Flow of Oldroyd-B fluid over a rotating disk with Cattaneo–Christov theory for heat and mass fluxes

Flow of Oldroyd-B fluid over a rotating disk with Cattaneo-Christov theory for heat and mass fluxes

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Flow of Oldroyd-B fluid over a rotating disk with Cattaneo-Christov theory for heat and mass fluxes Abdul Hafeez, Masood Khan, Jawad Ahmed PII: DOI: Reference:

S0169-2607(20)30080-8 https://doi.org/10.1016/j.cmpb.2020.105374 COMM 105374

To appear in:

Computer Methods and Programs in Biomedicine

Received date: Revised date: Accepted date:

10 January 2020 25 January 2020 27 January 2020

Please cite this article as: Abdul Hafeez, Masood Khan, Jawad Ahmed, Flow of Oldroyd-B fluid over a rotating disk with Cattaneo-Christov theory for heat and mass fluxes, Computer Methods and Programs in Biomedicine (2020), doi: https://doi.org/10.1016/j.cmpb.2020.105374

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Highlights • Three-dimensional stagnation point flow of Oldroyd-B fluid is investigated. • The flow is generated by a stretching and rotating disk. • The Cattaneo-Christov theory is used to perform heat and mass transfer mechanism. • The BVP Midrich technique is utilized in Maple software to acquire numerical solution.

1

Flow of Oldroyd-B fluid over a rotating disk with Cattaneo-Christov theory for heat and mass fluxes Abdul Hafeeza,1 , Masood Khana , Jawad Ahmeda,b a

b

Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan

Department of Basic Sciences, University of Engineering and Technology, Taxila 47050, Pakistan Abstract: Background: In general, heat transfer describes the flow of heat (thermal energy) due to

temperature differences. The phenomenon of transfer of heat from one body to another body or in same type bodies occurs because of the temperature difference. As heat transfer is a natural phenomenon, therefore much attentions have developed among the researchers to observe the heat transfer mechanism in the systems. To examine the heat and mass transport mechanism in the fluid, the Cattaneo-Christov theory is adopted instead of classical Fourier’s and Fick’s laws in the current study. Further, the stagnation point flow of Oldroyd-B fluid is explored. Here the flow is generated by a rotation of the disk. Additionally, the porous disk is considered here. The von Karman similarity variables are used to transform the partial differential equations (PDEs) into ordinary differential equations (ODEs).

Method: To handle the system of non-linear equations, we use a built-in technique (BVP Midrich) in Maple software to acquire numerical solution.

Results: The solution of the governing system of equations are presented graphically in the form of velocity fields, temperature and concentration distributions. It is noted that the higher values of thermal and solutal relaxation time parameters reduce the thermal and concentration distributions, respectively. Moreover, the comparison tables are presented to assure the validity of 1

Corresponding author: Electronic mail: [email protected] (Abdul Hafeez-Kakar)

2

our numerical results with the past outcomes.

Keywords: Rotating disk; Stagnation point flow; Cattaneo-Christov theory; Oldroyd-B fluid; Numerical solution.

1

Introduction

Phenomenon of transfer of heat from one body to another body or in same type bodies occurs because of the temperature difference. As heat transfer is a natural phenomenon, therefore much attentions have developed amoung the researchers to observe the heat transfer mechanism in the systems. Initially, the law of heat conduction was proposed in 1822 by Fourier [1]. But, this law is not enough to be fully articulated the characteristics of heat transfer phenomena and leads a parabolic type solution in the sense that any initial disruption immediately is detected in all part of the entire material. Practically, there are no such artifacts and materials that obey the Fourier’s law. To overcome the flaws of Fourier’s law, Cattaneo [2] proposed a model for heat flux by adding the thermal relaxation time in classical Fourier’s law which is also named by modified Fourier’s law. After that, Christov [3] improved the law by introducing the Oldroyd upper convected derivative with thermal relaxation time to attain the material invariant formulation. Furthermore, the structural stability and uniqueness for Cattaneo-Christov equation was discussed by Ciarletta et al. [4]. The Cattaneo-Christov heat flux model for viscoelastic fluid was analyzed by Han et al. [5]. Recently, Khan et al. [6] studied the swirling flow of Maxwell fluid with CattaneoChristov heat flux model and obtained the numerical solution of the problem. Moreover, the stagnation point flow with modified Fourier’s law by a stretching sheet is described by Sulti [7]. Some working based on Cattaneo-Christov theory can be reffered via Ref. [8 − 12]. Recent literature works have paid much attention to the study of magnetohydrodynamic 3

(MHD) fluid flow induced by rotating disk. This is because of the fact that, the fluid flows over a rotating disk is not just for theoretical interest, but in many cases, they are also of practical importance such as, rotating machinery, power generating systems, computer storage devices, medical equipment and specially aerodynamic applications. The liquid flows over a rotating disk was initially proposed by Karman [13] and the approximate solution of the problem is obtained using the integral method. It was then numerically and asymptotically investigated by Cochran [14] and Benton [15]. Moreover, many researchers analyzed the Karman’s renowned study with many physical aspects and features. Rashidi et al. [16] studied the fluid flow over a rotating disk by considering the entropy generation. The study of Bingham–Papanastasiou fluid flow by a rotating disk was explored by Khan and Sultan [17]. They used the von Karman similarity variables to transform the problem into ordinary differential equations and solved numerically. Additionally, the study of transient thin film flow of MHD Maxwell fluid over a rotating disk is reported by Ahmed et al. [18]. In this study, they used bvp4c technique in MATLAB to acquire the numerical solution of the problem. The latest articles designed on fluid flow by a rotating disk can be seen via Refs. [19 − 25]. The liturature review on the flow problems due to rotating disk reveal that Oldroyd-B fluid model in the regime of rotating disk geometry with Cattaneo-Christov heat and mass fluxes has not been discussed thoroughly. Therefore, this article focuses on the swirling flow of Oldroyd-B fluid caused by a rotating disk. The flow is generated by a stretching and rotating of the disk. The Cattaneo-Christov double diffusion theory is used instead of classical Fourier’s and Fick’s laws. To handle the governing system of equations, we first transform PDEs into non-dimensional ODEs by using von Karman similarity variables. The numerical solutions are obtained through BVP Midrich technique in Maple software. A 4

detail survey of the governing problem is presented through graphical structure in the form of velocities, temperature and concentration distributions.

2

Mathematical formulation

We assume steady, incompressible axisymmetric flow of Oldroyd-B fluid. The corresponding heat and mass transport features are performed by using Cattaneo-Christov double-diffusion theory instead of classical Fourier’s and Fick’s laws. The fluid motion is generated by stretching as well as rotation of the disk. Here the flow is assumed in the upper plane that is at z > 0. The disk rotates with the uniform angular velocity Ω. Furthermore, the disk is assumed to be porous with the mass flux velocity w0 (w0 < 0 for suction, w0 > 0 for injection). A physical geometry of the problem is given in Fig. 1. Additionally, the disk is kept at constant temperature (Tw ) and concentration (Cw ). The far away temperature and concentration from the disk are T∞ and C∞ , respectively. The nonlinear equations for the governing problem are [22 − 25]

5

∂u u ∂w + + = 0, ∂r r ∂z

(1)

  ∂u v 2 ∂u ∂ 2u ∂u ∂ue σ 2 u − +w = ν 2 + ue − B0 u + λ1 w − ue ∂r r ∂z ∂z ∂r ρ ∂z   2 2 2uv ∂v 2vw ∂v uv 2 v 2 ∂u ∂ 2u 2∂ u 2∂ u −λ1 u − − + 2 + +w + 2uw ∂r2 ∂z 2 ∂r∂z r ∂r r ∂z r r ∂r " #  2 1 ∂u ∂u ∂ 2 w ∂ 3u ∂ 3 u ∂u ∂ 2 u ∂u ∂ 2 u +νλ2 − −2 +u , +w 3 − − r ∂z ∂z ∂z 2 ∂z ∂r ∂z 2 ∂z ∂r∂z ∂r∂z 2

(2)

  ∂v uv ∂v ∂ 2v σ 2 ∂v u + +w = ν 2 − B0 v + λ1 w ∂r r ∂z ∂z ρ ∂z   2 2 2 ∂ v uv ∂u vw ∂u u2 v v 3 v 2 ∂v 2∂ v 2∂ v +w + 2uw +2 +2 −2 2 − 2 + −λ1 u ∂r2 ∂z 2 ∂r∂z r ∂r r ∂z r r r ∂r   3 3 2 2 2 ∂v ∂ w ∂ v 1 ∂u ∂v ∂v ∂ u v ∂ u ∂v ∂ 2 u u ∂ 2v ∂ v +νλ2 u −2 +w 3 − − + − − (3) , ∂r∂z 2 ∂z ∂z 2 ∂z r ∂z ∂z ∂r ∂z 2 r ∂z 2 ∂z ∂r∂z r ∂z 2  2  ∂T ∂ T ∂T +w =α u ∂r ∂z ∂z 2   2 2 ∂ 2T ∂w ∂T ∂u ∂T ∂u ∂T ∂w ∂T 2∂ T 2∂ T −ε0 u + 2uw +w +u +u +w +w ,(4) ∂r2 ∂r∂z ∂z 2 ∂r ∂z ∂r ∂r ∂z ∂r ∂z ∂z ∂C ∂ 2C ∂C +w = DB 2 ∂r ∂z ∂z   2 2 ∂ 2C ∂w ∂C ∂u ∂C ∂u ∂C ∂w ∂C 2∂ C 2∂ C + 2uw +w +u +u +w +w −ε1 u .(5) ∂r2 ∂r∂z ∂z 2 ∂r ∂z ∂r ∂r ∂z ∂r ∂z ∂z u

The boundary conditions are u = cr, u → ue = ar,

v = Ωr,

v → 0,

w = w0 ,

T = Tw ,

w → we = −2az,

C = Cw

T → T∞ ,

at z = 0,

C → C∞

as z → ∞.

(6)

Here u, v, w are the velocities in r, ϕ, z directions, respectively. Furthermore, (ν, c, ρ, σ, k, cp , λ1 , λ2 , DB , T, C) are the kinematic viscosity, the stretching rate of the disk, the density of the fluid, the electrical conductivity, the thermal conductivity, specific heat, the relaxation time, the retardation time, the diffusion coefficients, the temperature and the fluid concentration, respectively. 6

The similarity transformations are η=

r

√ Ω T − T∞ C − C∞ z, u = ΩrF, v = ΩrG, w = ΩvH, θ = , φ= . ν Tw − T∞ Cw − C∞

(7)

Where (F, G, H) are the dimensionless velocities in r, ϕ and z directions, respectively. (θ, φ) are the dimensionless temperature and concentration, respectively. Substituting the above mentioned transformations into governing Eqs. (1 − 5), we acquire H 0 + 2F = 0,

(8)

F 2 − G2 + F 0 H − F 00 + β 1 F 00 H 2 + 2F F 0 H − 2GG0 H





+β 2 2F 02 + 2F 0 H 00 − F 000 H + M (F + β 1 F 0 H − A) − A2 = 0, 2F G + G0 H − G00 + β 1 G00 H 2 + 2 (F G0 + F 0 G) H




−β 2 (G000 H − 2F 0 G0 − 2G0 H 00 ) + M (G + β 1 G0 H) = 0,   1 00 θ − Hθ0 − εt H 2 θ00 + HH 0 θ0 = 0, Pr   φ00 − ScHφ0 − εc Sc H 2 φ00 + HH 0 φ0 = 0.

(9)

(10) (11) (12)

The transformed boundary conditions are F (η) = R, G (η) = 1, H (η) = s, θ (η) = 1, φ (η) = 1 at η = 0, F (η) → A, G (η) → 0, θ (η) → 0, φ (η) → 0

as η → ∞,

(13)

here primes describe differentiation w.r.t. η. The dimensionless physical parameters involved in the above system of equations are as follow

7

 M =

σB02 ρΩ c Ω






is the magnetic field parameter, A =

a Ω




the velocity ratio parame-

the stretching parameter, β 1 (= λ1 Ω) the time relaxation time parame  o ter, β 2 (= λ2 Ω) the retardation time parameter, s = √wΩν the mass transfer parameter,

ter, R =

εt (= ε0 Ω) the thermal relaxation time parameter, εc (= ε1 Ω) solutal relaxation time param 
eter, Pr = αν the Prandtl number and Sc = DνB the Schmidt number.

3

Solution methodology

The system of ordinary differential equations in Eqs. (8) to (12) with boundary conditions (13) are highly non-linear in nature. Due to arising nonlinearity in the system, it is complex to compute the exact solutions. In order to find the solutions of the problem, we utilized a numerical procedure called BVP Midrich scheme for handling these non-linear differential equations. The solutions of the governing problem are obtained with the help of Maple software. The general procedure for the mid-point approach is described by Y ∗0 (t∗ ) = F (t∗ , Y ∗ (t∗ )) , Y ∗ (t∗0 ) = Y0∗ .

(14)

The term used for the modified Euler method is ∗ Yn+1

=

Yn∗

  h∗ h∗ ∗ ∗ ∗ ∗ + h F tn + , Yn + F (tn , Yn ) , 2 2 ∗

(15)

where h∗ refers the step size and t∗n = t∗0 + nh∗ . The implicit approach of the mid-point method strategy is articulated as ∗ Yn+1

=

Yn∗

 
h∗ 1 ∗ ∗ ∗ ∗ , + h F tn + , Yn + Y ,Y 2 2 n n+1 ∗

n = 0, 1, 2, .....

(16)

The local error at each step of the mid-point procedure is of O (h∗3 ) and the global error is of the order O (h∗2 ). With more computational intensive, the mid-point algorithm error declines more rapidly as h∗ → 0 and shall be a more stable solution. 8

4

Results and discussion

The current study describes the von Karman swirling flow of magnetized Oldroyd-B fluid subject to Cattaneo-Christov double-diffusion theory. The solution of the problem is obtained in the form of velocities, temperature and concentration distributions as shown in Figures 2 to 6. These figures are plotted with respect to different physical involved parameters. The arising controlling parameters of the physical current problem are magnetic field parameter M , suction/injection parameter s, velocity ratio parameter A, stretching parameter R, relaxation time parameter β 1 , retardation time parameter β 2 , the thermal relaxation time parameter εt , the solutal relaxation time parameter εc , Prandtl number Pr and Schmidt number Sc. To illustrate the solutions in the form of velocities, temperature and concentration distributions, we fixed M = 4.0, R = 0.8, s = 0.1, A = 0.1, β 1 = 0.05, β 2 = 0.05, εt = 0.1, εc = 0.1, Pr = 5.0 and Sc = 5.0 throughout the entire computation. A comparison table is made for limiting case, that is, in the absence of relaxation time parameter, retardation time parameter, magnetic parameter, stretching parameter, mass transfer parameter, velocity ratio parameter and thermal relaxation time parameter shown in tables 1 and 2. These tabulated results are the numerical values of F 0 (0), −G0 (0) and −θ0 (0) which are compared with the past published articles. In table 1, the results are shown for the comparison of {−θ0 (0)} between Sparrow and Gregg [26] , and the present study. Furthermore, the comparisons of numerical values between current literature and with those of Turkyilmazoglu [27] and Bachok et al. [28] are discussed in Table 2. These tables represent the validity of our scheme and problem. The behavior of stretching parameter R on velocity, temperature and concentration of the fluid is discussed via Figs. 2 (a − d). It is observed that the radial velocity profile F (η) enhances as stretching parameter R increases between 0.8 to 1.5. It is due to the 9

fact that the rotating parameter R is the ratio of stretch rate c to swirl Ω rates. Hence, by increasing the stretching parameter R, the velocity F (η) in radial direction boosts up because of the increment in the stretching rate of the disk. On the other hand, a converse behavior is noticed for the velocity G (η) in azimuthal direction. This is due to the reduction in the swirl rate of the disk. Additionally, the variation in temperature and concentration distributions on stretching parameter R is analyzed through graphically in Figs. 2 (c) and (d). It is scrutinized that the temperature as well as concentration of the Oldroyd-B fluid reduce as increases the stretching parameter R (= 0.8, 1.0, 1.3, 1.5). Figs. 3 (a) and (b) provide the variation of velocity profiles (F (η) , G (η)) on velocity ratio parameter A against η. All plots satisfy the far field boundary conditions asymptotically. These curves show that the velocity F (η) in radial direction declines with an enlargement of velocity ratio parameter A (= 0.0, 0.3, 0.5, 0.7). A similar behavior is seen for velocity in azimuthal direction G (η) (see Fig. 3 (b)). Figs. 4 (a) to (c) elucidate the impact of retardation time parameter β 2 on velocity field, temperature and concentration distributions, respectively with default values of other parameters. It is obvious from these figures that the velocity G (η) rises with an increment in retardation time parameter β 2 (= 0.05, 0.3, 0.6, 0.9). Similarly, the same behavior is clarifies for the temperature θ (η) and concentration φ (η) distributions (see Figs. 4 (b) and (c)). The variation of temperature θ (η) on thermal relaxation time parameter εt against η is captured in Fig. 5 (a). It reveals that the temperature of the liquid declines as thermal relaxation time parameter εt changes from 0.0 to 0.25. Basically, for εt = 0, the Cattaneo-Christov heat flux model reduces to the classical Fourier’s law that means the transport of heat energy instantaneously in the Oldroyd-B fluid. For non zero values of εt (= 0.1, 0.18, 0.25), the temperature declines as shown in Fig.

5 (a). It is because the Cattaneo-Christov 10

model control the heat transport rate in the given liquid. Similarly, Fig. 5 (b) accentuates the reduction in concentration distribution φ (η) on solutal relaxation parameter εt with default values of other parameters. The dependence of liquid temperature θ (η) on the Prandtl number Pr is plotted in Fig. 6 (a). As sketched in this figure, the temperature as well as thickness of thermal boundary layer diminish by increasing values of Prandtl number Pr (= 3.0, 5.0, 7.0, 9.0). It is basically the thermal diffusivity decreases by enlarging Prandtl number Pr which causes to decrease the temperature and corresponding thickness of boundary layer. Additionally, Fig. 6 (b) shows that the augmentation of the Schmidt number Sc (= 3.0, 5.0, 7.0, 9.0) results in decreasing the concentration distribution φ (η).

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Table 1: A comparison between Ref. [26] and present study for the −θ0 (0) on various values of Pr. Pr

Sparrow and Gregg [26]

Present study

1

0.39625

0.39682844

10

1.1341

1.13385776

100

2.6871

2.68677447

Table 2: A comparison of the F 0 (0), −G0 (0) and −θ0 (0) on fixed Pr = 6.2 with Refs. [27, 28]. Turkyilmazoglu [27]

Bachok et al. [28]

Present study

F 0 (0)

0.51023262

0.5102

0.51011626

−G0 (0)

0.61592201

0.6159

0.61584927

−θ0 (0)

0.93387794

0.9337

0.93369411

12

5

Conclusions

In this article, we have investigated the von Karman swirling flow of magnetized viscoelastic Oldroyd-B fluid caused by a rotating disk. The Cattaneo-Christov double-diffusion model is used for heat and mass transport phenomena. The von Karman transformations are used to convert the partial differential equation into ordinary differential equations. The BVP Midrich technique is utilized and obtained the numerical solution of the problem. In the last, some important results of the current problem is listed below: 1. The velocity in radial directions is enhanced by the higher values of stretching parameter while it reduces in azimuthal direction. 2. With the enlargement of velocity ratio parameter, both velocities in radial and azimuthal directions, respectively decline. 3. It is revealed that the temperature of the liquid drops down as thermal relaxation time parameter changes from 0.0 to 0.25. 4. It accentuates that the reduction in concentration distribution is seen on solutal relaxation time parameter.

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References [1] J.B.J. Fourier, Theorie analytique de la chaleur, Didot, Paris, (1822) 499-508. [2] C. Cattaneo, Sulla conduzione del calore, Atti Del Semin. Matem. E Fis. Della Univ. Modena, (1948) 3 83-101. [3] C.I. Christov, On frame indifferent formulation of the Maxwell-Cattaneo model of finite speed heat conduction, Mech. Res. Commun., 36 (2009) 481-486. [4] M. Ciarletta and B. Straughan, Uniqueness and structural stability for the CattaneoChristov equations, Mech. Res. Commun., 37 (2010) 445-447. [5] S. Han, L. Zheng, C. Li and X. Zhang, Coupled flow and heat transfer in viscoelastic fluid with Cattaneo-Christov heat flux model, Appl. Math. Lett., 38 (2014) 87-93. [6] M. Khan, J. Ahmed and L. Ahmed, Application of modified Fourier law in von K´arm´an swirling flow of Maxwell fluid with chemically reactive species, J. Braz. Soci. Mech. Sci. Eng., 40 (2018) 573. [7] F.A., Sulti, Impact of Cattaneo-Christov heat flux model on stagnation-point flow towards a stretching sheet with slip effects, J. Heat Transf., 141 (2019) 022003. [8] S.A.M. Haddad, Thermal instability in Brinkman porous media with Cattaneo–Christov heat flux, Int. J. Heat Mass Transf., 68 (2014) 659-668. [9] V. Tibullo and V. Zampoli, A uniqueness result for the Cattaneo–Christov heat conduction model applied to incompressible fluids, Mech. Res. Commun., 38 (2011) 77-79. [10] M.E. Ali and N. Sandeep, Cattaneo-Christov model for radiative heat transfer of magnetohydrodynamic Casson-ferrofluid: a numerical study, Results Phys., 7 (2017) 21-30. 14

[11] K.A. Kumar, J.V.R. Reddy, V. Sugunamma and N. Sandeep, Magnetohydrodynamic Cattaneo-Christov flow past a cone and a wedge with variable heat source/sink, Alex. Eng. J., 57 (2018) 435-443. [12] V. Nagendramma, C.S.K. Raju, B. Mallikarjuna, S.A. Shehzad and A. Leelarathnam, 3D Casson nanofluid flow over slendering surface in a suspension of gyrotactic microorganisms with Cattaneo-Christov heat flux, Appl. Math. Mech., 39 (2018) 623-638. [13] T. Von Karman, Uberlaminare und turbulente Reibung, ZAMM Zeitsch.Ang. Math. Mech., 1 (1921) 233-252. [14] W.G. Cochran, The flow due to a rotating disk, Math. Proceed. Camb. Phil. Soci., 30 (1934) 365-375. [15] E.T. Benton, On the flow due to a rotating disk, J. Fluid Mech., 24 (1966) 781-800. [16] M.M. Rashidi, N. Kavyani and S. Abelman, Investigation of entropy generation in MHD and slip flow over a rotating porous disk with variable properties, Int. J. Heat Mass Transf., 70 (2014) 892-917. [17] N.A. Khan and F. Sultan, Numerical analysis for the Bingham-Papanastasiou fluid flow over a rotating disk, J. Appl. Mech. Tech. Phys., 59 (2018) 638-644. [18] J. Ahmed, M. Khan and L. Ahmed, Transient thin film flow of nonlinear radiative Maxwell nanofluid over a rotating disk, Phys. Lett. A, 383 (2019) 1300-1305. [19] R. Ellahi, M.H. Tariq, M. Hassan and K. Vafaia, On boundary layer nano-ferroliquid flow under the influence of low oscillating stretchable rotating disk, J. Mol. Liq., 229 (2107) 339-345.

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[20] B. Mahanthesh, B.J. Gireesha, S.A. Shehzad, A. Rauf and P.B.S. Kumar, Nonlinear radiated MHD flow of nanoliquids due to a rotating disk with irregular heat source and heat flux condition, Phys. B: Cond. Matt., 537 (2018) 98-104. [21] M. Gholinia, K. Hosseinzadeh, H. Mehrzadi, D.D. Ganji, and A.A. Ranjbar, Investigation of MHD Eyring–Powell fluid flow over a rotating disk under effect of homogeneousheterogeneous reactions, Case Stud. Therm. Eng., 13 (2019) 100356. [22] J. Ahmed, M. Khan and L. Ahmed, Stagnation point flow of Maxwell nanofluid over a permeable rotating disk with heat source/sink, J. Mol. Liq., 287 (2019) 110853. [23] M. Khan, J. Ahmed and L. Ahmed, Chemically reactive and radiative von K´arm´an swirling flow due to a rotating disk, Appl. Math. Mech., 39 (2018) 1295-1310. [24] M. Khan, A. Hafeez and J. Ahmed, Impacts of non-linear radiation and activation energy on the axisymmetric rotating flow of Oldroyd-B fluid, Phys. A: Stat. Mech. Appl., (2020) 124085. [25] A. Hafeez, M. Khan and J. Ahmed, Stagnation point flow of radiative Oldroyd-B nanofluid over a rotating disk, Comp. Meth. Prog. Bio., (2020): 105342. [26] E.M. Sparrow and J.L. Gregg, Heat transfer from a rotating disk to fluids of any Prandtl number, ASME J. Heat Transf. 81 (1959) 249-251. [27] M. Turkyilmazoglu, Nanofluid flow and heat transfer due to a rotating disk, Comput. Fluids, 94 (2014) 139–146. [28] N. Bachok, A. Ishak and I. Pop, Flow and heat transfer over a rotating porous disk in a nanofluid, Phys., B: Cond. Matt., 406 (2011) 1767-1772.

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Figure 1: 1: A physical sketch of the problem. Figures captions Fig. 1: A physical sketch of the problem. Fig. 2: Effect of R on (a): F (η) (b): G (η) (c): θ (η) (d): φ (η). Fig. 3: Effect of A on (a): F (η) (b): G (η). Fig. 4: Effect of β 2 on (a): G (η) (b): θ (η) (c): φ (η). Fig. 5: (a): Effect of εt on θ (η). (b): Effect of εc on φ (η). Fig. 6: (a): Effect of Pr on θ (η). (b): Effect of Sc on φ (η).

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Figure 2: Effect of R on (a): F (η) (b): G (η) (c): θ (η) (d): φ (η).

18

Figure 3: Effect of A on (a): F (η) (b): G (η).

19

Figure 4: Effect of β 2 on (a): G (η) (b): θ (η) (c): φ (η).

20

Figure 5: (a): Effect of εt on θ (η). (b): Effect of εc on φ (η).

21

Figure 6: (a): Effect of Pr on θ (η). (b): Effect of Sc on φ (η).

22