Computational treatment of statistical declaration probable error for flow of nanomaterials with irreversibility

Abstract

The current investigation deals with entropy analysis for radiative flow of nanomaterials between two heated rotating disks. Titanium ( $\mathrm{Ti}{O}_2$ and $\mathrm{GO}$ ) and Graphene oxides are taken as nanoparticles. Water ( $H_{2}O$ ) is used as a conventional base liquid. Dissipation and radiation effects are incorporated in energy equation. Rotating disks have different angular velocities. Both disks have different stretching rates. Attention is focused for statistical declaration and probable error. Physical feature of entropy analysis is studied through thermodynamics second law. Nonlinear partial system (PDEs) is reduced to ordinary one (ODEs). Homotopy analysis technique (HAM) is used for convergent series solution. Features of sundry variables on entropy optimization, temperature, Bejan number, and velocity are discussed for both nanoparticles ( $\mathrm{Ti}{O}_2$ and $\mathrm{GO}$ ). Computational outcomes for velocity gradient and Nusselt number are addressed through tabulated values. For larger Reynold number the radial and axial velocities are decreased. Temperature is augmented for against higher Eckert number and radiation parameter. Bejan number and entropy rate are augmented versus radiation parameter. Bejan number and Entropy rate have opposite trend via Reynold number. Statistical declaration and probable error are deliberated via Tables.

Introduction

Nanoliquids are integrated through engineered colloidal suspended nano-size solid (1–100 nm) particles in base liquid (water, ethylene glycol, and oil). Nanomaterials have potential applications in industry, medicine, engineering and heat transfer technology. The commonly utilized nanoparticles consist of metals/CNTs, titania, stably suspended oxides, silica improvised thermo physical characteristics of base fluids. Obviously solid has more thermal conductivity in comparison to fluid. Transport rate of nanoparticles can develop augmentation of thermal conductivities. Due to this peculiarity characteristic the nanomaterials have wide range applications in microelectronics, nuclear plants, chemical engineering process, medical instruments, solar collectors, fabrication of medicines heat exchangers, and discourse of cancer and heat pipes. Titanium dioxide nanoparticles are similar to perfect semiconductors for photocatalysis. Titanium dioxide have cheaper coast, much safer and with high accuracy. Choi and Estman ¹ initiated the concept of nanomaterials. According to his study, addition of tiny particles into base liquid result in nanomaterial. Zangooee et al. ² investigated the hydromagnetic nanoparticles $(\mathrm{TiO}2-\mathrm{GO})$ flow between two rotating disks. Joule heating and radiation in energy equation are accounted. Ghadikolaei et al. ³ studied Carreau nanofluid by considering of $(\mathrm{TiO}2-\mathrm{GO})$ ethylene glycol. Rotating cone and non-linear thermal radiation are taken. Ijaz Khan et al. ⁴ discussed entropy analysis in hydromagnetic nanomaterials flow with homogeneous and heterogeneous reactions. The nanomaterials with $(\mathrm{TiO}-\mathrm{GO})$ are taken. Madaki et al. ⁵ studied unsteady squeezing nanofluid flow between two parallel plates. Heat generation/absorption and thermal radiation are present. Ganesh Kumar et al. ⁶ explored hydromagnetic in nanomaterials fluid flow through convergent/divergent channel. Here Joule heating and Darcy-Forchheimer effects are examined. Thermal and entropy analyses for radiative flow of nanomaterials with melting effect in flow by a stretching surface is illustrated by Khan et al. ⁷ Heat transfer analysis in hydromagnetic flow of hybrid (Ag) nanomaterials past a slender cylinder is studied by Patil and Kulkarni. ⁸ Khan et al. ⁹ reported thermal analysis for time-dependent flow of hybrid nanomaterial over a stretching sheet. Some recent studies regarding this topic are mentioned in Refs.^10–25

Recently many studies have been made about the applications and irreversibilities rates of the second law of thermo-dynamics. In heat energy, the concept of irreversibility is very frequently used due to the measure of irreversibilities of heat policy. Many researchers have made contributions. Irreversibility is key feature of physical action in nature. Initially Bejan^26,27 described the concept of irreversibilty analysis in convective flow with heat transfer problems. Mliki and Abbassi ²⁸ studied hydromagnetic flow hybrid nanomaterials with entropy analysis in an incinerator shaped cavity. Numerical outcomes of heat transfer analysis in hybrid (silver-water) nanoliquid with entropy rate in an annulus fin are investigated by Shahsavar et al. ²⁹ Few studies about entropy optimization have been discussed by Refs.^30–40

This investigation examines flow of Titanium oxide and Graphene oxide nanoparticles $(\mathrm{TiO}2-\mathrm{GO})$ base fluid flow with irreversibility. Heat transfer associated to viscous dissipation and thermal radiation is examined. Attempt is focused to statistical declaration and probable error. Computations for entropy rate are presented. Skin friction coefficient and Nusselt number for both disks (upper and lower) are computed with probable error and statistical declaration. Homotopic procedure^41–45 is adopted for the computations. Comparative studies are also presented through Table 4. The key findings of influential parameters are summarized.

Problem statement

We investigate axisymmetric boundary layer flow in presence of titanium oxide $(\mathrm{TiO}2)$ and Graphene oxide $(\mathrm{GO}).$ Water is taken as a base material. Viscous dissipation is present. The stretching disks are rotating with different angular velocities. The lower and upper disks rotates with ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$ namely. Both disks have different temperature. The upper and lower disks have temperatures $T_1$ and $T_2$ respectively. Flow sketch is highlighted in Figure 1.

Figure 1.

Flow sketch.

The components of velocity ( $u^{*}$ , $v^{*}$ , $w^{*}$ ) are

$$\frac{\partial u^{*}}{\partial r}+\frac{\partial w^{*}}{\partial z}+\frac{u^{*}}{r}=0,$$

(1)

$$\begin{array}{cc}\multicolumn{1}{c}{u^{*}\frac{\partial u^{*}}{\partial r}}& +w\frac{\partial u^{*}}{\partial z}-\frac{v^{*}}{r^2}=-\frac{1}{{\rho }_{\mathrm{nf}}}\frac{\partial p}{\partial r}\hfill \\ \multicolumn{1}{c}{}& +{\nu }_{\mathrm{nf}}(\frac{{\partial }^2{u}^{*}}{\partial r^2}+\frac{1}{r}\frac{\partial u^{*}}{\partial r}+\frac{{\partial }^2{u}^{*}}{\partial z^2}-\frac{u^{*}}{r^2}),\hfill \end{array}$$

(2)

$$\begin{array}{cc}\multicolumn{1}{c}{u^{*}\frac{\partial v^{*}}{\partial r}}& +w\frac{\partial v^{*}}{\partial z}+\frac{u^{*}{v}^{*}}{r}\hfill \\ \multicolumn{1}{c}{}& ={\nu }_{\mathrm{nf}}(\frac{{\partial }^2{v}^{*}}{\partial r^2}+\frac{1}{r}\frac{\partial v^{*}}{\partial r}+\frac{{\partial }^2{v}^{*}}{\partial z^2}-\frac{v^{*}}{r^2}),\hfill \end{array}$$

(3)

$$\begin{array}{cc}\multicolumn{1}{c}{u^{*}\frac{\partial w^{*}}{\partial r}}& +w\frac{\partial w^{*}}{\partial z}=-\frac{1}{{\rho }_{\mathrm{nf}}}\frac{\partial p}{\partial z}\hfill \\ \multicolumn{1}{c}{}& +{\nu }_{\mathrm{nf}}(\frac{{\partial }^2{w}^{*}}{\partial r^2}+\frac{{\partial }^2{w}^{*}}{\partial z^2}+\frac{1}{r}\frac{\partial w^{*}}{\partial r}),\hfill \end{array}$$

(4)

$$\begin{array}{c}{\left(\rho c_p\right)}_{\mathrm{nf}}(u^{*}\frac{\partial T^{*}}{\partial x}+w^{*}\frac{\partial T^{*}}{\partial r})=(k_{\mathrm{nf}}+\frac{16{\sigma }^{*}{T}_2^{*3}}{3k^{*}})\hfill \\ (+\frac{{\partial }^2{T}^{*}}{\partial r^2}\frac{1}{r}\frac{\partial T^{*}}{\partial r}+\frac{{\partial }^2{T}^{*}}{\partial z^2})+{\mu }_{\mathrm{nf}}\mathrm{\Phi },\hfill \end{array}$$

(5)

with

$$\begin{array}{cc}\multicolumn{1}{c}{\mathrm{\Phi }=}& 2[{\left(\frac{\partial u^{*}}{\partial z}\right)}^2+{\left(\frac{u^{*}}{r}\right)}^2+{\left(\frac{\partial w^{*}}{\partial z}\right)}^2]+\left[{\frac{\partial v^{*}}{\partial z}}^2\right]\hfill \\ \multicolumn{1}{c}{}& +{[\frac{\partial w^{*}}{\partial r}+\frac{\partial u^{*}}{\partial z}]}^2+{\left[r\frac{\partial }{\partial r}\left(\frac{v^{*}}{r}\right)\right]}^2.\hfill \end{array}$$

(6)

$$\begin{array}{c}{u}^{*}=r{d}_{a1,}{v}^{*}=r{\mathrm{\Omega }}_1,w^{*}=0,T^{*}=T_1^{*},\mathrm{at}z=0,\hfill \\ u^{*}=r{d}_{a2}{v}^{*}=r{\mathrm{\Omega }}_2,w^{*}=0,T^{*}=T_2^{*},\mathrm{at}z=h.\hfill \end{array}\}$$

(7)

In above expression $T$ fluid temperature $(r,\theta ,z)$ are cylindrical coordinates, $\left(\rho \right)$ density, $\left(p\right)$ pressure, $\left({\nu }_{\mathrm{nf}}\right)$ is kinematic viscosity, $k_{\mathrm{nf}}$ thermal conductivity, $\left({\sigma }^{*}\right)$ Stefan-Boltzman constant, $\left(k^{*}\right)$ mean absorption coefficient, $\left(d{a}_1\right)$ and $\left(d{a}_2\right)$ the stretching rates for lower and upper disks, $\left({\mu }_{\mathrm{nf}}\right)$ dynamic viscosity, and $\left(\rho c_p\right)$ heat capacitance. Nanofluid properties can be put in mathematical form as

$$\begin{array}{c}{\rho }_{\mathrm{nf}}=(1-\varphi ){\rho }_f+\varphi {\rho }_s,\hfill \\ {\mu }_{\mathrm{nf}}=\frac{{\mu }_f}{{(1-\varphi )}^{2.5}},\hfill \\ {(\rho c_p)}_{\mathrm{nf}}=(1-\varphi ){(\rho c_p)}_f+\varphi {(\rho c_p)}_s,\hfill \\ \frac{k_{\mathrm{nf}}}{k_f}=\frac{k_s+2k_f-2\varphi (k_f-k_s)}{k_s+2k_f+2\varphi (k_f-k_s)},\hfill \end{array}\}$$

(8)

where $\left(\mathrm{nf}\right)$ is for nanofluid, $\left(\varphi \right)$ is for nanoparticles volume fraction, and $\left(f\right)$ for base fluid.

Taking

$$\begin{array}{c}{u}^{*}=r{\mathrm{\Omega }}_{1}f\prime \left(\zeta \right),v^{*}=r{\mathrm{\Omega }}_{1}g\left(\zeta \right),w^{*}=-2h{\mathrm{\Omega }}_{1}f\left(\zeta \right),{\theta }^{*}=\frac{T^{*}-T_2^{*}}{T_1^{*}-T_2^{*}},\hfill \\ p={\rho }_f{\mathrm{\Omega }}_1{\nu }_f(P\left(\zeta \right)+\frac{1}{2}\frac{r^{*2}}{h^2}\epsilon ),\zeta =\frac{z}{h},\hfill \end{array}\}$$

(9)

one has

$$\begin{array}{c}\frac{1}{{(1-\varphi )}^{2.5}(1-\varphi +\varphi \frac{{\rho }_{\mathrm{nf}}}{{\rho }_f})}f″\prime +\mathrm{Rd}(2\mathrm{ff}″-f^{\prime 2}+g^2)\hfill \\ -\frac{\epsilon }{(1-\varphi +\varphi \frac{{\rho }_{\mathrm{nf}}}{{\rho }_f})}=0,\hfill \end{array}$$

(10)

$$\frac{1}{{(1-\varphi )}^{\frac{25}{10}}(1-\varphi +\varphi \frac{{\rho }_{\mathrm{nf}}}{{\rho }_f})}g″+\mathrm{Rd}(2\mathrm{fg}\prime -2f\prime g)=0,$$

(11)

$$\begin{array}{c}\frac{1}{(1-\varphi +\varphi \frac{{\rho }_{\mathrm{nf}}}{{\rho }_f})}P\prime =-4\mathrm{Rdff}\prime \hfill \\ -\frac{2}{{(1-\varphi )}^{\frac{25}{10}}(1-\varphi +\varphi \frac{{\rho }_{\mathrm{nf}}}{{\rho }_f})}f″=0,\hfill \end{array}$$

(12)

$$\begin{array}{c}(\frac{k_{\mathrm{nf}}}{k_f}+R)\theta ″+2\mathrm{Pr}\mathrm{Rd}(1-\varphi +\frac{{(\rho c_p}_{{)}_{\mathrm{nf}}}}{{(\rho c_p)}_f}\varphi )f\theta \prime +\frac{\mathrm{Pr}\mathrm{Ec}}{{(1-\varphi )}^{\frac{25}{10}}}12{f\prime }^2+\hfill \\ \multicolumn{1}{c}{\frac{\mathrm{Pr}\mathrm{Ec}}{{(1-\varphi )}^{\frac{25}{10}}}({g\prime }^2+{f″}^2)=0,}\end{array}\}$$

(13)

$$\begin{array}{c}\multicolumn{1}{c}{f\left(0\right)=0,f\left(1\right)=0,f\prime \left(0\right)=A_{1,a},f\prime \left(1\right)=A_{1,b},g\left(0\right)=1,g\left(1\right)=\mathrm{\Omega },}\\ \multicolumn{1}{c}{\theta \left(0\right)=1,\theta \left(1\right)=0,P\left(0\right)=0,}\end{array}\}$$

(14)

In above expressions indicate, Reynold number ( $\mathrm{Rd}$ ), stretching parameters $\left(\left(A_{1,a}\right)\mathrm{and}\left(A_{1,b}\right)\right)$ , Prandtl number $\left(\mathrm{Pr}\right)$ , rotational parameter $\left(\mathrm{\Omega }\right)$ , Eckert number $\left(\mathrm{Ec}\right)$ , Brinkman number $\left(\mathrm{Br}\right)$ , thermal radiation $\left(R\right)$ , and dimensionless constant $\left(\epsilon \right)$ . Mathematically we have

$$\begin{array}{c}\mathrm{Rd}=\frac{{\mathrm{\Omega }}_1{h}^2}{{\nu }_f},\mathrm{Pr}=\frac{{\left(\rho C_p\right)}_f{\nu }_f}{k_f},A_{1,a}=\frac{a_1}{{\mathrm{\Omega }}_1},A_{1,b}=\frac{a_2}{{\mathrm{\Omega }}_2},,\mathrm{\Omega }=\frac{{\mathrm{\Omega }}_1}{{\mathrm{\Omega }}_2},\hfill \\ \multicolumn{1}{c}{R=\frac{16{\sigma }^{*}{T}_2^3}{3k^{*}{k}_f},\mathrm{Ec}=\frac{{\mathrm{\Omega }}_1^2{r}^2}{(T_1-T_2)C_p},\mathrm{Br}=\frac{{\mu }_f{\mathrm{\Omega }}_1^2{r}^2}{k_f(T_1-T_2)}.}\end{array}\}$$

(15)

After differentiation of equation (10) with respect to $\xi$ we obtain

$$\frac{1}{{(1-\varphi )}^{2.5}(1-\varphi +\varphi \frac{{\rho }_{\mathrm{nf}}}{{\rho }_f})}{f}^{\mathrm{iv}}+\mathrm{Rd}(2\mathrm{ff}″\prime +2\mathrm{gg}\prime )=0.$$

(16)

On integration of equation (12) one has

$$\begin{array}{c}\epsilon =\frac{1}{{(1-\varphi )}^{2.5}}f″\prime \left(0\right)-\mathrm{Rd}(1-\varphi +\varphi \frac{{\rho }_{\mathrm{nf}}}{{\rho }_f})\hfill \\ [(f\prime {\left(0\right)}^2)-(g\prime {\left(0\right)}^2)].\hfill \end{array}$$

(17)

Integration of equation (12) gives rise to

$$\begin{array}{c}\frac{1}{(1-\varphi +\varphi \frac{{\rho }_{\mathrm{nf}}}{{\rho }_f})}P\hfill \\ =-2[\mathrm{Rd}\left(f^2\right)+\frac{1}{{(1-\varphi )}^{\frac{25}{10}}(1-\varphi +\varphi \frac{{\rho }_{\mathrm{nf}}}{{\rho }_f})}(f\prime -f\prime \left(0\right))],\hfill \end{array}$$

(18)

Physical quantities

Skin friction

In radial and tangential directions, the shear stress at lower disk satisfies

$$\begin{array}{c}\multicolumn{1}{c}{{\tau }_{\mathrm{zr}}={\mu }_{\mathrm{nf}}{\frac{\partial u^{*}}{\partial z}|}_{z=0}=\frac{{\mu }_{f}r{\mathrm{\Omega }}_{1}f″\left(0\right)}{{(1-\phi )}^{\frac{25}{10}}h},}\\ \multicolumn{1}{c}{{\tau }_{z\theta }={\mu }_{\mathrm{nf}}{\frac{\partial v^{*}}{\partial z}|}_{z=0}=\frac{{\mu }_{f}r{\mathrm{\Omega }}_{1}g\prime \left(0\right)}{{(1-\phi )}^{2\frac{25}{10}}h}.}\end{array}\}$$

(19)

Total shear stress satisfies

$${\tau }_w=\sqrt{({\tau }_{\mathrm{zr}}^2+{\tau }_{z\theta }^2)},$$

(20)

In dimensionless form

$$\begin{array}{c}\multicolumn{1}{c}{C_{f_0}=\frac{{{\tau }_w|}_{z=0}}{{\rho }_f{\left(r{\mathrm{\Omega }}_1\right)}^2}=\frac{1}{R{d}_r{(1-\varphi )}^{2.5}}{[{(f″\left(0\right))}^2+{(g\prime \left(0\right))}^2]}^{\frac{1}{2}},}\\ \multicolumn{1}{c}{C_{f_1}=\frac{{{\tau }_w|}_{z=h}}{{\rho }_f{\left(r{\mathrm{\Omega }}_1\right)}^2}=\frac{1}{R{d}_r{(1-\varphi )}^{2.5}}{[{(f″\left(1\right))}^2+{(g\prime \left(1\right))}^2]}^{\frac{1}{2}},}\end{array}\}$$

(21)

where $R{d}_r(=\frac{r{\mathrm{\Omega }}_{1}h}{{\nu }_f})$ is local Reynolds number.

Rates of heat transfer

Here

$$\begin{array}{c}N{u}_{x_0}={\frac{h{q}_w}{k_f(T_1^{*}-T_2^{*})}|}_{z=0},\hfill \\ \multicolumn{1}{c}{N{u}_{x_1}={\frac{h{q}_w}{k_f(T_1^{*}-T_2^{*})}|}_{z=h},}\end{array}\}$$

(22)

where heat flux $\left(q_w\right)$ satisfies

$$\begin{array}{c}\multicolumn{1}{c}{{q_w|}_{z=0}=-k_{\mathrm{nf}}\frac{\partial T}{\partial z}+{q_r|}_{z=0}=-\frac{(T_1^{*}-T_2^{*})}{h}(k_{\mathrm{nf}}+\frac{16{\sigma }^{*}{T}_2^3}{3k^{*}})\theta \prime \left(0\right),}\\ \multicolumn{1}{c}{{q_w|}_{z=h}=-k_{\mathrm{nf}}\frac{\partial T}{\partial z}+{q_r|}_{z=h}=-\frac{(T_1^{*}-T_2^{*})}{h}(k_{\mathrm{nf}}+\frac{16{\sigma }^{*}{T}_2^3}{3k^{*}})\theta \prime \left(1\right),}\end{array}\}$$

(23)

Nusselt number in dimensionless form

$$\begin{array}{c}\multicolumn{1}{c}{N{u}_{x_0}=-(\frac{k_{\mathrm{nf}}}{k_f}+R)\theta \prime \left(0\right),}\\ \multicolumn{1}{c}{N{u}_{x_1}=-(\frac{k_{\mathrm{nf}}}{k_f}+R)\theta \prime \left(1\right)}\end{array}\}$$

(24)

Entropy modeling

We have

$$S_G=\frac{k_f}{T_f^2}[\frac{k_{\mathrm{nf}}}{k_f}{\left(\frac{\partial T}{\partial z}\right)}^2+\frac{16{\sigma }^{*}{T}_2^3}{3k^{*}{k}_f}{\left(\frac{\partial T}{\partial z}\right)}^2]+\frac{{\mu }_{\mathrm{nf}}}{T_f}\mathrm{\Phi }.$$

(25)

we have

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{Sg}=\frac{k_f}{T_f^{*2}}[\frac{k_{\mathrm{nf}}}{k_f}{\left(\frac{\partial T^{*}}{\partial z}\right)}^2+\frac{16{\sigma }^{*}{T}_2^{*3}}{3k^{*}{k}_f}{\left(\frac{\partial T^{*}}{\partial z}\right)}^2]}\\ \multicolumn{1}{c}{+\frac{{\mu }_{\mathrm{nf}}}{T_f^{*}}\left[\begin{array}{c}2[{\left(\frac{\partial u^{*}}{\partial z}\right)}^2+{\left(\frac{u^{*}}{r}\right)}^2+{\left(\frac{\partial w^{*}}{\partial z}\right)}^2]+\hfill \\ \multicolumn{1}{c}{\left[{\left(\frac{\partial v^{*}}{\partial z}\right)}^2\right]+{[\frac{\partial w^{*}}{\partial r}+\frac{\partial u^{*}}{\partial z}]}^2+{\left[r\frac{\partial }{\partial r}\left(\frac{v^{*}}{r}\right)\right]}^2}\end{array}\right]\}}\end{array}$$

(26)

Dimensionless version is

$$\begin{array}{cc}\multicolumn{1}{c}{\mathrm{Ng}=}& (\frac{k_{\mathrm{nf}}}{k_f}+R)\theta {\prime }^2{\alpha }_1^{*}\hfill \\ \multicolumn{1}{c}{}& +\frac{\mathrm{Br}}{\mathrm{Rd}}\frac{1}{{(1-\varphi )}^{\frac{25}{10}}}(12{f\prime }^2+A_{1,c}{g\prime }^2+A_{1,c}{f″}^2).\hfill \end{array}$$

(27)

where $\left({\alpha }_1^{*}\right),$ $\left(\mathrm{Br}\right)$ , $\left(\mathrm{Ng}\right),$ and $\left(A_{1,c}\right)$ denote respectively the temperature difference variable, Brinkman number, and entropy. These parameters are

$$\begin{array}{c}{\alpha }_1=\frac{T_1^{*}-T_2^{*}}{T_f^{*}}=\frac{\mathrm{\Delta }{T}^{*}}{T_f^{*}},\mathrm{Br}=\frac{{\mu }_f{\mathrm{\Omega }}_1^2{h}^2}{k_f\mathrm{\Delta }{T}^{*}},\hfill \\ \mathrm{Ng}=\frac{T_f^{*}\mathrm{Sg}{\nu }_f}{k_f\mathrm{\Delta }{T}^{*}{\mathrm{\Omega }}_1},A_{1,c}=\frac{r^{*2}}{h^2},\hfill \end{array}$$

(28)

Bejan number is

$$\mathrm{Be}=\frac{(\frac{k_{\mathrm{CNTs}}}{k_f}+R_t){\theta \prime }^2{\alpha }_1}{(\frac{k_{\mathrm{nf}}}{k_f}+R){\theta \prime }^2{\alpha }_1+\frac{\mathrm{Br}}{\mathrm{Rd}}\frac{1}{{(1-\varphi )}^{\frac{25}{10}}}(12{f\prime }^2+A_{1,c}{g\prime }^2+A_{1,c}{f″}^2)}.$$

(29)

Homotopy analysis solution

The initial guesses and operators satisfy

$$\begin{array}{c}{f}_0\left(\eta \right)=(A_1\eta -2A_1{\eta }^2-A_2{\eta }^2+A_1{\eta }^3+A_2{\eta }^3),\hfill \\ \multicolumn{1}{c}{g_0\left(\eta \right)=1+(\mathrm{\Omega }-1)\eta ,}\\ \multicolumn{1}{c}{{\theta }_0\left(\eta \right)=(1-\eta ),}\end{array}\}$$

(30)

$${\hslash }_f=f^{\prime \prime \prime \prime },{\hslash }_g=g″,{\hslash }_{\theta }=\theta ″,$$

(31)

The properties satisfied by the above operators are

$$\begin{array}{c}{L}_f[c_1+c_2\xi +c_3^2\xi +c_4{\xi }^3]=0,\hfill \\ \multicolumn{1}{c}{L_g[c_5+c_6\xi ]=0,}\\ \multicolumn{1}{c}{L_{\theta }[c_7+c_8\xi ]=0,}\end{array}\}$$

(32)

in which $c_i(i=1-8)$ are the constants.

Series solution

The auxiliary parameters ${\hslash }_f,$ ${\hslash }_g$ , and ${\hslash }_{\theta }$ have key role in adjustment and convergence control of homotopic series solutions. Thus $\hslash -\mathrm{curves}$ for $11\mathrm{th}$ iteration are displayed. Figures 2 and 3 are displayed to show the $(\mathrm{TiO}2-\mathrm{GO})$ for lower and upper disk. Auxiliary parameter regarding $f″(0)$ , $g\prime (0)$ , and $\theta \prime (0)$ for $\left(\mathrm{TiO}2\right)$ water based nanofluid satisfy $-1.9\le {\hslash }_f\le -0.1,$ $-1.95\le {\hslash }_g\le 0.2$ , and $-0.85\le {\hslash }_g\le -0.3$ . However for $(\mathrm{GO})$ the water base nanofluid $-1.1\le {\hslash }_f\le 0.01,-1.1\le {\hslash }_g\le 0.01$ and $-1.1\le {\hslash }_{\theta }\le 0.01.$ Table 1 and 2 are constructed for the series solutions convergence $(\mathrm{TiO}2-\mathrm{GO})$ upto sixth digits. Table 2 shows that $f″(0)$ , $g\prime (0)$ , and $\theta \prime (0)$ for $\left(\mathrm{TiO}2\right)$ nanofluid converge at ${11}^{\mathrm{th}},{15}^{\mathrm{th}},{18}^{\mathrm{th}}$ , and ${20}^{\mathrm{th}}$ iterations respectively. Table 3 is prepared for $\left(\mathrm{GO}\right)$ water base nanofluid. It indicates that $f″(0)$ , $g\prime (0)$ , and $\theta \prime (0)$ converge at ${10}^{\mathrm{th}}$ , ${13}^{\mathrm{th}}$ , ${16}^{\mathrm{th}}$ , and ${22}^{\mathrm{th}}$ iterations is find.

Figure 2.

$\hslash -\mathrm{curves}$ $f″(0)$ , $g\prime (0)$ , and $\theta \prime (0)$ for $\left(\mathrm{TiO}2\right)$ .

Figure 3.

$\hslash -\mathrm{curves}$ $f″(0)$ , $g\prime (0)$ , and $\theta \prime (0)$ for $\left(\mathrm{GO}\right)$ .

Table 1.

Thermophysical properties of nanofluids.^3–6

$\mathrm{Fluids}$	$c_p(J/\mathrm{kgK})$	$k(W/\mathrm{mK})$	$\rho (\mathrm{kg}/m^3)$
$H_{2}O$	4179	997.1	0.613
$\mathrm{Ti}{O}_2$	686.2	4250	8.9538
$\mathrm{GO}$	717	1800	5000

Table 2.

Case of $(\mathrm{TiO}2).$

$\mathrm{No}\mathrm{of}\mathrm{iterations}$	$f″(0)$	$g\prime (0)$	$\theta \prime (0)$
1	−3.02998	−0.789610	−0.59239
5	−3.02809	−0.789169	−0.72332
10	−3.02809	−0.789169	−0.759680
15	−3.02809	−0.789169	−0.75495
20	−3.02809	−0.789169	−0.75495
25	−3.02809	−0.789169	−0.75495
35	−3.02809	−0.789169	−0.75495
45	−3.02809	−0.789169	−0.75495
50	−3.02809	−0.789169	−0.75495

Table 3.

Case for $\mathrm{GO}$ .

$\mathrm{No}\mathrm{of}\mathrm{iterations}$	$f″(0)$	$g\prime (0)$	$\theta \prime (0)$
1	−3.02998	−0.78960	−0.59368
5	−3.02050	−0.79258	−0.690479
10	−3.01977	−0.79259	−0.81241
15	−3.01979	−0.79260	−764685
20	−3.01979	−0.79260	−0.78328
25	−3.01979	−0.79260	−0.78328
35	−3.01979	−0.79260	−0.78328
45	−3.01979	−0.79260	−0.78328
50	−3.01979	−0.79260	−0.78328

Table 4.

Comparative study drag force with Khan et al. ⁴⁶

$\mathrm{\Omega }$	Khan et al. 46		Present results
	$f″\left(0\right)$	$g\prime \left(0\right)$	$f″\left(0\right)$	$g\prime \left(0\right)$
−1.0	0.06666	2.00095	0.06667	2.00093
−0.8	0.08399	1.80259	0.08398	1.80260
−0.3	0.10395	1.30443	0.10396	1.30444
0.0	0.09997	1.00428	0.09998	1.00429
0.5	0.0667	0.50261	0.06671	0.50263

Discussion

This section consists of predictions of velocity, temperature, entropy generation, Bejan number, skin friction, Nusselt number, statistical declaration, and probable error with variation of physical parameters ( $A1=0.2$ $\mathrm{Rd}=2.0$ , $A_{1,a}=2.0$ , $A_{1,b}=1.0$ , $\mathrm{\Omega }=1.0$ , $\mathrm{Pr}=6.2$ , $R=1.0$ , $\mathrm{Ec}=0.2$ , $\mathrm{Br}=0.3$ , ${\alpha }_1=1.3$ ).

Velocity

Figures 4–10 represented the impact of axial velocity $f\left(\zeta \right),$ radial velocity $f\prime \left(\zeta \right)$ , and tangential velocity $g\left(\zeta \right)$ with respect to various influential variables. Figures 4 and 5 accomplish features of $\left(A1\right)$ on axial $f\left(\zeta \right)$ and radial velocity $f\prime \left(\zeta \right).$ It is noted from Figure 4 that axial velocity $f\left(\zeta \right)$ is an increasing function of lower disk stretched parameter $\left(A1\right)$ . Radial velocity with respect to $A1$ is shown in Figure 5. Initially $f\prime \left(\zeta \right)$ upsurges for higher $\left(A1\right)$ and then it decreases when $\left(A1\right)$ approaches to maximum range. Figures 6 and 7 are portrayed for the salient features of $\left(\mathrm{Rd}\right)$ on axial $f\left(\zeta \right)$ and radial $f\prime \left(\zeta \right)$ . Clearly the axial velocity $f\left(\zeta \right)$ decays versus larger $\left(\mathrm{Rd}\right)$ . Physically the viscosity increases versus higher $\mathrm{Rd}$ and consequently axial velocity decays (see Figure 6). Dual behavior of $f\prime \left(\zeta \right)$ versus higher $\mathrm{Rd}$ is guaranteed in Figure 7. Physically, at lower disk the viscosity is increase at the surface and this factor slowly decrease when the fluid goes to the upper disk surface. In Figure 8 we have noticed that the higher value of $\left(\mathrm{\Omega }\right)$ variable in result axial velocity decline while in Figure 9 radial velocity has dual behave present. Here increasing behavior of tangential velocity is noticed for larger upper disk stretched parameter. Increasing impact of $\left(\mathrm{Rd}\right)$ on tangential velocity is reported in Figure 10.

Figure 4.

$f\left(\zeta \right)$ via $A1$ .

Figure 5.

$f\prime \left(\zeta \right)$ via $A1$ .

Figure 6.

$f\left(\zeta \right)$ via $\mathrm{Rd}$ .

Figure 7.

$f\prime \left(\zeta \right)$ via $\mathrm{Rd}$ .

Figure 8.

$f\left(\zeta \right)$ via $\mathrm{\Omega }$ .

Figure 9.

$f\prime \left(\zeta \right)$ via $\mathrm{\Omega }$ .

Figure 10.

$g\left(\zeta \right)$ via $\mathrm{Rd}$ .

Temperature

Figures 11–13 are portrayed to discuss the behavior of flow variables ( $A1=0.2$ , $\mathrm{Rd}=2.0$ , $A_{1,a}=2.0$ , $A_{1,b}=1.0$ , $\mathrm{\Omega }=1.0$ , $\mathrm{Pr}=6.2$ , $R=1.0$ , $\mathrm{Ec}=0.2$ , $\mathrm{Br}=0.3$ , ${\alpha }_1=1.3$ ) on temperature. Titanium dioxide and graphene oxide cases are considered. Figure 11 depicted for impact of $\left(\mathrm{Pr}\right)$ on temperature field. Clearly higher momentum diffusivity reduces thermal conductivity and consequently temperature field $\theta \left(\zeta \right)$ decays versus larger $\left(\mathrm{Pr}\right).$ Figures 12 and 13 are drawn for Brinkman $\left(\mathrm{Br}\right)$ and thermal radiation $\left(R\right)$ aspects on $\theta \left(\zeta \right)$ . A monotic increase for both parameters.

Figure 11.

$\theta \left(\zeta \right)$ via $\mathrm{Pr}$ .

Figure 12.

$\theta \left(\zeta \right)$ via $\mathrm{Br}$ .

Figure 13.

$\theta \left(\zeta \right)$ via $R$ .

Entropy analysis

Figures 14–19 are depicted to show effects of sundry variables ( $A1=0.2$ , $\mathrm{Rd}=2.0$ , $A_{1,a}=2.0$ , $A_{1,b}=1.0$ , $\mathrm{\Omega }=1.0$ , $\mathrm{Pr}=6.2$ , $R=1.0$ , $\mathrm{Ec}=0.2$ , $\mathrm{Br}=0.3$ , ${\alpha }_1=1.3$ ) on entropy generation and Bejan in presence of $(\mathrm{TiO}2-\mathrm{GO})$ outcomes. Figures 14 and 15 elucidate behavior of $\left(\mathrm{Br}\right)$ on entropy generation and Bejan number. Dual behavior is noticed against higher Brinkman number on entropy $\mathrm{Ng}\left(\zeta \right)$ and Bejan $\left(\mathrm{Be}\right)$ number. Through Figure 14 the irreversibility increases versus higher Brinkman $\left(\mathrm{Br}\right)$ while Bejan number is reduced (see Figure 15). Behavior of temperature difference ratio parameter $\left({\alpha }_1\right)$ on entropy generation $\mathrm{Ng}\left(\zeta \right)$ and Bejan $\left(\mathrm{Be}\right)$ number is depicted in Figures 16 and 17. Here $\mathrm{Ng}\left(\zeta \right)$ and $\mathrm{Be}$ are enhanced versus higher $\left({\alpha }_1\right).$ It is noticed that behavior of $\left(\mathrm{GO}\right)$ dominates than $\left(\mathrm{TiO}2\right).$ Impact of radiation $\left(R\right)$ on $\mathrm{Ng}\left(\zeta \right)$ and $\left(\mathrm{Be}\right)$ is shown in Figures 18 and 19. In fact for higher $\left(R\right)$ the inside source of energy increases and consequently kinetic energy of the particle enhances. Diffusion enhances inside the particles and so both $\mathrm{Ng}\left(\zeta \right)$ and $\left(\mathrm{Be}\right)$ are increased.

Figure 14.

$\mathrm{Ng}\left(\zeta \right)$ via $\mathrm{Br}$ .

Figure 15.

$\mathrm{Be}$ via $\mathrm{Br}$ .

Figure 16.

$\mathrm{Ng}\left(\zeta \right)$ via $\alpha 1$ .

Figure 17.

$\mathrm{Be}$ via $\alpha 1$ .

Figure 18.

$\mathrm{Ng}\left(\zeta \right)$ via $R$ .

Figure 19.

$\mathrm{Be}$ via $R$ .

Physical quantities

Tables 5 and 6 display the influence of physical variable ( $A1=0.2$ , $\mathrm{Rd}=2.0$ , $A_{1,a}=2.0$ , $A_{1,b}=1.0$ , $\mathrm{\Omega }=1.0$ , $\mathrm{Pr}=6.2$ , $R=1.0$ , $\mathrm{Ec}=0.2$ , $\mathrm{Br}=0.3$ , ${\alpha }_1=1.3$ ) on skin friction $\left(\mathrm{Cf}\right)$ for $\left(\mathrm{TiO}2\right)$ and $\left(\mathrm{GO}\right)$ . Outcomes of skin frictions for lower disk rise against higher $\mathrm{Rd}$ and $A2.$ However satisfies of $\left(\mathrm{\Omega }\right)$ for skin friction at lower disk is reverse. It is noticed that impact of $\mathrm{Cf}$ in case of $\left(\mathrm{Ti}{O}_2\right)$ is more than $\left(\mathrm{GO}\right)$ . Further the skin frictions at lower and upper disks for $\mathrm{Ti}{O}_2$ and $\mathrm{GO}$ cases have opposite impact. Tables 7 and 8 are prepared for behaviors of Reynold number, Brinkman $\left(\mathrm{Br}\right)$ and thermal radiation $\left(R\right)$ on $(\mathrm{Nu})$ for both disks in $\left(\mathrm{Ti}{O}_2\right)$ and $\left(\mathrm{GO}\right)$ cases. It is examined that $(\mathrm{Nu})$ is increased via $\left(\mathrm{Br}\right)$ and $\left(R\right)$ . Magnitude of $(\mathrm{Nu})$ against $\mathrm{Br}$ reduces at lower disk and it enhances for upper disk.

Table 5.

Numerical results for $\left(\mathrm{Cf}\right)$ for $(A2=0.1)$ .

$\mathrm{Rd}$	$A1$	$\mathrm{\Omega }$	$C{f}_0(\mathrm{Ti}{O}_2)$	$C{f}_1(\mathrm{Ti}{O}_2)$	$C{f}_0(\mathrm{GO})$	$C{f}_1(\mathrm{GO})$
0.7	0.4	0.2	3.59648	2.45031	3.55075	2.47043
0.8			3.61831	2.44077	3.56623	2.46359
0.9			3.64004	2.43131	3.58167	2.45681
0.7	0.5		4.26404	2.73703	4.21256	2.76022
	0.6		4.94634	3.03339	4.88896	3.05998
	0.4	0.3	3.51795	2.35853	3.47556	2.37848
		0.4	3.44561	2.27769	3.40706	2.29700

Table 6.

Skin friction for $(A2=0.6)$ .

$\mathrm{Rd}$	$A1$	$\mathrm{\Omega }$	$C{f}_0(\mathrm{Ti}{O}_2)$	$C{f}_1(\mathrm{Ti}{O}_2)$	$C{f}_0(\mathrm{GO})$	$C{f}_1(\mathrm{GO})$
0.7			5.17681	5.73496	5.15071	5.74329
0.8			5.18917	5.73089	5.15957	5.74050
0.9			5.20142	5.72676	5.16838	5.73769
0.7	0.5		5.87370	6.06751	5.84121	6.07783
	0.6		6.57660	6.39928	6.53740	6.41212
	0.4	0.3	5.12428	5.69224	5.10036	5.70151
		0.4	5.07600	5.65758	5.05471	5.66692

Table 7.

Nusselt number for $(A2=0.1).$

$R$	$\mathrm{Br}$	$\mathrm{Rd}$	$N{u}_0(\mathrm{TiO}2)$	$N{u}_1(\mathrm{TiO}2)$	$N{u}_0(\mathrm{GO})$	$N{u}_1(\mathrm{GO})$
0.4	03		0.52140	2.93525	0.68312	3.10698
0.5			0.62232	3.03512	0.78387	3.20688
0.6			0.72316	3.13500	0.88454	3.30679
	0.4		0.36128	3.03639	0.52404	3.20870
	05		0.20115	3.13752	0.36460	3.31042
	0.3	0.8	0.53388	2.92426	0.69572	3.09705
		0.9	0.54631	2.91349	0.70831	3.08726

Table 8.

Nusselt number for $(A2=0.6)$ .

$R$	$\mathrm{Br}$	$\mathrm{Rd}$	$N{u}_0(\mathrm{TiO}2)$	$N{u}_1(\mathrm{TiO}2)$	$N{u}_0(\mathrm{GO})$	$N{u}_1(\mathrm{GO})$
0.4	03	0.7	−0.92499	5.76586	−0.75516	5.90276
0.5			−0.82288	5.86225	−0.65350	5.99994
0.6			−0.72098	5.95897	−0.55199	6.09736
0.4	0.4		−1.21589	6.12943	−1.04538	6.26515
	0.5		−1.50679	6.49300	−1.33559	6.62754
	0.3	0.8	−0.93251	5.79810	−0.76130	5.93087
		0.9	−0.94020	5.83075	−0.76755	5.95926

Statistical approach

Tables 5–8 illustrate skin friction and Nusselt number for both disk when $(A2=0.1)$ and $(A2=0.6)$ . The cases of $(\mathrm{TiO}2-\mathrm{GO}).$ Tables 9 and 10 derived correlation coefficient. Value of correlation coefficient is between $(-1)$ to $(+1)$ .

Table 9.

Estimations of $P.E$ for skin friction.

$r_b$	$C{f}_0(\mathrm{TiO}2)$	$C{f}_0(\mathrm{TiO}2)$	$C{f}_1(\mathrm{TiO}2)$	$C{f}_1(\mathrm{TiO}2)$
	$A2=0.1$	$A2=0.6$	$A2=0.1$	$A2=0.6$
$\mathrm{Rd}$	0.98843627	0.98771116	0.98651511	0.98708052
$A1$	0.99868174	0.99499446	0.99325752	0.98346188
$\mathrm{\Omega }$	0.90727518	0.91346672	0.89891190	0.91505640
$r_b$	$C{f}_0(\mathrm{GO})$	$C{f}_0(\mathrm{GO})$	$C{f}_1(\mathrm{GO})$	$C{f}_1(\mathrm{GO})$
	$A2=0.1$	$A2=0.6$	$A2=0.1$	$A2=0.6$
$\mathrm{Rd}$	0.9881422	0.98757785	0.98665327	0.98712756
$A1$	0.99871652	0.99493677	0.99329760	0.98352614
$\mathrm{\Omega }$	0.90769355	0.91369010	0.89898741	0.91510807

Table 10.

Estimation of $P.E$ for Nusselt number.

$r_b$	$N{u}_0(\mathrm{Ti}{O}_2)$	$N{u}_0(\mathrm{Ti}{O}_2)$	$N{u}_1(\mathrm{Ti}{O}_2)$	$N{u}_1(\mathrm{Ti}{O}_2)$
	$A1=0.1$	$A1=0.6$	$A1=0.1$	$A1=0.6$
$\mathrm{Rt}$	0.99892488	−0.91658012	0.97798424	0.97269360
$\mathrm{Ec}$	0.67345954	−0.99992035	0.96397847	0.97233535
$\mathrm{Rd}$	0.99157676	−0.98885052	0.98645150	0.98835317
$r_b$	$N{u}_0(\mathrm{Ti}{O}_2)$	$N{u}_0(\mathrm{Ti}{O}_2)$	$N{u}_1(\mathrm{Ti}{O}_2)$	$N{u}_1(\mathrm{Ti}{O}_2)$
	$A1=0.1$	$A1=0.6$	$A1=0.1$	$A1=0.6$
$\mathrm{Rt}$	0.99610348	−0.90025687	0.97751026	0.97326646
$\mathrm{Ec}$	0.77368407	−0.99948974	0.96342578	0.97189186
$\mathrm{Rd}$	0.99068097	−0.98884708	0.98656496	0.98818905

Probable error

Tables 11 and 12 illustrate $\left(\mathrm{Cf}\right)$ and $\left(\mathrm{Nu}\right)$ . Probable is the function of $\left(r_b\right)$ . Theme of $(P.E)$ is to inspect the correct and exactness of approximations of correlation coefficient. By Saleh and Sundar ⁴⁰ the probable error satisfies

$$P.E\left(r_b\right)=\frac{1-r_b^2}{\sqrt{n}}*\frac{6745}{10000}$$

(33)

Table 11.

$P.E$ for skin friction.

$P.E\left(r_b\right)$	$C{f}_0(\mathrm{Ti}{O}_2)$	$C{f}_0(\mathrm{Ti}{O}_2)$	$C{f}_1(\mathrm{Ti}{O}_2)$	$C{f}_1(\mathrm{Ti}{O}_2)$
	$A2=0.1$	$A2=0.6$	$A2=0.1$	$A2=0.6$
$\mathrm{Rd}$	8.36902261 × 10−4	6.22612544 × 10−2	1.69581251 × 10−2	2.09770981 × 10−2
$A1$	2.12800940 × 10−1	6.20325745 × 10−5	2.75499113 × 10−2	2.12484504 × 10−3
$\mathrm{\Omega }$	6.53277276 × 10−3	8.63531298 × 10−3	1.04807052 × 10−2	9.01825661 × 10−2
$P.E\left(r_b\right)$	$C{f}_0(\mathrm{GO})$	$C{f}_0(\mathrm{GO})$	$C{f}_1(\mathrm{GO})$	$C{f}_1(\mathrm{GO})$
	$A2=0.1$	$A2=0.6$	$A2=0.1$	$A2=0.6$
$\mathrm{Rd}$	3.02887456 × 10−3	7.38089801 × 10−2	1.73190675 × 10−1	2.05429842 × 10−2
$A1$	1.56319341 × 10−1	3.97312319 × 10−4	2.79647466 × 10−2	2.15842284 × 10−2
$\mathrm{\Omega }$	7.22426555 × 10−3	8.63796233 × 10−3	1.03935297 × 10−1	914458150 × 10−3

Table 12.

Estimation of $P.E$ for Nusselt number.

$P.E\left(r_b\right)$	$N{u}_0(\mathrm{Ti}{O}_2)$	$N{u}_0(\mathrm{Ti}{O}_2)$	$N{u}_1(\mathrm{Ti}{O}_2)$	$N{u}_1(\mathrm{Ti}{O}_2)$
	$A2=0.1$	$A2=0.6$	$A2=0.1$	$A2=0.6$
$R$	8.36902261 × 10−4	6.22612544 × 10−2	1.69581251 × 10−2	2.09770981 × 10−2
$\mathrm{Br}$	2.12800940 × 10−1	6.20325745 × 10−5	2.75499113 × 10−2	2.12484504 × 10−3
$\mathrm{Rd}$	6.53277276 × 10−3	8.63531298 × 10−3	1.04807052 × 10−2	9.01825661 × 10−2
$P.E\left(r_b\right)$	$N{u}_0(\mathrm{GO})$	$N{u}_0(\mathrm{GO})$	$N{u}_1(\mathrm{GO})$	$N{u}_1(\mathrm{GO})$
	$A2=0.1$	$A2=0.6$	$A2=0.1$	$A2=0.6$
$R$	3.02887456 × 10−3	7.38089801 × 10−2	1.73190675 × 10−1	2.05429842 × 10−2
$\mathrm{Br}$	1.56319341 × 10−1	3.97312319 × 10−4	2.79647466 × 10−2	2.15842284 × 10−2
$\mathrm{Rd}$	7.22426555 × 10−3	8.63796233 × 10−3	1.03935297 × 10−1	914458150 × 10−3

Above equation $\left(n\right)$ denoted for observation number and utilized value is $\left(\frac{6745}{10000}\right)$ . We use ${\mu }_1\pm \left(\frac{6745}{10000}\right).$ Here $\left({\mu }_1\right)$ mean and $\left(\sigma \right)$ stand for standard deviation.

Finally

$$r_b>P.E\left(r_b\right)\mathrm{for}\mathrm{significance}\mathrm{correlation}$$

(34)

$$r_b<P.E\left(r_b\right)\mathrm{for}\mathrm{insignificance}\mathrm{correlation}$$

(35)

Statistical result

Tables 13 and 14 examined numerical results of $\left(\frac{r_b}{P.E\left(r_b\right)}\right)$ for $\left(\mathrm{Cf}\right)$ and $\left(\mathrm{Nu}\right)$ for both disks. Result is computed and compared all simulations of $(\frac{r_b}{P.E\left(r_b\right)}>6).$

Table 13.

Numerical results of $\left(\frac{r_b}{P.E(r_b)}\right)$ for $\left(\mathrm{Cf}\right).$

$\frac{r_b}{P.E(r_b)}$	$C{f}_0(\mathrm{Ti}{O}_2)$	$C{f}_0(\mathrm{Ti}{O}_2)$	$C{f}_1(\mathrm{Ti}{O}_2)$	$C{f}_1(\mathrm{Ti}{O}_2)$
	$A2=0.1$	$A2=0.6$	$A2=0.1$	$A2=0.6$
$\mathrm{Rd}$	110.386954	103.835167	94.567767	98.734915
$A1$	973.332147	255.862539	189.783065	76.988562
$\mathrm{\Omega }$	013.173717	014.166655	12.0251624	14.444892
$\frac{r_b}{P.E(r_b)}$	$C{f}_0(\mathrm{GO})$	$C{f}_0(\mathrm{GO})$	$C{f}_1(\mathrm{GO})$	$C{f}_1(\mathrm{GO})$
	$A2=0.1$	$A2=0.6$	$A2=0.1$	$A2=0.6$
$\mathrm{Rd}$	107.287557	102.711387	95.55343	99.098096
$A1$	999.725108	252.939912	190.921823	77.291418
$\mathrm{\Omega }$	13.236625	14.20513512	12.034684	14.454115

Table 14.

Values of $\frac{r_b}{P.E(r_b)}$ for Nusselt number.

$\frac{r_b}{P.E(r_b)}$	$N{u}_0(\mathrm{TiO}2)$	$N{u}_0(\mathrm{TiO}2)$	$N{u}_1(\mathrm{TiO}2)$	$N{u}_1(\mathrm{TiO}2)$
	$A2=0.1$	$A2=0.6$	$A2=0.1$	$A2=0.6$
$R$	1193.598018	−014.721517	57.6705405	46.3693116
$\mathrm{Br}$	03.1647395	−16119.2786	34.9902567	45.760296
$\mathrm{Rd}$	151.784976	−114.512412	94.1207181	19.5947047
$\frac{r_b}{P.E(r_b)}$	$N{u}_0(\mathrm{GO})$	$N{u}_0(\mathrm{GO})$	$N{u}_1(\mathrm{GO})$	$N{u}_1(\mathrm{GO})$
	$A2=0.1$	$A2=0.6$	$A2=0.1$	$A2=0.6$
$R$	328.8691758	−012.1971428	05.644127	47.377073
$\mathrm{Br}$	004.9493816	−02515.62736	34.451439	45.027871
$\mathrm{Rd}$	137.132413	−114.4768919	09.492107	108.062797

Conclusions

Main finding of present analysis are given below.

• A decrement in radial and axial velocity components is seen through Reynold number.

• A intensification in velocity is noted for rotation variable.

• An amplification in temperature is observed through radiation and Brinkman number.

• Larger estimation of Prandtl number diminishes the thermal field.

• An intensification in Bejan and entropy number is noticed for radiation variable.

• A decrement in Bejan number is noted through Brinkman number, while opposite effect holds for entropy.

• Higher approximation of thermal ratio variable has similar behavior on Bejan and entropy numbers.