Radiative heat transfer of second grade nanofluid flow past a porous flat surface: a single-phase mathematical model

The study of non-Newtonian liquids flow is becoming one of the major topics in the modern scientific research area because of its numerous usages in manufacturing and industrial sectors like plastics processing, lubrication, chemical engineering, and biomedical applications. The differential type liquids are non-Newtonian liquids, which are generally called as viscoelastic fluids. These fluids exhibit unique characteristics compared to that of viscous materials. The second-grade liquid is renowned as a subclass of viscoelastic non-Newtonian liquid. Currently, the study related to boundary layer flows of second-grade liquid has created much interest in the researchers. Recently, Khan et al [1] discussed the second-grade liquid stream past a cylinder. Hayat et al [2] elucidated the aspect of radiation effect on second-grade nanoliquid flow over a disk. Javaid et al [3] numerically explained the stream of a second-grade liquid past a cylinder. Khan et al [4] debriefed the second-grade liquid stream past an elastic surface. The levering of heat sink source and magnetic effect on a second grade nanoliquid stream past a stretchy geometry was scrutinized by Hayat et al [5].

The heat transference and viscous incompressible liquid stream over a stretching surface are captivating the attention of various investigators due to its engineering and manufacturing applications such as sedimentation, underground disposable of radioactive waste materials, blood rheology, exothermic and endothermic reactions, centrifugal separation of particles and flow through packed beds. Recently, in the existence of thermal radiation, and thermal dependent heat sink/source effects, Khedr et al [6] elucidated the stream of a micropolar liquid above a stretched semi-infinite porous surface. Chamkha et al [7] explicated the unsteady stream of a liquid above a stretchy sheet with chemical reaction and permeable medium. The stream of a micropolar liquid over an expanding geometry was explicated numerically by Waqas et al [8]. The heat transfer process in the stream of Carbon nanotubes past a stretched sheet was sophisticatedly investigated by Hayat et al [9]. Radhika et al [10] debriefed the impact of heat sink/source on dusty hybrid nanofluid stream above a melting surface. Qayyum et al [11] explored the radiation and viscous dissipation effects on the stream of Williamson liquid above a plane stretching sheet. Gowda et al [12] elucidated nanoliquid flow above an elastic sheet.

The work done through the fluid with shear forces on adjoining layers is transformed into heat energy, which is called as viscous dissipation. The impact of viscous dissipation on heat transference is much essential for higher velocity and viscous flows. Recently, the study of fluid flows in the existence of viscous dissipation is occupying a vital role in the modern scientific area. Maleki et al [13] elucidated the dissipative stream of a nanoliquid above a permeable plate. The significance of viscous dissipation on fluid stream above a shrinking surface was deliberated by Lund et al [14]. Khan and Khan [15] explored the nanoliquid stream with viscous dissipation effect. Kumar et al [16] exemplified the radiative convective stream of dissipative liquid above a plate. Xiong et al [17] explicated the dissipative Darcy-Forchheimer stream of cross nanoliquid past a thin needle.

The inspection of a thermal radiation effect on liquid stream past a stretched surface is one of the major sections for the researchers due to its usages in abundant manufacturing and physical processes such as propulsion devices, sun-oriented power innovation, polymer processing industry, atomic plants, chemical processes, aircraft and combustion chambers. Chamkha [18] examined the heat sink/source and radiation effects on the hydromagnetic stream above a permeable surface. Chamkha et al [19] elucidated the impact of Ohmic heating and radiation effects on convective flow of micropolar liquid past a porous plate. The significance of radiation effect on a nanoliquid flow above a cylinder was emphatically analyzed by Qayyum et al [20]. Basha et al [21] deliberated the radiation effect on the nanofluid stream above a plate and wedge. Jeffrey nanoliquid stream past an extending surface with radiation effect was explicated numerically by Hayat et al [22]. Rashid et al [23] exemplified the radiation impact on the stream of Maxwell liquid through an elongated sheet. Mabood et al [24] scrutinized the radiation effect on the stream of fluid above an elastic sheet with suspended nanoparticles. The rate type nanoliquid stream above a cylinder with the influence of radiation effect was inspected by Ifran et al [25].

A nanoliquid is an amalgamation of minute nanosubstance's and the traditional fluid. The minute nanoparticles may be nanopowder, metallic or non-metallic ions whereas traditional fluids are water, ethylene glycol and kerosene which are used for the synthesis of nanofluids. Nanofluid has innumerable applications in metal spinning, extrusion, power generation, lubricant, biomedicine, wire drawing, nuclear reactors, glass fiber manufacture, transportation, cooling process, and hot rolling. These noteworthy engineering and industrial applications motivated many researchers to explore the flow of nanofluids over different geometries. Chamkha and Aly [26] deliberated the heat production/absorption on convective flow of nanoliquid past an upright plate. Reddy et al [27] elucidated the chemically reacting nanoliquid flow above a spinning disk with porous medium. Sheikholeslami and Rokni [28] explored the nanoliquid stream over a plate with radiation effect. Hayat et al [29] examined the radiation impact on a nanoliquid flow instigated by a cylinder. The magnetic dipole impact on the stream of fluid suspended with more than one nanoparticle is explicated by Kumar et al [30]. Shehzad et al [31] elucidated Maxwell nanoliquid stream past a revolving disk.

The recent progression in the flow behavior of different liquids through the porous media have vital significance by virtue of its application in numerous industrial processes such as nuclear-based repositories, fermentation processes, nuclear waste disposal, drying of porous solid, welding and casting. Recently, Gorla and Chamkha [32] expounded the flow of nanoliquid above an upright plate in presence of porous medium. Shehzad et al [33] explicated the fluid flow above a sheet implanted in a permeable medium. Krishna and Chamkha [34] elucidated the slip effects on MHD stream of nanoliquid with porous medium. The salient aspects of viscous dissipation in the flow of ferroliquid past an extending sheet with permeable medium was explained by Khan et al [35]. The stream behavior of hybrid nanoliquid over an elastic sheet placed in a permeable media was discussed by Eid and Nafe [36].

According to the available literature, studies on unsteady permeable second-grade nanoliquids are extremely rare, and none of the published articles separately addressed the effects of the porous medium, heat source, viscous dissipation, and thermal radiation over the stretching sheet using the Tiwari and Das nanofluid model [37], which is the current work's novelty. The results only show the effects of Cu−EO and Al₂O₃ − EO nanofluids. To bridge the gap, the current research employs a numerical approach based on the shooting method to investigate the effect of key factors on the fluid characteristics of second-grade nanofluid within the boundary layer (BL).

The moving flat horizontal surface with non-uniform stretching velocity (figure 1) is represented mathematically as:

$$\begin{eqnarray}&&{U}_{w}\left(x,t\right)=\displaystyle \frac{bx}{1-\xi t}.\end{eqnarray} \tag{ 1 }$$

where b is initial rate of stretching of porous sheet. ${{\rm{T}}}_{w}\left(x,t\right)={{\rm{T}}}_{\infty }+\tfrac{b\,\ast \,x}{1-\xi t}$ is temperature of insulated sheet under consideration and for sake of ease, sheet's left end is supposed to be fixed at x = 0. Also, temperature Variation rate, temperature of wall and surroundings are represented by b*, T_w and (T_∞) respectively.

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2.1. Conditions and assumptions of the model

2.2. Second-grade fluid stress tensor

The stress tensor of Cauchy in a second-grade fluid is represented mathematically as (see for details, Shah et al [38] )

$$\begin{eqnarray*}&&{S}^{* }=\mu {A}_{\varsigma 1}+{\alpha }_{1}{A}_{\varsigma 2}+{\alpha }_{1}{{A}_{\varsigma 1}}^{2}-pI,\end{eqnarray*}$$

$$\begin{eqnarray*}&&{A}_{\varsigma 1}=\left({\rm{grad}}\,{\rm{V}}\right)+{\left({\rm{grad}}{\rm{V}}\right)}^{T},\end{eqnarray*}$$

$$\begin{eqnarray*}&&{A}_{\varsigma 2}=\displaystyle \frac{{\rm{d}}{A}_{\varsigma 1}}{{\rm{dt}}}+{A}_{\varsigma 1}\left({\rm{grad}}\,{\rm{V}}\right)+{A}_{\varsigma 1}{\left({\rm{grad}}{\rm{V}}\right)}^{T},\end{eqnarray*}$$

Where I is the identity tensor, pis the pressure, d/dt is the material time derivative, α₁and α₂ are material variables, μ is dynamic viscosity, A _ς1 and A _ς2 are Rivlin–Ericksen tensors described above and V is the fluid velocity. We confirm the Clausius–Duhem inequality. Additionally, we conclude that the Helmholtz free energy is the minimum in equilibrium for the fluid locally at rest when

$$\begin{eqnarray*}&&\mu \geqslant 0,{\alpha }_{1}\geqslant 0,{\alpha }_{1}+{\alpha }_{2}=0.\end{eqnarray*}$$

If α1 + α2 = 0 then the second grade fluid equation diminishes to viscous fluid.

2.3. Geometry of fluid flow

2.4. Model equations

The formative equations [38] of assumed flow are:

$$\begin{eqnarray}&&\displaystyle \frac{\partial u}{\partial x}+\displaystyle \frac{\partial v}{\partial y}=0,\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{\,\partial u}{\partial t}+u\displaystyle \frac{\partial u}{\partial x}+v\displaystyle \frac{\partial u}{\partial y}=\displaystyle \frac{{\alpha }_{1}}{{\rho }_{nf}}\left[\displaystyle \frac{\partial u}{\partial x}\left(\displaystyle \frac{{\partial }^{2}u}{\partial {y}^{2}}\right)+\left(\displaystyle \frac{{\partial }^{3}u}{\partial t\partial {y}^{2}}\right)+u\left(\displaystyle \frac{{\partial }^{3}u}{\partial x\partial {y}^{2}}\right)+\displaystyle \frac{\partial u}{\partial y}\left(\displaystyle \frac{{\partial }^{2}v}{\partial {y}^{2}}\right)+v\left(\displaystyle \frac{{\partial }^{3}u}{\partial {y}^{3}}\right)\right]\\ +\displaystyle \frac{{\mu }_{nf}}{{\rho }_{nf}}\left(\displaystyle \frac{{\partial }^{2}u}{\partial {y}^{2}}\right)-\displaystyle \frac{{\mu }_{nf}}{{\rho }_{nf}k}u,\end{array}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{\partial T}{\partial t}+u\displaystyle \frac{\partial T}{\partial x}+v\displaystyle \frac{\partial T}{\partial y}=\displaystyle \frac{{k}_{nf}}{{\left(\rho {C}_{p}\right)}_{nf}}\left(\displaystyle \frac{{\partial }^{2}T}{\partial {y}^{2}}\right)-\displaystyle \frac{1}{{\left(\rho {C}_{p}\right)}_{nf}}\left(\displaystyle \frac{\partial {q}_{r}}{\partial y}\right)\\ +\displaystyle \frac{1}{{\left(\rho {C}_{p}\right)}_{nf}}Q\left(T-{T}_{\infty }\right)+\displaystyle \frac{{\mu }_{nf}}{{\left(\rho {C}_{p}\right)}_{nf}}{\left(\displaystyle \frac{\partial u}{\partial y}\right)}^{2}.\end{array}\end{eqnarray} \tag{ 4 }$$

The relevant boundary conditions are:

$$\begin{eqnarray}&&u\left(x,0\right)={U}_{w}+{\mu }_{nf}\left(\displaystyle \frac{\partial u}{\partial y}\right),v\left(x,0\right)={V}_{w},-{k}_{0}\left(\displaystyle \frac{\partial T}{\partial y}\right)={h}_{f}\left({T}_{w}-T\right),\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&u\to 0,\displaystyle \frac{\partial u}{\partial y}\to 0,T\to {T}_{\infty }\,as\,y\to \infty .\end{eqnarray} \tag{ 6 }$$

Here, μ_nf is the nanofluid dynamic viscosity, κ_nf , (ρC_p )_nf and ρ_nf respectively denote the thermal conductivity, specfic heat capacity and density of nanofluid. V_w denotes how porous the stretching surface. u and v denote velocities in the respective directions of x and y. Q and q_r are defined to be the representative of the heat source and radiative heat flux, respectively. The other parameters k₀ and h_f are the thermal conducvity of solid and coefficient of heat transference rate, respectively.

2.5. Thermophysical features of the second grade nanofluid

The effective dynamic viscosity, density, heat capacitance and thermal conductivity of the nanofluids are defined as follow [39, 40].

$$\begin{eqnarray}&&{\mu }_{nf}={\mu }_{f}{\left(1-\phi \right)}^{-2.5}\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{\rho }_{nf}=\left(1-\phi \right){\rho }_{f}+\phi {\rho }_{s}\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{\left(\rho {C}_{p}\right)}_{nf}=\left(1-\phi \right){\left(\rho {C}_{p}\right)}_{f}+\phi {\left(\rho {C}_{p}\right)}_{s}\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{k}_{nf}}{{k}_{f}}=\left[\displaystyle \frac{\left({k}_{s}+\left(m-1\right){k}_{f}\right)-\left(m-1\right)\phi \left({k}_{f}-{k}_{s}\right)}{\left({k}_{s}+\left(m-1\right){k}_{f}\right)+\phi \left({k}_{f}-{k}_{s}\right)}\right]\end{eqnarray} \tag{ 10 }$$

Where, ϕ is the nanoparticle volumetric fraction coefficient. μ_f , ρ_f , κ_f , (C_p )_f and are dynamic viscosity, density, thermal conductivity and effective heat capacity of the basefluid correspondingly. Correspondingly, the other properties κ_s , ρ_s , (C_p )_s and are the thermal conductivity, density, and effective heat capacity of the nanoparticles.

2.6. Nanoparticles and base fluid features

2.7. Rosseland approximations

Using the Roseland approximation Brewster [45], one can write

$$\begin{eqnarray}&&{q}_{r}=-\displaystyle \frac{4{\sigma }^{* }}{3{k}^{* }}\displaystyle \frac{\partial {{\rm{T}}}^{4}}{\partial y},\end{eqnarray} \tag{ 11 }$$

Here k* is the absorption coefficient and σ* is the Stefan Boltzmann number .

Introducing stream functions ψ of the form

$$\begin{eqnarray}&&u=\displaystyle \frac{\partial \psi }{\partial y},v=-\displaystyle \frac{\partial \psi }{\partial x}.\end{eqnarray} \tag{ 12 }$$

and similarity variables of the form

$$\begin{eqnarray}&&\chi \left(x,y\right)=\sqrt{\displaystyle \frac{b}{{\nu }_{f}\left(1-\xi t\right)}}y,\psi \left(x,y\right)=\sqrt{\displaystyle \frac{{\nu }_{f}b}{\left(1-\xi t\right)}}xf\left(\chi \right),\theta \left(\chi \right)=\displaystyle \frac{T-{T}_{\infty }}{{T}_{w}-{T}_{\infty }}.\end{eqnarray} \tag{ 13 }$$

into equations (2)–(4). We get

$$\begin{eqnarray}&&f^{\prime\prime \prime} +{\phi }_{1}\left[{\phi }_{2}\left(f{f}^{{\prime\prime} }-{f}^{{\prime} 2}-A\left(\displaystyle \frac{\chi }{2}{f}^{{\prime\prime} }+{f}^{\prime }\right)\right)+\alpha \left(2{f}^{\prime }{f}^{{\prime\prime} ^{\prime} }+A\left(2{f}^{{\prime\prime} }+\displaystyle \frac{\chi }{2}{f}^{iv}\right)-{f}^{{\prime\prime} 2}-f{f}^{iv}\right)-\displaystyle \frac{1}{\phi 1}Kf^{\prime} \right]=0,\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&\theta ^{\prime\prime} \left(1+\displaystyle \frac{1}{{\phi }_{4}}{P}_{r}{N}_{r}\right)+{P}_{r}\displaystyle \frac{{\phi }_{3}}{{\phi }_{4}}\left[f\theta ^{\prime} -f^{\prime} \theta -A\left(\theta +\displaystyle \frac{\chi }{2}\theta ^{\prime} \right)+\theta \displaystyle \frac{Q}{{\phi }_{3}}+\displaystyle \frac{{E}_{c}}{{\phi }_{1}{\phi }_{3}}f{{\prime\prime} }^{2}\right]=0,\end{eqnarray} \tag{ 15 }$$

with

$$\begin{eqnarray}&&\left.\begin{array}{l}f\left(0\right)=S,f^{\prime} \left(0\right)=1+\displaystyle \frac{{\rm{\Lambda }}}{{\phi }_{1}}f^{\prime\prime} \left(0\right),\theta ^{\prime} \left(0\right)=-{B}_{\varsigma }\left(1-\theta \left(0\right)\right)\\ f^{\prime} \left(\chi \right)\to 0,f^{\prime\prime} \left(\chi \right)\to 0,\theta \left(\chi \right)\to 0,as\,\chi \to \infty 0\end{array}0\right\}\end{eqnarray} \tag{ 16 }$$

where ${\phi }_{i}^{{\prime} }s$ is 1 ≤ i ≤ 4 in equations (14)–(15) are as follow:

$$\begin{eqnarray}&&\left.\begin{array}{l}{\phi }_{1}={\left(1-\phi \right)}^{2.5},{\phi }_{2}=\left(1-\phi +\phi \displaystyle \frac{{\rho }_{s}}{{\rho }_{f}}\right),{\phi }_{3}=\left(1-\phi +\phi \displaystyle \frac{{\left(\rho {C}_{p}\right)}_{s}}{{\left(\rho {C}_{p}\right)}_{f}}\right)\\ {\phi }_{4}=\left(\displaystyle \frac{\left({k}_{s}+2{k}_{f}\right)-2\phi \left({k}_{f}-{k}_{s}\right)}{\left({k}_{s}+2{k}_{f}\right)+\phi \left({k}_{f}-{k}_{s}\right)}\right).\end{array}\right\}\end{eqnarray} \tag{ 17 }$$

3.1. Expression of parameters

3.2. Quantities for engineering interest

The skin friction (C_f ) alongside the local Nusselt number (Nu_x ) are stated as (refer Shah et al [38])

$$\begin{eqnarray}&&{C}_{f}=\displaystyle \frac{{\tau }_{w}}{{\rho }_{f}{U}_{w}^{2}},N{u}_{x}=\displaystyle \frac{x{q}_{w}}{{k}_{f}\left({T}_{w}-{T}_{\infty }\right)}\end{eqnarray} \tag{ 18 }$$

wherein τ_w and q_w represent the heat flux determined by

$$\begin{eqnarray}&&\left.\begin{array}{l}{\tau }_{w}={\left({\mu }_{nf}\displaystyle \frac{\partial u}{\partial y}+{\alpha }_{1}\left(\displaystyle \frac{{\partial }^{2}u}{\partial t\partial y}+u\displaystyle \frac{{\partial }^{2}u}{\partial x\partial y}+2\displaystyle \frac{\partial u}{\partial y}\displaystyle \frac{\partial u}{\partial x}+v\displaystyle \frac{{\partial }^{2}u}{{\partial }^{2}y}\right)\right)}_{y=0},\\ {q}_{w}=-{k}_{nf}\left(1+\displaystyle \frac{16}{3}\displaystyle \frac{{\sigma }^{* }{T}_{\infty }^{3}}{{\kappa }^{* }{\nu }_{f}{(\rho {C}_{p})}_{f}}\right){\left(\displaystyle \frac{\partial T}{\partial y}\right)}_{y=0}.\end{array}\right\}\end{eqnarray} \tag{ 19 }$$

By using dimensionless transformations (13), one attains

$$\begin{eqnarray}&&\left.\begin{array}{l}{C}_{f}R{e}_{x}^{\displaystyle \frac{1}{2}}=\left(\displaystyle \frac{{f}^{{\prime} ^{\prime} }\left(0\right)}{{\left(1-\phi \right)}^{2.5}}+\alpha \left(3f^{\prime\prime} \left(0\right){f}^{\prime }\left(0\right)-{f}^{{\prime} ^{\prime\prime} }\left(0\right)f\left(0\right)+\displaystyle \frac{A}{2}\left(\chi {f}^{{\prime} ^{\prime\prime} }\left(0\right)+3{f}^{{\prime} ^{\prime} }\left(0\right)\right)\right)\right)\\ N{u}_{x}R{e}_{x}^{-\displaystyle \frac{1}{2}}=-\displaystyle \frac{{k}_{nf}}{{k}_{f}}\left(1+Nr\right)\theta ^{\prime} \left(0\right).\end{array}\right\}\end{eqnarray} \tag{ 20 }$$

where C_f indicates the reduced skin friction and Nu_x denotes Nusselt number. $R{e}_{x}=\tfrac{{U}_{w}x}{{\nu }_{f}}$ is local Re depends upon the stretching velocity (U_w (x)).

For solving solutions to modelled equations, the shooting method [46] is used. The localized solution of the (14)–(15) equations, subject to (16) constraints, is found using shooting technique. The methodology of the shooting scheme (figure 2)is given as follows:

The initiative step of this method requires first order system of ODEs. To fulfill these criteria, conversion of (14)–(16) into first order system yields

$$\begin{eqnarray}&&{z}_{1}=f^{\prime} ,\end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray}&&{z}_{2}={z}_{1}^{{\prime} },\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&{z}_{3}={z}_{2}^{{\prime} },\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&{z}_{4}=\theta ^{\prime} ,\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}&&{z}_{2}^{{\prime} }+{\phi }_{1}\left[\begin{array}{c}{\phi }_{2}\left(f{z}_{2}-{{z}_{1}}^{2}-A\left(\displaystyle \frac{\chi }{2}{z}_{2}+{z}_{1}\right)\right)+\alpha \left(2{z}_{1}{z}_{2}^{{\prime} }+A\left(2{z}_{2}+\displaystyle \frac{\chi }{2}{z}_{3}^{{\prime} }\right)-{{z}_{2}}^{2}-f{z}_{3}^{{\prime} }\right)\\ -\displaystyle \frac{1}{\phi 1}K{z}_{1}=0,\end{array}\right]\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&{z}_{4}^{{\prime} }\left(1+\displaystyle \frac{1}{{\phi }_{4}}{P}_{r}{N}_{r}\right)+{P}_{r}\displaystyle \frac{{\phi }_{3}}{{\phi }_{4}}\left[f{z}_{4}-{z}_{1}\theta -A\left(\theta +\displaystyle \frac{\chi }{2}{z}_{4}\right)+\theta \displaystyle \frac{Q}{{\phi }_{3}}+\displaystyle \frac{{E}_{c}}{{\phi }_{1}{\phi }_{3}}{z}_{2}^{2}\right]=0,\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}&&f\left(0\right)=S,{z}_{1}\left(0\right)=1+\displaystyle \frac{{\rm{\Lambda }}}{{\phi }_{1}}{z}_{2}\left(0\right),{z}_{3}\left(0\right)=-{B}_{\varsigma }\left(1-\theta \left(0\right)\right),{z}_{1}\left(\infty \right)\to 0,{z}_{2}\left(\infty \right)\to 0,\,\theta (\infty )\to 0.\end{eqnarray} \tag{ 27 }$$

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We selected a discrete η_∞ value with care to ensure that far-field boundary constraints are fulfilled asymptotically. In this research, a relevant determinate value of η_∞ is taken to be η_∞  < 9 so that not only numerical solutions converge, but also boundary conditions specified at infinity satisfy asymptotically. For convergence, a tolerance for associated errors of up to 10⁻⁸ is considered and the step size is chosen as Δη = 0.001. The validity of the technique was determined by comparing the outcomes of the heat transference rate of current method with those found in the literature [47–50]. The comparison of consistencies found across the studies is summarised in table 2. However, The current work's findings are extremely accurate.

The current problem signifies the impact of heat source and radiative heat flux on viscous second grade nanofluid flow with porous media. The numerical solution of the nonlinear coupled ODEs is attained by applying shooting process. In this segment, we scrutinized the behavior of numerous dimensionless physical parameters on the velocity and thermal gradients by using suitable graphs. Figures are illustrated to get insight of the related variables on velocity and thermal gradients.

6.1. Velocity profile

Figure 3 displays the levering of K on velocity gradient. Here, escalation in K deteriorates the velocity gradient. The porosity parameter caused the liquid stream to be disturbed. As a result, it creates resistance to liquid stream, lowering the velocity gradient automatically. Further, the velocity gradient diminishes faster in Copper-engine oil second grade nanoliquid when compared to alumina-engine oil second grade nanoliquid. The power of A on velocity profile is illustrated in figure 4. The upsurge in A declines the velocity gradient. Furthermore, the velocity gradient diminishes faster in copper-engine oil second grade nanoliquid when compared to alumina-engine oil second grade nanoliquid for rise in values of A. The encouragement of Λ on velocity profile is depicted in figure 5. The inclined values of Λ declines the fluid velocity. Here, the velocity gradient diminishes slower in alumina-engine oil second grade nanoliquid when compared to copper -engine oil second grade nanoliquid for rise in values of Λ. The provocation of S on velocity gradient is demonstrated in figure 6. The boost up values of S declines the velocity gradient. Further, the velocity gradient diminishes faster in copper-engine oil second grade nanoliquid when compared to alumina-engine oil second grade nanoliquid. Figure 7 displays the fluctuation in velocity gradient for inclined values of α . The upsurge in α progresses the velocity of the fluid motion. Here, copper-engine oil second grade nanoliquid show less heat transfer features when compared to alumina-engine oil second grade nanoliquid for rising values of α.

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6.2. Thermal profile

The influence of N _r on heat transference is illustrated in figure 8. Here, gain in radiation parameter values improves the heat transfer. The inclination in radiation parameter produces inner heat which reasons for the augmentation of thermal gradient. Further, both liquids show similar increasing behaviour for boost up values of radiation parameter. The domination of Q on heat transference is represented in figure 9. The upsurge in Q declines the heat transfer. Here, the heat transfer diminishes faster in alumina-engine oil second grade nanoliquid when compared to copper-engine oil second grade nanoliquid. Figure 10 displays the variations in thermal gradient for diverse values of B _ς . Here, boost up values of B_ς improves the heat transfer. Further, copper-engine oil second grade nanoliquid show better heat transference for escalating values of B _ς when compared to alumina-engine oil second grade nanoliquid. Figure 11 portrays the power of Ec on thermal gradient. The escalation in Ec decreases the thermal profile. Furthermore, the heat transfer diminishes faster in alumina-engine oil second grade nanoliquid when compared to copper-engine oil second grade nanoliquid.

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6.3. Skin friction and rate of heat transfer

The computational studies of flow for Cu and Al₂O₃ non-Newtonian engine oil based nanoliquids above a permeable stretching sheet were conducted in this study. The arising equations for momentum, and energy were changed into ODEs using similarity transformations. The solutions are presented to record the encouragement of several pertinent parameters on velocity, and thermal gradient. Results concludes that, by rising up nanoparticle volume fraction parameter ϕ and second grade parameter α, a deterioration in the velocity gradient is detected. The variables have a massive effect on the heat transfer very close to the wall and are slightly away from the wall. An upsurge in the thermal gradient for the set of parameters α, K, N_r , Q, E_c , B_ς and S > 0 is seen whereas it is observed to decline with increase in parameter S < 0 and A. The comparative heat transference rate of second grade nanofluids (Cu−EO) gradually more upsurges as compared to (Al₂O₃ − EO) nanofluids. The increase in flow governing parameters increases the rate of heat transfer. Further, copper-engine oil second grade nanoliquid shows high rate of heat transference as compared to alumina-engine oil second grade nanoliquid.

The data generated and/or analysed during the current study are not publicly available for legal/ethical reasons but are available from the corresponding author on reasonable request.

The authors declare that there is no conflict of interest.