Thermo-Diffusion and Multislip Effects on MHD Mixed Convection Unsteady Flow of Micropolar Nanofluid over a Shrinking/Stretching Sheet with Radiation in the Presence of Heat Source

Abstract

The main purpose of this study is to investigate the multislip effects on the magneto-hydrodynamic (MHD) mixed convection unsteady flow of micropolar nano-fluids over a stretching/shrinking sheet along with radiation in the presence of a heat source. The consequences of multislip and buoyancy conditions have been integrated. By using the suitable similarity variables are used to solve the governing non-linear partial differential equations into a system of coupled non-linear ordinary differential equations. The transformed equations are solved numerically by using Runge–Kutta fourth-order method with shooting technique. The impacts of the several parameters on the velocity, temperature, micro-rotation, and concentration profiles as well as on the skin friction coefficient, Sherwood number, and Nusselt number are discussed with the help of graphs and tables.

1. Introduction

Nanofluids played an important role in recent years due to their vast variety and complex applications, such as in the medical applications, the petroleum industry, and food processing. Many researchers attracted by convective heat transfer using suspensions of nano-sized particles also the temperature of the base fluid increases by nano-particles which is the main source in the heat transfer performance [1,2]. The mixture of base fluids and nano-particles called nano-fluid and is the advanced category of heat transfer. Nano-fluid is created by the distribution of solid particles with no dimensions. The heat transfer capabilities are limited due to low thermal conductivity, such as water, engine oil, and ethylene glycol. On the other hand, metals have very high thermal conductivity. This is the most appropriate method for increasing the heat transfer coefficient. Choi and Eastman [3] firstly gave the concept of nano-fluid. They described that the thermal conductivity would increase after adding a small number of nanoparticles. Unsteady magneto-hydrodynamic (MHD) flow of Casson nanofluid over a moving vertical semi-infinite plate was numerically studied by Babu [4]. Ali et al. [5] investigated the effects of multiple slips with the presence of chemical reaction. Mohyud-Din et al. [6] examined the behavior of a flowing nano-fluid over a flat moving plate. They also described that when nano-particle increases in numbers thermal conductivity also increases. The same behavior of nano-particles was examined by Akbari et al. [7] and Hayat et al. [8]. Kamal et al. [9] investigated the MHD stagnation point flow of nano-fluid in the presence of a chemical reaction. The unsteady nano-fluid flow through the previous medium was analyzed by Kumar et al. [10].

In recent years, the theory of micro-fluids has found great attention, as traditional Newtonian fluids cannot properly identify the characteristic of liquids with suspended particles. A micropolar fluid assumes the key equations of the non-Newtonian fluid model. In the micropolar fluid model, a micro-rotation vector and a coupling parameter are gathered to analyze the kinematics of micro-transformation. Such types of fluid models can be applied to give the details of the flow of colloidal solutions, liquid crystals, liquids with additives, suspension solutions, animal blood, etc., A microscopic fluid is a fluid with internal structures during which the pairing between the spin and macroscopic velocity field of each particle is considered. It is a hydro-dynamic framework acceptable for angular systems because of macroscopic shaped particles. Mabood and Shateyi [11] studied the effects of multislip on the unsteady hydro-magnetic flow of heat transfer and mass transfer impact by radiation in a porous frame of reference. Magnetohydrodynamic boundary layer flow of incompressible fluid within the moving vertical plate involving heat source and the chemical reaction was investigated by Das and Dorjee [12]. Jena [13] and his coworkers studied the thermo effects on MHD viscoelastic fluid flow. They described that additional parameters rise by this phenomenon. Malarselvi et al. [14] accomplished the investigation on time-dependent MHD flow for several parameters over a flat surface. Nandeppanavar et al. [15] investigated that no stagnation point flow consisting of viscous fluid and heat transfer along with velocity due to the moving surface. Baag [16] and his coworkers described the flow analysis and heat transfer analysis past a porous medium in the presence of heat source. Imtiaz et al. [17] studied the unsteady MHD flow of a curved surface. Kempannagari et al. [18] analyzed the impact of heat transfer on MHD ferrofluid flow.

Pordanjani et al. [19] the effect of the presence of radiation on the convection heat transfer rate and the nano-fluid entropy generation within a diagonal rectangular chamber is investigated numerically in the presence of a magnetic field. Malvandi et al. [20] investigates the effects of nano-particle migration on hydromagnetic (MHD) mixed convection of alumina/water nano-fluid inside a vertical annular pipe. Karimipour et al. [21] simulate the nano-fluid flow in a micro-channel in the presence of a magnetic field. Abdollahzadeh et al. [22] investigates the effect of MHD forces on a two-phase boiling flow in a vertical path with nano-particles. Mishra et al. [23] examined the influence of the various physical parameters and the inertia effect of micropolar fluids. Sohaib et al. [24] analyzed the various effects of radiation on MHD flow of micropolar fluid due to the porous medium. Subba et al. [25] explained that the mixed convection entrenched in a porous medium. Liaqat et al. [26] investigated the impact of multiple slips on MHD unsteady Casson nano-fluid flow, heat, and mass transfer in the presence of heat source with thermo-diffusion effect over a stretching/shrinking sheet. Shahid Ali et al. [27] examined the multiple slips on hydromagnetics axisymmetric nano-fluid steady flow, heat and mass transfer influenced by radiation, buoyancy and chemical effect in a permeable frame of reference. Ilyas Khan et al. [28] focused on the MHD channel flow of electrically conducting nano-fluid in a porous medium with the suction and injection effect influence of a transverse magnetic field. Liaqat et al. [29] investigated the impact of multi-slip and solutal boundary conditions on MHD unsteady bioconvective micropolar nano-fluid restraining gyrotactic microorganism, heat and mass transfer effect over a stretching/shrinking sheet. Hayat et al. [30] studied that the velocity profile reduces as the values of the suction and stretching and the wall thickness parameters increase largely also the volume fraction and porosity parameters rises. In several manufacturing processes slip condition on stretching surfaces is important, the slip boundary condition is compulsory when the flow pressure is low. Slip conditions were used by Malarselvi et ai. [31]. Mozaffari et al. [32] investigated the effects of aging and S/B ratio on rheological properties of diluted bitumen and find the solutions at concentrations above and below the critical S/B. Darjani et al. [33] discussed an alternative approach for deriving the equation of state for a two-dimensional lattice gas which is based on arguments similar to those used in the derivation of the Langmuir–Szyszkowski equation of state for localized adsorption. Darjani et al. [34] investigated the possible mechanisms leading to the focusing and defocusing of colloidal nano-particle size distribution during the synthesis. Also, by using the in situ SAXS to measure the evolution of size, size distribution and concentration of Pd colloidal nano-particles during synthesis and population balance modeling (PBM). Li et al. [35] developed a methodology and a trimer-monomer ITC model to obtain the Gibbs free energy, enthalpy, and entropy for toluene and quantify the monomer content. Mozaffari et al. [36] focused on using continuum calculations to simulate the Stokes flow movement of (neutral) self-diffusiophoretically driven, spherical, and catalytic Janus locomotors in the vicinity of a planar wall to understand the fundamentals of boundary guidance. Mozaffari et al. [37] scrutinized that the characterizing of oil film rheology below and above critical S/B ratio at a length scale comparable to the film thickness where oil film separating two water droplets stops draining. Liu et al. [38] showed that all experimental features of the dilatational rheology of asphaltenes laden interfaces be qualitatively explained by diffusional relaxation models provided that the mixture nature of asphaltenes. Lok et al. [39] studied the unsteady flow of micropolar fluid. The impacts of an external magnetic field were numerically described by Sharma et al. [40]. Elahi et al. [41] numerically examined the solution of mixed convection heat transfer over a stretching sheet. The multiple solutions of MHD heat transfer flow with viscous dissipation were conducted by Dhanai et ai. [42].

In earlier research, several researchers revealed the effects of different operative parameters distinctly/collectively such as the Reynolds number, nano-particles volume fraction and slip effects. According to the author’s best knowledge, a numerical investigation of multi-slips effects on hydro-magnetic mixed convection unsteady flow of micropolar nano-fluid with radiation in the presence of heat source has not yet been considered. Numerical solutions are examined for some special cases, while the overall physical interpretation for the various parameters is investigated with the help of graphs. The main purpose of the present study was to extend the recently published work of Mabood and Shateyi [11]. The governing nonlinear PDEs are transformed into a set of particularly nonlinear odes with the support of appropriate similarity transformations and the nonlinear coupled odes are solved numerically with the R-K shooting technique. The impacts of the several physical parameters on the fluid velocity, temperature, solutal, and nano-particle volume fraction functions are examined in detail for some special cases. An exact solution of flow velocity, skin friction coefficient, and Nusselt number is compared with the numerical solution obtained by the R-K shooting method and also with numerical results available in the literature.

2. Mathematical Formulation

An unsteady two dimensional MHD flow of incompressible nano-fluid in the presence of thermal radiation over an electrically conducted shrinking/stretching sheet has been considered for investigation. The x-axis is chosen along the sheet and y-axis is normal to it as shown in Figure 1. The sheet is moving with non-uniform velocity $U(x,t)=ax/(1-\lambda t)$, where a is the stretching/shrinking rate along x-axis and $\lambda t$ is positive constant with property $\lambda t<1$. The transverse magnetic field is $B(x)=B_0{x}^{-1/2}$ with $B_0\ne 0$, where $B_0$ is the strength of magnetic field. The ambient temperature, ambient solutal concentration and ambient nano-particle concentration are $T_{\infty }$, $C_{\infty }$, and ${\chi }_{\infty }$ respectively. The temperature of the sheet, concentration, and the nano-particle volume fraction are $T_w(x,t)$, $C_w(x,t)$ and ${\chi }_w(x,t)$ at the surface respectively.

The $T_w(x,t)$, $C_w(x,t)$, and ${\chi }_w(x,t)$ assumed as of the following form (see [11]):

$$\begin{array}{c}{T}_w=T_{\infty }+T_0\left(\frac{ax}{2\nu {(1-\lambda t)}^2}\right)\\ C_w=C_{\infty }+C_0\left(\frac{ax}{2\nu {(1-\lambda t)}^2}\right)\\ {\chi }_w={\chi }_{\infty }+{\chi }_0\left(\frac{ax}{2\nu {(1-\lambda t)}^2}\right)\end{array}$$

where $T_0$,$C_0$ and ${\chi }_0$ are the reference temperature, reference solutal concentration and reference nano-particle concentration, respectively, such that $0\le T_0\le T_w$, $0\le C_0\le C_w$ and $0\le {\chi }_0\le {\chi }_w$. The expressions only valid if $(1-\lambda t)>0$.

By using the above assumptions, the governing boundary layer equations for the flow problem are as follows (see [8,9,43,44]):

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

(1)

$$\begin{gathered} \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=(\frac{\mu +k}{\rho })\frac{{\partial }^{2}u}{\partial y^2}+\frac{k}{\rho }\frac{\partial N}{\partial y}-\frac{\sigma B^2(x)u}{\rho } \\ \hspace{1em}\hspace{1em}+g{\beta }_T(T-T_{\infty })+g{\beta }_C(C-C_{\infty })+g{\beta }_{\chi }(\chi -{\chi }_{\infty }) \end{gathered}$$

(2)

$$\rho j(\frac{\partial N}{\partial t}+u\frac{\partial N}{\partial x}+v\frac{\partial N}{\partial y})=\gamma \frac{{\partial }^{2}N}{\partial y^2}-k(2N+\frac{\partial u}{\partial y})$$

(3)

$$\begin{array}{r}\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha \left(1+\frac{16T_{\infty }^3{\sigma }^{*}}{3k^{*}\kappa }\right)\frac{{\partial }^{2}T}{\partial y^2}+\tau D_B\frac{\partial C}{\partial y}\frac{\partial T}{\partial y}\\ +\tau \frac{D_T}{T_{\infty }}{\left(\frac{\partial T}{\partial y}\right)}^2+D_{TC}\frac{\partial C^2}{\partial y}+Q(T-T_{\infty })\end{array}$$

(4)

$$\frac{\partial C}{\partial t}+u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}=D_S\frac{{\partial }^{2}C}{\partial y^2}+D_{CT}\frac{{\partial }^{2}T}{\partial y^2}$$

(5)

$$\frac{\partial \chi }{\partial t}+u\frac{\partial \chi }{\partial x}+v\frac{\partial \chi }{\partial y}=D_B\frac{{\partial }^2\chi }{\partial y^2}+\frac{D_T}{T_{\infty }}\frac{{\partial }^{2}T}{\partial y^2}$$

(6)

and the boundary conditions are (see [11,44]):

$$u=U(x,t)+U_{slip},v=v_w,N=-m\frac{\partial u}{\partial y},T=T_w(x,t)+T_{slip},C=C_w(x,t)+C_{slip},$$

$$\chi ={\chi }_w(x,t)+{\chi }_{slip} at y=0$$

(7)

$$u\to 0,N\to 0,T\to T_{\infty },C\to C_{\infty },\chi \to {\chi }_{\infty },\hspace{1em}as\hspace{1em}y\to \infty ,$$

(8)

where u is the velocity component along x and v is the velocity component along y, respectively; $\mu$ is dynamic viscosity, k is vortex viscosity, $\rho$ is fluid density, N is micro-rotation vector, $\sigma$ is the electrical conductivity, $\alpha$ is thermal diffusivity, $\gamma$ is spin gradient viscosity, g is gravity acceleration. Further ${\beta }_T$, ${\beta }_C$ and ${\beta }_{\chi }$ are thermal expansion coefficient, solutal concentration expansion coefficient and nano-particle concentration expansion coefficient. T is the temperature, C is the solutal concentration, $\chi$ is the nano-particle concentration. $D_s$, $D_T$, $D_B$, $D_{CT}$, $D_{Tc}$ are the molecular diffusivity, thermal diffusivity, Brownian diffusivity, Soret diffusivity and Dufour diffusivity. ${\sigma }^{*}$ is the Stefan–Boltzmann constant, $k^{*}$ is the mean absorption coefficient, Q is the chemical reaction.

Generally, the stream function $\psi$ which is defined as $u=\frac{\partial \psi }{\partial y}$ and $v=-\frac{\partial \psi }{\partial x}$. By using the following similarity transformations to convert Equations (1)–(6) into system of ordinary differential equations [31,44].

$$\begin{array}{lll}\hspace{1em}\eta & =& \sqrt{\frac{a}{\nu (1-\lambda t)}}y,\hspace{1em}\psi =\sqrt{\frac{a\nu }{(1-\lambda t)}}xf(\eta ), N=\sqrt{\frac{a^3}{\nu {(1-\lambda t)}^3}}xh(\eta ),\hspace{1em}\theta (\eta )=\frac{T-T_{\infty }}{T_w-T_{\infty }},\\ \varphi (\eta )& =& \frac{C-C_{\infty }}{C_w-C_{\infty }},\hspace{1em}\gamma (\eta )=\frac{\chi -{\chi }_{\infty }}{{\chi }_w-{\chi }_{\infty }}\end{array}$$

(9)

By using similarity transformation Equation (9), the nonlinear partial differential Equations (1)–(6) transform into the system of nonlinear ODE’s:

$$(1+K)\frac{d^{3}f}{d{\eta }^3}+f\frac{d^{2}f}{d{\eta }^2}-{(\frac{df}{d\eta })}^2-\delta (\frac{\eta }{2}\frac{d^{2}f}{d{\eta }^2}+\frac{df}{d\eta })+K\frac{dh}{d\eta }-M\frac{df}{d\eta }+{\lambda }_1\theta +{\lambda }_2\varphi +{\lambda }_3\gamma =0,$$

(10)

$$A\frac{d{h}^2}{d{\eta }^2}+f\frac{dh}{d\eta }-\frac{df}{d\eta }h-\delta (\frac{\eta }{2}\frac{d^{2}f}{d{\eta }^2}+\frac{3}{2}\frac{df}{d\eta })-KB(2h+\frac{d^{2}f}{d{\eta }^2})=0$$

(11)

$$(1+R)\frac{1}{Pr}\frac{d^2\theta }{d{\eta }^2}-\frac{df}{d\eta }\theta +f\frac{d\theta }{d\eta }-\delta \left(\frac{\eta }{2}\frac{d\theta }{d\eta }+2\theta \right)+Nb\frac{d\varphi }{d\eta }\frac{d\theta }{d\eta }+Nt{(\frac{d\theta }{d\eta })}^2+Nd\frac{d^2\varphi }{d{\eta }^2}+\frac{dQ}{d\eta }\theta =0$$

(12)

$$\frac{1}{Sc}\frac{d^2\varphi }{d{\eta }^2}-\frac{df}{d\eta }\varphi +f\frac{d\varphi }{d\eta }-\delta \left(\frac{\eta }{2}\frac{d\varphi }{d\eta }+2\varphi \right)+S_r\frac{d^2\theta }{d{\eta }^2}=0$$

(13)

$$\frac{d^2\gamma }{d{\eta }^2}-Ln\left[\frac{df}{d\eta }\gamma -f\frac{d\gamma }{d\eta }+\delta \left(\frac{\eta }{2}\frac{d\gamma }{d\eta }+2\gamma \right)\right]+\frac{Nt}{Nb}\frac{d^2\theta }{d{\eta }^2}=0$$

(14)

and the transformed boundary conditions Equations (7) and (8) are:

$$f(0)=f_w,\frac{df(0)}{d\eta }=1+S_f\frac{d^{2}f(0)}{d{\eta }^2},h(0)=-m\frac{d^{2}f(0)}{d{\eta }^2},\theta (0)=1+S_{\theta }\frac{d\theta (0)}{d\eta },\varphi (0)=1+S_{\varphi }\frac{d\varphi (0)}{d\eta },$$

$$\gamma (0)=1+S_{\gamma }\frac{d\gamma (0)}{d\eta },$$

(15)

$$\frac{df(\infty )}{d\eta }\to 0,h(\infty )\to 0,\theta (\infty )\to 0,\varphi (\infty )\to 0,\gamma (\infty )\to 0,$$

(16)

The parameters in Equations (10)–(16) are defined as: $M=\frac{\sigma (1-\lambda t)}{\rho ax}{B}_0^2$, $\kappa =\frac{k}{\mu }$, $\delta =\frac{\lambda }{a}$, ${\lambda }_1=\frac{g{\beta }_T{T}_0}{2a\nu }$, ${\lambda }_2=\frac{g{\beta }_C{C}_0}{2a\nu }$, ${\lambda }_3=\frac{g{\beta }_{\chi }{\chi }_0}{2a\nu }$, $R=\frac{16{\sigma }^{*}{T}_{\infty }^3}{3k^{*}K}$, $Nt=\frac{\tau D_T(T_w-T_{\infty })}{\nu T_{\infty }}$, $Nb=\frac{\tau D_B({\chi }_w-{\chi }_{\infty })}{\nu }$, $Pr=\frac{\nu }{\alpha }$, $Nd=\frac{D_{Tc}{C}_0}{\nu T_0}$, $Sc=\frac{\nu }{D_S}$, $Sr=\frac{D_{CT}{T}_0}{\nu C_0}$, $Ln=\frac{\sigma }{D_B}$, $f_w=-\frac{v_0}{\sqrt{x}}(\sqrt{\frac{(1-\lambda t)}{\nu a}})$, $A=\frac{\gamma }{\mu j}$, $B=\frac{\nu (1-\lambda t)}{aj}$, $\frac{dQ}{d\eta }=\frac{Q(1-\lambda t)}{a}$, where M is the magnetic parameter, $\kappa$ is material parameter, $\delta$ is the unsteadiness parameter, ${\lambda }_1$, ${\lambda }_2$ and ${\lambda }_3$ are buoyancy parameters, $Nb$ is the Brownian motion parameter, $Pr$ is the Prandtl number, $Nt$ is the thermophoresis parameter, $Ln$ is the Lewis number, R is the thermal radiation parameter, $Nd$ is the Dufour parameter, $Sr$ is the Soret parameter, $Sc$ is the Schmidt number, $Q^{\prime }$ is the chemical reaction parameter, and $f_w$ is Suction/injection parameter.

3. Implementation of Method

By using the shooting method, numerically solve the system of ordinary differential Equations (10)–(14) with the boundary conditions (15) and (16). The Runge–Kutta method is more capable and proficient than other numerical methods as HPM, FDM, HPM, etc. Furthermore, it is seen that the Runge–Kutta method is employed in commercial software, such as ANSYS, ABAQUS, ADINA, and MATLAB. For the transformation of Equations (10)–(14) into a system of the first-order ordinary differential equations, introduce the new variables,

$y_1^{\prime }=y_2$

$y_2^{\prime }=y_3$

$y_3^{\prime }=\frac{1}{1+K}[y_2^2-y_1{y}_3-K{y}_5+M{y}_2+\delta (\frac{\eta }{2}{y}_3+y_2)-{\lambda }_1\theta -{\lambda }_2\varphi -{\lambda }_3\gamma ]$

$y_4^{\prime }=y_5$

$y_5^{\prime }=\frac{1}{A}[y_2{y}_4-y_1{y}_5+\delta (\frac{\eta }{2}{y}_3+\frac{3}{2}{y}_2)-KB(2y_4+y_3)]$

$y_6^{\prime }=y_7$

$y_7^{\prime }=\frac{Pr}{1+R}[y_2{y}_6-y_1{y}_7+\delta (\frac{\eta }{2}{y}_7+2y_6)-Nb{y}_7{y}_9-Nt{y}_7^2-Nd{y}_9^{\prime }-Q^{\prime }{y}_6]$

$y_8^{\prime }=y_9$

$y_9^{\prime }=Sc[y_2{y}_8-y_1{y}_9+\delta (\frac{\eta }{2}{y}_9+2y_8)-Sr{y}_7^{\prime }]$

$y_{10}^{\prime }=y_{11}$

$y_{11}^{\prime }=Ln(y_2{y}_{10}-y_1{y}_{11}+\delta (\frac{\eta }{2}{y}_{11}+2y_{10}))-\frac{Nt}{Nb}{y}_7^{\prime }$

The corresponding dimensionless boundary conditions are:

$$\begin{array}{c}{y}_1=f_w,y_2=1+y_3{S}_f,y_4=-m{y}_3,y_6=1+y_7{S}_{\theta },y_8=1+y_9{S}_{\eta },y_{10}=1+y_{11}{S}_{\gamma }at\eta =0\\ y_2\to 0,\hspace{1em}{y}_4\to 0,\hspace{1em}{y}_6\to 0,\hspace{1em}{y}_8\to 0,\hspace{1em}{y}_{10}\to 0\hspace{1em}as\hspace{1em}\eta \to \infty \end{array}$$

To explain the system of first-order ordinary differential equations with the help of the shooting technique, eleven initial conditions are required. Hence, we guess five unknown initial conditions $y_3(0)=a$, $y_5(0)=b$, $y_7(0)=c$, $y_9(0)=d$, $y_{11}(0)=e$. The appropriate guesses for these five missing unknown conditions are chosen such that the five known boundary conditions are nearly satisfied for $\eta \to \infty$. To develop the correctness of the missing initial conditions, Newton’s iterative structure is applied until the required approximation is seen. The calculations have been completed for the several developing parameters and for the suitable bounded domain $[0,{\eta }_{max}]$ instead of $[0,\infty )$ where ${\eta }_{max}$ is the positive real number and is chosen such that no significant variations appeared in the results for the values greater than ${\eta }_{max}$. The criteria for stopping the iteration process is

$$max\left\{\right|y_2({\eta }_{max})-0|,|y_4({\eta }_{max})-0|,|y_6({\eta }_{max})-0|,|y_8({\eta }_{max})-0|,|y_{10}({\eta }_{max})-0\left|\right\}<\xi .$$

where $\xi$ is real number which is positive and very small.

4. Results and Discussion

Our main goal is to examine the unsteadiness effects on the nano-fluid flow quantities. We consider the analysis of the thermo-diffusion and multislip effects on MHD mixed convection unsteady flow of micropolar nano-fluid over a shrinking/stretching sheet with radiation in the presence of a heat source.

Table 1. Comparison of $-{\theta }^{\prime }(0)$ for various values of $Pr$.

$\mathit{Pr}$	Liaqat et al.	Fazle et al.	Ishak et al.	Dulal et al.	Haile et al.	Ishak et al. [47]	Our Results	Error in %
	[29]	[11]	[47]	[45]	[46]	(a)	(b)	$|(\frac{\mathit{b}-\mathit{a}}{\mathit{a}})|\times 100$
0.72	0.8086	0.8088	-	-	-	0.8086313498	0.8086339299	0.0004
1.00	1.0000	1.0000	1.0000	1.0000	1.0004	1.000000000	1.0000080213	0.0008
3.00	1.9236	1.9237	1.9237	1.9236	1.9234	1.923682594	1.9236777221	0.0004
10.0	3.7206	3.7207	3.7207	3.7207	3.7205	3.720673901	3.7206681683	0.0002
100	12.2946	-	12.2941	12.2962	12.2962	12.294083260	12.294051659	0.0002

shows the comparison between the results of heat transfer which are obtained by R-K shooting technique and the previous studied numerical results [11,29,45,46] and exact solution of Ishak et al. [47]. There is 4 decimal accuracy between the results under special cases ($f_w=0$, $M=0$, $K=0$, $A=0$, $B=0$, $\delta =0$, ${\lambda }_1=0$, ${\lambda }_2=0$, ${\lambda }_3=0$, $R=0$, $S_{\theta }=0$, $Nd=0$, $S_f=0$).

Table 2. Comparison of $-f^{″}(0)$ for various values of M.

M	Gireesha et al. [48]	Mudassar et al. [49]	Bagh et al. [5]	Our Results	Error in %
	$\mathit{\beta }=0$	(a)		(b)	$|(\frac{\mathit{b}-\mathit{a}}{\mathit{a}})|\times 100$
0.0	1.000	1.000000	1.0000080	1.0000130	0.00130
0.2	1.095	1.095445	1.0954458	1.0954463	0.00013
0.5	1.224	1.224745	1.2247446	1.2247454	0.00003
1.0	1.414	1.414214	1.4142132	1.4142180	0.00002
1.2	1.483	1.483240	1.4832393	1.4832402	0.00001
1.5	1.581	1.581139	1.5811384	1.5811396	0.00003
2.0	1.732	1.732051	1.7320504	1.7320516	0.00003

shows the comparison of the skin friction coefficient $-f^{″}(0)$ with available numerical and exact results are done. In Table 2, we present a comparison between our results and the numerical results of Gireesha et al. [48], Bagh et al. [5] and the exact solution of Mudassar et al. [49] is done under the special cases ($\delta =0$, ${\lambda }_1=0$, ${\lambda }_2=0$, ${\lambda }_3=0$, $S_f=0$, $f_w=0$, $K=0$, $A=0$, $B=0$). The comparison of the results of local skin friction $-f^{″}(0)$ and couple stress $-g^{\prime }(0)$ with already published research work by Eldabe et al. [50] and Hsiao et al. [51] by using R-K shooting technique are shown in

Table 3. Comparison of $-f^{″}(0)$ and $-g^{\prime }(0)$ with our results.

M	K	$-{\mathit{f}}^{″}(0)$ [50]	$-{\mathit{f}}^{″}(0)$ [51]	Our Results	$-{\mathit{g}}^{\prime }(0)$ [50]	$-{\mathit{g}}^{\prime }(0)$ [51]	Our Results
0.0	0.2	0.9098	0.90976	0.909798	0.0950	0.09500	0.094895
0.5		1.1148	1011437	1.114378	0.1051	0.10509	0.105088
1.0		1.2871	1.28711	1.287148	0.1121	0.11212	0.112048
	0.0	1.4142	1.41423	1.414228	0.0000	0.00000	0.000000
	0.5	1.1408	1.14073	1.140772	0.2112	0.21116	0.211165
	2.0	0.7697	0.76958	0.769755	0.3586	0.35855	0.358646

.

The effect on velocity function of M without hydro-dynamic slip and with hydro-dynamic slip is shown in Figure 2. In both cases the component of velocity decreases as well as M increases. Physically the motion of the fluid slowed when M produced Lorentz force. However, the velocity boundary layer decreases in the presence of hydro-dynamic slip. The momentum boundary layer thickness reduced by $f_w$ as shown in Figure 2. The velocity profiles for different values of K as shown in Figure 3. As K increases the velocity profile increases in both cases. The velocity increases when the value of the buoyancy parameter increases without $s_f$ and with $s_f$ as shown in Figure 4. The growth of velocity depends on buoyancy parameters. It is noticed that the velocity profile enhanced when the radiation parameter R increased. Figure 5 shows that the velocity, boundary layer thickness increases with the increasing values of buoyancy parameter ${\lambda }_2$ and R without existence and existence of slip.

The influence of M on temperature with and without thermal slip shows in Figure 6. This shows that the temperature increases as the value of M increases. Physically warms up the liquid by applying the magnetic field. Thermal boundary layer thickness decreases due to $f_w$ and $S_{\theta }$, see Figure 6. It is observed that the behavior of $Nb$ is the same as the temperature profile see Figure 7. For small size nano-particles, $Nb$ will be more prominent and have large values. It is also observed that temperature has the same behavior when unsteady parameter increases. Similarly, the behavior of $Nt$ on thermal profile is shown in Figure 8. Clearly we can see that the increment in $Nt$ causes the increase in the thermal slip boundary layer and observed the behavior of $Nd$ is similar in Figure 8.

In Figure 9, the effect of soret parameter, $Sr$, on the solutal profile is examined, and several interesting interpretations of solutal distribution are seen. It is also clear from the figure that the solutal profile increases without and with solutal slip. Also, the solutal profile enhances with the escalating values of $Sc$. Figure 10 reveals that the impact of M on the solutal profile without and with solutal slip. Figure 10 shows that the solutal profile enhances with the increasing values of M in each case. Also, see the impact of $\delta$ on the solutal profile which is scrutinized that the solutal concentration enhances with the increment in $\delta$. Figure 11 explains the consequences of $Nt$ on the nano-particle volume fraction profile without and with concentration slip. The figure represents that the nano-particle volume fraction profile reduces with the increasing values of $Nt$ in each case. Also, it observes the effects of $Ln$ on the nano-particle volume fraction profile. It is clear that the nano-particle volume fraction profile reduces with increment in $Ln$ values. Figure 12 reveals that the impact of M on the nano-particle profile without and with concentration slip. Figure 12 describes that the nano-particle profile enhances with the increasing values of M in each case. Also, see the impact of $\delta$ on the nano-particle profile which is scrutinized that the nano-particle volume fraction profile enhances with the increment in $\delta$.

Figure 13 shows the effect of M on g without and with hydro-dynamic slip. It is observed in both cases that the micro-rotation decreases as the value of M increases. Influence of K and $f_w$ without and with hydro-dynamic slip-on g describes in Figure 14. The micro-rotation profile increases as an increase in the value of K. Figure 15 depicts the impact of M on g without and with thermal slip. It is observed in both cases that the micro-rotation decreases as the value of M increases. A similar effect of $f_w$ is observed.

Figure 16 shows the result of magnetic, unsteadiness, slip velocity on the skin friction coefficient. We see in this figure that the skin friction coefficient decreases along with increasing values of slip velocity $s_f$ and unsteady parameter $\delta$. However, it increases along with the increasing values of slip velocity $s_f$ and buoyancy parameter ${\lambda }_2$. Figure 17 represents the consequences of R, $fw$, $Nt$, concentration, and thermal slips on local Nusselt number. It is perceived that the local Nusselt number reduces on increasing R, $fw$, and $Nt$ with and without slip effects. Figure 18 depicts the impacts of $Sr$M, and $Nt$ on Sherwood number. It is perceived that the mass transfer rate enhances as M and $Nt$ enhances without and with the concentration slip and suction/injection parameter.

Table 4. Numerical values for different physical constraints M, ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$, $\delta$, $L_n$, $-f^{″}(0)$, $-g^{\prime }(0)$, $-{\theta }^{\prime }(0)$, $-{\varphi }^{\prime }(0)$.

M	${\mathit{\lambda }}_1$	${\mathit{\lambda }}_2$	${\mathit{\lambda }}_3$	$\mathit{\delta }$	${\mathit{L}}_{\mathit{n}}$	$-{\mathit{f}}^{″}(0)$	$-{\mathit{g}}^{\prime }(0)$	$-{\mathit{\theta }}^{\prime }(0)$	$-{\mathit{\varphi }}^{\prime }(0)$
0.5						0.945865	0.183972	0.708908	1.705238
1.0	0.1	0.1	0.1	0.2	5.0	1.033896	0.190324	0.694612	1.691997
3.0						1.271767	0.199565	0.659158	1.655360
0.5	0.1					0.945865	0.183972	0.708908	1.705238
	0.5	0.1	0.1	0.2	5.0	0.848947	0.174640	0.725500	1.719654
	1.0					0.737892	0.165374	0.742820	1.734753
0.5	0.1	0.1				0.945865	0.183972	0.708908	1.705238
		0.5	0.1	0.2	5.0	0.908433	0.180855	0.715054	1.710706
		1.0				0.863521	0.177375	0.722139	1.717023
0.5	0.1	0.1	0.1	0.2	5.0	0.945865	0.183972	0.708908	1.705238
			0.5			0.913618	0.182366	0.713483	1.709592
			1.0			0.874410	0.180529	0.718900	1.714746
0.5	0.1	0.1	0.1	0.2	5.0	0.945865	0.183972	0.708908	1.705238
				0.8		1.030296	0.145774	0.886881	1.852248
				1.0		1.054940	0.137761	0.931947	1.889387
0.5	0.1	0.1	0.1	0.2	5.0	0.945865	0.183972	0.708908	1.705238
					10.0	0.950153	0.184260	0.710358	2.175390
					13.0	0.951190	0.184309	0.711291	2.338004

illustrates the variation of physical parameters M, ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$, $\delta$, $L_n$ with ($Pr=1$, $R=0.5$, $Sc=8$, $Sr=0.4$, $S_f=S_{\gamma }=S_{\theta }=S_{\varphi }=0.3$, $f_w=B=0.5$, $Nb=Nt=Nd=K=0.1$, $A=Q=0.2$ on the local skin friction $-f^{″}(0)$, couple stress $-g^{\prime }(0)$, heat transfer coefficient $-{\theta }^{\prime }(0)$ and Sherwood number coefficient $-{\varphi }^{\prime }(0)$.

The following results are obtained from Table 4.

(i) Local skin friction $-f^{″}(0)$ and couple stress $-g^{\prime }(0)$ are increasing, while $-{\theta }^{\prime }(0)$ and $-{\varphi }^{\prime }(0)$ are decreasing at the increasing values of M. (ii) The increment in the values of buoyancy parameters ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$, causes the declining in values of $-f^{″}(0)$ and $-g^{\prime }(0)$ while the values of $-{\theta }^{\prime }(0)$ and $-{\varphi }^{\prime }(0)$ are increasing. (iii) For increasing values of $\delta$ the values of $-f^{″}(0)$, $-{\theta }^{\prime }(0)$ and $-{\varphi }^{\prime }(0)$ are increasing and the value of $-g^{\prime }(0)$ is decreasing. (iv) The values of $-f^{″}(0)$, $-g^{\prime }(0)$, $-{\theta }^{\prime }(0)$ and $-{\varphi }^{\prime }(0)$ increasing at the increment in values of $L_n$.

5. Conclusions

The present study investigates an unsteady magnetohydrodynamic micropolar nano-fluid flow and the heat transfer over a permeable stretching sheet with radiation in the presence of heat source. The study was conducted to investigate the multi-slip effects on magnetohydrodynamic micropolar nano-fluid flow over a stretching sheet. The mathematical model system of equations has been solved by using the Runge–Kutta method with shooting technique. The computations have been performed for velocity, temperature, micro-rotation, and nano-fluid volume fraction functions for the different values of physical parameters. For the given problem, the existing results in Table 1 show the accuracy of up to four decimal places. Moreover, the comparison of the existing results of the given model with previous studies has been discussed. A parametric study has been made to explore the effects of various parameters on the velocity, temperature, micro-rotation, concentration profiles, the skin friction coefficient, Sherwood number, and Nusselt number. Furthermore, the local skin friction, couple stress, Nusselt number, and Sherwood number are inspected graphically. The main findings from the present work are stated as,

• The fluid velocity, temperature solutal and nano-particle profile are seen to increase with an increment in unsteadiness parameter.
• The fluid velocity and micro-rotation declines while temperature shows opposite behavior with the enhancement in magnetic parameter, suction, hydro-dynamic, and thermal slips.
• The skin-friction coefficient decline with the increment of slip parameters, magnetic and unsteadiness parameter but shows the opposite effect for increasing values of hydrodynamic slip and thermal buoyancy.
• The reduced Nusselt number decreases with the enhancement in suction, radiation, thermophoresis parameter, thermal, and solutal slips.
• The Sherwood number increases with an increase in magnetic parameter, suction parameter, and hydro-dynamic slip.
• The fluid velocity and micro-rotation increase with the increment in K, $f_w$, and $S_f$.
• Temperature and solutal concentration increase with the increment in thermophoresis parameter, Schmidt number, Brownian motion parameter, Soret parameter, and thermal slip while the nano-particle concentration declines as the values of thermophoresis parameter, lewis number, and nano-particle slip increase.