1 Introduction

The study of non-Newtonian fluids has increased due to a substantial variety of engineering, industrial and commercial manufacturing applications. The study of Newtonian fluids is easy as compared to non-Newtonian fluids due to simple developed equations which can be handled numerically or analytically. The reason is that of a linear relationship between the rate of stress and strain in the case of Newtonian fluid, but this relation is not any more linear for non-Newtonian fluids. Various authors showed the applications of non-Newtonian fluids in their studies. Some significant examples of non-Newtonian fluids are cream, plastic melts, soap slurries, wet beach sand, mayonnaise, paper pulp, etc. [1]. Navier-Stokes equation is used to describe the flow behaviour of Newtonian fluid but there is no single equation to describe the flow behaviour of the non-Newtonian fluids with all the properties. Due to this reason, various empirical and semi-empirical non-Newtonian equations or models have been presented and used.

Ali et al. [2] studied an Eyring-Powell (EP) fluid under the effects of temperature-dependent viscosity cases in a pipe. They first convert their dimensional forms of momentum and energy equations with boundary conditions into non-dimensional form, and then discuss Reynolds and Vogel models for both momentum and energy equations. In their research work, they solved their equations with the help of both perturbation and numerical methods that match well. The flow of magnetohydrodynamics (MHD) EP nanofluid over an inclined surface is analysed by Khan et al. [3]. During their study, they assumed that the flow of fluid is incompressible and the fluid viscosity is varying exponentially. They expressed their results in the form of velocity and temperature profiles along with different parameters by plotting graphs and also in tabular form. Ellahi [4] examined the study of non-Newtonian nanofluid in the pipe and depicted the effects of magnetic and temperature-dependent viscosity. Akinshilo and Olaye [5] also analysed the flow of EP fluid in a pipe. They discussed the internal heat flow and temperature-dependent viscosity in their work. They solved the non-linear ordinary equation by using perturbation technique and examined the effect of fluidic parameters on velocity and temperature profiles. Akinbowale [6] studied the non-Newtonian flow of third-grade fluid in the horizontal channel. He solved an ordinary differential equation by using Adomian decomposition method and discussed the Vogel model for temperature-dependent viscosity and concluded that velocity distribution increases by decreasing the porosity parameter. Ellahi and Riaz [7] captured the magnetic effects in the flow of third-grade fluid in a pipe by taking variable viscosity. To calculate the analytical solution, the homotopy analysis method (HAM) was used. Aksoy and Pakdemirli [8] considered the non-Newtonian fluid with a porous medium between two parallel plates. They calculated the solutions of the non-linear differential equations by employing the perturbation technique. Three viscosity dependent cases were discussed i.e., (a) constant viscosity case; (b) Reynolds model; and (c) Vogel’s model. The validity range of the solution was also provided. Ellahi et al. [9] examined the third-grade fluid between the concentric cylinders and checked the effect of slip over this fluid. They discussed two cases, in the first one the outer cylinder is stationary and inner cylinder move and the second one the outer cylinder is moving and the inner cylinder is at rest position. They calculate the solutions for both conditions (with and without slip parameter). The flow of non-isothermal Poiseuille fluid was also examined by Farooq et al. [10]. They considered the fluid is flowing between two heated parallel inclined plates using coupled stress fluid which was incompressible. Perturbation techniques were used to calculate analytical solutions. In their paper, they discussed the Reynold’s model and obtained the expressions for temperature, velocity, dynamic pressure, shear stress and volume flow rate. Alharbi et al. [11] evaluated the effects of entropy generation in MHD EP fluid. They assumed that the flow is flowing over an oscillating porous stretching sheet and further assumed that the flow is unsteady. They analysed the behaviour of the heat source and thermal radiation in their study. They captured the effects of pertinent parameters on entropy generation rate, temperature, and velocity fields. They solved the governing equation by using HAM. The radiative effect on three-dimensional MHD flow of an EP fluid was described by Hayat et al. [12]. They calculated the series solution by non-linear analysis and also calculate the exact solution of the problem. Yurusoy [13] presented the heat transfer effects of third-grade fluid between two concentric cylinders. During his study, he considered that the fluid temperature is lower than the pipe temperature. He calculated the solution by using perturbation technique and discussed Reynolds model. Hayat et al. [14] explored the effect of porous medium of third-grade fluid. They solved the governing momentum and energy equations by employing HAM. They analysed the shear stress and velocity field at the plate. Massoudi and Christie [15] analysed the viscous dissipation and variable viscosity effects of third-grade fluid in a pipe and considered that the temperature of the fluid is lower than the temperature of the pipe. Huang [16] studied the heat generation effect of non-Newtonian fluid over a vertical permeable cone in a porous medium. The author used the Keller box method to find a solution of his problem and discussed the physical aspects of the problem. Analysis of EP nanofluid is selected by Hayat et al. [17]. They captured the effect of MHD flow over the non-linear stretching sheet and discussed the results for concentration and temperature profiles. Hayat and Nadeem [18] studied the three-dimensional flow of EP fluid and discussed the heat flux and mass quantities. They discussed the behaviour of temperature and Cattaneo-Christov heat flux theory.

From the above-cited studies, it is noted that there is no study available in literature to capture the effects of radiative heat flux and heat generation on EP fluid in a circular long pipe under various models of the viscosities. The considered fluid model (EP) is much complex and is advantageous over a Power-law model due to two aspects. The first one is that this model is not driven from an empirical relation like a Power-law model. The second one is, this fluid model can be reduced into a Newtonian fluid model for the low and high shear rates.Therefore, the objective of our investigation is to obtain the numerical and perturbation solutions of the one-dimensional flow of an EP fluid in a pipe under the combined effects of radiative and heat absorption. As various engineering processes only happen at moderate temperature, so the consideration of radiative heat flux cannot be ignored in these kind of processes namely space vehicles, gas turbines, propulsion devices, and nuclear plants [19], [20]. Futher, the study of heat generation (or absorption) in moving fluid is useful in the problems of chemical reaction. The flows in the porous media have numerious applications like, oil reservoir and geothermal [21], [22]. The absolute error in velocity and temperature will be presented. Three famous models will discuss which are listed below.

1. Constant viscosity model

2. Reynolds model

3. Vogel’s model

The numerical solution for all above-mentioned viscosity models will be obtained by the finite-difference method in our study. The effects of various physical parameters on viscosity and temperature profiles will be highlighted together with the validity of the perturbation solution. The current study can be useful to increase the heat transfer processes in energy conservation, medical process, fuel cells and micro-mixing. The applications of this problem are also beneficial in heat exchangers, film flow, catalytic reactors, paper production and polymer solutions. This study will be helpful in future to investigate the behaviour of velocity and temperature fields against pertinent prameters, whenever preliminary data of EP model is given.

2 Formulation of the Problem

Let us consider the one-dimensional, steady-state flow of non-Newtonian fluid in a pipe with the account of effects of radiation and heat generation. It is assuming that the flow is flowing due to applied pressure gradient and the viscosity is a function of temperature. The systematic flow behaviour is shown in Figure 1.

Figure 1:

Geometry of the flow problem.

The velocity and temperature fields for the present problem are defined by

The Cauchy stress tensor T[23], [24] is

where p is the pressure, I is the identity element. Now for the current situation, the extra stress tensor S is defined as

where

and

As V=[u,v,w] and in our case V=[0,0,w(r)], so in this case, we have

with the help of (10), we get

Putting these values in (4), (4) takes the following form:

Using X=k3k2,Y=k33k2 in the above (14) becomes

Now by substituting the value of A₁ in (15), we get

where S is defined as

Now using the value of S in (16)

(srrsrθsrzsθrsθθsθzszrszθszz)

(srrsrθsrzsθrsθθsθzszrszθszz)

=(00(μ+X−Y6(dwdr)2)dwdr000(μ+X−Y6(dwdr)2)dwdr00),

=(00(μ+X−Y6(dwdr)2)dwdr000(μ+X−Y6(dwdr)2)dwdr00),

From the above (18), only two components of extra stress are left which are:

The general form of the equation of motion in cylindrical coordinates system is:

Due to steady-state flow, we have

As the velocity components u = 0 and v = 0 so, left-hand side of general equations will vanish and only right-hand side will survive which is given by:

As, in the current investigation, the flow is flowing along z-direction only (one dimensional) so we can use the simple derivative instead of partial derivative symbols.

where ∂p∂z=c1.

Now for the derivation of the heat equation, first the general form of the heat equation is considered

The radiative heat flux H_R [25], [26] for the current situation is defined by

To obtain the values of θ⁴, we will expand θ⁴ in the Taylor series expansion about θ_w and overlooking upper order terms, yields

Using the value of θ⁴ into (34), we have

Use (32), (33) and (34) into (31), we get

The boundary conditions for (29) and (37) are:

Now introducing the dimensionless quantities

The dimensionless equations are given by:

where M=Xμ0,c=c1R∗2μ0w0,Γ=μ0w02k(θm−θw),δ=QR∗2k,R=16σ∗θw33kk∗ and Λ=Y32w026μ0R∗2

The dimensionless equations of motion and energy based on viscosity models are discussed in the next one.

3 Viscosity Models

Based on the viscosity of the fluid, three different models are selected. In the first case, viscosity is chosen as a constant i.e., viscosity does not depend on temperature and in second and third cases, viscosity is a function of fluid temperature and these models are known as Reynolds and Vogel’s models. In the last two models, viscosity is defined as an exponent form.

1. Constant Viscosity Model

   For this case, we set μ⌢=1, therefore dμ⌢dr⌢=0

   The new forms of the equation of motion and heat equation for the present case are:

   (43)1r⌢(dw⌢dr⌢+r⌢d2w⌢dr⌢2)+Mr⌢(dw⌢dr⌢+r⌢d2w⌢dr⌢2)−Λr⌢(dw⌢dr⌢)2[dw⌢dr⌢+3r⌢d2w⌢dr⌢2]=c,

   (44)(1+R)d2θ⌢dr⌢2+1r⌢dθ⌢dr⌢+δθ⌢+Γ(dw⌢dr⌢)2[1+M−Λ(dw⌢dr⌢)2]=0,

   (45)w⌢=0 at r⌢=1,θ⌢=0 atr⌢=1,anddw⌢dr⌢=0 at r⌢=0,dθ⌢dr⌢=0 at r⌢=0,

   To solve (43)–(45), let us assume to apply perturbation expansion on momentum and energy equation.

   (46)w⌢≅w⌢0+εw⌢1,θ⌢≅θ⌢0+εθ⌢1,Λ≅εσ,δ=εN1,

   where ϵ is the perturbation parameter and (0 < ϵ ≤ 1). By separating each order of we have the following systems:

   System of order (ϵ⁰) for velocity profile:

   (47)r⌢d2w⌢0dr⌢2+dw⌢0dr⌢=cr⌢(1+M),

   (48)w⌢0=0  at   r⌢=1,dw⌢0dr⌢=0  at  r⌢=0.

   System or order (ϵ1) for velocity profile:

   (49)r⌢d2w⌢1dr⌢2+dw⌢1dr⌢=σ(1+M)(dw⌢0dr⌢)3+3r(dw⌢0dr⌢)2d2w⌢0dr⌢2,

   (50)w⌢1=0  at   r⌢=1,dw⌢1dr⌢=0  at  r⌢=0.

   System of order (ϵ⁰) for temperature profile:

   (51)(1+R)d2θ⌢0dr⌢2+1r⌢dθ⌢0dr⌢=−Γ(1+M)(dw⌢0dr⌢)2,

   (52)θ⌢0=0  at   r⌢=1,dθ⌢0dr⌢=0  at  r⌢=0.

   System or order (ϵ¹) for temperature profile:

   (53)(1+R)d2θ⌢1dr⌢2+1r⌢dθ⌢1dr⌢=−2Γ(1+M)(dw⌢0dr⌢)(dw⌢1dr⌢)+Γσ(dw⌢0dr⌢)4−N1θ⌢0,

   (54)θ⌢1=0  at   r⌢=1,dθ⌢1dr⌢=0  at  r⌢=0.

   By solving each order of ϵ, we have the following results for velocity profile:

   (55)w⌢0=γ0(1−r⌢2).

   (56)w⌢1=γ1(1−r⌢4).

   Results for temperature profile:

   (57)θ⌢0=λ0 (1−r⌢4).

   (58)θ⌢1=λ6σ(1−r⌢6)+N1(λ3r⌢2+λ4r⌢6+λ5).

   The final expressions for temperature and velocity are:

   (59)w⌢=γ0(1−r⌢2)+γ1(1−r⌢4),

   (60)θ⌢=λ0 (1−r⌢4)+λ6Λ(1−r⌢6)+δ(λ3r⌢2+λ4r⌢6+λ5).

2. Reynolds Model

   For the case of variable viscosity model i.e., Reynolds model, we will set the values of viscosity in term of temperature as:

   (61)μ⌢=e−Lθ⌢⇒dμ⌢dr⌢=−Le−Lθ⌢dθ⌢dr⌢.

   In view of the above setting, the equation of motion and heat equation is defined by:

   (62)−Lr⌢e−Lθ⌢dθ⌢dr⌢dw⌢dr⌢+e−Lθ⌢(dw⌢dr⌢+r⌢d2w⌢dr⌢2)+M(dw⌢dr⌢+r⌢d2w⌢dr⌢2)−Λ(dw⌢dr⌢)2[dw⌢dr⌢+3r⌢d2w⌢dr⌢2]=cr⌢,

   (63)(1+R)d2θ⌢dr⌢2+1r⌢dθ⌢dr⌢+δθ⌢+Γ (dw⌢dr⌢)2[e−Lθ⌢+M−Λ (dw⌢dr⌢)2]=0,

   The perturbation expansions on momentum and energy equations for this case are as follows:

   (64)w⌢≅w⌢0+εw⌢1,θ⌢≅θ⌢0+εθ⌢1,Λ≅εσ,δ=εN1,L=εl1.

   System of order(ϵ0)for velocity profile:

   (65)r⌢d2w⌢0dr⌢2+dw⌢0dr⌢=cr⌢(1+M),

   (66)w⌢0=0  at   r⌢=1,dw⌢0dr⌢=0  at  r⌢=0.

   System or order(ϵ1)for velocity profile:

   (67)r⌢(1+M)d2w⌢1dr⌢2+(1+M)dw⌢1dr⌢=l1θ⌢0(dw⌢0dr⌢)+l1r⌢(dw⌢0dr⌢)(dθ⌢0dr⌢)+l1r⌢θ⌢0(d2w⌢0dr⌢2)+σ((dw⌢0dr⌢)3+3r⌢(dw⌢0dr⌢)2d2w⌢0dr⌢2)

   (68)w⌢1=0  at   r⌢=1,dw⌢1dr⌢=0  at  r⌢=0.

   System or order (ϵ⁰) for temperature profile:

   (69)(1+R)d2θ⌢0dr⌢2+1rdθ⌢0dr⌢=−Γ(1+M)(dw⌢0dr⌢)2,

   (70)θ⌢0=0  at   r⌢=1,dθ⌢0dr⌢=0  at  r⌢=0.

   System or order (ϵ1) for temperature profile:

   (71)(1+R)d2θ⌢1dr⌢2+1r(dθ⌢1dr⌢)=−2Γ(1+M)(dw⌢0dr⌢)(dw⌢1dr⌢)+Γσ(dw⌢0dr⌢)4+l1θ⌢0Γ(dw⌢0dr⌢)2−N1θ⌢0

   (72)θ⌢1=0  at   r⌢=1,dθ⌢1dr⌢=0  at  r⌢=0.

   By solving each order of ϵ, we have the following results for velocity profile:

   (73)w⌢0= γ0(1−r⌢2).

   (74)w⌢1=γ2l1(2−3r⌢2+r⌢6)+γ3σ(1−r⌢4).

   Results for temperature profile are:

   (75)θ⌢0=λ0 (1−r⌢4).

   (76)θ⌢1=λ7σ(−1+r⌢6)+N1(λ8+λ9r⌢2+λ10r⌢6)+l1(λ11+λ12r⌢4+λ13r⌢8).

   The final expression for temperature and velocity are:

   (77)w⌢=γ0(1−r⌢2)+γ2L(2−3r⌢2+r⌢6)+γ3Λ(1−r⌢4),

   (78)θ⌢=λ0 (1−r⌢4)+λ7Λ(−1+r⌢6)+δ(λ8+λ9r⌢2+λ10r⌢6)+L(λ11+λ12r⌢4+λ13r⌢8).

3. Vogel’s Model

   For Vogel’s model, we take the values of viscosity as

   μ⌢=e(AB+θ⌢−θ⌢w), or μ⌢=e(AB−θ⌢w)(1−Aθ⌢B2), and dμ⌢dr⌢=−e(AB−θ⌢w) (AB2)dθ⌢dr⌢.

   With the help of the above substitution, we get the equation of motion and heat equation in the following form:

   (79)(1+R)d2θ⌢dr⌢2+1r⌢dθ⌢dr⌢+δθ⌢+Γ(dw⌢dr⌢)2(1a(1−Aθ⌢B)+M−Λ(dw⌢dr⌢)2)=0,

   where δ=c1e(AB−θ⌢w) and a=1e(AB−θ⌢w).

   The perturbation expansions for this case are as follows:

   (80)w⌢≅w⌢0+εw⌢1,θ⌢≅εθ⌢0+ε2θ⌢1,Λ≅εσ,δ≅εN1,Γ=εγ,

   System of order (ϵ⁰) for velocity profile:

   (81)r⌢d2w⌢0dr⌢2+dw⌢0dr⌢=r⌢1+aM,

   (82)w⌢0=0  at   r⌢=1,dw⌢0dr⌢=0  at  r⌢=0.

   System of order (ϵ⁰) for temperature profile:

   (83)(1+R)d2θ⌢0dr⌢2+1rdθ⌢0dr⌢=−γ(1a+M)(dw⌢0dr⌢)2,

   (84)θ⌢0=0  at   r⌢=1,dθ⌢0dr⌢=0  at  r⌢=0.

   System or order (ϵ1) for velocity profile:

   (85)r⌢(1+Ma)d2w⌢1dr⌢2+(1+Ma)dw⌢1dr⌢=AB2dw⌢0dr⌢(θ⌢0+r⌢dθ⌢0dr⌢)+AB2r⌢θ⌢0d2w⌢0dr⌢2+aσ((dw⌢0dr⌢)3+3r⌢(dw⌢0dr⌢)2d2w⌢0dr⌢2),

   (86)w⌢1=0  at   r⌢=1,dw⌢1dr⌢=0  at  r⌢=0.

   System or order (ϵ1) for temperature profile:

   (87)(1+R)d2θ⌢1dr⌢2+1r⌢dθ⌢1dr⌢=−N1θ⌢0+1aB2(Aθ0γ(dw⌢0dr⌢)2)+γσ(dw⌢0dr⌢)4−2γdw⌢0dr⌢dw⌢1dr⌢(1a+M),

   (88)θ⌢1=0  at   r⌢=1,dθ⌢1dr⌢=0  at  r⌢=0.

   By solving each order of ϵ, we have the following results for velocity and temperature profiles:

   (89)w⌢0=γ4(1−r⌢2).

   (90)θ⌢0=λ14(1−r⌢4).

   (91)w⌢1=γ5A(2−3r⌢2+r⌢6)+γ6σ(1−r⌢4).

   (92)θ⌢1=λ15(−4r⌢4+r⌢8)+λ16(4r⌢4−r⌢8)+λ17(14r⌢4−3r⌢8)+λ18(−14r⌢4+3r⌢8)+λ19+σλ22(1−r⌢6)+N1(λ23r⌢2+λ24r⌢6+λ25).

   The final expression for temperature and velocity are as follows:

   (93)w⌢=γ4(1−r⌢2)+γ5A(2−3r⌢2+r⌢6)+γ6Λ(1−r⌢4),

   (94)θ⌢=λ14(1−r⌢4)+λ15(−4r⌢4+r⌢8)+λ16(4r⌢4−r⌢8)+λ17(14r⌢4−3r⌢8)+λ18(−14r⌢4+3r⌢8)+λ19+Λλ22(1−r⌢6)+δ(λ23r⌢2+λ24r⌢6+λ25).