Unsteady 3D heat transport in hybrid nanofluid containing brick shaped ceria and zinc-oxide nanocomposites with heat source/sink

Abstract

In the field of nano-composites, hybrid nano-fluids have noteworthy applications in aerospace, energy materials, thermal sensors, antifouling, etc. because of their ability to produce higher thermal conductivity than conventional nanofluids. Different combinations of nanocomposites have been found in the literature to develop the suitable hybrid-mixture, but no study has been found yet about the combined influence of ceria and zinc-oxide nanocomposites in the host liquid. In this article, unsteady 3 D transport of water driven hybrid nano-fluid with the consequences of brick shaped nanocomposites (ceria; $\mathrm{Ce}{\mathrm{O}}_2$ and zinc oxide; $\mathrm{ZnO}$) has deliberated with the thermal link of heat source/sink. Variable thermal conditions have been supplied at the surface with the effect of a magnetic environment. Similarity relations have been used to articulate the transport equations into solvable forms and then solved numerically via Keller-Box method. Nusselt number and skin-friction coefficients have also plotted with the wide ranges of involved parameters. Thermal setup has also briefly been discussed with the non-uniformity of surface temperature. Rate of heat transfer has significantly improved with the amounts of ceria (1 wt% to 10 wt%) and zinc-oxide (1 wt% to 10 wt%) nanocomposites. The Nusselt  number is reported in the range of 4.0 to 4.8 with the increasing amount of $\gamma$ from −0.6 to −0.2, whereas it is reported in the range of 3.1 to 3.9 with the varying amount of $\gamma$ from 0.2 to 0.6. Rate of heat transfer is observed higher for zinc-oxide nanoparticles as compared to ceria nanoparticles.

Graphical Abstract

Keywords:

• Brick shaped nanocomposites
• ceria
• heat source/sink
• hybrid nanofluid
• MHD
• unsteady 3D dynamics
• variable thermal conditions
• zinc-oxide

1. Introduction

The extension of nano-fluid with the composition of two or more different nanocomposites dissolves in the host liquid having higher thermal conductivity is termed as a hybrid nano-fluid. Hybrid nano-fluids are extensively applied in various fields of heat transfer such as coolant in machining, transformer and electronic cooling, drug reduction, biomedical and refrigeration, etc. with superior proficiency as compared to conventional nano-fluids. Waini et al. [1] incorporated heat transference and unsteady flow over elongating sheet in hybrid nano-fluid and elaborated that for a certain choice of unsteadiness constraint, dual solutions may exists. To determine stability of dual solutions, temporal stability analysis is performed and elaborated that one is stable and other is unstable. Abbasi and Farooq [2] glimpsed numerical simulation for transportation of hybrid nano-fluid in non-uniform vertical channel and deduced that current study is helpful in nano-medical technologies. Babazadeh et al. [3] analyzed the behavior of hybrid nano-fluid including radiation effects and Lorentz force within a porous cavity and depicted that higher values of Hartman factor enhances the conduction mode. Pourrajab et al. [4] experimentally deliberated collaborative effects of improvement in thermal conductivity of water driven hybrid nano-fluid containing $\mathit{MWCNTs}-\mathit{COOH}$ and $Ag$ nanocomposites. It was perceived that the hybrid nano-fluid’s thermal conductivity having 0.16 vol% $\mathit{MWCNTs}$ and 0.04 vol% $Ag$ nanocomposites was collectively improved by 47.3% in comparison with thermal conductivity of the water driven liquid. Shah et al. [5] reported the CFD simulation of water driven hybrid nano-fluid with nanocomposites (${Fe}_3{O}_4+\mathit{MWCNT})$ using free convection in permeable media and deduced that by enlargement of Hartmann number, temperature of heat source is diminished. Malika and Sonawane [6] disclosed consequences of nanocomposites mixed ratio on stability and thermo-physical characteristics of $\mathit{CuO}-\mathit{ZnO}$ water-driven hybrid nano-fluid. Size, shape and morphology of nanocomposites were characterized using DLS and Zeta analysis. Investigation of the properties of concentrated water driven $\mathit{AlN}+\mathit{ZnO}$ binary hybrid nano-fluid in heat pipe was presented by Çiftçi [7]. Properties of water driven graphene-silver ($Gr-Ag$) hybrid nano-fluid in recharging micro-channel (RMC) with temperature dependent properties numerically explored by Samal and Moharana [8] and deduced that use of RMC is advantageous for high heat flux. When pumping water is not constraint then utilization of water driven ($Gr-Ag$) hybrid nano-fluid has various advantageous. Tiwari et al. [9] experimentally investigated the influence of stabilization procedures on thermal conductivity and stability of $Ce{O}_2+\mathit{MWCNT} (80:20)$ water driven hybrid nano-fluid. Thermal conductivity of ${Al}_2{O}_3$ and $Ce{O}_2$ and their hybrid via ratio (50:50) by volume based deionized water nano-fluid experimentally incorporated by Kamel et al. [10]. Lahari et al. [11] examined heat transference characteristics of water driven $Ti{O}_2-\mathit{ZnO}$ nano-fluids. $Ti{O}_2$ and $\mathit{ZnO}$ nano-fluids prepared at 0.5%, 1.5% and 2.0% volume concentration and temperature range of $30-{70}^{0}C.$ It was noticed that substantial augmentation of both heat transfer rate and efficiency of heat exchanger with $Ti{O}_2$ nano-fluids at 2.0% volume concentration and 1.5% volume concentration of $Ti{O}_2-\mathit{ZnO}$ hybrid nano-fluids were achieved. Bhattad [12] inquired experimentally energy of plate heat exchanger via energy linked performance features like, coolant energy rate, coolant outlet temperature, non-dimensional energy, irreversibility rate, and second law efficiency. Coolant energy rate, coolant outlet temperature, non-dimensional energy and irreversibility rate increases by 2.5 wt%, 4.8 wt%, 7.5 wt% and 3.5 wt%, respectively, whereas the second law efficiency reduces using nanocomposites and increasing flow rate and decreases with the coolant inlet temperature. Yadav et al. [13] experimentally scrutinized rheological comportment of 13 nm ${Al}_2{O}_3-EG,$ 25 nm $Ce{O}_2-EG$ and the hybrid mixture $Ce{O}_2$–${Al}_2{O}_3$ (in 50:50 volume)/$EG$ over volumetric concentrations 0.05–1 vol% and temperature of 30–90 °C. It was concluded that performance of the hybrid mixture is similar to $EG$ which makes it advantageous in anti-freezing applications. Nano-structures of $\mathit{ZnO}$ with concrete based Jaffrey nano-fluid presented by Sheikh et al. [14] and conferred its applications in cement and concrete industry. Nfawa et al. [15] reported the use of tiny-particle $\mathit{MgO}$ for augmenting thermal conductivity of $\mathit{CuO}$-water nanofluid and used two-step method for synthesis of $\mathit{CuO}-\mathit{MgO}$-water hybrid nano-fluid. Some recent explorations related with the investigation of hybrid nanofluid in the view of various geometries are discussed via Refs. [16–19].

The nanocomposites shapes may have a significant effect on thermo-physical characteristics of the nano-fluids. Various experimental and numerical studies have been allocated to nanocomposites shapes effects. Saba et al. [20] analyzed water driven $Cu-{Al}_2{O}_3$ hybrid nano-fluid with different shapes and concluded that platelet shape nanocomposites offer better heat transference ability in comparison with other shapes. Convective heat transferal in horizontal pipe of hybrid nano-fluid considering different shapes effects were numerically studied by Benkhedda et al. [21] and revealed the idea that for platelet shaped nanocomposites, maximum value of friction factor has registered. Some more extraordinary considerations concerning the impact of nanocomposites shapes of various substances are discussed by the researchers via Refs. [22–28].

Magneto-hydro-dynamics (MHD) has gained extensive applications in the manufacturing and engineering industries such as crystal growth in liquid, cooling of nuclear reactor, food processing, electronic package, underground heat pumps, and solar technology. Mabood et al. [29] reported 3 D unsteady MHD flow of ${Fe}_3{O}_4$/graphene-water hybrid non-fluid with non-linear radiation effects. Non-Newtonian Casson hybrid nano-fluid with MHD mixed convection flow in stagnation link was investigated by Zahar et al. [30]. Sulochana and Apama [31] analyzed mass and heat transference mechanism of unsteady MHD thin film stream of $Al-Cu$–$H_{2}O$ hybrid nano-fluid subject to electromagnetic waves. Afterwards, numerous authors via Refs. [32, 33] exploited the magnetic effect on the flow of nanofluids passed/touching the various geometries. Heat transference can be controlled by external magnetic field which has several applications in different fields such as thermal engineering, aero-scope and electronic packing. Gorla et al. [34] explored numerically MHD effects of heat transferal on hybrid nano-fluid in square porous cavity. MHD mixed convection flow of $Cu-{Al}_2{O}_3-H_{2}O$ hybrid nano-fluid with impact of heat transferal near stagnation point is revealed by Jamaludin et al. [35] over moveable surface. Armaghani et al. [36] addressed the mixed convection of ${Al}_2{O}_3-Cu-H_{2}O$ hybrid nano-fluid by considering a heat source/sink for L-shaped cavity. Moreover, researchers [37–41] have been performed some novel studies about hybrid nano-fluid with heat source/sink.

Prescribed surface temperature and heat flux have plentiful applications in industrial and metallurgical processes such as glass fiber and paper production, wire drawing, extrusion of polymer sheets and drawing of plastic films etc. Shehzad et al. [42] provided the exact solution of Maxwell fluid over bidirectional elongating sheet with both VST and VHF cases. Hayat et al. [43] carried out the study of 3 D flow of an Oldroyd-B fluid over bi-directional enlarging surface with VST and VHF. Furthermore, Waini et al. [44] depicted permanence of flow and heat transferal over a stirring needle with VHF thermal supply. Natural convection mass and heat transferal of nano-fluids over vertical plate surrounded in a saturated Darcy porous medium subjected to VST and VHF was numerically deliberated by Noghrehabadi et al. [45]. Some latest contributions about the VST and VHF cases have been found in the Refs. [46–48].

From published literature, it is concluded that not much attention is given to the dynamics of hybrid nano-fluid with the effects of brick shaped nanocomposites. The objective of present study is to deliberate the unsteady 3 D magnetic transference of water driven hybrid nano-fluid containing brick shaped nanocomposites (ceria $Ce{O}_2$ and zinc-oxide $\mathit{ZnO}$) with heat source/sink. Thermophysical properties of ceria and zinc-oxide nanocomposites have been used to model the problem with the fluid properties of water. Hamilton-Crosser model has been used for composition and shape of the dispersed nanoparticles. The thermal supply is examined under the environment of VST and VHF modes. An innovative numerical approach, namely, Keller-Box method [49–52] is selected to solve the flow problem. Graphical illustrations are made to inspect the rate of heat transference, temperature fluctuation for VST and temperature fluctuation for VHF under the control of emerging involved parameters. To the best of our knowledge, no such contribution has been made in literature yet.

2. Mathematical formulation

Cartesian configuration is opted to enlighten the unsteady mathematical model for bidirectional dynamics of water-driven hybrid nano-fluid containing brick shaped nanocomposites (ceria; $Ce{O}_2$ and zinc oxide; $\mathit{ZeO}$) with heat source/sink. This combination of ceria and zinc oxide nanocomposites is not previously considered by any researcher. The distinctive property of ceria is its reversible conversation to a non-stoichiometric oxide and it has vast applications in the purification of elements as well as in the polishing processes. The distinctive property of zinc-oxide is its good transparency. Zinc-oxide has variety of applications in heat protecting windows, thin film transistors, light emitting diode, liquid crystal display, etc. To scrutinize the MHD (magneto hydrodynamics) effects with uniform strength $B_0=\frac{b_0}{\sqrt{1-ct}},$ mathematical relation with the deficiency of electric force is utilized. The considered nanocomposites are kept in thermal equilibrium. To keep the flow incompressible as well as laminar, the no-slip phenomenon is considered at the surface. The disturbance in the rest position of the nanofluid is observed by displacing the stretching device with velocity $u_w=\frac{ax}{1-ct};a>0,c>0$ along $x$- axis and with velocity $v_w=\frac{by}{1-ct};b>0$ along $y$- axis, however, the region covered by the hybrid nano-fluid is denoted by $0<z<\infty .$ Two types of thermal conditions, namely, VST and VHF are applied to provide a variable temperature environment at the surface. The graphical abstract of the mathematical modeling is included in the report via Figure 1. Table 1 is constructed to summarize the thermo-physical appearances of $H_{2}O,$ $Ce{O}_2$ and $\mathit{ZnO}.$

Figure 1. Graphical abstract of the present modeling.

Table 1. Thermophysical properties of ceria and zinc oxide with water as host liquid [9–11].

Thermophysical properties	Host liquid	Nanoparticles
	$H_{2}O$	$Ce{O}_2 ({\psi }_1)$	$\mathit{ZnO}$ $({\psi }_2)$
Density $\left(\rho \right):kg/m^3$	997.1	7600	5600
Thermal conductivity $\left(k\right):W/mK$	0.613	4.54	13
Specific heat $\left(C_p\right):J/\mathit{kgK}$	4179	390.0	495.2
Electrical conductivity: $\left(\sigma \right):\mathrm{S}/m$	0.05	10	0.01
Prandtl number: $Pr$	6.20	–	–

By keeping above mentioned measures in contact, the transport rheology of hybrid-powder along with boundary layer concept is established as [29, 46, 48]: (1) $$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0,$$(1) (2) $$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}=\frac{{\mu }_{\mathit{hnf}}}{{\rho }_{\mathit{hnf}}}\frac{{\partial }^{2}u}{\partial z^2}-\frac{{\sigma }_{\mathit{hnf}}}{{\rho }_{\mathit{hnf}}}{B_0}^{2}u,$$(2) (3) $$\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}=\frac{{\mu }_{\mathit{hnf}}}{{\rho }_{\mathit{hnf}}}\frac{{\partial }^{2}v}{\partial z^2}-\frac{{\sigma }_{\mathit{hnf}}}{{\rho }_{\mathit{hnf}}}{B_0}^{2}v,$$(3) (4) $$\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+w\frac{\partial T}{\partial z}={\alpha }_{\mathit{hnf}}\frac{{\partial }^{2}T}{\partial z^2}+\frac{Q}{{\left(\rho C_p\right)}_{\mathit{hnf}}}(T-T_{\infty }).$$(4)

The geometrical conditions for present circumstances are [42, 43, 46] (5) $$\begin{array}{c}z=0: u=u_w\left(x,t\right)=\frac{ax}{1-ct}, v=v_w\left(y,t\right)=\frac{by}{1-ct}, w=0\\ z\to \infty : u\to 0, v\to 0,\end{array},$$(5) (6) $$\begin{array}{c}\text{VST case}:\mathrm{ }z=0:T=T_w\left(x,y,t\right)=T_{\mathrm{\infty }}+T_0\left(\frac{x^r{y}^s}{1-ct}\right)\\ z\to \infty : T\to T_{\infty },\end{array},$$(6) (7) $$\begin{array}{c}\text{VHF case}:\mathrm{ }z=0:-k_{\mathit{hnf}}{\left(\frac{\partial T}{\partial z}\right)}_w=q_w\left(x,y,t\right)=T_1\left(\frac{x^r{y}^s}{1-ct}\right)\\ z\to \infty : T\to T_{\infty }.\end{array},$$(7)

Here, $Q=\frac{Q_0}{1-ct}$ indicates the heat source/sink ($Q>0$ represents source, whereas $Q<0$ represents sink), $C_{p_{\mathit{hnf}}}$ entitles the specific heat capacity, ${\rho }_{\mathit{hnf}}$ is chosen to label the density of the hybrid material, $T$ characterizes the temperature at the surface, $T_0$ and $T_1$ are dimensional constants, $(u,v,w)$ entitles velocity components along $x$-, $y$- and $z$- directions, respectively, $(r, s)$ are the indices that are used to control the heat distribution at the surface, time is suggested by $t,$ whereas, ${\mu }_{\mathit{hnf}},$ $k_{\mathit{hnf}},$ ${\alpha }_{\mathit{hnf}}$ and ${\sigma }_{\mathit{hnf}}$ are labeled to describe the effective viscosity, thermal conductivity, thermal diffusivity and influence of electrical conductivity of the hybrid nano-powder.

To familiarize brick-shaped nanocomposites, the mathematical relations for present development of hybrid nano-powder are communicated as [22]: (8) $${\rho }_{\mathit{hnf}}={\psi }_1{\rho }_{p1}+{\psi }_2{\rho }_{p2}+\left(1-{\psi }_1-{\psi }_2\right){\rho }_f,$$(8) (9) $${\left(\rho C_p\right)}_{\mathit{hnf}}={\psi }_1{\left(\rho C_p\right)}_{p1}+{\psi }_2{\left(\rho C_p\right)}_{p2}+\left(1-{\psi }_1-{\psi }_2\right){\left(\rho C_p\right)}_f,$$(9) (10) $$\frac{k_{\mathit{hnf}}}{k_{bf}}=\frac{\left(k_{p2}+2.72k_{bf}\right)+2.72{\psi }_2\left(k_{p2}-k_{bf}\right)}{\left(k_{p2}+2.72k_{bf}\right)-{\psi }_2\left(k_{p2}-k_{bf}\right)},$$(10) (11) $$\frac{k_{bf}}{k_f}=\frac{\left(k_{p1}+2.72k_f\right)+2.72{\psi }_1\left(k_{p1}-k_f\right)}{\left(k_{p1}+2.72k_f\right)-{\psi }_1\left(k_{p1}-k_f\right)},$$(11) (12) $$\frac{{\sigma }_{\mathit{hnf}}}{{\sigma }_{bf}}=1+\frac{3\left(\frac{{\sigma }_{p2}}{{\sigma }_{bf}}-1\right){\psi }_2}{\frac{{\sigma p}_2}{{\sigma }_{bf}}+2-\left(\frac{{\sigma }_{p2}}{{\sigma }_{bf}}-1\right){\psi }_2},$$(12) (13) $$\frac{{\sigma }_{bf}}{{\sigma }_f}=1+\frac{3\left(\frac{{\sigma }_{p1}}{{\sigma }_f}-1\right){\psi }_1}{\frac{{\sigma }_{p1}}{{\sigma }_f}+2-\left(\frac{{\sigma }_{p1}}{{\sigma }_f}-1\right){\psi }_1},$$(13) (14) $${\alpha }_{\mathit{hnf}}=\frac{k_{\mathit{hnf}}}{{\left(\rho C_p\right)}_{\mathit{hnf}}},$$(14) (15) $$\frac{{\mu }_{\mathit{hnf}}}{{\mu }_{bf}}=1+1.9{\psi }_2+471.4{{\psi }_2}^2,$$(15) (16) $$\frac{{\mu }_{bf}}{{\mu }_f}=1+1.9{\psi }_1+471.4{{\psi }_1}^2.$$(16)

Here, quantities of ceria and zinc oxide are expressed through ${\psi }_1$ and ${\psi }_2,$ respectively. The set of equations used to non-dimensionlize the above formulation are written as [50]: (17) $$u=\frac{ax}{1-ct}{f}^{\prime }\left(\eta \right), v=\frac{ay}{1-ct}{g}^{\prime }\left(\eta \right),w=-{\left(\frac{a{\vartheta }_f}{1-ct}\right)}^{1/2}\left[f\left(\eta \right)+g\left(\eta \right)\right],\eta ={\left(\frac{a}{{\vartheta }_f\left(1-ct\right)}\right)}^{1/2}z,$$(17) (18) $$\text{VST case}:\mathrm{ }\theta \left(\eta \right)=\frac{T(x,y,z,t)-T_{\infty }}{T_w(x,y,t)-T_{\infty }},\text{ VHF case}:\mathrm{ }T-T_{\infty }=\frac{T_1}{k_f}{\left(\frac{{\vartheta }_f}{a\left(1-ct\right)}\right)}^{1/2}{x}^r{y}^s\varphi \left(\eta \right).$$(18)

With the involvement of Eqs. (19)(19) $${\epsilon }_1{f}^{″\prime }-{f^{\prime }}^2+\left(f+g\right)f^{″}-S\left(f^{\prime }+\frac{\eta }{2}{f}^{″}\right)-{\epsilon }_2{M}^2{f}^{\prime }=0,$$(19) and (20)(20) $${\epsilon }_1{g}^{″\prime }-{g^{\prime }}^2+\left(f+g\right)g^{″}-S\left(g^{\prime }+\frac{\eta }{2}{g}^{″}\right)-{\epsilon }_2{M}^2{g}^{\prime }=0,$$(20) , the transport equations take the following forms (19) $${\epsilon }_1{f}^{″\prime }-{f^{\prime }}^2+\left(f+g\right)f^{″}-S\left(f^{\prime }+\frac{\eta }{2}{f}^{″}\right)-{\epsilon }_2{M}^2{f}^{\prime }=0,$$(19) (20) $${\epsilon }_1{g}^{″\prime }-{g^{\prime }}^2+\left(f+g\right)g^{″}-S\left(g^{\prime }+\frac{\eta }{2}{g}^{″}\right)-{\epsilon }_2{M}^2{g}^{\prime }=0,$$(20) (21) $$\text{VST case}:\mathrm{ }{\epsilon }_3\theta ″+\mathit{Pr}\left(\left(f+g\right){\theta }^{\prime }-\left(r{f}^{\prime }+s{g}^{\prime }\right)\theta -S\left(\theta +\frac{\eta }{2}{\theta }^{\prime }\right)+{\epsilon }_4\gamma \theta \right)=0,$$(21) (22) $$\text{VHF case}:\mathrm{ }{\epsilon }_3\varphi ″+\mathit{Pr}\left(\left(f+g\right){\varphi }^{\prime }-\left(r{f}^{\prime }+s{g}^{\prime }\right)\varphi -S\left(\varphi +\frac{\eta }{2}{\varphi }^{\prime }\right)+{\epsilon }_4\gamma \varphi \right)=0,$$(22) with geometrical conditions (23) $$f\left(0\right)+g\left(0\right)=0, f^{\prime }\left(0\right)=1, g^{\prime }\left(0\right)=\alpha ,f^{\prime }\left(\eta \to \infty \right)\to 0,g^{\prime }\left(\eta \to \infty \right)\to 0,$$(23) (24) $$\text{VST case}:\mathrm{ }\theta \left(0\right)=1,\theta \left(\eta \to \infty \right)\to 0,\text{ VHF case}:\mathrm{ }{\varphi }^{\prime }\left(0\right)=-\frac{k_f}{k_{\mathit{hnf}}}, ,\varphi (\eta \to \infty )\to 0.$$(24)

Here, $\gamma =Q_0/a{\left(\rho C_p\right)}_f$ indicates the heat source $(\gamma >0)$ or heat sink $(\gamma <0),$ $M={\left(\frac{{\sigma }_f}{a{\rho }_f}\right)}^{1/2}{b}_0$ and $S=\frac{c}{a}$ indicate the Hartman number and unsteady factor, respectively, elongation ratio is conveyed by $\alpha =\frac{b}{a},$ Prandtl factor is specified by $Pr=\frac{{\upsilon }_f}{{\alpha }_f}$ and $({\epsilon }_1,{\epsilon }_2,{\epsilon }_3,{\epsilon }_4)$ are the relations related with the inclusion of nanocomposites and these are mathematically conveyed as: (25) $${\epsilon }_1=\frac{\left(1+37.1{\psi }_2+612.6{{\psi }_2}^2\right)\left(1+37.1{\psi }_1+612.6{{\psi }_1}^2\right)}{\left({\psi }_1\frac{{\rho }_{p1}}{{\rho }_f}+{\psi }_2\frac{{\rho }_{p2}}{{\rho }_f}+\left(1-{\psi }_1-{\psi }_2\right)\right)},$$(25) (26) $${\epsilon }_2=\frac{\left(1+\frac{3\left(\frac{{\sigma }_{p2}}{{\sigma }_{bf}}-1\right){\psi }_2}{\frac{{\sigma }_{p2}}{{\sigma }_{bf}}+2-\left(\frac{{\sigma }_{p2}}{{\sigma }_{bf}}-1\right){\psi }_2}\right)\left(1+\frac{3\left(\frac{{\sigma }_{p1}}{{\sigma }_f}-1\right){\psi }_1}{\frac{{\sigma }_{p1}}{{\sigma }_f}+2-\left(\frac{{\sigma }_{p1}}{{\sigma }_f}-1\right){\psi }_1}\right)}{{\psi }_1\frac{{\rho }_{p1}}{{\rho }_f}+{\psi }_2\frac{{\rho }_{p2}}{{\rho }_f}+\left(1-{\psi }_1-{\psi }_2\right)},$$(26) (27) $${\epsilon }_3=\frac{\left(\frac{\left(k_{p2}+4.72k_{bf}\right)+4.72{\psi }_2\left(k_{p2}-k_{bf}\right)}{\left(k_{p2}+4.72k_{bf}\right)-{\psi }_2\left(k_{p2}-k_{bf}\right)}\right)\left(\frac{\left(k_{p1}+4.72k_f\right)+4.72{\psi }_1\left(k_{p1}-k_f\right)}{\left(k_{p1}+4.72k_f\right)-{\psi }_1\left(k_{p1}-k_f\right)}\right)}{{\psi }_1\frac{{\left(\rho C_p\right)}_{p1}}{{\left(\rho C_p\right)}_f}+{\psi }_2\frac{{\left(\rho C_p\right)}_{p2}}{{\left(\rho C_p\right)}_f}+\left(1-{\psi }_1-{\psi }_2\right)},$$(27) (28) $${\epsilon }_4=\frac{1}{{\psi }_1\frac{{\left(\rho C_p\right)}_{p1}}{{\left(\rho C_p\right)}_f}+{\psi }_2\frac{{\left(\rho C_p\right)}_{p2}}{{\left(\rho C_p\right)}_f}+\left(1-{\psi }_1-{\psi }_2\right)}.$$(28)

Reduced Nusselt number and skin-friction coefficients $(C_{fx}, C_{fy})$ are the most remarkable quantities for thermal and mechanical processes, respectively, and are incorporated here in the forms of Reynolds’s numbers ${Re}_x=\frac{x{u}_w}{{\vartheta }_f},$ ${Re}_y=\frac{y{v}_w}{{\vartheta }_f}$ as: (29) $$\begin{array}{c}Nu=\{\begin{array}{c}\frac{-x{k}_{\mathit{hnf}}{\left(\frac{\partial T}{\partial z}\right)}_{z=0}}{k_f(T_w-T_{\infty })}=-{{Re}_x}^{1/2}\frac{k_{\mathit{hnf}}}{k_f}{\theta }^{\prime }\left(0\right) \left(\mathit{VST} \mathit{case}\right)\\ \frac{-x{k}_{\mathit{hnf}}{\left(\frac{\partial T}{\partial z}\right)}_{z=0}}{k_f(T-T_{\infty })}={{Re}_x}^{1/2}\frac{1}{\varphi \left(0\right)} \left(\mathit{VHF} \mathit{case}\right)\end{array}\\ {{Re}_x}^{1/2}{C}_{fx}=\left(1+37.1{\psi }_1+612.6{{\psi }_1}^2\right)\left(1+37.1{\psi }_2+612.6{{\psi }_2}^2\right)f″\left(0\right),\\ {{Re}_y}^{1/2}{C}_{fy}={\alpha }^{-3/2}\left(1+37.1{\psi }_1+612.6{{\psi }_1}^2\right)\left(1+37.1{\psi }_2+612.6{{\psi }_2}^2\right)g″\left(0\right).\end{array},$$(29)

3. Keller-Box simulation

In the current era, researchers are interested to find the numerical solutions of the modeled problems because these solutions take the less computational time. Various numerical methods, namely, shooting method, RK-method, RKF-method, finite difference method and their combinations have been tested to solve the engineering problems. Among them the best method is implicit finite difference scheme, namely, Keller-Box method. In this method, higher order differential system is firstly transfigured into the first order differential system and then reduced differential system is reformed into a difference system by opting central difference approximations in the computed domain. The system of difference equations are then linearized along with boundary conditions by using the well-known Newton-Raphson method. The linearized system is then shaped into the matrix-vector form with tri-diagonal matrix as a coefficient of unknown vector. LU-decomposition scheme is opted to solve the formulated matrix-vector problem. The value of unknown vector will be the first approximation of the solution. The further approximations of the solutions are obtained by varying the numbers of grid-points $n_p.$ The final value of unknown vector is further used for graphics purpose as well as for the computation of heat transaction rate.

Table 2 provides the solution of the formulated problem by using the above mentioned steps of Keller-Box method. While obtaining the solution, the initial value of $\eta$ (say ${\eta }_0$) is opted zero and then increased gradually to obtain the final value of $\eta$ (i.e. ${\eta }_{\infty }=10$). The further increment in the value of ${\eta }_{\infty }$ did not affect the solution of the problem. The first approximation of the Keller-Box solution is obtained by considering $n_p=500$ with step-size $h=\frac{{\eta }_{\infty }-{\eta }_0}{n_p}$ and then the value of $n_p$ is increased gradually up-to 6500 in order to attain the convergent and stable numerical solution of the physical problem with accuracy $={10}^{-6}$ . It is inferred through Table 2 that five-hundred grid points are enough to obtain the convergent value of $f″(0),$ fifteen-hundred grid points are necessary to get the convergent amount of $g″(0),$ six-thousand grid points are enough to attain the convergence of $\theta \prime (0)$ for VST case and convergence of $\frac{1}{\varphi (0)}$ for VHF case as well. These values of $n_p$ that provide the convergent solution are further used for heat transfer analysis.

Table 2. Convergence of the Keller-Box simulation under the control of parameters $\alpha =S=M=0.5,\gamma =0.2,r=s=1.0,{\psi }_1=0.03,{\psi }_2=0.02.$

$n_p$	$-f″(0)$	$-g″(0)$	$-\theta \prime (0)$	$\frac{1}{\varphi (0)}$
500	1.114045	0.500372	3.341994	3.808766
1000	1.114045	0.500368	3.341426	3.808118
1500	1.114045	0.500367	3.341321	3.807998
2000	1.114045	0.500367	3.341284	3.807956
2500	1.114045	0.500367	3.341267	3.807937
3000	1.114045	0.500367	3.341257	3.807926
3500	1.114045	0.500367	3.341252	3.80792
4000	1.114045	0.500367	3.341248	3.807916
4500	1.114045	0.500367	3.341246	3.807913
5000	1.114045	0.500367	3.341244	3.807911
5500	1.114045	0.500367	3.341243	3.807909
6000	1.114045	0.500367	3.341242	3.807908
6500	1.114045	0.500367	3.341242	3.807908

4. Results and discussion

The effects of involved parameters on temperature distributions, reduced Nusselt  numbers and skin-friction coefficients are evaluated and discussed in this section via Figures 2–7 and Table 3. Effect of heat source/sink factor $\gamma$ on heat distribution $\theta (\eta )$ for VST state is assessed via Figure 2(a) and concluded that varying amount of $\gamma$ enhances the heat distribution $\theta (\eta )$ because of the reduction in the value of specific heat produced by water. Figure 2(b) explores the information about the effect of $\gamma$ on $\varphi (\eta )$ for VHF state and it is assessed that higher estimation of $\gamma$ develops the heat distribution $\varphi (\eta )$ because of the enhancement in the amount of heat source/sink connection $Q.$ The fluctuation in heat distributions is observed lower for heat sink state as compared to heat source state because of the transmission of heat from hotter region to colder region. Moreover, the width of thermal layer is compressed with the downfall in the amount of $\gamma$ and it is upgraded with the progression in the amount of $\gamma .$ Furthermore, the temperature for VHF state is observed less than the surface temperature because a huge amount of heat flux is produced at the stretchy surface.

Figure 2. (a, b). Heat estimations with the influence of $\gamma$ for VST state (pattern a) and for VHF state (pattern b).

Figure 3. (a, b). Heat estimations with the influence of $r$ for VST state (pattern a) and for VHF state (pattern b).

Figure 4. (a, b). Heat estimations with the influence of $s$ for VST state (pattern a) and for VHF state (pattern b).

Figure 5. (a-c). Nusselt number estimations with the pairs (a): ${\psi }_1$ (ceria) versus $\gamma ,$ (b): ${\psi }_2$ (zinc-oxide) versus ${\psi }_1$ (ceria) for heat sink situation, and ${\psi }_1$ (ceria) versus ${\psi }_2$ (zinc-oxide) for heat source situation.

Figure 6. (a-c). Skin-friction coefficients for ceria nanocomposites with the pairs (a): ${\psi }_1$ versus $S,$ (b): ${\psi }_1$ versus $\alpha ,$ and (c): ${\psi }_1$ versus $M.$

Figure 7. Comparison between rate of heat transfer by ceria $(1\mathit{\text{ wt\%}}, 3\mathit{\text{ wt\%}},5\mathit{\text{ wt\%}},7\mathit{\text{ wt\%}})$ and rate of heat transfer by zinc-oxide $(1\mathit{\text{ wt\%}}, 3\mathit{\text{ wt\%}},5\mathit{\text{ wt\%}},7\mathit{\text{ wt\%}})$ against the wide range of unsteady stretching parameter $S.$

Table 3. Calculation of Nusselt number for solid volume fractions ${\psi }_1$ (ceria) and ${\psi }_2$ (zinc-oxide) with $\alpha =S=M=0.5,\gamma =0.2,r=s=1.0.$

Volume percentages of nanocomposites		Nusselt number
		VST state		VHF state
		KBM	HAM	KBM	HAM
0.03	0.02	3.807908	3.807906	3.807908	3.807906
0.06	0.02	3.930894	3.930891	3.930894	3.930891
0.09	0.02	4.035562	4.035561	4.035562	4.035561
0.03	0.04	3.920627	3.920625	3.920627	3.920625
0.03	0.06	4.031697	4.031694	4.031697	4.031694
0.03	0.08	4.133501	4.133501	4.133501	4.133501

Figure 3(a) explores the influence of non-uniformity of the surface temperature along $x$- direction for VST state when the temperature along $y$- direction is fixed (i.e. $s=1.0$) and Figure 3(b) explains the role of the non-uniformity of the surface temperature along $x$- direction for VHF state when the temperature along $y$- direction is fixed (i.e. $s=1.0$). The heat distribution is reduced with the upraising value of $r$ for both the VST and VHF states. The width of thermal layer is also reduced with the higher estimation of $r$ because surface temperature is improved with the growth in the choice of $r.$ The temperature is higher than the surface temperature for $r=-2.0$ in the state of VHF and it is lower than the surface temperature for the values of $r$ greater than $-2$ because of huge heat flux production at the surface. The thermal space between the uppermost curve and the blue curve is very high because in between $r=-2.0$ and $r=-1.0$ the heat flux rate dominates in this region.

Figure 4(a) summaries the influence of non-uniformity of surface temperature along $y$- direction for VST state when the temperature along $x$- direction is kept fixed but non-uniform (i.e. $r=1.0$) and Figure 4(b) demonstrates the control of the non-uniformity of surface temperature along $y$- direction for VHF state when the temperature value along $x$- direction is kept constant but non-uniform (i.e. $r=1.0$). Here, we see that the width of the thermal layer is attained the identical behaviour with the progression of $s$ for VST state, whereas for VHF state it is initially developed and then comes to a uniform width for VHF state because a higher rate of heat flux is produced for this situation.

Figure 5(a) describes the graphical explanation of reduced Nusslet number for the ceria nanocomposites with varying choice of $\gamma .$ It is shown that higher loading of ceria nanocomposites develops a rate of heat transaction for both positive and negative values of gamma but the rate of heat transaction is higher for negative value of gamma than positive value of gamma. Physically, a negative value of gamma reduces the heat conduction process and positive value of gamma enhances the heat conduction process. Moreover, rate of heat transaction is reduced with the uprising in the amount of gamma. The higher rate of heat transaction is achieved for $\gamma =-0.6$ and lower rate of heat transaction is attained for $\gamma =0.6$ in the selected range of gamma. Figure 5(b) explains the behaviour of reduced Nusslet number for the zinc-oxide nanocomposites with various loading percentages of ceria nanocomposites. The rate of heat transaction is highly improved with the higher loading percentages of zinc-oxide nanocomposites. Moreover, Nusslet number is also improved with the various loading percentages of ceria nanocomposites. The higher loading percentages of ceria nanocomposites with the combination of zinc-oxide nanocomposites develop the Nusslet number because the thermal conductivity of zinc-oxide is almost triple to the thermal conductivity of ceria. Figure 5(c) enlightens the combined influence of nanocomposites (i.e. ceria and zinc-oxide) on the rate of heat transaction. It is worth to noticed that rate of heat transaction is dramatically improved with the combined influence of these nanocomposites. The smallest value of Nusslet number is achieved for 1 wt% inclusion of ceria nanocomposites with the 1 wt% inclusion of zinc-oxide nanocomposites and the highest value of Nusslet number is attained for 10 wt% ceria nanocomposites with the 10 wt% of zinc-oxide nanocomposites for the adopted percentages of nanocomposites.

Figure 6(a) communicates the influence of unsteady stretching factor $S$ on skin-friction coefficients $C_{fx}$ (represented by solid lines) and $C_{fy}$ (represented by dotted lines) with different amounts of ceria when the amount of zinc-oxide is kept constant (i.e. 4 wt%). Escalation in the choice of $S$ from 0.2 to 1.1 reduces the skin-friction coefficients because a higher rate of unsteady expansion is produced with the uprising of $S.$ Also, the value of $C_{fx}$ is higher than $C_{fy}$ because positive disturbance in the value of $S$ reduces the value of expansion rate $a$ along $x$- direction and hence expansion ratio $\alpha$ is improved. Mathematically, $\alpha$ is involved in the numerator of Eq. (29)(29) $$\begin{array}{c}Nu=\{\begin{array}{c}\frac{-x{k}_{\mathit{hnf}}{\left(\frac{\partial T}{\partial z}\right)}_{z=0}}{k_f(T_w-T_{\infty })}=-{{Re}_x}^{1/2}\frac{k_{\mathit{hnf}}}{k_f}{\theta }^{\prime }\left(0\right) \left(\mathit{VST} \mathit{case}\right)\\ \frac{-x{k}_{\mathit{hnf}}{\left(\frac{\partial T}{\partial z}\right)}_{z=0}}{k_f(T-T_{\infty })}={{Re}_x}^{1/2}\frac{1}{\varphi \left(0\right)} \left(\mathit{VHF} \mathit{case}\right)\end{array}\\ {{Re}_x}^{1/2}{C}_{fx}=\left(1+37.1{\psi }_1+612.6{{\psi }_1}^2\right)\left(1+37.1{\psi }_2+612.6{{\psi }_2}^2\right)f″\left(0\right),\\ {{Re}_y}^{1/2}{C}_{fy}={\alpha }^{-3/2}\left(1+37.1{\psi }_1+612.6{{\psi }_1}^2\right)\left(1+37.1{\psi }_2+612.6{{\psi }_2}^2\right)g″\left(0\right).\end{array},$$(29) with negative exponent and hence $C_{fy}$ must be less than $C_{fx}$ as observed through the Figure 6(a).

Figure 6(b) highlights the influence of $\alpha$ from 0.1 to 0.7 on skin-friction coefficients $C_{fx}$ (denoted by solid lines) and $C_{fy}$ (denoted by dashed lines). It is deduced that escalation in the choice of $\alpha$ reduces the worth of $C_{fx}$ and enhances the net worth of $C_{fy}$ because the expansion rate $b$ is improved with the escalation of $\alpha .$ Moreover, the enhancement in the value of $\alpha$ increases the flow rate of the hybrid nanofluid at the surface and hence the combined influence of $C_{fy}$ is improved. The equality between skin-friction coefficients exists when $\alpha =1$ (i.e. axisymmetric case) as the dotted curves and dashed curves are being closer to each other when $\alpha \to 1.0.$ Furthermore, higher percentage involvement of ceria nanocomposites diminishes the skin-friction coefficients because the specific heat of ceria is lower than the specific heat of zinc-oxide, whereas the density of ceria is higher than the density of zinc-oxide. Figure 6(c) describes the impact of Hartman number $M$ on skin-friction coefficients with the varying amounts of ceria and fixed amount of zinc-oxide (i.e. 4 wt%). It is summarized that higher choice of $M$ diminishes the skin-friction coefficients as the electrical conductivity of the host liquid is much less than the density of the host liquid. Moreover, the higher estimation of $M$ produces a strong Lorentz force and hence a reduction in the skin-friction coefficient is obtained.

Figure 7 constructs the comparison between the rate of heat transaction by ceria (1 wt%, 3 wt%, 5 wt% and 7 wt%) represented by solid lines and the rate of heat transaction by zinc-oxide (1 wt%, 3 wt%, 5 wt% and 7 wt%) represented by dotted lines. Figure 7 declares that the rate of heat transaction by zinc-oxide is higher than the rate of heat transaction by ceria and the difference between their rates of heat transaction is lower for smaller amounts of nanocomposites than higher contents of nanoparticles. Physically, the thermal conductivity of zinc-oxide is almost triple the thermal conductivity of ceria and the above spectacular outcomes are achieved.

Table 3 addresses the Nusslet number for both the VST and VHF states against the different amounts of nanocomposites (ceria and zinc-oxide) for constant choices of other involved parameters. The results obtained through Keller-Box method (KBM) are also compared with the homotopy analysis method (HAM) for the validity of the numerical code. The value of Nusslet number is tremendously improved with the increasing nanocomposite contents i.e. 3 wt%, 6 wt% and 9 wt%. The Nusslet number for both thermal conditions is similar as the relation for Nusslet number for VST state via Eq. (29)(29) $$\begin{array}{c}Nu=\{\begin{array}{c}\frac{-x{k}_{\mathit{hnf}}{\left(\frac{\partial T}{\partial z}\right)}_{z=0}}{k_f(T_w-T_{\infty })}=-{{Re}_x}^{1/2}\frac{k_{\mathit{hnf}}}{k_f}{\theta }^{\prime }\left(0\right) \left(\mathit{VST} \mathit{case}\right)\\ \frac{-x{k}_{\mathit{hnf}}{\left(\frac{\partial T}{\partial z}\right)}_{z=0}}{k_f(T-T_{\infty })}={{Re}_x}^{1/2}\frac{1}{\varphi \left(0\right)} \left(\mathit{VHF} \mathit{case}\right)\end{array}\\ {{Re}_x}^{1/2}{C}_{fx}=\left(1+37.1{\psi }_1+612.6{{\psi }_1}^2\right)\left(1+37.1{\psi }_2+612.6{{\psi }_2}^2\right)f″\left(0\right),\\ {{Re}_y}^{1/2}{C}_{fy}={\alpha }^{-3/2}\left(1+37.1{\psi }_1+612.6{{\psi }_1}^2\right)\left(1+37.1{\psi }_2+612.6{{\psi }_2}^2\right)g″\left(0\right).\end{array},$$(29) differs from the Nusslet number for VHF state by the factor $\frac{k_{\mathit{hnf}}}{k_f},$ which has been adjusted in the thermal boundary condition i.e. ${\varphi }^{\prime }\left(0\right)=-\frac{k_f}{k_{\mathit{hnf}}}.$ Mathematically, for the present heat transfer problem the temperature distribution remained different for VST and VHF states whereas the rate of heat transfer is remained the same for both the thermal states because Eq. (21)(21) $$\text{VST case}:\mathrm{ }{\epsilon }_3\theta ″+\mathit{Pr}\left(\left(f+g\right){\theta }^{\prime }-\left(r{f}^{\prime }+s{g}^{\prime }\right)\theta -S\left(\theta +\frac{\eta }{2}{\theta }^{\prime }\right)+{\epsilon }_4\gamma \theta \right)=0,$$(21) and Eq. (22)(22) $$\text{VHF case}:\mathrm{ }{\epsilon }_3\varphi ″+\mathit{Pr}\left(\left(f+g\right){\varphi }^{\prime }-\left(r{f}^{\prime }+s{g}^{\prime }\right)\varphi -S\left(\varphi +\frac{\eta }{2}{\varphi }^{\prime }\right)+{\epsilon }_4\gamma \varphi \right)=0,$$(22) are the same for both the thermal cases but boundary conditions are slightly different.

5. Conclusions

A mathematical solution for unsteady bidirectional dynamics of water driven hybrid nano-fluid (i.e. the combination of nanocomposites ceria $Ce{O}_2$ and zinc-oxide $\mathit{ZnO}$ having brick shapes) with VST and VHF thermal conditions has been reported. The magnetic impact and internal heat source/sink impact have also incorporated in the momentum balance and energy balance equations, respectively. The heat fluctuation was reduced with the higher estimations of heat controlling indices along $x$- and $y$- directions, whereas it is significantly improved with the increasing of heat source/sink factor $\gamma .$ Increasing amounts of ceria and zinc-oxide developed the rate of heat transaction, whereas skin-friction coefficients were slightly reduced with the positive tendencies in these percentage amounts. The amount of reduced Nusslet number doubled for zinc-oxide nanocomposites compared to ceria nanocomposites. It is hoped that the present heat transport model will not only contribute to the development of nanocomposites but also will set a new roadmap for researchers working in this field.

Acknowledgments

We are grateful to honorable Editor-in-Chief and worthy reviewers for their useful recommendations which helped us in order to improve the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Notes on contributors

Iftikhar Ahmad

Iftikhar Ahmad currently working as a professor in the Department of Mathematics at Azad Jammu and Kashmir University, Muzaffarabad, Pakistan. He completed his PhD in mathematics from Quaid-i-Azam University, Islamabad, Pakistan. His research is in the field of fluid dynamics focusing on heat and mass transfer problems. Other research areas address modeling of engineering problems, solution of differential equations, numerical analysis, etc.

Muhammad Faisal

Muhammad Faisal is a Faculty Member in the Department of Mathematics, Faculty of Science, Azad Jammu & Kashmir University, Muzaffarabad, Pakistan. He received his PhD, MPhil and MSc degrees from Department of Mathematics UAJK in 2021, 2013 and 2010, respectively. His research is in the field of computational fluid dynamics, thermal engineering, engineering mathematics and physics, nanocomposites, computational methods, nanofluids, hybrid nanofluids, etc.

Qazi Zan-Ul-Abadin

Qazi Zan-Ul-Abadin is a PhD scholar at the Department of Mathematics, Faculty of Science, Azad Jammu & Kashmir University, Muzaffarabad, Pakistan. He received his MSc and MPhil degrees from Department of Mathematics UAJK in 2013 and 2016, respectively. He is working in the field of nanocomposites and hybrid nanofluids.

Tariq Javed

Tariq Javed is currently working as a professor in the Department of Mathematics and Statistics at International Islamic University, Islamabad, Pakistan. He completed his PhD in mathematics from Quaid-i-Azam University, Islamabad, Pakistan. His research is in the field of computational fluid dynamics focusing on cavity flow problems. Other research areas address modeling of engineering problems, solution of differential equations, numerical analysis, boundary layer flows, stagnation point flows, etc.

K. Loganathan

K. Loganathan is currently working as researcher in the Department of Mathematics and Statistics at Manipal University Jaipur, Rajasthan, India. He received MSc and MPhil degrees in 2014 and 2016, respectively. His areas of interest are fluid dynamics, CFD, convective heat and mass transfer, nanofluids, bioconvection, microorganisms, porous media, etc.