Unique Outcomes of Internal Heat Generation and Thermal Deposition on Viscous Dissipative Transport of Viscoplastic Fluid over a Riga-Plate

Commun. Theor. Phys. 69 2018 68 76 Vol. 69, No. 1, January 1, 2018 Unique Outcomes of Internal Heat Generation and Thermal Deposition on Viscous Dissipative Transport of Viscoplastic Fluid over a Riga-Plate Z. Iqbal, Ehtsham Azhar, Zaffar Mehmood, and E. N. Mara Department of Mathematics, Faculty of Sciences, HITEC University Taxila, Pakistan Received October 9, 2017; revised manuscript received November 20, 2017 Abstract Boundary layer stagnation point flow of Casson fluid over a Riga plate of variable thickness is investigated in present article. Riga plate is an electromagnetic actuator consists of enduring magnets and gyrated aligned array of alternating electrodes mounted on a plane surface. Physical problem is modeled and simplified under appropriate transformations. Effects of thermal radiation and viscous dissipation are incorporated. These differential equations are solved by Keller Box Scheme using MATLAB. Comparison is given with shooting techniques along with Range- Kutta Fehlberg method of order 5. Graphical and tabulated analysis is drawn. The results reveal that Eckert number, radiation and fluid parameters enhance temperature whereas they contribute in lowering rate of heat transfer. The numerical outcomes of present analysis depicts that Keller Box Method is capable and consistent to solve proposed nonlinear problem with high accuracy. PACS numbers: 44.25.+f, 47.10.ad, 47.50.-d DOI: 10.1088/0253-6102/69/1/68 Key words: Riga-plate, Keller Box method, Casson fluid, variable thickness, stagnation point flow, internal heat generation 1 Introduction Radiation effects on flow and heat transfer is very important in the scaffold of space technology and processes involving high temperature. Visible light and infrared light emitted by an incandescent light bulb, the infrared radiation emitted by animals that is detectable with an infrared camera, and the cosmic microwave background radiation are some examples of thermal radiation. In manufacturing industries radiative heat transfer flow is very significant for depiction of reliable equipment, nuclear power plants, gas turbines and different propulsion devices for satellites, missiles, aircraft and space vehicles. Viscous and non-newtonian fluids through various aspects and thermal radiation have been examined by many researchers. Pramanik [1] discussed numerical solutions for steady boundary layer flow and heat transfer for a Casson fluid over an exponentially permeable stretching surface in the presence of thermal radiation. He concluded that temperature as well as thermal boundary layer enhances due to thermal radiation. Hydromagnetic mixed convection heat and mass transfer flow of an incompressible Boussinesq fluid past a vertical porous plate with constant heat flux and thermal radiation is analyzed by Makinde. [2] The effects of thermal radiation over a stretching sheet under different flow geometries have been reported by several researchers. [3 5] Impact of viscous dissipation is usually neglected but its presence become noteworthy when liquid viscosity is high. It changes the temperature distributions by playing a role like an energy source, which leads to affect heat Corresponding author, E-mail: 12-phd-mt-007@hitecuni.edu.pk c 2018 Chinese Physical Society and IOP Publishing Ltd transfer rates. Viscous dissipation is of interest for several applications such as notable temperature rises are witnessed in polymer processing, inection molding or extrusion at high rates. The effect of viscous dissipation was initially considered by Brickman. [6] According to him temperature distribution of Newtonian fluid in straight circular tube and interprets result that the effects were produced in the close region. Chand et al. [7] studied effects of viscous dissipation and radiation on unsteady flow of electrically conducting fluid through a porous stretching surface. He examined that Eckert number boost temperature profile. Hayat et al. [8] presented MHD stagnation point flow of Jeffrey fluid by a radially stretching surface with viscous dissipation and Joule heating. Hayat et al., [9] Barik and Dash [10] also studied the flow of Newtonian fluid under the combined effects of thermal radiation and viscous dissipation. Some notable recent articles are cited in Refs. [11 19] and many therein. In nature, some non- Newtonian fluids behave like elastic solid that is, no flow occurs with small shear stress. Casson fluid is one of such fluids. The examples of Casson fluid are of the type as follows: elly, tomato sauce, honey, soup and concentrated fruit uices. Human blood can also be treated as Casson fluid. In 1959, Casson introduced this fluid model for the prediction of flow behavior of pigment-oil suspensions. [20] Animasaun [21] has studied MHD dissipative Casson fluid flow with suction and n-th order of chemical reaction. He depicts that temperature and concentration are decreasing function of Casson fluid parameter. Nadeem et al. [22] put his contribution on Casson fluid past a linearly stretching http://www.iopscience.iop.org/ctp http://ctp.itp.ac.cn

No. 1 Communications in Theoretical Physics 69 sheet with convective boundary condition. Casson fluid model under different circumstances like thermal radiation, slip condition etc. is studied by many researchers see Refs. [23 25]. All of the above studies on Casson fluid had reported the boundary layer flow over stretching sheet in the absence of electrically conducting fluids over a Riga plate. This plate is used to produce magnetic field due to which Lorentz force is generated which controls the fluid flow. Riga plate is electromagnetic plate consisting of periodic pairs of electrodes proposed by Gailitis and Lielausis. [17] After identifying Lorentz force theoretically [26] and experimentally [27] as a resourceful mediator to reduce the skin friction; few recent studies of laminar fluid flow over a Riga plate has been reported. Pantokratoras and Magyari [28] explained EMHD free-convection boundary-layer flow from a Riga-plate. Later on, aiding and opposing mixed convection flows over the Rigaplate was investigated by Magyari and Pantokratoras. [29] Pantokratoras [30] considered Sakiadis and Blasius flow for Riga-plate by using finite difference method. After reviewing above literature, it predicts that effect of viscous dissipation and thermal radiation for Casson fluid over an electrically conducted Riga plate incorporating with stagnation point has not been considered. Keeping this in mind, in present article we have studied the influences of viscous dissipation on boundary layer flow of a Casson fluid over a Riga plate in presence of activation energy near a stagnation point. Governing nonlinear ordinary differential equations are then solved by using finite difference approach named as Keller-Box scheme. To validate proposed scheme results comparison is provided with well establish, stable numerical procedure. 2 Problem Formulation and Governing Equations Present article focuses on study steady state, incompressible, two-dimensional flow of an electrically conducting Casson fluid over a stretchable Riga-plate. The plate is considered to have variable thickness δ, such that thickness is small relative to plate length. It is worth mentioning here that an alternate arry of electrodes and permanent magnets fixed on a plane surface constructs a Riga-plate see Fig. 1. In addition, heat transfer phenomenon is examined under the influence of viscous dissipation and thermal radiation. Temperature at the wall T w and ambient temperature T are taken to be constant. Moreover, free stream velocity is taken to be U e x = U x + b n and sheet is assumed to be stretched with velocity U w x = U 0 x + b n, where U, U 0 > 0 are dimensional constants. Schematic diagram for both stretching and boundary-layer flows is presented in Fig. 2. Fig. 2 Fig. 1 Riga plate. Physical flow diagram. Now, governing boundary layer equations see Refs. [23], [30], and [31] of stagnation-point flow and heat transfer with internal heat generation of Casson fluid over Riga-plate are: u x + v y = 0, u u x + v u y = v 1 + 1 u T x + v T y = with associated boundary conditions 2 u y 2 + U du e e dx + π 0M 0 exp π 8ρ a y, 2 k 2 T ρc p y 2 + Q 0 T T + v 1 + 1 ρc p C p u = U w x = U 0 x + b m, v = 0, T = T w, at y = δx + b 1 m/2, 1 u 2 16σ T + 3 2 T y 3k ρc p y 2, 3 u U e x = U x + b m, T T, as y, 4 here u and v are velocity components along flow x-direction and normal to flow y-directions, ν denotes kinematic

70 Communications in Theoretical Physics Vol. 69 viscosity, is Casson fluid parameter, 0 stands for applied current density in electrodes, M 0 is magnetization of permanents magnets mounted on Riga plate surface with a denotes width of magnets between electrodes, k denotes fluid thermal conductivity, k is mean absorption coefficient, σ stands for Stefan-Boltzmann constant, C p is specific heat, ρ represents fluid density and Q 0 is heat generation/absorption parameter. We introduce similarity transformation see Ref. [23]. 2 m + 1 ψ = νu U0 0x + b m+1 x + b F η, η = y m 1, u = U 0 x + b m F η, 2 ν 2 v = [ νu 0x + b m 1 2 Equation 1 is identically satisfied and Eqs. 2 4 yield 1 + 1 F + F F F 2 + F η + η m 1 F η 2 A + 1 1 + 4 Θ + F Θ + 2 1 P r 3Nr λθ + Ec + 1 ], Θη = T T T w T. 5 Q exp 1η = 0, 6 F 2 = 0, 7 F α = α 1 m 1 + m, F α = 1, Θα = 1, F A, Θ 0, 8 here differentiation with respect to η is denoted by prime and α = δ[m+1u 0 /2ν] 1/2. Letting F η = fη α = fξ we have 1 + 1 f + ff f 2 + A + Q exp 1ξ + α = 0, 9 1 1 + 4 θ + fθ + 2 1 P r 3Nr λθ + Ec + 1 f 2 = 0, 10 whereas corresponding boundary conditions become f0 = α 1 m 1 + m, f 0 = 1, θ0 = 1, f A, θ 0. 11 Furthermore, Prandtl number P r, stretching rate ratio parameter A, heat source parameter λ, modified Hartmann number Q, dimensionless parameter 1, Eckert number Ec and radiation parameter Nr are defined as A = U U 0, λ = Q 0x + b ρc p U w, 1 = π/a /2U0 x + b m 1 /ν, P r = µc p k, Q = π 0M 0 x + b 8ρUw 2, Nr = 4σ T 3 k k, Ec = U 2 0 x + b ρt w T. 12 Significant physical quantities such as skin friction coefficient C f and Nusselt number Nu x are defined as C f = τ w ρu w 2, Nu xq w x = k f T w T, 13 in which q w is heat flux at wall and τ w is wall shear stress are expressed as τ w = µ 1 + 1 u y, y=δx+b 1 m/2 T q w = k y y=δx+b1 m/2, 14 after applying similarity variables above expressions take the form: Re 1/2 x C f = Nu x Re 1/2 x 1 + 1 f 0, 2 = θ 0, 15 2 where Re x = U 0 x + b m+1 /ν denotes local Reynolds number. 3 Computational Scheme and Accuracy In order to solve nonlinear system 9 and 10 subect to boundary conditions 11, Keller box technique has been applied by incorporating new independent variables ux, ξ, vx, ξ and tx, ξ such that f = u, u = v and θ = t, so that Eqs. 9 10 simplify to first order ODE i.e. 1 + 1 v + fv 1 + 4 3Nr u2 + A 2 + 2 Q e 1ξ+α = 0, 16 t + P rft + 2P rλ θ + EcP r 1 + 1 v 2 = 0. 17 Moreover, domain discretize in x-η plane and according to mesh points, net points can be expressed as x 0 = 0, x i = x i 1 + k i, i = 1, 2,..., J, ξ 0 = 0, ξ = ξ 1 + h, = 1, 2,..., J, where k i and h are the x and ξ-spacing. Here i and

No. 1 Communications in Theoretical Physics 71 are ust sequences of numbers that indicate the coordinate location. Using central difference formulation at midpoint x i, ξ 1/2 we arrived at f i f i 1 h 2 ui + u i 1 = 0, 18 Central differencing of Eqs. 16 and 17 at point x i 1/2, ξ 1/2 is visualized as 1 + 1 v i v i 1 + hf 1/2 i vi 1/2 h 1 + 4 3Nr in which R 1/2 = 1+ 1 T 1/2 = 1 + 4 t i t i 1 + hp rf i 1/2 ti 1/2 + h 2P rλ v i 1 3Nr with boundary conditions v i 1 i 1 1 hf 1/2 vi 1 1/2 +h u i 1 1/2 2 h t i 1 t i 1 i 1 1 hp rf 1/2 ti 1 u i u i 1 h 2 vi + v i 1 = 0, 19 θ i θ i 1 h 2 ti + t i 1 = 0. 20 u i 1/2 2 = R 1/2, 21 2P rλ 1/2 h θ 1/2 i + hecp r 1 + 1 v 1/2 i 2 = T 1/2, 22 2 A 2 h θ i 1 1/2 hecp r 1 + 1 Q e 1ξi 1 1/2 +α, 23 v i 1 1/2 2, 24 f i 0 = α 1 m 1 + m, ui 0 = 1, u i J = A, θ i 0 = 1, θ i J = 0. 25 Linearization of Eqs. 20 and 21 is performed by Newton s method and can be expressed as f i+1 = f i + δf i, u i+1 = u i + δu i, v i+1 = v i + δv i, θ i+1 = θ i + δθ i, t i+1 = t i + δt i. 26 Incorporating above expressions in Eqs. 21 22 and then dropping quadratic and higher order terms in δf i, δu i,, δθ i, δti, this procedure yields following tri-diagonal system δv i [A 1 ] [C 2 ] [B 2 ] [A 2 ] [C 2 ]...... [B J 1 ] [A J 1 ] [C J 1 ] [B J ] [A J ] [δ 1 ] [δ 2 ] [δ J 1 ] [δ J ] = [r 1 ] [r 2 ] [r J 1 ] [r J ], 27 where the elements are 0 0 1 0 0 h /2 0 1 0 0 h 1 /2 0 0 h 1 /2 0 1 0 0 h /2 0 [A 1 ]= 0 h 1 /2 0 0 h 1 /2, [A J ]= 0 1 0 0 h /2, 2 J, a 2 1 0 a 5 1 a 1 1 0 a 4 0 a 5 a 1 0 b 2 1 b 6 1 b 3 1 b 1 1 b 5 1 0 b 8 b 3 b 1 b 5 0 0 1 0 0 h /2 0 0 0 0 0 0 0 h /2 0 1 0 0 0 0 [B J ] = 0 0 0 0 h /2, 2 J, [C J ] = 0 1 0 0 0, 2 J. 28 0 0 a 6 a 2 0 a 3 0 0 0 0 0 0 b 4 b 2 b 6 0 b 7 0 0 0 Block tri-diagonal matrix is solved by means of LU factorization. The value of δ is calculated repeatedly until δv i 0 ϵ 1. 29 is attained where ϵ 1 = 10 6 is small prescribed value. 4 Theoretical Results Description This section provides detailed study on graphical aspects of velocity f η and temperature θη profiles for noteworthy parameters. Figure 3 demonstrates behavior of Casson fluid velocity for stretching parameter A. It is revealed that an inverted boundary layer appeared for A < 1 and A > 1. Moreover, thickness of boundary layer decreases with increase in A < 1. Internal forces like adhesive forces contribute in lowering velocity of fluid. Wall thickness maximizes these forces acting on fluid and hence velocity decreases. The effect of wall thickness parameter α on velocity profile is displayed in Fig. 4. From this diagram it is depicted that velocity falls as wall thickness parameter α increases. Moreover, viscous boundary layer thickness has no remarkable affects with an increas-

72 Communications in Theoretical Physics Vol. 69 ing value of α however, thickness decays for higher values of α. Fig. 6 Variation of 1 on f η. Fig. 3 Variation of A on f η. Fig. 7 Variation of m on f η. Fig. 4 Variation of α on f η. Fig. 8 Variation of Q on f η. Fig. 5 Variation of on f η. Figure 5 demonstrates variation of velocity f η for various values of fluid parameter. Flow velocity depends directly on yield stress. Fluid parameter adds in reducing yield stress for its greater values. Due to this fact velocity f η and associated momentum boundary layer thickness diminishes significantly with rise of fluid parameter. Physical analysis of velocity f η profile for dimensionless parameter 1 is given in Fig. 6. It is obvious that fluid velocity declines with mount in value of 1. It is also viewed that viscous boundary layer descends with ascending values of 1. Figure 7 exhibits behavior of power index m on velocity f η distribution. This analysis reveals that with an increase in value of m, f η drops near sheet and opposite trend is observed far from sheet. The fact

No. 1 Communications in Theoretical Physics 73 is that power index m contributes in varying stretching velocity positively and hence fluid behavior is ultimately changed, so for growing values of m, flow velocity at and near sheet boosts. Electric field strength contributes in varying magnetic behavior in flow field. This results in modifying flow pattern. On increasing external electric force field, Hartmann/magnetic number Q lessens which results in enhancing velocity distribution. Impact of modified Hartmann number Q on velocity f η is shown in Fig. 8. Heat transfer depends on thickness of wall. The thicker wall results in less heat transfer in it. heat source parameter λ. By definition, λ is a source to provide heat to the system. Fig. 11 Variation of Ec on θη. Fig. 9 Variation of α on θη. Fig. 12 Variation of λ on θη. Fig. 10 Variation of on θη. Figure 9 portraits temperature θη with variation of wall thickness parameter α. Temperature distribution is almost self-governing of Casson parameter in several cases. For higher values of, it results in raise temperature of fluid. Plot of temperature distribution θη for fluid parameter is displayed in Fig. 10. Eckert number is an illustration of heat dissipation in flow. Kinetic energy of flow is increased for increasing values of Eckert number and therefore, temperature rises. The impact of Eckert number Ec on temperature profile θη is sketched in Fig. 11. Temperature θη is an increasing function of Fig. 13 Variation of m on θη. It results in increasing thermal energy of fluid, so temperature enhances with greater values of heat source parameter λ. This fact is analyzed through Fig. 12. Influence of power index m on temperature profile θη is discussed in Fig. 13. On increasing stretching velocity, kinetic energy of fluid grows significantly which enhances

74 Communications in Theoretical Physics Vol. 69 heat transfer rate. Because of this, temperature and associated thermal boundary layer grows up for higher values of m. associated to thermal diffusivity of fluid and accordingly fall downs temperature distribution. P r contributes in lowering temperature θη of fluid. Figure 15 is a graphical representation of this fact. Figures 16 and 17 represent the streamlines for various values of 1 and show the Casson fluid flow pattern. Fig. 14 Variation of Nr on θη. Fig. 16 Stream pattern when 1 = 0.0. Fig. 15 Variation of P r on θη. Radiation parameter amplifies fluid thermal capability. Hence temperature rises with thermal radiation parameter Nr as shown in Fig. 14. Prandtl number P r is inversely Fig. 17 Stream pattern when 1 = 0.2. Table 1 Tabulated skin friction values for distinct values of several parameters. m Q 1 α A Re 1/2 x C f Shooting algorithm Finite difference Scheme 0.1 0.5 0.1 1.0 1.0 0.5 1.6625 1.6668 0.2 1.2330 1.2370 0.1 1.0 2.1420 2.1456 1.5 2.5294 2.5325 0.5 0.4 1.4047 1.4217 0.7 1.1475 1.1771 0.5 2.0 1.5040 1.5278 3.0 1.5771 1.6017 1.0 1.5 1.3582 1.3795 2.0 1.3982 1.4196 1.0 1.0 0.4190 0.3983 1.5 2.7327 2.7128

No. 1 Communications in Theoretical Physics 75 Further Table 1 presents numerical results of rate of shear stress at surface for several influential parameters. We suggest from this tabulated outcomes that magnitude of skin friction coefficient increases with an increase in m, 1, and α. It can be observed that fluid parameter and Hartmann numbers result in lessening stress rate at surface. Table 2 shows heat transfer rate with respect to pertinent parameters. It is obvious from this table that P r, α and Q related directly with Nusselt number Nu x and inversely related to index m, fluid parameter, stretching parameter A radiation parameter Nr and Eckert number Ec. Table 2 Numerical values of local heat flux for notable parameters. m Q 1 α λ P r Nr Ec A Nusselt number Shooting algorithm Finite difference Scheme 0.1 0.5 0.5 1.0 1.0 0.1 2.0 2.0 0.2 0.5 0.5084 0.5080 0.2 0.5081 0.5075 0.1 1.0 0.4684 0.4688 1.5 0.4510 0.4607 0.5 1.0 0.5173 0.5166 1.5 0.5248 0.5239 0.5 2.0 0.5019 0.5017 3.0 0.5001 0.4999 1.0 1.5 0.5642 0.5638 2.0 0.6217 0.6213 1.0 0.3 0.3907 0.3902 0.5 0.2443 0.2436 0.1 3.0 0.6546 0.6542 4.0 0.7857 0.7853 2.0 4.0 0.3638 0.3635 6.0 0.2940 0.2938 2.0 0.5 0.4609 0.4601 1.0 0.3817 0.3803 0.2 1.0 0.5719 0.5719 1.5 0.5492 0.5497 5 Summary and Novelty of Article Boundary layer stagnation point Casson flow problem influenced by thermal radiation and viscous dissipation is investigated over a Riga plate of variable thickness. Governing physical problem is tackled numerically using Keller Box Method. Moreover, a comparative numerical analysis is drawn with shooting method. This analysis will help in developing more understanding to boundary layer flow over an electromagnetic plate for viscoelastic fluids. Such type of problems are encounter in electronic and electric devices manufacturing. Graphical and numerical results are shown for various pertinent parameters on velocity and temperature distribution. It is concluded that velocity f η, skin friction C f and rate of heat flux Nu x is a decreasing function of Casson rheological parameter and distribution of temperature θη grows up with. Further, Eckert number contributes in enhancing temperature of fluid. Modified Hartmann number Q results in increasing fluid velocity. Radiation parameter Nr and index numbers m enhance temperature. Rise in Eckert number leads to raise temperature and reduce Nusselt number at the sheet. Finally, comparative analysis reveals that Keller Box Method converges more rapid as compare to shooting method. Hence proposed method is reliable and efficient to solve nonlinear differential equations. References [1] S. Pramanik, Ain Shams Engg. J. 5 2014 205. [2] O. D. Makinde, Chem. Engg. Commun. 198 2011 590. [3] A. Y. Bakier, Int. Commun. Heat Mass Transf. 28 2001 119. [4] I. Zahmatkesh, Emir. J. Engg. Res. 12 2007 47. [5] S. Nadeem, S. Zaheer, and T. Fang, Num. Algo. 57 2011 187. [6] H. C. Brinkman, Appl. Sci. Res. A 2 1951 120. [7] G. Chand and R. N. Jat, Therm. Energy Power Engg. 3 2014 266. [8] T. Hayat, M. Waqas, S. A. Shehzad, and A. Alsaedi, J. Hydrol. Hydromech. 63 2015 311. [9] T. Hayat, S. Asad, and A. Alsaedi, Appl. Math. Mech. 35 2014 717.

76 Communications in Theoretical Physics Vol. 69 [10] R. N. Barik and G. C. Dash, Appl. Math. Comp. 226 2014 423. [11] K. Maqbool, A. Sohail, N. Manzoor, and R. Ellahi, Commun. Theor. Phys. 66 2016 547. [12] M. Nawaz, A. Zeeshan, R. Ellahi, et al., Int. J. Numer. Meth. Heat Fluid Flow 25 2015 665. [13] A. Zeeshan, A. Maeed, and R. Ellahi, J. Mol. Liq. 215 2016 549. [14] A. Maeed, A. Zeeshan, and R. Ellahi, J. Mol. Liq. 223 2016 528. [15] T. Hayat, R. Saad, T. Muhammad, et al., Results Phys. 7 2017 535. [16] T. Hayat, R. Saad, R. Ellahi, et al., J. Mol. Liq. 240 2017 209. [17] R. Ellahi, M. Hassan, and A. Zeeshan, J. Mech. Adv. Mat. Struct. 24 2017 1231. [18] T. Hayat, R. S. Saif, R. Ellahi, et al., Results Phys. 7 2017 2601. [19] R. S. Saif, T. Hayat, R. Ellahi, et al., Results Phys. 7 2017 2821. [20] N. Casson, A Flow Equation for the Pigment Oil Suspensions of the Printing Ink Type, in Rheology of Disperse Systems, Pergamon, New York 1959 84 102. [21] I. Animasaun, J. Nigerian Math. Soc. 34 2015 11. [22] S. Nadeem, R. Mehmood, and N. Akbar, Int. J. Therm. Sci. 78 2014 90. [23] N. Akbar and Z. Khan, J. Magn. Magn. Mater. 378 2015 320. [24] K. Ramesh and M. Devakar, Ain Shams Engg. J. 6 2015 967. [25] S. Pramanik, Ain Shams Engg. J. 5 2014 205. [26] A. Gailitis and O. Lielausis, Appl. Magnetohydrodyn. 12 1961 143. [27] T. W. Berger, J. Kim, C. Lee, and J. Lim, Phys. Fluids 12 2000 631. [28] J. Pang and K. S. Choi, Phys. Fluids 16 2004 35. [29] A. Pantokratoras and E. Magyari, J. Engg. Math. 64 2009 303. [30] E. Magyari and A. Pantokratoras, Commun. Nonlinear Sci. Numer. Simul. 16 2011 3158. [31] A. Pantokratoras, Prog. Comp. Fluid Dyn. 11 2011 329. [32] T. Hayat, M. I. Khan, M. Farooq, et al., Int. J. Heat Mass Transf. 99 2016 702.