A Riga plate is an electromagnetic actuator made up of permanent magnets and alternating electrodes assembled on a horizontal surface1. A Riga plate concept was invented half a century ago by Gailitis and Lielausis at the Physics Institute in Riga, Latvia2. An arrangement of alternating electrodes and permanent magnets arranged spanwise on a flat surface constituted as a flow control device, produced wall-parallel Lorentz force3. This electromagnetic actuator may effectively minimize the friction and hydraulic drag in submarines through boundary layer (BL) separation mitigation and turbulence suppression. The Riga plate have potential uses including the aviation industry to reduce drag on airplane wings, in naval engineering to reduce hull friction in ships and submarines, in automotive systems to controlee the BL in high-speed automobiles, in MHD power generation for regulating flow in plasma channels, and turbomachinery for improving compressors and turbine productivity4,5,6,7. Due to its various applications in modern engineering it has been a core subject among modern researchers. In a recent study, the thermal hydrodynamic performance of Maxwell fluid with precise heat and mass transfer over a Riga plate was elaborated by Alrihieli et al.8. Their results demonstrated that when the Hartmann number is increased the Maxwell fluid displays an extremely high velocity distribution. Their study highlighted the main physical findings of improved electromagnetic influence and a reduction in the flow obstruction of the system. Sharma and Gorai9 examined the flow and heat transfer over a Riga squeezing plate taking melting heat at the surface. They proposed that the repercussions permit the anti-relation between melting heat and changed Hartman number in fluid flow. Magnetic behavior drives the thermal BL thickness of the Riga plate parameter expand. The dissipative nanofluid (NF) flow over a stretching Riga plate was elaborated by Ahmed et al.10. They further analyzed the heat and mass transfer features and shape effects. Shah11 investigated the bioconvection micropolar NF flow near stagnation point over tilted Riga plate implanting Keller-box approach. He suggested that velocity is increasing function of Hartman number.
The scientific community are currently interested in NF because of its importance in many applications. NF primarily involves dispersion or mixing nanoparticles with fluids, is utilized in a wide range of real-world heat transfer applications, such as cooling systems, electronic devices, biomaterials, transportation, and the food industries12,13. Since NF differ from their solid counterparts in terms of their physical and chemical properties, their uses in biomedical technologies has grown recently. Because of these characteristic, NF can be used for a number of biological activities, such as the splitting of magnetic cells, improving the contrast of magnetic resonance imaging, administering medication, curing infections, and causing hyperthermia, among others14. Tri-hybrid NF (Tri-HNF) have attracted a lot of interest because of their improved thermophysical characteristics, which result from the synergistic effects of integrating three different kinds of nanomaterials in a base fluid. These NFs are especially appealing for heat exchange applications in sectors like electronics cooling, energy generation and aerospace because they have better heat transmission capabilities, improved thermal conductivity, and enhanced stability when compared to single-phase fluids or hybrid NFs. Three different nanoparticles \(A{l}_{2}{O}_{3}\), \(CuO\) and \(Ti{O}_{2}\) are dispersed in a water base fluid to create the tri-HNF used in this investigation. These particular nanoparticles are carefully chosen with the goal of utilizing their respective thermophysical characteristics to produce synergistic performance15. The nanomaterial \(CuO\) offers much better thermal conductivity to dramatically increase the heat transfer capability, \(Ti{O}_{2}\) helps to the stability and distribution of the mixture, and \(A{l}_{2}{O}_{3}\) offers a balance between superior thermal conductivity and affordability16,17. An equal volumetric percentage of 5% is taken into account for each type of nanoparticle in the investigation. This method is frequently used in early research to methodically examine the combined effects of the triple hybrid composition. The enhanced performance over mono or HNF is predicted using existing models, including the Hamilton-Crosser model for thermal conductivity and the Gharesim model for viscosity.
The effect of Tri-HNF on the temporal thermal performance of inclined merge fins was investigated by Pavan et al.18, with an emphasis on internal heat production. Babu et al.19 used a statistical technique to simulate the squeezed flow of HNF made of PEG and water over a magnetic sensor surface, providing information on the scattering of nanoparticles in intelligent sensing systems. Rauf et al.20 investigated convective heat transfer in a non-Newtonian Tri-HNF with non-local kernel conditions and rising temperatures. The Tri-HNF technique performs better than hybrid and traditional NF in terms of energy transfer and momentum profile, according to their numerical and graphical evaluations. On flat and thin sheets, Olabode et al.21 thoroughly examined Tri-HNF flow with temperature-dependent thermophysical characteristics. Using Keller box analysis, Shah11 investigated the stagnation-point flow of micropolar bioconvective nanofluids over a tilted Riga plate, highlighting the significance of magnetohydrodynamic (MHD) phenomena in microscopic heat transfer. A growing curiosity in simulating and optimizing tri-HNF systems under diverse physical conditions is highlighted by these research taken together.
Density variations brought on by the temperatures, chemical, and concertation gradients driven flow occurs in many natural transportation mechanisms22,23. Therefore, it is significant to investigate flow caused by concentration variations separately or in conjunction with temperature variations. The Dufour effect is a physical phenomenon arises due to heat flux brought on by the chemical composition gradient (diffusion-thermo). However, if temperature gradients are the source of mass fluxes, this phenomenon is known as the thermal-diffusion (Soret effect)24. In most cases, these effects are only an insignificant order of magnitude. When these effects are taken jointly they are referred to as cross-diffusion effects. The cross-diffusion effects in a three dimensional mixed convection flow of Maxwell fluid embedded in a Darcy-Forchheimer medium were elaborated by Zhnag et al.25. They verified that when diffusion thermal effects and thermos-diffusion are increased, the temperature and concentration behave in opposing ways. Estimation of cross diffusion phenomenon on the chemically reacting micropolar fluid flow over an extending sheet was explored by Salawu26. It is clearly demonstrated that the concentration profile decreases with chemical reaction, while the cross diffusion influences enhance the system heat dispersion. An unsteady bioconvective fluid flow across concentric elongating cylinders with cross-diffusion effects were scrutinized by Shaheen et al.27. They concluded the conclusion that the thermal field expands by increasing the thermal conjugate parameter and Dufour number. When the Soret and Dufour numbers are amplified, the heat and mass flux shows reverse behavior. The computational investigation of cross-diffusion effects of chemically reactive magneto-NF over an oscillating absorbent plate were explored by Reddy et al.28. As cross diffusion and radiation effects increase, the temperature distribution widens. The concentration distribution is expanded by the thermo-diffusion effect. Zhao et al.29 examined the free-convective flow of NF due to titled surface taking cross-diffusion effects and chemical reaction. The cross-diffusion and thermal radiation effects on MHD non-Newtonian BLF over two different morphologies were investigated by Dharmaiah et al.30. They discovered that the temperature field improves with an increase in the diffusion thermal parameter. Temperature and velocity increase and concentration falls as radiation absorption increases. The radiation intake parameter decreases the concentration and increases temperatures closer to the porous boundary layer (BL).
Sensitivity analysis (SA) together with Response Surface Methodology (RSM) is commonly employed in aerodynamics for the optimization of drag and lift coefficients and in fluid dynamics to enhance parameters such as Reynolds number, concentration of nanoparticles, and strength of magnetic field in NF heat transfer systems. Beyond fluid mechanics, RSM-based SA propels advances in medical device engineering such as delivery of medications, materials science composite to optimize material design, and energy systems for fuel cell performance tuning, allowing for data-driven decision-making at low computational cost. In this regard, an active parametric SA to develop electrical and thermal exergy/energy proficiency of PVT system using NF was elaborated by Jabeen et al.31. They found that the goal of simultaneously achieving optimal electrical and thermal power is positively impacted by multiple optimization objectives with the ideal composite acceptability values, which illustrates the impact of input design parameters. Lin et al.32 performed SA for an investigation of hybrid thermal/photovoltaic system based on intense NF. They concluded that the solar radiation is among the most sensitive factor for electrical performance, with R and F-values of 10.729 and 2934.770, respectively. The R-value and F-value for solar radiation are 27.620 and 13,689.811, respectively, for energy efficiency, demonstrating its substantial impact. The heat transfer optimization using SA in buoyancy driven flow of Williamson tri-HNF over a thin needle was conducted by Bouzidi et al.33. The found that the higher load 5% of nanomaterial is more sensitive to heat transfer. Huang et al.34 performed SA for thermal conductivity of Aluminum-water based NF. According to their findings, the utilization of sphere-shaped particles enables the NF thermal conductivity to be stable over time, and the sensitive of volume fraction section can result in thermal conductivity fluctuating across a wide range ranges from 2.5% and 5.5%. A novel neural network and SA for forecasting the thermal confrontation of heat pipes with NF was proposed by Wang et al.35. They arrived at the conclusion that this work offers clear guidelines for creating a highly accurate and universal forecasting model for heat pipes using NF.
Numerous researchers have used single-objective heat transfer enhancement under the impacts of MHD and heat radiation in a BLF over a Riga sensor. However, these models usually fails inadequately in capturing interaction effects such as cross-diffusion effects, activation energy, and bi-convection effects in a BLF. The BLF of tri-HNF and heat transfer optimization using regression and sensitivity analysis, which is significant in engineering technologies and thermal management using tri-HNF, is examined for the first time. A tri-HNF containing three distinct different nanoparticles \(A{l}_{2}{O}_{3}\), \(CuO\) and \(Ti{O}_{2}\) are used in the study to create an improved cooling fluid with improved heat transfer capabilities. The Riga plate sensor is a popular technique used in marine and aeronautical engineering to manipulate the BLF in order to increase efficiency and decrease drag. The use of statistical and computational methods (SA and RSM) are used to develop a framework to forecast and optimize the system performance. The current study combines the RSM with a computational method RK-4 to formulate and prediction framework that assesses skin friction, heat transfer, and mass transfer performance under various physical parameters simultaneously. A state-of-the-art development in thermal fluid technology, this numerical solution for the Riga plate-induced momentum, thermal, and concentric BL, this study grasp a critical research gap. These solutions are informative for future Riga plate research, not only for verification purposes but also for detecting novel physical properties because of their simplicity and elegance in the wall shear, heat, and mass transfer expressions.
Problem statement and analysis.
The aim of this analysis is to model and investigate the electro-magneto-hydrodynamic mechanisms in an unsteady BLF of water-based tri-HNF across a Riga plate. A laminar time-dependent fluid flow is generated across a convectively heated Riga plate is described in a Cartesian coordinate system \(\left(x,y\right).\) An infinite Riga plate stretches in \(x-\) direction and \(y-\) axis is normal to it, with \(u\) and \(v\) denoting velocity components in \(x\) and \(y-\) directions, respectively. The Riga plate electromagnetic field induces an external parallel Lorentz force. As the distance normal to the plate increases, this force decreases significantly. As illustrated in Fig. 1, the Riga-plate is constructed from a permanent magnet and an alternating arrangement of electrodes positioned on a horizontal surface separated by a distance \(d\). Only a Lorentz force \(f=J\times B\), aligned to the arrangement accelerate the flow over the horizontal plate with \({\overline{\overline{u}}}_{w}=bx\), where \(b>0\) is constant. There are no additional driving forces (such as an expanding wall or pressure gradient) besides to this electromagnetic body force. The velocity \({\overline{\overline{u}}}_{e}\to cx\) describes the fluid that is far from the plate. The plate face comes into contact with water-based tri-HNF, at temperature \({\overline{T} }_{w}\) and concentration \({\overline{C} }_{w}\), at the surface with heat and mass transfer coefficient \({H}_{f}\) and \({H}_{m}\), respectively. Assuming zero typical flux through the plate dynamically regulates the concentration of nanoparticles there. The cross-diffusion effects, activation energy, and bi-convection effects are assumed in the model.
Fig. 1
Illustration of proposed model and coordinates system.
The governing BL equations can be expressed in the following ways by using the Oberbeck-Boussinesq approximation5,36,37,38:
$$\frac{{\partial \overline{u}}}{\partial x} + \frac{{\partial \overline{v}}}{\partial y} = 0$$
(1)
$$\left( {\frac{{\partial \overline{u}}}{\partial t} + \overline{u}\frac{{\partial \overline{u}}}{\partial x} + \overline{v}\frac{{\partial \overline{u}}}{\partial y}} \right) = \frac{{\partial \overline{\overline{u}}_{e} }}{\partial t} + \frac{{\partial \overline{\overline{u}}_{e} }}{\partial x} + \frac{{\mu_{tri – HNF} }}{{\rho_{tri – HNF} }}\frac{{\partial^{2} \overline{u}}}{{\partial y^{2} }} + \frac{{\pi J_{0} M_{0} }}{{8\rho_{tri – HNF} }}e^{{ – \left( {\frac{\pi }{d}} \right)y}}$$
(2)
$$\left( {\frac{{\partial \overline{T}}}{\partial t} + \overline{u}\frac{{\partial \overline{T}}}{\partial x} + \overline{v}\frac{{\partial \overline{T}}}{\partial y}} \right) = \frac{{k_{tri – HNF} }}{{\left( {\rho c_{p} } \right)_{tri – HNF} }}\frac{{\partial^{2} \overline{T}}}{{\partial y^{2} }} + \frac{{16\sigma^{*} }}{{3\left( {\rho c_{p} } \right)_{tri – HNF} k^{*} }}\frac{{\partial^{2} \overline{T}}}{{\partial y^{2} }} + \frac{{D_{m} K_{T} }}{{\left( {c_{p} } \right)_{tri – HNF} C_{S} }}\frac{{\partial^{2} \overline{C}}}{{\partial y^{2} }}$$
(3)
$$\left( {\frac{{\partial \overline{C}}}{\partial t} + \overline{u}\frac{{\partial \overline{C}}}{\partial x} + \overline{v}\frac{{\partial \overline{C}}}{\partial y}} \right) = D_{tri – HNF} \frac{{\partial^{2} \overline{C}}}{{\partial y^{2} }} + \frac{{D_{m} K_{T} }}{{T_{S} }}\frac{{\partial^{2} \overline{T}}}{{\partial y^{2} }} – K_{1} \left( {\overline{C} – \overline{C}_{\infty } } \right)^{m} \left( {\frac{{\overline{T}}}{{\overline{T}_{\infty } }}} \right)exp\left( {\frac{{ – E_{a} }}{{k_{tri – HNF} \overline{T}}}} \right)$$
(4)
Subject to boundary conditions39,40:
At
$$t = 0:\overline{T} = \overline{T}_{\infty } ,\overline{C} = \overline{C}_{\infty } ,\overline{u} = 0 = \overline{v}$$
(5)
$$t \ge 0:\overline{u} = \overline{\overline{{u_{w} }}} = bx,\overline{v} = 0,k_{tri – HNF} \frac{{\partial \overline{T}}}{\partial y} = H_{f} \left( {\overline{T} – \overline{T}_{w} } \right)D_{tri – HNF} \frac{{\partial \overline{C}}}{\partial y} = H_{m} \left( {\overline{C} – \overline{C}_{w} } \right)at \, y = 0$$
(6)
$$\overline{u} = \overline{\overline{u}}_{e} \to cx,\overline{T} \to \overline{T}_{\infty } ,\overline{C} \to \overline{C}_{\infty } ,at \, y = 0$$
(7)
where, \(\overline{u }\) and \(\overline{v }\) are the velocity constituents, \({\overline{T} }_{\infty }, {\overline{T} }_{w}\) and \({\overline{C} }_{\infty }, {\overline{C} }_{w}\) are the ambient, wall temperature and concertation respectively. Furthermore, \({J}_{0}\) is the Current density, \({M}_{0}\) is the magnetization of permanent magnets, \(d\) is the width between magnets and electrodes. Additionally, \({\overline{\overline{u}}}_{e}\), \(\rho\), \({c}_{p}\), \({\sigma }^{*}\), \({k}^{*}\), \({k}_{f}\), \({D}_{tri-HNF}\), \({K}_{T}\), \({T}_{S}\), \({K}_{1}\), \({E}_{a}\), \({H}_{f}\) and \({H}_{m}\) are the far field velocity, density, heat capacity, Stefan Boltzmann, mean absorption, thermal conductivity, diffusion, thermal diffusion, chemical reaction, activation energy, energy and mass transmission coefficient, respectively.
Introducing the proper similarity transformation as suggested by9,41,42,43:
$$u = \frac{{axf^{\prime}\left( \eta \right)}}{1 – \beta t},v = – \sqrt {\frac{{av_{f} }}{1 – \beta t}} f\left( \eta \right),\eta = y\sqrt {\frac{a}{{v_{f} 1 – \beta t}}} ,\Theta \left( \eta \right) = \frac{{\overline{T} – \overline{T}_{\infty } }}{{\overline{T}_{w} – \overline{T}_{\infty } }},\Phi \left( \eta \right) = \frac{{\overline{C} – \overline{C}_{\infty } }}{{\overline{C}_{w} – \overline{C}_{\infty } }}$$
(8)
The dimensionless equations in the view of Eq. (8), becomes
$$\frac{{\mu_{tri – HNF} }}{{\mu_{f} }}f^{\prime\prime\prime} + \frac{{\rho_{tri – HNF} }}{{\rho_{f} }}\left( {ff^{\prime\prime} – f^{^{\prime}2} } \right) – A\left( {\frac{\eta }{2}f^{^{\prime\prime}} + f^{\prime} – 1} \right) + \frac{{\rho_{tri – HNF} }}{{\rho_{f} }}M_{r} exp\left( { – \delta \eta } \right) + A^{2} = 0$$
(9)
$$\left( {\frac{{k_{tri – HNF} }}{{k_{f} }} + Nr} \right)\Theta^{\prime\prime} + \frac{{\left( {\rho c_{p} } \right)_{tri – HNF} }}{{\left( {\rho c_{p} } \right)_{f} }}\Pr \left( {\left( {f^{\prime}\Theta – f\Theta^{\prime}} \right) – \frac{\eta }{2}A\Theta^{\prime}} \right) + \frac{{D_{tri – HNF} }}{{D_{f} }}D_{f} \Psi^{\prime\prime} = 0$$
(10)
$$\frac{{D_{tri – HNF} }}{{D_{f} }}\Psi^{\prime\prime} + Scf\Psi^{\prime} – \frac{\eta }{2}\Psi^{\prime} – Sc\chi \left( {1 + \omega \Theta } \right)^{m} \exp \left( { – \frac{{E_{A} }}{{\left( {1 + \omega \Theta } \right)}}} \right) + \frac{{k_{tri – HNF} }}{{k_{f} }}Sr\Theta^{\prime\prime} = 0$$
(11)
$$f^{\prime}\left( 0 \right) = 1, f\left( 0 \right) = 1, \Theta^{\prime}\left( 0 \right) = – \frac{{k_{tri – HNF} }}{{k_{f} }}Bi_{1} \left( {1 – \Theta \left( 0 \right)} \right)\Psi^{^{\prime\prime}} (0) = – \frac{{D_{tri – HNF} }}{{D_{f} }}Bi_{2} (1 – \Psi (0))$$
(12)
$$f{\prime} (\infty ) = A,\Theta \left( \infty \right) = 0,\Psi (\infty ) = 0$$
(13)
where,
\({M}_{r}=\frac{\pi {J}_{0}{M}_{0}}{8{\rho }_{f}}\frac{1-\beta t}{{u}_{w}^{2}}\)
Modified Hartman number
\(A=\frac{c}{b}\)
Stagnation point
\(\delta =\frac{\pi }{\sqrt{a/{v}_{f}\left(1-\beta t\right)}}\)
Width of magnets and electrodes
\(Pr=\frac{{\left(\mu {c}_{p}\right)}_{f}}{{k}_{f}}\)
Prandtl number
\({D}_{f}=\frac{{D}_{f}{K}_{T}\left({C}_{w}-{C}_{\infty }\right)}{{\nu }_{f}{C}_{S}{c}_{p}\left({T}_{w}-{T}_{\infty }\right)}\)
Dufour number
\({S}_{r}=\frac{{D}_{f}{K}_{T}\left({T}_{w}-{T}_{\infty }\right)}{\nu {T}_{S}\left({C}_{w}-{C}_{\infty }\right)}\)
Soret number
\(Sc=\frac{{\nu }_{f}}{{D}_{m}}\)
Schmidt number
\(Nr=\frac{4{\sigma }^{*}{T}_{\infty }^{3}}{{k}^{*}{k}_{f}}\)
Radiative parameter
\({E}_{A}=\frac{{E}_{a}}{{k}_{f}{T}_{\infty }}\)
Activation energy
\(\chi =\frac{{K}_{1}}{b}\)
Chemical reaction parameter
\(\omega =\frac{{\overline{T} }_{w}-{\overline{T} }_{\infty }}{{\overline{T} }_{\infty }}\)
Temperature difference parameter
\({Bi}_{1}=\sqrt{\frac{a}{{v}_{f}1-\beta t}}\frac{{H}_{f}}{{k}_{f}}\)
Thermal Biot number
\({Bi}_{2}=\sqrt{\frac{a}{{v}_{f}\left(1-\beta t\right)}}\frac{{H}_{m}}{{D}_{f}}\)
Solutal Biot number
The quantity of engineering concern are the skin friction \({C}_{f}\), Nusselt \({Nu}_{x}\) and Sherwood number \({Sh}_{x}\) are defined as
$$C_{f} = \frac{{\tau_{w} }}{{\rho_{tri – HNF} u_{w}^{2} }},Nu_{x} = \frac{{xq_{w} + q_{r} }}{{k_{tri – HNF} \left( {\overline{T}_{w} – \overline{T}_{\infty } } \right)}},and \, Sh_{x} = \frac{{xq_{m} }}{{D_{tri – HNF} \left( {\overline{C}_{w} – \overline{C}_{\infty } } \right)}} \cdot$$
(14)
where, \({\tau }_{w}\), \({q}_{w}\) and \({q}_{m}\) are the wall stresses, heat and mass fluxes, respectively. Mathematically
$$\tau_{w} = \mu_{tri – HNF} \left. {\left( {\frac{\partial u}{{\partial y}}} \right)} \right|_{y = 0} ,q_{w} = – k_{tri – HNF} \left. {\left( {\frac{\partial T}{{\partial y}}} \right)} \right|_{y = 0} , q_{m} = – D_{tri – HNF} \left. {\left( {\frac{\partial C}{{\partial y}}} \right)} \right|_{y = 0}$$
(15)
In dimensionless form, the physical quantities becomes
$${\text{Re}}_{x}^{(1/2)} C_{f} = \frac{{\mu_{tri – HNF} }}{{\mu_{f} }}f^{^{\prime\prime}} (0),{\text{Re}}_{x}^{( – 1/2)} Nu_{x} = – (\frac{{k_{tri – HNF} }}{{k_{f} }} + Nr)\Theta{\prime} (0),Sh_{x} = – \frac{{D_{tri – HNF} }}{{D_{f} }}\Psi^{\prime}\left( 0 \right)$$
(16)
where, \({Re}_{x}^{1/2}=\frac{{u}_{x}x}{{\nu }_{f}}\) is the local Reynold number.
Thermo-physical and rheological features of tri-HNF
The mathematical expressions for density of mono, hybrid and tri-HNF are defined as44,45:
$$\frac{{\rho_{nf} }}{{\rho_{f} }} = \left( {1 – \phi_{1} } \right) + \phi_{1} \frac{{\rho_{bf} }}{{\rho_{f} }}$$
(17)
$$\frac{{\rho_{HNF} }}{{\rho_{nf} }} = \left[ {\left\{ {\left( {1 – \phi_{1} } \right) + \frac{{\phi_{1} \rho_{np1} }}{{\rho_{nf} }}} \right\}\left( {1 – \phi_{2} } \right) + \frac{{\phi_{2} \rho_{np2} }}{{\rho_{nf} }}} \right]$$
(18)
$$\rho_{tri – HNF} = \left( {1 – \phi_{3} } \right)\left[ {\left\{ {\left( {1 – \phi_{1} } \right)\rho_{bf} + \phi_{1} \rho_{np1} } \right\}\left( {1 – \phi_{2} } \right) + \phi_{2} \rho_{np2} } \right] + \phi_{3} \rho_{np3}$$
(19)
And
$$\frac{{\rho_{tri – HNF} }}{{\rho_{f} }} = \left( {1 – \phi_{3} } \right)\left[ {\left\{ {\left( {1 – \phi_{1} } \right) + \phi_{1} \frac{{\rho_{np1} }}{{\rho_{f} }}} \right\}\left( {1 – \phi_{2} } \right) + \phi_{2} \frac{{\rho_{np2} }}{{\rho_{f} }}} \right] + \phi_{3} \frac{{\rho_{np3} }}{{\rho_{f} }}$$
(20)
The mathematical expressions for viscosity of mono, hybrid and tri-HNF are defined as46,47:
$$\mu_{nf} = \frac{{\mu_{f} }}{{\left( {1 – \phi_{1} } \right)^{2.5} }}$$
(21)
$$\mu_{HNF} = \frac{{\mu_{f} }}{{\left( {1 – \phi_{1} } \right)^{2.5} \left( {1 – \phi_{2} } \right)^{2.5} }}$$
(22)
$$\mu_{tri – HNF} = \frac{{\mu_{f} }}{{\left( {1 – \phi_{1} } \right)^{2.5} \left( {1 – \phi_{2} } \right)^{2.5} \left( {1 – \phi_{3} } \right)^{2.5} }}$$
(23)
The mathematical expressions for heat capacity of mono, hybrid and tri-HNF are defined as46:
$$(C_{p} )_{nf} = \left( {1 – \phi_{1} } \right)(C_{p} )_{bf} + \phi_{1} \left( {C_{p} } \right)_{p1}$$
(24)
$$(C_{p} )_{HNF} = \left\{ {\left( {1 – \phi_{1} } \right)(C_{p} )_{bf} + \phi_{1} \left( {C_{p} } \right)_{np1} } \right\}\left( {1 – \phi_{2} } \right) + \phi_{1} \left( {C_{p} } \right)_{np2}$$
(25)
$$(C_{p} )_{tri – HNF} = \left[ {\left\{ {\left( {1 – \varphi_{1} } \right)(C_{p} )_{f} + \varphi_{1} \left( {C_{p} } \right)_{p1} } \right\}\left( {1 – \varphi_{2} } \right) + \varphi_{1} \left( {C_{p} } \right)_{p2} } \right]\left( {1 – \varphi_{3} } \right) + \varphi_{3} \left( {C_{p} } \right)_{p3}$$
(26)
And
$$\frac{{(C_{p} )_{tri – HNF} }}{{(C_{p} )_{f} }} = \left[ {\left\{ {\left( {1 – \phi_{1} } \right) + \phi_{1} \frac{{\left( {C_{p} } \right)_{np1} }}{{(C_{p} )_{f} }}} \right\}\left( {1 – \phi_{2} } \right) + \frac{{\phi_{1} \left( {C_{p} } \right)_{np2} }}{{(C_{p} )_{f} }}} \right]\left( {1 – \phi_{3} } \right) + \phi_{3} \frac{{\left( {C_{p} } \right)_{np3} }}{{(C_{p} )_{f} }}$$
(27)
The mathematical expressions for thermal conductivity of mono, hybrid and tri-HNF are defined as48
$$\frac{{k_{nf} }}{{k_{f} }} = \frac{{\left( {k_{np1} + 2k_{bf} } \right) – \left( {k_{bf} – k_{np1} } \right)2\phi_{1} }}{{\left( {k_{np1} + 2k_{bf} } \right) + \left( {k_{bf} – k_{np1} } \right)\phi_{1} }}$$
(28)
$$\frac{{k_{HNF} }}{{k_{nf} }} = \frac{{\left( {k_{np2} + 2k_{nf} } \right) – \left( {k_{nf} – k_{np2} } \right)2\phi_{2} }}{{\left( {k_{np2} + 2k_{nf} } \right) + \left( {k_{nf} – k_{np2} } \right)\phi_{2} }}$$
(29)
$$\frac{{k_{tri – HNF} }}{{k_{HNF} }} = \frac{{\left( {k_{np3} + 2k_{HNF} } \right) – \left( {k_{HNF} – k_{np3} } \right)2\phi_{3} }}{{\left( {k_{pn3} + 2k_{HNF} } \right) + \left( {k_{HNF} – k_{np3} } \right)\phi_{3} }}$$
(30)
And
$$\frac{{k_{tri – HNF} }}{{k_{bf} }} = \left[ {\frac{{\left( {k_{np3} + 2k_{HNF} } \right) – \left( {k_{HNF} – k_{np3} } \right)2\phi_{3} }}{{\left( {k_{np3} + 2k_{HNF} } \right) + \left( {k_{HNF} – k_{np3} } \right)\phi_{3} }}} \right]\frac{{k_{HNF} }}{{k_{bf} }}$$
(31)
The mathematical expressions for nanomaterial diffusion of mono, hybrid and tri-HNF are defined as48
$$\frac{{D_{nf} }}{{D_{f} }} = \left( {1 – \phi_{1} } \right)$$
(32)
$$\frac{{D_{HNF} }}{{D_{nf} }} = \left( {1 – \left( {\phi_{1} + \phi_{2} } \right)} \right)$$
(33)
$$\frac{{D_{tri – HNF} }}{{D_{HNF} }} = \left( {1 – \left( {\phi_{1} + \phi_{2} } \right)} \right)$$
(34)
In above expressions, the symbols \({\phi }_{1}\), \({\phi }_{2}\), and \({\phi }_{3}\) showing the nanomaterials load. In Table 1 the thermo-chemo-physical features of water and three nanomaterials are given.
Table 1 Thermo-chemo-physical characteristic of water and three nanomaterials49,50,51.
Numerical framework.
The procedure for resolving this BLF problem is explained in this section. The fundamental concept behind RK-4 fourth-order technique is not satisfied by the dimensional partial differential form of the governing Eq. (2)-Eq. (4). First, we create non-dimensional ODEs Eq. (9)-(11) with boundary conditions (BCs) Eq. (12) and (13) by applying the proper transformation Eq. (8). Seventh order equations with seven BCs are produced by non-dimensional equations. Initial conditions are obtained from BCs using the shooting approximation. By calculating the difference between two computations, which guides the step size modification, the correctness of this method is assessed. Appropriate initial estimations are crucial to the shooting method’s successful use. These starting values have a significant impact on the overall result. Iterations continue with a set step size of 0.001 until the answer converges within a predetermined error margin.
Introducing new variables
$$Z_{1} = f,Z_{2} = f^{\prime},Z_{2} = f^{\prime\prime},Z^{\prime}_{3} = f^{\prime\prime\prime},Z_{4} = \Theta ,Z_{5} = \Theta^{\prime},Z^{\prime}_{5} = \Theta^{\prime\prime},Z_{6} = \Psi ,Z_{7} = \Psi^{\prime},Z^{\prime}_{7} = \Psi^{\prime\prime}$$
(35)
$${\text{\rm Z}}_{3}{\prime} = \frac{1}{{\frac{{\mu_{tri – HNF} }}{{\mu_{f} }}}}\left( { – \frac{{\rho_{tri – HNF} }}{{\rho_{f} }}\left( {{\text{\rm Z}}_{1} {\text{\rm Z}}_{3} – {\text{\rm Z}}_{2}^{2} } \right) + A\left( {\frac{\eta }{2}{\text{\rm Z}}_{3} + {\text{\rm Z}}_{2} – 1} \right) – \frac{{\rho_{tri – HNF} }}{{\rho_{f} }}M_{r} exp\left( { – \delta \eta } \right) – A^{2} } \right)$$
(36)
$${\text{\rm Z}}_{5}{\prime} = – \frac{1}{{\left( {\frac{{k_{tri – HNF} }}{{k_{f} }} + Nr} \right)}}\left( {\frac{{\left( {\rho c_{p} } \right)_{tri – HNF} }}{{\left( {\rho c_{p} } \right)_{f} }}Pr\left( {\left( {{\text{\rm Z}}_{2} {\text{\rm Z}}_{4} – {\text{\rm Z}}_{1} {\text{\rm Z}}_{5} } \right) – \frac{\eta }{2}A{\text{\rm Z}}_{5} } \right) + \frac{{D_{tri – HNF} }}{{D_{f} }}D_{f} {\text{\rm Z}}_{7}{\prime} } \right)$$
(37)
$${\text{\rm Z}}_{7}{\prime} = – \frac{1}{{\frac{{D_{tri – HNF} }}{{D_{f} }}}}\left( {Sc{\text{\rm Z}}_{1} {\text{\rm Z}}_{7} – \frac{\eta }{2}{\text{\rm Z}}_{7} – Sc\chi \left( {1 + \omega {\text{\rm Z}}_{4} } \right)^{m} exp\left( { – \frac{{E_{A} }}{{\left( {1 + \omega {\text{\rm Z}}_{4} } \right)}}} \right) + \frac{{k_{tri – HNF} }}{{k_{f} }}Sr{\text{\rm Z}}_{5}{\prime} } \right)$$
(38)
$${\rm Z}_{2} \left( 0 \right) = 1, {\rm Z}_{1} \left( 0 \right) = \lambda_{1} , {\rm Z}_{5} \left( 0 \right) = – \lambda_{2} \left( {\frac{{k_{tri – HNF} }}{{k_{f} }}Bi_{1} \left( {1 – {\rm Z}_{4} \left( 0 \right)} \right)} \right),{\rm Z}_{7} \left( 0 \right) = – \lambda_{3} \left( {\frac{{D_{tri – HNF} }}{{D_{f} }}Bi_{2} \left( {1 – {\rm Z}_{6} \left( 0 \right)} \right)} \right)$$
(39)
$${\rm Z}_{2} \left( \infty \right) = A,{\rm Z}_{4} \left( \infty \right) = 0,{\rm Z}_{6} \left( \infty \right) = 0.$$
(40)
where \({\lambda }_{1}\), \({\lambda }_{3}\), and \({\lambda }_{3}\) are unknown which are computed using the Newton procedure from additional BCs. A built-in mathematical program in MATLAB 2019b is used to carry out this complete process. The current model is validated with previous models in a limiting scenarios, setting \({M}_{r}=0\) is shown in Table 2.
Table 2 Model validation with previous models.