Mixed convection MHD hybrid nanofluid flow between two parallel rotating discs with joule heating and chemical reactions using bvp4c

Abstract

Several characteristics of the resulting fluid are influenced by the nanoparticle suspension. Understanding the heat transfer mechanism in nanofluids is necessary for many production and manufacturing applications. The current study examines the effects of mixed convection MHD and Joule heating on the flow of (TiO₂–GO/water) hybrid nanofluid and (TiO₂/water) nanofluids in the porous media between two parallel, infinitely spinning discs when radiation occurs. Using the “bvp4c” function in MATLAB, the governing equations are numerically solved. A graphic is used to show how important parameters affect the velocity, temperature, and concentration of nanoparticles. Finally, a table depicting the interactions between several important factors and skin friction, the Nusselt number, and the Sherwood number at the upper and lower discs is created. The findings show that while the local skin fraction drops with an increase in the mixed convection parameter, the heat transmission rate at both discs increases. Additionally, the rate of heat transmission at both the top and lower discs is reduced as the radiation and magnetic parameters increase. This study’s findings will be helpful to numerous nanofluid-based medicinal applications, architectural design systems, better oil recovery systems, and transportation procedures, among other sectors.

Introduction

Hybrid nanofluids are frequently used in experimental and numerical fluid dynamics research due to their importance in thermal and energy performance. Like regular NFs, hybrids NFs are usually created by mixing nanoparticles with common fluids like water, oil, and ethylene glycol. The distinction is that hybrid NFs has a solid phase made up of two or more different kinds of nanoparticles, one of which is commonly graphene, carbon nanotubes, or a nanoparticle with extremely high heat conductivity. Due to the inclusion of two or more different types of particles in hybrid NFs, the fluid’s thermal and rheological characteristics alter significantly. The key attributes that change to improve thermal performance as a result of the interaction between various nanoparticles are those related to thermophysics. Abderrahmane et al. ¹ investigated the flow of a hybrid nanofluid in a cubic hollow under a magnetic field in three dimensions. The flow of an incompressible, viscous, electrically conducting hybrid nanofluid over a vertical surface that is accelerated exponentially was studied by Krishna and Chamkha. ² The Yamada-Ota model and the Xue model were extended versions that were used to show the magnetohydrodynamic stationary point flow in hybrid nanofluid in Ishtiaq et al. ³ In the temperature range of 5°C–55°C and under various laboratory settings, Esfe et al. ⁴ examined the rheological behavior of a particular kind of hybrid nanofluid (HNF). Gamachu and Ibrahim ⁵ investigated the fluid flow, temperature, and entropy formation in a partially heated cavity saturated with porous media for buoyancy-driven flow of hybrid nanofluid. For various physical characteristics and surfaces, such as a moving plate, the investigation of numerical solutions in hybrid nanofluid flow was investigated,^6–12 cylinder,^13–17 disc,^18–23 and thin needle.^24–28

According to theory, joule heating is the generation of heat as a result of resistive losses during the conversion of electric to thermal state energy. The arrangement of electrical and electronic devices frequently uses this method. According to research by Reddy and Reddy, ²⁹ this control parameter is employed to raise the temperature of the nanofluid from the perspective of boundary layer flow. Maskeen et al. ³⁰ and Khashi’ie et al. ³¹ examined the impact of Joule heating in Cu-Al₂O₃/water along a stretching and contracting cylinder, respectively. According to Yan et al. ³² and Khashi’ie et al., ³¹ the dimensionless Eckert number (derived from the Joule heating) has no effect on prolonging the point of separation. According to Chamkha et al., ³³ MHD hybrid nanofluid flow with Joule heating and thermal radiation between two parallel plates was examined. Li et al. ³⁴ conducted an analysis of the effect of Joule heating on the flow of nanofluid through a permeable conduit. While this was going on, Saranya et al. ³⁵ noticed how the hybrid ferrofluid flow through a contracting cylinder was affected by viscous dissipation and Joule heating. They employ a base fluid mixture of water (50%) and ethylene glycol (50%) along with magnetite-cobalt ferrite hybrid nanoparticles in their study. The combined impact of an angled magnetic field and a radiation parameter on the Marangoni hybrid nanofluid flow was examined by Al-Mdallal et al. ³⁶ The following remarkable articles on the energy transmission of nanofluids were also discussed in Refs.^37–42

Magnetic fields can induce currents to flow in a mobile conductive fluid, and rotating MHD flows make use of this possibility. Liquid metals, plasmas, and electrolytes are notable examples of MHD fluids. MHD flows can change a number of flow regimes by employing a Lorentzian drag force. The study of MHD flow and its effect on entropy production in related systems has recently received considerable interest due to its importance in many fields.^43–46 In magnetohydrodynamic (MHD) flow on a spinning disc, Arikoglu et al. ⁴⁷ investigated the impact of slip on the generation of entropy. The mixed convection Darcy-Forchheimer flow of ZnO-SAE50 oil nanolubricant across a rotating disc that was inclined and subject to a uniform magnetic field was studied by Nayak et al. ⁴⁸ The dynamics of a rotating disc model with a stretchable flow field were investigated by Mehmood et al. ⁴⁹ Barman et al. ⁵⁰ investigated the entropy generated and the flow of a radiative magneto-hybrid nanofluid across a spinning sphere. Mahanthesh et al.^51–54 explored a nonlinear radiated MHD flow of nanoliquid generated by a rotating disc with an unequal heat source and heat flux condition.

Since just a few researches have been published to examine the significance of HNF between two spinning discs, it can be concluded from the discussion that came before it. To examine the HNF flow (TiO₂-Go/water flow) and NF flow (TiO₂/water flow) between two parallel, infinitely rotating discs, authors have created a model. Additionally, the significance of mixed convection, joule heating, thermal radiation, magnetic field, and chemical reaction are also researched as a novelty. Additionally, this study seeks to examine the mass transmission rate (MTR) and the HTR of the HNF flow (TiO₂-Go/water flow) and NF flow (TiO₂/water flow) at both discs. The bvp4c solver in MATLAB is used to look for the numerical solution. According to the author’s knowledge, no one has ever reported a comparison of the HNF flow (TiO₂-Go/water) and NF flow (TiO₂/water) between two parallel, infinite spinning discs, and the findings are fresh and original.

Problem formulation

Consider the axisymmetric, incompressible, and steady three-dimensional flow of the hybrid nanofluid $(\mathrm{Ti}{O}_2-\mathrm{GO}/\mathrm{water})$ and nanofluid $(\mathrm{GO}/\mathrm{water})$ between the two spinning infinite discs (see Figure 1). For the cylindrical coordinate system, the notation $(r,\theta ,z)$ has been employed (see Figure 1). In the z-direction, a magnetic field $B_0$ is present and exposed to the flow. Mixed convection and joule heating are utilized as a new porous medium in the development of the model. Thermal radiation is additionally utilized to study hybrid nanofluid transcends (HTR) and mass transfer rate (MTR). The chemical reaction is also represented by the final term in the concentration equation (equation (6)).The lower and upper disc are placed at $z=0$ and $z=h$ (see equation (7)). The lower and upper discs’ radial shrinkage rates are $A_1$ and $A_2$ which represented in the boundary conditions. Additionally, the angular velocities of the bottom and higher discs, respectively, are assumed to be ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$ . It is considered that hot liquids with surface temperatures $T_1$ (lower disc) and away from the surface, temperature $T_2$ (upper disc) with a heat transfer coefficient of $h_1$ (lower disc) are convectively heating the outer surfaces of the discs. $C_1$ and $C_2$ also stand for the concentration of NPs at the lower and upper discs, respectively. The $(r,\theta ,z)$ cylindrical coordinate system is used to create the following mathematical model.^52–54

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

(1)

$$\begin{array}{c}\multicolumn{1}{c}{u\frac{\partial u}{\partial r}+w\frac{\partial u}{\partial z}-\frac{v^2}{r}=-\frac{1}{{\rho }_{\mathrm{hnf}}}\frac{\partial p}{\partial r}+{\nu }_{\mathrm{hnf}}(\frac{{\partial }^{2}u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{{\partial }^{2}u}{\partial z^2}-\frac{u}{r^2})}\\ -\frac{{\sigma }_{\mathrm{hnf}}}{{\rho }_{\mathrm{hnf}}}{B}_0^{2}u+\frac{{\left(\rho \beta \right)}_{\mathrm{hnf}}}{{\rho }_{\mathrm{hnf}}}g(T-T_{\infty }),\hfill \end{array}$$

(2)

$$u\frac{\partial v}{\partial r}+w\frac{\partial v}{\partial z}+\frac{\mathrm{uv}}{r}={\nu }_{\mathrm{hnf}}(\frac{{\partial }^{2}v}{\partial r^2}+\frac{1}{r}\frac{\partial v}{\partial r}+\frac{{\partial }^{2}v}{\partial z^2}-\frac{v}{r^2})-\frac{{\sigma }_{\mathrm{hnf}}}{{\rho }_{\mathrm{hnf}}}{B}_0^{2}v,$$

(3)

$$u\frac{\partial w}{\partial r}+w\frac{\partial w}{\partial z}=-\frac{1}{{\rho }_{\mathrm{hnf}}}\frac{\partial p}{\partial z}+{\nu }_{\mathrm{hnf}}(\frac{{\partial }^{2}w}{\partial r^2}+\frac{1}{r}\frac{\partial w}{\partial r}+\frac{{\partial }^{2}w}{\partial z^2}),$$

(4)

$$\begin{array}{c}\multicolumn{1}{c}{(u\frac{\partial T}{\partial r}+w\frac{\partial T}{\partial z})=\frac{1}{{\left(\rho c_p\right)}_{\mathrm{hnf}}}(k_{\mathrm{hnf}}+\frac{16{\sigma }^0{T}_2^3}{3k^0})(\frac{1}{r}\frac{\partial T}{\partial r}+\frac{{\partial }^{2}T}{\partial r^2}+\frac{{\partial }^{2}T}{\partial z^2})}\\ +\frac{{\sigma }_{\mathrm{hnf}}}{{\left(\rho c_p\right)}_{\mathrm{hnf}}}{B}_0^2(u^2+v^2),\hfill \end{array}$$

(5)

$$u\frac{\partial C}{\partial r}+w\frac{\partial C}{\partial z}=D(\frac{{\partial }^{2}C}{\partial r^2}+\frac{1}{r}\frac{\partial C}{\partial r}+\frac{{\partial }^{2}C}{\partial z^2})-k_1(C-C_{\infty }),$$

(6)

$$\begin{array}{c}\multicolumn{1}{c}{\left\{\begin{array}{c}u=b_{1}r,v=r{\mathrm{\Omega }}_1,w=0,k_{\mathrm{hnf}}\frac{\partial T}{\partial z}=-h_1(T_1-T),\hfill \\ \multicolumn{1}{c}{C=C_1,\mathrm{at}\mathrm{lower}\mathrm{disk}z=0,}\\ \multicolumn{1}{c}{u=b_{2}r,v=r{\mathrm{\Omega }}_2,w=0,T=T_2,C\to C_2,}\\ \multicolumn{1}{c}{\mathrm{at}\mathrm{upper}\mathrm{disk}z=h,}\end{array}\right\}}\end{array}$$

(7)

Figure 1.

Flow sketch.

where $r$ and $\theta$ are the radial and tangential axes of the cylindrical system, respectively, and $z$ is its vertical axis. $p$ denotes pressure, $T$ denotes temperature, $T_1$ and $T_2$ denote lower and upper disc temperatures, respectively, and $k_0^{*}$ is the mean absorption coefficient and ${\sigma }_0^{*}$ stefan Boltzmann constant (Table 1).

Table 1.

Titanium di oxide and graphene oxide thermophysical characteristics with ordinary fluid water.^55,58

Properties	$k(w/\mathrm{mK})$	$\sigma {\left(\mathrm{Um}\right)}^{-1}$	$c_p(J/\mathrm{kgK})$	$\rho (\mathrm{kg}/m^3)$
Titanium di oxide $\left(\mathrm{Ti}{O}_2\right)$	8.9538	2.6 $e^6$	686.2	4250
Graphene oxide $\left(\mathrm{GO}\right)$	5000	1.1 $e^{-5}$	717	1800
Water $\left(H_{2}O\right)$	0.613	0.05	4179	997.1

The effective hybnanofluids density ${\rho }_{\mathrm{hnf}}$ , $k_{\mathrm{hnf}}$ thermal conductivity, ${\left(\rho c_p\right)}_{\mathrm{hnf}}$ heat capacitance, ${\mu }_{\mathrm{hnf}}$ dynamic viscosity, and ${\sigma }_{\mathrm{hnf}}$ electrical conductivity can be introduced as follow (Table 2):

Table 2.

Thermophysical models of hybrid nanofluids.^55–59

Properties	Hybrid nanofluid
Density	${\rho }_{\mathrm{hnf}}={\rho }_f(1-{\varphi }_b)(1-{\varphi }_a)+{\varphi }_a{\rho }_{\mathrm{sa}}+{\varphi }_b{\rho }_{\mathrm{sb}}.$
viscosity	${\mu }_{\mathrm{hnf}}=\frac{{\mu }_f}{{(1-{\varphi }_a)}^{2.5}{(1-{\varphi }_b)}^{2.5}}.$
Heat capacity	$\begin{array}{l}{\left(\rho c_p\right)}_{hnf}={\left(\rho c_p\right)}_f\left(1-{\varphi }_b\right)\left(\left(1-{\varphi }_a\right)+\frac{{\varphi }_a{\left(\rho c_p\right)}_{sa}}{{\left(\rho c_p\right)}_f}\right)\\ +{\varphi }_b{\left(\rho c_p\right)}_{sb}.\end{array}$
Thermal conductivity ratio	$\begin{array}{c}\frac{k_{\mathrm{hnf}}}{k_{\mathrm{bf}}}=\left(\frac{(k_{\mathrm{sb}}+2k_{\mathrm{bf}})-2{\varphi }_b(k_{\mathrm{bf}}-k_{\mathrm{sb}})}{(k_{\mathrm{sb}}+2k_{\mathrm{bf}})+{\varphi }_b(k_{\mathrm{bf}}-k_{\mathrm{sb}})}\right)\hfill \\ \mathrm{where}\frac{k_{\mathrm{bf}}}{k_f}=\frac{k_{\mathrm{sa}}+2k_f-2{\varphi }_a(k_f-k_{\mathrm{sa}})}{k_{\mathrm{sa}}+2k_f+{\varphi }_a(k_f-k_{\mathrm{sa}})}\hfill \end{array}$
Electrical conductivity ratio	$\begin{array}{c}\frac{{\sigma }_{\mathrm{hnf}}}{{\sigma }_{\mathrm{bf}}}=\left(\frac{{\sigma }_{\mathrm{sb}}+2{\sigma }_{\mathrm{bf}}-2{\varphi }_b({\sigma }_{\mathrm{bf}}-{\sigma }_{\mathrm{sb}})}{{\sigma }_{\mathrm{sb}}+2{\sigma }_{\mathrm{bf}}+{\varphi }_b({\sigma }_{\mathrm{bf}}-{\sigma }_{\mathrm{sb}})}\right)\hfill \\ \mathrm{where}\frac{{\sigma }_{\mathrm{bf}}}{{\sigma }_f}=\frac{{\sigma }_{\mathrm{sa}}+2{\sigma }_f-2{\varphi }_a({\sigma }_f-{\sigma }_{\mathrm{sa}})}{{\sigma }_{\mathrm{sa}}+2{\sigma }_f+{\varphi }_a({\sigma }_f-{\sigma }_{\mathrm{sa}})}.\hfill \end{array}$
Buoyancy coefficient	$\begin{array}{l}{\left(\rho \beta \right)}_{hnf}={\left(\rho \beta \right)}_f\left(1-{\varphi }_b\right)\left(1-{\varphi }_a\right)+{\varphi }_a{\left(\rho \beta \right)}_{sa}\\ +{\varphi }_b{\left(\rho \beta \right)}_{sb}.\end{array}$

Here ${\beta }_T,k,C_p,v,\mu ,\rho ,{\varphi }_a,{\varphi }_b,\sigma ,\mathrm{sa},\mathrm{sb},f$ and $\mathrm{hnf}$ are thermal expansion coefficient, thermal conductivity, specific heat, dynamic viscosity, density, nanoparticle volume fraction of first nanoparticle, nanoparticle volume fraction of second nanoparticle, electric conductivity, first solid nanoparticle, second solid nanoparticles, base fluid, and hybrid nanofluid, respectively.

It is taken into consideration though using the Karman transformation:

$$\left\{\begin{array}{c}u=r{\mathrm{\Omega }}_{1}f\prime \left(\xi \right),v=r{\mathrm{\Omega }}_{1}g\left(\xi \right),w=-2h{\mathrm{\Omega }}_{1}f\left(\xi \right),\theta =\frac{T-T_2}{T_1-T_2}\hfill \\ \multicolumn{1}{c}{p={\rho }_f{\mathrm{\Omega }}_1{\nu }_f(p\left(\xi \right)+\frac{1}{2}\frac{r^2}{h^2}\epsilon ),\varphi =\frac{C-C_2}{C_1-C_2},\xi =\frac{z}{h}}\end{array}\right\}$$

(8)

Using equation (8) the governing non-linear partial differential equations (1)–(6) together with the boundary conditions (7) reduce to

The continuity equation is satisfied and other equations (2)–(6) are transformed to:

$$\frac{Z_1}{Z_2}f″\prime +\mathrm{Re}(2\mathrm{ff}″-{f\prime }^2+g^2\frac{Z_3}{Z_2}\mathrm{Mf}\prime )-\frac{\epsilon }{Z_2}+\frac{Z_6}{Z_2}\mathrm{Ri}.\theta =0,$$

(9)

$$\frac{Z_1}{Z_2}g″+\mathrm{Re}(2\mathrm{fg}\prime -2f\prime g-M\frac{Z_3}{Z_2}g)=0,$$

(10)

$$P\prime \left(\xi \right)+4\frac{Z_2}{Z_3}\mathrm{Re}\mathrm{ff}\prime +2f″=0,$$

(11)

$$\frac{1}{\mathrm{Pr}}(Z_5+R)\theta ″+2\mathrm{Re}{Z}_{2}f\theta \prime +\mathrm{Ec}\mathrm{Re}M{Z}_3({f\prime }^2+g^2)=0,$$

(12)

$$\varphi ″+2\mathrm{Sc}\mathrm{Re}f\varphi \prime -\mathrm{ScCr}\varphi =0,$$

(13)

Where,

$$\begin{array}{c}\{Z_1=\frac{{\mu }_{\mathrm{hnf}}}{{\mu }_f},Z_2=\frac{{\rho }_{\mathrm{hnf}}}{{\rho }_f},Z_3=\frac{{\sigma }_{\mathrm{hnf}}}{{\sigma }_f},Z_4=\frac{{\left(\rho c_p\right)}_{\mathrm{hnf}}}{{\left(\rho c_p\right)}_f},\hfill \\ Z_5=\frac{k_{\mathrm{hnf}}}{k_f},Z_6=\frac{{\left(\rho \beta \right)}_{\mathrm{hnf}}}{{\rho }_f{\beta }_f}\}\hfill \end{array}$$

The transfer boundary conditions are

$$\begin{array}{c}\multicolumn{1}{c}{\left\{\begin{array}{c}f\left(0\right)=0,f\left(1\right)=0,f\prime \left(0\right)=A_1,f\prime \left(1\right)=A_2,g\left(0\right)=1,\hfill \\ \multicolumn{1}{c}{g\left(1\right)=\mathrm{\Omega },\varphi \left(0\right)=1,}\\ \multicolumn{1}{c}{\theta \prime \left(0\right)=-\frac{1}{Z_5}{B}_1(1-\theta \left(0\right)),\theta \left(1\right)=0,\varphi \left(1\right)=0,}\end{array}\right\}}\end{array}$$

(14)

$$\begin{array}{c}\multicolumn{1}{c}{\left\{\begin{array}{c}\mathrm{Re}=\frac{{\mathrm{\Omega }}_1{h}^2}{{\nu }_f},M=\frac{B_0^2{\sigma }_f}{{\rho }_f{\mathrm{\Omega }}_1},\mathrm{Pr}=\frac{{\left(\rho c_p\right)}_f{\nu }_f}{k_f},\mathrm{Ec}=\frac{r^2{\mathrm{\Omega }}_1^2}{{\left(c_p\right)}_f(T_1-T_2)},\hfill \\ \multicolumn{1}{c}{\mathrm{Cr}=\frac{K_1{h}^2}{{\nu }_f}}\\ \multicolumn{1}{c}{\mathrm{Sc}=\frac{{\nu }_f}{D},A_1=\frac{a_1}{{\mathrm{\Omega }}_1},A_2=\frac{a_2}{{\mathrm{\Omega }}_1},\mathrm{\Omega }=\frac{{\mathrm{\Omega }}_2}{{\mathrm{\Omega }}_1},B_1=\frac{h{h}_1}{k_f},}\\ \multicolumn{1}{c}{\mathrm{Ri}=\frac{\mathrm{Gr}}{{\mathrm{Re}}^2},\mathrm{Gr}=\frac{g{\rho }_f{\beta }_f(T_1-T_2)h^3}{{{\nu }_f}^2}}\end{array}\right\}}\end{array}$$

where $\mathrm{Ri}$ stand for mixed convection, $\mathrm{Re}$ stands for Reynolds number, $\mathrm{Gr}$ expresses the Grashof number, $\mathrm{Pr}$ stands for Prandtl number, $M$ stand for magnetic field, $\mathrm{Sc}$ stands for Schmidt number, scaled stretching parameters $A_1$ and $A_2$ , and $\mathrm{\Omega }$ stand for rotation number, respectively. The diffusion coefficients of Chemical species $\mathrm{Cr}$ and $B_1$ convective parameter.

For making simpler form of equation (9) and removing $\epsilon$ , it can be differentiated with respect to $\xi$

$$\frac{Z_1}{Z_2}{f}^{\mathrm{iv}}+\mathrm{Re}(2\mathrm{ff}″\prime -2\mathrm{gg}\prime -M\frac{Z_3}{Z_2}f″)=0,$$

(15)

Equations (9)–(14) defined the $\epsilon$ (pressure variable) as follow:

$$\begin{array}{c}\multicolumn{1}{c}{\epsilon =Z_{1}f″\prime \left(0\right)+\mathrm{Re}{Z}_2(2f\left(0\right)f″\left(0\right)-{f\prime }^2\left(0\right)+g^2\left(0\right)\frac{Z_3}{Z_2}\mathrm{Mf}\prime \left(0\right))}\\ +Z_6\mathrm{Ri}.\theta \left(0\right),\hfill \end{array}$$

(16)

For pressure, solve equation (11) by taking integration from 0 to $\xi$ and obtained this form:

$$P=-2\{f\prime \left(\xi \right)-f\prime \left(0\right)+\frac{Z_2}{Z_1}\mathrm{Re}{f}^2\left(\xi \right)\}=0$$

(17)

At lower disc the shear stress in radial and tangential direction is described by:

$${\tau }_{\mathrm{zr}}={\mu }_{\mathrm{nf}}{\frac{\partial u}{\partial z}|}_{z=0}=\frac{{\mu }_{f}r{\mathrm{\Omega }}_{1}f″\left(0\right)}{{(1-\varphi )}^{2.5}h},{\tau }_{z\theta }={\mu }_{\mathrm{nf}}{\frac{\partial v}{\partial z}|}_{z=0}=\frac{{\mu }_{f}r{\mathrm{\Omega }}_{1}g\prime \left(0\right)}{{(1-\varphi )}^{2.5}h},$$

(18)

Therefore total difference is defined as:

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

(19)

The skin fraction coefficients $C{f}_1$ and $C{f}_2$ in lower and upper discs is illustrated by:

$$C{f}_1=\frac{{\tau }_{{w|}_{z=0}}}{{\rho }_f{\left(r{\mathrm{\Omega }}_1\right)}^2}=\frac{1}{{\mathrm{Re}}_r{(1-\varphi )}^{2.5}}{[{(f″\left(0\right))}^2+{(g\prime \left(0\right))}^2]}^{1/2},$$

(20)

$$C{f}_2=\frac{{\tau }_{{w|}_{z=h}}}{{\rho }_f{\left(r{\mathrm{\Omega }}_1\right)}^2}=\frac{1}{{\mathrm{Re}}_r{(1-\varphi )}^{2.5}}{[{(f″\left(1\right))}^2+{(g\prime \left(1\right))}^2]}^{1/2},$$

(21)

In which local Reynolds number ${\mathrm{Re}}_x=\frac{r{\mathrm{\Omega }}_{1}h}{{\nu }_f}$

Nusselt number for lower and upper discs can be expressed as:

$$N{u}_{x1}={\frac{h{q}_w}{k_f(T_1-T_2)}|}_{z=0},N{u}_{x2}={\frac{h{q}_w}{k_f(T_1-T_2)}|}_{z=h},$$

(22)

In which surface heat flux is

$${q_w|}_{z=0}=-k_{\mathrm{nf}}\frac{\partial T}{\partial z}+{q_r|}_{z=0}=-\frac{(T_1-T_2)}{h}(k_{\mathrm{nf}}+\frac{16{\sigma }_0{T}_2^3}{3k^0})\theta \prime \left(0\right),$$

(23)

$${q_w|}_{z=h}=-k_{\mathrm{nf}}\frac{\partial T}{\partial z}+{q_r|}_{z=h}=-\frac{(T_1-T_2)}{h}(k_{\mathrm{nf}}+\frac{16{\sigma }_0{T}_2^3}{3k^0})\theta \prime \left(1\right),$$

(24)

Therefore Nusselt number can be written as:

$$N{u}_{x1}=-B_1[Z_5+R]\theta \prime \left(0\right),N{u}_{x2}=-B_1[Z_5+R]\theta \prime \left(1\right).$$

(25)

Sherwood numbers for lower and upper discs can be expressed as

$$S{h}_{x1}={\frac{J_w}{D(C_1-C_2)}|}_{z=0},S{h}_{x2}={\frac{J_w}{D(C_1-C_2)}|}_{z=h}$$

(26)

In which $J_w$ is surface mass flex is

$${J_m|}_{z=0}=-D{\frac{\partial C}{\partial z}|}_{z=0},{J_m|}_{z=h}=-D{\frac{\partial C}{\partial z}|}_{z=h}$$

(27)

$$S{h}_{x1}=-\varphi \prime \left(0\right),S{h}_{x2}=-\varphi \prime \left(1\right)$$

(28)

Appendix solution methodology

In this section of the paper, we demonstrate the mathematical approach of the investigated numerical scheme in detail, as well as the accuracy of the code for the given flow problems. After applying the similarity variables, the updated similarity set of ordinary differential equations (9)–(13) and the boundary condition (15) take on a highly nonlinear form that is very challenging to solve analytically. Thus, using a finite difference approach known as bvp4c, the following numbers of highlighted equations are approximately solved.^60–62

$$\begin{array}{c}\multicolumn{1}{c}{\left\{\begin{array}{c}f\left(\xi \right)=u\left(1\right),f\prime \left(\xi \right)=u\left(2\right),f″\left(\xi \right)=u\left(3\right),f″\prime \left(\xi \right)=u\left(4\right),\hfill \\ \multicolumn{1}{c}{f^{\mathrm{iv}}\left(\xi \right)=u\prime \left(4\right),}\\ \multicolumn{1}{c}{g\left(\xi \right)=u\left(5\right),g\prime \left(\xi \right)=u\left(6\right),g″\left(\xi \right)=u\prime \left(6\right),\theta \left(\xi \right)=u\left(7\right),}\\ \multicolumn{1}{c}{\theta \prime \left(\xi \right)=u\left(8\right),}\\ \multicolumn{1}{c}{\theta ″\left(\xi \right)=u\prime \left(8\right),\varphi \left(\xi \right)=u\left(9\right),\varphi \prime \left(\xi \right)=u\left(10\right),\varphi ″\left(\xi \right)=u\prime \left(10\right),}\end{array}\right\}}\end{array}$$

(29)

$$\begin{array}{c}f\prime \left(\xi \right)=u\left(2\right)\hfill \\ \multicolumn{1}{c}{u\prime \left(2\right)=u\left(3\right)}\\ \multicolumn{1}{c}{u\prime \left(3\right)=u\left(4\right)}\\ \multicolumn{1}{c}{u\prime \left(4\right)=\frac{Z_1}{Z_2}(-\mathrm{Reu}\left(1\right)u\left(4\right)-2\mathrm{u}\left(5\right)u\left(6\right)-\mathrm{M}\frac{{\mathrm{Z}}_3}{{\mathrm{Z}}_2}u\left(3\right)-Z_6\mathrm{Riu}\left(8\right)),}\\ \multicolumn{1}{c}{u\prime \left(5\right)=u\left(6\right)}\\ \multicolumn{1}{c}{u\prime \left(6\right)=\frac{Z_1}{Z_2}(-2\mathrm{Re}\mathrm{u}\left(1\right)u\left(6\right)-2\mathrm{u}\left(2\right)u\left(5\right)-\mathrm{M}\frac{{\mathrm{Z}}_3}{{\mathrm{Z}}_2}u\left(5\right)),}\\ \multicolumn{1}{c}{u\prime \left(7\right)=u\left(8\right)}\\ \multicolumn{1}{c}{u\prime \left(8\right)=\frac{\mathrm{Pr}}{(Z_5+\mathrm{Rd})}(-2{\mathrm{Z}}_4\mathrm{Reu}\left(1\right)u\left(8\right))-\mathrm{Re}\mathrm{EcM}{Z}_3(u{\left(2\right)}^2+u{\left(2\right)}^2),}\\ \multicolumn{1}{c}{u\prime \left(9\right)=u\left(10\right)}\\ \multicolumn{1}{c}{u\prime \left(10\right)=-2\mathrm{ScReu}\left(1\right)u\left(10\right)-\mathrm{CrScu}\left(9\right),}\end{array}\}$$

(30)

With boundary condition

$$\begin{array}{c}\multicolumn{1}{c}{\left\{\begin{array}{c}{u}_a\left(1\right)=0,u_a\left(2\right)={\mathrm{A}}_1,u_b\left(1\right)=0,u_b\left(1\right)={\mathrm{A}}_2,\hfill \\ \multicolumn{1}{c}{{\mathrm{u}}_a\left(5\right)=1,u\left(5\right)=\mathrm{\Omega },}\\ \multicolumn{1}{c}{u_a\left(8\right)=-\frac{{\mathrm{B}}_1}{{\mathrm{Z}}_5}(1-{\mathrm{u}}_a\left(7\right)),{\mathrm{u}}_b\left(7\right)=0,u_a\left(9\right)=1,}\\ \multicolumn{1}{c}{u_b\left(9\right)=0}\end{array}\right\}}\end{array}$$

(31)

Results and discussion

This section focuses on the numerical outcomes and graphical and tabular results are interpreted physically. Authors have used the following general values in the numerical computations: $\mathrm{Pr}=6.2,M=\mathrm{Re}=\mathrm{Sc}=\mathrm{Ec}=\mathrm{Ri}=\mathrm{Cr}=\mathrm{Rd}=1,B_1=0.5,A_1=A_2=\mathrm{\Omega }=1.5$ and ${\phi }_1={\phi }_2=0.3$ . The analysis is performed for magnetic parameter $(1\le M\le 3)$ , Reynolds numbers $(1\le \mathrm{Re}\le 3)$ , Schmidt number $(1\le \mathrm{Sc}\le 3)$ , Eckert number $(1\le \mathrm{Ec}\le 3)$ , Mixed convection parameter $(1\le \mathrm{Ri}\le 3)$ , chemical reaction parameter $(1\le \mathrm{Cr}\le 3)$ , thermal radiation parameter $(1\le \mathrm{Rd}\le 3)$ , convective parameter $(0.5\le B_1\le 0.7)$ , nanoparticle volume fraction $(0.3\le \phi \le 0.9)$ , and rotation parameter $(1.5\le \mathrm{\Omega }\le 1.7)$ . The effect numerous parameter on velocity profile, temperature profile, and concentration profile of hybrids nanofluid $(\mathrm{Ti}{O}_2-\mathrm{GO})$ /water and nanofluid $\mathrm{Ti}{O}_2$ /water is graphically represented. The effect of the magnetic field parameter $M$ on axial velocity of hybrids nanofluid $(\mathrm{Ti}{O}_2-\mathrm{GO})$ /water and nanofluid $\mathrm{Ti}{O}_2$ /water is depicted in Figure 2. It can be seen in the figure that the velocity profile exhibits a declining trend. Physically, the Lorentz force, which is what provides resistance to liquid motion and causes axial velocity decay, increases with the value of the $M$ . Figure 3 depicts how the magnetic field $M$ affects radial velocity $F\prime \left(\xi \right)$ of hybrids nanofluid $(\mathrm{Ti}{O}_2-\mathrm{GO})$ /water and nanofluid $\mathrm{Ti}{O}_2$ /water. As the value increases, a similar pattern emerges. In terms of physics, as increases, the momentum boundary layer’s radial thickness drops and the thermal boundary layer of the flow thickens. The effect of the magnetic field $M$ on tangential velocity $G\left(\xi \right)$ of hybrids nanofluid $(\mathrm{Ti}{O}_2-\mathrm{GO})$ /water and nanofluid $\mathrm{Ti}{O}_2$ /water is depicted in Figure 4. With a higher value of $M$ , $G\left(\xi \right)$ has a stronger impact. By physically decreasing the thickness of the momentum boundary layer in the tangential direction, raising the thermal boundary layer of flow increases the $M$ . Therefore, the $G\left(\xi \right)$ displayed rising behavior as the thermal boundary layer expanded in the tangential direction. The effect of magnetic parameters $M$ on the temperature profile $\theta \left(\xi \right)$ of hybrids nanofluid $(\mathrm{Ti}{O}_2-\mathrm{GO})$ /water and nanofluid $\mathrm{Ti}{O}_2$ /water is depicted in Figure 5. A higher value of $M$ is seen to improve the $\theta \left(\xi \right)$ . Physically, as $M$ rises, the resistance of fluid particles rises as well, raising the $\theta \left(\xi \right)$ . The effect of the Reynolds number $\mathrm{Re}$ on the temperature profile $\theta \left(\xi \right)$ of hybrids nanofluid $(\mathrm{Ti}{O}_2-\mathrm{GO})$ /water and nanofluid $\mathrm{Ti}{O}_2$ /water is depicted in Figure 6. It is clear that the $\theta \left(\xi \right)$ decreases as the $\mathrm{Re}$ increases. Physically, the thickness of the thermal boundary layer decreases as $\mathrm{Re}$ increases. The effect of the $\mathrm{Re}$ on tangential velocity $G\left(\xi \right)$ of hybrids nanofluid $(\mathrm{Ti}{O}_2-\mathrm{GO})$ /water and nanofluid $\mathrm{Ti}{O}_2$ /water is depicted in Figure 7. It has been found that a higher value of $\mathrm{Re}$ causes them $G\left(\xi \right)$ to increase. Physically, bigger $\mathrm{Re}$ makes sure that the frequency of harsh collisions declines and that less resistance to liquid motion is offered. Figure 8 depicts the influence of Schmidt number $\mathrm{Sc}$ on the concentration field $\psi \left(\xi \right)$ of hybrids nanofluid $(\mathrm{Ti}{O}_2-\mathrm{GO})$ /water and nanofluid $\mathrm{Ti}{O}_2$ /water. With an increase in the $\mathrm{Sc}$ , a decrease in the $\psi \left(\xi \right)$ is seen because of a decrease in mass diffusivity. Figure 9 shows how the radiation parameter $\mathrm{Rd}$ affects the temperature field $\theta \left(\xi \right)$ of hybrids nanofluid $(\mathrm{Ti}{O}_2-\mathrm{GO})$ /water and nanofluid $\mathrm{Ti}{O}_2$ /water. The temperature-increasing tendency is depicted in this figure. Since the ratio of conduction heat transfer to thermal radiation is the $\mathrm{Rd}$ . Therefore, it follows that rising $\mathrm{Rd}$ temperature causes them $\theta \left(\xi \right)$ to rise, as is clear. The distribution of the mixed convection $\mathrm{Ri}$ over the profile of $F\left(\xi \right)$ , radial $F\prime \left(\xi \right)$ , and tangential $G\left(\xi \right)$ velocities of hybrids nanofluid $(\mathrm{Ti}{O}_2-\mathrm{GO})$ /water and nanofluid $\mathrm{Ti}{O}_2$ /water is shown in Figures 10 to 12. Physically, $\mathrm{Ri}$ increases fluid $F\left(\xi \right)$ while reducing boundary layer thickness. Because of forces and pressure, convection causes the fluid density to drop, which causes the fluid’s particles to flow. The fluid velocities in the boundary layer region also tend to increase with an increase in the values of $\mathrm{Ri}$ . It is obvious that $\mathrm{Ri}$ is the buoyancy to inertial forces ratio. Therefore, a rise in the $\mathrm{Ri}$ values leads to an increase in the buoyant forces in the boundary layer region, which quickens the flow of fluid. The effects of $\mathrm{Ri}$ on the temperature $\theta \left(\xi \right)$ and concentration profile $\psi \left(\xi \right)$ of hybrids nanofluid $(\mathrm{Ti}{O}_2-\mathrm{GO})$ /water and nanofluid $\mathrm{Ti}{O}_2$ /water are depicted in Figures 13 and 14. It is obvious that $\mathrm{Ri}$ has a decreasing effect on $\theta \left(\xi \right)$ and an increasing effect on $\psi \left(\xi \right)$ . The rate of heat transfer is really accelerated by an increase in the values of the mixed convection parameter in the flow region, which causes buoyant forces to dominate inertial forces. A $\theta \left(\xi \right)$ and its boundary layer thickness grow up decreasing as a result. Additionally, a weaker buoyancy force is applied to concentration growth due to its support of the pressure gradient. The effect of the Eckert number $\mathrm{Ec}$ on the temperature profile $\theta \left(\xi \right)$ of hybrids nanofluid $(\mathrm{Ti}{O}_2-\mathrm{GO})$ /water and nanofluid $\mathrm{Ti}{O}_2$ /water is shown in Figure 15. It is obvious that temperature increases with $\mathrm{Ec}$ . $\mathrm{Ec}$ is a crucial physical variable for analyzing the thermal behavior of the fluid flow. The outcome (Figure 15) demonstrates that as the $\mathrm{Ec}$ rises, the temperature distribution within the boundary layer does as well. As a result, the heat dissipation decreases, and the thermal boundary layer thickens. The effect of chemical reaction parameters $\mathrm{Cr}$ on the concentration profile $\psi \left(\xi \right)$ of hybrids nanofluid $(\mathrm{Ti}{O}_2-\mathrm{GO})$ /water and nanofluid $\mathrm{Ti}{O}_2$ /water is depicted in Figure 16. The figure makes it very evident that a higher value of $\mathrm{Cr}$ raises the $\psi \left(\xi \right)$ . Chemical molecular diffusivity physically decreases with increasing $\mathrm{Cr}$ due to its consumption in the process. Additionally, as the $\mathrm{Cr}$ parameter grows, the number of solute molecules performing $\mathrm{Cr}$ rises as well, which causes a drop in the $\psi \left(\xi \right)$ . Thus, a destructive $\mathrm{Cr}$ significantly reduces the thickness of the solutal boundary layer. This result and Chamkha ⁶² are nearly identical. The effect of the stretching ratio parameter on the profile of axial $F\left(\xi \right)$ , radial $F\prime \left(\xi \right)$ , and tangential $G\left(\xi \right)$ velocities of hybrids nanofluid $(\mathrm{Ti}{O}_2-\mathrm{GO})$ /water and nanofluid $\mathrm{Ti}{O}_2$ /water is depicted in Figures 17 to 19. It has been found that higher values of $A_1$ improve velocity profiles. Physically, when $A_1$ rises, the surface of the disc is stretched more. With an increasing amount of $A_1$ , the $F\prime \left(\xi \right)$ and $G\left(\xi \right)$ velocities exhibit dual behavior. The effect of the stretching ratio parameter on the profile of axial $F\left(\xi \right)$ , radial $F\prime \left(\xi \right)$ , and tangential $G\left(\xi \right)$ velocities of hybrids nanofluid $(\mathrm{Ti}{O}_2-\mathrm{GO})$ /water and nanofluid $\mathrm{Ti}{O}_2$ /water is depicted in Figures 20 to 22. It has been shown that the velocity profiles fall as $A_2$ increases. Physically, as $A_2$ rises, the higher disc gets more stretching force while the lower disc gets less. With an increasing amount of $A_2$ , the $F\prime \left(\xi \right)$ and $G\left(\xi \right)$ velocities exhibit dual behavior. The effects of the rotation parameter $\mathrm{\Omega }$ on the temperature profile $\theta \left(\xi \right)$ and concentration profiles $\psi \left(\xi \right)$ of hybrids nanofluid $(\mathrm{Ti}{O}_2-\mathrm{GO})$ /water and nanofluid $\mathrm{Ti}{O}_2$ /water are depicted in Figures 23 and 24. The $\mathrm{\Omega }$ values that are increasing indicate a larger impact on both profiles. Physically, rising value of the $\mathrm{\Omega }$ , results in a thickening of the thermal boundary layer. The effects of rotational parameters $\mathrm{\Omega }$ on the profile of $F\prime \left(\xi \right)$ and $F\left(\xi \right)$ velocities of hybrids nanofluid $(\mathrm{Ti}{O}_2-\mathrm{GO})$ /water and nanofluid $\mathrm{Ti}{O}_2$ /water are depicted in Figures 25 and 26. The function of $\mathrm{\Omega }$ is becoming more dependent on $F\prime \left(\xi \right)$ and $F\left(\xi \right)$ velocities. Physically, The Coriolis force, which operates orthogonally to the velocity field and the rotational axis in a rotating channel, opposes the fluid’s motion. Therefore, for $F\prime \left(\xi \right)$ , a two-fold impression is observed. When the velocity is amplified, the area near the lower disc experiences an increase in $F\prime \left(\xi \right)$ , while the higher half of the disc gets a regression of this trend. It appears that the velocity $F\left(\xi \right)$ degrades for increasing values of 1. The effect of the rotation parameter $\mathrm{\Omega }$ on tangential velocity $G\left(\xi \right)$ of hybrids nanofluid $(\mathrm{Ti}{O}_2-\mathrm{GO})$ /water and nanofluid $\mathrm{Ti}{O}_2$ /water is depicted in Figure 27. The rotation parameter has an increasing relationship with $G\left(\xi \right)$ . Physically, higher $\mathrm{\Omega }$ values cause the angular velocity to increase, and $G\left(\xi \right)$ , therefore, has an increasing effect. The effects of the convective parameter $B_1$ on the temperature profile $\theta \left(\xi \right)$ of hybrids nanofluid $(\mathrm{Ti}{O}_2-\mathrm{GO})$ /water and nanofluid $\mathrm{Ti}{O}_2$ /water are depicted in Figure 28. The more is worth, the more $\theta \left(\xi \right)$ . Physically, the thickness of the thermal boundary layer and the hydrodynamic boundary layer both increase as $B_1$ rises. Figures 29 to 31 shows the impact of nanoparticle volume fraction $\phi$ of hybrids nanofluid $(\mathrm{Ti}{O}_2-\mathrm{GO})$ /water and nanofluid $\mathrm{Ti}{O}_2$ /water on $F\left(\xi \right)$ , $F\prime \left(\xi \right)$ , and $G\left(\xi \right)$ . On increasing volume fraction of each nanoparticle $\mathrm{Ti}{O}_2\left({\phi }_1\right)$ and $\mathrm{GO}\left({\phi }_2\right)$ in equal quantity, the $F\left(\xi \right)$ and $F\prime \left(\xi \right)$ shows greater impact whereas tangential velocity shows decreasing impact. The increasing nanoparticle volume fraction in base fluid causes faster for the motion in axial and radial direction, because the addition of nanoparticle causes the outward movement of fluid, hence the velocities $F\left(\xi \right)$ and $F\prime \left(\xi \right)$ increases. On contrary, tangential velocity increase causes the hindrance for the motion in the tangential direction. Figure 32 shows how nanoparticle effect on temperature profile $\theta \left(\xi \right)$ of hybrids nanofluid $(\mathrm{Ti}{O}_2-\mathrm{GO})$ /water and nanofluid $\mathrm{Ti}{O}_2$ /water. On increasing volume fraction of each nanoparticle $\mathrm{Ti}{O}_2\left({\phi }_1\right)$ and $\mathrm{GO}\left({\phi }_2\right)$ in equal quantity, the $\theta \left(\xi \right)$ shows greater impact. The rise in temperature can be attributed to the enhancement in the thermal conduction of hybrid nanofluid or nanofluid due to increase in the nanoparticles in the base fluid. Figure 33 shows how nanoparticle effect on concentration profile $\psi \left(\xi \right)$ . On increasing volume fraction of each nanoparticle $\mathrm{Ti}{O}_2\left({\phi }_1\right)$ and $\mathrm{GO}\left({\phi }_2\right)$ in equal quantity, the $\psi \left(\xi \right)$ raise with $\mathrm{Ti}{O}_2\left({\phi }_1\right)$ nanofluid but due to mixing of $\mathrm{Ti}{O}_2\left({\phi }_1\right)$ and $\mathrm{GO}\left({\phi }_2\right)$ in base fluid, $\psi \left(\xi \right)$ falls. The reason for this behavior can be attributed to the fact the mixing of more nanoparticles causes the hybrid nanofluid or nanofluid to be more viscous and flow is resisted, hence $\psi \left(\xi \right)$ of nanoparticles falls. Significant effect of various parameters on $C{f}_{x1}$ and $C{f}_{x2}$ , $N{u}_{x1}$ and $N{u}_{x2}$ , and $S{h}_{x1}$ and $S{h}_{x2}$ for Titanium oxide $\left(\mathrm{Ti}{O}_2\right)$ and Graphene oxide $\left(\mathrm{GO}\right)$ are examined in Tables 3 to 6. Table 3 examine the numerical values of the gradients of velocity $C{f}_{x1}$ and $C{f}_{x2}$ at both discs with $\varphi ,\mathrm{Re},\mathrm{\Omega },\mathrm{Ri}$ and $A_2$ parameters. Clearly, $C{f}_{x1}$ and $C{f}_{x2}$ are augments with $\varphi$ and decreases with $\mathrm{Re},\mathrm{Ri},\mathrm{\Omega }$ at both disc while, $C{f}_{x1}$ increases with $A_2$ at lower disc but decreases at upper disc. Table 4 examined the numerical values of the temperature $N{u}_{x1}$ and $N{u}_{x2}$ at both discs using a verity of parameters. Clearly, $N{u}_{x1}$ and $N{u}_{x2}$ are decreased at $M,\mathrm{\Omega },B_1,\mathrm{Rd}$ while, increased with $\mathrm{Ri}$ at both disc. Table 5 examined the numerical values of rates of change of concentration $S{h}_{x1}$ and $S{h}_{x2}$ at both discs using a verity of parameters. Clearly, $S{h}_{x1}$ and $S{h}_{x2}$ are increased at $\mathrm{Sc}$ and $\mathrm{Cr}$ while decreased with $\mathrm{Ri}$ at both disc. By contrasting the numerical values of $f″\left(0\right)$ and $-g\prime \left(0\right)$ in the limiting situation and changing the values of $\mathrm{\Omega }$ , Table 6 is created to demonstrate the validity of our problem. Table 6 shows that there is a great deal of agreement between our numerical estimates and earlier literature Rawat et al. ⁶³ and Khan et al. ⁶⁴

Figure 2.

The impact of $M$ over $F(\xi )$ .

Figure 3.

The impact of $M$ over $F\prime (\xi )$ .

Figure 4.

The impact of $M$ over $G(\xi )$ .

Figure 5.

The impact of $M$ over $\theta (\xi )$ .

Figure 6.

The impact of $\mathrm{Re}$ over $\theta (\xi )$ .

Figure 7.

The impact of $\mathrm{Re}$ over $G(\xi )$ .

Figure 8.

The impact of $\mathrm{Sc}$ over $\psi (\xi )$ .

Figure 9.

The impact of $\mathrm{Rd}$ over $\theta (\xi )$ .

Figure 10.

The impact of $\mathrm{Ri}$ over $F(\xi )$ .

Figure 11.

The impact of $\mathrm{Ri}$ over $F\prime (\xi )$ .

Figure 12.

The impact of $\mathrm{Ri}$ over $G(\xi )$ .

Figure 13.

The impact of $\mathrm{Ri}$ over $\theta (\xi )$ .

Figure 14.

The impact of $\mathrm{Ri}$ over $\psi (\xi )$ .

Figure 15.

The impact of $\mathrm{Ec}$ over $\theta (\xi )$ .

Figure 16.

The impact of $\mathrm{Cr}$ over $\psi (\xi )$ .

Figure 17.

The impact of $A_1$ over $F(\xi )$ .

Figure 18.

The impact of $A_1$ over $F\prime (\xi )$ .

Figure 19.

The impact of $A_1$ over $G(\xi )$ .

Figure 20.

The impact of $A_2$ over $F(\xi )$ .

Figure 21.

The impact of $A_2$ over $F\prime (\xi )$ .

Figure 22.

The impact of $A_1$ over $G(\xi )$ .

Figure 23.

The impact of $\mathrm{\Omega }$ over $\theta (\xi )$ .

Figure 24.

The impact of $\mathrm{\Omega }$ over $\psi (\xi )$ .

Figure 25.

The impact of $\mathrm{\Omega }$ over $F\prime (\xi )$ .

Figure 26.

The impact of $\mathrm{\Omega }$ over $F(\xi )$ .

Figure 27.

The impact of $\mathrm{\Omega }$ over $G(\xi )$ .

Figure 28.

The impact of $B_1$ over $\theta (\xi )$ .

Figure 29.

The impact of $\varphi$ over $F(\xi )$ .

Figure 30.

The impact of $\varphi$ over $F\prime (\xi )$ .

Figure 31.

The impact of $\varphi$ over $G(\xi )$ .

Figure 32.

The impact of $\varphi$ over $\theta (\xi )$ .

Figure 33.

The impact of $\varphi$ over $\psi (\xi )$ .

Table 3.

Different values of important parameter effects on the numerical results of $C{f}_{x1}$ and $C{f}_{x2}$ .

$\varphi$	$\mathrm{Re}$	$\mathrm{\Omega }$	$A_2$	$\mathrm{Ri}$	$C{f}_{x1}$	$C{f}_{x2}$
					$\begin{array}{l}\left(Ti{O}_2-GO\right)/\\ H_{2}O\end{array}$	$\begin{array}{l}\left(Ti{O}_2-GO\right)/\\ H_{2}O\end{array}$
1.0					1.905723	1.603434
1.2					1.906534	1.608744
1.4					1.908445	1.612345
	0.2				1.890889	1.468723
	0.3				1.890865	1.461251
	0.4				1.890843	1.458723
		0.2			1.608343	0.545667
		0.4			1.524901	0.456743
		0.6			1.429834	0.334765
			0.3		1.347189	1.325664
			0.4		1.548946	0.768763
			0.5		1.750234	0.345400
				0.3	0.876534	0.876530
				0.4	0.876233	0.876226
				0.5	0.876021	0.876018

Table 4.

Different values of important parameter effects on the numerical results of $N{u}_{x1}$ and $N{u}_{x2}$ .

$M$	$B_1$	$\mathrm{Rd}$	$\mathrm{\Omega }$	$\mathrm{Ri}$	$N{u}_{x1}$	$N{u}_{x2}$
					$\begin{array}{l}\left(Ti{O}_2-GO\right)/\\ H_{2}O\end{array}$	$\begin{array}{l}\left(Ti{O}_2-GO\right)/\\ H_{2}O\end{array}$
1.5					1.890807	1.930805
1.6					1.890768	1.930769
1.7					1.890685	1.930688
	0.5				2.105689	1.105678
	0.6				1.905676	0.805667
	0.7				1.123409	0.113413
		1.1			0.564345	0.562022
		1.2			0.564219	0.561776
		1.3			0.564188	0.561476
			0.5		1.742133	1.862145
			0.6		1.705454	1.605423
			0.7		1.606565	1.603454
				0.5	1.447143	1.447143
				0.6	1.648943	1.648954
				0.7	1.850244	1.850244

Table 5.

Different values of important parameter effects on the numerical results of $S{h}_{x1}$ and $S{h}_{x2}$ .

$\mathrm{Ri}$	$\mathrm{Sc}$	$\mathrm{Cr}$	$S{h}_{x1}$	$S{h}_{x2}$
			$\begin{array}{l}\left(Ti{O}_2-GO\right)/\\ H_{2}O\end{array}$	$\begin{array}{l}\left(Ti{O}_2-GO\right)/\\ H_{2}O\end{array}$
1.2			1.896576	1.613454
1.3			1.896543	1.608754
1.4			1.896456	1.597623
	0.5		1.908354	1.437645
	0.6		1.908465	1.438745
	0.7		1.908845	1.448776
		0.5	0.477165	0.355654
		0.6	0.605489	0.798755
		0.7	1.810254	1.345454

Table 6.

The comparison of values $f″\left(0\right)$ and $-g\prime \left(0\right)$ for various values of $\mathrm{\Omega }$ and the rest parameters are $\mathrm{Pr}=6.2,{\varphi }_1={\varphi }_2=M=B_1=A_1=A_2=0$ .

$\mathrm{\Omega }$	Khan et al. 64 $f″\left(0\right)$	Rawat et al. 63 $f″\left(0\right)$	Present $f″\left(0\right)$	Khan et al. 64 $-g\prime \left(0\right)$	Rawat et al. 63 $-g\prime \left(0\right)$	Present $-g\prime \left(0\right)$
−1	0.06666314	0.06666303	0.06666301	2.0009522	2.00095213	2.00095209
−0.8	0.83942070	0.8394201	0.8394200	1.8025885	1.8025846	1.8025846
−0.3	0.1395088	0.1395088	0.1395087	1.3044236	1.30442355	1.30442321
0	0.09997221	0.09997221	0.09997221	1.0042776	1.0042756	1.00427551
0.5	0.06663419	0.06663416	0.06663412	0.5026135	0.50261344	0.502613432

Conclusion

The current problem concerns the influence of a mixed convection MHD flow of a hybrid nanofluid (TiO₂-Go/water) as well as nanofluid (TiO₂/water) between two parallel and infinite spinning discs with joule heating and mixed convection in a porous medium in the presence of the magnetic field and radiation. The numerical solution is deduced by employing the “bvp4c” function in MATLAB. The influence of various key parameters, such as $\varphi ,M,A_1,A_2,B_1,\mathrm{Re},\mathrm{Ri},\mathrm{Sc},\mathrm{\Omega }\mathrm{and}\mathrm{Cr}$ . Some vital conclusions, is listed below.

• Radial velocity shows increasing behavior near the lower and upper disc for increasing values of the magnetic parameter and the mix convection parameter.

• The temperature profile rises as the magnetic, rotational, and convective parameters rise, as do the Eckert number and radiation parameter, while the mixed convection parameter falls.

• The concentration profile rises with the rotation parameter while falling with the chemical reaction parameter and Schmidt number.

• Increasing the radiation and magnetic parameters causes a decline in the heat transmission rate at both the upper and lower plates.

• The lower disc has a higher rate of heat transmission and mass transmission.