Darcy resistant of Soret and Dufour impact of radiative induced magnetic field sutterby fluid flow over stretching cylinder

Abstract

The incompressible two-dimensional steady flow of Sutterby fluid over a stretching cylinder is taken into account. The magnetic Reynolds number is not deliberated low in the present analysis. Radiation and variable thermal conductivity are considered to debate the impact on the cylindrical surface. The Dufour and Soret impacts are considered on the cylinder. The mathematical model is settled by employing boundary layer approximations in the form of differential equations. The system of differential equations becomes dimensionless using suitable transformations. The dimensionless nonlinear differential equations are solved through a numerical scheme(bvp4c technique). The flow parameters of physical effects on the velocity, temperature, heat transfer rate, and friction between surface and liquid are presented in tabular as well as graphical form. The velocity function declined by improving the values of the Sponginess parameter. The fluid temperature is reduced by increment in curvature parameter.

Nomenclature

$l$ (m)

Length of stretching sheet

${\alpha }_f$ ($m^2/s$)

Thermal diffusivity

($u,v$)($m/s$)

velocity components

$b^2$

Consistency index

$T_w$ (K)

Wall temperature

$\mathit{Pr}$ (1)

Prandtl number

$U_w$ ($m/s$)

Wall Velocity

$Sc\left(1\right)$

Schmidt number

(r, x)(m)

Coordinates

${\gamma }_1$ (1)

Curvature parameter

$\beta \left(1\right)$

Magnetic field parameter

$Rd\left(1\right)$

Thermal radiation

$Ec\left(1\right)$

Eckert number

$Du$ (1)

Dufour parameter

$D_m$ ($m^2/s$)

Molecular diffusivity of the species concentration

$c_s$

Concentration susceptibility

$T_m\left(K\right)$

Mean fluid temperature

${Re}_x$ (1)

Local Reynold number

ρ($kg/m^3$)

Fluid density

τ($J/kgK$)

Heat capacitance and base fluid ratio

${\beta }_2\left(1\right)$

Darcy resistant parameter

$C_p$ ($J/kgK$)

Specific heat capacitance

$T\left(K\right)$

Prescribed temperature

ν($m^2/s$)

Kinematic viscosity

${\beta }_1\left(1\right)$

Sutterby fluid parameter

$k_f$ ($W/m^{2}K$)

Thermal conductivity of fluid

$T_{\infty }\left(K\right)$

Ambient temperature

$ϵ\left(1\right)$

Variable thermal conductivity

${\lambda }_0\left(1\right)$

Magnetic Prandtl number

$n$

Temperature index

$Sr$ (1)

Soret parameter

$k_t$

Ratio of thermal diffusion

$c_p\left(\mathrm{J}/\text{kg}\mathrm{K}\right)$

Heat capacity

$q_r\left(\mathrm{W}/m^2\right)$

Radiative heat flux

1. Introduction

In the past, several researchers developed ideas using the stretching sheet in two-dimensional flow and heat transfer phenomena, but not much more analysis about the complex problems of the flow over-stretching cylinder. The problem developed having a large radius of the cylinder compared with the thickness of the boundary layer. There are several applications in hot rolling, fiber or wire drawing, and so on. Crane [1] deliberated on the impact of stretching cylinders due to boundary layer flow. Wang [2] debated the stretching cylinder using the fluid flow numerically. Bourgoin et al. [3] premeditated the induced magnetic field flow at the stretching cylinder. Daniel [4] debated the magnetic hydrodynamic steady flow at the porous surface and achieved results analytically. Daniel [5] studied the convective slip of laminar flow at a flat surface and studied the consequences analytically. Daniel [6] considered the magnetic hydrodynamic flow model with slip phenomena at permeable surfaces and solved it analytically. Daniel et al. [7] settled the numerical outcomes of time-dependent magnetic hydrodynamic fluid flow of nanomaterial at the stretchable surface. Meenakumari et al. [8] highlighted the flow impression of chemical reaction with convective Prandtl liquid with induced magnetic field at a stretching surface. Ramamoorthy and Pallavarapu [9] initiated the work on the radiative hall impact for Williamson liquid at the stretching surface. Kotha et al. [10] studied the inspiration of bioconvection laminar flow of nanofluid under the presence of gyrotactic microorganisms. The numerical outcomes are debated in their analysis. Gangadhar and Chamkha [11] revealed the impression of a couple stresses of Boussinesq liquid flow under heat generation at a stretchable surface. Meenakumari and Lakshminarayana [12] emphasized the influence of the magnetic hydrodynamic flow of the Powell eyring liquid model using the stretchable surface. Sherma and Wakif [13] comprehensively studied the viscous fluidic media under the convective instabilities. Abbas et al. [14] initiated the influence of non-viscous fluid at the stretching surface. Manzoor et al. [15] worked on the flow of Oldroyd-B nanomaterial fluid under the bioconvective impression. Recently, several authors discussed the boundary layer flow at the stretchable surface under several flow assumptions see Refs. [[16], [17], [18], [19]].

Magnetic field dynamics is the name given to the interaction of a magnetic field with fluid mechanics. The fluids involved in this interaction are referred to as electrically conducting fluids. The dynamics of magnetic field flow drawn the attention of eager scientists from all over the world due to its wide-ranging applications, including astronomical physics, geophysics, and the movement of the earth's layers. The applications of flow problems with magnetic field characteristics in engineering and commercial industry, such as plasma confinement, solar physics, and many more, are also supported by several researchers. The experimental analyst has also noted that when fluid flows are subjected to an external magnetic field, both an electric and magnetic field are generated. Due to the low magnetic Reynold assumption, the induced magnetic field receives most of the energy while the induced electric field is ignored. This generated magnetic field is essential for fluid flow analysis to be accurately represented. The following authors reported preliminary investigations on electrically conducting fluid flows under diverse geometrical situations. Few are mentioned here for the sake of conciseness. Gailitis et al. [20] investigated the influence of induced magnetic field flow using the Riga surface. Mekheimer [21] studied the induced magnetic field flow of peristaltic couple stress liquid at a stretching surface. The effects of pressure gradient, induced magnetic field flow along the axial direction, and temperature gradient are exposed. Daniel et al. [22] highlighted the entropy analysis for magnetic hydrodynamic nanomaterial fluid flow under the impression of radiative chemical reaction and viscous dissipations numerically. Daniel et al. [23] evaluated the flow of MHD using the nanofluid model under mixed convection. Daniel [24] scrutinized the flow of MHD using the nanofluid model under a partial slip impression at a porous surface. Daniel et al. [25] deliberated the flow of MHD using the velocity slip with the nanofluid model under a convective role. Daniel et al. [26] deliberated the time-dependent magnetic hydrodynamic fluid flow of nanomaterial at a stretchable sheet. Daniel et al. [27] considered the thermal stratification flow of magnetic hydrodynamics using the nanofluid model at a stretchable sheet. Mulinti and Pallavarapu [28] considered the flow of UCM fluid under magnetic hydrodynamic and thermal radiation at a porous surface with a chemical reaction. Wakif et al. [29] deliberated an exponentially stretchable surface to analyze the inspiration of MHD and nanofluid. Reddy et al. [30] debated on the Cattaneo-Christov model under Maxwell nanofluid with radiative MHD impression at stretching sheet. Yahaya et al. [31] emphasized the inspiration of magnetic hydrodynamic stagnation region at the stretchable sheet. Scholars have recently deliberated on the influence of the magnetic hydrodynamic model under some assumptions (see Refs. [[32], [33], [34], [35], [36]]).

Applications of the radiative effects in physics and engineering procedures are significant. The temperature processes and space technology heavily rely on the effects of radiation from heat transfer on various flows. However, little is known about how radiation affects the boundary layer. Thermal radiation effects may be a significant problem in the polymer processing sector, as the quality of the completed product depends partly on variables governing heat transport. Daniel and Daniel [37] deliberated the impression of radiative and buoyancy for magnetic hydrodynamic viscous fluid in a porous sheet. They took the results of developing flow problem analytically. Daniel et al. [38] developed the numerical outcomes of radiative and slip influence of nanomaterial liquid flow at porous stretchable surface. Daniel et al. [39] considered the variable thickness of nanomaterial liquid flow at a stretchable sheet. Daniel et al. [40] deliberated the thermal stratification of nanomaterial liquid flow at nonlinear stretchable surface. Daniel et al. [41] reflected the thermal radiation time-dependent flow of nanomaterial fluid at a nonlinear stretchable surface. Gangadhar et al. [42] calculated the couple stress fluid of magnetized flow at a stretchable surface. Reddy and Lakshminarayana [43] worked on the Cattaneo-Christov under Williamson nanofluid with radiative MHD inspiration at stretching sheet. Gangadhar et al. [44] planned the radiation on the flow of Walter's B nanofluid at a rotating surface. Bhargavi et al. [45] debated the radiation for Maxwell hybrid nanofluid. Gangadhar et al. [46] highlighted the impact of the thermal feature of radiative hybrid nanoliquid material flow at rotating cylinder. Recently, researchers have been studied the influence of radiation under several assumptions (see Refs. [[47], [48], [49], [50]]).

From the above literature, the gap of the research available has been filled as sutterby fluid flow over a stretching cylinder in the presence of Dufour and Soret impacts. The induced magnetic field has been considered under the assumption of a high Reynold number. The variable thermal conductivity and radiation impacts have been discussed. Developing a mathematical model under flow assumptions has been considered and solved through the numerical technique bvp4c using MATLAB software. The results have been presented in graphs and tabular form. Such results are unique and have not still discussed before.

2. Mathematical formulation

The Darcy resistance flow of Sutterby fluid is considered in this analysis over a stretching cylinder. The flow over the stretching cylinder is presented in Fig. 1. The temperature-dependent flow properties are analyzed at the surface of the cylinder. The radiations are applied using the viscous dissipations of the flow model. The Dufour and Soret impacts are considered on the cylindrical surface. The induced magnetic field is applied along the $r$ direction, which is normal to $x-$ direction. The magnetic Reynolds number is considered large of the induced magnetic field. The stretching velocity of the cylinder is $U_w$, and the temperature of the surface is $T_w$. The ambient fluid temperature and free stream of induced magnetic hydrodynamics are $T_{\infty }$ and $H_e$. The assumptions of the problem are as follows:

• •Induced magnetic field
• •Dufour and Soret effects
• •Stretching cylinder
• •Sutterby fluid flow
• •Radiation
• •Viscous dissipation and variable thermal conductivity

Fig. 1.

Flow analysis of induced Sutterby fluid over a stretching cylinder.

The governing model is proposed and defined as (see Refs. 28–30,60, 61):

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

$$\frac{\partial H_2}{\partial r}+\frac{H_2}{r}+\frac{\partial H_1}{\partial x}=0\text{,}$$ (2)

$$u\frac{\partial u}{\partial x}+v\frac{\partial v}{\partial r}-\frac{{\mu }_e}{4\pi {\rho }_f}\left(H_1\frac{\partial H_1}{\partial x}+H_2\frac{\partial H_1}{\partial r}\right)=\left(\frac{{\upsilon }_f}{2}\right)\frac{{\partial }^{2}u}{\partial r^2}+\left(\frac{{\upsilon }_f}{2}\right)\frac{1}{r}\frac{\partial u}{\partial r}-\left(\frac{{\upsilon }_{f}m{a}^2}{4}\right){\left(\frac{\partial u}{\partial r}\right)}^2\frac{{\partial }^{2}u}{\partial r^2}+\frac{R_z}{{\rho }_f}\text{,}$$ (3)

$$u\frac{\partial H_1}{\partial x}+v\frac{\partial H_1}{\partial r}-H_1\frac{\partial u}{\partial x}-H_2\frac{\partial u}{\partial r}={\eta }_0\left(\frac{{\partial }^2{H}_1}{\partial r^2}+\frac{1}{r}\frac{\partial H_1}{\partial r}\right)\text{,}$$ (4)

$$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial r}=\frac{1}{\rho c_p}\frac{1}{r}\frac{\partial }{\partial r}\left(rK\left(T\right)\frac{\partial T}{\partial r}\right)-\frac{1}{r}\frac{\partial \left({rq}_r\right)}{\partial r}+\frac{1}{\rho c_p}\left(1-\left(\frac{{\upsilon }_{0}m{a}^2}{4}\right){\left(\frac{\partial u}{\partial r}\right)}^2\right){\left(\frac{\partial u}{\partial r}\right)}^2+\frac{D_m{k}_t}{c_s{c}_p}\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial C}{\partial r}\right)\text{,}$$ (5)

$$u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial r}=D_m\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial C}{\partial r}\right)+\frac{D_m{k}_t}{T_m}\frac{1}{r}\frac{\partial }{\partial r}\left(\frac{\partial T}{\partial r}r\right)\text{.}$$ (6)

with BC(boundary conditions):

$$u=U_w=\frac{U_{0}x}{l},v=0\text{,}{H}_1=0\text{,}{H}_2=0\text{,}T=T_w,C=C_w,atr\to R,u\to 0\text{,}{H}_1\to H_e,T\to T_{\infty },C\to C_{\infty },asr\to \infty ..$$ (7)

The suitable transformations are

$$u=U_w{F}^{\prime }\left(\zeta \right),v=-\frac{R}{r}\sqrt{\frac{{U_0\upsilon }_f}{l}}F\left(\zeta \right),H_1=\frac{H_{0}x}{l}{G}^{\prime }\left(\zeta \right),H_2=-\frac{R{H}_0}{r}\sqrt{\frac{{\upsilon }_f}{l{U}_0}}G\left(\zeta \right),\zeta =\frac{r^2-R^2}{2R}\sqrt{\frac{U_0}{l{\upsilon }_f}},T=T_{\infty }+\left(T_w-T_{\infty }\right)\Theta \left(\zeta \right),C=C_{\infty }+\left(C_w-C_{\infty }\right)\phi \left(\zeta \right)..$$ (8)

Using the appropriate transfo rmations and partial differential equations above, we obtain the following ordinary differential equations:

$$\left(1+2\zeta {\gamma }_1\right)\frac{{\partial }^{3}F}{{\partial \zeta }^3}+2{\gamma }_1\frac{{\partial }^{2}F}{{\partial \zeta }^2}+2\frac{{\partial }^{2}F}{{\partial \zeta }^2}\frac{\partial F}{\partial \zeta }-{\left(\frac{\partial F}{\partial \zeta }\right)}^2-\frac{{\beta }_1}{2}\left(1+2\zeta {\gamma }_1\right){\left(\frac{{\partial }^{2}F}{{\partial \zeta }^2}\right)}^2\left(\left(1+2\zeta {\gamma }_1\right)\frac{{\partial }^{3}F}{{\partial \zeta }^3}+{\gamma }_1\frac{{\partial }^{2}F}{{\partial \zeta }^2}\right)+\frac{{\lambda }_1}{2}\frac{\partial F}{\partial \zeta }+\frac{1}{12}{\beta }_2{\left(\frac{{\partial }^{2}F}{{\partial \zeta }^2}\right)}^2\frac{\partial F}{\partial \zeta }+\beta \left(\frac{\partial G}{\partial \zeta }\frac{\partial G}{\partial \zeta }-G\left(\zeta \right)\frac{{\partial }^{2}G}{{\partial \zeta }^2}\right)=0\text{,}$$ (9)

$${\lambda }_0\left(1+2\zeta {\gamma }_1\right)\frac{{\partial }^{3}G}{{\partial \zeta }^3}+2{\lambda }_0{\gamma }_1\frac{{\partial }^{2}G}{{\partial \zeta }^2}+G\left(\zeta \right)\left(\frac{{\partial }^{2}F}{{\partial \zeta }^2}\right)-\frac{{\partial }^{2}G}{{\partial \zeta }^2}F\left(\zeta \right)=0\text{,}$$ (10)

$$\frac{1}{\mathit{Pr}}\left(1+ϵ\Theta \left(\zeta \right)+\frac{4}{3}{R}_d\right)\left(1+2\zeta {\gamma }_1\right)\frac{{\partial }^2\Theta }{{\partial \zeta }^2}+\left(PrF\left(\zeta \right)+ϵ\left(1+2\zeta {\gamma }_1\right)\Theta \left(\zeta \right)+\left(1+2\zeta {\gamma }_1\right)\frac{4}{3}{R}_d{\gamma }_1+2{\gamma }_1\right)\frac{\partial \Theta }{\partial \zeta }-nPr\Theta \left(\zeta \right)\frac{\partial F}{\partial \zeta }+PrEc\left(1-\frac{{\beta }_1}{6}{F}^{″}\left(\zeta \right)F^{″}\left(\zeta \right)\right)F^{″}\left(\zeta \right)F^{″}\left(\zeta \right)+Du\left(\left(1+2\zeta {\gamma }_1\right)\frac{{\partial }^2\phi }{{\partial \zeta }^2}+2{\gamma }_1\frac{\partial \phi }{\partial \zeta }\right)=0\text{,}$$ (11)

$$\left(1+2\zeta {\gamma }_1\right)\frac{{\partial }^2\phi }{{\partial \zeta }^2}+2{\gamma }_1\frac{\partial \phi }{\partial \zeta }+Sc\left(F\left(\zeta \right)\frac{\partial \phi }{\partial \zeta }-n\phi \left(\zeta \right)\frac{\partial F}{\partial \zeta }\right)+ScSr\left(\left(1+2\zeta {\gamma }_1\right)\frac{{\partial }^2\Theta }{{\partial \zeta }^2}+2{\gamma }_1\frac{\partial \Theta }{\partial \zeta }\right)=0\text{.}$$ (12)

with relevant BC(boundary conditions):

$$F\left(0\right)=0\text{,}\left(0\right)=1\text{,}{F}^{\prime }\left(\infty \right)=0\text{,}{F}^{\prime }\left(0\right)=1\text{,}\phi \left(0\right)=1\text{,}G\left(0\right)=0\text{,}\Theta \left(\infty \right)=0\text{,}{G}^{\prime }\left(\infty \right)=1\text{,}{G}^{\prime }\left(0\right)=0\text{,}\phi \left(\infty \right)=0\text{.}$$ (13)

Where, ${\gamma }_1$ (Curvature parameter), ${\lambda }_1$ (Sponginess parameter), ${\beta }_1$ (Sutterby fluid parameter), ${\beta }_2$ (Darcy resistant parameter), $R_d$ (Thermal radiation), $Ec$ (Eckert number), ${\lambda }_0$ (Magnetic Prandtl number), $\beta$ (Magnetic field parameter), $Du$ (Dufour), $Sr$ (Soret), $Sc$ (Schmidt) and $\mathit{Pr}$ (Prandtl number). Where,

$${\gamma }_1\left(\frac{1}{R}\sqrt{\frac{{\nu }_{f}l}{U_0}}\right),{\beta }_1\left(\frac{m{a}^2{U_0}^3{x}^2}{{\nu }_f{l}^3}\right),{\beta }_2\left(\frac{m^{*}{a}^2{U_0}^2{x}^2}{k^{*}{l}^2}\right),{\lambda }_1\left(\frac{{\nu }_{f}l}{k^{*}{U}_0}\right)\text{,}$$

$$Sc\left(\frac{{\nu }_f}{D_B}\right),Ec\left(\frac{{U_0}^2}{c_p\left(T_w-T_{\infty }\right)}\right),R_d\left(\frac{4{\sigma }^{*}{T_{\infty }}^3}{k{k}^{*}}\right),\mathit{Pr}\left(\frac{{\nu }_f}{\alpha }\right)\text{,}$$

$$Du\left(\frac{D_m{k}_T\left(C_w-C_{\infty }\right)}{c_p{c}_s\left(T_w-T_{\infty }\right)}\right),Du\left(\frac{D_m{k}_T\left(C_w-C_{\infty }\right)}{c_p{c}_s\left(T_w-T_{\infty }\right)}\right),{\lambda }_0\left(\frac{{\eta }_0}{{\nu }_f}\right)\text{and}\beta \left(\frac{{\mu }_e}{4\pi {\rho }_f}{\left(\frac{H_{0}l}{U_0}\right)}^2\right)\text{.}$$

The coefficients of physical quantities are presented as $C_f$ (Skin friction) and $N_u$ (Heat transfer). The performance of these quantities are well-defined as

$$C_f=\frac{2{\left[{\tau }_w\right]}_{r=R}}{\rho {\left(U_w\right)}^2},N_u=\frac{x{\left(1+\frac{4}{3}{R}_d\right)\left[\frac{\partial T}{\partial r}\right]}_{r=R}}{k\left(T-T_w\right)},{\tau }_w=\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial r}\right)-\frac{\mu \omega b^2}{12}\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial r}\right)\left(2{\left({\left(\frac{\partial u}{\partial x}\right)}^2+{\left(\frac{\partial v}{\partial r}\right)}^2+{\left(\frac{u}{r}\right)}^2\right)+\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial r}\right)}^2\right)\text{.}$$ (14)

The dimensionless form is presented as

$${C_f}^S={\left(\frac{{\partial }^{2}F}{{\partial \zeta }^2}-\frac{{\beta }_1}{6}\left[3{\left(\frac{\partial F}{\partial \zeta }\right)}^2+3\left(1+2\zeta {\gamma }_1\right)\left(F^2\right)-2\left(1+2\zeta {\gamma }_1\right)F\frac{\partial F}{\partial \zeta }+\left(1+2\zeta {\gamma }_1\right){\left(\frac{{\partial }^{2}F}{{\partial \zeta }^2}\right)}^2\right]\right)}_{\zeta \to 0},{N_u}^n=-{\left(1+\frac{4}{3}{R}_d\right)\frac{\partial \Theta }{\partial \zeta }}_{\zeta \to 0}\text{.}$$ (15)

3. Numerical method procedure

The integral component of solving a BVP is making an initial guess at the solution, and the accuracy of that guess may even determine whether the computation is successful. The boundary conditions define the relationship between the solution values at two points along the interval of integration. The system of differential equations is solved numerical bvp4c technique using the Matlab software package. The numerical procedure of the differential equations is transformed into the first-order differential equations. The selection of finite value is $\zeta \to \infty$ as $\zeta =10\text{,}$ which shows that the present results are corrected asymptotically for numerical technique. The ${10}^{-6}$ is tolerance error of convergence criteria and process is adopted as follows:

$$F\left(\zeta \right)=S\left(1\right);\frac{{\partial }^{2}F}{{\partial \zeta }^2}=S\left(3\right);\frac{\partial F}{\partial \zeta }=S\left(2\right);\frac{{\partial }^{3}F}{{\partial \zeta }^3}=SS1\text{;}G\left(\zeta \right)=S\left(4\right);\frac{{\partial }^{2}G}{{\partial \zeta }^2}=S\left(6\right);\frac{\partial G}{\partial \zeta }=S\left(5\right);\frac{{\partial }^{3}G}{{\partial \zeta }^3}=SS2;;\Theta \left(\zeta \right)=S\left(7\right);\frac{\partial \Theta }{\partial \zeta }=S\left(8\right);\frac{{\partial }^2\Theta }{{\partial \zeta }^2}=SS3;;\phi \left(\zeta \right)=S\left(9\right);\frac{\partial \phi }{\partial \zeta }=S\left(10\right);\frac{{\partial }^2\phi }{{\partial \zeta }^2}=SS4\text{;}$$ (16)

$$SS1=-{\left(\left(1+2\zeta {\gamma }_1\right)-\frac{{\beta }_1}{2}{\left(\left(1+2\zeta {\gamma }_1\right)\right)}^2{\left(S\left(3\right)\right)}^2\right)}^{-1}\left(2{\gamma }_{1}S\left(3\right)-\frac{{\lambda }_1}{2}S\left(2\right)-\frac{{\beta }_1}{2}\left(1+2\zeta {\gamma }_1\right){\left(S\left(3\right)\right)}^2\left({\gamma }_{1}S\left(3\right)\right)+\frac{1}{12}{\beta }_2{\left(S\left(3\right)\right)}^{2}S\left(2\right)+\beta \left(S\left(5\right)S\left(5\right)-S\left(1\right)S\left(6\right)\right)+2S\left(3\right)S\left(2\right)-{\left(S\left(2\right)\right)}^2\right)\text{;}$$ (17)

$$SS2=-{\left({\lambda }_0\left(1+2\zeta {\gamma }_1\right)\right)}^{-1}\left(2{\lambda }_0{\gamma }_{1}S\left(6\right)+S\left(4\right)S\left(3\right)-S\left(1\right)S\left(6\right)\right)\text{;}$$ (18)

$$SS3=\frac{\mathit{Pr}}{\left(1+ϵS\left(7\right)+\frac{4}{3}{R}_d\right)\left(1+2\zeta {\gamma }_1\right))}\left(\left(\mathit{Pr}S\left(1\right)+ϵ\left(1+2\zeta {\gamma }_1\right)S\left(7\right)+\left(1+2\zeta {\gamma }_1\right)\frac{4}{3}{R}_d{\gamma }_1+2{\gamma }_1\right)S\left(8\right)-nPrS\left(7\right)S\left(2\right)+PrEc\left(1-\frac{{\beta }_1}{6}S\left(3\right)S\left(3\right)\right)S\left(3\right)S\left(3\right)+Du\left(\left(1+2\zeta {\gamma }_1\right)SS4+2{\gamma }_{1}S\left(10\right)\right)\right)\text{;}$$ (19)

$$SS4=-{\left(\left(1+2\zeta {\gamma }_1\right)\right)}^{-1}\left(2{\gamma }_{1}S\left(10\right)+Sc\left(S\left(1\right)S\left(10\right)-nS\left(9\right)S\left(2\right)\right)+ScSr\left(\left(1+2\zeta {\gamma }_1\right)SS3+2{\gamma }_1\left(8\right)\right)\right)\text{;}$$ (20)

Relevant boundary conditions are

$$S0\left(1\right);S_{\infty }\left(2\right);S0\left(2\right)-1\text{;}S0\left(4\right);S_{\infty }\left(5\right)-1\text{;}S0\left(5\right);S0\left(7\right)-1\text{;}S0\left(9\right)-1\text{;}{S}_{\infty }\left(9\right);S_{\infty }\left(7\right)\text{;}$$ (21)

The interval of integration is divided into smaller intervals by the collocation technique using a mesh points. The boundary conditions and collocation conditions imposed on each subinterval cause a global system of algebraic equations, which the solver solves to produce a numerical solution. The solver then calculates the error of the numerical solution for each subinterval. If the solution doesn't meet the tolerance standards, the solver adapts the mesh and repeats the procedure. A first approximation of the solution at the mesh points must be given along with the initial mesh points. The boundaries of residual are publicized:

$${\tilde{Y}}_1=\left|S_2\left(\infty \right)-\widehat{S_2}\left(\infty \right)\right|,{\tilde{Y}}_2=\left|S_5\left(\infty \right)-\widehat{S_5}\left(\infty \right)\right|,{\tilde{Y}}_3=\left|S_7\left(\infty \right)-\widehat{S_7}\left(\infty \right)\right|,{\tilde{Y}}_3=\left|S_9\left(\infty \right)-\widehat{S_9}\left(\infty \right)\right|\text{.}$$

Here, $\hat{S}\left(\infty \right)$, $\widehat{S_5}\left(\infty \right)$, $\widehat{S_7}\left(\infty \right)$ and $\widehat{S_7}\left(\infty \right)$ are calculated boundary values.

4. Results and discussion

The differential equations are interpreted by numerical patterns with the help of MATLAB software packages. The impression of involving physical parameters has been presented through graphs and tabular form. Fig. 2, Fig. 3, Fig. 4, Fig. 5 reported the impression Sponginess parameter (${\lambda }_1$), Sutterby fluid parameter (${\beta }_1$), Darcy resistant parameter (${\beta }_2$) and Curvature parameter on the velocity ($F^{\prime }\left(\zeta \right)$). Fig. 2 publicized the variation of the Sponginess parameter and velocity. Velocity declined by improving the values of the Sponginess parameter. Physically, the permeability of the porous material generates a rise in resistance, which lowers the fluid velocity. The fluid velocity and Sutterby fluid parameter variation are offered in Fig. 3 is publicized that fluid velocity deteriorated against the increment in the Sutterby fluid parameter. Because of the increment in the Sutterby fluid parameter, which increment in fluid viscosity, ultimately, fluid velocity declined. The impact of the Darcy resistant parameter indicated the velocity results offered in Fig. 4. The fluid velocity achieved greater when values of the Darcy resistant parameter improved. Fig. 5 is shown the velocity function results for different values of Curvature. The velocity function accelerated for higher values of Curvature. The curvature of the cylindrical surface enhanced, which enhanced fluid velocity. Fig. 6 is revealed the Curvature parameter's impression on the magnetic profile. The curves of the magnetic profile revealed higher than higher for boosting values of the Curvature parameter. The influence of magnetic Prandtl number on the magnetic profile is revealed in Fig. 7. The curves of the magnetic profile publicized higher than higher due to enlarging values of. Fig. 8, Fig. 9, Fig. 10, Fig. 11, Fig. 12 publicized the effect of Curvature, Prandtl number, Dufour, Eckert number, and radiation on temperature. The impression of temperature and radiation offered in Fig. 8. Temperature revealed larger than larger due to greater values of radiation. The impression of curvature parameter and fluid temperature is publicized in Fig. 9. As the curvature of the cylinder increases, the surface becomes flat, so heat transfer on the surface of the cylinder decreases, and the liquid temperature decreases. The variation of Dufour impact and temperature is publicized in Fig. 10. The temperature enlarged due to augmentation of Dufour parameter because energy flux generation boosted up, which improved the temperature. Fig. 11 shows the variation of fluid temperature and Prandtl factor. Since the Prandtl factor organizes the thickness of the thermosensitive layer, increasing the value of the Prandtl factor decreases the liquid temperature. The impression of Eckert factor and temperature is presented in Fig. 12. The Eckert factor values improved by improving the temperature at the cylindrical surface. Due to the increment in Eckert factor, which boosted up kinetic energy as well as heat transfer enhancement. So, the fluid temperature improved at the surface of the cylinder. Fig. 13, Fig. 14, Fig. 15, Fig. 16 debated the influence of Soret, Schmidt, and Dufour on the concentration. Fig. 13 presented the impression of Soret on concentration. The concentration curves revealed declining due to upper values of Soret parameter. Impression of Schmidt factor on the concentration function is stated in Fig. 14. Concentration curves declined due to larger values of Schmidt parameter. Impression of the Dufour parameter on the concentration function is reported in Fig. 15. Concentration curves declined due to larger values of Dufour parameter. Fig. 16 reported the impact of $ϵ$ on the temperature profile. The temperature boosted due to enlarging values of $ϵ$. The thermal conductivity values extended ultimately heat transfer phenomena enlarging as well as temperature enlarged.

Fig. 2.

Impression of ${\lambda }_1$ on $F^{\prime }\left(\zeta \right)$.

Fig. 3.

Impression of ${\beta }_1$ on $F^{\prime }\left(\zeta \right)$.

Fig. 4.

Impression of ${\beta }_2$ on $F^{\prime }\left(\zeta \right)$.

Fig. 5.

Impression of ${\gamma }_1$ on $F^{\prime }\left(\zeta \right)$.

Fig. 6.

Impression of ${\gamma }_1$ on $G^{\prime }\left(\zeta \right)$.

Fig. 7.

Impression of ${\lambda }_0$ on $G^{\prime }\left(\zeta \right)$.

Fig. 8.

Impression of $R_d$ on $\Theta \left(\zeta \right)$.

Fig. 9.

Impression of ${\gamma }_1$ on $\Theta \zeta )$.

Fig. 10.

Impression of $Du$ on $\Theta \left(\zeta \right)$.

Fig. 11.

Impression of $\mathit{Pr}$ on $\Theta \left(\zeta \right)$.

Fig. 12.

Impression of $Ec$ on $\Theta \left(\zeta \right)$.

Fig. 13.

Impression of $Sr$ on $\phi \left(\zeta \right)$.

Fig. 14.

Impression of $Sc$ on $\phi \left(\zeta \right)$.

Fig. 15.

Impression of $Du$ on $\phi \left(\zeta \right)$.

Fig. 16.

Impression of $ϵ$ on $\Theta \left(\zeta \right)$.

Table 1 revealed the impression of ${\lambda }_1$, ${\beta }_2$, $R_d$ (Thermal radiation), $ϵ$, $Ec$ (Eckert number), ${\lambda }_0$ (Magnetic Prandtl number), $\beta$ (Magnetic field parameter), $\mathit{Pr}$ (Prandtl number), ${\gamma }_1$ (Curvature parameter), $Du$ (Dufour parameter), $Sc$ (Schmidt), ${\beta }_1$ (Sutterby fluid parameter) and $Sr$ (Soret parameter) on the ${C_f}^S$ and ${N_u}^n$. The heat transfer and friction force revealed lesser than lesser by increment of curvature parameter. The curvature of cylindrical surface expanded and heat transfer phenomena reduced due to large surface area covered. But the skin friction is reduced because increment in curvature of cylindrical surface, the surface becomes flat and friction force is reduced. By raising the values of the Sutterby fluid factor, the heat transfer and friction force improved. The viscosity of liquid enhanced due to larger values of Sutterby fluid parameter ultimately declining the heat and friction near the surface. Heat transfer rate and friction force publicized improve due to different values of Sponginess parameter. Physically, the permeability of the porous material generates an increase in resistance, which boosts the values of friction between surface and fluid. The Darcy resistant parameter improved which declined the friction and movement of heat transfer. The magnetic field boosted up which declined the friction and movement of heat between surface and liquid. The induced magnetic field applied on the fluid which reduced heat transfer rate and friction forces. The magnetic Prandtl number declined due to increasing values of movement of heat and friction. The heat transfer rate exposed lesser due to boost up value of radiation but friction between surface and liquid. Nusselt number revealed boost up for improving the values of Prandtl number while friction force remains fixed due to changes values of Prandtl number. Nusselt number exposed boost up for improving the values of Eckert number while friction force remains fixed due to changes values of Eckert factor. Dufour parameter improved gradually which declined heat transfer at cylindrical surface. Schmidt parameter improved gradually which declined heat transfer at cylindrical surface. Soret parameter improved gradually which enlarged heat transfer at the cylindrical surface. The nusselt number boosted while skin friction remained constant due to enlarging values of variable thermal conductivity. Table 2 shows the results of Rangi and Ahmad [51] compared with the actual results for various $ϵ$ values for ${\mathrm{\Theta }}^{\prime }\left(0\right)$ when the rest of the physical parameters are taken as zero. Note that our results are in good agreement with Rangi and Ahmad [51] when with ${\beta }_1={\lambda }_1={\beta }_2=\beta ={\lambda }_0=R_d=Ec=Du=Sr=Sc=0$ and $\mathit{Pr}=1.0$.

Table 1.

Nusselt number and Skin friction for various physical parameters.

${\gamma }_1$	${\beta }_1$	${\lambda }_1$	${\beta }_2$	$\beta$	${\lambda }_0$	$R_d$	$\mathit{Pr}$	$Ec$	$Du$	$Sc$	$Sr$	$ϵ$	${N_u}^n$	${C_f}^S$
0.1	0.2	0.3	0.4	0.2	1.5	0.1	2.5	0.1	0.3	0.3	0.1	0.3	1.874262	−1.861863
0.2	–	–	–	–	–	–	–	–	–	–	–	–	1.776790	−1.200218
0.3	–	–	–	–	–	–	–	–	–	–	–	–	1.667783	−1.191969
0.4	–	–	–	–	–	–	–	–	–	–	–	–	1.574710	−1.167122
0.2	0.0	–	–	–	–	–	–	–	–	–	–	–	1.778245	−1.070026
–	0.2	–	–	–	–	–	–	–	–	–	–	–	1.77679	−1.200218
–	0.4	–	–	–	–	–	–	–	–	–	–	–	1.775119	−1.338146
–	0.6	–	–	–	–	–	–	–	–	–	–	–	1.77313	−1.489551
–	0.2	0.1	–	–	–	–	–	–	–	–	–	–	1.8949099	−1.140464
–	–	0.3	–	–	–	–	–	–	–	–	–	–	1.7767900	−1.200218
–	–	0.5	–	–	–	–	–	–	–	–	–	–	1.7497780	−1.255118
–	–	0.7	–	–	–	–	–	–	–	–	–	–	1.7227960	−1.309786
–	–	0.3	0.2	–	–	–	–	–	–	–	–	–	1.776342	−1.20331
–	–	–	0.4	–	–	–	–	–	–	–	–	–	1.77679	−1.200218
–	–	–	0.6	–	–	–	–	–	–	–	–	–	1.777237	−1.197144
–	–	–	0.8	–	–	–	–	–	–	–	–	–	1.777683	−1.194088
–	–	–	0.4	0.2	–	–	–	–	–	–	–	–	1.77679	−1.200218
–	–	–	–	0.4	–	–	–	–	–	–	–	–	1.83809	−1.130643
–	–	–	–	0.6	–	–	–	–	–	–	–	–	1.977124	−0.9223671
–	–	–	–	0.8	–	–	–	–	–	–	–	–	2.17182	−0.7605157
–	–	–	–	0.2	1.5	–	–	–	–	–	–	–	1.77679	−1.200218
–	–	–	–	–	2.0	–	–	–	–	–	–	–	1.750116	−1.228851
–	–	–	–	–	2.5	–	–	–	–	–	–	–	1.742192	−1.237634
–	–	–	–	–	3.0	–	–	–	–	–	–	–	1.737946	−1.240094
–	–	–	–	–	1.5	0.1	–	–	–	–	–	–	1.77679	−1.200218
–	–	–	–	–	–	0.3	–	–	–	–	–	–	1.905318	−1.200218
–	–	–	–	–	–	0.5	–	–	–	–	–	–	2.024203	−1.200218
–	–	–	–	–	–	0.7	–	–	–	–	–	–	2.135149	−1.200218
–	–	–	–	–	–	0.1	1.0	–	–	–	–	–	1.920884	−1.200218
–	–	–	–	–	–	–	1.5	–	–	–	–	–	1.856346	−1.200218
–	–	–	–	–	–	–	2.0	–	–	–	–	–	1.727922	−1.200218
–	–	–	–	–	–	–	2.5	–	–	–	–	–	1.77679	−1.200218
–	–	–	–	–	–	–	2.5	0.0	–	–	–	–	1.833323	−1.200218
–	–	–	–	–	–	–	–	0.1	–	–	–	–	1.77679	−1.200218
–	–	–	–	–	–	–	–	0.2	–	–	–	–	1.720333	−1.200218
–	–	–	–	–	–	–	–	0.3	–	–	–	–	1.663952	−1.200218
–	–	–	–	–	–	–	–	0.1	0.1	–	–	–	2.131304	−1.200218
–	–	–	–	–	–	–	–	–	0.3	–	–	–	1.77679	−1.200218
–	–	–	–	–	–	–	–	–	0.5	–	–	–	1.446937	−1.200218
–	–	–	–	–	–	–	–	–	0.7	–	–	–	1.148901	−1.200218
–	–	–	–	–	–	–	–	–	0.3	0.1	–	–	1.879545	−1.200218
–	–	–	–	–	–	–	–	–	–	0.3	–	–	1.77679	−1.200218
–	–	–	–	–	–	–	–	–	–	0.5	–	–	1.688214	−1.200218
–	–	–	–	–	–	–	–	–	–	0.7	–	–	1.608465	−1.200218
–	–	–	–	–	–	–	–	–	–	0.3	0.1	–	1.77679	−1.200218
–	–	–	–	–	–	–	–	–	–	–	0.2	–	1.784157	−1.200218
–	–	–	–	–	–	–	–	–	–	–	0.3	–	1.791611	−1.200218
–	–	–	–	–	–	–	–	–	–	–	0.4	–	1.799154	−1.200218
–	–	–	–	–	–	–	–	–	–	–	0.1	0.0	1.51928	−1.200218
–	–	–	–	–	–	–	–	–	–	–	–	0.3	1.77679	−1.200218
–	–	–	–	–	–	–	–	–	–	–	–	0.6	2.012539	−1.200218
–	–	–	–	–	–	–	–	–	–	–	–	0.9	2.231421	−1.200218

Table 2.

Comparative outcomes of current analysis with Rangi and Ahmad [51] for various values of ${\gamma }_1$ with ${\beta }_1={\lambda }_1={\beta }_2=\beta ={\lambda }_0=R_d=Ec=Du=Sr=Sc=0$ and $\mathit{Pr}=1.0$.

	Present results		Rangi and Ahmad [51]
${\gamma }_1$	$ϵ=0.0$	$ϵ=0.2$	$ϵ=0.0$	$ϵ=0.2$
0.0	−0.989453	−0.8708143	−0.985286	−0.862122
0.25	−1.087542	−0.9512472	−1.079447	−0.949659
0.50	−1.183420	−1.0437821	−1.173899	−1.037605
0.75	−1.272102	−1.1312764	−1.267214	−1.124397
1.0	−1.362015	−1.2110952	−1.359308	−1.209949

5. Conclusion

Induced magnetic field sutterby fluid flow over a stretching cylinder is deliberated under Darcy resistance. Radiation and thermal slip are considered to debate the impact on the cylindrical surface. The Dufour and Soret impacts are considered on the cylinder. The main achievements are highlighted as follows:

• •The skin friction and Nusselt number revealed less values by an increment of curvature parameter. The curvature of the cylindrical surface expanded, which reduced heat transfer phenomena due to the large surface area covered. However, skin friction is reduced because of the increment in the curvature of the cylindrical surface; the surface becomes flat, and the friction force is reduced.
• •The heat transfer and friction force improved by enlarging the values of the Sutterby fluid factor. The viscosity of liquid enhanced due to larger values of the Sutterby fluid factor ultimately upgrading the heat and friction near the surface.
• •Heat transfer rate and friction force improved due to different values of Sponginess parameter. Physically, the permeability of the porous material generates an increase in resistance, which boosts the values of friction between fluid and surface.
• •The Darcy resistant parameter improved, reducing the friction and movement of heat transfer. The magnetic field boosted up, which declined the friction and movement of heat between the surface and liquid.

Data availability

No data was used for the research described in the article.

Funding

No funding

CRediT authorship contribution statement

Nadeem Abbas: Writing – review & editing, Writing – original draft, Methodology, Investigation, Conceptualization. Zead Mustafa: Writing – review & editing, Methodology, Data curation. Kamaleldin Abodayeh: Visualization, Validation. Taqi A.M. Shatnawi: Investigation, Formal analysis. Wasfi Shatanawi: Validation, Supervision, Software, Formal analysis.

Declaration of competing interest

The authors declare no conflict of interest.