Eur. Phys. J. Plus (2017) 132: 550 DOI 10.1140/epjp/i2017-11822-0
THE EUROPEAN PHYSICAL JOURNAL PLUS
Regular Article
Numerical solution of chemically reactive non-Newtonian fluid flow: Dual stratification Khalil Ur Rehman1,a , M.Y. Malik1 , Abid Ali Khan1 , Iffat Zehra2 , Mostafa Zahri3 , and M. Tahir4 1 2 3 4
Department Department Department Department
of of of of
Mathematics, Mathematics, Mathematics, Mathematics,
Quaid-i-Azam University, Islamabad 44000, Pakistan Air University, PAF Complex E-9, Islamabad 44000, Pakistan College of Sciences, University of Sharjah, Sharjah 27272, United Arab Emirates Faculty of Basic Sciences, HITEC University Taxila, Taxila, Pakistan
Received: 10 October 2017 / Revised: 27 November 2017 c Societ` Published online: 28 December 2017 – a Italiana di Fisica / Springer-Verlag 2017 Abstract. We have found that only a few attempts are available in the literature relatively to the tangent hyperbolic fluid flow induced by stretching cylindrical surfaces. In particular, temperature and concentration stratification effects have not been investigated until now with respect to the tangent hyperbolic fluid model. Therefore, we have considered the tangent hyperbolic fluid flow induced by an acutely inclined cylindrical surface in the presence of both temperature and concentration stratification effects. To be more specific, the fluid flow is attained with the no slip condition, which implies that the bulk motion of the fluid particles is the same as the stretching velocity of a cylindrical surface. Additionally, the flow field situation is manifested with heat generation, mixed convection and chemical reaction effects. The flow partial differential equations give a complete description of the present problem. Therefore, to trace out the solution, a set of suitable transformations is introduced to convert these equations into ordinary differential equations. In addition, a self-coded computational algorithm is executed to inspect the numerical solution of these reduced equations. The effect logs of the involved parameters are provided graphically. Furthermore, the variations of the physical quantities are examined and given with the aid of tables. It is observed that the fluid temperature is a decreasing function of the thermal stratification parameter and a similar trend is noticed for the concentration via the solutal stratification parameter.
1 Introduction The study of boundary layer flows in both Newtonian and non-Newtonian fluids brought by stretching surfaces with both heat and mass transfer characteristics plays a vital role in industrial and engineering areas. It can be seen, from the literature, that researchers have studied the flow characteristics of both Newtonian and non-Newtonian fluid models in the presence of various physical effects [1–15]. In non-Newtonian fluids, we do not have a linear relationship between deformation rates and stress, which leads to uncertainty making it difficult to inspect the features of all non-Newtonian fluid models. In particular, the shear thinning characteristics were delineated via some well-known non-Newtonian fluid models, e.g., power law model (1923), Williamson fluid model (1929), cross model (1965), carreau model (1972), tangent hyperbolic fluid and Ellis fluid model, which are always difficult to handle. In general, the Newtonian behaviour cannot describe most of physiological fluids, like blood. Therefore, different fundamental non-Newtonian fluid models are proposed by scientists to envision the characteristics of physiological fluids. Here, the tangent hyperbolic fluid (THF) model closely describes the flow diversity of blood as described by prolific researchers like Akbar et al. [16], who discussed the numerical solutions of the magneto hydrodynamic boundary layer flow of THF via a stretching flat surface. One can refer to the remarkable work on the THF flow characteristics in refs. [17,18]. Stratification phenomena include temperature stratification as well as concentration stratification and originate due to variation in the temperature and concentration differences of the medium. Temperature and concentration stratification are also known as thermal and solutal stratification, respectively. The non-Newtonian fluids flow in the presence of thermal and concentration stratification is a topic of great interest for present researchers and scientists. The environment and geophysical flows where heat rejection occurs, such as lakes, seas, rivers, and systems for storing thermal energy, like solar ponds, reflect the role of stratification involvements. Further, astrophysics, agriculture, oceanography, a
e-mail:
[email protected] (corresponding author)
Page 2 of 15
Eur. Phys. J. Plus (2017) 132: 550
and various chemical processes admit the properties of temperature and concentration stratification phenomena. In fact, stratification plays a dynamic role in many natural and industrial phenomena. Therefore, remarkable attention has been given by investigators towards double stratification occurrences due to their several applications. Several (experimental and analytical) attempts have been made regarding stratification effects. To mention just a few, Yang et al. [19] studied the laminar flow with free convection effects in a thermally stratified medium, whereas the influence of natural convection subject to simple bodies in temperature stratification was studied by Chen and Eichhorn [20]. Narayana and Murthy [21] offered the free convection effects on the power-law fluid in a porous thermal and solutal stratified medium. The boundary flow towards a stratified medium along with mixed convection was considered by Ishak et al. [22]. Cheng [23] pointed out the flow properties of a power-law fluid in a double stratified medium. The characteristics of the nanofluid flow over a vertical plate in a double stratified medium were reported by Ibrahim and Makinde [24]. The effects of stratification in a MHD micropolar fluid were addressed by Skrinvasacharya and Upendra [25]. Plenty of researchers (see recent studies, refs. [26–30]) addressed the stratification phenomena by altering the physical effects. The novelty of the article is the exploration of the flow characteristics of the THF when both thermal and solutal stratification effects are present. The fluid flow is due to an inclined stretching cylindrical surface in the presence of some pertinent effects, namely, heat generation, mixed convection and chemically reactive species. To be more specific, the researchers considered stratification effects (either it is single stratification or double stratification) on flat surfaces. As of now, few attempts are available with accuracy on a cylindrical surface, but they are still insufficient. In particular, the stratification effects on the THF flow towards an inclined cylindrical surface in the presence of heat generation, mixed convection and chemical reaction are not found in the literature. Therefore, the key objective of the present study is to study the temperature and concentration effects on the THF flow in the presence of heat generation, mixed convection and chemical reaction. The flow field situation in a concerned constrain is translated in terms of partial differential equations (PDEs). These PDEs are converted into ordinary differential equations and then a computational algorithm is executed to yield the numerical solution. The effects logs of an involved parameters on dimensionless quantities are discussed in detail by way of graphs. Further, the skin friction coefficient (SFC), heat transfer rate (HTR) and mass transfer rate (MTR) are presented with the help of tables.
2 Mathematical formulation The two-dimensional flow of a THF due to a stretching inclined cylindrical surface is considered. The flow field is manifested with thermal and solutal stratification. Moreover, the role of chemical reaction, heat generation and mixed convection is also considered. The boundary layer approximation reduces the continuity, momentum, heat and mass equations to ∂(ru) ∂(rv) + = 0, (1) ∂x ∂r 2 √ ∂u ∂ 2 u ∂u ∂ 2 u nΓ ∂u ∂u 1 ∂u = ν (1−n) 2 + √ +n 2Γ +gβT (T −T∞ ) cos α+gβc (C −C∞ ) cos α, u +v +(1−n) ∂x ∂r ∂r r ∂r ∂r ∂r2 2r ∂r (2) 2 ∂T k Q0 ∂ T 1 ∂T ∂T +v = + + (T − T∞ ), (3) u ∂x ∂r ρcp ∂r2 r ∂r ρcp 2 ∂C ∂ C 1 ∂C ∂C +v =D − R0 (C − C∞ ). + (4) u ∂x ∂r ∂r2 r ∂r The x-axis is supposed as the axial axis of the cylinder while the r-axis is perpendicular to it. The velocity component u is in the x-direction and v is in the r-direction, whereas Γ > 0, g, ρ, Q0 , n, k, βT , ν, βc , α, D and cp denote time constant, gravity, fluid density, heat generation coefficient, power-law index, thermal conductivity, coefficient of thermal expansion, kinematic viscosity, coefficient of solutal expansion, inclination, mass diffusivity and specific heat, respectively. The corresponding endpoint conditions of problem are given as u(x, R) = U (x) =
U0 x, L
v(x, R) = 0,
dx , L ex C(x, ∞) → C∞ (x) = C0 + . L C(x, R) = Cw (x) = C0 +
T (x, R) = Tw (x) = T0 +
u(x, ∞) → 0,
bx , L
T (x, ∞) → T∞ (x) = T0 +
cx , L (5)
Eur. Phys. J. Plus (2017) 132: 550
Page 3 of 15
The stream function (ψ) satisfies eq. (1) and is described as −1 ∂ψ 1 ∂ψ , v= . u= r ∂r r ∂x
(6)
To attain dimensionless form we use U0 x f (η), u= L T (η) =
T − T∞ , Tw − T0
R v=− r
C(η) =
U0 ν f (η), L
C − C∞ , Cw − C0
1 r2 − R2 U0 2 η= , 2R νL 1 U0 νx2 2 ψ= Rf (η), L
(7)
where f (η) represents the dimensionless variable, so that f (η) is the THF velocity, U0 is the free stream velocity, L is the reference length, and Tw (x), T∞ (x), T0 , Cw , C∞ and C0 denote surface temperature, variable ambient temperature, reference temperature, surface concentration, variable ambient concentration and reference concentration, respectively, where b, c, d and e are positive constants. One obtains 2 2 d2 f (η) d f (η) d2 f (η) d3 f (η) + 2f (η) − 2 + 4K(1 − n) dη 3 dη 2 dη 2 dη 2 2 2 2 3 2 1 3 d f (η) d f (η) d f (η) + 3(1 + 2Kη) 2 Kλ + 2λn(1 + 2Kη) 2 + λm (T (η) + N C(η)) cos α = 0, dη 2 dη 2 dη 3 dT (η) df (η) d2 T (η) dT (η) df (η) + Pr f (η) − T (η) − k1 + DH T (η) = 0, (1 + 2Kη) + 2K dη 2 dη dη dη dη dC(η) df (η) d2 C(η) dC(η) df (η) + Sc f (η) − C(η) − k2 − Rc C(η) = 0, (1 + 2Kη) + 2K dη 2 dη dη dη dη df (η) = 1, T (η) = 1 − k1 , C(η) = 1 − k2 , as η → 0, f (η) = 0, dη df (η) → 0, T (η) → 0, C(η) → 0, as η → ∞, dη 2(1 − n)(1 + 2Kη)
(8) (9) (10)
(11)
where Rc , k2 , Sc, DH , k1 , Pr, N , λm , λ, and K denote chemical reaction parameter, solutal stratification parameter, Schmidt number, heat generation parameter, thermal stratification parameter, Prandtl number, ratio of concentration to thermal buoyancy forces, mixed convection parameter, Weissenberg number and curvature parameter, respectively, and are given as follows: μcp Gr Gr∗ 2U 3 1 ν , λ=Γ , λm = , Pr = , , N = K= 2 R a νx Gr K Rex c LQ0 ν e R0 L k1 = , DH = , k2 = , Rc = , Sc = , (12) b U0 ρcp D d U0 where Gr, Gr∗ denote the Grashof number due to temperature and concentration, respectively, and are defined as Gr =
gβT (Tw − T0 )x3 , ν2
Gr∗ =
gβT (Cw − C0 )x3 . ν2
(13)
The skin friction coefficient (SFC) is described as τw Cf = U 2 , ρ 2
∂u nΓ +√ τw = μ (1 − n) ∂r 2
∂u ∂r
2 ,
where μ and τw denote viscosity of the fluid and the shear stress. The dimensionless form of SFC is given by Cf Rex = (1 − n)f (0) + nλ(f (0))2 , with Rex =
U0 x2 νL
as the local Reynolds number.
(14)
r=R
(15)
Page 4 of 15
Eur. Phys. J. Plus (2017) 132: 550
The local Nusselt and Sherwood numbers are given by xqw ∂T N ux = , qw = −k , k(Tw − T0 ) ∂r r=R In dimensionless form we have
Nu √ x = −T (0), Rex
xjw , Sh = D(Cw − C0 )
jw = −D
∂C ∂r
.
(16)
r=R
Sh √ = −C (0). Rex
(17)
For instance, in the absence of concentration equation, if we neglect mixed convection and heat generation effects, eqs. (8) and (9) reduce to the problem reported by Rehman et al. [29].
3 Numerical scheme To implement the shooting method, the system given by eqs. (8)–(10) with endpoint conditions given by eqs. (11) is firstly converted into a first-order system of differential equations. For this purpose, the useful substitution is y2 = f ,
y3 = y3 = f ,
y5 = T ,
y7 = C .
One can get 1
y1 = y2 ,
y2 = y3 ,
y3 =
(y2 )2 − y1 y3 − (2K)(1 − n)y3 − 32 λK(1 + 2Kη) 2 y32 − λm (y4 + N y6 ) cos α 3
(1 − n)(1 + 2Kη) + nλ(1 + 2Kη) 2 y3
y4 = y5
y5 =
Pr(y2 y4 + k1 y2 − y1 y5 − DH y4 ) − 2Ky5 , 1 + 2Kη
y6 = y7
y7 =
Sc(y2 y6 + k2 y2 − y1 y7 + Rc y6 ) − 2Ky7 , 1 + 2Kη
y1 (0) = 0,
y2 (0) = 1,
y4 (0) = 1 − k1 , y6 (0) = 1 − k2
,
(18)
y3 (0) = unknown,
y5 (0) = unknown, y7 (0) = unknown.
(19)
It is important to note that the reduced system (eqs. (18) and (19)) is an initial value problem. To integrate, we need the numeric values for y3 (0), i.e. f (0), y5 (0) i.e. T (0) and y7 (0) implies C (0). Moreover, the initial conditions, y3 (0), y5 (0) and y7 (0), are not given but we have additional endpoint conditions: y2 (∞) = 0,
y4 (∞) = 0,
y6 (∞) = 0.
(20)
The integration is carried in such a way that the conditions given by eq. (20) hold fairly.
4 Analysis Tables 1–3 are constructed to examine the impact of the involved parameters on physical quantities, namely, SFC, HTR and MTR. In detail, table 1 shows the SFC variations (in absolute sense) for positive values of K, n, λ and Pr. It can be seen that the SFC near the cylindrical surface is an increasing function of both K and Pr, while an opposite trend is seen for n and λ. The negative sign with the values of the skin friction coefficient physically shows that the cylindrical surface exerts drag force on fluid particles. Table 2 is designed to provide the variations of HTR to K, Pr, k1 and λ. It is found that, when we increase the values of both K and Pr, the HTR shows an increasing behaviour but opposite trends are noticed for higher values of k1 and λ. The influence of K, k2 and Sc is reported on MTR by means of table 3. It is noticed that the mass transfer rate reflects higher values for K, k2 and Sc. It is noticed that, in the absence of concentration equation, thermal stratification, mixed convection and heat generation effects, we find the problem reported by Akbar et al. [16]. Table 4 provides comparative values of the skin friction coefficient with respect to both power-law index and Weissenberg number. An excellent match is found with the existing values, which confirms the execution of computational algorithm.
Eur. Phys. J. Plus (2017) 132: 550
Page 5 of 15 Table 1. Variation in SFC via K, n, λ and Pr. √ n λ Pr Cf Rex = (1 − n)f (0) + nλf (0)2 0.1 0.1 2.0 −1.2108 – – – −1.9104 – – – −1.9443 0.1 – – −1.2108 0.3 – – 1.1017 0.5 – – −1.0023 – 0.1 – −1.2108 – 0.3 – −1.1710 – 0.5 – −1.1083 – – 2.0 −1.2108 – – 2.2 −1.2129 – – 2.4 −1.2147
K 0.4 0.6 0.8 – – – – – – – – –
Table 2. Variation in HTR via K, Pr, k1 and λ. K 0.4 0.6 0.8 – – – – – – – – –
Pr 2.0 – – 2.0 2.2 2.4 – – – – – –
k1 0.1 – – – – – 0.1 0.3 0.5 – – –
−T (0) 0.8350 0.9625 1.0444 0.8345 0.9015 0.9824 0.8350 0.7892 0.6905 0.8350 0.7108 0.6012
λ 0.1 – – – – – – – – 0.1 0.3 0.5
Table 3. Variation in MTR via K, k2 and Sc. K 0.4 0.6 0.8 – – – – – –
k2 0.1 – – 0.1 0.3 0.5 – – –
Sc 0.1 – – – – – 0.1 0.3 0.5
−C (0) 1.8081 1.9901 2.015 1.8081 1.9655 2.0008 1.8081 1.8252 1.8434
Table 4. Comparison of SFC with existing work when K = 0. n
λ
Akbar et al. [16]
Rehman et al. [29]
Present outcomes
0.0 0.0 0.0 0.1 0.1 0.1 0.2 0.2
0.0 0.3 0.5 0.0 0.3 0.5 0.0 0.3
1.00000 1.00000 1.00000 0.94868 0.94248 0.93826 0.89442 0.88023
1.00000 1.00000 1.00000 0.94916 0.94321 0.93801 0.89447 0.88056
1.0000 1.0000 1.0000 0.9491 0.9432 0.9380 0.8944 0.8805
Page 6 of 15
Eur. Phys. J. Plus (2017) 132: 550
Fig. 1. Streamline pattern for (a) K = 0.1; (b) K = 0.2; (c) K = 0.5; (d) K = 0.9.
Figures 1(a)–(d) is drawn to examine the streamline patterns against the curvature parameter. It is seen that for higher values of the curvature parameter the fluid velocity enhances. The accelerated magnitude is witnessed by streamlines because, for high curvature parameter the adjacent distance among the streamlines decreases. Figures 2–7 are plotted to show how f (η) is affected by flow controlling parameters, namely, curvature, mixed convection, thermal stratification parameter, inclination, Weissenberg number and power-law index. In detail, fig. 2 is plotted to report the impact of K on f (η). It is noticed that the THF velocity varies directly, that is for lager values of K the THF velocity enhances. Actually, a large variation in K implies a decrease in the radius of the curvature. The contact surface area reduces and less resistance is faced by the THF fluid particles; as a result, the fluid flow accelerates. Figure 3 shows the impact of α on f (η). It is seen that for increasing values of α, f (η) decreases. The fact behind that is that an
Eur. Phys. J. Plus (2017) 132: 550
Page 7 of 15
Fig. 2. Effect of K on f (η).
Fig. 3. Effect of α on f (η).
increase in α about the x-axis reduces the effect of gravity, which causes a decrease in f (η) within the momentum boundary layer. The effect of λm on f (η) is shown in fig. 4. It is found that, for higher values of λm , f (η) increases within the boundary layer. Physically, this is because of the increase in the buoyancy force. Figure 5 shows that there exists an inverse relation between k1 and f (η) because, for positive values of k1 , the THF velocity decreases as well as the concerned momentum boundary layer. Figure 6 shows the impact of λ on f (η). It is noticed that, for higher
Page 8 of 15
Eur. Phys. J. Plus (2017) 132: 550
Fig. 4. Effect of λm on f (η).
Fig. 5. Effect of k1 on f (η).
values of λ, f (η) shows a declining trend because, increasing the values of λ, the relaxation time of the THF fluid increases, which offers resistance to the THF fluid particles; as a result f (η) decreases. Figure 7 shows the impact of n on f (η). It is concluded that there exists an inverse relation between n and f (η) because, for large positive values of n, the THF velocity shows decreasing curves.
Eur. Phys. J. Plus (2017) 132: 550
Page 9 of 15
Fig. 6. Effect of λ on f (η).
Fig. 7. Effect of n on f (η).
Figures 8–13 are plotted to show the impact of K, DH , k1 , Pr, α, and λ on the THF temperature. In detail, the influence of K on T (η) is examined and given with the aid of fig. 8. It is noticed that for higher values of K the THF temperature increases. The Kelvin temperature is evaluated as an average kinetic energy of fluid particles, therefore, with an increase in K of the cylinder, the velocity of the fluid increases, which causes an increase in the kinetic energy,
Page 10 of 15
Eur. Phys. J. Plus (2017) 132: 550
Fig. 8. Effect of K on T (η).
Fig. 9. Effect of DH on T (η).
which corresponds to an increase in the THF temperature. Figure 9 illustrates the effect of DH on T (η). It is noticed that an increase in DH results in an increase in T (η). This fact is due to the significant amount of heat produced via increasing the values of DH , so that the temperature of the THF increases. Figure 10 shows the impact of Pr on T (η). It is seen that there exists an inverse relation between Pr and T (η). Moreover, fluids with high Pr correspond to a weaker diffusion energy. So, an increase in Pr results in a strong reduction in T (η), which is the cause of a thinner
Eur. Phys. J. Plus (2017) 132: 550
Page 11 of 15
Fig. 10. Effect of Pr on T (η).
Fig. 11. Effect of α on T (η).
thermal boundary layer. Figure 11 reports the direct relation between α and T (η) because, for higher values of α, T (η) increases. The fact behind that is that with an increase in α about the x-axis, the effect of gravity is less, which results in an increase in T (η). Figure 12 shows the impact of k1 on T (η) and it is seen that, with an increase in k1 , the temperature of the THF decreases because, for higher values of k1 , the convective potential between the cylindrical surface and the ambient fluid drops. The drop of convective potential results in a decrease in T (η) and the related
Page 12 of 15
Eur. Phys. J. Plus (2017) 132: 550
Fig. 12. Effect of k1 on T (η).
Fig. 13. Effect of λ on T (η).
thermal boundary layer. Figure 13 is used to examine the effect of λ on T (η). We found an indirect relation between λ and T (η), because large values of λ lead to an increase in the relaxation time of THF particles, due to which resistance increases. An increase in the resistance causes a decease in the velocity. The drop in the average kinetic energy leads to a declining trend in the THF temperature.
Eur. Phys. J. Plus (2017) 132: 550
Page 13 of 15
Fig. 14. Effect of Sc on C(η).
Fig. 15. Effect of k2 on C(η).
Figures 14–16 show the influence of Sc, k2 and Rc on the THF concentration. In detail, fig. 14 is plotted to study the impact of Sc on C(η). The mass diffusivity varies inversely via Sc so larger values of Sc bring thinning in the concentration boundary layer; as a result C(η) decreases. Figure 15 shows that with an increase in k2 , the concentration distribution decreases. This fact is similar to the relation of k1 against T (η). Figure 16 is drawn to examine the impact of Rc on C(η). It is clearly seen that for large values of Rc the concentration profile shows a decreasing trend.
Page 14 of 15
Eur. Phys. J. Plus (2017) 132: 550
Fig. 16. Effect of Rc on C(η).
5 Concluding remarks The key findings of present analysis are itemized as follows: 1) THF velocity increases via λm and K while it shows an inverse relation against k1 , α, n and λ. 2) Temperature of THF is an increasing function of K, DH , and α but it reflects an opposite trend towards k1 , Pr and λ. 3) Concentration of THF decreases for positive values of k2 , Rc and Sc. 4) In absolute sense, the SFC increases for greater values of K and Pr, while an inverse relation is observed in the case of n and λ. 5) The HTR enhances via Pr and K while an inverse relation is seen for both k1 and λ. 6) The MTR increases for large values of K, Pr and Sc. 7) Streamline patterns of the THF flow due to stretching cylindrical surface (see figs. 1(a)–(d)) increase for positive values of K.
Conflict of interest The authors declare that they have no conflict of interest.
References 1. 2. 3. 4. 5. 6.
K.T. Yang, J.L. Novotny, Y.S. Cheng, Int. J. Heat Mass Transfer 15, 1097 (1972). Y. Jaluria, B. Gebhart, J. Fluid Mech. 66, 593 (1974). C.C. Chen, R. Eichhorn, ASME J. Heat Transf. 98, 446 (1976). A. Ishak, R. Nazar, I. Pop, Int. J. Heat Mass Transfer 51, 3693 (2008). R. Ellahi, S. Aziz, A. Zeeshan, J. Porous Media 16, 205 (2013). M. Naseer, M.Y. Malik, S. Nadeem, A. Rehman, Alex. Eng. J. 53, 747 (2014).
Eur. Phys. J. Plus (2017) 132: 550 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.
Page 15 of 15
N.S. Akbar, M. Raza, R. Ellahi, Eur. Phys. J. Plus 129, 185 (2014). S. Nadeem, H. Sadaf, N.S. Akbar, Alex. Eng. J. 54, 843 (2015). S.U. Rahman, R. Ellahi, S. Nadeem, Q.Z. Zia, J. Mol. Liq. 218, 484 (2016). R. Ellahi, E. Shivanian, S. Abbasbandy, T. Hayat, Int. J. Numer. Methods Heat Fluid Flow 26, 1433 (2016). T. Hayat, T. Abbas, M. Ayub, M. Farooq, A. Alsaedi, J. Mol. Liq. 222, 854 (2016). A.A. Khan, S. Muhammad, R. Ellahi, Q.Z. Zia, J. Mag. 21, 273 (2016). M. Mamourian, K.M. Shirvan, R. Ellahi, A.B. Rahimi, Int. J. Heat Mass Transfer 102, 544 (2016). S. Bilal, K.U. Rehman, M.Y. Malik, Results Phys. 7, 690 (2017). M. Awais, K.U. Rehman, M.Y. Malik, A. Hussain, T. Salahuddin, Eur. Phys. J. Plus 132, 392 (2017). N.S. Akbar, S. Nadeem, R.U. Haq, Z.H. Khan, Indian J. Phys. 87, 1121 (2013). M.A. Abbas, Y.Q. Bai, M.M. Bhatti, M.M. Rashidi, Alex. Eng. J. 55, 653 (2016). K.U. Rehman, A.A. Malik, M.Y. Malik, N.U. Saba, Case Stud. Therm. Eng. 10, 244 (2017). K.T. Yang, J.L. Novotny, Y.S. Cheng, Int. J. Heat Mass Transfer 15, 1097 (1972). C.C. Chen, R. Eichhorn, ASME J. Heat Transf. 98, 446 (1976). P.A. Lakshmi Narayana, P.V.S.N. Murthy, Int. J. Fluid Mech. Res. 36, 524 (2007). A. Ishak, R. Nazar, I. Pop, Int. J. Heat Mass Transfer 51, 3693 (2008). C.Y. Cheng, Int. Commun. Heat Mass 36, 351 (2009). W. Ibrahim, O.D. Makinde, Comput. Fluids 86, 433 (2013). D. Srinivasacharya, M. Upendar, J. Egypt. Math. Soc. 21, 370 (2013). K.U. Rehman, M.Y. Malik, T. Salahuddin, M. Naseer, AIP Adv. 7, 075112 (2016). T. Mahmood, J. Ahmed, A. Shahzad, R. Ali, Z. Iqbal, Bulgarian Chem. Commun. 48, 506 (2016). M.Y. Malik, S. Bilal, M. Bibi, Results Phys. 7, 482 (2017). K.U. Rehman, A.A. Khan, M.Y. Malik, A. Hussain, Eur. Phys. J. Plus 132, 389 (2017). K.U. Rehman, M.Y. Malik, O.D. Makinde, A.A. Malik, Eur. Phys. J. Plus 132, 427 (2017).
