A Note on Heat Transport with Aspect of Magnetic Dipole and Higher Order Chemical Process for Steady Micropolar Fluid

Abstract

Heat transfer through non-uniform heat source/sink is the most significant aspect in view of many physical problems. Heat sink/source with heat transfer help to change the energy distribution in fluids, which consequently disturbs the particle deposition rate like as nuclear reactors, semiconductors and electronic devices. Further, also, the vital role of heat transfer is to enhance the thermal conductivity of micro sized solid particles in fluid. This study scrutinizes the heat transport of steady micropolar fluid via non-uniform heat sink/ source and mass transfer is scrutinized through higher order chemical reaction over a stretching surface with variable heat flux. Moreover, the velocity of micropolar fluid is studied by considering aspects of magnetic dipole and Newtonian heating; velocity slip conditions are also examined. The numerical results have been performed by using the well-known numerical shooting technique and comparison is performed with the Matlab built-in solver bvp4c. Geometrically explanation reveals the properties of numerous parameters that are the system parts. The observed outcomes show that the local skin-friction coefficient and Sherwood number values goes up with the increase of chemical reaction rate parameters and Schmidt numbers. Chemical reaction based parameters boosts up the rate of heat as well as mass transfer. The stress of wall couple increased by increasing the Schmidt and chemical parameters. Moreover, the plots of dimensionless parameters have been drawn, as well as some parameter results are tabulated.

1. Introduction

Magnetohydrodynamic (MHD) flow possesses real world applications for example, in the extrusion of a polymer sheet procedure, several product properties and significant control of cooling rate [1, 2, 3] and it controls under investigated physical state of the problem. Due to these important features of MHD flow, many researchers put their active attention towards the study of MHD and derived several numerical and theoretical results in mechanism of fluid flow. Andersson [4] did investigated about MHD flow of viscoelastic fluid with geometry of stretching surface. Thermal radiation and MHD flow are interrelated and have massive applications in industries to know impact of thermal radiation on MHD flow explored by Raptis et al. [5]. Khan et al. [6] examined the MHD flow of boundary layer using the electric behavior of the fluid due to the cause of stretching in an elastic plane surface along the magnetic field. Abergel et al. [7] did work on different problems related to different physical situations and existence of such solutions to control problems, like boundary and boundary control in a channel. Moreover, he provided basic numerical algorithms named as conjugate gradient and steepest descent methods. Agrawal et.al [8] did research to investigate the transfer of heat with MHD flow by applying an unvarying suction over stretching surface. These investigations were classified to the Newtonian fluids. However, a novel phase to evaluate the theory of fluid using MHD flow is categorized by Aliakbar et al. [9]. Chemical reactive flow and its relationship with magnetic dipole impact on Cross model is investigated by khan et al. [10]. Mahanthesh et al. [11] explore preparation of numerical results related to MHD nanofluid flow with bidirectional linear stretching surface. Babu et al. [12] published his study about MHD slip flow of nanofluid with mass transfer via thermophoresis and Brownian motion. Nadeem et al. [13] examined time dependent MHD three- dimensional flow due to stretching/shrinking sheet.

Class of fluid that exhibits microscopic effects arising the phenomenon of micromotion of fluid particles is called micropolar fluid. This fluid consists of rigid macromolecule of individual motion that supports stress, body moment by spin inertia. These fluids contain micro-constituents that are capable of undergoing the rotation and have several practical applications in different areas, like depicting the attitude of exotic lubricants, turbulent shear flow, colloidal suspensions of nanofluid flow, human and animal blood, exotic lubricants, additive suspensions, colloidal fluids, liquid crystal, real fluids with interruptions and so forth [14]. Soundalgekar et al. [15] described an outstanding analysis of the micropolar fluids (MPFs) along with its applications. In another work, Soundalgekar et al. [16] explored the suction/injection effects in the flow passing over a semi-infinite porous plate using (MHD) MPF flow. Hady et al. [17] obtained the analytical results for the heat transfer model to a MPF using a non-stretching sheet. Ishak et al. [18] worked on the heat transfer over a stretching surface together with variable or uniform surface in (MHD) MPFs. Hassanien et al. [19] numerically discussed the suction/blowing effects on the heat/flow transfer using the MPF on a stretching surface. Hayat et al. [20] studied the two-dimensional mixed convection steady and stagnation point flow using (MHD) MPF on a stretching surface. Sajid et al. [21, 22] studied the true results for thin-film flows using MPF.

Vital role of thermal transport in various engineering problems, like as nuclear reactor cooling, metallurgical processes and continuous strips in which the performance of machine strongly dependent on the heat transfer rate and many hydrodynamic methods. Heat transfer through heat source/sink is most noteworthy aspect in view of many physical models. Heat generation/absorption can help to change the distribution energy in the fluid that consequently disturbs the particle deposition rate in the network like as semiconductors, nuclear reactors and electronic devices. Heat source/sink is assumed to be constant, temperature or space dependent. In this study, contains non-uniform heat sink/ source, i.e., temperature and space dependent heat source/sink. Motivated by the submissions of heat transfer with non-uniform heat source, numerous theoretical soundings have been discussed the heat transfer phenomenon in flows close to the stagnation point region [23]. Mabood et al. [24] used a shooting approach by considering the effects of thermal conductivity and variable viscosity using the MHD flow together with the transfer of heat in MPF through non uniform heat sink/source on a stretching sheet. The induction of flow is noticed because of an elastic sheet that is stretched back as well as forward. Reddy et al. [25] explored the heat transport via heat generation and mass transfer effects on MHD flow together with inclined porous plate. Ravidran et al. [26] implemented the non-uniform single/double effects of slot suction/injection into an unstable mixed convection of an electrically conducting. Ghadikolaei et al. [27] expressed detailed study with MHD flow heat transport inspired by the thermal radiations as well as heat generation over a porous stretching sheet. Sandeep et al. [28] scrutinized the non-uniform heat source/sink influences, chemical reaction and mass transfer on the mixed convection flow using a MPF along with viscous dissipation.

Most significance aspect of daily life is chemical reaction, and without chemical reaction there is no concept of daily life because chemical reactions appear in biomedical field, agriculture, photosynthesis, reproduction system, chemical industry, even earth is fertile with the chemical reactions and used in many engineering applications. Chemical reaction helps to transport the mass of fluid flow and many researchers did work on transport of mass with taking different models of fluid like Cross, Carreau, sisko and Maxwell model of non-Newtonian fluid. Numerical interpretation related mass transfer of 3D Cross fluid with chemical reaction is made by [29]. First order chemical reaction impact in MPF is studied by damesh et al. [30]. Das et al. [31] published their investigations on chemical reaction and thermal radiation on heat and mass transfer flow of MHD micropolar fluid with rotating frame of reference Further Magyari et al. [32] depicted that Combined effect of heat generation or absorption and first-order chemical reaction on micropolar fluid flows over a uniformly stretched permeable surface. Effects of higher order chemical reaction on micropolar fluid and Influence of thermophoresis and chemical reaction on MHD micropolar fluid flow is discussed by [33]. Sajid et al. investigate the effects of variable molecular diffusivity, nonlinear thermal radiation, convective boundary conditions, momentum slip, and variable molecular diffusivity on Prandtl fluid past a stretching sheet [34].

From many year scientists did a lot of work with different fluid models to investigate heat/ mass transport through heat generation/ absorption and activation energy respectively. But here in this manuscript we deal transport of heat of MPF via non uniform heat sink/ source because this transportation deal disposition rate of particle (space dependent) and mass transfer is carried out through chemical reaction. Further linear velocity of MPF is scrutinized by magnetic dipole aspect and also angular movement of said fluid is made in this struggle.

The remaining parts of the paper are organized as: Section 2 shows the problem formulation, Section 3is designed the methodology, Section 4 shows the results and discussion, while conclusion is drawn in the final Section.

2. Problem formulation

The problem is formulated by using the equations based on conservation of mass, momentum, angular momentum, energy equation and concentration equations, written as:

$$\nabla .V=0,$$E1

$$\rho \left[\nabla .V\right]V=-\nabla .p+\left(\mu +\chi \right){\nabla }^{2}V+k\nabla \times \mathrm{\Omega }+\mathit{\rho f},$$E2

$$\mathit{\rho j}\left[\nabla .V\right]\mathrm{\Omega }=\left({\alpha }_0+{\beta }_0+{\gamma }_0\right)\nabla \left[\nabla \mathrm{\Omega }\right]-{\gamma }_0\nabla \times \left[\nabla \times \mathrm{\Omega }\right]+\chi \nabla \times V-2k\mathrm{\Omega }+\mathit{\rho l},$$E3

$$\rho c_p\left[V.\nabla \right]T=k{\nabla }^{2}T-\phi +Q$$E4

$$\left[\nabla .j\right]=D{\nabla }^{2}C.$$E5

Consider a mixed convective steady state incompressible MPF passing over a porous plate that shrinks and stretches towards the velocity $u={\mathit{su}}_w\left(x\right)$, whereas, $u_w\left(x\right)=\mathit{ax}$, a is a dimensional constant towards the flow axis (Figure 1). The sheet is stretched with the speed that vary with distance $\mathrm{\ell }$, further consider the flow region is y > 0 and $B_0$ is the magnetic field towards they-axis. Presence of boundary layer heat transport, non-uniform source of heat, viscous dissipation along with magnetic field is presented. The components of micro-rotation and velocity components are (0, 0, w) and (u, v, 0), respectively. The basic equations are accounted in the presence of heat source and viscous dissipation. The flow positions are x and y − axis along with the slip flow model, i.e., u-slip as well as the Newtonian heating are conditions as:

Figure 1.

Geometry of the problem.

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0,$$E6

$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\left(\frac{\mu +\chi }{\rho }\right)\frac{{\partial }^{2}u}{\partial y^2}+\frac{\chi }{\rho }\frac{\partial w}{\partial y}-\frac{{\mathit{\sigma \beta }}_0^2}{\rho }u+g_1{\beta }_C\left(C-C_{\infty }\right)+g_1{\beta }_T\left(T-T_{\infty }\right),$$E7

$$u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}=\left(\frac{\gamma }{{\rho }_j}\right)\frac{{\partial }^{2}w}{\partial y^2}+\frac{\chi }{{\rho }_j}\left(2w+\frac{\partial w}{\partial y}\right),$$E8

$$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{k}{\rho C_p}\frac{{\partial }^{2}T}{\partial y^2}+\frac{\mu }{\rho C_p}{\left(\frac{\partial T}{\partial y}\right)}^2+\frac{q^{‴}}{\rho C_p},$$E9

$$u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}=D_m\frac{{\partial }^{2}C}{\partial y^2}+D_m\frac{K_T}{T_m}\frac{{\partial }^{2}T}{\partial y^2}-\xi \left(C-C_{\infty }\right).$$E10

To consider the spin gradient viscosity as

$$\gamma =\left(\mu +\frac{\chi }{2}\right)j=\mu \left(1+\frac{K}{2}\right)j,K=\frac{\chi }{\mu }\text{and}j=\frac{\nu }{a}$$E11

are the material parameter as well as micro inertia density, and $\nu =\frac{\mu }{\rho }$. Here $q‴$ is the non-uniform heat source and defined as

$$q‴=\left({\mathit{kU}}_w\left(x\right)/\mathit{xv}\right)\left[A^{\ast }\left(T_{\infty }\right)f^{\prime }\left(\eta \right)+B^{\ast }\left(T-T_{\infty }\right)\right]$$E12

Moreover, $v_0>0$ and $v_0<0$ indicate the velocities based on suction as well as injection of the permeable plate.

The associated conditions are:

$$u={\mathit{su}}_w\left(x\right)+u_{\mathit{\text{slip}}},v=v_0,w=-n\frac{\partial u}{\partial y},k\frac{\partial T}{\partial y}=-h_{s}T,C=C_w,\mathrm{at}y=0,$$E13

$$u\to 0,w\to 0,C\to C_{\infty },T\to T_{\infty },\mathrm{as}y\to \infty .$$E14

where n is constant and can be $0\le n\le 1$, when, n = 0 leads to the concentration based micro-elements in MPF close to sheet that are do not rotate for $w=0$, while n = 1 implies the turbulent flow. Moreover, $u_{\mathit{\text{slip}}}$ represents the slip velocity and given as:

$$u_{\mathit{\text{slip}}}=\frac{2}{3}\left[\frac{3-\epsilon l^3}{\epsilon }-\frac{3}{2}\frac{l-l^2}{k_n}\right]d\frac{\partial u}{\partial y}-\frac{1}{4}\left[l^4+\frac{2}{k_n^2}\left(1-l^2\right)\right]d^2\frac{{\partial }^{2}u}{\partial y^2},$$E15

$$u_{\mathit{\text{slip}}}=A\frac{\partial u}{\partial y}+B\frac{{\partial }^{2}u}{\partial y^2},$$E16

where $l$ is $min\left[\left(\frac{1}{k_n}\right),1\right]$ that goes to $0<l\le 1$. The Knudsen number is $k_n$ and $\epsilon$ represents the coefficient of momentum that lies in the range of $0<\epsilon \le 1$. The term d remains positive and shows the mean molecular free path, while $B$ is accordingly negative. By presenting the following suitable transformations as:

$$\mathrm{\Psi }\left(x,y\right)=\sqrt{\mathit{a\nu }}\mathit{xf}\left(\eta \right),\varphi \left(\eta \right)=\frac{C-C_{\infty }}{C_w-C_{\infty }},\theta \left(\eta \right)=\frac{T-T_{\infty }}{T_{\infty }},\eta =y\sqrt{\frac{a}{\nu }},$$E17

$$w=\mathit{axh}\left(\eta \right)\sqrt{\frac{a}{\nu }},u=\frac{\mathrm{\partial \Psi }}{\partial y}=\mathit{ax}{f}^{\prime }\left(\eta \right),v=-\frac{\mathrm{\partial \Psi }}{\partial x}=-\sqrt{\mathit{a\nu }}f\left(\eta \right),$$E18

where $\mathrm{\Psi }$ is called stream function. The momentum and heat equations are transformed to the governing momentum and heat transfer equations into the coupled ordinary differential equations as:

$$\left(1+K\right)f‴+f{f}^{″}-{f^{\prime }}^2+K{h}^{\prime }-M{f}^{\prime }+\mathit{\lambda \theta }+\mathit{\lambda A\varphi }=0,$$E19

$$\left(1+\frac{K}{2}\right)h^{″}+f{h}^{\prime }-f^{\prime }h-K\left(f^{″}+2h\right)=0,$$E20

$${\theta }^{″}+Prf{\theta }^{\prime }+\mathit{Ec}{\left(f^{″}\right)}^2+A^{\ast }{f}^{\prime }+B^{\ast }\theta =0,$$E21

$${\varphi }^{″}+\mathit{Sc}\left[f{\theta }^{\prime }+\mathit{Sr}{\theta }^{″}+C_m{\varphi }^{\prime }\right]=0,$$E22

The boundary conditions are

$$\left\{\begin{array}{l}f=f_w,f^{\prime }=s+\alpha f^{″}+\beta f^{‴},h=-n{f}^{″},{\theta }^{\prime }=-\delta \left(1+\theta \right),\varphi =1\mathrm{at}\eta =0,\\ f^{\prime }\left(\eta \right)\to 0,h\left(\eta \right)\to 0,\theta \left(\eta \right)\to 0,\varphi \left(\eta \right)\to 0,\mathrm{as}\eta =\infty .\end{array}\right.$$E23

Where the dimensionless parameters are defined as:

$K=\frac{\chi }{\mu },$$Pr=\frac{\mu C_p}{k},$$\mathit{Ec}=\frac{a^2{x}^2}{c_p},$$M=\frac{\sigma B_0^2}{\mathit{\rho a}},$$\lambda =\frac{{\mathit{Gr}}_x}{{\mathit{Re}}_x},$${\mathit{Gr}}_x=\frac{g{\beta }_T{T}_{\infty }x}{\mathit{a\nu }},$$C_m=\frac{\xi }{a},$$\beta =\frac{h_s}{k}\sqrt{\frac{\upsilon }{a}},$$\mathit{Sr}=\frac{D_m{k}_T\left(T_w-T_{\infty }\right)}{T_m\nu \left(C_w-C_{\infty }\right)},$$\mathit{Du}=\frac{D_m{k}_T\rho c_p\left(C_w-C_{\infty }\right)}{c_s{c}_{p}k\left(T_{\infty }\right)},$$f_w=-{\left[\mathit{a\upsilon }\right]}^{-\frac{1}{2}}{\nu }_0$, ${Re}_x=\frac{{\mathit{ax}}^2}{\upsilon }$, $\mathrm{\Lambda }=\frac{g{\beta }_c\left(C_w-C_{\infty }\right)}{{\beta }_T{T}_{\infty }}$, $a=A\left(\frac{a}{\upsilon }\right)>0$ and $\beta =\left(\frac{\mathit{Ba}}{\upsilon }\right)<0$.

The physical quantities based on skin-friction coefficient $c_f$ is $c_{\mathit{fx}}=-\frac{{\tau }_w}{\rho U_w^2}$ the wall shear stress, ${\tau }_w$ is given as:

$${\tau }_{\mathit{wx}}={\left|\left(\mu +\chi \right)\frac{\partial u}{\partial y}+\mathit{\chi w}\right|}_{y=0},$$E24

The value of $c_f$ is given as:

$$c_{\mathit{fx}}{{Re}_x}^{1/2}={\left.-\left(1+K-\mathit{nK}\right)f^{″}\left(\eta \right)\right|}_{y=0},$$E25

Here ${\mathrm{R}}_{\mathrm{e}x}$ is the Reynolds number. The skin-friction coefficient defined in Eq. (18) does not contain the micro rotation term. In the temperature field, the heat transfer rate is defined as:

$${\mathit{Nu}}_x=-\frac{{\mathit{xq}}_w}{T-T_{\infty }},\text{where}{q}_w={\left(\frac{\partial T}{\partial T}\right)}_{y=0}.$$E26

The local Nusselt number is shown as:

$${\mathit{Nu}}_x{\left({Re}_x\right)}^{-0.5}=\delta {\left(1+\frac{1}{\theta \left(\eta \right)}\right)}_{\eta =0}.$$E27

The couple stress is given as:

$$M_x=\frac{-m_w}{\mathit{\rho x}{\left(\mathit{ax}\right)}^2},m_w=\left(\mu +\frac{\chi }{2}\right)j{\left[\frac{\partial w}{\partial y}\right]}_{y=0},M_x{Re}_x=-\left(1+\frac{K}{2}\right)h^{\prime }\left(0\right).$$E28

Furthermore, the mass diffusion flux and Sherwood number become as:

$${\mathit{Sh}}_x=\frac{{\mathit{xS}}_m}{\left(C-C_w\right)},$$E29

$$S_m={\left(\frac{\partial C}{\partial y}\right)}_{y=0}.$$E30

Finally, Sherwood number becomes as:

$${\mathit{Sh}}_x/{\left({\mathrm{R}}_{\mathrm{e}x}\right)}^{-1/2}={\left(-\varphi \left(\eta \right)\right)}_{\eta =0}.$$E31

The results of the above nonlinear equations have been performed by using a well-known shooting technique and comparison is performed with the bvp4c. The shooting technique is fast convergent scheme and have been used to solve the extensive applications of fluid dynamics [35, 36, 37, 38, 39, 40]. For the implementations of the shooting scheme, the boundary value system has been converted into the initial value equations. The nonlinear Eqs. (19)-(22) take the form as:

$$\left\{\begin{array}{l}f=n_1,f^{\prime }=n_2,f^{″}=n_3,\\ n_3^{\prime }=\frac{1}{1+K}\left[{\mathit{Mn}}_2-n_1{n}_3+n_2^2-{\mathit{Kn}}_5-\lambda n_6-\lambda \mathrm{\Lambda }{n}_8\right],\\ n_4=h,n_4^{\prime }=n_5,\\ n_5^{\prime }=\frac{1}{\left(1+\frac{K}{2}\right)}\left[n_2{n}_4-n_1{n}_5+K\left(2n_4+n_3\right)\right],\\ n_6=\theta ,{\theta }^{\prime }=n_7,\\ n_7^{\prime }=\left[-Pr{n}_1{n}_7-{\mathit{Ecn}}_3^2-\left(A^{\ast }{n}_2+B^{\ast }{n}_6\right)\right],\\ n_8=\varphi ,{\varphi }^{\prime }=n_9,\\ n_9^{\prime }=\left[-\mathit{Sc}\left(n_1{n}_7+\mathit{Sr}{n}_7^{\prime }+C_m{n}_8\right)\right].\end{array}\right.$$E32

The concerned initial conditions are written as:

$$\left\{\begin{array}{l}{n}_1=f_w,n_2=s+\alpha n_3+\beta n_3^{\prime },n_4=-{\mathit{nn}}_3,n_7=-\delta \left(1+n_6\right),n_8=1\mathrm{at}\eta =0,\\ n_2=n_4=n_6=n_8=0\mathrm{as}\eta =\infty .\end{array}\right.$$E33

Matlab bvp4c technique procedure is given as

For the satisfaction of the results, Table 1 is provided is based on the literature results as well as the shooting and bvp4c for $f^{″}\left(0\right)$ using numerous value of $\alpha$ by putting K = 0 and M = 0. The matching of the shooting and bvp4c results with the literature results [38, 39] depicts the satisfaction and validity of the scheme.

$f_w$	$\alpha$	$\beta$	-$f^{″}\left(0\right)$	-$f^{″}\left(0\right)$	-$f^{″}\left(0\right)$	-$f^{″}\left(0\right)$
			Ref [27]	Ref [28]	Bvp4c	Shooting
2.0	0.5	−1	0.341213	0.3412	0.341214	0.341214
2.0	0.5	−2	0.203824	0.2038	0.203825	0.203825
2.0	1.0	−1	0.290548	0.2905	0.29057	0.29057
2.0	1.0	−2	0.184657	0.1846	0.18463	0.18463
3.0	0.5	−1	0.262681	0.2626	0.26281	0.26281
3.0	0.5	−2	0.147012	0.1470	0.14712	0.14712
3.0	1.0	−1	0.232017	0.2320	0.2314	0.2314
3.0	1.0	−2	0.136905	0.1369	0.13605	0.13605

Table 1.

Comparison value of $f^{″}\left(0\right)$ for different values of the $f_w$.

3. Results and discussions

In this section, the detail of the numerical results is presented to solve the system of nonlinear equations using shooting scheme. The effects of velocity profile, with physical parameters $K,M,A$, are examined, while $Pr,\mathit{Ec},A^{\ast },B^{\ast }$, are checked on temperature profile. Moreover, the influences of $\mathit{Sc},\mathit{Sr},C_m$ is drawn on concentration profile through Figures 2–14 as well as evaluation of physical quantities like $c_f$, $\mathit{Nu}$, $M_x$, ${\mathit{Sh}}_x$ are provided.

Figure 2.

$s$ effects on $f$ profile.

Figure 3.

$n$ effects on $f$profile.

Figure 4.

$M$ effects on $f$ profile.

Figure 5.

$K$ effects on $f$ profile.

Figure 6.

$f_w$ effects on $f$ profile.

Figure 7.

Pr effects on $\theta$ profile.

Figure 8.

$E_c$ effects on $\theta$ profile.

Figure 9.

$A^{\ast }$ effects on $\theta$ profile.

Figure 10.

$B^{\ast }$ effects on $\theta$ profile.

Figure 11.

Pr effects on $h$ profile.

Figure 12.

$n$ effects on $h$ profile.

Figure 13.

$M$ effects on $h$ profile.

Figure 14.

$K$ effects on $h$ profile.

3.1 Physical interpretation of parameters with velocity

The effects of parameter $K,M,A$, are examined by on the velocity profile and presented by Figures 2–6. Each parameter has its own impact for all the profiles along with its physical significance. As s is increasing, the sheet stretches due to this velocity increases $n\in \left(0,1\right)$. When n = 0, then the MPF flow get closer to the sheet that are inept to rotate, likewise for n = 1 indicates the turbulent flow. The values of material constant ‘K’ enhance the velocity due to its materialistic properties. The velocity state decreases due to the Lorentz force by increasing the values of ‘M’. Influence of the suction/injection parameter ‘f_w’ on the velocity component of the sheet shows an increment in the suction factor shows a decrease in the velocity together with the increment of the similarity values of the variable.

3.2 Physical interpretation of parameters with energy

The effects of different parameters are drawn on temperature profile are noticed in Figures 7–12. Temperature is decreasing for growing value of $Pr$, and $B^{\ast }$ because of $Pr$ reduces thermal conductivity and $B^{\ast }$ is internal heat generation so negative values reduces the temperature. For also positive value of $A^{\ast }$ temperature grows up. The energy intemperance exhibits a considerable increase with the wall temperature. This is reliable with the physical state because of elastic deformation work, ohmic and frictional heating are considered that become the cause of incrementing the thermal based boundary layer.

3.3 Physical interpretation of parameters with concentration

Figures 11–14 depicts the angular velocity of MPF with attached parameters, for increasing value of $Pr$, $n$, $M$ angular velocity increases. For growing $Pr$ temperature loses due to this movement of particle slows down due to this rotation gets down, and n is constant and there is no rotation when n = 0, so increase in n results growth in angular velocity. Similarly, a greater value of M produces Lorentz force due to this linear velocity downs but angular velocity uplifts. As the material parameter increases, it is observed that the boundary layer thickness increases due to this fact angular velocity decreases. The concentration profile became down with increment of $Pr$, $\mathit{Sc}$, $\mathit{Sr}$, $f_w$ and $C_m$. The increasing values of ‘Pr’ depicts that temperature decreases, as a result decrement is noticed in concentration. The concentration along with thickness of boundary layer decreases by enhancing the ‘Sc’. The Soret term, i.e., ‘Sr’ shows the temperature gradients effects on the profile of concentration. it is noticed that increment in ‘Sr’, temperature together with concentration increases (Figures 15–19).

Figure 15.

$S_r$ effects on $\varphi$ profile.

Figure 16.

$\mathit{Sc}$ effects on $\varphi$ profile.

Figure 17.

$s$ effects on $\varphi$ profile.

Figure 18.

$f_w$ effects on $\varphi$ profile.

Figure 19.

$\mathit{Cm}$ effects on $\varphi$ profile.

3.4 Physical quantities interpretation

Skin friction is called the rate of heat transfer that shows the increment for greater value of $K$, $\mathit{Sc}$, $C_m$ and decreases for growing values of $A$ and $f_w$. Couple stress $M_x{Re}_x$ values vary with parameter K. In this study, an increment is found in the couple stress $M_x{Re}_x$ for growing value of K. Local mass diffusion flux namely ${\mathit{Sh}}_x$ goes down for rising value of $K$ and numerical result of the present study in Table 2 is listed.

$K$	$A$	${Sc}$	$C_m$	$f_w$	$c_{{fx}}{{Re}_x}^{1/2}$	${{Nu}}_x{\left({\mathrm{R}}_{\mathrm{e}x}\right)}^{-1/2}$	$M_x{Re}_x$	${{Sh}}_x/{\left({\mathrm{R}}_{\mathrm{e}x}\right)}^{-1/2}$
0					0.3455454	1.658233	0.164237	0.7418682
0.5					0.4135856	1.668674	0.193939	0.7471964
1.0					0.482298	1.678695	0.2210438	0.7525225
1.5					0.5488359	1.678766	0.2460082	0.7575476
2.0					0.6130726	1.688377	0.2691283	0.7622267
0.5	0.5				0.4189335	1.668688	0.1939364	0.9283288
	1.0				0.41838978	1.688139	0.193138	0.9291135
	1.5				0.4178619	1.7081801	0.192349	0.9298976
	2.0				0.4173480	1.738553	0.191564	0.9306813
	0.5	0.5			0.4135858	1.668739	0.162426	0.7471966
		1.0			0.4319673	1.668286	0.169963	1.148792
		1.5			0.4431084	1.668264	0.174907	1.488861
		2.0			0.4507155	1.6687357	0.1785145	1.800042
		0.5	0.0		0.4014076	1.6683189	0.157579	0.1215778
			0.5		0.4135857	1.668763	0.162424	0.7471969
			1.0		0.4216537	1.6685654	0.1657039	1.148795
			1.5		0.4274917	1.668875	0.1681314	1.488867
			2.0	1.0	0.431996	1.748467	0.170059	1.800044
			0.5	0.5	0.420988	1.288242	0.212292	0.799846
				0.0	0.4189334	1.998349	0.193938	0.842233
				−0.5	0.4147433	1.808545	0.174625	0.885127
				−1.0	0.3915482	1.69878	0.1535629	0.9283398

Table 2.

Behavior of skin friction, Nusselt number, couple stress, and Sherwood number.

4. Concluding remarks

The key purpose of the current work is to discuss the effects of heat transportation and source/sink of heat with magnetic effect on boundary layer MPF. This study elaborates that angular velocity and linear velocity of MPF with different facts which impact on temperature and concentration of said fluid. For the numerical purpose, the shooting scheme has been implemented and comparison of the results with bvp4c is presented. Moreover, main key points of study are provided as:

1. Velocity of flow raises for growing of $s$, $n$, $K$.

2. Velocity of flow downs for growing of $M$, $f_w$.

3. For also positive value of $A^{\ast }$ and increasing value of Ec temperature grows up

4. Mass field and corresponding boundary layers thickness downs by increasing the ‘Sc’.

5. An increasing in ‘Sr’ causes a increase in the concentration and temperature through the boundary layer.

Nomenclature of parameters and short terms

$f$

Body force

$l$

Body couple

V

translational vector

$\mathrm{\Omega }$

micro-rotation vector

K

Thermal conductivity

$p$

Pressure

$\begin{array}{l}{\alpha }_0,{\beta }_0,{\gamma }_0,\chi ,\\ K\end{array}$

Material constants

$c_s$

Concentration susceptibility

M

Magnetic parameter

$j$

Micro-inertia

$\mu$

dynamic viscosity

$\phi$

dissipation function

$\tau$

Ratio parameter

C

Concentration of fluid.

T

Temperature of the fluid

V

translational vector

$\mathrm{\Omega }$

micro-rotation vector

$T_{\infty }$

Infinite temperature

A

Usual constant

A

Constant

$a$

first order slip flow parameter

T_w

Temperature of the plate

$c_p$

Specific heat

$\rho$

Fluid density

$D_m$

coefficient of mass diffusivity

$\alpha$

Thermal diffusivity

$\mathit{Du}$

Dufour number,

$\beta$

Second order slip flow parameter

MPF

Micropolar fluid

$q\text{'}\text{'}\text{'}$

Non-uniform heat source

A*, B*

Coefficients of space and temperature

$D$

Mass diffusivity

$K_T$

Thermal diffusion ratio

$D_m$

Coefficient of mass diffusivity

$T_m$

Mean fluid temperature.

N

Constant

${\alpha }^{\ast }$

Slip coefficient

$w$

Micro rotation component

$h$

Heat transfer coefficient

$Pr$

Prandtl number

$G$

Micro rotation parameter

$\mathit{Ec}$,

Eckert number

$\sigma$

Reaction rate parameter

${\beta }_o$

Strength of magnetic field.

$\mathit{Sc}$

Schmidt number

${\lambda }^A$

Activation energy parameter

${\mathit{Bi}}_{\theta }$

Thermal Biot number

$\alpha$

slip parameter

${\gamma }_1$

Thermal concentration parameter.

$\mathit{Sr}$

Soret number

$f_w$

Suction or injection parameter

${\mathit{Sh}}_x$

Sherwood number

$\nu$

Kinematic viscosity

$c_f$

Skin-friction coefficient

${\mathrm{R}}_{\mathrm{e}x}$

Local Reynold number

$N_{\mathit{ux}}$

Nusselt number

$m_w$

couple-stress

$\delta$

Newtonian heating parameter

MHD

Magnetohydrodynamic