Darcy–Forchheimer flow of micropolar nanofluid between two plates in the rotating frame with non-uniform heat generation/absorption

Abstract

This article studies the Darcy–Forchheimer flow of three-dimensional micropolar nanofluid between parallel and horizontal plates in a rotating system. The micropolar nanofluid in permeable media is described by assuming the Darcy–Forchheimer model, where drenching the permeable space obeys the Darcy–Forchheimer expression. The significant influence of Brownian motion and thermophoresis has been taken in the nanofluids model. The thermal radiation impact is taken to be varying in terms of non-uniform absorption/generation for the purpose to see the concentration as well as the temperature modifications between the nanofluid and the surfaces. The leading equations are converted into a system of differential nonlinear equations and then homotopic method is used for solving the modeled equations. The other physical impacts, that is, skin friction, heat flux, and mass flux, have been studied through tables. The impacts of the porosity, rotation, and inertia coefficient analysis have been mainly focused in this research. It is observed that the higher value of Fr decay the velocity profile, while it increases the transverse velocity, and the increase in the porosity parameter $\gamma$ increases the porous space, which creates resistance in the flow path and reduces the flow motion. Skin friction coefficient is observed to be larger for the strong concentration $k=0$ , as compared to the case of weak concentration $k=0.5$ . Impact of strong and weak concentrations on Nusselt and Sherwood numbers seems to be similar in a quantitative sense.

Introduction

The fluids with microstructure are called micropolar fluids. Micropolar fluids belong to a class of fluids with non-symmetric stress tensor, named polar fluids. The theory of micropolar fluid has been a field of very active research because it takes into consideration the microscopic influences arising from the local structure and micromotions of the fluid elements. The theory is expected to provide a mathematical model, which can be utilized to describe the behavior of non-Newtonian fluids such as liquid crystals, polymeric fluids, paints, animal blood, ferroliquids, and colloidal fluids. Eringen^1,2 was the pioneer of the basic idea regarding micropolar fluid. Also, he has found applications in physiological and engineering problems. Keeping in view the depth and wide range of uses, Lukaszewicz ³ has briefly studied the micropolar fluids. Generally, these fluids indicate fluids covering unpredictably oriented liquids suspended in a viscid medium. A micropolar fluid by the influence of magnetic induction on a flat plate was primarily deliberated by Mohammeadein and Gorla. ⁴ Kasivishwanathan and Gandhi. ⁵ have examined magnetohydrodynamic (MHD) micropolar fluid with exact solutions. Agarwal and Dhanapal ⁶ have inspected micropolar fluid with convective conditions in parallel absorbent vertical plates. Bhargava et al. ⁷ have scrutinized varied convective micropolar fluid. The stagnation point micropolar fluid with stretched sheet has been examined in Nazar and colleagues.^8,9 The other relevant studies with micropolar fluid can be seen in the literature.^10–13 Recently, Shah and colleagues^14,15 have studied micropolar nanofluid flow with Hall effect in rotating parallel plates with impacts of thermal radiation.

Many environmental and industrial systems like system of geothermal energy and system of heat exchanger design include the convection flow subject to permeable medium. The adapted form of classical Darcian model is the non-Darcian porous medium, which contains the inertia and boundary topographies. The standard Darcy’s law is effective under constrained range of small permeability and little velocity. Forchheimer ¹⁶ has predicted the inertia and boundary features by including a square velocity term to the countenance of Darcian velocity. Muskat ¹⁷ has entitled this term as “Forchheimer term,” which is permanently operative for large Reynolds number. Flow through a porous medium has numerous practical applications and usage especially in geophysical fluid dynamics. Some common examples of natural porous media are sandstone, beach sand, the human lung, limestone, bile duct, and gall bladder with stones in small blood vessels. The detail of porous media with application can be seen in the literature.^18,19

Pal and Mondal ²⁰ have investigated the MHD flow of inconstant viscosity liquid in a permeable medium by employing the Darcy–Forchheimer theory. Hayat and colleagues^21,22 have explored the Darcy–Forchheimer flow considering Maxwell fluid with variable heated surface. The idea of liquid motion and heat transmission analysis with rotating situation shows a very leading role in the petrochemical industry, geophysics, meteorology, aeronautics, and oceanography. A Dawar et al. ²³ have investigated Eyring–Powell fluid flow over an unsteady oscillatory porous stretching surface with impact of thermal radiation and heat source/sink. Recently, Ishaq et al. ²⁴ have studied nanofluid thin film flow with thermal radiation effect on an unsteady porous stretching sheet with entropy generation.

With the advent of nanoscience, nanofluids have become a focus of attention in the study of fluid flow in the presence of nanoparticles. Nanofluids are the fluids that are arranged by scattering-nanometer-sized (10⁻⁹) substances such as nanoparticles, nanotubes, nanofibers, and droplets in fluids. Actually, nanofluids are nanoscale-shattered suspensions involving concise nanometer-sized materials. These are two period systems: the first one is solid phase and the second one is liquid phase. Nanofluids can be used to increase the thermal conductivity of the fluids and are more stable fluids having better writing, spreading, and dispersion properties on solid surfaces.^25,26 Nanoscience can be offered applicable working fluid to improve thermal behavior and convective heat transfer enhancement.^27–31 The most current investigational and theoretical research was given by Sheikholeslami^32–34 on nanofluids using dissimilar phenomena, with modern application, possessions, and properties with usages of diverse approaches. Sheikholeslami and colleagues^35,36 have shown importance of nanofluid in nanotechnology. Yadav and colleagues^37–48 have studied the onset of MHD nanofluid using different phenomena and further studied the numerical simulation, heat transfer enhancement, linear and nonlinear, stability and instability of nanofluid.

The study of nanofluid flow in rotating parallel plates have gigantic applications in aerospace technology, rotation of machinery, chemical industries and engineering, generating systems of thermal power, rotor–stator systems, medical apparatus, electronic and computer storing appliances, crystal growing phenomena, machines of air cleaning, food processing technologies, turbo machinery, and many others. Greenspan and Howard ⁴⁹ have studied the rotating flow behavior of viscous fluid filled in the axisymmetric closed container. Nazar et al. ⁵⁰ have derived the analytical results for unsteady flow problem of rotating fluid. The impact of rotational scheme on nanoliquid flow was studied by Mustafa et al., ⁵¹ and they have analyzed that the influence of rotational variable decreases the heat transfer coefficient. Recently, Shah and colleagues^52,53 have investigated rotating nanofluid flow between parallel plates. The recent investigation of nanofluid in rotating system can be studied in the literature.^54–56 Qayyum and colleagues^57–62 have studied nanofluid flow under the influence of chemical reaction and thermal radiation using different phenomena. Hayat and colleagues^63,64 investigated the nanofluid flow under the influence of nonlinear thermal radiation through a stretching sheet.

The purpose of this research is to examine the Darcy–Forchheimer flow of three-dimensional micropolar nanofluid between parallel and horizontal plates in a rotating system. Homotopy analysis method (HAM) is used in this work for solving the modeled equations which are nonlinear and coupled. The effect of all embedding parameters has been studied graphically.

Problem formulation

Consider three-dimensional thermally conducting micropolar nanofluid flows between two plates which are horizontal as well as parallel. The distance between them is taken as h. The coordinate system for the flow of micropolar Casson nanofluid is nominated that both plate and fluid rotate about the $y-\mathrm{axis}$ with angular velocity $\mathrm{\Omega }$ . Here, two forces are assumed having equal magnitude, but opposite in direction to stretch the lower plate along $x-\mathrm{axis}$ , which keep the origin constant. The upper plate is subject to a uniform wall suction/injection velocity ${\tilde{v}}_0$ . The micropolar nanofluid saturates the porous space characterizing Darcy–Forchheimer model. The hot base nanofluid is also retained at lower surface of the lower plate such that $T>T_h$ and $C>C_h$ , where $T\mathrm{and}C$ are temperature and volume friction of micropolar nanofluid at lower plate and $T_h\mathrm{and}{C}_h$ are temperature and volume friction upper plate, respectively. Moreover, the thermal radiation is taken to be non-uniform in absorption/generation. Using all the above discussed assumptions, the governing equations of micropolar nanofluid are reduces as^10–21

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

(1)

$$\begin{array}{l}u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+2w\mathrm{\Omega }=-\frac{1}{{\rho }_f}\frac{\partial \widehat{p}}{\partial x}\\ +\left(\nu +\frac{\stackrel{⌢}{\kappa }}{{\rho }_f}\right)\left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right)+\frac{\stackrel{⌢}{\kappa }}{{\rho }_f}\frac{\partial N}{\partial y}-\frac{\nu }{{\rho }_f{\mathrm{K}}^{\ast }}u-\frac{1}{{\rho }_f}F{u}^2\end{array}$$

(2)

$$u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}=-\frac{1}{{\rho }_f}\frac{\partial \hat{p}}{\partial y}+(\frac{{\partial }^{2}v}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2})+\frac{\stackrel{⌢}{\kappa }}{{\rho }_f}\frac{\partial N}{\partial x}$$

(3)

$$\begin{array}{c}u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}-2u\mathrm{\Omega }=\hfill \\ (\nu +\frac{\stackrel{⌢}{\kappa }}{{\rho }_f})(\frac{{\partial }^{2}w}{\partial x^2}+\frac{{\partial }^{2}w}{\partial y^2})-\frac{\nu }{{\rho }_f{\mathrm{K}}^{*}}w-\frac{1}{{\rho }_f}F{w}^2\hfill \end{array}$$

(4)

In equations (1)–(4), $\upsilon$ and $\mu$ characterize the kinematic and dynamic viscosities coefficient, respectively; ${\rho }_f$ represents the nanofluid density; N represents microrotation velocity; $\mathrm{\Omega }$ means angular velocity; $\stackrel{⌢}{\kappa }$ is vertex viscosity; ${\mathrm{K}}^{*}$ represents the permeability of medium; and F signifies the variable inertia coefficient of permeable medium where $F=C_b({\mathrm{K}}^{*}{)}^{-1/2}{x}^{-1}$ , $C_b$ is used for drag coefficients^10–15

$$\begin{array}{c}u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+w\frac{\partial T}{\partial z}=\hfill \\ {\alpha }^{*}(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}+\frac{{\partial }^{2}T}{\partial z^2})-\frac{1}{{\left(\rho c\right)}_p}\frac{\partial }{\partial y}{q}_{\mathrm{rd}}+\frac{1}{{\left(\rho c\right)}_p}(q″\prime )\hfill \\ +{\tau }^{*}[D_B\{\frac{\partial C}{\partial x}\frac{\partial T}{\partial x}+\frac{\partial C}{\partial y}\frac{\partial T}{\partial y}+\frac{\partial C}{\partial z}\frac{\partial T}{\partial z}\}+(D_T/T_c)\{{\left(\frac{\partial T}{\partial x}\right)}^2+{\left(\frac{\partial T}{\partial y}\right)}^2+{\left(\frac{\partial T}{\partial z}\right)}^2\}]\hfill \end{array}$$

(5)

$$\begin{array}{c}u\frac{\partial N}{\partial x}+v\frac{\partial N}{\partial y}=\hfill \\ -\frac{\gamma }{{\rho }_j}(\frac{{\partial }^{2}N}{\partial x^2}+\frac{{\partial }^{2}N}{\partial y^2})-\frac{2\stackrel{⌢}{\kappa }N}{{\rho }_j}+\frac{\stackrel{⌢}{\kappa }}{{\rho }_j}(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y})\hfill \end{array}$$

(6)

$$\begin{array}{c}u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}+w\frac{\partial C}{\partial z}=\hfill \\ \frac{D_T}{T_0}\{\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}+\frac{{\partial }^{2}T}{\partial z^2}\}+D_B(\frac{{\partial }^{2}C}{\partial x^2}+\frac{{\partial }^{2}C}{\partial y^2}+\frac{{\partial }^{2}C}{\partial z^2})\hfill \end{array}$$

(7)

In equation (5), T symbolizes temperature; ${\alpha }^{*}$ signifies thermal diffusivity; $D_B$ and $D_T$ are the coefficients of Brownian diffusion and the coefficient of thermophoretic diffusion, respectively; $c_p$ represents specific heat; C is fluid concentration of the fluids particles; ${\tau }^{*}$ is ratio between nanoparticles and heat capacity; $q_{\mathrm{rd}}$ indicates the radioactive heat flux, which is given by Rosseland approximation as $q_r=-(16\phi /3k^{*})(\partial T^4/\partial y)$ , where $\phi$ is known as Stefan–Boltzmann constant and $\stackrel{⌢}{k}$ denoted the coefficient of mean absorption. By Taylor series, expanding $T^4$ and sampling it, we get

$$\frac{\partial q_r}{\partial y}=-\frac{16T_C^3\phi }{3K}\frac{{\partial }^{2}T}{\partial y^2}$$

(8)

The term $q″\prime$ is the non-uniform heat source/sink and is defined as^55–58

$$q″\prime =\frac{\kappa \tilde{u}}{\mathrm{xv}}(\mathrm{A}(T-T_h)f\prime +(T-T_h)\mathrm{B})$$

(9)

Here, A and B represent the coefficients of temperature- and space-dependent heat source and sink term, respectively. When $\mathrm{A}>0\&\mathrm{B}>0$ , links to inner heat source, and when $\mathrm{A}<0\&\mathrm{B}<0$ , links the inner heat sink. The subjected boundary conditions are

$$\begin{array}{c}u=\alpha x,v=0,w=0,T=T_h,\hfill \\ N=-k\frac{\partial u}{\partial y},C=C_h\mathrm{at}y=0\hfill \\ u=v=0,w=0,T=T_0,\hfill \\ N=k\frac{\partial u}{\partial y},C=C_0\mathrm{at}y=h\hfill \end{array}$$

(10)

Here, k specifies the boundary parameter. The non-dimensional variables are given as

$$\begin{array}{c}u=\alpha \mathrm{xf}\prime (\eta ),v=-\alpha \mathrm{hf}(\eta ),w=\alpha \mathrm{xg}(\eta ),\hfill \\ N=-\frac{\alpha \mathrm{xG}(\eta )}{h},\theta (\eta )=\frac{T-T_h}{T_0-T_h}\hfill \\ \varphi (\eta )=\frac{C-C_h}{C_0-C_h},\mathrm{where}\eta =\frac{y}{h}\hfill \end{array}$$

(11)

Substituting equation (11) into equations (1)–(7) and (10), equation on satisfied identically and the other equations are converted to the following form

$$\begin{array}{c}(1+N_1)f^{\mathrm{iv}}-\mathrm{Re}(f\prime f″-\mathrm{ff}″\prime )-2\mathrm{Krg}\prime \hfill \\ +N_{1}G″-\gamma f\prime -2\mathrm{Frf}\prime f″=0\hfill \end{array}$$

(12)

$$\begin{array}{c}(1+N_1)g″+\mathrm{Re}(\mathrm{fg}\prime -\mathrm{gf}\prime )+2\mathrm{Krf}\prime \hfill \\ -\gamma g-\mathrm{Fr}{\left(g\right)}^2=0\hfill \end{array}$$

(13)

$$\begin{array}{c}(1+\frac{4}{3}\mathrm{Rd})\theta ″+\mathrm{Pr}(\mathrm{Ref}\theta \prime +\mathrm{N}b\varphi \prime \theta \prime +\mathrm{Nt}{(\theta \prime )}^2)\hfill \\ -({\mathrm{A}}_{1}f\prime +{\mathrm{B}}_1\theta )=0\hfill \end{array}$$

(14)

$$N_{2}G″-N_1(2G+f″)-N_3\mathrm{Re}(\mathrm{Gf}\prime -G\prime f)=0$$

(15)

$$\varphi ″+\mathrm{Re}\mathrm{Scf}\varphi \prime +\frac{\mathrm{Nt}}{\mathrm{Nb}}\theta ″=0$$

(16)

$$\begin{array}{c}f(0)=0,f\prime (0)=1,g(0)=0,\theta (0)=1,\hfill \\ G(0)=-\mathrm{kf}″(0),\varphi (0)=1\hfill \\ f(1)=\lambda ,f\prime (1)=0,g(1)=0,\theta (1)=0,\hfill \\ G(1)=\mathrm{kf}″(0),\varphi (1)=0\hfill \end{array}$$

(17)

where $\mathrm{Kr}=2\mathrm{\Omega }{h}^2/\nu$ represents rotation parameter, $\gamma =h^2\nu /a{\mathrm{K}}^{*}$ is porosity parameter, $\mathrm{Fr}=h^2{C}_b({\mathrm{K}}^{*}{)}^{-\frac{1}{2}}/\rho$ is the coefficient of inertia, $N_1=\stackrel{⌢}{\kappa }/\mu$ denotes the coupling parameter, $\delta ={\nu }_o/h$ signifies transpiration parameter, $N_2={\nu }_s/\nu h^2$ signifies viscosity gradient parameter, $\mathrm{Re}=a{h}^2/\nu$ is the Reynolds number, $N_3=j/h^2$ denotes constant of micropolar fluid, $\mathrm{Pr}=\mu /{\rho }_f\alpha$ is the Prandtl number, $\mathrm{Rd}=4T_c^3\phi /((\rho c_p{)}_f\stackrel{⌢}{k}K)$ represents radiation parameter, $\mathrm{N}b=(\rho c{)}_p{D}_B{C}_h/{\alpha }^{*}(\rho c{)}_f$ signifies the parameter of the Brownian movement of the fluid molecules, $\mathrm{Sc}=\mu /D{\rho }_f$ is Schmidt number, and $(\rho c{)}_p{D}_T{T}_0/(\rho c{)}_f{T}_C=\mathrm{Nt}$ denotes thermophoretic parameter.

Physical quantities of interest

The physical quantities of interest such as skin friction, heat flux, and mass flux have abundant applications in the field of engineering. For micropolar nanofluid flow problem, skin friction is defined as

$$C_f=\frac{{\left({\tau }_{\mathrm{xy}}\right)}_{y=0}}{{\rho }_f{\tilde{u}}_w^2}$$

(18)

And here

$${\tau }_{\mathrm{xy}}=((\mu +\stackrel{⌢}{\kappa })\frac{\partial u}{\partial y}+\stackrel{⌢}{\kappa }N)$$

(19)

The Nusselt number is defined as $\mathrm{Nu}=h{Q}_w/\hat{k}(T_0-T_h)$ , where $Q_w$ is the heat flux and $Q_w=-\hat{k}(\partial T/\partial y{)}_{y=0}$ . The Sherwood number is defined as $\mathrm{Sh}=h{J}_w/D_B(C_0-C_h)$ , where $J_w$ is the mass flux and $J_w=-D_B(\partial C/\partial y{)}_{y=0}$ . The dimensionless form of $C_f$ , Nu, and Sh are obtained as

$$\begin{array}{c}{C}_f\sqrt{R{e}_x}=(1+N_1)f″(0)+N_{1}G(0),\hfill \\ \mathrm{Nu}=-(1+\frac{4}{3}\mathrm{Rd})\mathrm{\Theta }\prime (0),\mathrm{Sh}=-\mathrm{\Phi }\prime (0)\hfill \end{array}$$

(20)

where $R{e}_x$ is called local Reynolds number and defined as $R{e}_x={\tilde{u}}_{w}x/\nu$ .

Solution by HAM

HAM is one of the substitute methods and mostly used to solve the nonlinear differential equations without discretization and linearization. Liao^65,66 is the pioneer to derive it using the basic concept of topology called homotopy, for the derivation of this HAM. For this purpose, he used two homotopic functions.

For the basic idea, let ${\Psi }_1$ and ${\Psi }_2$ are two functions which are continuous, and $\overline{X}$ and $\overline{Y}$ are two topological spaces where ${\Psi }_1$ and ${\Psi }_2$ map from $\overline{\mathrm{X}}$ to $\overline{Y}$ then ${\Psi }_1$ is said to be homotopic to ${\Psi }_2$ if there produce a continuous function $\psi$

$$\psi :\overline{\mathrm{X}}\times [0,1]\to \overline{Y}$$

(21)

Such that $\forall ,\overline{x}\in \overline{X}$

$$\xi [\overline{x},0]={{\Psi }_1}_1\left(\overline{x}\right)\mathrm{and}\psi [\overline{x},1]={\xi }_2\left(\overline{x}\right)$$

(22)

Then, this mapping $\psi$ is called homotopic. This technique has several advantages, some of them are as follows: (1) it is free from the values of the parameters which may be small or large. (2) It guarantees the convergence of the solution. (3) It is self-determining for an assortment of base function and linear operator. The solutions of equations (12)–(16) with the dependable boundary conditions equation (17) are obtained by HAM. The initial guesses for equations (12)–(16) are calculated as

$$\begin{array}{c}{\hat{f}}_0(\eta )=\eta -2{\eta }^2+{\eta }^3+3\delta {\eta }^2-2\delta {\eta }^3\hfill \\ {\hat{\mathrm{g}}}_0(\eta )=0\hfill \\ {\hat{\mathrm{\Theta }}}_0(\eta )=1-\eta \hfill \\ {\hat{G}}_0(\eta )=4k-8k\eta -6k\delta +12k\delta \eta \hfill \\ {\hat{\mathrm{\Phi }}}_0(\eta )=1-\eta \hfill \end{array}$$

(23)

where the linear operator for the equations

$$\begin{array}{c}{L}_f(f)=\frac{{\partial }^{4}f}{\partial {\eta }^4},L_g(g)=\frac{{\partial }^{2}g}{\partial {\eta }^2},L_{\mathrm{\Theta }}(\mathrm{\Theta })=\frac{{\partial }^2\mathrm{\Theta }}{\partial {\eta }^2},\hfill \\ L_G(G)=\frac{{\partial }^{2}G}{\partial {\eta }^2},L_{\mathrm{\Phi }}(\mathrm{\Phi })=\frac{{\partial }^2\mathrm{\Phi }}{\partial {\eta }^2}\hfill \end{array}$$

(24)

The initial solution form is

$$\begin{array}{c}{L}_f({\kappa }_1+{\kappa }_2\eta +{\kappa }_3{\eta }^2+{\kappa }_4{\eta }^3)=0\hfill \\ L_g({\kappa }_5+{\kappa }_6\eta )=0\hfill \\ L_{\theta }({\kappa }_7+{\kappa }_8\eta )=0\hfill \\ L_G({\kappa }_9+{\kappa }_{10}\eta )=0\hfill \\ L_{\varphi }({\kappa }_{11}+{\kappa }_{12}\eta )=0\hfill \end{array}$$

(25)

Here, ${\displaystyle \sum _{m=1}^{12}{\kappa }_m}$ , where $m=1,2,3,\dots$ are arbitrary constants. Zeroth-order deformation to the following form with boundary conditions is

$$(1-\tau )L_f[\stackrel{⌢}{f}(\eta ,\tau )-f_0(\eta )]=\tau {\hslash }_f{\mathrm{N}}_f[\stackrel{⌢}{f}(\eta ,\tau ),\stackrel{⌢}{g}(\eta ,\tau ),\stackrel{⌢}{G}(\eta ,\tau )]$$

(26)

$$(1-\tau )L_g[\stackrel{⌢}{g}(\eta ,\tau )-g_0(\eta )]=\tau {\hslash }_g{\mathrm{N}}_g[\stackrel{⌢}{f}(\eta ,\tau ),\stackrel{⌢}{g}(\eta ,\tau )]$$

(27)

$$(1-\tau )L_{\mathrm{\Theta }}[\stackrel{⌢}{\mathrm{\Theta }}(\eta ,\tau )-{\mathrm{\Theta }}_0(\eta )]=\tau {\hslash }_{\mathrm{\Theta }}{\mathrm{N}}_{\mathrm{\Theta }}[\stackrel{⌢}{\mathrm{\Theta }}(\eta ,\tau ),\stackrel{⌢}{\mathrm{\Phi }}(\eta ,\tau )]$$

(28)

$$(1-\tau )L_G[\stackrel{⌢}{G}(\eta ,\tau )-G_0(\eta )]=\tau {\hslash }_G{\mathrm{N}}_G[\stackrel{⌢}{f}(\eta ,\tau ),\stackrel{⌢}{G}(\eta ,\tau )]$$

(29)

$$(1-\tau )L_{\mathrm{\Phi }}[\stackrel{⌢}{\mathrm{\Phi }}(\eta ,\tau )-{\mathrm{\Phi }}_0(\eta )]=\tau {\hslash }_{\mathrm{\Phi }}{\mathrm{N}}_{\mathrm{\Phi }}[\stackrel{⌢}{\mathrm{\Phi }}(\eta ,\tau ),\stackrel{⌢}{\mathrm{\Theta }}(\eta ,\tau )]$$

(30)

$$\begin{array}{c}\stackrel{⌢}{f}(0,\tau )=0,\stackrel{⌢}{f}\prime (0,\tau )=0,\stackrel{⌢}{g}(0,\tau )=0,\stackrel{⌢}{\mathrm{\Theta }}(0,\tau )=0\hfill \\ \stackrel{⌢}{\mathrm{\Phi }}(0,\tau )=0,\stackrel{⌢}{G}(0;\tau )=-k\stackrel{⌢}{f}″(0,\tau ),\stackrel{⌢}{f}(1,\tau )=0,\stackrel{⌢}{f}\prime (1,\tau )=0\hfill \\ \stackrel{⌢}{g}(1,\tau )=0,\stackrel{⌢}{\mathrm{\Theta }}(1,\tau )=0,\stackrel{⌢}{\mathrm{\Phi }}(1,\tau )=0,\stackrel{⌢}{G}(0;\tau )=k\stackrel{⌢}{f}″(1,\tau )\hfill \end{array}$$

(31)

The substantial nonlinear operators $N_f,N_g,N_{\mathrm{\Theta }},N_{\mathrm{\Phi }}$ , and $N_G$ are planned as

$$\begin{array}{c}{N}_f[\stackrel{⌢}{f}(\eta ;\tau ),\stackrel{⌢}{g}(\eta ;\tau ),\stackrel{⌢}{G}(\eta ;\tau )]\hfill \\ =(1+N_1){\stackrel{⌢}{f}}_{\eta \eta \eta \eta }(\eta ;\tau )\hfill \\ -\mathrm{Re}({\stackrel{⌢}{f}}_{\eta \eta }(\eta ;\tau ){\stackrel{⌢}{f}}_{\eta }(\eta ;\tau )-{\stackrel{⌢}{f}}_{\eta \eta \eta }(\eta ;\tau )\stackrel{⌢}{f}(\eta ;\tau ))\hfill \\ -2\mathrm{Kr}{\stackrel{⌢}{g}}_{\eta }(\eta ;\tau )+N_1{\stackrel{⌢}{G}}_{\eta \eta }(\eta ;\tau )-\gamma {\stackrel{⌢}{f}}_{\eta }(\eta ;\tau )\hfill \\ -2\mathrm{Fr}{\stackrel{⌢}{f}}_{\eta \eta }(\eta ;\tau ){\stackrel{⌢}{f}}_{\eta }(\eta ;\tau )\hfill \end{array}$$

(32)

$$\begin{array}{c}{N}_g[\stackrel{⌢}{f}(\eta ;\tau ),\stackrel{⌢}{g}(\eta ;\tau )]\hfill \\ =(1+N_1){\stackrel{⌢}{g}}_{\eta \eta }(\eta ;\tau )\hfill \\ +\mathrm{Re}(\stackrel{⌢}{f}(\eta ;\tau ){\stackrel{⌢}{g}}_{\eta }(\eta ;\tau )-\stackrel{⌢}{g}(\eta ;\tau ){\stackrel{⌢}{f}}_{\eta }(\eta ;\tau ))\hfill \\ +2\mathrm{Kr}{\stackrel{⌢}{f}}_{\eta }(\eta ;\tau )-\gamma \stackrel{⌢}{g}(\eta ;\tau )-\mathrm{Fr}(\stackrel{⌢}{g}(\eta ;\tau )\stackrel{⌢}{g}(\eta ;\tau ))\hfill \end{array}$$

(33)

$$\begin{array}{c}{N}_{\mathrm{\Theta }}[\mathrm{\Theta }(\eta ;\tau ),\mathrm{\Phi }(\eta ;\tau )]\hfill \\ =(1+\frac{4}{3}\mathrm{Rd}){\mathrm{\Theta }}_{\eta \eta }(\eta ;\tau )+\mathrm{Pr}(\mathrm{Re}\stackrel{⌢}{f}(\eta ;\tau ){\mathrm{\Theta }}_{\eta }(\eta ;\tau ))\hfill \\ +\mathrm{Pr}(\mathrm{Nb}{\mathrm{\Theta }}_{\eta \eta }(\eta ;\tau ){\mathrm{\Phi }}_{\eta }(\eta ;\tau )+\mathrm{Nt}{({\mathrm{\Theta }}_{\eta \eta }(\eta ;\tau ))}^2-({\mathrm{A}}_1{f}_{\eta }(\eta ;\tau )+{\mathrm{B}}_1\mathrm{\Theta }(\eta ;\tau )))\hfill \end{array}$$

(34)

$$\begin{array}{c}{N}_G[\stackrel{⌢}{f}(\eta ;\tau ),\stackrel{⌢}{G}(\eta ;\tau )]\hfill \\ =N_2{\stackrel{⌢}{G}}_{\eta \eta }(\eta ;\tau )-N_1(2\stackrel{⌢}{G}(\eta ;\tau )+{\stackrel{⌢}{f}}_{\eta \eta }(\eta ;\tau ))\hfill \\ -N_3\mathrm{Re}({\stackrel{⌢}{f}}_{\eta }(\eta ;\tau )\stackrel{⌢}{G}(\eta ;\tau )-{\stackrel{⌢}{G}}_{\eta }(\eta ;\tau )\stackrel{⌢}{f}(\eta ;\tau ))\hfill \end{array}$$

(35)

$$\begin{array}{c}{N}_{\mathrm{\Phi }}[\stackrel{⌢}{\mathrm{\Phi }}(\eta ;\tau ;\stackrel{⌢}{\mathrm{\Theta }}(\eta ;\tau );\stackrel{⌢}{f}(\eta ;\tau ),)]\hfill \\ ={\mathrm{\Phi }}_{\eta \eta }(\eta ;\tau )+\mathrm{Re}\mathrm{Sc}{\mathrm{\Phi }}_{\eta }(\eta ;\tau )\stackrel{⌢}{f}(\eta ;\tau )+\frac{\mathrm{Nt}}{\mathrm{Nb}}{\mathrm{\Theta }}_{\eta \eta }(\eta ;\tau )\hfill \end{array}$$

(36)

when $\tau$ changes from $0\mathrm{to}1,$ then expanding with boundary conditions

$$\begin{array}{c}\stackrel{⌢}{f}(\eta ;0)=\stackrel{⌢}{\mathrm{\Theta }}(\eta ;1),\stackrel{⌢}{f}(\eta ;1)=f(\eta ),\stackrel{⌢}{g}(\eta ;0)=g_0(\eta ),\hfill \\ \stackrel{⌢}{g}(\eta ;1)=g(\eta )\stackrel{⌢}{\mathrm{\Theta }}(\eta ;0)={\mathrm{\Theta }}_0(\eta ),\stackrel{⌢}{\mathrm{\Theta }}(\eta ;1)=\mathrm{\Theta }(\eta ),\hfill \\ \stackrel{⌢}{G}(\eta ;0)=G_0(\eta ),\stackrel{⌢}{G}(\eta ;1)=G(\eta )\hfill \\ \stackrel{⌢}{\mathrm{\Phi }}(\eta ;0)={\mathrm{\Phi }}_0(\eta ),\stackrel{⌢}{\mathrm{\Phi }}(\eta ;1)=\mathrm{\Phi }(\eta )\hfill \end{array}$$

(37)

$$\begin{array}{c}{f}_i(\eta )={\frac{1}{i!}{\hat{f}}_{\eta }(\eta ,\tau )|}_{\tau =0},g_i(\eta )={\frac{1}{i!}{\hat{g}}_{\eta }(\eta ,\tau )|}_{\tau =0}\hfill \\ {\mathrm{\Theta }}_i(\eta )={\frac{1}{i!}{\hat{\mathrm{\Theta }}}_{\eta }(\eta ,\tau )|}_{\tau =0},G_i(\eta )={\frac{1}{i!}\frac{\partial G(\eta ;\tau )}{\partial \eta }|}_{\tau =0}\hfill \\ {\mathrm{\Phi }}_i(\eta )={\frac{1}{i!}{\hat{\mathrm{\Phi }}}_{\eta }(\eta ,\tau )|}_{\tau =0}\hfill \end{array}$$

(38)

At $\tau =1$ , we obtain

$$\begin{array}{c}f(\eta )=f_0(\eta )+\sum _{i=1}^{\infty }{f}_i(\eta ),g(\eta )=g_0(\eta )+\sum _{i=1}^{\infty }{g}_i(\eta )\hfill \\ \mathrm{\Theta }(\eta )={\mathrm{\Theta }}_0(\eta )+\sum _{i=1}^{\infty }{\mathrm{\Theta }}_i(\eta ),G(\eta )=G_0(\eta )+\sum _{i=1}^{\infty }{G}_i(\eta )\hfill \\ \mathrm{\Phi }(\eta )={\mathrm{\Phi }}_0(\eta )+\sum _{i=1}^{\infty }{\mathrm{\Phi }}_i(\eta )\hfill \end{array}$$

(39)

Differentiating zeroth-order equations, the ith-order deformation equations with boundary conditions can be reduced as

$$\begin{array}{c}{L}_f\{f_i(\eta )-{\xi }_i{f}_{i-1}(\eta )\}={\hslash }_f{\Re }_i^f(\eta ),\hfill \\ L_g\{g_i(\eta )-{\xi }_i{g}_{i-1}(\eta )\}={\hslash }_g{\Re }_i^g(\eta )\hfill \\ L_{\mathrm{\Theta }}\{{\mathrm{\Theta }}_i(\eta )-{\xi }_i{\mathrm{\Theta }}_{i-1}(\eta )\}={\hslash }_{\mathrm{\Theta }}{\Re }_i^{\mathrm{\Theta }}(\eta ),\hfill \\ L_G\{G_i(\eta )-{\chi }_i{G}_{i-1}(\eta )\}={\hslash }_G{R}_i^G(\eta )\hfill \\ L_{\mathrm{\Phi }}\{{\mathrm{\Phi }}_i(\eta )-{\xi }_i{\mathrm{\Phi }}_{i-1}(\eta )\}={\hslash }_{\mathrm{\Phi }}{\Re }_i^{\mathrm{\Phi }}(\eta )\hfill \end{array}$$

(40)

$$\begin{array}{c}{f}_i={f\prime }_i=g_i={\mathrm{\Theta }}_i={\mathrm{\Phi }}_i=0,G_i=-k{f″}_i\mathrm{at}\eta =0\hfill \\ {\hat{f}}_i=\hat{f}\prime ={\hat{g}}_i={\hat{\mathrm{\Theta }}}_i={\hat{\mathrm{\Phi }}}_i=0,G_i=k{f″}_i\mathrm{at}\eta =1\hfill \end{array}$$

(41)

$$\begin{array}{c}{\Re }_i^f(\eta )=\hfill \\ (1+N_1)f_{i-1}^{\mathrm{iv}}-\mathrm{Re}(\sum _{j=0}^{i-1}{f\prime }_{i-1-j}{f″}_k-\sum _{j=0}^{i-1}{f}_{i-1-j}{f″\prime }_k)\hfill \\ -2\mathrm{Kr}{g\prime }_{i-1}+N_1{g″}_{i-1}-\gamma f\prime -2\mathrm{Fr}\sum _{j=0}^{i-1}{f\prime }_{i-1-j}f″\hfill \end{array}$$

(42)

$$\begin{array}{c}{\Re }_i^g(\eta )=\hfill \\ (1+N_1){g″}_{i-1}+\mathrm{Re}(\sum _{j=0}^{i-1}{f}_{i-1-j}{g\prime }_j-\sum _{k=0}^{i-1}{f\prime }_{i-1-j}{g}_j)\hfill \\ +2\mathrm{Kr}{f\prime }_{i-1}-\gamma g_{i-1}-\mathrm{Fr}\left(g_{i-1}{g}_{i-1}\right)\hfill \end{array}$$

(43)

$$\begin{array}{c}{\Re }_i^{\mathrm{\Theta }}(\eta )=(1+\frac{4}{3}\mathrm{Rd}){\mathrm{\Theta }″}_{i-1}\hfill \\ +\mathrm{Pr}(\mathrm{Re}\sum _{j=0}^{i-1}{f}_{i-1-j}{\mathrm{\Theta }\prime }_j+\mathrm{Nb}\sum _{j=0}^{i-1}{\mathrm{\Phi }\prime }_{i-1-j}{\mathrm{\Theta }\prime }_j+\mathrm{Nt}\sum _{j=0}^{i-1}{\mathrm{\Theta }\prime }_{i-1-j}{\mathrm{\Theta }\prime }_j)\hfill \\ -({\mathrm{A}}_1{f}_{i-1}+{\mathrm{B}}_1{\mathrm{\Theta }}_j)\hfill \end{array}$$

(44)

$$\begin{array}{c}{\Re }_i^G(\eta )=N_2{G″}_{i-1}-N_1(G_{i-1}+{f″}_{i-1})\hfill \\ -N_3\mathrm{Re}(\sum _{j=0}^{i-1}{f}_{i-1-j}{G\prime }_j-\sum _{j=0}^{i-1}{f\prime }_{i-1-j}{G}_j)\hfill \end{array}$$

(45)

$${\Re }_i^{\mathrm{\Phi }}(\eta )={\mathrm{\Phi }″}_{i-1}+\mathrm{Re}\mathrm{Sc}\sum _{j=0}^{i-1}{f}_{i-1-j}\mathrm{\Phi }{\prime }_j+\frac{\mathrm{Nt}}{\mathrm{Nb}}{\mathrm{\Theta }″}_{i-1}$$

(46)

where

$${\tau }_i=\{\begin{array}{cc}1,\hfill & i>1\hfill \\ \multicolumn{1}{c}{0,}& i\le 1\hfill \end{array}$$

(47)

The overall homotopic solutions in general form are specified as

$$\begin{array}{c}{f}_i\left(\eta \right)={\hat{f}}_i\left(\eta \right)+{\kappa }_1+{\kappa }_2\eta +{\kappa }_3{\eta }^2+{\kappa }_4{\eta }^3\hfill \\ g_i\left(\eta \right)={\hat{g}}_i\left(\eta \right)+{\kappa }_5+{\kappa }_6\eta \hfill \\ {\mathrm{\Theta }}_i\left(\eta \right)={\hat{\mathrm{\Theta }}}_i\left(\eta \right)+{\kappa }_7+{\kappa }_8\eta \hfill \\ G_i\left(\eta \right)={\hat{G}}_i\left(\eta \right)+{\kappa }_9+{\kappa }_{10}\eta \hfill \\ {\mathrm{\Phi }}_i\left(\eta \right)={\hat{\mathrm{\Phi }}}_i\left(\eta \right)+{\kappa }_{11}+{\kappa }_{12}\eta \hfill \end{array}$$

(48)

Here, ${\hat{f}}_i(\eta ),{\hat{g}}_i(\eta ),{\hat{\mathrm{\Theta }}}_i(\eta ),{\hat{G}}_i(\eta )\mathrm{and}{\hat{\mathrm{\Phi }}}_i(\eta )$ are the particular solutions of the equations.

Analysis

Here, our interest is to analyze the analytical solution of obtaining system of ordinary differential equations by HAM. When the solutions of equations (12)–(16) with the dependable boundary conditions (equation (17)) are computed in series form using HAM, the embedding parameters ${\hslash }_f,{\hslash }_g,{\hslash }_{\mathrm{\Theta }},{\hslash }_G$ , and ${\hslash }_{\mathrm{\Phi }}$ appeared, which are responsible for adjusting of convergence of solutions. The convergence of HAM solutions for the model equations at different approximation using dissimilar values of parameters is shown in Table 1, showing that homotopy analysis technique is a fast convergent technique. The velocity profiles $f\prime (\eta )$ and $g(\eta )$ , microrotation velocity profiles $G(\eta )$ , temperature profiles $\mathrm{\Theta }(\eta )$ , and concentration profile $\mathrm{\Phi }(\eta )$ are concentrated by dissimilar micropolar nanofluid flow parameters. Attention is explicitly given to upshot coefficient of inertia porosity parameter Fr, porosity parameter $\gamma$ , rotation parameter Kr, coupling parameter $N_1$ , Reynolds number Re, viscosity gradient parameter $N_2$ , coefficients of temperature source/sink term $A_1$ , space-dependent heat source/sink term $B_1$ , Brownian motion parameter Nb, thermophoretic parameter Nt, Prandtl number Pr, radiation parameter Rd , and Schmidt number $\mathrm{Sc}.$

Table 1.

The convergence of HAM up to 15th-order approximations when $\mathrm{Rd}=A_1=B_1=0.5,\lambda =\mathrm{Nt}=\mathrm{Nb}=\mathrm{Fr}=\gamma =0.1,\mathrm{Kr}=\mathrm{Sc}=\mathrm{Pr}=\mathrm{Re}=1$ .

Order of approximation	$f″(0)$	$g\prime (0)$	$\theta \prime (0)$	$G\prime (0)$	$\varphi \prime (0)$
1	3.41575	−0.01393	−0.99782	−0.68008	−1.00331
3	3.42436	−0.02152	−0.99687	−0.68023	−1.00783
5	3.42570	−0.02268	−0.99722	−0.68036	−1.00961
7	3.42598	−0.02286	−0.99717	−0.68048	1.01013
9	3.42598	−0.02288	−0.99314	−0.68059	−1.01032
11	3.42598	−0.02289	−0.99314	−0.68053	−1.01032
13	3.42598	−0.02289	−0.99314	−0.68053	−1.01032
15	3.42598	−0.02289	−0.99314	−0.68053	−1.01032

Results and discussion

In this segment, we discuss the physical assets of different embedding parameters of the modeled problems and their influence on velocity, temperature, microrotation, and concentration profile which are showed in Figures 2–25. The graphical view of problem is presented in Figure 1.

Figure 1.

Sketch of the micropolar nanofluid between parallel plates.

Velocity profile $f\prime (\eta )$ and $g(\eta )$

The impact of physical parameters Fr, $\gamma$ , Kr, $N_1$ and Re on velocity profiles $f\prime (\eta )$ and $g(\eta )$ are presented in Figures 2–11. The effect of Fr on $f\prime (\eta )$ and $g(\eta )$ are shown in Figures 2 and 3. It is observed that the greater value of Fr decay the velocity profile $f\prime (\eta )$ . Actually, the higher value of inertial coefficient produces resistance in the flow path, which in turn, eases the nanofluid motion. In case of z-direction, the result is found opposite, that is, the large value of inertial coefficient increases the velocity $g(\eta )$ (Figure 3). The impact of $\gamma$ on $f\prime (\eta )$ and $g(\eta )$ is shown in Figures 4 and 5, respectively. Increasing $\gamma$ increases the porous space, which creates resistance in the flow path and reduces the flow motion of nanoparticles. It is perceived that an increment in porosity variable leads to improvement in the velocity field $f\prime (\eta )$ . Physically, it is due to the fact that the porous surface influence on the boundary layers progress is important owing to rise in the viscosity of the thermal boundary layers. So, it is predictable that an increment in the porosity of surface leads to accelerate the motion of the liquid on it. Basically, when the dumps of the permeable surface become large, the opposition of the porous surface may be ignored. The opposite trend is found in case of z-direction, that is, the enormous value of $\gamma$ decreases $g(\eta ).$ Figures 6 and 7 present the impact of rotational parameter Kr on x-component of $f\prime (\eta )$ and $g(\eta )$ , respectively. Here, inspection of graphs validate that with the enhancement of rotational parameter Kr, the velocity distribution $f\prime (\eta )$ shows a decreasing trend for micropolar nanofluids. Physically, the rotation parameter Kr drops its linear velocity; as a result, the liquid is flowing and rotating, and its velocity reduced the rotational friction force made by flow. In case of large magnitude of rotation parameter Kr, frequency of rotation becomes leading as compared to stretching frequency and it provides resistance to the fluid flow, so velocity $f\prime (\eta )$ is a decreasing function of Kr. The impact of $N_1$ on $f\prime (\eta )\mathrm{and}g(\eta )$ is displayed in Figures 8 and 9, respectively. It is observed that the coupling parameter $N_1$ drops velocity profile when it is close to the lower plate and increases it from the center toward the upper plate (Figure 8). The impact of Re on $f\prime (\eta )$ and $g(\eta )$ is shown in Figures 10 and 11, respectively. It is noticed that the accelerating value of the Re drops the $f\prime (\eta )$ (Figure 10). The conflicting consequence is seen in Figure 11, that is, greater values of Re surges the $g(\eta )$ .

Figure 2.

The impact of Fr on velocity profile $f(\eta )$ .

Figure 3.

The impact of Fr on velocity profile $g(\eta )$ .

Figure 4.

The impact of $\gamma$ on velocity profile $f(\eta )$ .

Figure 5.

The impact of $\gamma$ on velocity profile $g(\eta )$ .

Figure 6.

The impact of Kr on velocity profile $f(\eta )$ .

Figure 7.

The impact of Kr on velocity profile $g(\eta )$ .

Figure 8.

The impact of $N_1$ on velocity profile $f(\eta )$ .

Figure 9.

The impact of $N_1$ on velocity profile $g(\eta )$ .

Figure 10.

The impact of Re on velocity profile $f(\eta )$ .

Figure 11.

The impact of Re on velocity profile $g(\eta )$ .

Microrotation velocity profile $G(\eta )$

The impact of physical parameters $N_1\mathrm{and}{N}_2$ and Re on microrotation velocity profiles $G(\eta )$ is presented in Figures 12–14. The impact of $N_1$ on $G(\eta )$ is displayed in Figure 12. The opposite conduct is perceived in Figure 12 from Figure 8, that is, the higher value of the coupling parameter $N_1$ rises the microrotation profile $G(\eta )$ . The effect of $N_2$ on $G(\eta )$ is shown in Figure 13. Same effect is seen for $N_2$ , while the opposite tend is found for spin gradient viscosity on the profile $G(\eta )$ . The higher value of $N_2$ drops $G(\eta )$ . The influence of Re on $G(\eta )$ is displayed in Figure 14. It is perceived that augmented value of Re improves $G(\eta )$ between parallel plates.

Figure 12.

The impact of $N_1$ on microrotation $G(\eta )$ .

Figure 13.

The impact of $N_2$ on microrotation $G(\eta )$ .

Figure 14.

The impact of Re on microrotation $G(\eta )$ .

Temperature profile $\mathrm{\Theta }(\eta )$

The impact of physical parameters $A_1$ , $B_1$ , Nb, Nt, Pr, Rd, and Re on temperature profiles $\mathrm{\Theta }(\eta )$ is presented in Figures 15–21. Figure 15 displays the effect of space-dependent parameter $A_1$ on nanofluid flow field of temperature $\mathrm{\Theta }(\eta )$ . It is observed that increasing $A_1$ rises the temperature of the nanofluid between parallel plates. Figure 16 displays the influence of temperature-dependent coefficient $B_1$ . Same effect is observed for $B_1$ on temperature field $\mathrm{\Theta }(\eta )$ . Actually, the increase in $A_1$ and $B_1$ increase the thermal layer of the boundary produces internal energy which gives rise in temperature profile while, in turn the small value it drop the temperature. The impact of Nb on $\mathrm{\Theta }(\eta )$ is shown in Figure 17. It is detected from Figure 17 that augmented value of Nb rises $\mathrm{\Theta }(\eta )$ . In fact, increasing Nb increases the kinetic energy of the nanofluid inside the fluid with which the rate of internal heat transmission, microrotation velocity, and boundary layer thickness rise leading to increase in the temperature profile. The consequence of thermophoresis parameter Nt on $\mathrm{\Theta }(\eta )$ is demonstrated in Figure 18. From Figure 18, it is clear that Nt increases $\mathrm{\Theta }(\eta )$ when it is augmented. Actually, Nt depends on the temperature gradient in the surrounding nanofluid molecules. Increasing Nt increases the random motion of the nanoparticles, which in turn increases the internal energy, hence the temperature profile $\mathrm{\Theta }(\eta )$ increases. The influence of Pr on $\mathrm{\Theta }(\eta )$ is given in Figure 19. It is vibrant that $\mathrm{\Theta }(\eta )$ declines with bulky value of Pr and rises for lesser values. Actually, nanofluids with lesser value of Pr have large thermal diffusivity and this consequence is conflicting for greater Pr. Therefore, high value of Pr causes the thermal boundary layer to drop. Figure 20 shows the influence of Re on $\mathrm{\Theta }(\eta )$ ; the higher values of Re strengthens the inertial forces and tends to decrease the temperature field. Opposite effect is observed of Re on the $\mathrm{\Phi }(\eta ).$ The impact of Rd on $\mathrm{\Theta }(\eta )$ is presented in Figure 21. When we increase Rd, then it is perceived that Rd augments the temperature in the boundary layer area in the nanofluid layer. This increase leads to drop in the rate of cooling of micropolar nanofluid flow.

Figure 15.

The impact of $A_1$ on temperature $\mathrm{\Theta }(\eta )$ .

Figure 16.

The impact of $B_1$ on temperature $\mathrm{\Theta }(\eta )$ .

Figure 17.

The impact of Nb on temperature $\mathrm{\Theta }(\eta )$ .

Figure 18.

The impact of Nt on temperature $\mathrm{\Theta }(\eta )$ .

Figure 19.

The impact of Pr on temperature $\mathrm{\Theta }(\eta )$ .

Figure 20.

The impact of Re on temperature $\mathrm{\Theta }(\eta )$ .

Figure 21.

The impact of Rd on temperature $\mathrm{\Theta }(\eta )$ .

Concentration profile $\mathrm{\Phi }(\eta )$

The impact of physical parameters Nb, Nt, Sc, and Re on concentration profile $\mathrm{\Phi }(\eta )$ are presented in Figures 22–25.

Figure 22.

The impact of Re on temperature $\Phi (\eta )$ .

Figure 23.

The impact of Nb on temperature $\mathrm{\Phi }(\eta )$ .

Figure 24.

The impact of Nt on temperature $\mathrm{\Phi }(\eta )$ .

Figure 25.

The impact of Sc on temperature $\mathrm{\Phi }(\eta )$ .

Figure 22 shows the influence of Re on $\mathrm{\Phi }(\eta )$ . It is observed that higher value Re increases $\mathrm{\Phi }(\eta )$ . Actually, higher values of Re reinforce the inertial forces of nanofluids, resulting the concentration field to increase. The impact of Nb on $\mathrm{\Phi }(\eta )$ is shown in Figure 23. The opposite trend has been found for concentration distribution as from temperature field, that is, increasing Nb reduces the concentration profile. This is because of the rises of Brownian motion diminish the boundary layer thicknesses, which leads to decrease in the concentrations. The characteristics of Nt on $\mathrm{\Phi }(\eta )$ is shown in Figure 24. Increasing Nt increases $\mathrm{\Phi }(\eta )$ . This is due to that Nt depends on the temperature gradient of the nanofluids molecules. Due to increase of Nt kinetic energy of the nanofluids increase, which resulting in the increase in the concentration profile. Figure 25 labels the effect of Schmidt number Sc. The Schmidt number is a dimensionless number, defined as the ratio of momentum diffusivity to mass diffusivity. It is apparent that increasing Schmidt number decreases the concentration profile, which results to reduce the boundary layer thickness.

Skin friction, Nusselt number, and Sherwood number

The numerical values of coefficient of skin friction $C_f$ , Nusselt number Nu, and Sherwood number Sh are calculated in Tables 2–4 for large concentration and small concentration. Table 3 observes the effect of $N_1,\mathrm{Fr},\gamma ,\mathrm{Kr}$ and Re on $C_f$ . It seems that $N_1,\mathrm{Fr},\gamma ,\mathrm{Kr}$ , and Re has positive impact on $C_f$ at the lower plate. It is also seen that the higher performance is more projecting in case of $k=0$ . Table 3 studies the stimuli of $A_1,\mathrm{Nb},\mathrm{Nt},\mathrm{Re}$ , and Rd on Nu. It is alleged that greater values of $N_1$ and Re rise, while the greater values of $\mathrm{Nb},\mathrm{Nt}$ , and Rd decline the heat flux Nu. Table 4 scrutinizes the impacts of $\mathrm{Nb},\mathrm{Nt},\mathrm{Re}$ and Sc on Sh. Increasing Nb reduces the mass flux, while Nt increases the mass flux. The higher value of Re reduces the Sh, while it increases with increase in the value of Sc. The present results are compared with the previous results of Nadeem et al., ¹³ and an excellent agreement is found between both current and previous results.

Table 2.

Variation in skin friction coefficient for weak and strong concentrations when $\mathrm{Rd}=A_1=B_1=0.5,\lambda =\mathrm{Nt}=\mathrm{Nb}=0.1,\mathrm{Sc}=\mathrm{Pr}=1$ .

$N_1$	Fr	$\gamma$	Kr	Re	$\begin{array}{c}f″(0)\hfill \\ k=0\hfill \end{array}$ Nadeem et al. 13 results	$\begin{array}{c}f″(0)\hfill \\ k=0\hfill \end{array}$ Present results	$\begin{array}{c}f″(0)\hfill \\ k=0.5\hfill \end{array}$ Nadeem et al. 13 results	$\begin{array}{c}f″(0)\hfill \\ k=0.5\hfill \end{array}$ Present results
0.0	0.1	0.1	1.0	0.5	−3.46067	−3.46167	−3.46067	−3.77449
0.1					−3.79947	−4.15194	−3.62896	−3.86341
0.5					−5.13213	−3.66174	−4.31894	−6.46908
0.1	0.0				−3.79947	−3.66174	−3.62881	−3.56334
	0.3				−3.79947	−3.56341	−3.62924	−3.15194
	0.5				−3.79947	−4.56368	−3.62953	−3.15194
	0.1	0.0			−3.78723	−3.15194	−3.61727	−3.55663
		0.5			−3.84813	−3.59054	−3.67540	−3.14516
		1.0			−3.90829	−3.79532	−3.73284	−3.29532
		0.1	0.0		−3.79942	−3.21299	−3.62890	−3.56341
			0.5		−3.80078	−3.56341	−3.63020	−3.15194
			1.0		−3.80486	−3.23427	−3.63410	−3.15194
			0.1	0.5	−3.79947	−3.56341	−3.62896	−3.15194
				1.0	−3.84769	−3.56341	−3.67514	−4.61094
				1.5	−3.89581	−3.61094	−3.72123	−4.17564

Table 3.

Variation in Nusselt number weak and strong concentrations when $\mathrm{Rd}=0.5,\lambda =\mathrm{Fr}=\gamma =0.1,\mathrm{Sc}=\mathrm{Pr}=1$ .

$A_1$	Nb	Nt	Re	$B_1$	$\begin{array}{c}\mathrm{\Theta }\prime (0)\hfill \\ k=0\hfill \end{array}$ Present results	$\begin{array}{c}\mathrm{\Theta }\prime (0)\hfill \\ k=0\hfill \end{array}$ Nadeem et al. 13 results	$\begin{array}{c}\mathrm{\Theta }\prime (0)\hfill \\ k=0.5\hfill \end{array}$ Present results	$\begin{array}{c}\mathrm{\Theta }\prime (0)\hfill \\ k=0.5\hfill \end{array}$ Nadeem et al. 13 results
0.0	0.1	0.1	1.0	0.5	1.61458	0.92042	0.92042	1.61458
0.1					1.61461	0.92044	0.92043	1.61460
0.3					1.61458	0.92049	0.92043	1.61460
0.1	0.5				−1.51457	0.71075	0.71074	1.55455
	0.8				1.45208	0.57831	0.57831	1.47208
	1.2				1.11875	0.43240	0.43240	1.30208
	0.1	0.5			1.72408	0.71285	0.71284	1.71458
		0.8			1.71458	0.58125	0.58124	1.62454
		1.2			−1.51324	0.43578	0.43575	0.92271
		0.1	0.5		1.72434	0.92044	0.92043	1.72435
			1.0		1.76667	0.95655	0.95653	1.78761
			1.5		1.81875	0.99309	0.99306	1.85141
			0.1	0.5	1.67292	0.92078	0.92077	1.67395
				1.0	1.34082	0.92044	0.92043	1.34085
				1.5	−1.01125	0.92010	−0.92009	1.00771

Table 4.

Variation in Sherwood number in case of weak and strong concentrations when $\mathrm{Rd}=0.5,\lambda =\mathrm{Fr}=\gamma =0.1,\mathrm{Sc}=\mathrm{Pr}=1$ .

$A_1$	Nb	Nt	Re	$B_1$	$\begin{array}{c}\mathrm{\Phi }\prime (0)\hfill \\ k=0\hfill \end{array}$ Nadeem et al. 13 results	$\begin{array}{c}\mathrm{\Phi }\prime (0)\hfill \\ k=0\hfill \end{array}$ Present results	$\begin{array}{c}\mathrm{\Phi }\prime (0)\hfill \\ k=0.5\hfill \end{array}$ Nadeem et al. 13 results	$\begin{array}{c}\mathrm{\Phi }\prime (0)\hfill \\ k=0.5\hfill \end{array}$ Present results
0.0	0.1	0.1	1.0	0.5	1.09115	1.03125	1.09058	1.05752
0.1					1.08604	1.01128	1.08571	1.02625
0.3					1.08064	1.00625	1.08030	1.00242
0.1	0.5				1.30437	1.99947	1.30476	1.13868
	0.8				2.47999	2.22991	2.48122	2.33261
	1.2				4.41360	3.47801	4.41532	4.03809
	0.1	0.5			1.11270	1.01121	1.11274	1.01191
		0.8			1.10926	1.06250	1.10935	1.81922
		1.2			1.10493	1.06437	1.10509	2.01922
		0.1	0.5		1.09558	1.06649	1.09580	2.31213
			1.0		1.11270	1.80417	1.11274	2.39456
			1.5		1.12995	2.00345	1.12980	2.80021
			0.1	0.5	1.09115	1.09558	1.09058	1.05752
				1.0	1.08604	1.11270	1.08571	1.02625
				1.5	1.08064	1.12995	1.08030	1.00242

Conclusion

In this article, the Darcy–Forchheimer flow of three-dimensional micropolar nanofluid between two parallel plates in a rotating system has been investigated. The micropolar nanofluid flow in permeable media is described by taking the Darcy–Forchheimer model, where drenching permeable space obeys Darcy–Forchheimer expression. The thermal radiation impact is taken to be varying in absorption/generation for the purpose to see the concentration as well as the temperature modifications between the nanofluid and the surfaces. The central concluded points given as follows:

• It is observed that the higher value of Fr decay the velocity profile, while it increases the transverse velocity.

• It is observed that increasing the porosity parameter $\gamma$ increases the porous space, which creates resistance in the flow path and reduces the flow motion.

• The higher value of Re reduces $f\prime (\eta )$ and increases $g(\eta )$ .

• The coupling parameter $N_1$ and spin gradient viscosity parameter $N_2$ having the key rule to change the microrotation profile $G(\eta )$ . Moreover, it is observed that rises in the coupling parameter $N_1$ increases $G(\eta )$ while $N_2$ reduces it.

• Increasing Nt and Nb consequences in a conflicting behavior on $\mathrm{\Phi }(\eta )$ .

• It is observed the temperature field enhances with larger values of Nb and Nt for $k=0$ and $k=0.5$ .

• The greater values of $N_1$ and Re increase the heat flux, while the higher values of $\mathrm{Nb}\mathrm{and}\mathrm{Nt}$ reduces the heat flux $(\mathrm{Nu}=-\mathrm{\Theta }\prime (0))$ for $k=0$ and $k=0.5$ .

• Thermal boundary layer thickness reduces with rise of Rd and Nusselt number Nu increases with large value of radiation parameter Rd.

• It is detected that $\mathrm{\Theta }(\eta )$ and $\mathrm{\Phi }(\eta )$ fall with higher value of Pr and rise for lesser values of Pr.

• The convergence of the HAM method with the variation of physical parameters has been observed numerically and best result is obtained.