Motion of micropolar and Walters-B nanofluids towards a stretching sheet with the significance of heat generation, thermal radiation and Soret–Dufour mechanisms

Abstract

The influences of thermal radiation, Heat Generation and Soret–Dufour on the boundary layer flow of Micropolar and Walters-B non-Newtonian fluid with MHD heat mass transfer are investigated, The formulated model resulted to set of partial differential equations $(\mathit{PDEs}).$ The PDEs were simplified using suitable similarity variable to obtain total differential eqautions. The transformed equations were solve by employing spectral homotopy analysis method (SHAM). SHAM uses the basic concept of both Chebyshev pseudospectral method and homotopy analysis method. The outcomes for encountered flow parameters for temperature, velocity and concentration are presented with the aid of figures. An increase in the viscoelastic parameter is observed to enhance both the velocity and entire hydrodynamics boundary layer thickness. Very close to the wall the micro-rotation parameter is noticed to increase the velocity profile and negligible very far from the wall. The Prandtl number and thermal radiation parameter are noticed to increase both velocity and temperature as their values increases. As applied in various engineering processes, we hope the present study will contribute to a better understanding of boundary layer flow of non-Newtonian fluids, such as Micropolar and Walters-B. The local skin friction, local Nusselt number and local Sherwood number are also computed and investigated for different embedded parameters in the problem statements, i.e., all results are presented graphically and all physical quantities are computed and tabulated.

Keywords:

• dimensionless equations
• magnetohydrodynamics (MHD)
• micropolar fluid
• Soret–Dufour
• spectral homotopy analysis method (SHAM
• Walters-B liquid)

1. Introduction

The study of non-Newtonian fluid flow and heat transfer under the influence of variable viscosity and thermal conductivity has gained much interest due to its numerous engineering applications such as glass blowing, food processing, crystal growth, etc. Mostafa et al. studied the MHD flow and heat transport of a micropolar liquid past a stretching surface. Their equations were solved numerically using the Chebyshev spectral method. Hady (1996) examined the problem of heat transfer to micropolar fluid from a non-isothermal, permeable stretching sheet. Ishak, Nazar, and Pop (2008) investigated heat transfer over a stretching surface, including the uniform or variable surface heat flux in micropolar fluids. Hayat, Javed, and Abbas (2009) also examined the two-dimensional stagnation-point flow of an incompressible micropolar fluid over a nonlinear stretching surface. The study of Ashmawy explained fully developed nature of convective micropolar fluid flow. Sajid, Abbas, and Hayat (2009) examined the boundary layer liquid flow of a micropolar fluid that passes through a porous channel with the aid of exact solution as a method of solution. Macha et al. worked on the unsteady flow of a micropolar flow nanoliquid past a stretching surface with MHD and thermal radiation impacts. According to Hayat, Abbas, and Javed (2008), using HAM as the method of solution, a steady two-dimensional mixed convection flow of a micropolar fluid over a nonlinear stretching surface could be obtained analytically. Daniel et al. (2018) studied the impact of the thermal radiation flow past a vertical stretching sheet. Ashraf Bilal, Hayat, and Alsulami (2016) elucidate the mechanism of Soret–Dufour on the mixed convection flow oldroyd-B fluid. Javaherdeh, Mirzaei Nejad, and Moslemi (2015) In MHD fluid flow past moving vertical plates in a porous medium, natural convection of heat transfer was observed. Vijaya et al. (2018) Researched unsteady flow of Casson fluid through a porous vertical channel influenced by radiation and Soret. Heat transfer in boundary layer flow of free convection flow with magnetism as well as buoyancy forces has gained much attention due to its applications in rubber and plastic manufacturing, glass, fibre, metal extrusion and materials handling conveyors. Because of these numerous applications, Salawu and Dada (2016) explored variable thermal conductivity and viscosity on radiative heat transfer under influence of magnetic field with dissipative effect. Jayachandra Babu et al. (2017) investigated free convective MHD flow of Cattaneo–Christov past three different geometries with Brownian motion and thermophoresis. Ramana Reddy, Anantha Kumar, Sugunamma, and Sandeep (2018) studied cross diffusion effect on MHD non-Newtonian. Fluids flow past a stretching sheet. Ayegbusi examined MHD, Thermal radiation, thermophoresis and Soret–Dufour mechanism contributions on unsteady motion. Ayegbusi, Onwubuoya, and Falodun (2020) examined the MHD heat problem over a moveable plate alongside mass transfer convective flow. Fagbade, Falodun, and Omowaye (2018) examined the effect of MHD on natural convection viscoelastic liquid flow that past an accelerating permeable surface under heat source/sink and thermal radiation influence. Abbas, Begum Jeelani, Alnahdi, and Ilyas (2022a) examine the behaviour of MHD Williamson nanofluid flow and heat transfer over a non-linear stretching sheet embedded in a porous medium using MTLAB built-in solver bvp4c as a method of solution, in their work we discovered that Eckert number Ec increase leads to an appreciation in temperature and mass concentration profile. Abbas, Begum Jeelani, and Hussain Alharthi (2022b) investigated MHD effects on heat and mass transfer phenomena in third-grade fluid past an inclined exponentially stretching sheet fixed in a porous medium with Darcy–Forchheimer law influence and discovered that their third-grade fluid parameter is appreciating while the velocity profile increases, but the temperature field and mass concentration are depreciating. Irfan et al. (2021) have studied the performance of new mass flux theory on Carreau nanofluid using the thermal aspects of convective heat transport as well as the nonlinear properties of mixed convection. The shear thinning–thickening properties were explained with the aid of bvp4c algorithm as an influence on variables. They found out that the thermophoresis and Brownian motion parameters have conflicting influences on concentration field.

Shoaib Anwar and Rasheed (2017) have examined Heat transfer at microscopic level in a MHD fractional inertial flow confined between non-isothermal boundaries. In their study, Governing partial differential equations with appropriate initial and boundary conditions are solved by employing a finite-difference scheme to discretize the fractional time derivative and finite-element discretization for spatial variables. They used fractional calculus techniques to model the boundary layer and Forchheimer medium flows that involve catalytic converters and gas turbines can be modelled in a similar manner.

Bilal, Muhammad, Amer, and Muhammad (2020) have examined the MHD Maxwell flow modelled by fractional derivatives with chemical reaction and thermal radiation. The governing equations are non-linear, coupled and Boussinesq approximation is assumed for the formulation of the momentum equation. To solve the derived model numerically, the spatial variables are discretized by employing the finite element method and the Caputo-time derivatives are approximated using finite difference approximations. It reveals that the fractional derivative strengthens the flow field. We also observe that the magnetic field and relaxation time suppress the velocity.

The Soret with Dufour impact has really gained interest in Newtonian and Non-Newtonian convective heat and mass transfer. The impacts are very significant especially when density differences surface in the flow regime, during the investigation of heat plus mass transfer effect, Dufour or diffusion-thermal defined as the thermal energy flux resulting from concentration gradients while Soret or thermo-diffusion effect is the contribution to the mass fluxes as a result of temperature gradient. Soret and Dufour impacts are essential for instant molecular weight gases in coupled heat and mass transfer in a binary system, mostly come across in chemical process, engineering, as well as in high-speed aerodynamics, Soret and Dufour effects are influential whenever there is chemical reaction species at a surface in the domain of the fluid with lower density more than the surrounding fluid by Makinde and Olanrewaju , Oorthy and Senthilvadivy examined heat together with mass transport of natural convection flow in a saturated porous medium with Soret–Dufour and considered the variability of the viscosity along-side with their work, Pal and Mondal (2011) studied the impact of Soret Dufour that has to do with chemical reaction as well as thermal radiation on MHD non-Darcy unsteady and they discovered that thermal radiation have an impact over temperature profile that usually lead to an increase in the thermal boundary. Noor et al. (2012) investigated the system of equation that has to do with free convective thermophoretic which flow over a radiative isothermal plate. In their work, they adopted one of numerical approach as a method of solution, i.e., shooting method. Uwanta and Usman (2014) studied the behaviour of Soret with Dufour in the heat and mass transport problem in the presence of viscous dissipation as well as constant suction. Aruna, Varma, and Raju (2015) discussed detail effects of adding Soret with Dufour on the problem of unsteady mixed convection MHD heat together with mass transport flow. Tella, Vijaykumar, and Naidu (2015) also investigated Soret together with Dufour phenomenon in their study of viscous as well as chemically reacting fluid. Sharma, Yadav, Mishra, and Chaudhary (2012) investigate unsteady mixed convection along a radiatively porous vertical plate using Soret’s work in conjunction with Dufour’s work. In their study of Soret–Dufour coupled with ohmic dissipation effects on mixed convection flows, Seini and Makinde (2013) used numerical methods to derive solutions. Ayegbusi et al. (2020) studied Soret along-side with Dufour mechanism which during an electrically conducting nanofluid flow above half porous plate. The study concluded that when Dufour number is appreciate it leads to the deterioration of skin friction coefficient and rate of heat transfer by Omowaye, Fagbade, and Ajayi (2015).

The target of this study is to explore the impact of thermal radiation, heat generation and Soret–Dufour on the boundary layer flow of Micropolar and Walters-B non-Newtonian fluid with MHD heat mass transfer employing a numerical method known as (SHAM). This has not been examined in literature based on our knowledge. This study is very important due to various applications it has in engineering and sciences such as nuclear power plants, devices for aircraft, gas turbines, MHD accelerators etc.

2. Mathematical analysis

This study considered a two- dimensional, steady, laminar and incompressible boundary layer flow of non-Newtonian fluids flow past a stretching sheet under the influence of magnetic field. The fluid flow across the plate with velocity Ux = bx along x-axis while the y-axis are taking as the horizontal sheet in a region y$>0$ and the medium is taking to be permeable over an infinite surface. The fluids Micropolar and Walters-B liquid are considered to start flowing from the slot to the stretching sheet. A magnetic field of constant strength is imposed on their flow transversely. As shown in Figure 1, $(T_w)$ is the temperature while $(C_w)$ is the wall concentration. The free stream temperature and concentration are $T_{\infty }$ and $C_{\infty }$ respectively.

Figure 1. The fluid flow over a stretching surface.

2.1. Physical configuration

Hence, the present study examined a case where by $T_w<T_{\infty }$ and $C_w<C_{\infty }$ which correspond to cooled stretching surface and heat generation, thermal radiation and Soret–Dufour mechanisms etc. A study by Mehmood et al. (2008) took as an example the following equation of motion which relates the Cauchy stress tensor to Walters-B liquid: (1) $$S=-pT+\tau$$(1) (2) $$\tau =2{\eta }_{0}e-2k_0\frac{\delta e}{\delta t}$$(2) $p$ represents the fluid pressure, $I$ represents the identity tensor and also represents the rate of strain tensor $e,$ which is defined below (3) $$2e=▽\left(v\right)+▽(v{)}^T,$$(3)

$v,$ $▽$ and $\frac{\delta }{\delta t},$ is velocity vector, gradient operator, differentiation of a tensor quantity in relation to the material in motion. Given below is the convected differentiation which represents the rate of strain tensor (4) $$\frac{\delta e}{\delta t}=\frac{\partial e}{\partial t}+v.▽\left(e\right)-e.▽\left(v\right)-(▽(v){)}^T.e$$(4) ${\eta }_0,$ $k_0$ is known as limiting viscosity when rate are small and coefficient for the Walters-B fluid, (5) $${\eta }_0={\int }_0^{\infty }\lambda \left(\xi \right)d\xi$$(5) (6) $$k_0={\int }_0^{\infty }\tau \lambda \left(\xi \right)d\xi$$(6)

The above model determines the Walters’-B liquid fluid based on Walters (1962) $\lambda (\xi )$ as a relaxation spectrum. Taking into account short memories, the above model is able to determine Walters’-B liquid fluid, the above model determines Walters’-B liquid fluid and the terms involving ${\int }_0^{\infty }{\tau }^n\lambda (\tau )d\tau , n\ge 2$ are neglected. Due to the above relational Eqs. (1)–(6)(6) $$k_0={\int }_0^{\infty }\tau \lambda \left(\xi \right)d\xi$$(6) , the stress components are represented below: (7) $$\tau =\left[\begin{array}{c}{\tau }_{xx}\mathrm{ }{\tau }_{xy}\\ {\tau }_{yx}\mathrm{ }{\tau }_{yy}\end{array}\right]$$(7) (8) $${\tau }_{xx}=2{\mu }_0\frac{\partial u}{\partial x}-2k_0[u\frac{{\partial }^{2}u}{\partial x^2}+v\frac{{\partial }^{2}u}{\partial x\partial y}-2\left({\left(\frac{\partial u}{\partial x}\right)}^2+\frac{1}{2}\frac{\partial u}{\partial y}\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)\right)]$$(8) (9) $$\begin{array}{c}{\tau }_{yx}={\tau }_{xy}={\mu }_0\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)\\ -2k_0[\frac{1}{2}u\left(\frac{{\partial }^{2}u}{\partial x\partial y}+\frac{{\partial }^{2}v}{\partial x^2}\right)+\frac{1}{2}v(\left(\frac{{\partial }^{2}u}{\partial y^2}+\frac{{\partial }^{2}v}{\partial x\partial y}\right)-\left(\frac{\partial u}{\partial x}\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\frac{\partial v}{\partial y}\right))]\end{array}$$(9) (10) $${\tau }_{yy}=2{\mu }_0\frac{\partial v}{\partial y}-2k_0[\frac{{\partial }^{2}v}{\partial t\partial x}+u\frac{\partial v}{\partial x\partial y}+v\frac{{\partial }^{2}v}{\partial y^2}-2\left(\frac{1}{2}\frac{\partial v}{\partial x}\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)+{\left(\frac{\partial v}{\partial y}\right)}^2\right)]$$(10)

Where ${\tau }_{xx},{\tau }_{xy},{\tau }_{yx}$ and ${\tau }_{yy}$ are components. The stress tensor above becomes as below after differentiating (11) $$\rho \left(\frac{\partial {\tau }_{xx}}{\partial x}+\frac{\partial {\tau }_{xy}}{\partial y}\right)={\mu }_0\left(\frac{{\partial }^{2}u}{\partial y^2}\right)-k_0\left(v\frac{{\partial }^{3}u}{\partial y^3}+u\frac{{\partial }^{3}u}{\partial x\partial y^2}-2\frac{\partial u}{\partial y}\frac{{\partial }^{2}u}{\partial x\partial y}\right)$$(11)

Based on Alao, Fagbade, and Falodun (2016) report, Rosseland diffusion approximation is preferred to simplify the radiative heat flux on the flow and such that: (12) $$q_r=-\frac{4{\sigma }_s}{3k_e}\frac{\partial T^4}{\partial y}$$(12)

A constant ${\sigma }_s$ and $k_e$ mean absorption coefficient known as the Stefan-Boltzman constant.

Using Taylor series to expand about, and ignoring higher order terms, the temperature differences within the flow can be expressed linearly. (13) $$T^4\approxeq 4T_{\infty }^{3}T-3T_{\infty }^4$$(13) substituting (24)(24) $$f^{‴}+\mathrm{\Delta }{f}^{\prime }+\frac{1}{M}(1+f^{\prime })+Nr(1-f^{{\prime }^2})+f{f}^{″}-\alpha (2\eta f^{\prime }{f}^{iv}-f^{\prime }f″\prime -\eta f^{″}{f}^{‴})=0$$(24) into (23)(23) $$\eta =(\frac{U_0}{2\nu x}{)}^{\frac{1}{2}}\gamma ,\theta (\eta )=\frac{T-T_{\infty }}{T_w-T_{\infty }}, \varphi (\eta )=\frac{C-C_{\infty }}{C_w-C_{\infty }}$$(23) and get (14) $$-\frac{\partial q_r}{\partial y}=\frac{16{\sigma }_s{T}_{\infty }^3}{3k_e}\frac{{\partial }^{2}T}{\partial y^2}$$(14)

In Alam et al. (2008), the thermophoretic velocity (TV) $V_T$ in Eq. (4)(4) $$\frac{\delta e}{\delta t}=\frac{\partial e}{\partial t}+v.▽\left(e\right)-e.▽\left(v\right)-(▽(v){)}^T.e$$(4) given below (15) $$V_T=-k\nu \frac{▽T}{T_{\mathit{ref}}}=-\frac{k\nu }{T_{\mathit{ref}}}\frac{\partial T}{\partial y}$$(15) where $k$ is know as the (TV) explain below (16) $$=\frac{2C_s\left(\frac{{\lambda }_g}{{\lambda }_p}+C_t{K}_n\right)\left[1+K_n\left(C_1+C_2{e}^{\frac{-C_3}{K_n}}\right)\right]}{\left(1+3C_m{K}_n\right)\left(1+2\frac{{\lambda }_g}{{\lambda }_p}+2C_t{K}_n\right)}$$(16) $C_1,C_2,C_3,C_m,C_s,C_t$ are constants, ${\lambda }_g$ and ${\lambda }_p$ are the thermal conductivity and diffused particles, $K_n$ is known as Knudsen number.

Based on the assumptions above, the governing equations becomes (17) $$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$$(17) (18) $$\begin{array}{c}{\rho }_m\left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\right)=\nu \frac{{\partial }^{2}u}{\partial y^2}+K_1\frac{\partial \sigma }{\partial y}+\frac{\nu \phi }{K}(U-u)+C_r(U^2-u^2)-K_0\left(\nu \frac{{\partial }^{3}u}{\partial y^3}+u\frac{{\partial }^{3}u}{\partial x\partial y^2}-2\frac{\partial u}{\partial y}\frac{{\partial }^{2}u}{\partial x\partial y}\right)\\ G_1\frac{{\partial }^2\sigma }{\partial y^2}-2\sigma -\frac{\partial u}{\partial y}=0\end{array}$$(18) (19) $${\rho }_m{C}_p\left(u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}\right)=\frac{k}{\rho cp}\frac{{\partial }^{2}T}{\partial y^2}+\frac{16{\sigma }^{\mathrm{*}}}{3\rho c_p{K}^{\mathrm{*}}}\frac{{\partial }^{2}T}{\partial y^2}+\frac{\nu D_m{K}_T}{T_m{c}_s{c}_p}\frac{{\partial }^{2}C}{\partial y^2}-Q(T-T_{\mathrm{\infty }})+\tau \left[D_B\frac{\partial C^{\mathrm{*}}}{\partial y^{\mathrm{*}}}\frac{\partial T^{\mathrm{*}}}{\partial y^{\mathrm{*}}}+\frac{D_T}{T_{\mathrm{\infty }}}{\left(\frac{\partial T^{\mathrm{*}}}{\partial y^{\mathrm{*}}}\right)}^2\right]$$(19) (20) $$u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}=D_m\frac{{\partial }^{2}C}{\partial z^2}+\frac{D_m{K}_T}{T_m}\frac{{\partial }^{2}T}{\partial y^2}-K_o(C-C_{\mathrm{\infty }})-\frac{\partial \left(V_{T}C\right)}{\partial y}$$(20) subject to: (21) $$u={Bx=U}_0,\mathrm{ }v=0,\sigma =0,T=T_w,C=C_w\mathrm{ }at\mathrm{ }y=0$$(21) (22) $$\mathrm{u}\to 0,\mathrm{ }\mathrm{T}⟶T_{\infty }, \mathrm{C}⟶C_{\infty }\mathit{\text{ at y}}⟶\infty$$(22)

The similarity variables are (Bilal et al. (2022)) $$\psi (x,y)=(2\nu \mathit{Uox}{)}^{\frac{1}{2}}{U}_{0}g(\eta ),u=\frac{\partial \psi }{\partial y},v=\frac{-\partial \psi }{\partial x},\sigma =(\frac{U_0}{2\nu x}{)}^{\frac{1}{2}}{U}_{0}g\left(\eta \right).$$ (23) $$\eta =(\frac{U_0}{2\nu x}{)}^{\frac{1}{2}}\gamma ,\theta (\eta )=\frac{T-T_{\infty }}{T_w-T_{\infty }}, \varphi (\eta )=\frac{C-C_{\infty }}{C_w-C_{\infty }}$$(23)

Using the similarity variables defined above, the transformed dimensionless equations becomes: (24) $$f^{‴}+\mathrm{\Delta }{f}^{\prime }+\frac{1}{M}(1+f^{\prime })+Nr(1-f^{{\prime }^2})+f{f}^{″}-\alpha (2\eta f^{\prime }{f}^{iv}-f^{\prime }f″\prime -\eta f^{″}{f}^{‴})=0$$(24) (25) $$G_r{g}^{″}-2(2g+f^{″})=0$$(25) (26) $$(4+3R){\theta }^{″}+\mathit{Prf}{\theta }^{\prime }+P_r{Q}_{0}Q+\mathit{PrDu}{\theta }^{″}+N_b\varphi \prime \theta \prime +N_t{\theta }^{\prime 2}=0.$$(26) (27) $${\varphi }^{″}+S_{c}f\varphi \prime +S_c{S}_r\theta ″-\gamma ({\theta }^{\prime }\varphi \prime +\varphi {\theta }^{″})=0.$$(27) subject to: (28) $$f^{\prime }\left(0\right)=g\left(0\right)=1,f\left(0\right)=f_w, \theta \left(0\right)=\varphi \left(0\right)=1\mathit{\text{ at }}\eta =0$$(28) (29) $$f^{\prime }\left(\mathrm{\infty }\right)⟶0,\mathrm{ }\theta ⟶0,\mathrm{ }\varphi ⟶0\mathrm{ }\mathrm{at}\mathrm{ }y⟶\mathrm{\infty }$$(29)

The quantity of engineering interesst are the Skin friction, Nusselt number and Sherwood number. Thus quantities are defined as follows: $$C_f=\frac{{\tau }_w}{\rho \nu w^2},$$ where $${\tau }_w=\frac{\partial u}{\partial y}{|}_{\eta }=0,$$ $$(Re{)}^{\frac{1}{2}}{C}_f=f″(0),$$

The heat transfer coefficient of the heat flux between the fluid and the wall (Nusselt number) is specified $Nu=\frac{q_w}{\frac{\gamma }{b}(T_w-T_{\infty })}=\theta \prime (0),$where $$q_w=k{\left(\frac{\partial T}{\partial y}\right)}_{y=0}=\frac{\gamma }{b}(T_w-T_{\infty })\theta \prime \left(0\right).$$

The mass transfer rate leads to the equation provided by Sherwood $$Sh=\frac{d_w}{\frac{D}{b}(C_w-C_{\infty })}=\varphi \prime \left(0\right),$$ where $$d_w=D{\left(\frac{\partial C}{\partial y}\right)}_{y=0}=\frac{D}{b}(C_w-C_{\infty })\varphi \prime \left(0\right).$$ where $\Delta =\frac{K_c}{j}$ = Coupling parameter, M = $\frac{K}{\vartheta \phi }$ = Permeability term, Nr = cr $\phi$ = Inertial force, α = $\frac{k_{0u_0}}{4\vartheta k}$ = viscoelastic parameter, Gr = $\frac{G_1{u}_0}{\vartheta x}$ = micro—rotational term, R = $\frac{16{\sigma }^{*}{T}_{\infty }^3}{3k^{*}ke}$ = Thermal Radiation Term, Pr = $\frac{\mathit{\text{ρcpϑ}}}{k}$ = Prandtl number, $Q_0$ = $\frac{Q}{u_0\rho cp}$ = Heat generation term, Du=$\frac{Dm{K}_T(Cw-C\mathrm{\infty })}{T_{m{C}_S{C}_P}(Tw-T\mathrm{\infty })}$ = Dufour number, Nb = $\frac{\tau DB(c_{\omega -C_{\mathrm{\infty }})}}{\vartheta }$ = Brownian motion, Nt = $\frac{\tau D_B(T_{\omega }-T_{\infty }) }{T_{\infty }\vartheta }$ = Thermophoresis Term., Sc = $\frac{\cup }{Dm}$ = Schmidt number, Sr = $\frac{Dm{K}_T(Tw-T\mathrm{\infty })}{T_{m\mathrm{ }}\vartheta (Cw-C\mathrm{\infty })}$ = Soret term, $\gamma \mathrm{ }$= = $\frac{k(T_{\omega }-T_{\mathrm{\infty }})\mathrm{ }}{\mathit{Tref}}$ = Thermophoresis (soft) term.

3. Spectral homotopy analysis method (SHAM)

In this section of the paper, we will explain how to solve the dimensionless set of Eqs. (9)–(11)(11) $$\rho \left(\frac{\partial {\tau }_{xx}}{\partial x}+\frac{\partial {\tau }_{xy}}{\partial y}\right)={\mu }_0\left(\frac{{\partial }^{2}u}{\partial y^2}\right)-k_0\left(v\frac{{\partial }^{3}u}{\partial y^3}+u\frac{{\partial }^{3}u}{\partial x\partial y^2}-2\frac{\partial u}{\partial y}\frac{{\partial }^{2}u}{\partial x\partial y}\right)$$(11) that are subject to (13)(13) $$T^4\approxeq 4T_{\infty }^{3}T-3T_{\infty }^4$$(13) and (14)(14) $$-\frac{\partial q_r}{\partial y}=\frac{16{\sigma }_s{T}_{\infty }^3}{3k_e}\frac{{\partial }^{2}T}{\partial y^2}$$(14) . We will employ the SHAM to solve these couple nonlinear partial differential equations. A version of HAM, called SHAM, can be used to solve numerical problems related to heat and mass transport. It is Liao [34] who is credited with proposing HAM and provides detailed instructions on how to use it. Motsa, Sibanda, and Shateyi (2010) proposed SHAM as a decomposition method to solve nonlinear differential equations when analytical solutions are not possible. Therefore, SHAM combines CSC with HAM for solving boundary layer equations. To develop an algorithm that selects the complete linear part of the flow equations, SHAM employed a linear operator. It results in a tedious sequence of linear ODEs that can be numerically solved using SHAM. This method aids in iteratively solving the set of equations by linearizing and decoupling the systems of coupled partial differential equations. (30) $$\begin{array}{c}f\left(\eta \right)=f\left(ϵ\right)+f_0\left(\eta \right),\theta \left(\eta \right)=\theta \left(ϵ\right)+{\theta }_0\left(\eta \right),\mathrm{ }\\ g\left(\eta \right)=g\left(ϵ\right)+g_0\left(\eta \right),\varphi \left(\eta \right)=\varphi \left(ϵ\right)+{\varphi }_0\left(\eta \right).\end{array}$$(30)

The following initial guess is defined to satisfy the boundary conditions: (31) $$f_0\left(\eta \right)=f_w+1-e^{-\eta },g_0\left(\eta \right)=e^{-\eta },{\theta }_0\left(\eta \right)=e^{-\eta },{\varphi }_0\left(\eta \right)=e^{-\eta }$$(31) (32) $$f^{″\prime }+\mathrm{\Delta }{g}^{\prime }-\frac{1}{M}-\frac{1}{M}{f}^{\prime }+a_1{f}^{\prime }+a_{2}f+a_3{f}^{″}+a_4{f}^{\prime }+a_5{f}^{\prime v}+a_{6}f+a_7{f}^{\prime v}+a_8{f}^{\prime }+a_9{f}^{″\prime }+a_{10}{f}^{″}+a_{11}{f}^{″}=H_1\left(\eta \right)$$(32) (33) $$G_r{g}^{″}-4g-2f^{″}=H_2\left(\eta \right)$$(33) (34) $$(4+3R){\theta }^{″}+P_{r}f{\theta }^{\prime }+b_{1}f+b_2{\theta }^{\prime }-P_r{Q}_0\theta ++P_{r}Du{\varphi }^{″}+b_3{\varphi }^{\prime }+b_4{\theta }^{\prime }+b_5{\theta }^{\prime }=H_3\left(\eta \right)$$(34) (35) $${\varphi }^{″}+c_1{\varphi }^{\prime }+c_{2}f+S_c{S}_r{\theta }^{\prime }+c_3{\theta }^{\prime }+c_4{\varphi }^{\prime }-\gamma \varphi {\theta }^{″}+c_5\varphi +c_6{\theta }^{″}=H_4\left(\eta \right)$$(35) where $$H_1\left(\eta \right)=-f_0^{‴}\text{-}\mathrm{\Delta }{g}_0^{\prime }+\frac{1}{M}{f}_0^{\prime }+N_r{f}_0^{\prime 2}-f_0^{\prime }{f}_0^{″}+2\alpha \eta f_0^{\prime }{f}_0^{iv}-f_0{f}_0^{\prime v}-\alpha f_0^{\prime v}{f}_0^{‴}-\alpha \eta f_0^{″}{f}_0^{‴}.$$ $$H_2\left(\eta \right)=-(4+3R){\theta }_0^{″}-P_r{f}_0{\theta }_0^{\prime }-P_r{Q}_0{Q}_0-P_{r}Du{\varphi }_0^{″}+N_b\varphi {\theta }_0^{\prime }-N_b{\theta }_0^{\prime 2}.$$ $$H_3\left(\eta \right)=-(4+3R){\theta }_0^{″}-P_r{f}_0{\theta }_0^{\prime }-P_r{Q}_0{Q}_0-P_{r}Du{\varphi }_0^{″}+N_b\varphi {\theta }_0^{\prime }-N_b{\theta }_0^{\prime 2}.$$ $$H_4\left(\eta \right)=-{\varphi }_0^{″}-S_c{f}_0{\varphi }_0^{\prime }+\gamma {\theta }^{\prime }\varphi \prime 0_0+\gamma {\varphi }_0{\theta }_0^{″}$$

The coefficient parameters are: $a_1=-2N_r{f}_0^{\prime },$ $a_2=f_0^{″},$ $a_3=f_0,$ $a_4=-2\alpha \eta f_0^{\prime v},$ $a_5=-2\alpha \eta f_0^{\prime },$ $a_6=\alpha f_0^{\prime v},$ $a_7=\alpha f_0\mathrm{ }{a}_8=\alpha f_0^{‴},$ $a_9=\alpha f_0^{\prime },$ $a_{10}=\alpha \eta f_0^{‴},$ $a_{11}=\alpha \eta f_0^{‴},$

$b_1=P_r{\theta }_0^{\prime },$ $b_2=P_r{f}_0,$ $b_3=N_b{\theta }_0^1,$ $b_4=N_b{\varphi }_0^{\prime },$ $b_5=2N_t{\theta }_0^{\prime }\mathrm{ }{c}_1=S_c{f}_0,$ $c_2=S_c{\varphi }_0^{\prime },$ $c_3=-\gamma {\varphi }_0^{\prime },\mathrm{ }{c}_4=-\gamma {\varphi }_0^{\prime }{c}_5=-\gamma {\varphi }_0^{″}{S}_c{\theta }_0^{\prime }\mathrm{ }{c}_6=-\gamma {\theta }_0{S}_c{\theta }_0^{\prime }$

We used the Chebyshev pseudospectral approach (CPM) to provide solutions to Eqs. (17)–(20)(20) $$u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}=D_m\frac{{\partial }^{2}C}{\partial z^2}+\frac{D_m{K}_T}{T_m}\frac{{\partial }^{2}T}{\partial y^2}-K_o(C-C_{\mathrm{\infty }})-\frac{\partial \left(V_{T}C\right)}{\partial y}$$(20) , and the unknown functions $f(\xi ),\theta (\xi )$ and $\varphi (\xi ).$ (36) $$\begin{array}{c}f\left(\xi \right)\approxeq \sum _{k=0}^N\mathrm{\hspace{0.17em}}{f}_k{T}_k\left({\xi }_j\right), \theta \left(\xi \right)\approxeq \sum _{k=0}^N\mathrm{\hspace{0.17em}}{\theta }_k{T}_k\left({\xi }_j\right),\\ \varphi \left(\xi \right)\approxeq \sum _{k=0}^N\mathrm{\hspace{0.17em}}{\varphi }_k{T}_k\left({\xi }_j\right), j=0,1,2,\dots ,N\end{array}$$(36)

Here ${\xi }_0,{\xi }_1,{\chi }_2,\dots ,{\xi }_N$ are Gauss-Lobatto collocation points which is defined as ${\xi }_j=\cos \frac{\pi j}{N}, j=0,1,2,\dots ,N,$ $T_k$ is the $k^{th}$ Chebyshev polynomial and $N+1$ which is the number of collocation points and the derivatives of the functions $f(\eta ),\theta (\eta )$ and $\varphi (\eta )$ at the collocation points are below (37) $$\frac{d^{r}f}{d{\eta }^r}=\sum _{k=0}^N\mathrm{\hspace{0.17em}}{D}_{kj}^{r}f\left(\xi \right),\mathrm{ }\frac{d^r\theta }{d{\eta }^r}=\sum _{k=0}^N\mathrm{\hspace{0.17em}}{D}_{kj}^r\theta \left(\xi \right),\mathrm{ }\frac{d^r\varphi }{d{\eta }^r}=\sum _{k=0}^N\mathrm{\hspace{0.17em}}{D}_{kj}^r\varphi \left(\xi \right)$$(37)

Details can be found in Canuto, Hussaini, Quarteroni, and Zang (1998) and Trefethen (2000).

4. Result and discussion

The dimensionless Eqs. (24)–(27)(27) $${\varphi }^{″}+S_{c}f\varphi \prime +S_c{S}_r\theta ″-\gamma ({\theta }^{\prime }\varphi \prime +\varphi {\theta }^{″})=0.$$(27) subject to (28)(28) $$f^{\prime }\left(0\right)=g\left(0\right)=1,f\left(0\right)=f_w, \theta \left(0\right)=\varphi \left(0\right)=1\mathit{\text{ at }}\eta =0$$(28) and (29)(29) $$f^{\prime }\left(\mathrm{\infty }\right)⟶0,\mathrm{ }\theta ⟶0,\mathrm{ }\varphi ⟶0\mathrm{ }\mathrm{at}\mathrm{ }y⟶\mathrm{\infty }$$(29) has been solve using SHAM. To give detailed explanation on the physics of the problem, we obtained results for all flow parameter such as, viscoelastic parameter, Dufour number, micro-rotation parameter, Brownian motion parameter, thermophoresis parameter, Prandtl number, thermal radiation parameter, and Eckert number, The present paper examined effect of all this flow parameters using the following default values $Du = 1.0 Gm = 2.0, Pr= 0.71, Sc = 0.2, R = 0.5, M = 0.1, So = 0.5, Ec = 0.01, K = 0.5, Na = 0.7, Nb = 0.5, Ln = 5$to obtain an insight to the physical problem. Hence, all plotted graphs and tables used the above values unless stated.

The effect of the viscoelastic parameter on the velocity profile is shown in Figure 2. When there’s an increase in viscoelastic parameter this lead to an appreciation in the velocity profile, i.e., an incremental value is observed in the velocity profile when viscoelastic is increase. A velocity increase near the plate is observed within the flow domain. When there is deformation, viscoelasticity has both viscous and elastic properties. Viscosity opposes fluid flow because the parameter’s viscous properties drag the hydrodynamic boundary layer when both viscosity and thermal conductivity remain constant.

Figure 2. The impact of viscoelastic parameter on the velocity plot.

Figure 3 indicates the effects of the Dufour parameter on velocity and temperature profiles. An increase in Dufour values is observed to increase the velocity profile and temperature profiles. The Dufour effects represent the energy flux caused by a gradient in mass concentration. This effect is most noticeable when irreversible processes are combined. As a result, the temperature profile is altered as a result of concentration.

Figure 3. Impact of Dufour parameter on the velocity and temp. plots.

Figure 4 shows the impact of micro-rotation term on the velocity profile. The micro-rotation term is notice to appreciate the velocity graph. Rotation parameter declines its linear velocity while the viscoelastic fluid is rotating. However, the velocity decrease the rotational friction force produced due to the flow.

Figure 4. Impact of micro-rotation parameter on the vel. plot.

Figure 5 shows the impact of Brownian motion parameter (Nb) on the velocity and concentration profile. When there’s an appreciation in Brownian motion it leads to velocity depreciation also a decrease in concentration profile is observed due to appreciation of the Brownian motion term. Physically, the Brownian motion made the fluid particles motion to be constant. Due to this fact, the particles will settle down and hereby stable the colloidal solutions.

Figure 5. Effect of Brownian motion term on the vel. and concentr. plots.

Figure 6 indicates the effect of thermophoresis term (Nt) on both velocity and temperature profiles. Higher value of $Nt$ is noticed to increase the velocity and temperature profiles. Since viscosity causes a resistance to the flow liquid, therefore the viscous properties of the Walters-B term coupled with the nanofluid particles causes a drastic appreciate in the fluid velocity and temperature respectively.

Figure 6. Effect of thermophoresis term on the vel. and temp. plots.

Figure 7 demonstrates the impact of Prandtl number $(Pr)$ on the velocity and temperature graph. A higher value of $Pr$ is detected to degenerate the velocity and temperature profiles. Practically, $Pr$ is used in coordinating both thermal and momentum diffusivity in the transport process of mass and heat. For $Pr\ll 1$ the flow behaviour is moderated by thermal diffusivity while $Pr\gg 1$ means the transport behaviour is moderated by momentum diffusivity. Increasing Pr results in a drop in both the momentum and thermal layers. The thickness of the momentum and thermal boundary layers reduces as a result. This means that when Pr1 is reached, the fluid particles within the boundary layer become conducive. Physically, increase in the value of Pr reduces thermal diffusivity, resulting in fluid temperature degeneration.

Figure 7. Effect of Prandtl number on the vel. and temp. plots.

The effect of thermal radiation on the velocity and temperature profiles is depicted in Figure 8. An increase in the value of thermal radiation is observed to increase hydrodynamics and thermal boundary layer thickness by improving velocity and temperature profiles. In practice, the presence of thermal radiation in the fluid transport phenomenon raises the temperature of the fluid, thus increasing its thermal condition.

Figure 8. Effect of thermal radiation parameter on the vel. and temp. plots.

Figure 9 indicates the effects of the Schmidt number on the velocity and concentration profiles. An increase in Schmidt values is observed to appreciate the velocity profile while concentration buoyancy effects have every tendency to reduce the fluid velocity, which is indicated in the graph, i.e., concentration is reduces when there is an increase in Schmidt value.

Figure 9. Effect of Schmidt number on the vel. and concentr. plots.

The significance of relevant flow terms on the local skin friction, Nusselt, and Sherwood numbers is shown in Table 1. The presence of the permeability term (M) indicates a decrease in local skin friction. Appreciation in M has no effect on the Nusselt and Sherwood numbers. An increase in the value of the viscoelastic parameter is observed to reduce the local skin friction and Nusselt number. An enhancement in the local skin friction and Nusselt number is observed due to increases in thermal radiation (R), Prandtl number, and Dufour number. In Table 1, a depreciation in local skin friction is noticed because of an appreciation value in the Schmidt number. A depreciation in skin friction and Sherwood is noticed due to a higher thermophoresis parameter.

Table 1. The computational values that represent the coefficient of skin friction $(C_f),$ Nusselt number $(Nu),$ sherwood number $(Sh)$ for difference values of, $M\mathrm{ }\alpha ,\mathrm{ }{G}_r,$ $R,$ $Pr,$ $Sc,$ $D_u,$ $S_r,$ $Nb,\mathrm{ }Nt,$ $Cf,$ $Nu$ and $Sh.$

$M$	$\alpha$	$G_r$	$R$	$Pr$	$Sc$	$D_u$	$S_r$	$Nb$	$Nt$	$Cf$	$Nu$	$Sh$
0.0										0.4258	0.1412	0.5355
0.5										0.3312	0.1412	0.5353
0.1										0.2162	0.1412	0.5353
	0.2									0.7676	0.3371	0.7977
	0.4									0.6621	0.3371	0.7977
	0.6									0.4220	0.3371	0.7977
		1.0								0.4343	0.2812	0.6112
		2.0								0.8710	0.3897	0.6112
		3.0								1.2396	0.4858	0.6112
			0.0							0.0845	0.3391	0.7177
			0.5							0.6731	0.3471	0.7177
			1.0							1.3052	0.3427	0.7177
				0.71						0.6621	0.3371	0.5002
				1.00						1.2168	0.3380	0.5002
				7.00						2.3040	0.4116	0.5002
					0.22					0.3255	0.4168	0.6401
					0.31					0.6820	0.4168	0.8034
					0.61					1.1911	0.4168	1.0274
						1.0				0.1042	0.1066	0.5125
						2.0				0.4431	0.2712	0.5125
						3.0				0.9905	0.4359	0.5125
							1.0			0.1864	0.5181	0.9776
							2.0			0.4541	0.2812	0.9776
							3.0			0.8710	0.3897	0.9776
								2.0		1.9194	0.7141	1.4814
								3.0		1.8826	0.7141	1.6772
								4.0		1.6342	0.7141	1.7184
									2.0	0.6621	0.3371	0.6005
									3.0	7126	0.7630	0.6005
									4.0	2.9503	1.4573	0.6005

4.1. Validation of the code

The correctness as well as accuracy of the present analysis is checked by comparing the present analysis with the previous study of Bilal et al. (2022) when $Nb=Nt=\delta \gamma =cr=\alpha =0.$

The validations as shown in Table 2 are observed to be in agreement. We can conclude that the outcome of the present analysis is good for future studies in simultaneous flow of double non-Newtonian fluid is a stretching sheet.

Table 2. Comparing the present study with an existing work.

		Bilal et al. (2022)	Present study			Bilal et al. (2022)	Present study
R	Pr	Nux	Nux	$R$	Pr	Nux	Nux
0.3	0.7	0.245742	0.245743	0.3	0.7	0.245756	0.245755
0.3	0.7	0.241842	0.241844	0.3	0.7	0.241852	0.241851
0.3	0.7	0.234106	0.234108	0.3	0.7	0.234108	0.234108
0.3	0.7	0.245742	0.245741	0.3	0.7	0.245746	0.245748
0.3	0.7	0.324886	0.324885	0.3	0.7	0.324890	0.324892
0.3	0.7	0.402525	0.402524	0.3	0.7	0.402528	0.402530

Bilal et al. (2022) when $Nb=Nt=\delta \gamma =cr=\alpha =0.$

5. Conclusion

The present exploration is based on the numerical scrutinization of Micropolar and Walters-B non-Newtonian fluids motion under the contribution of heat generation, Soret–Dufour, magnetic field and thermal radiation. The mass and heat transfer analysis resulted into coupled nonlinear partial differential equations. A well-known numerical method called SHAM is utilized on the transformed equations. The calculations of quantities of engineering interest are tabulated while effects of different parameters are presented in graphs.

The current result shows that as the viscoelastic term rises, the fluid velocity significantly reduces. Furthermore, this research demonstrates that an increase in the Dufour term enhances fluid velocity and temperature. This is because the concentration gradient’s energy flux is inversely proportional to velocity.

The key findings of this study are, as the value of viscoelastic term $\beta$ is raised, the fluid viscosity increases and hereby causes resistance to the transport phenomena then a higher value of Prandtl $Pr$ is found to degenerate the velocity along-side the temperature. A higher value of the Dufour number appears to increase the velocity and temperature profile while the chemical reaction is observed to produce a destructive reaction by reducing the velocity alongside concentration profile and Larger values of thermal viscosity parameter created an enhancement in velocity and angular velocity profile due to higher thermal conductivity of the liquid while Soret parameter improves the velocity and concentration field because the mass flux created by the temperature gradient is inversely proportional to the mean temperature.