The MHD graphene−CMC−water nanofluid past a stretchable wall with Joule heating and velocity slip impact: Coolant application

The heat transport mechanism has an engrossing application in effective heat management for the automobile industry and the biomedical industry. The analysis of the MHD graphene−carboxymethyl cellulose (CMC) solution−water nanofluid past a stretchable wall with Joule heating and velocity slip impact is performed in this regard. A graphene-based nanofluid is considered. The dynamic model is used to simplify the complicated ordinary differential equations into non-dimensional forms, which are then evaluated analytically. Numerical data and graphs are produced to analyze the consequences of a physical entity with the aid of Maple 17. Moreover, the velocity field is decreased, while the magnitude of the magnetic parameter is increased. A decrease in θ(η) is observed as a result of an increase in ϕ. It is noted that a rise in the magnetic parameter causes a fall in the temperature distribution. It is perceived that −f′′(0) is decreased with an augmentation in β_s, and an opposite trend is shown for ϕ. The velocity profile is the growing function of M_gn, β_s, and K_ve, with the reversed mode shown in case of ϕ. The temperature profile is the declining function of Pr, E_crt, ϕ, and χ, with a contradictory trend observed for M_gn and β_s. The flow regime is displayed against the viscoelastic parameter.

1 Introduction

In recent times, numerous fascinating energy applications have made use of nanotechnology to provide more effective and eco-friendly products and services. Moreover, one crucial aspect that is necessary to bring down the cost of maintenance or installation is the effective heat transport distribution across a power network. The optimal transmission of thermal qualities, such as density, specific heat, viscosity, and thermal conductivity, is essential for the optimum thermal efficiency. In this context, we consider the graphene nanoparticles to improve the thermal characteristics of the host fluid (CMC−water). In order to increase the rate of heat exchange, the flow patterns of nanofluids across thermal energy technologies are crucial. Several nanofluids have very engrossing applications in several fields, such as electronics cooling, automotive engine cooling, heat exchangers, boiling heat transport, solar collectors, medicine, and nuclear system cooling [1]. For the first time, [2] introduced the idea of nanofluids and empirically supported his concept. Gamachu and Ibrahim [3] investigated the mixed convection hybrid viscoelastic nanofluid. The heat exchange of the viscoelastic nanofluid past the stretched surface with many impacts was studied by [4]. In [5], a corrugated channel was considered to study the heat transport of the graphene nanofluid. [6] examined the influence of viscous dissipation on the viscoelastic nanofluid due to a circular cylinder. Numerical examination of the graphene nanofluid because of the stretchable wall was conducted by [7]. [8] directed the enhanced thermal progress of the graphene hybrid nanofluid past a stretched sheet. The impact of zero nanoparticle flux and hall current regarding the movement of the nanofluid above a stretched wall was discussed by [9]. Several other researchers studied the graphene nanofluid in [10–15]. Recently, [16] examined the MHD nanofluid past a convectively heated Riga plate placed horizontally, embedded in the porous (Darcy−Forchheimer) medium. The EMHD flow of a heterogeneous micropolar mixture with different concentrations of water, ethylene glycol, and copper oxide nanoparticles was discussed by [17]. Due to the distributions of the polymer/CNT structure nanocomposite material, morphological nanolayers have an influence on the movement of hybrid nanofluids which was studied by [18]. [19] analyzed the tetragonal nanoparticles with a variety of density and conductivity characteristics flowing in 3D using non-linear Boussinesq and Rosseland estimations. Recent research investigates how the presence of nanoparticles can significantly improve heat exchange in a variety of physical geometries [20–22]. [23] examined the analytical answers for a thermal conductor force peristaltic flow to temperature-dependent nanofluid viscosity. Utilizing two distinct methods for nanofluid analysis, the silver−water nanofluid electro-osmotic movement that is controlled by peristalsis was investigated by [24]. [25] studied the electro-osmotically assisted peristaltic pressurization of the MoS₂ Rabinowitsch nanofluid and produced randomness. The use of novel nanofluids throughout clinical isolates to fight Staphylococcus aureus was investigated by [26]. Many researchers talked about boundary layer problems in various media [27–29].

The investigation of the convective boundary layer flows across a stretched surface has a broad range of uses in the polymer production sector. For example, it is employed in manufacturing fibers, polymeric extruder, and fiber glass, sheet manufacturing, condensing of aqueous films, food production, growth of crystals, and fabrication of artificial films [4]. The idea of a boundary layer flow above a stretching sheet was promoted by [30]. [31] examined the velocity slip impact on the MHD nanofluid because of the stretched sheet. The study of the magnetite nanofluid provoked by a stretching wall was observed by [32]. [33] introduced the system disorder inside the nanofluid past a stretchable wall. The MHD nanofluid with two different nanoparticles due to a heated stretched wall was analyzed by [34]. [35] revealed the linearly stretched sheet-induced double-diffusive time-dependent magnetohydrodynamic flow of the nanofluid involving convection boundary constraints. Optimizing entropy production with an unstable stagnating Casson nanofluid flow having dual chemical changes across a stretched surface was discussed by [36]. The investigation of the nanofluid with many effects above a stretching sheet was presented in [37–39].

From the potential applications mentioned previously, we analyzed the MHD graphene−CMC−water nanofluid past a stretchable wall with Joule heating and velocity slip impact. The closed-form solutions were derived analytically. To understand the behavior of various physical parameters on the fluid temperature, Nusselt number, velocity, and skin friction, the numeric tables and graphs were portrayed.

2 Problem statement

In framing the model, the flow of an incompressible and steady MHD viscoelastic nanofluid (Walter’s liquid B type) across a stretchable sheet in two dimensions is investigated. The graphene and carboxymethyl cellulose (CMC) solution−water are taken as nanoparticles and host fluid, respectively. The fluid is made up of the region y > 0, where the x-axis runs alongside the stretched surface in a flow pattern and the y-axis runs perpendicular to the flow. With a velocity of $\underline{u}=cx+\beta \frac{\partial \underline{u}}{\partial y}$, the surface is pulled in the x dimension. Furthermore, the magnetic field B₀ is supplied normally toward the flowing fluid, and the consequences of thermal radiation are also utilized. The basic equations regulating the flow are as follows [6]:

$\frac{\partial \underline{u}}{\partial x}+\frac{\partial \underline{v}}{\partial y}=0,(1)$

$\underline{u}\frac{\partial \underline{u}}{\partial x}+\underline{v}\frac{\partial \underline{u}}{\partial y}=\frac{{\mu }_{\mathit{gnf}}}{{\rho }_{\mathit{gnf}}}\frac{{\partial }^2\underline{u}}{\partial y^2}-k_{\mathit{0in}}\left(\underline{u}\frac{{\partial }^3\underline{u}}{\partial x\partial y^2}+\underline{v}\frac{{\partial }^3\underline{u}}{\partial y^3}+\frac{\partial \underline{u}}{\partial x}\frac{{\partial }^2\underline{u}}{\partial y^2}-\frac{{\partial }^2\underline{u}}{\partial x\partial y}\frac{\partial \underline{u}}{\partial y}\right)-\frac{{\sigma }_{\mathit{gnf}}B{\left(x\right)}^2}{{\rho }_{\mathit{gnf}}}\underline{u}.(2)$

The problem’s appropriate boundary criteria are as follows [40]:

$\left.\begin{array}{cc}\hfill \hfill & \hfill \underline{u}=cx+\beta \frac{\partial \underline{u}}{\partial y},\underline{v}=0\text{at}y=0,\hfill \\ \hfill \hfill & \hfill \underline{u}=0\text{as}y\to \infty ,\hfill \end{array}\right\}(3)$

where c is the stretching rate, β is the slip factor, ν_gnf is the kinematic viscosity, k_0in is the initial relaxation time distribution function, ρ_gnf is the density, B(x) is the magnetic parameter, $\underline{u}$ and $\underline{v}$ are the coordinates of the flowing fluid, μ_gnf is the dynamic viscosity, and α_gnf is the thermal diffusivity. The nanofluid is represented by the notation gnf. The thermal attributes are represented as follows [40]:

$\left.\begin{array}{cc}\hfill \hfill & \hfill {\alpha }_{\mathit{gnf}}=\frac{k_{\mathit{gnf}}}{{\left(\rho c_p\right)}_{\mathit{gnf}}},{\mu }_{\mathit{gnf}}=\frac{{\mu }_{fc}}{{\left(1-\varphi \right)}^{2.5}},\hfill \\ \hfill \hfill & \hfill {\left(\rho c_p\right)}_{\mathit{gnf}}=\varphi {\left(\rho c_p\right)}_{\mathit{gnf}}+\left(1-\varphi \right){\left(\rho c_p\right)}_{fc},\hfill \\ \hfill \hfill & \hfill {\rho }_{\mathit{gnf}}=\varphi \left({\rho }_{\mathit{gnf}}\right)+\left(1-\varphi \right){\rho }_{fc},{\nu }_{\mathit{gnf}}=\frac{{\mu }_{\mathit{gnf}}}{{\rho }_{\mathit{gnf}}},\hfill \\ \hfill \hfill & \hfill k_{\mathit{gnf}}=\frac{\left(k_{\mathit{gnf}}{k}_{fc}+2k_{fc}^2\right)-\left(k_{fc}^2-k_{fc}{k}_{\mathit{gnf}}\right)2\varphi }{\left(k_{\mathit{gnf}}+2k_{fc}\right)+\varphi \left(k_{fc}-k_{\mathit{gnf}}\right)},\hfill \\ \hfill \hfill & \hfill {\sigma }_{\mathit{gnf}}=\frac{3\varphi {\sigma }_{fc}\left(\frac{{\sigma }_{\mathit{gnf}}}{{\sigma }_{fc}}-1\right)}{-\varphi \left(\frac{{\sigma }_{\mathit{gnf}}}{{\sigma }_{fc}}-1\right)+\left(\frac{{\sigma }_{\mathit{gnf}}}{{\sigma }_{fc}}+2\right)}+{\sigma }_{fc}.\hfill \end{array}\right\}(4)$

In Eq. 4, ρ_fc is the density, ${\left(\rho c_p\right)}_{fc}=$ is the effective heat capacity, ϕ is the particle volume ratio, k_fc is the thermal conductivity of the base liquid, and k_gnf is the nanoparticle thermal conductivity. The accompanying similarity variables have been proposed to non-dimensionalize the basic equations [40]:

$\underline{u}=cx{f}^{\prime }\left(\eta \right),\underline{v}=-{\left(\nu c\right)}^{1/2}f\left(\eta \right),\eta =y{\left(\frac{c}{\nu }\right)}^{1/2},\theta \left(\eta \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }}.(5)$

Applying Eqs 5, 2 and Eq. 3 become as:

$f‴-{\tau }_2{\tau }_1{f^{\prime }}^2-{\tau }_1{M}_{gn}{f}^{\prime }+{\tau }_2{\tau }_{1}f{f}^{″}-{\tau }_2{\tau }_1{K}_{ve}\left(-f{f}^{iv}+2f^{\prime }f‴-{f^{″}}^2\right)=0,(6)$

$\left.\begin{array}{cc}\hfill \hfill & \hfill f\left(\eta \right)=0,f^{\prime }\left(\eta \right)=1+{\beta }_s{f}^{″}\left(0\right),\text{at}\eta =0,\hfill \\ \hfill \hfill & \hfill f^{\prime }\left(\eta \right)\to 0\text{as}\eta \to \infty .\hfill \end{array}\right\}(7)$

Here, the slip parameter is β_s = β(c/ν)^1/2, τ₁ = (1 − ϕ)^2.5, ${\tau }_2=\left(\frac{{\rho }_{\mathit{gnf}}}{{\rho }_{fc}}\varphi -\varphi +1\right)$, $K_{ve}=\frac{c{k}_{\mathit{0in}}}{{\nu }_{fc}}$ is the viscoelastic parameter, and $M_{gn}=\frac{2{\beta }_s{\sigma }_{fc}{B}_0^2}{\rho }$ is the Hartmann number. The solution of Eq. 6 in a closed form is as follows [41]:

$f\left(\eta \right)={\gamma }_1+{\gamma }_2{e}^{-\mathrm{\Upsilon }\eta }.(8)$

Using Eq. 7, we determine the answer to Eq. 6, as shown as follows:

$f\left(\eta \right)=\frac{1}{{\beta }_s{\mathrm{\Upsilon }}^2+\mathrm{\Upsilon }}-\frac{e^{-\mathrm{\Upsilon }\eta }}{{\beta }_s{\mathrm{\Upsilon }}^2+\mathrm{\Upsilon }}.(9)$

Applying Eq. 6 and Eq. 9 together, we gain

$\begin{array}{ll}\hfill \mathrm{\Upsilon }& =\frac{\sqrt[3]{{\lambda }_1+12\sqrt{3}\sqrt{{\tau }_1\left({\lambda }_2-{\lambda }_3\right)}{\beta }_s-8}}{6{\beta }_s}+\frac{2{\lambda }_4}{3{\beta }_s\sqrt[3]{{\lambda }_5+12\sqrt{3}\sqrt{{\tau }_1\left({\lambda }_6-{\lambda }_7\right)}{\beta }_s-24}}\hfill \\ \hfill & +\frac{{\tau }_1{\tau }_2{K}_{\mathit{\text{ve}}}-1}{3{\beta }_s},\hfill \end{array}(10)$

where

${\lambda }_1=8{{\tau }_1}^3{{\tau }_2}^3{K_{\mathit{\text{ve}}}}^3+36{{\tau }_1}^2{\tau }_2{K}_{\mathit{\text{ve}}}{{\beta }_s}^2{M}_{\mathit{\text{gn}}}-24{{\tau }_1}^2{{\tau }_2}^2{K_{\mathit{\text{ve}}}}^2+72{\tau }_1{M}_{\mathit{\text{gn}}}{{\beta }_s}^2+108{\tau }_1{\tau }_2{{\beta }_s}^2+24{\tau }_1{\tau }_2{K}_{\mathit{\text{ve}}},(11)$

${\lambda }_2=-{K_{\mathit{\text{ve}}}}^2{M_{\mathit{\text{gn}}}}^2{{\beta }_s}^2{{\tau }_1}^3{{\tau }_2}^2+4{K_{\mathit{\text{ve}}}}^3{M}_{\mathit{\text{gn}}}{{\tau }_1}^3{{\tau }_2}^3+4{K_{\mathit{\text{ve}}}}^3{{\tau }_1}^3{{\tau }_2}^4-4{M_{\mathit{\text{gn}}}}^3{{\beta }_s}^4{{\tau }_1}^2+20K_{\mathit{\text{ve}}}{M_{\mathit{\text{gn}}}}^2{{\beta }_s}^2{{\tau }_1}^2{\tau }_2+18K_{\mathit{\text{ve}}}{M}_{\mathit{\text{gn}}}{{\beta }_s}^2{{\tau }_1}^2{{\tau }_2}^2,(12)$

$\begin{array}{ll}\hfill {\lambda }_3& =-12{K_{\mathit{\text{ve}}}}^2{M}_{\mathit{\text{gn}}}{{\tau }_1}^2{{\tau }_2}^2-12{K_{\mathit{\text{ve}}}}^2{{\tau }_1}^2{{\tau }_2}^3+8{M_{\mathit{\text{gn}}}}^2{{\beta }_s}^2{\tau }_1+36M_{\mathit{\text{gn}}}{{\beta }_s}^2{\tau }_1{\tau }_2\hfill \\ \hfill & +27{{\beta }_s}^2{\tau }_1{{\tau }_2}^2+12K_{\mathit{\text{ve}}}{M}_{\mathit{\text{gn}}}{\tau }_1{\tau }_2+12K_{\mathit{\text{ve}}}{\tau }_1{{\tau }_2}^2-4M_{\mathit{\text{gn}}}-4{\tau }_2,\hfill \end{array}(13)$

${\lambda }_4={{\tau }_1}^2{{\tau }_2}^2{K_{\mathit{\text{ve}}}}^2+3{\tau }_1{M}_{\mathit{\text{gn}}}{{\beta }_s}^2-2{\tau }_1{\tau }_2{K}_{\mathit{\text{ve}}}+1,(14)$

${\lambda }_5=8{{\tau }_1}^3{{\tau }_2}^3{K_{\mathit{\text{ve}}}}^3+36{{\tau }_1}^2{\tau }_2{K}_{\mathit{\text{ve}}}{{\beta }_s}^2{M}_{\mathit{\text{gn}}}-24{{\tau }_1}^2{{\tau }_2}^2{K_{\mathit{\text{ve}}}}^2+72{\tau }_1{M}_{\mathit{\text{gn}}}{{\beta }_s}^2+108{\tau }_1{\tau }_2{{\beta }_s}^2+24{\tau }_1{\tau }_2{K}_{\mathit{\text{ve}}},(15)$

$\begin{array}{ll}\hfill {\lambda }_6& =-{K_{\mathit{\text{ve}}}}^2{M_{\mathit{\text{gn}}}}^2{{\beta }_s}^2{{\tau }_1}^3{{\tau }_2}^2+4{K_{\mathit{\text{ve}}}}^3{M}_{\mathit{\text{gn}}}{{\tau }_1}^3{{\tau }_2}^3+4{K_{\mathit{\text{ve}}}}^3{{\tau }_1}^3{{\tau }_2}^4-4{M_{\mathit{\text{gn}}}}^3{{\beta }_s}^4{{\tau }_1}^2\hfill \\ \hfill & +20K_{\mathit{\text{ve}}}{M_{\mathit{\text{gn}}}}^2{{\beta }_s}^2{{\tau }_1}^2{\tau }_2+18K_{\mathit{\text{ve}}}{M}_{\mathit{\text{gn}}}{{\beta }_s}^2{{\tau }_1}^2{{\tau }_2}^2-12{K_{\mathit{\text{ve}}}}^2{M}_{\mathit{\text{gn}}}{{\tau }_1}^2{{\tau }_2}^2,\hfill \end{array}(16)$

${\lambda }_7=12{K_{\mathit{\text{ve}}}}^2{{\tau }_1}^2{{\tau }_2}^3+8{M_{\mathit{\text{gn}}}}^2{{\beta }_s}^2{\tau }_1+36M_{\mathit{\text{gn}}}{{\beta }_s}^2{\tau }_1{\tau }_2+27{{\beta }_s}^2{\tau }_1{{\tau }_2}^2+12K_{\mathit{\text{ve}}}{M}_{\mathit{\text{gn}}}{\tau }_1{\tau }_2+12K_{\mathit{\text{ve}}}{\tau }_1{{\tau }_2}^2-4M_{\mathit{\text{gn}}}-4{\tau }_2,(17)$

where γ₁, γ₂, and ϒ are the constants.

3 Heat transfer analysis

This section describes the heat exchange investigation under the influence of Joule heating and thermal radiation phenomenon. The following is the elementary equation [42]:

$\underline{u}\frac{\partial T}{\partial x}+\underline{v}\frac{\partial T}{\partial y}={\alpha }_{\mathit{gnf}}\frac{{\partial }^{2}T}{\partial y^2}-\frac{1}{{\left(\rho c_p\right)}_{\mathit{gnf}}}\frac{\partial q_{\mathit{rad}}}{\partial y}+\frac{{\sigma }_{\mathit{gnf}}B{\left(x\right)}^2}{{\left(\rho c_p\right)}_{\mathit{gnf}}}{\underline{u}}^2,(18)$

where

$q_{\mathit{rad}}=-\frac{{\sigma }^{*}}{3k^{*}}\frac{\partial T^4}{\partial y}.(19)$

Putting Eq. 19 into Eq. 18, we acquire the following:

$\underline{u}\frac{\partial T}{\partial x}+\underline{v}\frac{\partial T}{\partial y}={\alpha }_{\mathit{gnf}}\frac{{\partial }^{2}T}{\partial y^2}+\frac{4^2{\sigma }^{*}{T}_{\infty }^3}{k^{*}3{\left(\rho c_p\right)}_{\mathit{gnf}}}\frac{{\partial }^{2}T}{\partial y^2}+\frac{{\sigma }_{\mathit{gnf}}B{\left(x\right)}^2}{{\left(\rho c_p\right)}_{\mathit{gnf}}}{\underline{u}}^2,(20)$

and the boundary restrictions are as follows:

$\left.\begin{array}{cc}\hfill \hfill & \hfill T=T_w=T_{\infty }+T_{\mathit{0cr}}{\left(x/c\right)}^2\text{at}y=0,\hfill \\ \hfill \hfill & \hfill T\to T_{\infty }\text{as}y\to \infty .\hfill \end{array}\right\}(21)$

Here, T is the temperature field, c is the characteristic length, T_0cr is the constant reference temperature, α_gnf is the thermal diffusivity, ${\left(c_p\right)}_{\mathit{gnf}}$ is the specific heat, T_w is the wall temperature, k* is the mass absorption coefficient, T_∞ = is the free stream temperature, and σ* is the Stefan−Boltzmann constant. After applying Eq. 5 and Eq. 21 together, the corresponding non-dimensional energy equation is obtained as follows:

$\frac{\mathrm{\Omega }}{\mathit{Pr}}{\theta }_{\eta \eta }-2f_{\eta }\theta +f{\theta }_{\eta }+\frac{E_{\mathit{crt}}{M}_{gn}}{{\tau }_4}{f}_{\eta }^2=0,(22)$

where

$\begin{array}{ll}\hfill & E_{\mathit{crt}}=\frac{{\underline{u}}^2}{\mathrm{\Delta }Tcp},\mathit{Pr}=\frac{{\nu }_{fc}}{{\alpha }_{fc}},\chi =\frac{k^{*}{k}_{fc}}{2^2{\sigma }^{*}{T}_{\infty }^3},\mathrm{\Omega }=\left(\frac{{\tau }_3}{{\tau }_4}\frac{3\chi {\tau }_3+4}{3\chi {\tau }_3}\right)\hfill \\ \hfill & {\tau }_3=\frac{\left(k_{\mathit{gnf}}+2k_{fc}\right)-2\varphi \left(k_{fc}-k_{\mathit{gnf}}\right)}{\left(k_{\mathit{gnf}}+2k_{fc}\right)+2\varphi \left(k_{fc}-k_{\mathit{gnf}}\right)},{\tau }_4=\left(1-\varphi +\varphi \frac{{\left(\rho cp\right)}_{\mathit{gnf}}}{{\left(\rho cp\right)}_{fc}}\right).\hfill \end{array}(23)$

Here, χ is the radiation entity, E_crt is the Eckert number, and Pr is the Prandtl number. The modified boundary constraints are as follows:

$\left.\begin{array}{cc}\hfill \hfill & \hfill \theta \left(\eta \right)=1\text{at}\eta =0,\hfill \\ \hfill \hfill & \hfill \theta \left(\eta \right)\to 0\text{as}\eta \to \infty .\hfill \end{array}\right\}(24)$

Consequently, it is simple to obtain by entering Eq. 9 into Eq. 24, expressed as

$\frac{\mathrm{\Omega }}{\mathit{Pr}}{\theta }_{\eta \eta }-2\left(\frac{e^{-\mathrm{\Upsilon }\eta }}{{\beta }_s\mathrm{\Upsilon }+1}\right)\theta +\left(\frac{1}{{\beta }_s{\mathrm{\Upsilon }}^2+\mathrm{\Upsilon }}-\frac{e^{-\mathrm{\Upsilon }\eta }}{{\beta }_s{\mathrm{\Upsilon }}^2+\mathrm{\Upsilon }}\right){\theta }_{\eta }+\frac{E_{\mathit{crt}}{M}_{gn}}{{\tau }_4}{\left(\frac{e^{-\mathrm{\Upsilon }\eta }}{{\beta }_s\mathrm{\Upsilon }+1}\right)}^2=0.(25)$

To convert Eq. 26 into Kummer’s ordinary differential equation, a latest expression is established:

$\zeta =-\frac{\mathit{Pr}{e}^{-\mathrm{\Upsilon }\eta }}{\mathrm{\Omega }{\mathrm{\Upsilon }}^2\left({\beta }_s\mathrm{\Upsilon }+1\right)}.(26)$

As a consequence, Eq. 26 is converted into Kummer’s ordinary differential equation, which is as follows:

$\zeta \frac{{\partial }^2\theta }{\partial {\zeta }^2}+\left(\kappa -\zeta \right)\frac{\partial \theta }{\partial \zeta }+2\theta =-\frac{E_{\mathit{crt}}{M}_{gn}}{{\tau }_4}{\left(\frac{e^{-\mathrm{\Upsilon }\eta }}{{\beta }_s\mathrm{\Upsilon }+1}\right)}^2,(27)$

where κ = (1 − κ₁) and ${\kappa }_1=\frac{\mathit{Pr}}{\mathrm{\Omega }{\mathrm{\Upsilon }}^2({\beta }_s\mathrm{\Upsilon }+1)}$. The updated boundary requirements are as follows:

$\theta \left(\zeta \right)=1,\theta \left(0\right)=0.(28)$

With respect to Kummer’s functions [43], the closed-form solution of Eq. 28 accompanying Eq. 29 is:

$\theta (\zeta )=\frac{-(\zeta {\mathrm{\Psi }}_1{\mathrm{\Psi }}_2{\mathit{\text{E}}}_{\mathit{\text{crt}}},{\mathrm{\Upsilon }}^2)\mathrm{\Omega }}{2\mathit{Pr}{\tau }_4{\mathrm{\Psi }}_4{\mathrm{\Psi }}_5}+\frac{{\left(-\frac{\zeta {\mathrm{\Psi }}_6{\mathrm{\Upsilon }}^2\mathrm{\Omega }\left({\beta }_s\mathrm{\Upsilon }+1\right)}{\mathit{Pr}}\right)}^{{\mathrm{\Psi }}_7}{\mathrm{\Psi }}_2{\mathrm{\Psi }}_9}{2{{\mathrm{\Psi }}_6}^{{\mathrm{\Psi }}_7}{\mathrm{\Psi }}_3{\mathrm{\Omega }}^2{\mathrm{\Upsilon }}^4{\tau }_4{\mathrm{\Psi }}_4{\mathrm{\Psi }}_5}+\frac{\zeta M_{gn}{\mathit{\text{E}}}_{\mathit{\text{crt}}}{\mathrm{\Upsilon }}^2\mathrm{\Omega }}{2\mathit{Pr}{\tau }_4},(29)$

where

${\mathrm{\Psi }}_1={\left(1-\frac{\mathit{Pr}}{\left({\beta }_s\mathrm{\Upsilon }+1\right)\mathrm{\Omega }{\mathrm{\Upsilon }}^2}\right)}^2+\left(2\zeta +1\right)\left(1-\frac{\mathit{Pr}}{\left({\beta }_s\mathrm{\Upsilon }+1\right)\mathrm{\Omega }{\mathrm{\Upsilon }}^2}\right)+{\zeta }^2+2\zeta ,$

${\mathrm{\Psi }}_2=M\left(-2+\frac{\mathit{Pr}}{\left({\beta }_s\mathrm{\Upsilon }+1\right)\mathrm{\Omega }{\mathrm{\Upsilon }}^2},1+\frac{\mathit{Pr}}{\left({\beta }_s\mathrm{\Upsilon }+1\right)\mathrm{\Omega }{\mathrm{\Upsilon }}^2},-\zeta \right),$

${\mathrm{\Psi }}_3=M\left(-2+\frac{\mathit{Pr}}{\left({\beta }_s\mathrm{\Upsilon }+1\right)\mathrm{\Omega }{\mathrm{\Upsilon }}^2},1+\frac{\mathit{Pr}}{\left({\beta }_s\mathrm{\Upsilon }+1\right)\mathrm{\Omega }{\mathrm{\Upsilon }}^2},-\frac{\mathit{Pr}}{\left({\beta }_s\mathrm{\Upsilon }+1\right)\mathrm{\Omega }{\mathrm{\Upsilon }}^2}\right),$

${\mathrm{\Psi }}_4=1-\frac{\mathit{Pr}}{\left({\beta }_s\mathrm{\Upsilon }+1\right)\mathrm{\Omega }{\mathrm{\Upsilon }}^2},{\mathrm{\Psi }}_5=2-\frac{\mathit{Pr}}{\left({\beta }_s\mathrm{\Upsilon }+1\right)\mathrm{\Omega }{\mathrm{\Upsilon }}^2},{\mathrm{\Psi }}_6=-\frac{\mathit{Pr}}{\left({\beta }_s\mathrm{\Upsilon }+1\right)\mathrm{\Omega }{\mathrm{\Upsilon }}^2},$

${\mathrm{\Psi }}_7=\frac{\mathit{Pr}}{\left({\beta }_s\mathrm{\Upsilon }+1\right)\mathrm{\Omega }{\mathrm{\Upsilon }}^2},{\mathrm{\Psi }}_8=\frac{-2\zeta {\mathit{\text{E}}}_{\mathit{\text{crt}}}{M}_{gn}{\mathrm{\Upsilon }}^4{\mathrm{\Omega }}^2\mathit{Pr}}{\left({\beta }_s\mathrm{\Upsilon }+1\right)\mathit{Pr}},$

${\mathrm{\Psi }}_9=2{\mathrm{\Omega }}^2{\mathrm{\Upsilon }}^4{\tau }_4{{\mathrm{\Psi }}_4}^2+2{\mathrm{\Omega }}^2{\mathrm{\Upsilon }}^4{\tau }_4{\mathrm{\Psi }}_4+{\mathrm{\Psi }}_4{\mathrm{\Psi }}_8-{\mathrm{\Psi }}_8+\frac{\zeta {\mathit{\text{E}}}_{\mathit{\text{crt}}}{M}_{gn}\mathit{Pr}{\mathrm{\Upsilon }}^2\mathrm{\Omega }}{{\left({\beta }_s\mathrm{\Upsilon }+1\right)}^2}.$

Here, M is the confluent hypergeometric function. The solution of the energy equation is as follows:

$\theta \left(\eta \right)=\frac{{\omega }_1{\mathrm{\Psi }}_2{\mathit{\text{E}}}_{\mathit{\text{crt}}}{\mathrm{e}}^{-\mathrm{\Upsilon }\eta }}{\left(2{\beta }_s\mathrm{\Upsilon }+2\right){\tau }_4{\mathrm{\Psi }}_4{\mathrm{\Psi }}_5}+\frac{{\left({\mathrm{\Psi }}_6{\mathrm{e}}^{-\mathrm{\Upsilon }\eta }\right)}^{{\mathrm{\Psi }}_7}{\omega }_2{\omega }_9}{2{{\mathrm{\Psi }}_6}^{{\mathrm{\Psi }}_7}{\mathrm{\Psi }}_3{\mathrm{\Omega }}^2{\mathrm{\Upsilon }}^4{\tau }_4{\mathrm{\Psi }}_4{\mathrm{\Psi }}_5}-\frac{{\mathit{\text{E}}}_{\mathit{\text{crt}}}{M}_{gn}{\mathrm{e}}^{-\mathrm{\Upsilon }\eta }}{\left(2{\beta }_s\mathrm{\Upsilon }+2\right){\tau }_4},(30)$

where

${\omega }_1={\left(1-\frac{\mathit{Pr}}{\left({\beta }_s\mathrm{\Upsilon }+1\right)\mathrm{\Omega }{\mathrm{\Upsilon }}^2}\right)}^2+\left(2\frac{\mathit{Pr}{\mathrm{e}}^{-\mathrm{\Upsilon }\eta }}{\left({\beta }_s\mathrm{\Upsilon }+1\right)\mathrm{\Omega }{\mathrm{\Upsilon }}^2}+1\right)\left(1-\frac{\mathit{Pr}}{\left({\beta }_s\mathrm{\Upsilon }+1\right)\mathrm{\Omega }{\mathrm{\Upsilon }}^2}\right)+\frac{{\mathit{Pr}}^2{\left({\mathrm{e}}^{-\mathrm{\Upsilon }\eta }\right)}^2}{{\left({\beta }_s\mathrm{\Upsilon }+1\right)}^2{\mathrm{\Omega }}^2{\mathrm{\Upsilon }}^4}+2\frac{\mathit{Pr}{\mathrm{e}}^{-\mathrm{\Upsilon }\eta }}{\left({\beta }_s\mathrm{\Upsilon }+1\right)\mathrm{\Omega }{\mathrm{\Upsilon }}^2},$

${\omega }_2=M\left(-2+\frac{\mathit{Pr}}{\left({\beta }_s\mathrm{\Upsilon }+1\right)\mathrm{\Omega }{\mathrm{\Upsilon }}^2},1+\frac{\mathit{Pr}}{\left({\beta }_s\mathrm{\Upsilon }+1\right)\mathrm{\Omega }{\mathrm{\Upsilon }}^2},-\frac{\mathit{Pr}{\mathrm{e}}^{-\mathrm{\Upsilon }\eta }}{\left({\beta }_s\mathrm{\Upsilon }+1\right)\mathrm{\Omega }{\mathrm{\Upsilon }}^2}\right),$

${\omega }_8=2\frac{{\mathit{\text{E}}}_{\mathit{\text{crt}}}{M}_{gn}{\mathrm{e}}^{-\mathrm{\Upsilon }\eta }{\mathrm{\Upsilon }}^2\mathrm{\Omega }\mathit{Pr}}{{\left({\beta }_s\mathrm{\Upsilon }+1\right)}^2},$

${\omega }_9=2{\mathrm{\Omega }}^2{\mathrm{\Upsilon }}^4{\tau }_4{{\mathrm{\Psi }}_4}^2+2{\mathrm{\Omega }}^2{\mathrm{\Upsilon }}^4{\tau }_4{\mathrm{\Psi }}_4+{\mathrm{\Psi }}_4{\omega }_8-{\omega }_8-\frac{{\mathit{\text{E}}}_{\mathit{\text{crt}}}{M}_{gn}{\mathrm{e}}^{\mathrm{\Upsilon }\eta }{\mathit{Pr}}^2}{{\left({\beta }_s\mathrm{\Upsilon }+1\right)}^3}.$

${\theta }_{\eta }^{\prime }\left(0\right)=\frac{-A_3{M}_{gn}{\mathit{\text{E}}}_{\mathit{\text{crt}}}}{\left(2{\beta }_s\mathrm{\Upsilon }+2\right){\tau }_4{A}_1{A}_2}+\frac{A_4{M}_{gn}{\mathit{\text{E}}}_{\mathit{\text{crt}}}\mathrm{\Upsilon }}{\left(2{\beta }_s\mathrm{\Upsilon }+2\right){\tau }_4{A}_1{A}_2}+\frac{A_5\mathit{Pr}{A}_8{A}_6}{2\left({\beta }_s\mathrm{\Upsilon }+1\right){\mathrm{\Omega }}^3{\mathrm{\Upsilon }}^5{A}_5{A}_8{\tau }_4{A}_1{A}_2}-A_{10},(31)$

where

$\begin{array}{ll}\hfill & A_1=1-\frac{\mathit{Pr}}{\left({\beta }_s\mathrm{\Upsilon }+1\right)\mathrm{\Omega }{\mathrm{\Upsilon }}^2},A_2=2-\frac{\mathit{Pr}}{\left({\beta }_s\mathrm{\Upsilon }+1\right)\mathrm{\Omega }{\mathrm{\Upsilon }}^2},A_3=\frac{-2pr{A}_1}{\mathrm{\Upsilon }\left({\beta }_s\mathrm{\Upsilon }+1\right)\mathrm{\Omega }}-\frac{{2\mathit{Pr}}^2}{{\left({\beta }_s\mathrm{\Upsilon }+1\right)}^2{\mathrm{\Omega }}^2{\mathrm{\Upsilon }}^3}\hfill \\ \hfill & -\frac{2\mathit{Pr}}{\mathrm{\Upsilon }\left({\beta }_s\mathrm{\Upsilon }+1\right)\mathrm{\Omega }},A_4={A_1}^2+\left(2\frac{\mathit{Pr}}{\left({\beta }_s\alpha +1\right)\mathrm{\Omega }{\mathrm{\Upsilon }}^2}+1\right)A_1+\frac{{\mathit{Pr}}^2}{{\left({\beta }_s\mathrm{\Upsilon }+1\right)}^2{\mathrm{\Omega }}^2{\mathrm{\Upsilon }}^4}+\frac{2pr}{\left({\beta }_s\mathrm{\Upsilon }+1\right)\mathrm{\Omega }{\mathrm{\Upsilon }}^2},\hfill \\ \hfill & A_5={\left(-\frac{\mathit{Pr}}{\left({\beta }_s\mathrm{\Upsilon }+1\right)\mathrm{\Omega }{\mathrm{\Upsilon }}^2}\right)}^{1-A_1},A_6=\frac{-2E_{\mathit{crt}}{M}_{gn}{A}_1{\mathrm{\Upsilon }}^2\mathrm{\Omega }\mathit{Pr}}{{\left({\beta }_s\mathrm{\Upsilon }+1\right)}^2}+2{\mathrm{\Omega }}^2{\mathrm{\Upsilon }}^4{\tau }_4{A_1}^2-\frac{2E_{\mathit{crt}}{M}_{gn}{\mathrm{\Upsilon }}^2\mathrm{\Omega }\mathit{Pr}}{{\left({\beta }_s\alpha +1\right)}^2}\hfill \\ \hfill & +2{\mathrm{\Omega }}^2{\mathrm{\Upsilon }}^4{A}_4{A}_1-\frac{{\mathit{\text{E}}}_{\mathit{\text{crt}}}{M}_{gn}{\mathit{Pr}}^2}{{\left({\beta }_s\mathrm{\Upsilon }+1\right)}^3},A_7=2\frac{{\mathit{\text{E}}}_{\mathit{\text{crt}}}{M}_{gn}{\mathrm{\Upsilon }}^3{A}_1\mathrm{\Omega }\mathit{Pr}}{{\left({\beta }_s\mathrm{\Upsilon }+1\right)}^2}+2\frac{{\mathit{\text{E}}}_{\mathit{\text{crt}}}{M}_{gn}{\mathrm{\Upsilon }}^3\mathrm{\Omega }\mathit{Pr}}{{\left({\beta }_s\mathrm{\Upsilon }+1\right)}^2}+\frac{{\mathit{\text{E}}}_{\mathit{\text{crt}}}{M}_{gn}\mathrm{\Upsilon }\mathrm{e}{\mathit{Pr}}^2}{{\left({\beta }_s\mathrm{\Upsilon }+1\right)}^3},\hfill \\ \hfill & A_8=M\left(-A_2,1+\frac{\mathit{Pr}}{\left({\beta }_s\mathrm{\Upsilon }+1\right)\mathrm{\Omega }\mathrm{\Upsilon }},A_1-1\right),A_9=M\left(-A_1,2+\frac{\mathit{Pr}}{\left({\beta }_s\mathrm{\Upsilon }+1\right)\mathrm{\Omega }{\mathrm{\Upsilon }}^2},A_1-1\right),\hfill \\ \hfill & A_{10}=-\frac{A_9\mathit{Pr}{A}_6}{{\mathrm{\Upsilon }}^5\left({\beta }_s\alpha +1\right){\mathrm{\Omega }}^3{A}_8{\tau }_4{A}_1}{\left(2+2\frac{\mathit{Pr}}{\left({\beta }_s\mathrm{\Upsilon }+1\right)\mathrm{\Omega }{\mathrm{\Upsilon }}^2}\right)}^{-1}-A_7\frac{1}{2{\mathrm{\Omega }}^2{\mathrm{\Upsilon }}^4{\tau }_4{A}_1{A}_2}.\hfill \end{array}$

4 Skin friction and the local Nusselt number

The expression for the local skin friction is as follows:

$C_f=\frac{{\tau }_w}{\rho {\underline{u}}_w^2}=\frac{R{e}_x^{-1/2}}{{\tau }_1}{f}^{″}\left(0\right),{\tau }_1{C}_{f}R{e}_x^{-1/2}=f^{″}\left(0\right),(32)$

where ${\tau }_w={\mu }_{\mathit{gnf}}{\left(\frac{\partial \underline{u}}{\partial y}\right)}_{y=0}$ and $R{e}_x=\frac{x{\underline{u}}_w}{\nu }$ are the stress at the wall and the Reynolds number, respectively.

The local Nusselt number is defined as follows:

$Nu=\frac{-k_{\mathit{gnf}}x}{k_{fc}\left(T_w-T_{\infty }\right)}{\left(\frac{\partial T}{\partial y}\right)}_{y=0}=-\frac{k_{\mathit{gnf}}}{k_{fc}}R{e}_x^{1/2}{\theta }_{\eta }\left(0\right).(33)$

It is obtained as follows in the present study:

$\frac{k_{fc}}{k_{\mathit{gnf}}}N{u}_{x}R{e}_x^{-1/2}=-{\theta }_{\eta }\left(0\right).(34)$

5 Results and discussion

We have analyzed the fluid flow and energy transport of the MHD viscoelastic nanofluid past a stretching surface with the effects of velocity slip and thermal radiation entity. The influence of several emerging parameters such as ϕ, M_gn, β_s, K_ve, E_crt, Pr, and χ on f′(η), θ(η), −f′′(0), and −θ′(0) is shown. In this connection, we have drawn the graphs and the corresponding numerical tables. The physical perspective of the considered model is depicted in Figure 1. Figures 2–5 are created to discuss the influence of these parameters on the velocity field. In Figure 2, the variation in ϕ above the velocity field is illustrated. It is noted that the magnitude of the velocity distribution is enhanced by an increment in the values of ϕ. Figure 3 depicts the variation in M_gn with β_s, K_ve, and ϕ. The velocity field diminishes as the M_gn increases. Physically, the Lorentz forces are dominant, which causes the velocity field to de-escalate while increasing the magnitude of M_gn. The influence of β_s on f′(η) is portrayed in Figure 4. It is perceived that increasing the amount of β_s causes the velocity distribution to drop rapidly. Actually, when the velocity slip occurs, the stretching surface velocity is faster than the fluid velocity, resulting in a decrease in the velocity of the nanofluid. The influence of K_ve on the velocity field is delineated in Figure 5. Physically, viscoelasticity creates a tensile force that opposes fluid movement. Due to viscous and elastic factors, escalating values of K_ve cause a reduction in the flow velocity.

FIGURE 1

FIGURE 1. Model’s physical perspective.

FIGURE 2

FIGURE 2. Impact of ϕ with M_gn = 2, β_s = 1, and K_ve = 1 on velocity profiles.

FIGURE 3

FIGURE 3. Impact of M_gn with ϕ = 0.1, β_s = 1, and K_ve = 1 on velocity profiles.

FIGURE 4

FIGURE 4. Impact of β_s with M_gn = 2, ϕ = 0.1, and K_ve = 1 on velocity profiles.

FIGURE 5

FIGURE 5. Impact of K_ve with M_gn = 2, ϕ = 0.1, and β_s = 1 on velocity profiles.

Figures 6–11 investigate the effects of ϕ, M_gn, β_s, E_crt, Pr, and χ on θ(η). Figure 6 shows the temperature field against the nanoparticle ratio. It implies that the temperature field is enhanced with an enlargement in the magnitude of ϕ. Physically, the thermal conductivity of the fluid is enhanced by adding nanoparticles. Consequently, the temperature decreases. The behavior of the temperature field against M_gn is studied in Figure 7. It is revealed that a rise in M_gn causes the fall in temperature distribution. It is due to friction between the fluid and nanoparticles by Lorentz forces that leads to an increase in the temperature field. The development of Pr on θ(η) is shown in Figure 8. The temperature field is observed to rise, increasing the magnitude of Pr. The Prandtl number, in terms of physics, is the fraction of momentum diffusivity to thermal diffusivity. Thermal diffusivity is enhanced by adding nanoparticles, which causes a decrement in temperature distribution. Figure 9 illustrates the impact of E_ert over θ(η). It is discovered that increasing E_ert increases the magnitude of θ(η). The influence of the thermal radiation parameter on θ(η) is depicted in Figure 10. It demonstrates that the magnitude of θ(η) de-escalates, while the magnitude of χ increases. Figure 11 shows how decreasing the amount of χ increases the value of θ(η).

FIGURE 6

FIGURE 6. Impact of ϕ with M_gn = 1, β_s = 1, K_ve = 1, E_crt = 0.3, Pr = 6.2, and χ = 0.3 on the temperature profile.

FIGURE 7

FIGURE 7. Impact of M_gn with ϕ = 0.1, β_s = 1, K_ve = 1, E_crt = 0.1, Pr = 6.2, and χ = 0.3 on the temperature profile.

FIGURE 8

FIGURE 8. Impact of Pr with ϕ = 0.1, β_s = 1, K_ve = 1, E_crt = 0.1, M_gn = 0.5, and χ = 1 on the temperature profile.

FIGURE 9

FIGURE 9. Impact of E_crt with ϕ = 0.1, β_s = 1, K_ve = 1, Pr = 6.2, M_gn = 0.5, and χ = 1 on the temperature profile.

FIGURE 10

FIGURE 10. Impact of χ with ϕ = 0.1, β_s = 1, K_ve = 1, Pr = 6.2, M_gn = 0.5, and E_crt = 0.1 on the temperature profile.

FIGURE 11

FIGURE 11. Impact of β_s with ϕ = 0.1, χ = 1, K_ve = 1, Pr = 6.2, M_gn = 0.5, and E_crt = 0.1 on the temperature profile.

Figures 12–14 investigate the change in M_gn, β_s, and ϕ over −f′′(0). The local skin friction factor is decreased with the growing magnitude of M_gn as shown in Figure 12. Physically, magnetic fields and electric forces produce Lorentz forces that create resistance forces because of which the skin friction on the wall increases. Figure 13 and Figure 14 show a plot of −f′′(0) versus β_s and ϕ, respectively. It is perceived that the local skin friction field is decreased with an augmentation in β_s, and an opposite trend is shown for ϕ. Figures 15–18 scrutinize the variation in M_gn, E_crt, Pr, and χ over −θ′(0). The impact of M_gn on −θ′(0) against ϕ is presented in Figure 15, which shows a decrement in the amount of −θ′(0). Figure 16 depicts the effects of the Eckert number on the dimensionless temperature variation, −θ′(0). It can be seen that when the Eckert number increases, −θ′(0) also increases. The Eckert number E_crt is expressed as the ratio of advective transmission to energy-dissipated capability in physical terms. Furthermore, a rise in the rate of heat transport suggests that the energy loss is accompanied by a fall in the Eckert number. Figure 17 and Figure 18 express the influence of Pr and χ on the heat exchange ratio. It is discovered that θ′(0) is augmented while the strengths of Pr and χ are increased.

FIGURE 12

FIGURE 12. Impact of M_gn with β_s = 0.5 and K_ve = 0.5 on −f′′(0).

FIGURE 13

FIGURE 13. Impact of β_s with ϕ = 0.1 and K_ve = 0.5 on −f′′(0).

FIGURE 14

FIGURE 14. Impact of ϕ with β_s = 0.5 and K_ve = 0.5 on −f′′(0).

FIGURE 15

FIGURE 15. Impact of M_gn with E_crt = 0.3, β_s = 0.5, K_ve = 0.5, Pr = 6.2, and χ = 0.5 on −θ′(0).

FIGURE 16

FIGURE 16. Impact of E_crt with ϕ = 0.1, β_s = 0.5, K_ve = 0.5, Pr = 6.2, and χ = 0.5 on −θ′(0).

FIGURE 17

FIGURE 17. Impact of Pr with ϕ = 0.1, E_crt0, β_s = 0.5, K_ve = 0.5, and χ = 0.5 on −θ′(0).

FIGURE 18

FIGURE 18. Impact of χ with E_crt = 0.3, Pr = 6.2, β_s = 0.5, K_ve = 0.5, and ϕ = 0.1 on −θ′(0).

The thermophysical characteristics of graphene and CMC−water are displayed in Table 1. Tables 2, 3 show the numerical values of local skin friction coefficient and local Nusselt number respectively. Figures 19–24 indicate the streamline characteristics. It has been shown that the boundary-layer thickness dramatically shrinks when K_ve is increased.

TABLE 1

TABLE 1. Host fluid (carboxymethyl cellulose−water) and nanoparticle (graphene) thermophysical characteristics [5].

TABLE 2

TABLE 2. Numerical table of −f′′(0) with ϕ = 0.1.

TABLE 3

TABLE 3. Numerical table of −θ′(0) with K_ve = 0.5, ϕ = 0.1, and χ = 0.5.

FIGURE 19

FIGURE 19. Impact of ϕ = 0 with M_gn = 0.5, β_s = 0.5, and K_ve = 0.5 on the flow regime.

FIGURE 20

FIGURE 20. Impact of ϕ = 0.1 with M_gn = 0.5, β_s = 0.5, and K_ve = 0.5 on the flow regime.

FIGURE 21

FIGURE 21. Impact of ϕ = 0.2 with M_gn = 0.5, β_s = 0.5, and K_ve = 0.5 on the flow regime.

FIGURE 22

FIGURE 22. Impact of K_ve = −15 with β_s = 0.5, ϕ = 0.1, and M_gn = 0.5 on the flow regime.

FIGURE 23

FIGURE 23. Impact of K_ve = 0 with M_gn = 0.5, β_s = 0.5, and ϕ = 0.1 on the flow regime.

FIGURE 24

FIGURE 24. Impact of K_ve = 15 with M_gn = 0.5, β_s = 0.5, and ϕ = 0.1 on the flow regime.

6 Conclusion

In this study, the analysis of the MHD graphene−CMC−water nanofluid past a stretchable wall with Joule heating and velocity slip impact was performed. We arrived at the following conclusions:

• The velocity profile is the growing function of M_gn, β_s, and K_ve, with the reversed mode shown in case of ϕ.

• The temperature profile is the declining function of Pr, E_crt, ϕ, and χ, while a contradictory trend is observed for M_gn and β_s.

• Increasing the magnitude of M_gn increases local skin friction, while β_s and ϕ reveal an opposing trend.

• An increase in E_crt, Pr, and χ is shown to be accountable for an increase in the fraction of heat exchange. As M_gn increases, the heat transmission rate decreases.

• An enhancement in K_ve decreases the boundary-layer thickness which is examined in the flow regime.

Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding authors.

Author contributions

IR initiated the fluid model and methodology. TZ and IR produced numerical data and graphs using software. SI completed the write-up. MA and SE assisted in numerical data and write up.

Acknowledgments

The authors are grateful to the University of Management and Technology of Lahore, and HEC Pakistan for facilitating this research under research project No 15911 (NRPU).

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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