are illustrated through graphs and table. The boundary layer separation points along the surface of cylinder are also calculated with/without radiation, and a comparison is shown. The presence of radiation helps to reduce the skin friction coefficient in opposing flow case and enhances it for assisting flow case. The increase in value of radiation-conduction parameter helps increase the value of skin friction coefficient and Nusselt number for viscoelastic fluids. The boundary layer separation delays due to thermal radiation. Key words: viscoelastic fluid, circular cylinder, mixed convection, boundary layer flow, thermal radiation, numerical solution.

Introduction Mixed convection is significant in many engineering situations including the heating or cooling of heat exchangers, in chemical processes, nuclear power technology and some aspects of electronic cooling. It is an aspect of heat transfer in which fluid flows generated by buoyancy forces are comparable with those by freestream velocity. Mixed convection flows over horizontal circular cylinders are very important in many physical and engineering situations often faced in the cases of geothermal power generation and drilling operation. This area of research has become the pursuit of many researchers for the last three decades. The author of [1] was the first one who gave comprehensive analysis of mixed convection boundary layer flow over a horizontal circular cylinder. He investigated boundary layer separation point along the surface of cylinder. Later on, mixed convection heat transfer from an isothermal horizontal circular cylinder was studied in the work [2], and the work [3] dealt with mixed convection boundary layer flow from a horizontal circular cylinder with constant surface temperature. In recent years, the flow of viscoelastic fluids has gained considerable interest due to their applications in engineering and several manufacturing processes e.g., petroleum drilling, manufacturing of food, paper, paints, coating, inks and jet fuels etc. The viscoelastic fluid is of H. Ahmad , T. Javed, and A. Ghaffari, 2017

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H. Ahmad , T. Javed, and A. Ghaffari

second grade nature. A comprehensive discussion on second and third order fluid was done in the work [4]. Moreover, the authors of [5−7] also studied viscoelastic fluids in different geometries. It is necessary to mention the works [8−11] on second grade fluids. The authors of [12] investigated the steady mixed convection boundary layer flow of a viscoelastic fluid over a horizontal circular cylinder with constant heat flux. The study of convective heat transfer with thermal radiation has great importance especially in the processes involving high temperature such as gas turbines, nuclear power plants and thermal energy storage etc. The thermal radiation effects using the Rooseland diffusion approximation on mixed convection along vertical plate with uniform freestream velocity and surface temperature were discussed in the work [13]; the thermal radiation of a gray fluid which emits and absorbs radiation in a non-scattering medium was investigated in [14]. The authors of [15] discussed radiative flow in the presence of a magnetic field. The influence of thermal radiation on the boundary layer flow due to an exponentially stretching sheet was investigated in [16]. The authors of [17] studied the influence of thermal radiation on MHD flow of a second grade fluid. The effect of radiation on natural convection laminar flow from a horizontal circular cylinder was considered in the work [18]. It was concluded that increase in radiation causes increase in velocity and thermal boundary layer thicknesses. From the available literature, it appears that radiation effect on mixed convection flow of viscoelastic fluids over an isothermal circular cylinder has not yet been considered and the present study demonstrates this problem. In present study, we investigate thermal radiation effects on mixed convection boundary layer of a viscoelastic fluid over an isothermal horizontal circular cylinder by considering the Rosseland diffusion approximation. The governing equations are transformed into nondimensional forms which are solved numerical by an accurate and efficient finite difference scheme namely Keller box method. The results are compared for K = 0 and Rd = 0 with those given in [3]. It is found that our results are in excellent concordance with the previous studies. 1. Mathematical formulation We consider the mixed convection boundary layer flow past a horizontal circular cylinder embedded in a second grade viscoelastic fluid in the presence of heat radiation. The horizontal circular cylinder is of radius a that is maintained at a constant surface temperature Tw and is placed in a constant freestream temperature T∞ of viscoelastic fluid. The physical model and the corresponding coordinates system are drawn in Fig. 1. As in the work [1], it is assumed that constant freestream velocity is U ∞ 2 which is vertically upwards so that the freestream velocity for the boundary layer is ue ( x ) = U ∞ sin( x a ). Here Tw > T∞ corresponds to heated cylinder (assisting flow) while Tw < T∞ corresponds to cooled cylinder (opposing flow). The length of circular cylinder is considered long enough to neglect end-point effects. With these assumptions, Boussinesq and boundary layer approximations, the continuity, momentum and energy equations are (see [3] and [12])

∂u ∂x

+

∂v ∂y

= 0,

Fig. 1. Physical model of the problem. 116

(1)

Thermophysics and Aeromechanics, 2017, Vol. 24, No. 1

u

∂u ∂x

+v

∂u ∂ 2 u ko ∂ ∂ 2 u ∂ 3 u ∂u ∂ 2 u du e = ue +ν + + + + u v 2 ρ ∂ x ∂ y 2 ∂ y 3 ∂ y ∂ x∂ y ∂y dx ∂y x + g β (T − T∞ ) sin , α

u

∂T ∂x

+v

∂T ∂y

=

1 ∂qr k ∂ 2T , ⋅ − ⋅ ρC p ∂ y2 ρC p ∂ y

(2)

(3)

where x and y are the Cartesian coordinates measured along and normal to the surface of the cylinder respectively. Here x is measured from the lower stagnation point of the cylinder, u and v are components of velocity in the x and y directions respectively, ν is the kine-

matic viscosity, ρ is the density of the fluid, T is the temperature, C p is the specific heat, k is the thermal conductivity of the fluid, and k0 is the viscoelastic material parameter (the case when k0 = 0 corresponds to Newtonian fluid), g is acceleration due to gravity, qr is radiative heat flux, α and β are thermal diffusivity and thermal expansion coefficient, respectively. To fulfill thermodynamics requirements as suggested by the authors of the work [4], k0 is considered positive. Since the governing equations for viscoelastic fluid are one order higher than those of Newtonian fluids, therefore, we need an extra boundary condition ∂u ∂ y → 0 as

y → ∞, which was suggested in the work [19] in order to solve partial differential equations (1−3) numerically. The boundary conditions for the considered problem are given by

x ≥ 0. ∂u u → u e ( x), → 0, T → T∞ as y → ∞, ∂y = u 0,= v 0,= T Tw at= y 0,

(4)

The radiative heat flux qr is simplified by the Rosseland diffusion approximation [20] as

4σ ∂T 4 qr = , − ⋅ 3(α r + σ s ) ∂ y

(5)

where σ is the Stefan−Boltzmann constant, α r is the Rosseland mean absorption coefficient and σ s is the scattering coefficient. Let us assume that the fluid-phase temperature differences within the flow are sufficiently small as reported in the work [15] so that T 4 may be expanded as a linear function of temperature T and neglecting higher-order terms as

T 4 ≈ 4T∞3T − 3T∞4 .

(6)

Now expression (5) reduces to the form

16σ T∞3 ∂T − ⋅ qr = . 3(α r + σ s ) ∂ y

(7)

It is worth mentioning here that the use of the Rosseland diffusion approximation is valid in the interior of a medium but it is not employed near the boundaries. It is good only for an optically thick boundary layer. Since expression (7) does not contain any term for the radiation from the boundary surface, therefore is not valid to predict a complete description of this 117

H. Ahmad , T. Javed, and A. Ghaffari

physical situation near the surface. In other words, the boundary surface effects are negligible in the interior of an optically thick boundary layer region, which is due to the fact that the radiation from the boundaries becomes very weak before reaching the interior [18]. The non-dimensional variables are introduced as follows: 1 1 x x= / a ), u u / = a , y Re 2 ( y = U ∞ , v Re 2 (v / U ∞ ), = ue ( x ) = u e ( x) U ∞ , θ = T − T∞ Tw − T∞ ,

(8)

where Re = aU ∞ /ν is the Reynolds number. After substituting Eqs. (7) and (8) into Eqs. (1-3) we arrive at: ∂u ∂v (9) + = 0, ∂x ∂y

u

∂ ∂ 2u du ∂u ∂u ∂ 2u ∂ 3u ∂u ∂ 2 u + v = ue e + 2 + K u 2 + v 3 + + λθ sin( x), ∂x ∂y dx ∂y ∂y ∂x∂y ∂y ∂x ∂y

u

∂θ ∂θ 1 4 +v = 1 + Rd ∂x ∂y Pr 3

(10)

2

∂ θ 2, ∂y

(11)

where Pr is the Prandtl number, K is the dimensionless viscoelastic parameter, λ is the constant mixed convection parameter, and Rd is radiation-conduction parameter or Planck number, which are defined as µC p k0U ∞ 4σ T∞3 Gr R Pr = , K , λ = , , = = (12) d k k (α r + σ s ) ρ aν Re 2 with = Gr g β (Tw − T∞ )a 3 /ν 2 being the Grashof number. The mixed convection parameter λ in terms of Gr indicates that λ > 0 corresponds to assisting flow (Tw > T∞ ), λ < 0 corre-

sponds to opposing flow (Tw < T∞ ) , and λ = 0 corresponds to the forced convection case of the problem. For K = 0 , we get the case for viscous (Newtonian) fluids. The boundary conditions (4) become

= = u 0,= v 0,= θ 1 при y 0, x ≥ 0. u → U e ( x), ∂u ∂y → 0, θ → 0 при y → ∞

(13)

To solve Eqs. (9−11) subject to the boundary conditions (13), we assume that ue ( x) = sin x, as given in the work [1] and introduce the following variables:

= ψ xF = ( x, y ), θ θ ( x, y ),

(14)

where ψ is the stream function given by

u= ∂ψ ∂y , v = − ∂ψ ∂x .

(15)

Using Eqs. (14) and (15) in Eqs. (10) and (11), we get: ∂3 F ∂y 3

+F

2 4 3 2 ∂F sin x cos x sin x F ∂ F − 2 ∂F ⋅ ∂ F + ∂ F K − + + − λ θ 2 4 3 ∂y 2 ∂y x x ∂y ∂y ∂y ∂y

∂2 F

2

+

4 2 3 2 3 ∂F ∂ 2 F ∂F ∂ 2 F ∂F ∂ 4 F ∂F ∂ F ∂ F ∂ F ∂ F ∂ F x + x ⋅ 4 − ⋅ 3 + 2 ⋅ − ⋅ = ⋅ − ⋅ ∂y ∂x∂y ∂x ∂y 2 ∂x∂y ∂y ∂y ∂x∂y 2 ∂y ∂x∂y 3 ∂x ∂y

118

,

(16)

Thermophysics and Aeromechanics, 2017, Vol. 24, No. 1

∂F ∂θ ∂F ∂θ 1 4 ∂ 2θ ∂θ 1 + Rd 2 + F = x ⋅ − ⋅ . Pr 3 ∂y ∂y ∂y ∂x ∂x ∂y

(17)

The boundary conditions will become

F = 0, ∂F ∂y= 0, θ = 1 at y= 0, x ≥ 0, (18) sin x ∂ 2 F , → 0, θ → 0 as y → ∞, x ≥ 0, 2 x ∂y where prime denote differentiation with respect to y. The physical quantities of principal interest are the shearing stress and the rate of heat transfer in terms of the skin-friction coefficient C f and the Nusselt number Nu, respectively.

∂F ∂y →

For the present problem, these are given as

a (qw + qr ) y 0 1 (τ w ) y 0= = −1 2 = C f Re = , Nu Re 2 , 2 k (Tw − T∞ ) ρU ∞

(19)

where τ w and qw are the skin friction and surface heat flux respectively, which are defined by

∂2 u ∂T ∂2 u ∂u ∂u + k0 u +v 2 +2 ⋅ −k , qw = . ∂ y ∂ x∂ y ∂ x ∂ y y =0 ∂y = y 0 y =0

∂u

τw = µ ∂y

(20)

Using Eqs. (4) and (14) , Eq. (19) becomes

∂2 F C f =x 2 ∂y

4 , Nu =−(1 + Rd )θ ′( x, 0). 3 y =0

(21)

At the lower stagnation point of the cylinder, i.e., at x ≈ 0, the partial differential equations (16)−(17) with the boundary conditions (18) reduce to the following ordinary differential equations (22) f ′′′ + ff ′′ − f ′ 2 + 1 + λθ − K ( ff iv − 2 f ′f ′′′ + f ′′ 2 ) =0, 1 4 (1 + Rd )θ ′′ + f θ ′ = 0 Pr 3

with the boundary conditions

= 0, f ′(0) = 0, θ (0) = 1; f ′(∞ = = = f (0) ) 1, f ′′(∞ ) 0, θ (∞ ) 0,

(23)

(24)

where the prime denotes the differentiation with respect to y. The skin friction Cf and the Nusselt number Nu are given by Cf =x f ′′(0), Nu =−(1 +

4 Rd )θ ′(0). 3

(24)

2. Results and discussion The systems of partial differential equations (16) and (17) subject to the boundary conditions (18), and the ordinary differential equations (22) and (23) subject to the boundary conditions (24) are solved numerically by using an implicit finite difference scheme, known as Keller-box method. The method is very well explained in the monograph [21] and in the work [22]. The step size ∆y in y and the edge of the boundary layer y∞ is adjusted for different values of the parameters like λ, K and Rd to maintain accuracy in the results. Therefore, the step size

π 180 has been chosen in present numerical study. First, the numerical ∆y =0.02 and ∆x =

119

H. Ahmad , T. Javed, and A. Ghaffari

Fig. 2. Comparison of skin friction Cf and Nusselt number Nu with the data of [3] (circles) at various values of λ. K = 0 (Newtonian fluid); Pr = 1; Rd = 0 (1), 0.5 (2).

solution is obtained at the lower stagnation point of the cylinder x ≈ 0 and proceeds round the cylinder up to the separation point. The results for the skin friction coefficient C f and Nusselt number Nu are obtained for some values of the mixed convection parameter λ, viscoelastic parameter K and radiation conduction parameter Rd with Pr = 1. The comparison of the values of skin friction coefficient C f and the Nusselt number Nu with those reported in the work [3] for a Newtonian fluid (K = 0) is illustrated in Figs. 2a and 2b. The comparison shows an excellent rapport of our results with those of the work [3]. The effect of thermal radiation on the skin friction coefficient C f as well as on the Nusselt number Nu is also presented in the same figure. For cooled cylinder case (λ < 0) , a small decrease in skin friction is depicted while for heated cylinder case (λ > 0) , a substantial increase in its value has been noticed on account of the radiation. A significant increase in Nusselt number has been observed due to the radiation in both cooled and heated cylinder cases. The curves drawn in Figs. 2a and 2b show that a positive value of mixed convection parameter λ induces a supporting pressure gradient which helps increasing the values of skin friction coefficient C f and Nusselt number Nu. Figures 3a and 3b show the graphs of the skin friction coefficient C f and the Nusselt number Nu for viscoelastic fluid by putting K = 0.2. The curves are drawn for different values of mixed convection parameter λ in the absence as well as presence of radiation effects. The presence of radiation reduces the skin friction for λ < 0, and for λ > 0 , the radiation

Fig. 3. Variation of skin friction Cf and Nusselt number Nu against curvature parameter x at various values of λ. K = 0.2 (viscoelastic fluid); Pr = 1; Rd = 0 (1), 0.5 (2).

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Thermophysics and Aeromechanics, 2017, Vol. 24, No. 1

Fig. 4. Variation of skin friction Cf and Nusselt number Nu against curvature parameter x for the different values of Rd K = 1; Pr = 1; λ = −1 (cooled cylinder) (1), 4 (heated cylinder) (2).

effect increases the skin friction. The Nusselt Nu increases due to the effect of radiation for various values of mixed convection parameter λ in both cases (λ < 0) and (λ > 0). The figures also show that there is a critical value of λ = λc(K) depending upon the viscoelastic parameter K below which a boundary layer solution is not possible. The same effects were reported in the work [1] for the Newtonian case, in that for sufficiently cooled cylinder (λ < 0) , the natural convection would start at upper stagnation point ( x ≈ π ), the flow of stream upwards cannot overcome the motion of fluid next to cylinder in downward direction under the action of buoyancy forces that oppose the development of boundary layer. Figures 4a and 4b show the effect of radiations for= K 1,= Pr 1 , λ = −1 and 4 on skin friction coefficient C f and Nusselt number Nu, respectively. For λ < 0 , the skin friction coefficient Cf decreases with the increase in value of radiation-conduction parameter Rd but interestingly, for λ > 0 , the skin friction coefficient Cf increases with the increase in value of radiation-conduction parameter Rd. On the other hand, with increase in Rd , the Nusselt number Nu increases in both cases of cooled and heated cylinder, i.e., at λ < 0 and λ > 0. In Fig. 5, the variation of boundary layer separation point xs with λ is shown for Pr = 1, K = 0.2, and for some values of radiation-conduction parameter Rd : Rd = 0, 0.5, 1 and 2. It is

λ λ0 (< 0), below shown that for different values of Rd , there exist some critical values of = which boundary layer solution does not exist. It is also noticed that by increasing the value of radiation-conduction parameter Rd , the critical value of λ = λ0 increases, and boundary layer separation delays. The effect of radiation-conduction parameter Rd on the development of streamlines and isotherms are illustrated in Figs. 6 and 7, which are plotted for K = 1, Rd = 0, 0.5, Pr = 1, and λ = 2. Fig. 6a indicates that without effect of radiation (i.e., Rd = 0), the value of ψ max within the computational domain is about 8.6 near the boundary layer separation point ( x ≈ 1.5). But Fig. 6b shows that for Rd = 0.5, the value of ψ max is about 9.0 near the boundary layer separation point. So, we conclude that the application of thermal radiation results in an increase in the flow flux in the boundary layer. Figures 7a and 7b show Fig. 5. Variation of boundary layer separation point xs with λ for Pr = 1 and K = 0.2. Rd.= 0 (1), 0.5 (2), 1 (3), 2 (4).

121

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Fig. 6. Streamlines for Rd = 0 (a), 0.5 (b) while K = 1, Pr = 1, and λ = 2.

the significant increase in boundary layer thickness due to thermal radiation effect. The same phenomenon has been observed in Figs. 8 and 9 for the case K = 1, Rd = 0, 0.5, Pr = 7 and

λ = 2. We observe from the Table that the skin friction coefficient C f and Nusselt number Nu increase by increasing the value of radiation-conduction parameter Rd. It is also observed that the skin friction coefficient C f increases along the surface of cylinder from lower stagnation point to separation point but the value of Nusselt number Nu decreases along the surface of the cylinder from lower stagnation point to the point of separation. Conclusion The influence of radiation on mixed convection boundary layer flow of a viscoelastic fluid past an isothermal horizontal circular cylinder with constant surface temperature has been studied. The boundary layer equations governing the flow and heat transfer are transformed to nondimensional, nonlinear system of partial differential equations which are then solved numerically using Keller-box method. We have discussed the effect of radiation-conduction parameter Rd on flow, heat transfer and the boundary layer separation point xs on the surface of the cylinder. The present investigation helps conclude that Table Values of skin friction coefficient Cf and Nusselt number Nu for different values of radiation-conduction parameter Rd for Pr = 1, λ = 2, K = 0.2. x 0° 10° 20° 30° 40° 50° 60° 70° 80° 90° 100°

122

Rd = 0.0 Cf 0 0.3160 0.6211 0.9047 1.1574 1.3711 1.5394 1.6581 1.7253 1.7416 −

Nu 0.6187 0.6171 0.6123 0.6042 0.5931 0.5789 0.5618 0.5419 0.5195 0.4947 −

Rd = 0.5 Cf 0 0.3253 0.6396 0.9324 1.1944 1.4174 1.5950 1.7230 1.7993 1.8245 −

Nu 0.8508 0.8487 0.8422 0.8315 0.8166 0.7977 0.7750 0.7230 1.7993 1.6864 −

Rd = 1.0 Cf 0 0.3311 0.6512 0.9499 0.2177 1.4465 1.6299 1.7637 1.8458 1.8765 −

Nu 1.0461 1.0435 1.0357 1.0228 1.0048 0.9820 1.9546 0.9229 0.8872 0.8481 −

Rd = 2.0 Cf 0 0.3384 0.6659 0.9720 1.2472 1.4834 1.6742 1.8152 1.9045 1.9522 1.9311

Nu 1.3749 1.3716 1.3615 1.3448 1.3217 1.2923 1.2571 1.2164 1.1708 1.1208 1.0671

Thermophysics and Aeromechanics, 2017, Vol. 24, No. 1

Fig. 7. Isotherms for Rd = 0 (a), 0.5 (b) while K = 1, Pr = 1, and λ = 2.

Fig. 8. Streamlines for Rd = 0 (a), 0.5 (b) while K = 1, Pr = 7, and λ = 2.

Fig. 9. Isotherms for Rd = 0 (a), 0.5 (b) while K = 1, Pr = 7, and λ = 2. 123

H. Ahmad , T. Javed, and A. Ghaffari

— the presence of radiation increases the skin friction Cf for heated cylinder (λ > 0) case, but it decreases the skin friction for the cooled cylinder (λ < 0) case. — the radiation increases the Nusselt number Nu in both cases (λ > 0, λ < 0). — an increase in the value of radiation-conduction parameter R d , leads to increase in the value of both skin friction Cf and Nusselt number Nu. — the increase in radiation-conduction parameter Rd results in delay of boundary layer separation point xs. References 1. J.H. Merkin, Mixed convection from a horizontal circular cylinder, Int. J. Heat Mass Transfer, 1977, Vol. 20, P. 73–77. 2. H.M. Badr, A theoretical study of laminar mixed convection from a horizontal cylinder in a cross stream, Int. J. Heat Mass Transfer, 1983, Vol. 26, P. 639–653. 3. I. Anwar, S. Amin, and I. Pop, Mixed convection boundary layer flow of a viscoelastic fluid over a horizontal circular cylinder, Int. J. Non-Linear Mech., 2008, Vol. 43, P. 814–821. 4. J.E. Dunn and K.R. Rajagopal, Fluids of differential type: critical review and thermodynamic analysis, Int. J. Engng Sci., 1995, Vol. 33, P. 689–729. 5. P.D. Ariel, Stagnation point flow of a viscoelastic fluid towards a moving plate, Int. J. Engng Sci., 1995, Vol. 33, P. 1679–1687. 6. K.R. Rajagopal, M. Renardy, Y. Renardy, and A.S. Wineman, Flow of viscoelastic fluids between plates rotating about distinct axes, Rheol. Acta, 1986, Vol. 25, P. 459–467. 7. K.R. Rajagopal, Flow of viscoelastic fluids between rotating disks, Theor. Comput. Fluid Dyn., 1992, Vol. 3, P. 185–206. 8. R.A. Cortell, Note on flow and heat transfer of a viscoelastic fluid over a stretching sheet, Int. J. Non-Linear Mech., 2006, Vol. 41, P. 78–85. 9. M.S. Abel, S.K. Khan, and K.V. Prasad, Study of visco-elastic fluid flow and heat transfer over a stretching sheet with variable viscosity, Int. J. Non-Linear Mech., 2002, Vol. 37, P. 81–88. 10. T. Hayat, Z. Abbas, and T. Javed, Mixed convection flow of a micropolar fluid over a nonlinear stretching sheet, Physics Letters A, 2008, Vol. 372, No. 5, P. 637–647. 11. M. Sajid, Z. Abbas, T. Javed, and N. Ali, Boundary layer flow of an Oldroyd B fluid in the Region of stagnation point over a stretching sheet, Canadian J. Phys., 2010, Vol. 88, P. 635–640. 12. A.R.M. Kasim, N.F. Muhammad, S. Shafie, and I. Pop, Constant heat flux solution for mixed convection boundary layer viscoelastic fluid, Int. J. Heat Mass Transfer, 2013, Vol. 49, P. 163−171. 13. M.A. Hossain and H.S. Takhar, Radiation effects on mixed convection along a vertical plate with uniform surface temperature, Int. J. Heat Mass Transfer, 1996, Vol. 31, P. 243–248. 14. M.A. Hossain, M.A. Alim, and D. Rees, The effect of radiation on free convection from a porous vertical plate, Int. J. Heat Mass Transfer, 1999, Vol. 42, P. 181–191. 15. A. Raptis, C. Perdikis, and H.S. Takhar, Effect of thermal radiation on MHD flow, Appl. Math. Comp., 2004, Vol. 153, P. 645–649. 16. M. Sajid and T. Hayat, Influence of thermal radiation on the boundary layer flow due to an exponentially stretching sheet, Int. Commun. Heat and Mass Transfer, 2008, Vol. 35, P. 347–356. 17. T. Hayat, Z. Abbas, M. Sajid, and S. Asghar, The influence of thermal radiation on MHD flow of a secondgrade fluid, Int. J. Heat Mass Transf., 2007, Vol. 50, P. 931–941. 18. M.M. Molla, S.C. Saha, M.A.I. Khan, and M.A. Hossain, Radiation effects on natural convection laminar flow from a horizontal circular cylinder, Desalination and Water Treatment, 2011, Vol. 30, P. 89–97. 19. V.K. Garg and K.R. Rajagopal, Stagnation point flow of a non-Newtonian fluid, Mech. Res. Commun., 1990, Vol. 17, P. 415–421. 20. S. Rosseland, Theoretical Astrophysics, Oxford University Press, London, 1936. 21. T. Cebeci and P. Bradshaw, Physical and Computational Aspects of Convective Heat Transfer, Springer, New York, 1984. 22. H.B. Keller and T. Cebeci, Numerical methods in boundary layer theory, Annual Rev. Fluid Mech., 1978, Vol. 10, P. 417–433.

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