Thermal analysis of radiative Sutterby nanofluid flow over stretching curved surface

Abstract

In this analysis, Sutterby fluid model over a curved surface is considered. The main mechanisms that contribute to the improvement of the convection characteristics of the nanofluid are categorized as Brownian motion and thermophoresis. The radiation and slip mechanism have been studied at curved stretchable surface. The suction/injection impacts also studied. The partial differential equations are converted into ordinary differential equations through transformations.

The numerical solution of the specified mathematical model is obtained using the built-in bvp4c tool in MATLAB. The effects of various parameters related to the system of ordinary differential equations are illustrated in the graphs. The influence of some intended parameters through the physical quantities are presented through tabular form.

Nomenclature

$\omega$	Section/injection parameter	$N_B$	Brownian motion
$R_d$	Radiation	$\varpi$	Sutterby fluid
$\delta$	Thermal slip	$\varsigma$	Curvature
${\delta }_1$	Concentration slip	$Ec$	Eckert number
$\nabla$	Gradient operator	${\rho }_f$	Density of fluid
${\mu }_f$	Dynamic viscosity	${\left(C_p\right)}_f$	Fluid heat capacity
$K\left(T\right)$	Variable thermal conductivity	${\left(C_p\right)}_s$	Heat capacity of particle
${\rho }_s$	Density of particles	$T$	Temperature
$C$	Concentration	$T_{\infty }$	Temperature of Ambient
$C_{\infty }$	Concentration of Ambient	$D_B$	Brownian diffusion
$D_T$	Thermophoretic	$B$	Dipole of Magnetic
$E_c$	Activation energy dimensionless	$\mathit{Pr}$	Prandtl number

1. Introduction

The fact that non-Newtonian fluids are presented in real life which has led to some applications in the engineering, biological, pharmaceutical, and industrial fields. The rheological properties of complex fluid models are paints, industrial oils, polymeric solutions, and so on belong to the power-law fluid model. For the first time, Sutterby [1] initiated the idea to combine study of dilatant and pseudoplastic fluid properties. This idea has attracted much attention from researchers due to its several applications in real-life problems. The uses of such types of fluids played an essential role in engineering and industries. Akbar and Nadeem [2] prolonged the ideas of Sutterby [1] for peristaltic model having asymmetric channel inclindely. The analytical results have developed to present the effects of slip. Ahmad et al. [3] disputed the flow of Sutterby fluid analytically. The impact of chemical, radiation, and mixed convection on a squeezed channel are considered. Khan et al. [4] considered the reaction of heterogeneous-homogeneous on Sutterby fluid at a rotating desk. The Cattaneo–Christov model is executed on the Sutterby fluid model to attain the major results. Sajid et al. [5] reflected the Maxwell Sutterby fluid flow at an inclined exponential stretching surface. The MHD impacts with temperature-dependent properties have been studied in their analysis. Activation energy and heat sink/source influence have been debated on the exponential stretching surface. Mabood et al. [6] contested the dynamic characteristic of Sutterby fluid with Fourier and Fick law at spinning stretching disk. Abdal et al. [7] initiated to analyze the nanobioflim by Sutterby fluid at stretching sheet. Hanif et al. [8] tinted the model of Sutterby fluid. Hussain et al. [9] considered the Sutterby fluid the impressions of ferromagnetic field and magnetic dipole. Hussain et al. [10] contested the inspiration of Sutterby fluid flow model under the activation energy. Abdelsalam et al. [11] scrutinized the effects of Sutterby fluid in the occurrence of an entropy model. Recently, few authors have developed ideas about the Sutterby fluid model using various assumptions and stretchable surfaces (see Refs. [12,13]).

The boundary layer flow under the stretching of surface has been discussed in detail due to the number of applications in the real fields but stretchable sheets have one of the most important examples of procedure of manufacturing. Several types of stretching velocities are defined namely: nonlinear, linear, and exponentially stretching velocities. Several stretching velocities applications are hot rolling, glass blowing, rubber sheet, the product of paper, spinning of fiber, fiberglass, polymer sheet, and so on. In the early times, the boundary layer flow at stretchable surface was studied by Crane [14]. Crane [14] considered the cylindrical moving surface and debated the impacts of fluid at motion or rest. The impact boundary layer at faraway and close to the orifice surface. Vajravelu and Cannon [15] worked on the Crane [14] assumptions. Vajravelu and Cannon [15] considered the nonlinear stretchable surface to develop the analytical results. Sajid et al. [16] considered the viscous model at curved surface. In the early times, the results have been achieved analytically. Abbas et al. [17] deliberated the results of PST and PHF at stretching curved surface. The MHD is applied to the normal flow of viscous fluid and analytical results are achieved. Akbar et al. [18] initiated the Eyring power fluid at a stretchable surface with MHD influence. Sanni et al. [19] pondered the numerical outcomes of the power law stretchable curved sheet. Saif et al. [20] deliberated the Jeffrey fluid model at a curved sheet. The Buongiorno model with heat generation is debated at a curved sheet. Kumar et al. [21] deliberate the nanomaterial flow over curved sheet. The thermal and viscous impacts have been considered in their exploration. Bhatti and Abdelsalam [22] highlighted the influence of stretching surface impacts using the ferromagnetic flow of nanofluid. Nawaz et al. [23] discussed the third-grade fluid model using the stretchable parallel surface for mixed convection. Abbas et al. [24] pondered the influence of the radiative flow of Sutterby fluid at a nonlinear stretchable cylinder. Few researchers have studied the curved sheet under several fluid models with external flow assumptions (see Refs. [[25], [26], [27]]).

Nanofluids is a new class that was just established. A nanofluid is made up of base fluid with nanosized particles. The field of nanotechnologies and others benefited greatly from this development. Buongiorno [28] was initiated the comprehensive analysis of the convective transport of nanofluid based at MIT (Massachusetts Institute of Technology). He considered the seven slip mechanisms between base fluid and nanoparticles having relative velocities. Two of them are the most important found such as thermophoresis and Brownian diffusion. Rana et al. [29] used the finite element technique to investigate the nanofluid at a stretching surface. The time-dependent flow of MHD (Magneto-hydro-dynamics) is debated in their analysis. Goyal and Bhargava [30] debated the viscoelastic fluid with the Buongiorno model at a stretchable sheet. Boundary layer flow of double-diffusive using the finite element technique to achieve the results. Akbar [31] highlighted the impact of the Buongiorno model and Sutterby fluid under the natural convection of peristaltic flow in a porous channel. Hayat et al. [32] considered the satterby nanofluid under the chemical reactive flow and heat absorption. Sohail and Naz [33] studied the sutterby nanofluid at a stretchable cylinder. Khan et al. [34] discussed the inspiration for the magnetized flow of nanofluid with sutterby fluid model at a wedge. Abdelsalam and Zaher [35] discussed the spermatic fluid in the attendance of electroosmotic forces. Aldabesh et al. [36] deliberated the inspiration of nanofluid under the microorganism phenomena. Nawaz et al. [37] discussed the heat transfer for micropolar fluid model. Few authors have developed ideas about the Buongiorno model under the flow mechanism (see Refs. [[38], [39], [40]]).

The literature presents an analysis of the Sutterby fluid model flowing over a curved surface, considering linear stretching velocity. The study includes the effects of suction/injection, Brownian motion, and thermophoresis on the curved surface. Thermal and concentration slip mechanisms are also accounted for. The mathematical model is cracked numerically by the bvp4c tool in MATLAB, and the impacts of various parameters related to the system of ordinary differential equations (ODEs) are reported in both graphical and tabular forms. The results illustrate how certain parameters influence physical quantities, making them applicable to real-life problems.

2. Mathematical formulation

Steady flow of incompressible sutterby nanofluid at stretchable curved surface which revealed in Fig. 1(a). Thermophoresis and Brownian motion mechanism are deliberated on the curved surface. Thermal and concentration slip mechanism is considered in present analysis. The fluid velocity invented as $u_w=ax$ (linear stretching) where $a>0$ at curved surface. The ($x,r$) are the curvilinear coordinates and $x-$ along axial and $r-$ along radial directions. The radiation influence have been taken into account. The $T_w$ (wall temperature) at surface and $T_{\infty }$ (ambient temperature) is for away from surface. The $C_w$ (wall concentration) at surface and $C_{\infty }$ (ambient concentration) is for away from surface. $S$ is the difference length between origin and surface such as the surface become flat if $S\to \infty$. The boundary approximation are implemented on the heat and mass transfer of Sutterby fluid and the physical model become in the differential form:

$$\frac{S}{S+r}\frac{\partial u}{\partial x}+\frac{1}{S+r}\frac{\partial }{\partial r}\left(v\left(r+S\right)\right)=0\text{,}$$ (1)

$$\frac{u^2}{S+r}=\frac{1}{{\rho }_f}\frac{\partial p}{\partial r}\text{,}$$ (2)

$$\left(v\frac{\partial u}{\partial r}+\frac{Su}{S+r}\frac{\partial u}{\partial x}+\frac{vu}{S+r}\right)+\frac{1}{{\rho }_f}\frac{S}{S+r}\frac{\partial p}{\partial r}=\frac{\nu }{2}\left(\frac{{\partial }^{2}u}{{\partial }^{2}r}+\frac{1}{S+r}\frac{\partial u}{\partial r}-\frac{u}{{\left(S+r\right)}^2}-\frac{m{N}^2}{2}{\left(\frac{\partial u}{\partial r}-\frac{u}{S+r}\right)}^2\left(\frac{{\partial }^{2}u}{{\partial }^{2}r}-\frac{1}{3\left(S+r\right)}\frac{\partial u}{\partial r}+\frac{u}{{3\left(S+r\right)}^2}\right)\right)\text{,}$$ (3)

$$\left(v\frac{\partial T}{\partial r}+\frac{Su}{S+r}\frac{\partial T}{\partial x}\right)-\frac{16{\sigma }^{*}{T}_m^3}{3k^{*}\rho C_p}\left(\frac{{\partial }^{2}T}{{\partial }^{2}r}+\frac{1}{S+r}\frac{\partial T}{\partial r}\right)={\alpha }_f\left(\frac{{\partial }^{2}T}{{\partial }^{2}r}+\frac{1}{S+r}\frac{\partial T}{\partial r}\right)+\tau \left(D_B\left(\frac{\partial C}{\partial r}\right)\left(\frac{\partial T}{\partial r}\right)+\left(\frac{D_T}{T_{\infty }}\right){\left(\frac{\partial T}{\partial r}\right)}^2\right)+\frac{{\mu }_0}{{\rho C}_p}\left(1-\frac{m{N}^2}{6}{\left(\frac{\partial u}{\partial r}+\frac{u}{S+r}\right)}^2\right){\left(\frac{\partial u}{\partial r}+\frac{u}{S+r}\right)}^2\text{,}$$ (4)

$$\left(v\frac{\partial C}{\partial r}+\frac{Su}{S+r}\frac{\partial C}{\partial s}\right)-\left(\frac{D_T}{T_{\infty }}\right)\left(\frac{{\partial }^{2}T}{{\partial }^{2}r}+\frac{1}{S+r}\frac{\partial T}{\partial r}\right)=D_B\left(\frac{{\partial }^{2}C}{{\partial }^{2}r}+\frac{1}{S+r}\frac{\partial C}{\partial r}\right)\text{.}$$ (5)

Fig. 1(a).

Flow pattern of Sutterby nanofluid over stretching curved surface.

The respected boundary conditions are,

$$\begin{array}{c}u=U_w+{\lambda }_2\left(\frac{\partial u}{\partial r}-\frac{u}{R+r}\right),v=V_w,T=T_w+{\lambda }_1\left(\frac{\partial T}{\partial r}\right),C=C_w+{\lambda }_0\left(\frac{\partial C}{\partial r}\right),atr\to 0\text{,}\\ u\to 0,\frac{\partial u}{\partial r}\to 0,T\to T_{\infty ,}C\to C_{\infty ,}atr\to \infty \text{.}\end{array}$$ (6)

The similarity variables are presented as

$$\begin{array}{c}T=T_{\infty }+\left(T_w{-T}_{\infty }\right)\theta \left(\mathrm{{\rm Y}}\right),\mathrm{{\rm Y}}=\sqrt{\frac{a}{{\nu }_f}}r,u=ax{F}^{\prime }\left(\mathrm{{\rm Y}}\right)\text{,}\\ v=-\frac{S}{r+S}\sqrt{a{\nu }_f}F\left(\mathrm{{\rm Y}}\right),C=C_{\infty }+\left(C_w{-C}_{\infty }\right)\phi \left(\mathrm{{\rm Y}}\right),P=\rho a^2{x}^{2}p\left(\mathrm{{\rm Y}}\right)\text{.}\end{array}$$ (7)

Eq. (7) of dimensionless terms are utilized on Eqs. (1), (2), (3), (4), (5), (6) than become dimensionless form as:

$$P^{\prime }=\frac{{F^{\prime }}^2}{\mathrm{{\rm Y}}+\varsigma }\text{,}$$ (8)

$$\frac{2\varsigma }{\mathrm{{\rm Y}}+\varsigma }P=\left(\frac{1}{2}\right)\left[F^{‴}+\frac{F^{″}}{\mathrm{{\rm Y}}+\varsigma }-\frac{F^{\prime }}{{\left(\mathrm{{\rm Y}}+\varsigma \right)}^2}\right]-\frac{\varsigma }{\mathrm{{\rm Y}}+\varsigma }{F^{\prime }}^2+\frac{\varsigma }{\mathrm{{\rm Y}}+\varsigma }{F}^{\prime }{F}^{″}+\frac{\varsigma }{{\left(\mathrm{{\rm Y}}+\varsigma \right)}^2}F{F}^{\prime }-\left(\frac{\varpi }{4}\right)\left({F^{″}}^2{F}^{‴}-\frac{{6F}^{″}{F}^{‴}{F}^{\prime }+{F^{″}}^3}{3\left(\mathrm{{\rm Y}}+\varsigma \right)}-\frac{{F^{\prime }}^2{F}^{″}}{3{\left(\mathrm{{\rm Y}}+\varsigma \right)}^3}+\frac{{F^{\prime }}^3}{3{\left(\mathrm{{\rm Y}}+\varsigma \right)}^4}+\frac{{F^{″}}^2{F}^{\prime }+{F^{\prime }}^2{F}^{‴}}{3{\left(\mathrm{{\rm Y}}+\varsigma \right)}^2}\right)\text{,}$$ (9)

$$\frac{\left(1+R\right)}{\mathit{Pr}}\left({\theta }^{″}+\frac{1}{\mathrm{{\rm Y}}+\varsigma }{\theta }^{\prime }\right)+\frac{\varsigma }{\mathrm{{\rm Y}}+\varsigma }F{\theta }^{\prime }+\left(N_B{\theta }^{\prime }{\phi }^{\prime }+N_T{\theta }^{\prime }{\theta }^{\prime }\right)+Ec\left({F^{″}}^2-\frac{2F^{\prime }{F}^{″}}{\mathrm{{\rm Y}}+\varsigma }+\frac{{F^{\prime }}^2}{{\left(\mathrm{{\rm Y}}+\varsigma \right)}^2}-\left(\frac{\varpi }{4}\right)\left(\frac{2{F^{\prime }}^2{F^{″}}^2}{{\left(\mathrm{{\rm Y}}+\varsigma \right)}^2}+{F^{″}}^4+\frac{{F^{\prime }}^4}{{\left(\mathrm{{\rm Y}}+\varsigma \right)}^4}-\frac{4{F^{″}}^3{F}^{\prime }}{\mathrm{{\rm Y}}+\varsigma }-4\frac{{F^{\prime }}^3{F}^{″}}{{\left(\mathrm{{\rm Y}}+\varsigma \right)}^3}+4\frac{{F^{″}}^2{F^{\prime }}^2}{{\left(\mathrm{{\rm Y}}+\varsigma \right)}^2}\right)\right)=0\text{,}$$ (10)

$$\left({\phi }^{″}+\frac{1}{\zeta +K_0}{\phi }^{\prime }\right)+\frac{K_0}{\zeta +K_0}F{\phi }^{\prime }+\frac{N_B}{N_T}\left({\theta }^{″}+\frac{1}{\zeta +K_0}{\theta }^{\prime }\right)=0\text{,}$$ (11)

The pressure profile can be computed from Eq. (9) as follows:

$$P\left(\zeta \right)=-\frac{F^{\prime }}{{4\varsigma \left(\mathrm{{\rm Y}}+\varsigma \right)}^3}+\frac{F^{″}}{4\varsigma }-\left(\frac{\varpi \left(\mathrm{{\rm Y}}+\varsigma \right)}{8\varsigma }\right)\left({F^{″}}^2{F}^{‴}-\frac{{F^{″}}^3+6F^{″}{F}^{‴}{F}^{\prime }}{3\left(\mathrm{{\rm Y}}+\varsigma \right)}-\frac{{F^{\prime }}^3}{{3\left(\mathrm{{\rm Y}}+\varsigma \right)}^4}-\frac{{F^{\prime }}^2{F}^{″}}{{\left(\mathrm{{\rm Y}}+\varsigma \right)}^3}+\frac{F^{‴}{F^{\prime }}^2+F^{\prime }{F^{″}}^2}{{\left(\mathrm{{\rm Y}}+\varsigma \right)}^2}\right)+\left(\frac{\left(\mathrm{{\rm Y}}+\varsigma \right)}{2\varsigma }\right)\left(\frac{\varsigma \left(F{F}^{″}-{F^{\prime }}^2\right)}{\left(\mathrm{{\rm Y}}+\varsigma \right)}+\frac{\varsigma F{F}^{\prime }}{{\left(\mathrm{{\rm Y}}+\varsigma \right)}^2}\right)+\left(\frac{\left(\mathrm{{\rm Y}}+\varsigma \right)}{4\varsigma }\right)F^{‴}\text{,}$$ (12)

Solving Eq. (8) and Eq. (12), while dismissing the term of pressure:

$$-\frac{F^{\prime }-2\varsigma F{F}^{\prime }}{{\left(\mathrm{{\rm Y}}+\varsigma \right)}^3}+F^{⁗}-\left(\frac{\varpi }{2}\right)\left(\frac{{F^{\prime }}^2{F}^{⁗}+4F^{″}{F}^{‴}{F}^{\prime }}{2}+{F^{″}}^2{F}^{⁗}+2{F^{‴}}^2{F}^{″}-\frac{{F^{\prime }}^3}{{\left(\mathrm{{\rm Y}}+\varsigma \right)}^5}-\frac{2{F^{″}}^2{F}^{‴}}{\mathrm{{\rm Y}}+\varsigma }-\frac{2{F^{\prime }}^2{F}^{‴}}{{\left(\mathrm{{\rm Y}}+\varsigma \right)}^3}-\frac{{3F^{″}}^2{F}^{\prime }}{{\left(\mathrm{{\rm Y}}+\varsigma \right)}^3}+\frac{3{F^{\prime }}^2{F}^{″}}{{\left(\mathrm{{\rm Y}}+\varsigma \right)}^4}+\frac{{F^{″}}^3}{{\left(\mathrm{{\rm Y}}+\varsigma \right)}^2}-2\frac{F^{\prime }{F^{″}}^2+F^{⁗}{F}^{\prime }{F}^{″}}{\mathrm{{\rm Y}}+\varsigma }\right)+\frac{2F^{‴}+2\varsigma \left(F{F}^{‴}-F^{″}{F}^{\prime }\right)}{\mathrm{{\rm Y}}+\varsigma }-\frac{2\varsigma {F^{\prime }}^2+F^{″}+2\varsigma F{F}^{″}}{{\left(\mathrm{{\rm Y}}+\varsigma \right)}^2}=0\text{,}$$ (13)

$$\frac{1}{\mathit{Pr}}\left(1+\frac{4}{3}{R}_d\right)\left({\theta }^{″}+\frac{1}{\mathrm{{\rm Y}}+\varsigma }{\theta }^{\prime }\right)+\frac{\varsigma }{\mathrm{{\rm Y}}+\varsigma }F{\theta }^{\prime }+\left(N_B{\theta }^{\prime }{\phi }^{\prime }+N_T{\theta }^{\prime }{\theta }^{\prime }\right)+Ec\left({F^{″}}^2-\frac{2F^{\prime }{F}^{″}}{\mathrm{{\rm Y}}+\varsigma }+\frac{{F^{\prime }}^2}{{\left(\mathrm{{\rm Y}}+\varsigma \right)}^2}-\left(\frac{\varpi }{4}\right)\left(\frac{2{F^{\prime }}^2{F^{″}}^2}{{\left(\mathrm{{\rm Y}}+\varsigma \right)}^2}+{F^{″}}^4+\frac{{F^{\prime }}^4}{{\left(\mathrm{{\rm Y}}+\varsigma \right)}^4}-\frac{4{F^{″}}^3{F}^{\prime }}{\mathrm{{\rm Y}}+\varsigma }-4\frac{{F^{\prime }}^3{F}^{″}}{{\left(\mathrm{{\rm Y}}+\varsigma \right)}^3}+4\frac{{F^{″}}^2{F^{\prime }}^2}{{\left(\mathrm{{\rm Y}}+\varsigma \right)}^2}\right)\right)=0\text{,}$$ (14)

$$\left({\phi }^{″}+\frac{1}{\mathrm{{\rm Y}}+\varsigma }{\phi }^{\prime }\right)+\frac{\varsigma }{\mathrm{{\rm Y}}+\varsigma }F{\phi }^{\prime }+\frac{N_B}{N_T}\left({\theta }^{″}+\frac{1}{\mathrm{{\rm Y}}+\varsigma }{\theta }^{\prime }\right)=0\text{,}$$ (15)

having boundary conditions are

$$\begin{array}{c}F\left(0\right)=\omega ,F^{\prime }\left(0\right)=1,{\delta }_1{\phi }^{\prime }\left(0\right)+1=\phi \left(0\right)\text{,}\\ \delta {\theta }^{\prime }\left(0\right)+1=\theta \left(0\right),\theta \left(\infty \right)=F^{\prime }\left(\infty \right)=\phi \left(\infty \right)=F^{″}\left(\infty \right)=0\text{.}\end{array}$$ (16)

The engineering quantities of the fluid flow at the surface are defined as:

$$C_f=\frac{{\tau }_{rx}}{{\rho }_f{u}_w^2},{Nu}_n=\frac{x{q}_w}{k_f\left(T_w-T_{\infty }\right)},{Sh}_n=\frac{x{h}_m}{D_B\left(C_w-C_{\infty }\right)}\text{.}$$ (17)

Where, the quantities (${\tau }_{rs}$ and $q_w$) are revealed as

$$\begin{array}{c}{\tau }_{rs}={\frac{\mu }{2}\left(1-\frac{m{N}^2}{6}\left({\left(\frac{u}{S+r}\right)}^2-2\frac{1}{\left(S+r\right)}\frac{\partial u}{\partial r}+{\left(\frac{\partial u}{\partial r}\right)}^2\right)\right)\left(\frac{\partial u}{\partial r}-\frac{u}{S+r}\right)|}_{r=0}\text{,}\\ q_w=-{\left(\frac{\partial T}{\partial r}\right)}_{r=0},h_m=-{\left(\frac{\partial C}{\partial r}\right)}_{r=0}\text{.}\end{array}$$ (18)

Using Eq. (7) in Eqs. (17), (18), we obtain Eqs. (19), (20), (21) as follows:

$${Re}_s^{-1/2}{C}_f=F^{″}\left(0\right)-\frac{\varpi }{6}\left(-\frac{3}{\varsigma }{F}^{″}\left(0\right)+\frac{3}{{\varsigma }^2}{F}^{″}\left(0\right)-\frac{1}{{\varsigma }^3}+{F^{″}\left(0\right)}^3\right)-\frac{1}{\varsigma }\text{,}$$ (19)

$${Re}_s^{-1/2}{Nu}_n=-\left(\left(1+\frac{4}{3}{R}_d\right)\right){\theta }^{\prime }\left(0\right)\text{,}$$ (20)

$${Re}_s^{-1/2}{Sh}_n=-{\phi }^{\prime }\left(0\right)$$ (21)

3. Numerical procedure

The dimensionless differential equations (13), (14), (15), (16) are transformed into first order differential equation as initial value problem and the procedure are defined in detail. The convergent criteria of the present model is ${10}^{-6}$ and the finite range is considered in ${\mathrm{{\rm Y}}}_{\infty }=10$. The finite range ${\mathrm{{\rm Y}}}_{\infty }=10$ are shown correct results and satisfied the numerical results asymptotically. The procedure are mentioned in detail as below:

$$Y\left(1\right)=F\left(\mathrm{{\rm Y}}\right);Y\left(2\right)=F^{\prime }\left(\mathrm{{\rm Y}}\right);Y\left(3\right)=F^{″}\left(\mathrm{{\rm Y}}\right);Y\left(4\right)=F^{‴}\left(\mathrm{{\rm Y}}\right);YY1=F^{⁗}\left(\mathrm{{\rm Y}}\right)\text{;}$$

$$YY1={\left(1-\left(\frac{\varpi }{2}\right)\left(\frac{Y\left(2\right)Y\left(2\right)}{2}+Y\left(3\right)Y\left(3\right)-2\frac{Y\left(3\right)Y\left(2\right)}{\mathrm{{\rm Y}}+\varsigma }\right)\right)}^{-1}\left(-\frac{Y\left(2\right)-2\varsigma Y\left(1\right)Y\left(2\right)}{{\left(\mathrm{{\rm Y}}+\varsigma \right)}^3}-\left(\frac{\varpi }{2}\right)\left(\frac{4Y\left(2\right)Y\left(3\right)Y\left(4\right)}{2}+2Y\left(3\right)Y\left(4\right)Y\left(4\right)-\frac{Y\left(2\right)Y\left(2\right)Y\left(2\right)}{{\left(\mathrm{{\rm Y}}+\varsigma \right)}^5}-\frac{2Y\left(3\right)Y\left(4\right)Y\left(3\right)}{\mathrm{{\rm Y}}+\varsigma }-\frac{2Y\left(2\right)Y\left(2\right)Y\left(4\right)}{{\left(\mathrm{{\rm Y}}+\varsigma \right)}^3}-\frac{3Y\left(2\right)Y\left(3\right)Y\left(3\right)}{{\left(\mathrm{{\rm Y}}+\varsigma \right)}^3}+\frac{3Y\left(2\right)Y\left(2\right)Y\left(3\right)}{{\left(\mathrm{{\rm Y}}+\varsigma \right)}^4}+\frac{Y\left(3\right)Y\left(3\right)Y\left(3\right)}{{\left(\mathrm{{\rm Y}}+\varsigma \right)}^2}-2\frac{Y\left(3\right)Y\left(3\right)Y\left(2\right)}{\mathrm{{\rm Y}}+\varsigma }\right)+\frac{2Y\left(4\right)+2\varsigma \left(Y\left(1\right)Y\left(4\right)-Y\left(3\right)Y\left(2\right)\right)}{\mathrm{{\rm Y}}+\varsigma }-\frac{2\varsigma Y\left(2\right)Y\left(2\right)+Y\left(3\right)+2\varsigma Y\left(1\right)Y\left(3\right)}{{\left(\mathrm{{\rm Y}}+\varsigma \right)}^2}\right)\text{;}$$

$$Y\left(5\right)=\theta \left(\mathrm{{\rm Y}}\right);Y\left(6\right)={\theta }^{\prime }\left(\mathrm{{\rm Y}}\right);YY2={\theta }^{″}\left(\mathrm{{\rm Y}}\right)\text{;}$$

$$YY2=\frac{\mathit{Pr}}{\left(1+\frac{4}{3}R\right)}\left(\frac{\varsigma }{{\rm Y}+\varsigma }Y\left(6\right)Y\left(1\right)+\left(N_{B}Y\left(8\right)Y\left(6\right)+N_{T}Y\left(6\right)Y\left(6\right)\right)+Ec\left(Y\left(3\right)Y\left(3\right)-\frac{2Y\left(2\right)Y\left(3\right)}{{\rm Y}+\varsigma }+\frac{Y\left(2\right)Y\left(2\right)}{{\left({\rm Y}+\varsigma \right)}^2}-\left(\frac{\varpi }{4}\right)\left(\frac{2Y\left(2\right)Y\left(2\right)Y\left(3\right)Y\left(3\right)}{{\left({\rm Y}+\varsigma \right)}^2}+Y\left(3\right)Y\left(3\right)Y\left(3\right)Y\left(3\right)+\frac{Y\left(2\right)Y\left(2\right)Y\left(2\right)Y\left(2\right)}{{\left({\rm Y}+\varsigma \right)}^4}-\frac{4Y\left(3\right)Y\left(3\right)Y\left(3\right)Y\left(2\right)}{{\rm Y}+\varsigma }-4\frac{Y\left(2\right)Y\left(2\right)Y\left(2\right)Y\left(3\right)}{{\left({\rm Y}+\varsigma \right)}^3}+4\frac{Y\left(2\right)Y\left(2\right)Y\left(3\right)Y\left(3\right)}{{\left({\rm Y}+\varsigma \right)}^2}\right)\right)\right)-\frac{\left(1+\frac{4}{3}R\right)}{\mathit{Pr}}\frac{1}{{\rm Y}+\varsigma }Y\left(6\right)\text{;}$$

$$Y\left(8\right)={\phi }^{\prime }\left(\mathrm{{\rm Y}}\right);Y\left(7\right)=\phi \left(\mathrm{{\rm Y}}\right);YY3={\phi }^{″}\left(\mathrm{{\rm Y}}\right)\text{;}$$

$$YY3=-\left(\frac{\varsigma }{{\rm Y}+\varsigma }Y\left(8\right)Y\left(1\right)+\frac{N_B}{N_T}\left(YY2+\frac{1}{{\rm Y}+\varsigma }Y\left(6\right)\right)\right)-\frac{1}{{\rm Y}+\varsigma }Y\left(8\right)\text{;}$$

having boundary conditions are

$$Y0\left(1\right)-\omega ;Y0\left(2\right)-1;Yinf\left(2\right);Yinf\left(3\right);{\delta }_{1}Y0\left(8\right)+1-Y0\left(7\right)\text{;}$$

$$Yinf\left(5\right);Yinf\left(7\right);\delta Y0\left(6\right)+1-Y0\left(5\right)\text{;}$$

${\tilde{R}}_1,{\tilde{R}}_2,{\tilde{R}}_3$ are the residual of current model and the system of differential equations are cracked by implementing the numerical scheme RK-4th order. The solution converge criteria defined that the tolerance error is less than ${10}^{-6}$. Residuals are revealed as:

$${\tilde{R}}_1=\left|{\mathrm{Y}}_2\left(\infty \right)-\widehat{{\mathrm{Y}}_2}\left(\infty \right)\right|\text{;}$$

$${\tilde{R}}_2=\left|{\mathrm{Y}}_5\left(\infty \right)-\widehat{{\mathrm{Y}}_5}\left(\infty \right)\right|\text{;}$$

$${\tilde{R}}_3=\left|{\mathrm{Y}}_7\left(\infty \right)-\widehat{{\mathrm{Y}}_7}\left(\infty \right)\right|\text{;}$$

Hence, $\widehat{Y_2}\left(\infty \right),\widehat{Y_5}\left(\infty \right),\widehat{Y_7}\left(\infty \right)$ are calculated boundary values. The residual error of the present work is highlighted in Fig. 1(b).

Fig. 1(b).

Residual error of the present work.

4. Results and discussion

The results of the concerning parameters: Eckert number ($\text{Ec}$), Brownian motion (${\mathrm{N}}_B$), Thermophoresis (${\mathrm{N}}_T$) and radiation (${\mathrm{R}}_d$) on the temperature ($\theta \left(\mathrm{{\rm Y}}\right)$) are presented in Fig. 2, Fig. 3, Fig. 4, Fig. 5 for the both cases of suction/injection. Fig. 2 debated the influence of Eckert number on the temperature ($\theta \left(\mathrm{{\rm Y}}\right)$). The temperature increased due to a rise in the Eckert number. The higher Eckert number enhanced the kinetic energy, which then converted into internal energy. This internal energy counteracted the viscous stress of the fluid, ultimately leading to growth in the temperature. The difference of temperature ($\theta \left(\mathrm{{\rm Y}}\right)$) and Brownian motion (${\mathrm{N}}_B$) which presented in Fig. 3. The relation between temperature ($\theta \left(\mathrm{{\rm Y}}\right)$) and Brownian motion (${\mathrm{N}}_B$) have directly proportion. Brownian motion enriched which improved the temperature ($\theta \left(\mathrm{{\rm Y}}\right)$) because energy (K. E) increased the motion of the particle of the fluid ultimately, fluid temperature improved. Fig. 4 presented the impression of thermophoresis on the fluid temperature ($\theta \left(\mathrm{{\rm Y}}\right)$). The relationship with thermophoresis and fluid temperature ($\theta \left(\mathrm{{\rm Y}}\right)$) have directly proportion. Impression of radiation $\left({\mathrm{R}}_d\right)$ on fluid temperature ($\theta \left(\mathrm{{\rm Y}}\right)$) exhibited in Fig. 5. Radiation improved fluid temperature ($\theta \left(\mathrm{{\rm Y}}\right)$) because stronger radiation is more heated the surface. Influence of curvature ($\varsigma$) on fluid temperature ($\theta \left(\mathrm{{\rm Y}}\right)$) which revealed in Fig. 6. Temperature ($\theta \left(\mathrm{{\rm Y}}\right)$) revealed the curves declined by boosting values of curvature ($\varsigma$) because surface become flat and temperature spread, so heat reduced. Influence of curvature ($\varsigma$), thermophorsis (${\mathrm{N}}_T$), sutterby fluid ($\varpi$) and Brownian motion (${\mathrm{N}}_B$) on the concentration ($\phi \left(\mathrm{{\rm Y}}\right)$) which exhibited in Fig. 7, Fig. 8, Fig. 9, Fig. 10. Impression of curvature ($\varsigma$) and concentration ($\phi \left(\mathrm{{\rm Y}}\right)$) presented in Fig. (7). It is noted that concentration ($\phi \left(\mathrm{{\rm Y}}\right)$) curves enriched due to boosting values of curvature ($\varsigma$). Fig. 8 reported the influence of thermophorsis (${\mathrm{N}}_T$) on the concentration ($\phi \left(\mathrm{{\rm Y}}\right)$). The concentration ($\phi \left(\mathrm{{\rm Y}}\right)$) curves condensed by larger values of thermophorsis (${\mathrm{N}}_T$). Impression of sutterby fluid ($\varpi$) and fluid concentration ($\phi \left(\mathrm{{\rm Y}}\right)$) presented in Fig. (9). It is noted that fluid concentration ($\phi \left(\mathrm{{\rm Y}}\right)$) curves enriched due to boosting values of sutterby fluid ($\varpi$). Fig. (10) reported the influence of Brownian motion (${\mathrm{N}}_B$) on the fluid concentration ($\phi \left(\mathrm{{\rm Y}}\right)$). The fluid concentration ($\phi \left(\mathrm{{\rm Y}}\right)$) curves improved by larger values of Brownian motion (${\mathrm{N}}_B$). Table 1 stated the influence of Brownian motion ($N_B$), thermophoresis ($N_T$), sutterby fluid ($\varpi$), curvature ($\varsigma$), Eckert number ($Ec$), radiation ($R_d$), Prandtl number ($\mathit{Pr}$), thermal slip ($\delta$) and concentration slip (${\delta }_1$) on absolute value of $C_f{Re}^{1/2}$, ${Nu}_n{Re}^{-1/2}$ and ${Sh}_n{Re}^{-1/2}$. Nusselt number (${Nu}_n{Re}^{-1/2}$) boosted up while sherwood number (${Sh}_n{Re}^{-1/2}$) declined for larger values of Brownian motion ($N_B$) and skin friction remain constant. The relation between thermophoresis ($N_T$) and Nusselt number (${Nu}_n{Re}^{-1/2}$) have inversely while with Sherwood number (${Sh}_n{Re}^{-1/2}$) have directly. The values of sutterby fluid ($\varpi$) boosted which resist to increasing values of absolute value of skin friction ($C_f{Re}^{1/2}$). Because, viscosity enriched due to increment in sutterby fluid increase ultimately, skin friction increased at the surface. Variation of Nusselt number (${Nu}_n{Re}^{-1/2}$) and sutterby fluid ($\varpi$) presented in Table 1. The relation between Nusselt number (${Nu}_n{Re}^{-1/2}$) and sutterby fluid ($\varpi$) have found to be directly which inversely relationship with Sherwood (${Sh}_n{Re}^{-1/2}$) because heat transfer improved and mass transfer reduced. The curvature ($\varsigma$) increased which increased the absolute value of skin friction ($C_f{Re}^{1/2}$) because the values of curvature increase which surface become flat so skin friction increased. The curvature ($\varsigma$) and Nusselt number (${Nu}_n{Re}^{-1/2}$) found to be inversely relation. The surface become flat and heat reduced due to larger values of curvature ($\varsigma$) while same behavior found for Sherwood (${Sh}_n{Re}^{-1/2}$) have increasing. Relationship between Eckert number ($Ec$) and Nusselt number (${Nu}_n{Re}^{-1/2}$) have opposite but same relationship between Eckert number ($Ec$) and Sherwood (${Sh}_n{Re}^{-1/2}$). Relationship between radiation ($R_d$) and Nusselt number (${Nu}_n{Re}^{-1/2}$) have directly but same relationship between radiation ($R_d$) and sherwood (${Sh}_n{Re}^{-1/2}$) have increasing. Radiation improved which boosted up the heat transfer rate and also boosted up mass transfer. The heat transfer enhanced due to greater values of Prandtl number ($\mathit{Pr}$) while mass transfer condensed as well as Prandtl number ($\mathit{Pr}$) increased. Relationship between thermal slip ($\delta$) and Nusselt number (${Nu}_n{Re}^{-1/2}$) have opposing behavior. The relationship between thermal slip ($\delta$) and sherwood (${Sh}_n{Re}^{-1/2}$) have opposing behavior. The concentration slip (${\delta }_1$) increased which declined the Nusselt number (${Nu}_n{Re}^{-1/2}$). The concentration slip (${\delta }_1$) increased which declined the sherwood (${Sh}_n{Re}^{-1/2}$). Table 2 shows our results compared to Okechi et al. [41], with our results being more accurate and approaching literature more closely.

Fig. 2.

Variation of $\text{Ec}$ and $\theta \left(\mathrm{{\rm Y}}\right)$.

Fig. 3.

Variation of ${\mathrm{N}}_B$ and $\theta \left(\mathrm{{\rm Y}}\right)$.

Fig. 4.

Variation of ${\mathrm{N}}_T$ and $\theta \left(\mathrm{{\rm Y}}\right)$.

Fig. 5.

Variation of ${\mathrm{R}}_d$ and $\theta \left(\mathrm{{\rm Y}}\right)$.

Fig. 6.

Variation of $\mathrm{\varsigma }$ and $\theta \left(\mathrm{{\rm Y}}\right)$.

Fig. 7.

Variation of $\mathrm{\varsigma }$ and $\phi \left(\mathrm{{\rm Y}}\right)$.

Fig. 8.

Variation of ${\mathrm{N}}_T$ and $\phi \left(\mathrm{{\rm Y}}\right)$.

Fig. 9.

Variation of $\mathrm{\varpi }$ and $\phi \left(\mathrm{{\rm Y}}\right)$.

Fig. 10.

Variation of ${\mathrm{N}}_B$ and $\phi \left(\mathrm{{\rm Y}}\right)$.

Table 1.

Variation of physical parameters on the $C_f{Re}^{1/2}$, ${Nu}_n{Re}^{-1/2}$ and ${Sh}_n{Re}^{-1/2}$.

$N_B$	$N_T$	$\varpi$	$\varsigma$	$Ec$	$R_d$	$\mathit{Pr}$	$\delta$	${\delta }_1$	$C_f{Re}^{1/2}$	${Nu}_n{Re}^{-1/2}$	${Sh}_n{Re}^{-1/2}$
0.1	0.4	0.5	4.0	0.2	0.5	3.0	0.3	0.2	−5.2993	1.1243	0.4209
0.3									−5.2993	1.1329	0.1597
0.5									−5.2993	1.1479	−0.1182
0.7									−5.2993	1.2141	−0.4622
0.5	0.2								−5.2993	1.4653	−1.2139
	0.4								−5.2993	1.1479	−0.1182
	0.6								−5.2993	1.0424	0.1633
	0.8								−5.2993	0.9809	0.2869
	0.4	0.1							−5.1626	0.5609	0.4818
		0.3							−5.2543	0.6239	0.4188
		0.5							−5.2993	1.1479	−0.1182
		0.7							−5.8488	1.3787	−0.3160
		0.5	3.0						−4.3645	1.5110	−0.0742
			4.0						−5.2993	1.1479	−0.1182
			5.0						−6.3437	1.0784	−0.1358
			6.0						−7.3174	1.0925	−0.1620
			4.0	0.0					−5.2993	1.2890	−0.2309
				0.2					−5.2993	1.1479	−0.1182
				0.4					−5.2993	1.0077	−0.0962
				0.6					−5.2993	0.8685	0.0451
				0.2	0.0				−5.2993	0.6345	−2.6361
					0.5				−5.2993	1.1479	−0.1182
					1.0				−5.2993	1.2367	0.0775
					1.5				−5.2993	1.3166	0.1831
					0.5	2.0			−5.2993	0.8380	0.1096
						3.0			−5.2993	1.1479	−0.1182
						4.0			−5.2993	1.4382	−0.3428
						5.0			−5.2993	4.3908	−2.6361
						3.0	0.1		−5.2993	1.1479	−0.1182
							0.3		−5.2993	1.0677	−0.0911
							0.5		−5.2993	1.0360	−0.0802
							0.7		−5.2993	1.0057	−0.0706
							0.1	0.1	−5.2993	1.1479	−0.1182
								0.2	−5.2993	1.1481	−0.1165
								0.3	−5.2993	1.1480	−0.1142
								0.4	−5.2993	1.1479	−0.1119

Table 2.

Comparing the results of our analysis to the literature, assuming the other physical parameters are zero.

$\varsigma$	Current analysis	Okechi et al. [41]
5.00	1.4206532	1.41960
10.00	1.3454691	1.34670
20.00	1.3109850	1.31350
30.00	1.2997415	1.30280
40.00	1.2942970	1.29750
50.00	1.2908963	1.29440
100.0	1.2844968	1.28810
200.0	1.2812471	1.28500
1000.0	1.2783552	1.28180

5. Final remarks

Steady incompressible two-dimensional satterby nanofluid flow at a curved surface is considered. Thermal and concentration slip phenomena under radiation effects are taken into account. The thermophoresis and Brownian motion process were exhibited on the curved surface. The numerical solution of the specified mathematical model is obtained using the built-in bvp4c tool in MATLAB. The effects of various parameters related to the system of ordinary differential equations are illustrated in the graphs and tabular form. The main results are presented below:

• •Temperature boosted up for improving the Eckert number. The Eckert number increased which improved the K. E. The K. E transformed into to I. E which work against the viscous of fluid stress ultimately, temperature of fluid boosted.
• •The relation between fluid temperature ($\theta \left(\mathrm{{\rm Y}}\right)$) and Brownian motion (${\mathrm{N}}_B$) have directly proportion. Brownian motion enriched which improved the fluid temperature ($\theta \left(\mathrm{{\rm Y}}\right)$) because energy (K. E) increased the motion of the particle of the fluid ultimately, fluid temperature improved.
• •Temperature ($\theta \left(\mathrm{{\rm Y}}\right)$) revealed the curves declined by boosting values of curvature ($\varsigma$) because surface become flat and temperature spread, so heat reduced.
• •Radiation improved fluid temperature ($\theta \left(\mathrm{{\rm Y}}\right)$) because stronger radiation is more heated the surface. It is noted that fluid concentration ($\phi \left(\mathrm{{\rm Y}}\right)$) curves enriched due to boosting values of sutterby fluid ($\varpi$).
• •The heat and mass transfer in case of section achieved lesser than that of injection case.

Consent for publication

Consent for publication ensures that authors give explicit, informed permission to publish their data and maintain ethical standards.

Availability of data and material

Data will be made available on request.

CRediT authorship contribution statement

Nadeem Abbas: Writing – review & editing, Writing – original draft, Methodology, Conceptualization. Wasfi Shatanawi: Validation, Supervision, Investigation. Fady Hasan: Writing – original draft, Methodology, Formal analysis, Data curation. Zead Mustafa: Validation, Methodology, Formal analysis, Data curation.

Declaration of competing interest

All the authors have no declaration of interest in the manuscript.