Three-dimensional MHD flow of hybrid material between rotating disks with heat generation

Abstract

In this study, we investigate the flow of electrically conducting hybrid nanofluid $\left(Ag+Cu/H_{2}O\right)$, due to rotating disks, along with thermal slip, heat generation, and viscous dissipation. The nonlinear differential system is modelled and transformed into dimensionless partial differential equations using suitable dimensionless variables. To obtain solutions for the considered model, a finite difference toolkit is implemented, and numerical solutions are achieved. Graphical results are presented to display the influences of different dimensionless variables on flow velocity and temperature. This research contributes to a better understanding of hybrid nanofluid flows and can inform the design of cooling systems and other practical applications.

1. Introduction

Nanofluids, consisting of a host fluid and solid-state nanoparticles, have shown promise in enhancing heat transfer in various engineering systems. Early research by Choi and Eastman [1] paved the way for the development of nanofluids with greater thermal conductivities. As a result, the properties and behavior of different types of nanofluids have been extensively studied. Utilizing nanofluids can significantly improve the performance of heat exchangers, electronic cooling, heating systems in pipes, solar heating, and nuclear plants. Additionally, the use of these fluids can help reduce costs. In recent years, several researchers have made significant contributions to the field of nanofluid flow. Reddy et al. [2] presented boundary layer hybrid nanofluid flow, including chemical reactions, past a rotating porous surface. Sreedevi and Reddy [3] investigated the impact of carbon nanotubes on Maxwell nanofluid model. Sreedevi et al. [4] studied the effect of homogeneous-heterogeneous reactions on Maxwell nanofluid past a horizontally stretched cylinder. Hayat et al. [5] analyzed energy transport in the slip flow of a hybrid nanomaterial through thermal radiation and Newtonian heating. Additionally, Hayat et al. [6] inquired melting heat distribution in the stagnation-point flow. Hayat et al. [7] also carried out investigation on the nonlinear mixed convection in radiative nanofluid flow with entropy generation. Further discussion on nanofluids is in Refs. [[8], [9], [10], [11], [12]].

Magnetohydrodynamic (MHD) flow is a field in which the interaction between a magnetic field and an electrically conductive fluid is studied. Its applications involve energy storage, cooling systems, accelerators, MHD generators, flow meters, and pumps. Recently, several researchers have contributed to this field. Sheikholeslami et al. [13] researched the thermal transport features of MHD flow in a concentric annulus. Rashidi et al. [14] explored the thermally radiative nanofluids over a stretching surface, while. Hayat et al. [15,16] also made significant contributions to the field of fluid mechanics, particularly in the thermodynamics of melting in the flow of Oldroyd-B material and the melting effect in MHD stagnation point flow. Bilal et al. [17] analyzed Sutterby fluid immersed over a linearly stretched surface. These studies have improved our understanding of fluid mechanics and have practical implications for various applications, including cooling systems, nuclear reactors, flow meters, and pumps.

The flow among two rotating surfaces has diverse applications in various fields, including energy storage devices, electronic cooling, medical equipment, power generating systems, manufacturing industries, and aerodynamic engineering. Turkyilmazoglu [18] investigated the effects of MHD flow, while Attia [19] examined the flow behavior past rotating surfaces with injection and suction. Sharma et al. [20] and Shehzad et al. [21] extended the work of Turkyilmazoglu [22] by considering viscous dissipation and thermos diffusion, and diffusion thermos effects, respectively. The significance of nanofluid and variable thermal conductivity in non-Newtonian fluid was demonstrated by Mabood et al. [23]. Jayadevamurthy et al. [24] investigated the irreversibility for hybrid nanofluids and swimming of gyrotactic microorganisms over a rotating disk. The diverse research in this field has contributed to the advancement of understanding of fluid dynamics and the practical implications for various applications, which are given in Refs. [[25], [26], [27], [28]].

The motivation for this study is to investigate the behavior of magnetohydrodynamic hybrid nanofluid $\left(Ag+Cu/H_{2}O\right)$ with heat generation and thermal slip effects. Using a finite difference scheme, the model was solved numerically, and the results were presented graphically. The study analyzed the variations of the parameters against the temperature and velocity fields and drew its conclusions in the final section. The findings of this study could have significant applications in various fields, including energy storage, nuclear reactors, electronic cooling, aerospace engineering, and chemical industries. Specifically, the study's results could contribute to the development of more efficient heat transfer systems and improved performance of energy storage devices. The insights gained from this research could also help in the design of more effective cooling systems for electronic equipments, enhancing their overall efficiency and lifespan.

2. Problem modeling

This study investigates the characteristics of hybrid nanofluid with thermal slip due to electrically conducting rotating disks. Hybrid nanofluid is developed by immersing silver $\left(Ag\right)$ and copper $\left(Cu\right)$ nanoparticles into water $\left(H_{2}O\right)$, which functions as the base fluid. The disks used in the experiment are arranged in a vertical configuration, with the lower disk positioned at $\left(z=0\right)$ and the upper disk situated at a distance of $\left(z=h_0\right)$ from the origin. The lower disk is stretched at a rate of $a_1$, while the upper disk is stretched at a rate of ($a_2$). Magnetic field with a strength of ($B_0$) is applied, with corresponding angular velocities (${\mathrm{\Omega }}_2$ and ${\mathrm{\Omega }}_1$) assigned to the upper and lower disks, as shown in Fig. 1. Thermal slip effects are accounted for, and the energy equation is supplemented with heat generation. Table 1 provides the expressions for the hybrid fluid and nanofluid, which have been derived using the Hamilton-Crosser model. For this investigation, cylindrical shaped nanoparticles are the focus of interest, thus $n=6$ has been selected. Table 2 presents the experimentally verified values of the nanoparticles and base fluid.

Fig. 1.

Geometry of the flow.

Table 1.

Hamilton-Crosser model for hybrid fluid and nanofluid [5].

For Hybrid Nanofluid	For Nanofluid
${\mu }_{hnf}=\frac{{\mu }_f}{{\left(1-{\phi }_{Ag}\right)}^{2.5}{\left(1-{\phi }_{Cu}\right)}^{2.5}},{\upsilon }_{hnf}=\frac{{\mu }_{hnf}}{{\rho }_{hnf}}\text{,}$	${\mu }_{nf}=\frac{{\mu }_f}{{\left(1-{\phi }_{Ag}\right)}^{2.5}},{\upsilon }_{nf}=\frac{{\mu }_{nf}}{{\rho }_{nf}}\text{,}$
$\begin{array}{r}{\left(\rho c_p\right)}_{hnf}={\phi }_{Ag}{\left(\rho c_p\right)}_{Ag}+{\phi }_{Cu}{\left(\rho c_p\right)}_{Cu}+\\ \left(1-{\phi }_{Cu}\right)\left(1-{\phi }_{Cu}\right){\left(\rho c_p\right)}_f\text{,}\end{array}$	${\left(\rho c_p\right)}_{nf}={\phi }_{Ag}{\left(\rho c_p\right)}_{Ag}+\left(1-{\phi }_{Ag}\right){\left(\rho c_p\right)}_f\text{,}$
${\rho }_{hnf}={\phi }_{Ag}{\rho }_{Ag}+{\phi }_{Cu}{\rho }_{Cu}+\left(1-{\phi }_{Cu}\right)\left(1-{\phi }_{Ag}\right){\rho }_f\text{,}$	${\rho }_{nf}={\phi }_{Ag}{\rho }_{Ag}+\left(1-{\phi }_{Ag}\right){\rho }_f\text{,}$
$\frac{k_{hnf}}{k_{nf}}=\frac{k_{Cu}+\left(n-1\right)k_{nf}-\left(n-1\right)\left(k_{nf}-k_{Cu}\right){\phi }_{Cu}}{k_{Cu}+\left(n-1\right)k_{nf}+\left(k_{nf}-k_{Cu}\right){\phi }_{Cu}}\text{,}$	$\frac{k_{nf}}{k_f}=\frac{k_{Ag}+\left(n-1\right)k_f-\left(n-1\right)\left(k_f-k_{Ag}\right){\phi }_{Ag}}{k_{Ag}+\left(n-1\right)k_f+\left(k_f-k_{Ag}\right){\phi }_{Ag}}\text{,}$
$\frac{{\sigma }_{hnf}}{{\sigma }_{nf}}=\frac{{\sigma }_{Cu}+\left(n-1\right){\sigma }_{nf}-\left(n-1\right)\left({\sigma }_{nf}-{\sigma }_{Cu}\right){\phi }_{Cu}}{{\sigma }_{Cu}+\left(n-1\right){\sigma }_{nf}+\left({\sigma }_{nf}-{\sigma }_{Cu}\right){\phi }_{Cu}}\text{.}$	$\frac{{\sigma }_{nf}}{{\sigma }_f}=\frac{{\sigma }_{Ag}+\left(n-1\right){\sigma }_f-\left(n-1\right)\left({\sigma }_f-{\sigma }_{Ag}\right){\phi }_{Ag}}{{\sigma }_{Ag}+\left(n-1\right){\sigma }_f+\left({\sigma }_f-{\sigma }_{Ag}\right){\phi }_{Ag}}\text{.}$

Table 2.

Numerical values of $Ag,Cu$ and $H_{2}O$.

	Ag	Cu	Water
$\rho$	10,490	632	997.1
$k$	429	765	0.63
$\sigma$	$6.3\times {10}^7$	$5.96\times {10}^7$	0.05
$c_p$	$18.9\times {10}^{-6}$	$16.7\times {10}^{-6}$	$210\times {10}^{-6}$
Pr	235	531.8	6.2

The corresponding governing equations for the system is given below [24]:

$$\frac{u}{r}+\frac{\partial w}{\partial z}+\frac{\partial u}{\partial r}=0,$$ (1)

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

$$\frac{\partial v}{\partial t}+w\frac{\partial v}{\partial z}+\frac{vu}{r}+u\frac{\partial v}{\partial r}={\nu }_{hnf}\left(\frac{{\partial }^{2}v}{\partial r^2}+\frac{1}{r}\frac{\partial v}{\partial r}+\frac{{\partial }^{2}v}{\partial z^2}-\frac{v}{r^2}\right)-\frac{{\sigma }_{hnf}}{{\rho }_{hnf}}{B}_0^{2}v\text{,}$$ (3)

$$\frac{\partial w}{\partial t}+u\frac{\partial w}{\partial r}+w\frac{\partial w}{\partial z}={\nu }_{hnf}\left(\frac{{\partial }^{2}w}{\partial z^2}+\frac{1}{r}\frac{\partial w}{\partial r}+\frac{{\partial }^{2}w}{\partial r^2}\right)\text{,}$$ (4)

$${\left(\rho c_p\right)}_{hnf}\left(\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial r}+w\frac{\partial T}{\partial z}\right)=k_{hnf}\left(\frac{1}{r}\frac{\partial T}{\partial r}+\frac{{\partial }^{2}T}{\partial r^2}+\frac{{\partial }^{2}T}{\partial z^2}\right)+Q_0\left(T-T_w\right)\text{,}$$ (5)

with associated conditions

$$\begin{array}{l}\mathrm{I}\mathrm{C}\mathrm{s}:u=0,v=0,w=0,T=1,\mathrm{a}\mathrm{t}t=0\text{.}\\ u=\frac{r{a}_1}{1-ct}+{\lambda }_1\frac{\partial u}{\partial z},v=\frac{r{\mathrm{\Omega }}_1}{1-ct}+{\lambda }_2\frac{\partial v}{\partial z},w=0,T=\frac{T_0}{1-ct}+{\lambda }_3\frac{\partial T}{\partial z},\mathrm{a}\mathrm{t}z=0\text{,}\\ u=\frac{r{a}_2}{1-ct}-{\lambda }_1\frac{\partial u}{\partial z},v=\frac{r{\mathrm{\Omega }}_2}{1-ct}-{\lambda }_2\frac{\partial v}{\partial z},w=0,T=\frac{T_1}{1-ct}-{\lambda }_3\frac{\partial T}{\partial z},\mathrm{a}\mathrm{t}z=h_0\text{,}\end{array}\}$$ (6)

Utilizing the following non-dimensional variables in Eqs. (1), (2), (3), (4), (5) (see Ref. [7])

$$\begin{array}{l}f\left(\tau ,\xi ,\eta \right)=\frac{L}{\nu }u,g\left(\tau ,\xi ,\eta \right)=\frac{L}{\nu }v,h\left(\tau ,\xi ,\eta \right)=\frac{L}{\nu }w,\theta \left(\tau ,\xi ,\eta \right)=\frac{T-T_1}{T_0-T_1}\text{,}\\ where\xi =\frac{r}{L},\eta =\frac{z}{L},\tau =\frac{\nu }{L^2}t\text{.}\end{array}\}$$ (7)

We have

$$\frac{\partial f}{\partial \xi }+\frac{\partial h}{\partial \eta }+\frac{f}{\xi }=0\text{,}$$ (8)

$$\frac{\partial f}{\partial \tau }+\frac{\partial f}{\partial \xi }f+\frac{\partial f}{\partial \eta }h-\frac{g^2}{\xi }=\frac{A_{11}}{A_{12}}\left(\frac{{\partial }^{2}f}{\partial {\eta }^2}+\frac{{\partial }^{2}f}{\partial {\xi }^2}+\frac{1}{\xi }\frac{\partial f}{\partial \xi }-\frac{f}{{\xi }^2}\right)-Ha\frac{A_{13}}{A_{12}}f\text{,}$$ (9)

$$\frac{\partial g}{\partial \tau }+\frac{\partial g}{\partial \xi }f+\frac{\partial g}{\partial \eta }h+\frac{fg}{\xi }=\frac{A_{11}}{A_{12}}\left(\frac{{\partial }^{2}g}{\partial {\xi }^2}+\frac{{\partial }^{2}g}{\partial {\eta }^2}-\frac{1}{\xi }\frac{\partial g}{\partial \xi }-\frac{g}{{\xi }^2}\right)-Ha\frac{A_{13}}{A_{12}}g\text{,}$$ (10)

$$\frac{\partial h}{\partial \tau }+\frac{\partial h}{\partial \xi }f+\frac{\partial h}{\partial \eta }h=\frac{A_{11}}{A_{12}}\left(\frac{{\partial }^{2}h}{\partial {\xi }^2}+\frac{{\partial }^{2}h}{\partial {\eta }^2}+\frac{1}{\xi }\frac{\partial h}{\partial \xi }\right)\text{,}$$ (11)

$$\frac{\partial \theta }{\partial \tau }+f\frac{\partial \theta }{\partial \xi }+h\frac{\partial \theta }{\partial \eta }=\frac{A_{14}}{A_{15}}\left(\frac{{\partial }^2\theta }{\partial {\xi }^2}+\frac{{\partial }^2\theta }{\partial {\eta }^2}+\frac{1}{\xi }\frac{\partial \theta }{\partial \xi }\right)+\frac{Q^{*}}{A_{15}}\theta \text{,}$$ (12)

where

$$A_{11}=\frac{{\mu }_{hnf}}{{\mu }_f},A_{12}=\frac{{\rho }_{hnf}}{{\rho }_f},A_{13}=\frac{{\sigma }_{hnf}}{{\sigma }_f},A_{14}=\frac{k_{hnf}}{k_f},A_{15}=\frac{{\left(\rho c_p\right)}_{hnf}}{{\left(\rho c_p\right)}_f}\text{.}$$ (13)

The dimensionless version of Eq. (6) is

$$\mathrm{I}\mathrm{C}\mathrm{s}:f=0,g=0,h=0,\theta =1,\mathrm{a}\mathrm{t}\tau =0\text{,}$$ (14)

$$f-{\lambda }^{*}\frac{\partial f}{\partial \eta }-M\xi =0,g-{\lambda }^{**}\frac{\partial g}{\partial \eta }-M{\mathrm{\Omega }}^{*}\xi =0,h=0,\theta -{\lambda }^{**}\frac{\partial \theta }{\partial \eta }-\beta =0,\mathrm{a}\mathrm{t}\eta =0\text{,}$$ (15)

$$f+{\lambda }^{*}\frac{\partial f}{\partial \eta }-M\xi =0,g+{\lambda }^{**}\frac{\partial g}{\partial \eta }-M{\mathrm{\Omega }}^{**}\xi =0,h=0,\theta +{\lambda }^{***}\frac{\partial \theta }{\partial \eta }+\beta =0,\mathrm{a}\mathrm{t}\eta =1\text{.}$$ (16)

The dimensionless numbers are defined below

$$\begin{array}{l}Ha=\frac{B_0^2{L}^2}{{\nu }_f},M=\frac{a_1{L}^2}{{\upsilon }_f-c\tau L^2},{\lambda }^{*}=\frac{{\lambda }_1}{L\left({\upsilon }_f-c\tau L^2\right)},{\lambda }^{**}=\frac{{\lambda }_2}{L\left({\upsilon }_f-c\tau L^2\right)},{\lambda }^{***}=\frac{{\lambda }_3}{L\left({\upsilon }_f-c\tau L^2\right)}\text{,}\\ Q^{*}=\frac{Q_0{L}^2}{{\upsilon }_f{\left(\rho c_p\right)}_f},{\mathrm{\Omega }}^{*}=\frac{{\mathrm{\Omega }}_1}{a_1},{{\mathrm{\Omega }}^{*}}^{*}=\frac{{\mathrm{\Omega }}_2}{a_2},\beta =\frac{T_0-T_1\left({\upsilon }_f-c\tau L^2\right)}{\left(T_1-T_0\right)\left({\upsilon }_f-c\tau L^2\right)}\text{.}\end{array}\}$$ (17)

The skin friction coefficients ($C{f}_0$) and ($C{f}_1$) is defined as below

$$C{f}_0=\frac{{{\tau }_w|}_{z=0}}{{\rho }_f{\left(\frac{r{a}_1}{1-ct}\right)}^2}\text{,}$$ (18)

$$C{f}_1=\frac{{{\tau }_w|}_{z=h_0}}{{\rho }_f{\left(\frac{r{a}_2}{1-ct}\right)}^2}\text{,}$$ (19)

where

$${\tau }_w=\sqrt{{\tau }_{zr}^2+{\tau }_{z\theta }^2}\text{,}$$ (20)

After implementing the similarity variables, we have the dimensionless form as below

$$C{f}_0=\frac{{\left(1-{\phi }_{Ag}\right)}^{-2.5}{\left(1-{\phi }_{Cu}\right)}^{-2.5}}{M^2{\xi }^2}\sqrt{{{\left(\frac{\partial f}{\partial \eta }\right)}^2|}_{z=0}+{\left(\frac{\partial g}{\partial \eta }\right)|}_{z=0}}\text{,}$$ (21)

$$C{f}_1=\frac{{\left(1-{\phi }_{Ag}\right)}^{-2.5}{\left(1-{\phi }_{Cu}\right)}^{-2.5}}{M_1^2{\xi }^2}\sqrt{{{\left(\frac{\partial f}{\partial \eta }\right)}^2|}_{z=1}+{\left(\frac{\partial g}{\partial \eta }\right)|}_{z=1}}\text{,}$$ (22)

where

$$M_1=\frac{a_2{L}^2}{{\upsilon }_f-c\tau L^2}\text{.}$$ (23)

The heat transfer rates ($N{u}_0$) and ($N{u}_1$) are given by

$$N{u}_0={\frac{z{q}_w}{k_f\left(T_0-T_1\right)}|}_{z=0},with{q_w|}_{z=0}=-k_{hnf}{\frac{\partial T}{\partial z}|}_{z=0}\text{,}$$ (24)

$$N{u}_1={\frac{z{q}_w}{k_f\left(T_0-T_1\right)}|}_{z=h_0},with{q_w|}_{z=h_0}=-k_{hnf}{\frac{\partial T}{\partial z}|}_{z=1}\text{.}$$ (25)

We get the dimensionless form after utilizing the variables in Eq. (7)

$$N{u}_0=-A_{14}{\frac{\partial \theta }{\partial \eta }|}_{\eta =0},N{u}_1=-A_{14}{\frac{\partial \theta }{\partial \eta }|}_{\eta =1}\text{.}$$ (26)

3. Problem solution

To find the solutions of our dimensionless partial differential system in Eqs. (8), (9), (10), (11), (12) we have utilized Finite difference method. The basic scheme here is to construct finite difference toolkit for our system of partial differential equations [7].

$$\frac{\partial f}{\partial \tau }=\frac{f_{n,m}^{p+1}-f_{n,m}^p}{\Delta \tau },\frac{\partial f}{\partial \xi }=\frac{f_{n+1,m}^p-f_{n,m}^p}{\Delta \xi },\frac{\partial f}{\partial \eta }=\frac{f_{n,m+1}^p-f_{n,m}^p}{\Delta \eta }\text{,}$$ (27)

$$\frac{\partial g}{\partial \tau }=\frac{g_{n,m}^{p+1}-g_{n,m}^p}{\Delta \tau },\frac{\partial g}{\partial \xi }=\frac{g_{n+1,m}^p-g_{n,m}^p}{\Delta \xi },\frac{\partial g}{\partial \eta }=\frac{g_{n,m+1}^p-g_{n,m}^p}{\Delta \eta }\text{,}$$ (28)

$$\frac{\partial h}{\partial \tau }=\frac{h_{n,m}^{p+1}-h_{n,m}^p}{\Delta \tau },\frac{\partial h}{\partial \eta }=\frac{h_{n,m+1}^p-h_{n,m}^p}{\Delta \eta },\frac{\partial h}{\partial \xi }=\frac{h_{n+1,m}^p-h_{n,m}^p}{\Delta \xi }\text{,}$$ (29)

$$\frac{\partial \theta }{\partial \xi }=\frac{{\theta }_{n+1,m}^p-{\theta }_{n,m}^p}{\Delta \xi },\frac{\partial \theta }{\partial \tau }=\frac{{\theta }_{n,m}^{p+1}-{\theta }_{n,m}^p}{\Delta \tau },\frac{\partial \theta }{\partial \eta }=\frac{{\theta }_{n,m+1}^p-{\theta }_{n,m}^p}{\Delta \eta }\text{,}$$ (30)

$$\frac{{\partial }^{2}f}{\partial {\eta }^2}=\frac{f_{n,m+2}^p-2f_{n,m+1}^p+f_{n,m}^p}{{\left(\Delta \eta \right)}^2},\frac{{\partial }^{2}f}{\partial {\xi }^2}=\frac{f_{n+2,m}^p-2f_{n+1,m}^p+f_{n,m}^p}{{\left(\Delta \xi \right)}^2}\text{,}$$ (31)

$$\frac{{\partial }^{2}g}{\partial {\eta }^2}=\frac{g_{n,m+2}^p-2g_{n,m+1}^p+g_{n,m}^p}{{\left(\Delta \eta \right)}^2},\frac{{\partial }^{2}g}{\partial {\xi }^2}=\frac{g_{n+2,m}^p-2g_{n+1,m}^p+g_{n,m}^p}{{\left(\Delta \xi \right)}^2}\text{,}$$ (32)

$$\frac{{\partial }^{2}h}{\partial {\eta }^2}=\frac{h_{n,m+2}^p-2h_{n,m+1}^p+h_{n,m}^p}{{\left(\Delta \eta \right)}^2},\frac{{\partial }^{2}h}{\partial {\xi }^2}=\frac{h_{n+2,m}^p-2h_{n+1,m}^p+h_{n,m}^p}{{\left(\Delta \xi \right)}^2}\text{,}$$ (33)

$$\frac{{\partial }^2\theta }{\partial {\eta }^2}=\frac{{\theta }_{n,m+2}^p-2{\theta }_{n,m+1}^p+{\theta }_{n,m}^p}{{\left(\Delta \eta \right)}^2},\frac{{\partial }^2\theta }{\partial {\xi }^2}=\frac{{\theta }_{n+2,m}^p-2{\theta }_{n+1,m}^p+{\theta }_{n,m}^p}{{\left(\Delta \xi \right)}^2}\text{.}$$ (34)

Implementing these FD-approximations in dimensionless version of partial differential system, we get

$$\frac{f_{n+1,m}^p-f_{n,m}^p}{\Delta \xi }+\frac{h_{n,m+1}^p-h_{n,m}^p}{\Delta \eta }+\frac{f_{n,m}^p}{\xi }=0\text{,}$$ (35)

$$\begin{array}{l}\frac{f_{n,m}^{p+1}-f_{n,m}^p}{\Delta \tau }+f_{n,m}^p\left[\frac{f_{n+1,m}^p-f_{n,m}^p}{\Delta \xi }\right]+h_{n,m}^p\left[\frac{f_{n,m+1}^p-f_{n,m}^p}{\Delta \eta }\right]-\frac{{\left[g_{n,m}^p\right]}^2}{\xi }=-Ha\frac{A_{13}}{A_{12}}\left[f_{n,m}^p\right]\\ A_{11}\left(\frac{f_{n,m+2}^p-2f_{n,m+1}^p+f_{n,m}^p}{{\left(\Delta \eta \right)}^2}+\frac{f_{n+2,m}^p-2f_{n+1,m}^p+f_{n,m}^p}{{\left(\Delta \xi \right)}^2}+\frac{1}{\xi }\left[\frac{f_{n+1,m}^p-f_{n,m}^p}{\Delta \xi }\right]-\frac{f_{n,m}^p}{{\xi }^2}\right)\text{,}\end{array}\}$$ (36)

$$\begin{array}{l}\frac{g_{n,m}^{p+1}-g_{n,m}^p}{\Delta \tau }+f_{n,m}^p\left[\frac{g_{n+1,m}^p-g_{n,m}^p}{\Delta \xi }\right]+h_{n,m}^p\left[\frac{g_{n,m+1}^p-g_{n,m}^p}{\Delta \eta }\right]+\frac{f_{n,m}^p\left[g_{n,m}^p\right]}{\xi }=-Ha\frac{A_{13}}{A_{12}}\left[g_{n,m}^p\right]\\ A_{11}\left(\frac{g_{n+2,m}^p-2g_{n+1,m}^p+g_{n,m}^p}{{\left(\Delta \xi \right)}^2}+\frac{g_{n,m+2}^p-2g_{n,m+1}^p+g_{n,m}^p}{{\left(\Delta \eta \right)}^2}-\frac{1}{\xi }\left[\frac{g_{n+1,m}^p-g_{n,m}^p}{\Delta \xi }\right]-\frac{g_{n,m}^p}{{\xi }^2}\right)\text{,}\end{array}\}$$ (37)

$$\begin{array}{l}\frac{h_{n,m}^{p+1}-h_{n,m}^p}{\Delta \tau }+f_{n,m}^p\left[\frac{h_{n+1,m}^p-h_{n,m}^p}{\Delta \xi }\right]+h_{n,m}^p\left[\frac{h_{n,m+1}^p-h_{n,m}^p}{\Delta \eta }\right]=\\ A_{11}\left(\frac{h_{n+2,m}^p-2h_{n+1,m}^p+h_{n,m}^p}{{\left(\Delta \xi \right)}^2}+\frac{h_{n,m+2}^p-2h_{n,m+1}^p+h_{n,m}^p}{{\left(\Delta \eta \right)}^2}+\frac{1}{\xi }\left[\frac{h_{n+1,m}^p-h_{n,m}^p}{\Delta \xi }\right]\right)\text{,}\end{array}\}$$ (38)

$$\frac{{\theta }_{n,m}^{p+1}-{\theta }_{n,m}^p}{\Delta \tau }+f_{n,m}^p\left[\frac{{\theta }_{n+1,m}^p-{\theta }_{n,m}^p}{\Delta \xi }\right]+h_{n,m}^p\left[\frac{{\theta }_{n,m+1}^p-{\theta }_{n,m}^p}{\Delta \eta }\right]=\frac{Q^{*}}{A_{15}}\left[{\theta }_{n,m}^p\right]+\frac{A_{14}}{A_{15}}\left(\frac{{\theta }_{n+2,m}^p-2{\theta }_{n+1,m}^p+{\theta }_{n,m}^p}{{\left(\Delta \xi \right)}^2}+\frac{{\theta }_{n,m+2}^p-2{\theta }_{n,m+1}^p+{\theta }_{n,m}^p}{{\left(\Delta \eta \right)}^2}+\frac{1}{\xi }\left[\frac{{\theta }_{n+1,m}^p-{\theta }_{n,m}^p}{\Delta \xi }\right]\right)\text{,}\}$$ (39)

with initial and boundary conditions

$$\begin{array}{l}{f}_{n,m}^0=0,g_{n,m}^0=0,{\theta }_{n,m}^0=1,h_{n,m}^0=0\text{,}\\ f_{n,0}^p-{\lambda }^{*}\left[\frac{f_{n,m+1}^p-f_{n,m}^p}{\Delta \eta }\right]-M\xi =0,g_{n,0}^p-{{\lambda }^{*}}^{*}\left[\frac{g_{n,m+1}^p-g_{n,m}^p}{\Delta \eta }\right]-M{\mathrm{\Omega }}^{*}\xi =0\text{,}\\ h_{n,0}^p=0,{\theta }_{n,0}^p-{\lambda }^{***}\left[\frac{{\theta }_{n,m+1}^p-{\theta }_{n,m}^p}{\Delta \eta }\right]-\beta =0,f_{n,1}^p+{\lambda }^{*}\left[\frac{f_{n,m+1}^p-f_{n,m}^p}{\Delta \eta }\right]-M\xi =0\text{,}\\ g_{n,1}^p+{{\lambda }^{*}}^{*}\left[\frac{g_{n,m+1}^p-g_{n,m}^p}{\Delta \eta }\right]-M{\mathrm{\Omega }}^{*}\xi =0,h_{n,1}^p=0,{\theta }_{n,1}^p+{\lambda }^{***}\left[\frac{{\theta }_{n,m+1}^p-{\theta }_{n,m}^p}{\Delta \eta }\right]-\beta =0\text{.}\end{array}\}$$ (40)

4. Discussion

In this section, an investigation is conducted on the impacts of the dimensionless variables defined in Eq. (13), and the outcomes are illustrated. The illustrations showcase the effects of dimensionless parameters on the hybrid nanofluid $\left(Ag+Cu/H_{2}O\right)$ concerning velocities $\left(f\left(\tau ,\xi ,\eta \right),g\left(\tau ,\xi ,\eta \right)\right)$ and temperature $\left(\theta \left(\tau ,\xi ,\eta \right)\right)$. Here, Fig. 2 represents variations of ${\gamma }^{*}$ over $f\left(\tau ,\xi ,\eta \right)$. Clearly, it is noticed that velocity decays when values of ${\gamma }^{*}$ increases. Fig. 3 displays the influence of $Ha$ on $f\left(\tau ,\xi ,\eta \right)$, where it can be observed that $f\left(\tau ,\xi ,\eta \right)$ decreases as $Ha$ increases. Fig. 4 shows influence of $M$ for $f\left(\tau ,\xi ,\eta \right)$. As the value of $M$ increases, the $f\left(\tau ,\xi ,\eta \right)$ enhances. Fig. 5, Fig. 6 depict the behavior of $f\left(\tau ,\xi ,\eta \right)$ against ${\mathrm{\Omega }}^{*}$ and ${\mathrm{\Omega }}^{**}$. Here, in both cases $f\left(\tau ,\xi ,\eta \right)$ intensifies against ${\mathrm{\Omega }}^{*}$ and ${\mathrm{\Omega }}^{**}$. The variation for velocity $\left(g\left(\tau ,\xi ,\eta \right)\right)$ against ${\gamma }^{**}$ can be observed in Fig. 7. The velocity $\left(g\left(\tau ,\xi ,\eta \right)\right)$ shows an increasing behavior against ${\gamma }^{**}$. Outcomes of ${\mathrm{\Omega }}^{*}$ and ${\mathrm{\Omega }}^{**}$ over velocity field $\left(g\left(\tau ,\xi ,\eta \right)\right)$ are given in Fig. 8, Fig. 9. Again, the velocity field $\left(g\left(\tau ,\xi ,\eta \right)\right)$ intensifies for ${\mathrm{\Omega }}^{*}$ and ${\mathrm{\Omega }}^{**}$. Fig. 10 displays the impact of Hartmann number over velocity field $\left(g\left(\tau ,\xi ,\eta \right)\right)$. Clearly, velocity field $\left(g\left(\tau ,\xi ,\eta \right)\right)$ decays when the values of Hartmann number increases. Fig. 11 depicts the behavior of $\left(M\right)$ over velocity field $\left(g\left(\tau ,\xi ,\eta \right)\right)$. It is obvious from the plot that velocity field $\left(g\left(\tau ,\xi ,\eta \right)\right)$ enhances with an increment in $\left(M\right)$. The variations of temperature difference parameter $\left(\beta \right)$ over temperature $\left(\theta \left(\tau ,\xi ,\eta \right)\right)$ is presented in Fig. 12. An escalation in $\beta$ corresponds to a rise in $\theta \left(\tau ,\xi ,\eta \right)$. Fig. 13 display the outcomes of (${\gamma }^{***}$) over temperature distribution. An enhancement in temperature is observed in temperature when (${\gamma }^{***}$) increases. Fig. 14 shows the impact of $Ha$ over $\theta \left(\tau ,\xi ,\eta \right)$. Temperature $\left(\theta \left(\tau ,\xi ,\eta \right)\right)$ increases for higher $Ha$. Fig. 15 gives the distinctions of $\left(M\right)$ against temperature $\left(\theta \left(\tau ,\xi ,\eta \right)\right)$. Temperature shows a decreasing behavior against increasing values of $\left(M\right)$. Fig. 16 display outcomes for $\left({\mathrm{\Omega }}^{**}\right)$ over temperature field. Here, temperature $\left(\theta \left(\tau ,\xi ,\eta \right)\right)$ improves with an enhancement in $\left({\mathrm{\Omega }}^{**}\right)$. Fig. 17 shows behavior of $\theta \left(\tau ,\xi ,\eta \right)$ against higher values of ${\mathrm{\Omega }}^{*}$. Here, $\theta \left(\tau ,\xi ,\eta \right)$ grows because of ${\mathrm{\Omega }}^{*}$. Fig. 18 shows behavior of temperature against higher $Q^{*}$. Temperature $\left(\theta \left(\tau ,\xi ,\eta \right)\right)$ is observed to be an increasing function of $Q^{*}$.

Fig. 2.

${\gamma }^{*}$ over velocity $\left(f\left(\tau ,\xi ,\eta \right)\right)$.

Fig. 3.

Ha over velocity $\left(f\left(\tau ,\xi ,\eta \right)\right)$.

Fig. 4.

M over velocity $\left(f\left(\tau ,\xi ,\eta \right)\right)$.

Fig. 5.

${\mathrm{\Omega }}^{*}$ over velocity $\left(f\left(\tau ,\xi ,\eta \right)\right)$.

Fig. 6.

${\mathrm{\Omega }}^{**}$ over velocity $\left(f\left(\tau ,\xi ,\eta \right)\right)$.

Fig. 7.

${\gamma }^{**}$ over velocity $\left(g\left(\tau ,\xi ,\eta \right)\right)$.

Fig. 8.

${\mathrm{\Omega }}^{*}$ over velocity $\left(g\left(\tau ,\xi ,\eta \right)\right)$.

Fig. 9.

${\mathrm{\Omega }}^{**}$ over velocity $\left(g\left(\tau ,\xi ,\eta \right)\right)$.

Fig. 10.

Ha over velocity $\left(g\left(\tau ,\xi ,\eta \right)\right)$.

Fig. 11.

$M$ over velocity $\left(g\left(\tau ,\xi ,\eta \right)\right)$.

Fig. 12.

$\beta$ over temperature $\left(\theta \left(\tau ,\xi ,\eta \right)\right)$.

Fig. 13.

${\gamma }^{***}$ over temperature $\left(\theta \left(\tau ,\xi ,\eta \right)\right)$.

Fig. 14.

Ha over temperature $\left(\theta \left(\tau ,\xi ,\eta \right)\right)$.

Fig. 15.

$M$ over temperature $\left(\theta \left(\tau ,\xi ,\eta \right)\right)$.

Fig. 16.

${\mathrm{\Omega }}^{**}$ over temperature $\left(\theta \left(\tau ,\xi ,\eta \right)\right)$.

Fig. 17.

${\mathrm{\Omega }}^{*}$ over temperature $\left(\theta \left(\tau ,\xi ,\eta \right)\right)$.

Fig. 18.

$Q^{*}$ over temperature $\left(\theta \left(\tau ,\xi ,\eta \right)\right)$.

5. Concluding remarks

To summarize, this inquiry scrutinized characteristics of hybrid nanofluid $\left(Ag+Cu/H_{2}O\right)$, wherein thermal slip arises owing to heat generation effects induced by rotating disks. The outcomes were depicted pictorially to visually represent the influence of various dimensionless variables. The main outcomes are:

• ❖Velocity increased against both rotation parameters $\left({\mathrm{\Omega }}^{*}\right)$ and $\left({\mathrm{\Omega }}^{**}\right)$.
• ❖Velocity field decayed against higher values of $\left({\gamma }^{*}\right)$.
• ❖Velocity field enhanced due to increment in the value of $\left(M\right)$.
• ❖Velocities reduced against higher estimations of Hartman number $\left(Ha\right)$.
• ❖Temperature enhanced against higher values of $\left(\beta \right)$ and $\left({\gamma }^{***}\right)$.
• ❖Temperature improved with an enhancement in rotation parameters $\left({\mathrm{\Omega }}^{*}\right)$ and $\left({\mathrm{\Omega }}^{**}\right)$.
• ❖Temperature showed decreasing behavior against increasing values of Reynolds number $\left(M\right)$.
• ❖Temperature was observed to be a monotonically increasing function concerning heat generation $\left(Q^{*}\right)$ and the Hartmann number $\left(Ha\right)$.

Author contribution statement

Khursheed Muhammad, Syed Irfan Shah: Conceived and designed the analysis; Wrote the paper.

Inayatullah: Analyzed and interpreted the data; Wrote the paper.

Taghreed A. Assiri: Contributed analysis tools or data.

Ibrahim E. Elseesy: Conceived and designed the analysis; Contributed analysis tools or data.

Data availability statement

Data will be made available on request.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Nomenclature

$u,v,w$

Components of velocity

$k$

Thermal conductivity

$\sigma$

Electrical conductivity

$\rho$

Density

$hnf$

Hybrid nanofluid $\left(Ag+Cu/H_{2}O\right)$

$f$

Fluid

${\phi }_{Ag}$

Concentration of $Ag$

$\mathbf{V}$

Velocity field of fluid

$\tau ,\xi ,\eta$

Dimensionless variables

$B_0$

Magnetic field of strength

$\mathrm{Pr}$

Prandtl number

${\mathrm{\Omega }}_1,{\mathrm{\Omega }}_2$

Angular velocities rates

$a_1,a_2$

Stretching rate

${\lambda }^{*},{\lambda }^{**}$

Slip velocity parameter along $u$ and $v$ axis

${\lambda }^{***}$

Thermal slip parameter

$Q^{*}$

Heat generation parameter

$M,M_1$

Reynolds number along $u$ and $v$ axis

$C{f}_0,C{f}_1$

Coefficients of skin friction

$N{u}_0,N{u}_1$

Heat transfer rates

$z,r$

Cylindrical coordinates

$\nu$

Kinematic viscosity

$c_p$

Specific heat capacity

$\mu$

Dynamic viscosity

$nf$

Nanofluid $\left(Ag/H_{2}O\right)$

${\phi }_{Cu}$

_{Concentration of} $Cu$

$n$

Shape factor

$T$

Temperature of fluid

$f,g,h,\theta$

Dimensionless velocities and temperature

$Ha$

Hartmann number

$Sc$

Schmidt number

${\mathrm{\Omega }}^{*},{\mathrm{\Omega }}^{**}$

Rotation parameters

$h_0$

Distance from the origin

${\lambda }_1,{\lambda }_2$

Slip velocity coefficient along $u$ and $v$ axis

${\lambda }_3$

Thermal slip coefficient

$Q_0$

Uniform volumetric heat generation constant

$\beta$

Temperature difference parameter

${\tau }_w$

Wall shear stress

$q_w$

Heat flux