Three-dimensional magnetohydrodynamic nanofluid thin-film flow with heat and mass transfer over an inclined porous rotating disk

Abstract

In the present study, the three-dimensional Darcy–Forchheimer magnetohydrodynamic thin-film nanofluid containing flow over an inclined steady rotating plane is observed. Nanofluid thin-film flows are taken thermally radiated and suction/injection effect is also considered. By similarity variables, the partial differential equations are transformed into a set of first-ordinary differential equations (ODES). By Homotopy Analysis Method, the required ODES is solved. The boundary layer over an inclined steady rotating plane is plotted and observed in detail for the velocity, $\theta (\eta )$ , and $\varphi (\eta )$ profiles. The influence of various embedded parameters such as variable thickness, $\mathrm{Sc},\lambda ,\mathrm{Nb},$ Pr, and thermophoretic parameter on velocity, $\theta (\eta )$ , and $\varphi (\eta )$ profile. The influence of many parameters is explained by graphs for the velocity, $\theta (\eta )$ , and $\varphi (\eta )$ . The crucial terms of Nusselt number and Sherwood number have also been observed numerically and physically for $\theta (\eta )$ and $\varphi (\eta )$ . Radiation phenomena is the cause of energy to the liquid system. For more rotation parameters, the thermal boundary-layer thickness is reduced.

Introduction

In chemical and mechanical engineering progressions, the elimination from cooling of a fluid condensate, saturated vapor, is significant. Numerous scientists demonstrated this problem at dissimilar conditions. The deferrals of nanoparticles in fluids display vital enrichment of their possessions at modest nanoparticle concentrations. Numerous researchers’ work on nanofluids appreciate their behaviors so that they can be applied where straight heat transfer enrichment is vital as in numerous industrialized uses, transportation, and nuclear reactors. Nanofluid as a smart fluid, where heat transfer can be decreased or increased at will, has also been described. This research work aims at deliberating the wide-ranging present and future uses of nanofluids, emphasizing their enriched heat transfer possessions that are governable and the definite features that these nanofluids preserve that make them suitable for such uses. Moreover, a nanofluid is a new kind of energy transference fluid, which is the suspension of a base fluid and nanoparticles. For cooling rate requirements, usual heat transfer liquids cannot be used, due to their lesser thermal conductivity. By implanting nanoparticles into normal fluids, their thermal enactment can be enriched considerably. Bhatti et al. ¹ have explored simultaneous impacts of varying magnetic field of Jeffrey nanofluid. Ellahi ² has deliberated the magnetohydrodynamic (MHD) non-Newtonian nanofluid with temperature-dependent viscosity flow through a pipe. The flow exploration cooled by Cu water nanofluid of microchannel heat sink applying least square method and porous media approach observed by Hatami et al.^3,4 have explored nanofluid laminar flow between rotating disks with heat transfer. Shamshuddin et al. ⁵ have deliberated numerical study of non-Fourier heat flux model in a rotating disk in the presence of viscosity and heat transfer. Khan and Pop ⁶ have deliberated boundary-layer nanofluid flow through a stretching surface. Khanafer et al. ⁷ have described two-dimensional buoyancy driven with enhancement heat transfer enclosure utilizing nanofluids. Bég et al. ⁸ have studied solar magnetic nanopolymer fabrication simulation, by applying the modeling of magnetic nanopolymer flow with induction. Rashidi et al. ⁹ have explored flow of nanofluid on a steady porous rotating disk with entropy generation and MHD. Rashidi and Dinarvand ¹⁰ have investigated three-dimensional (3-D) film condensation on steady inclined rotating disk. Saleh et al. ¹¹ have studied carbon-nanotubes suspended nanofluids flow with convective condition using Laplace transform. Sheikholeslami et al. ¹² have studied flow of nanofluid in a semi-annulus enclosure with heat transfer and MHD effects. Sheikholeslami et al. ¹³ have deliberated flow of MHD nanofluid in a semi-porous channel. Later investigators ¹⁴ have deliberated unsteady nanofluid flow through a stretched sheet. Hayat et al. ¹⁵ have explored boundary-layer flow of nanofluid. Malik et al. ¹⁶ have examined MHD flow through a stretching of Erying–Powell nanofluid. Nadeem et al. ¹⁷ have examined flow of Maxwell liquid with nanoparticle through a vertical stretching sheet. Raju et al. ¹⁸ have examined flow with free convective heat transfer through a cone of MHD nano liquid. Rokni et al. ¹⁹ have explored flow through the plates of nanofluid with heat transfer. Nadeem et al. ²⁰ have deliberated flow on stretching sheet of nano non-Newtonian liquid. Shehzad et al. ²¹ have investigated convective boundary conditions of Jeffrey nano liquid flow. Sheikholeslami et al. ²² have explored flow with magnetic field and heat transfer of nano liquid. Mahmoodi and Kandelousi ²³ have examined flow for cooling application of nanonfluid with heat transfer. Shah et al.^24–27 have deliberated nanofluid flow through a rotating system with Hall current and thermal radiation effects. The further present theoretical and investigational examine of Sheikholeslami using different phenomena for nanofluids, with present usages and possessions with applications of numerous methods can be deliberate in previous studies.^28–32

The thin-film flow exploration have achieved substantial presentation due to its frequent usages in the field of industries and technology, and engineering in certain years. Investigation of thin liquid flow has practical uses such as cable, fibber undercoat. Several well-known uses of thin film are fluidization of the devices, elastic sheets drawing, and constant forming. Sandeep and Malvandi ³³ have studied non-Newtonian nano liquids thin-film fluid flow. Wang ³⁴ has detected unsteady flow of thin-film fluid through stretching sheet. Usha and Sridharan ³⁵ have deliberated unsteady finite thin liquid past through a stretching surface. Liu and Andersson ³⁶ have deliberated film flow with heat transfer on a stretching surface. Aziz et al. ³⁷ have perceived flow of thin fluid film on a stretching sheet for the production of inner heat. Tawade et al. ³⁸ have examined fluid flow with thermal radiation and heat transmission of thin film. Fluid film flow on stretching sheet with heat transfer have deliberated by Anderssona et al. ³⁹ Also investigators^40–43 have deliberated unsteady flow of liquid film on stretching surface for further dissimilar cases. Hatami et al. ⁴⁴ have examined 3-D nanofluid flow on a steady rotating disk. Jawad et al. ⁴⁵ have examined Darcy–Forchheimer nanofluid thin-film flow with MHD effect. Jawad et al. ⁴⁶ have studied 3-D Single-Wall Carbon Nanotubes rotating flow with impact of non-linear thermal radiation and the viscous dissipation. Bég et al. ⁴⁷ have studied flow on an inclined surface in the presence of heat generation effects and Soret diffusion with heat and mass transfer. Kadir et al. ⁴⁸ have deliberated flow of von Karman swirling bioconvection in a deformable rotating disk. Bég et al. ⁴⁹ have discussed improved rheology and lubricity of drilling fluid enhanced with nanoparticles. Bég et al. ⁴⁷ have investigated steady state flow along an inclined surface with MHD heat and mass transfer.

Problem formulation

Suppose flow over rotating disk of a steady 3-D nanofluid thin film. The rotation of the disk is due to the angular velocity $(\mathrm{\Omega })$ in its own plane as displayed in Figure 1. An angle $\beta$ is observed by the inclined disk with horizontal axis; $h$ denotes the film thickness of nanofluid, and $W$ represents the spraying velocity. The radius of the disk is very large as matched to the liquid film thickness and hence the termination influence is unnoticed. The $\overline{g}$ is gravitational acceleration, the temperature at the film surface is represented by $T_0$ , while $T_w$ denotes the surface temperature of the disk. Likewise $C_0$ , $C_h$ is the concentration at the film and on the disk surfaces, respectively. Pressure is a function of z-axis only and the ambient pressure $(P_0)$ at sheet of the film is kept constant. The equations for steady state of continuity, momentum, concentration, and energy are shown in equations (1)–(6)

$$u_x+u_y+u_z=0$$

(1)

$$\begin{array}{cc}\multicolumn{1}{c}{u{u}_x}& +v{u}_y+w{u}_z=\frac{{\mu }_{\mathrm{nf}}}{{\rho }_{\mathrm{nf}}}(u_{\mathrm{xx}}+u_{\mathrm{yy}}+u_{\mathrm{zz}})\hfill \\ \multicolumn{1}{c}{}& +\overline{g}\sin \frac{\beta }{\mathrm{\Omega }\prime }-\frac{\sigma B_0^{2}u}{\rho }-\frac{{\nu }_{\mathrm{nf}}}{k}u\hfill \end{array}$$

(2)

$$u{v}_x+v{v}_y+w{v}_z=\frac{{\mu }_{\mathrm{nf}}}{{\rho }_{\mathrm{nf}}}(v_{\mathrm{xx}}+v_{\mathrm{yy}}+v_{\mathrm{zz}})-\frac{\sigma B_0^{2}v}{\rho }-\frac{{\nu }_{\mathrm{nf}}}{k}v$$

(3)

$$\begin{array}{cc}\multicolumn{1}{c}{u{w}_x}& +v{w}_y+w{w}_z=\frac{{\mu }_{\mathrm{nf}}}{{\rho }_{\mathrm{nf}}}(w_{\mathrm{xx}}+w_{\mathrm{yy}}+w_{\mathrm{zz}})\hfill \\ \multicolumn{1}{c}{}& -\overline{g}\cos \frac{\beta }{\mathrm{\Omega }\prime }-\frac{P_z}{{\rho }_{\mathrm{nf}}}\hfill \end{array}$$

(4)

$$\begin{array}{cc}\multicolumn{1}{c}{u{T}_x}& +v{T}_y+w{T}_z=\frac{k_{\mathrm{nf}}}{{\left({\rho }_{\mathrm{cp}}\right)}_{\mathrm{nf}}}(T_{\mathrm{xx}}+T_{\mathrm{yy}}+T_{\mathrm{zz}})\hfill \\ \multicolumn{1}{c}{}& -\frac{1}{{\rho }_{\mathrm{cp}}}{q}_z^{\mathrm{rd}}+\frac{1}{{\rho }_{\mathrm{cp}}}q‴\hfill \end{array}$$

(5)

$$\begin{array}{cc}\multicolumn{1}{c}{u{C}_x}& +v{C}_y+w{C}_z=D_{\beta }(C_{\mathrm{xx}}+C_{\mathrm{yy}}+C_{\mathrm{zz}})\hfill \\ \multicolumn{1}{c}{}& +\left(\frac{D_T}{T_0}\right)(T_{\mathrm{xx}}+T_{\mathrm{yy}}+T_{\mathrm{zz}})\hfill \end{array}$$

(6)

Figure 1.

Geometry of the problem.

In the above equations, the velocity components in the $x-,y-$ , and $z$ -axes are represented by $u,v$ , and $w$ individually, where BCs are as follows

$$\begin{array}{c}u=-\mathrm{\Omega }y,v=\mathrm{\Omega }x,\mathrm{w}=0,T=T_w,C=C_h\mathrm{at}z=0\hfill \\ u_z=v_z=0,\mathrm{w}=0,T=T_w,C=C_0,P=P_0\mathrm{at}z=h\hfill \end{array}$$

(7)

Consider Similarity transformation are

$$\begin{array}{c}u=-\mathrm{\Omega }\mathrm{yg}\left(\eta \right)+\mathrm{\Omega }\mathrm{xf}\prime \left(\eta \right)+\overline{g}k\left(\eta \right)\sin \frac{\beta }{\mathrm{\Omega }\prime }\hfill \\ v=\mathrm{\Omega }\mathrm{xg}\left(\eta \right)+\mathrm{\Omega }\mathrm{yf}\prime \left(\eta \right)+\overline{g}s\left(\eta \right)\sin \frac{\beta }{\mathrm{\Omega }\prime }\hfill \\ w=-2\sqrt{\mathrm{\Omega }{v}_{\mathrm{nf}}}f\left(\eta \right),T=(T_0-T_w)\theta \left(\eta \right)+T_w\hfill \\ \eta \varphi \left(\eta \right)=\frac{C-C_w}{C_0-C_w},\eta =z\sqrt{\frac{\mathrm{\Omega }}{v_{\mathrm{nf}}}}\hfill \end{array}$$

(8)

The transformations identify in equation (8) is introduced in equations (2)–(7) such that equation (1) is proved identically and equations (2)–(6) are obtained in resulting form as

$$f‴-f{\prime }^2+g^2+2\mathrm{ff}″+f\prime (M-\gamma )=0$$

(9)

$$k″+\mathrm{gs}-\mathrm{kf}\prime +2\mathrm{kf}+1-k(M+\gamma )=0$$

(10)

$$g″-2\mathrm{gf}\prime +2g\prime f-g(M+\gamma )=0$$

(11)

$$s″-\mathrm{kg}-\mathrm{sf}\prime +2s\prime f-s(M+\gamma )=0$$

(12)

If $\theta (\eta )$ and $\varphi (\eta )$ are a function of $z$ only, equations (5) and (6) become

$$(1-\frac{4}{3}R)\theta ″+2\mathrm{Pr}\frac{A_2{A}_3}{A_1{A}_4}(I_{1}f\prime +j_1)\theta =0$$

(13)

$$\varphi ″+2\mathrm{Scf}\varphi \prime +\frac{\mathrm{Nt}}{\mathrm{Nb}}\theta ″+\frac{S}{2}(\eta \varphi \prime +{\eta }^2\varphi ″)=0$$

(14)

$$\begin{array}{c}f\left(0\right)=0,f\prime \left(0\right)=0,f″\left(\delta \right)=0\hfill \\ g\left(0\right)=0,g\prime \left(\delta \right)=0\hfill \\ k\left(0\right)=0,k\prime \left(\delta \right)=0\hfill \\ s\left(0\right)=0,s\prime \left(\delta \right)=0\hfill \\ \theta \left(0\right)=0,\theta \prime \left(\delta \right)=1\hfill \\ \varphi \left(0\right)=0,\varphi \left(\delta \right)=1\hfill \end{array}$$

(15)

Now $\mathrm{Pr}$ , $\mathrm{Sc}$ , $\mathrm{Nb}$ , $\mathrm{Nt}$ are identified as

$$\begin{array}{c}\mathrm{Pr}=\frac{{\rho }_{\mathrm{cp}}{\nu }_{\mathrm{nf}}}{k},\mathrm{Sc}=\frac{\mu }{\rho \mathrm{fD}},\mathrm{Nb}=\frac{\tau D_B}{\nu }(C_0-C_w)\hfill \\ \mathrm{Nt}=\frac{\tau D_T}{\nu T_w}(T_0-T_w),S=\frac{\alpha }{\mathrm{\Omega }},M=\frac{\sigma {\beta }_0^2}{\mathrm{\Omega }\rho },K=\frac{\nu \prime \mathrm{\Omega }}{{\nu }_{\mathrm{nf}}^2}\hfill \end{array}$$

(16)

The normalized thickness constant is presented as

$$\delta =h\sqrt{\frac{\mathrm{\Omega }}{v_{\mathrm{nf}}}}$$

(17)

The condensation velocity is defined as

$$f\left(\delta \right)=\frac{W}{2\sqrt{\mathrm{\Omega }\nu }}=\alpha$$

(18)

The pressure can be attained by integration equation (4).

For $\mathrm{Pr}=0$ , by using $\theta (\delta )=1$ , the exact solution is

$$\theta \prime \left(0\right)=\frac{1}{\delta }$$

(19)

An asymptotic limit for small $\delta$ is defined in equation (17). The reduction of $\theta \prime (0)$ for rising $\delta$ is not monotonic, so $\mathrm{Nu}$ is denoted as

$$\mathrm{Nu}=\frac{k_{\mathrm{nf}}}{k_f}\frac{{\left(T_z\right)}_w}{(T_0-T_w)}=A_4\delta \theta \prime \left(0\right)$$

(20)

The Sherwood number is denoted as

$$\mathrm{Sh}=\frac{{\left(C_z\right)}_w}{C_0-C_w}=\delta \varphi \prime \left(0\right)$$

(21)

Solution by HAM

Optimal approach is used for solution process. Equations (9)–(14) with boundary conditions (15) are solved by Homotopy Analysis Method (HAM). Mathematica software is used for this aim. Basic derivation of the model equation through HAM is given in detail below.

Linear operators are denoted as $L_{\hat{f}},L_{\hat{\theta }}$ , and $L_{\hat{\varphi }}$ , and represented as

$$\begin{array}{c}{L}_{\hat{f}}\left(\hat{f}\right)=\hat{f}‴,L_{\hat{k}}\left(\hat{k}\right)=k″,L_{\hat{g}}\left(\hat{g}\right)=g″,\hfill \\ L_{\hat{s}}\left(\hat{s}\right)=\mathrm{s}″,L_{\hat{\theta }}\left(\hat{\theta }\right)=\hat{\theta }″,L_{\hat{\varphi }}\left(\hat{\varphi }\right)=\varphi ″\hfill \end{array}$$

(22)

which is shown as

$$\begin{array}{c}{L}_{\hat{f}}(e_1+e_2\eta +e_3{\eta }^2)=0,L_{\hat{k}}(e_5+e_6\eta )=0,\hfill \\ L_{\hat{g}}(e_7+e_8\eta )=0,\hfill \\ L_{\hat{s}}(e_9+e_{10}\eta )=0,L_{\hat{\theta }}(e_{11}+e_{12}\eta )=0,\hfill \\ L_{\hat{\varphi }}(e_{13}+e_{14}\eta )=0\hfill \end{array}$$

(23)

where the representation of coefficients is included in the general solution by $e_i(i=1\to 7)$ .

The corresponding non-linear operators are sensibly chosen as $N_{\hat{f}},N_{\hat{k}},N_{\hat{g}},N_{\hat{s}},N_{\hat{\theta }},\mathrm{and}{N}_{\hat{\varphi }}$ , and identified in the form

$$N_{\hat{f}}[\hat{f}(\eta ;\zeta ),\hat{g}(\eta ;\zeta )]={\hat{f}}_{\eta \eta \eta }-\frac{{\beta }_0^2}{\lambda }{\hat{f}}_{\eta }^2+{\hat{g}}^2+2\hat{f}{\hat{f}}_{\eta \eta }+{\hat{f}}_{\eta }(M-\gamma )$$

(24)

$$\begin{array}{c}{N}_{\hat{k}}[\hat{k}(\eta ;\zeta ),\hat{g}(\eta ;\zeta ),\hat{f}(\eta ;\zeta ),\hat{s}(\eta ;\zeta )]\hfill \\ ={\hat{k}}_{\eta \eta }+\hat{g}\hat{s}-\hat{k}{\hat{f}}_{\eta }+2\hat{k}\hat{f}+1-\hat{k}(M+\gamma )\hfill \end{array}$$

(25)

$$N_{\hat{g}}[\hat{g}(\eta ;\zeta ),\hat{f}(\eta ;\zeta )]={\hat{g}}_{\eta \eta }-2\hat{g}{\hat{f}}_{\eta }+2{\hat{g}}_{\eta }\hat{f}-\hat{g}(M+\gamma )$$

(26)

$$\begin{array}{c}{N}_{\hat{s}}[\hat{s}(\eta ;\zeta ),\hat{g}(\eta ;\zeta ),\hat{f}(\eta ;\zeta ),\hat{k}(\eta ;\zeta )]\hfill \\ ={\hat{s}}_{\eta \eta }-2\hat{k}\hat{g}-\hat{s}{\hat{f}}_{\eta }+2{\hat{s}}_{\eta }\hat{f}-\hat{s}(M+\gamma )\hfill \end{array}$$

(27)

$$\begin{array}{c}{N}_{\hat{\theta }}[\hat{\theta }(\eta ;\zeta ),\hat{f}(\eta ;\zeta )]\hfill \\ =(1-\frac{4}{3}R){\hat{\theta }}_{\eta \eta }+2\mathrm{Pr}\frac{A_2{A}_3}{A_1{A}_4}(I_1{\hat{f}}_{\eta }+j_1)\hat{\theta }\hfill \end{array}$$

(28)

$$\begin{array}{c}{N}_{\hat{\varphi }}[\hat{\varphi }(\eta ;\zeta ),\hat{f}(\eta ;\zeta ),\hat{\theta }(\eta ;\zeta )]={\hat{\varphi }}_{\eta \eta }+2\mathrm{Sc}\hat{f}{\hat{\varphi }}_{\eta }\hfill \\ +\frac{\mathrm{Nt}}{\mathrm{Nb}}{\hat{\theta }}_{\eta \eta }+\frac{S}{2}(\eta {\hat{\varphi }}_{\eta }+{\eta }^2{\hat{\varphi }}_{\eta \eta })\hfill \end{array}$$

(29)

For equations (8)–(10), the 0th-order scheme takes the form

$$(1-\zeta )L_{\hat{f}}[\hat{f}(\eta ;\zeta )-{\hat{f}}_0\left(\eta \right)]=p{\hslash }_{\hat{f}}{N}_{\hat{f}}[\hat{f}(\eta ;\zeta ),\hat{g}(\eta ;\zeta )]$$

(30)

$$\begin{array}{c}(1-\zeta )L_{\hat{k}}[\hat{k}(\eta ;\zeta )-{\hat{k}}_0\left(\eta \right)]\hfill \\ =p{\hslash }_{\hat{k}}{N}_{\hat{k}}[\hat{k}(\eta ;\zeta ),\hat{g}(\eta ;\zeta ),\hat{f}(\eta ;\zeta ),\hat{s}(\eta ;\zeta )]\hfill \end{array}$$

(31)

$$(1-\zeta )L_{\hat{g}}[\hat{g}(\eta ;\zeta )-{\hat{g}}_0\left(\eta \right)]=p{\hslash }_{\hat{g}}{N}_{\hat{g}}[\hat{f}(\eta ;\zeta ),\hat{g}(\eta ;\zeta )]$$

(32)

$$\begin{array}{c}(1-\zeta )L_{\hat{s}}[\hat{s}(\eta ;\zeta )-{\hat{s}}_0\left(\eta \right)]\hfill \\ =p{\hslash }_{\hat{s}}{N}_{\hat{s}}[\hat{s}(\eta ;\zeta ),\hat{g}(\eta ;\zeta ),\hat{f}(\eta ;\zeta ),\hat{k}(\eta ;\zeta )]\hfill \end{array}$$

(33)

$$(1-\zeta )L_{\hat{\theta }}[\hat{\theta }(\eta ;\zeta )-{\hat{\theta }}_0\left(\eta \right)]=p{\hslash }_{\hat{\theta }}{N}_{\hat{\theta }}[\hat{\theta }(\eta ;\zeta ),\hat{f}(\eta ;\zeta )]$$

(34)

$$\begin{array}{c}(1-\zeta )L_{\hat{\varphi }}[\hat{\varphi }(\eta ;\zeta )-{\hat{\varphi }}_0\left(\eta \right)]\hfill \\ =p{\hslash }_{\hat{\varphi }}{N}_{\hat{\varphi }}[\hat{\varphi }(\eta ;\zeta ),\hat{f}(\eta ;\zeta ),\hat{\theta }(\eta ;\zeta )]\hfill \end{array}$$

(35)

The boundary constrains are

$$\begin{array}{c}{\hat{f}(\eta ;\zeta )|}_{\eta =0}=0,{\frac{\partial \hat{f}(\eta ;\zeta )}{\partial \eta }|}_{\eta =0}=0,{\frac{{\partial }^2\hat{f}(\eta ;\zeta )}{\partial {\eta }^2}|}_{\eta =\delta }=0\hfill \\ {\hat{g}(\eta ;\zeta )|}_{\eta =0}=0,{\frac{\partial \hat{g}(\eta ;\zeta )}{\partial \eta }|}_{\eta =\delta }=0,\hfill \\ {\hat{k}(\eta ;\zeta )|}_{\eta =0}=0,{\frac{\partial \hat{k}(\eta ;\zeta )}{\partial \eta }|}_{\eta =\delta }=0,\hfill \\ {\hat{s}(\eta ;\zeta )|}_{\eta =0}=0,{\frac{\partial \hat{s}(\eta ;\zeta )}{\partial \eta }|}_{\eta =\delta }=0,\hfill \\ {\hat{\theta }(\eta ;\zeta )|}_{\eta =0}=0,{\frac{\partial \hat{\theta }(\eta ;\zeta )}{\partial \eta }|}_{\eta =\delta }=1,\hfill \\ {\hat{\varphi }(\eta ;\zeta )|}_{\eta =0}=0,{\hat{\varphi }(\eta ;\zeta )|}_{\eta =\delta }=1\hfill \end{array}$$

(36)

where the embedding restriction is $\zeta \in [0,1]$ , to normalize for the solution convergence ${\hslash }_{\hat{f}},{\hslash }_{\hat{k}},{\hslash }_{\hat{g}},{\hslash }_{\hat{s}}$ , ${\hslash }_{\hat{\theta }}$ , and ${\hslash }_{\hat{\varphi }}$ are used. When $\zeta =0\mathrm{and}\zeta =1$ , we get

$$\begin{array}{c}\hat{f}(\eta ;1)=\hat{f}\left(\eta \right),\hat{k}(\eta ;1)=\hat{k}\left(\eta \right),\hat{g}(\eta ;1)=\hat{g}\left(\eta \right),\hfill \\ \hat{s}(\eta ;1)=\hat{s}\left(\eta \right),\hat{\theta }(\eta ;1)=\hat{\theta }\left(\eta \right),\hat{\varphi }(\eta ;1)=\hat{\varphi }\left(\eta \right)\hfill \end{array}$$

(37)

Expand the $\hat{f}(\eta ;\zeta ),\hat{k}(\eta ;\zeta ),\hat{g}(\eta ;\zeta ),\hat{s}(\eta ;\zeta ),\hat{\theta }(\eta ;\zeta )$ , and $\hat{\varphi }(\eta ;\zeta )$ through Taylor’s series for $\zeta =0$

$$\begin{array}{c}\hat{f}(\eta ;\zeta )={\hat{f}}_0\left(\eta \right)+\sum _{n=1}^{\infty }{\hat{f}}_n\left(\eta \right){\zeta }^n\hfill \\ \hat{k}(\eta ;\zeta )={\hat{k}}_0\left(\eta \right)+\sum _{n=1}{\hat{k}}_n(\eta ){\zeta }^n\hfill \\ \hat{g}(\eta ;\zeta )={\hat{g}}_0\left(\eta \right)+\sum _{n=1}^{\infty }{\hat{g}}_n\left(\eta \right){\zeta }^n\hfill \\ \hat{s}(\eta ;\zeta )={\hat{s}}_0\left(\eta \right)+\sum _{n=1}^{\infty }{\hat{s}}_n\left(\eta \right){\zeta }^n\hfill \\ \hat{\theta }(\eta ;\zeta )={\hat{\theta }}_0\left(\eta \right)+\sum _{n=1}^{\infty }{\hat{\theta }}_n\left(\eta \right){\zeta }^n\hfill \\ \hat{\varphi }(\eta ;\zeta )={\hat{\varphi }}_0\left(\eta \right)+\sum _{n=1}^{\infty }{\hat{\varphi }}_n\left(\eta \right){\zeta }^n\hfill \end{array}$$

(38)

$$\begin{array}{c}{\hat{f}}_n\left(\eta \right)={\frac{1}{n!}\frac{\partial \hat{f}(\eta ;\zeta )}{\partial \eta }|}_{p=0},{\hat{k}}_n\left(\eta \right)={\frac{1}{n!}\frac{\partial \hat{f}(\eta ;\zeta )}{\partial \eta }|}_{p=0},\hfill \\ {\hat{g}}_n\left(\eta \right)={\frac{1}{n!}\frac{\partial \hat{f}(\eta ;\zeta )}{\partial \eta }|}_{p=0},\hfill \\ {\hat{s}}_n\left(\eta \right)={\frac{1}{n!}\frac{\partial \hat{f}(\eta ;\zeta )}{\partial \eta }|}_{p=0},{\hat{\theta }}_n\left(\eta \right)={\frac{1}{n!}\frac{\partial \hat{\theta }(\eta ;\zeta )}{\partial \eta }|}_{p=0},\hfill \\ {\hat{\varphi }}_n\left(\eta \right)={\frac{1}{n!}\frac{\partial \hat{g}(\eta ;\zeta )}{\partial \eta }|}_{p=0}\hfill \end{array}$$

(39)

The boundary constrains are

$$\begin{array}{c}\hat{f}\left(0\right)=0,\hat{f}\prime \left(0\right)=0,\hat{f}″\left(\delta \right)=0\hfill \\ \hat{g}\left(0\right)=0,\hat{g}\prime \left(\delta \right)=0\hfill \\ \hat{k}\left(0\right)=0,\hat{k}\prime \left(\delta \right)=0\hfill \\ \hat{s}\left(0\right)=0,\hat{s}\prime \left(\delta \right)=0\hfill \\ \hat{\theta }\left(0\right)=0,\hat{\theta }\prime \left(\delta \right)=1\hfill \\ \hat{\varphi }\left(0\right)=0,\hat{\varphi }\left(\delta \right)=1\hfill \end{array}$$

(40)

Here

$$\begin{array}{cc}\multicolumn{1}{c}{{\Re }_n^{\hat{f}}\left(\eta \right)}& =\hat{f}″{\prime }_{n-1}-\hat{f}{\prime }_{n-1}^2+{\hat{g}}_{n-1}^2\hfill \\ \multicolumn{1}{c}{}& +2\sum _{j=0}^{w-1}{\hat{f}}_{w-1-j}{\hat{f}″}_j+\hat{f}{\prime }_{n-1}(M-\gamma )\hfill \end{array}$$

(41)

$$\begin{array}{cc}\multicolumn{1}{c}{{\Re }_n^{\hat{k}}\left(\eta \right)}& ={\hat{k}″}_{n-1}+\sum _{j=0}^{w-1}{\hat{g}}_{w-1-j}{\hat{s}}_j-\sum _{j=0}^{w-1}{\hat{k}}_{w-1-j}\hat{f}{\prime }_j\hfill \\ \multicolumn{1}{c}{}& +2\sum _{j=0}^{w-1}{\hat{k}}_{w-1-j}{\hat{f}}_j+1-{\hat{k}}_{n-1}(M-\gamma )\hfill \end{array}$$

(42)

$$\begin{array}{c}{\Re }_n^{\hat{g}}\left(\eta \right)={\hat{g}″}_{n-1}+\sum _{j=0}^{w-1}{\hat{g}}_{w-1-j}f{\prime }_j\hfill \\ +2\sum _{j=0}^{w-1}\hat{g}{\prime }_{w-1-j}{\hat{f}}_j-{\hat{g}}_{n-1}(M+\gamma )\hfill \end{array}$$

(43)

$$\begin{array}{cc}\multicolumn{1}{c}{{\Re }_n^{\hat{s}}\left(\eta \right)=}& {\hat{s}″}_{n-1}-\sum _{j=0}^{w-1}{\hat{k}}_{w-1-j}{\hat{g}}_j-\sum _{j=0}^{w-1}{\hat{s}}_{w-1-j}\hat{f}{\prime }_j\hfill \\ \multicolumn{1}{c}{}& +2\sum _{j=0}^{w-1}\hat{s}{\prime }_{w-1-j}{\hat{f}}_j+1-{\hat{s}}_{n-1}(M+\gamma )\hfill \end{array}$$

(44)

$$R_n^{\hat{\theta }}\left(\eta \right)=(1-\frac{4}{3}R)\left({\hat{\theta }″}_{n-1}\right)+2\mathrm{Pr}\frac{A_2{A}_3}{A_1{A}_4}(I_1\hat{f}{\prime }_{n-1}+j_1){\hat{\theta }}_{n-1}$$

(45)

$$\begin{array}{cc}\multicolumn{1}{c}{{\Re }_n^{\hat{\varphi }}\left(\eta \right)=}& {\hat{\varphi }″}_{n-1}+2\mathrm{Sc}\sum _{j=0}^{w-1}{\hat{f}}_{w-1-j}\hat{\varphi }{\prime }_j+\frac{\mathrm{Nt}}{\mathrm{Nb}}{\hat{\theta }″}_{n-1}\hfill \\ \multicolumn{1}{c}{}& +\frac{S}{2}(\eta \varphi {\prime }_{n-1}+{\eta }^2{\varphi ″}_{n-1})\hfill \end{array}$$

(46)

where

$${\chi }_n=\{\begin{array}{c}0,\mathrm{if}\zeta \le 1\hfill \\ \multicolumn{1}{c}{1,\mathrm{if}\zeta >1}\end{array}$$

(47)

Result and discussion

The flow of 3-D thin-film nanofluid through a steady rotating inclined surface with mass and heat transmission has been examined. The influence of the embedded parameters $M,\gamma ,R,\mathrm{Pr},\mathrm{Nb},\mathrm{Nt},\mathrm{Sc}$ , and $S$ has been investigated for axial velocity $f(\eta )$ , radial velocity $k(\eta )$ , drainage flow $g(\eta )$ , and induced flow $s(\eta )$ , $\theta (\eta )$ and $\varphi (\eta )$ profile, respectively. The effect of $M$ on $f(\eta )$ , $k(\eta )$ , $g(\eta )$ , and $s(\eta )$ are shown in Figures 2–5. Figures 2 and 5 display the impact of $M$ on $f(\eta )$ and $s(\eta )$ . With increases in $M$ axial velocity, the induced flow of fluid film increases. It is also detected that rise in $M$ results inside liquid velocity of nanofluid and layer thickness to decrease suggestively. The purpose behind such influence of $M$ is the stimulation Lorentz force. To both fields, the action of this force is vertical. Figures 3 and 4 displays that, the decreased values of $M$ increases radial velocity and drainage flow. As $M$ recommends the ratio of viscous and hydromagnetic body forces, larger value of $M$ specifies a more hydromagnetic body force due to which the fluid flow is reduced. Lorentz force theory states that $M$ has an inverse effect on velocity function. Figures 6–9 display the influence of $\gamma$ on $f(\eta )$ , $k(\eta )$ , $g(\eta )$ , and $s(\eta )$ . Figures 6 and 7 display that rising $\gamma$ rises the porous space which generates resistance in the flow path and decreases the flow motion of nanoparticles. It is observed that an increase in porosity variable leads to decrease in the $f(\eta )$ and $g(\eta )$ . Physically, it is due to this fact that the porous surface effect on the boundary layers progress is significant owing to increase in the viscosity of the thermal boundary layers. Also Figures 8 and 9 display that rising values of $\gamma$ reduce $k(\eta )$ and $s(\eta )$ . So it is expected that, an increase in the porosity leads to decrease in the motion of the liquid on it. Basically, when the dumps of the permeable surface come to be large, the opposition of the porous surface may be ignored. The opposite is found in case of z-direction, that is, the enormous value of $\gamma$ decreases the $k(\eta )$ and $s(\eta )$ . The influence of $\mathrm{Pr}$ on $\theta (\eta )$ is displayed in Figure 10. It is expected that $\theta (\eta )$ regressions with huge value of $\mathrm{Pr}$ increases for smaller values. Thermal diffusivity of nanofluids is greater by reducing $\mathrm{Pr}$ and this significance is inconsistent for larger $\mathrm{Pr}$ . Hence, the greater value of $\mathrm{Pr}$ drops in thermal boundary layer. The influence of $R$ on $\theta (\eta )$ is presented in Figure 11. It is observed if we rise $R$ , then in the boundary-layer area $\theta (\eta )$ augment. The effect of $\mathrm{Nb}$ on $\varphi (\eta )$ is displayed in Figure 12. The converse influence has been created for $\varphi (\eta )$ as for $\theta (\eta )$ , which is augmenting $\mathrm{Nb}$ as $\varphi (\eta )$ decreases. The boundary-layer thickness is decreased due to the rises of $\mathrm{Nb}$ as a result to reduce the $\varphi (\eta )$ . The features of $\mathrm{Nt}$ on $\varphi (\eta )$ is presented in Figure 13. Enhancing $\mathrm{Nt}$ rises $\varphi (\eta )$ . Due to that $\mathrm{Nt}$ depends on the temperature gradient of the nanofluids. Kinetic energy of the nanofluids rises with rise of $\mathrm{Nt}$ , and as a result rises the $\varphi (\eta )$ . The influence of the $S$ on the $\varphi (\eta )$ is shown in Figure 14. It is seen that $\varphi (\eta )$ directly varies with $S$ . Enhancing $S$ rises the temperature, which in consequence rises kinetic energy of the liquid, so the fluid film speed rises. Figure 15 identifies the influence of $\mathrm{Sc}.$ The dimensionless number $\mathrm{Sc}$ is stated as the ratio of momentum and mass diffusivity. It is obvious that amassed $\mathrm{Sc}$ reduces the $\varphi (\eta )$ and as a result decreases the boundary-layer thickness.

Figure 2.

The effect of $M$ on $f(\eta )$ when ${\beta }_0=0.6,\rho =1,\sigma =0.5,k=1,{\nu }_{\mathrm{nf}}=0.2,\gamma =0.6$ .

Figure 3.

The effect of $M$ on $g(\eta )$ when ${\beta }_0=0.6,\rho =1,\sigma =0.5,k=1,{\nu }_{\mathrm{nf}}=0.2,\gamma =0.1$ .

Figure 4.

The influence of $M$ on $k(\eta )$ when ${\beta }_0=0.6,\rho =1,\sigma =0.5,k=1,{\nu }_{\mathrm{nf}}=0.2,\gamma =0.7$ .

Figure 5.

The effect of $M$ on $s(\eta )$ when ${\beta }_0=0.6,\rho =1,\sigma =0.5,k=1,{\nu }_{\mathrm{nf}}=0.2,\gamma =0.7$ .

Figure 6.

The effect of $\gamma$ on $f(\eta )$ when $\mathrm{\Omega }=1,\rho =1,\sigma =0.5,k=1,{\nu }_{\mathrm{nf}}=0.2,M=1$ .

Figure 7.

The influence of $\gamma$ on $g(\eta )$ when $\mathrm{\Omega }=1,\rho =1,\sigma =0.5,k=1,{\nu }_{\mathrm{nf}}=0.2,M=1$ .

Figure 8.

The influence of $\gamma$ on $k(\eta )$ when $\mathrm{\Omega }=1,\rho =1,\sigma =0.5,k=1,{\nu }_{\mathrm{nf}}=0.2,M=1$ .

Figure 9.

The influence of $\gamma$ on $s(\eta )$ when $\mathrm{\Omega }=1,\rho =1,\sigma =0.5,k=1,{\nu }_{\mathrm{nf}}=0.2,M=1$ .

Figure 10.

The effect of $\mathrm{Pr}$ on $\theta (\eta )$ when $R=0.3,A_1=A_2=A_3=A_4=0.3,I_1=0.1,j_1=0.1$ .

Figure 11.

The effect of $R$ on $\theta (\eta )$ when $\mathrm{Pr}=0.5,A_1=A_2=A_3=A_4=0.3,I_1=0.1,j_1=0.1$ .

Figure 12.

The effect of $\mathrm{Nb}$ on $\varphi (\eta )$ when $\mathrm{Nt}=0.6,\mathrm{Sc}=0.6,S=0.7$ .

Figure 13.

The influence of $\mathrm{Nt}$ on $\varphi (\eta )$ when $\mathrm{Nb}=0.6,\mathrm{Sc}=0.7,S=0.7$ .

Figure 14.

The influence of unsteadiness parameter $(S)$ on $\varphi (\eta )$ when $\mathrm{Nb}=0.6,\mathrm{Sc}=0.7,\mathrm{Nt}=0.5$ .

Figure 15.

The effect of Schmidt number $(\mathrm{Sc})$ on $\varphi (\eta )$ when $\mathrm{Nb}=0.6,S=0.7,\mathrm{Nt}=0.5$ .

Table 1 observes the influence of $\mathrm{Pr},R,A_1$ , and $k$ on $\mathrm{Nu}$ . Figures 16 and 17 label the effect of $\mathrm{Pr},R$ . It is seeming that $\mathrm{Pr},R$ has positive impact on $\mathrm{Nu}$ . It is also realized that the rising values of $\mathrm{Pr}$ and $R$ increase the $\mathrm{Nu}$ . Figures 18 and 19 identify that the $\mathrm{Nu}$ reduces for the amassed values of $A_1$ and $k$ . Table 2 observes the effect of $\mathrm{Nb},\mathrm{Nt},S$ , and $\mathrm{Sc}$ on $\mathrm{Sh}$ . The $\mathrm{Sh}$ increases for increased value of $\mathrm{Nb}$ and $S$ . For increasing values of $\mathrm{Nt}\mathrm{and}\mathrm{Sc}$ , $\mathrm{Sh}$ reduces. Variation in heat flux and mass flux are shown in Tables 1 and 2 numerically.

Table 1.

Nusselt number for various values of embedded parameters.

$\mathrm{Pr}$	$R$	$A_1$	$\sigma$	$A_4\delta \theta \prime (0)$
0.1	0.1	0.4	1.6	$0.12254$
0.3				$0.12792$
0.5				$0.13371$
0.7	0.1			$0.12254$
	0.3			$0.13325$
	0.5			$0.13593$
	0.7	0.4		$0.12254$
		0.5		$0.18750$
		0.6		$0.17220$
		0.7	1.6	$0.63549$
			1.1	$0.63548$
			0.6	$0.63451$
			0.1	$0.63422$

Table 2.

Sherwood number for various values of embedded parameters.

$\mathrm{Nb}$	$\mathrm{Nt}$	$S$	$\mathrm{Sc}$	$\delta \varphi \prime (0)$
0.5	0.6	0.7	1.6	$0.48195$
0.6				$0.48976$
0.7				$0.49561$
0.8	0.6			$0.5001$
	0.7			$0.48195$
	0.8			$0.47101$
	0.9	0.7		$0.46008$
		0.6		$0.47302$
		0.5		$0.43121$
		0.4	1.6	$0.51653$
			1.1	$0.48195$
			0.6	$0.44959$
			0.1	$0.41931$

Figure 16.

The influence of Prandtl number $(\mathrm{Pr})$ on Nusselt number when $R=0.3,A_1=A_2=A_3=A_4=0.3,I_1=0.1,j_1=0.1$ .

Figure 17.

The influence of radiation parameter $(R)$ on Nusselt number when $\mathrm{Pr}=0.5,A_1=A_2=A_3=A_4=0.3,I_1=0.1,j_1=0.1$ .

Figure 18.

The influence of $A_1$ on Nusselt number when $R=0.3,A_1=A_2=A_3=A_4=0.3,I_1=0.1,j_1=0.1$ .

Figure 19.

The influence of $k$ on Nusselt number when $R=0.3,A_1=A_2=A_3=A_4=0.3,I_1=0.1,j_1=0.1$ .

Conclusion

In this article, the 3-D Darcy–Forchheimer MHD thin-film nanofluid containing flow over a steady inclined rotating plane is observed. Nanofluid thin-film flow is taken thermally radiated and suction/injection effect is also considered. By similarity variables, the partial differential equations are transformed into a set of first-ordinary differential equations (ODES). The required ODES is solved by HAM with association of mathematica program. In this exploration, flow through a steady inclined rotating disk of nanofluid 3-D condensation film is solved by HAM.

• A higher e value of the S parameter delivers to cool the boundary layer and the $\mathrm{Nu}$ rises.

• A greater value of the Sc decreases the Sh as a result to enhancement in the concentration and kinematic viscosity of the fluids.

• For a larger amount of Pr, the $\mathrm{Nu}$ decreases.

• Rising nanofluid volume fraction leads to rise in $\theta (\eta )$ and the maximum temperature was achieved in this situation for Al₂O₃-water.