Computational and regression analysis of EMHD Casson tri-hybrid nanofluid flow over a radiant Riga plate: Viscous dissipation,suction,and shape factor considerations

Abstract

Riga plates serve as electromagnetic actuators originally designed to minimize hydrodynamic drag and pressure. Their utility has since broadened to drag reduction in submarines, micro-coolers, biomedical flow control, and thermal reactors. This study computationally simulates and statistically analyzes the energy transfer efficiency of an electro-magnetohydrodynamic (EMHD) Casson trihybrid nanofluid flowing past a vertical radiant Riga plate in scenarios involving natural and forced convection, considering the effects of viscous dissipation, suction, and the nanomaterials’ shape factor. The governing model is solved using a hybrid spectral technique, with numerical results validated against benchmark data. According to the results of the current work, suction augmentation diminishes the tri-hybrid Casson nanofluid’s velocity and drag forces. Elevating the values of mixed convection or Casson factors reduces the Casson tri-hybrid nanofluid’s temperature and enhances its energy transfer. The incorporation of non-spherical nanoparticles reduces the Nusselt number by 0.23%–5.3% and increases skin friction by 1.6%–14% in comparison to using spherical nanoparticles. Multiple regression indicates that thermal radiation is the most positive contributor in boosting energy transfer rates. These insights may advance predictive capabilities for thermal management in energy systems and guide next-generation EMHD reactor design.

Introduction

Electric field, or electrohydrodynamic (EHD), is one of the external factors that regulates fluid movement and produces the required boundary layer disturbances, resulting in substantial improvements in the ability to control energy transport rate and flow characteristics.^1–4 Similar to the electric field, the magnetic field, commonly referred to as magnetohydrodynamics (MHD), is another substantial external factor in regulating the required level of the rate of energy transport. The use of electric and magnetic fields as a means of controlling heat transfer has found wide-ranging applications in diverse fields such as heat exchanger performance improvement, electronics cooling, nuclear reactors, and MHD pump applications.^5–7 The Riga plate provides a practical tool for investigating the interplay between EHD and MHD, or what is known as the electromagnetohydrodynamic (EMHD) force, its influence on thermal efficiency, and the flow characteristics of electrically conducting fluids. Electrodes and magnets are utilized to create Riga plates, which function as electromagnetic actuators affixed to a flat object. Initially developed to reduce pressure and drag force in hydrodynamics, they have since expanded their applications to include biomedical areas and various technological advancements. The multi-application phenomena present in Riga plate flows have gained a significant amount of interest in the mathematical simulation area. A numerical simulation conducted by Vaidya et al. ⁸ on combined convection over a stretched Riga surface revealed that increased levels of the modified Hartmann number can enhance fluid velocity while simultaneously reducing temperature. In their numerical analysis of convective liquid flow on a stretching Riga surface, Shamshuddin et al. ⁹ revealed that raising the radiative factor significantly boosts the Nusselt number, while a larger combined convection factor or wider Riga surface electrodes enhance boundary layer thickness. Li et al. ¹⁰ numerically simulated the influence of alumina nanosolids on the heat transfer efficiency of a liquid as it flowed across a vertical Riga surface. They revealed that as the volume fraction factors grow, the original liquid’s temperature and its ability to transfer heat improve. According to Bilal et al.’s numerical research ¹¹ on combined convective nanoliquid as it flows over a vertical Riga surface, a combined convection factor with a positive value promotes flow, while one with a negative value restricts flow. Additionally, a higher Eckert number results in a stronger thermal field.

The investigation of fluid viscosity is crucial for enhancing thermal performance in heat exchanger systems. At elevated flow velocities or with highly viscous fluids, viscous dissipation significantly influences the heat transfer dynamics and thermal distribution within the fluid. This phenomenon has diverse applications, notably in scenarios where substantial temperature increases occur. For instance, in polymer processing techniques like injection molding and extrusion at high flow rates, as well as within the thin boundary layers surrounding high-speed aircraft, viscous dissipation raises surface temperatures. Additionally, its effect on rising temperatures in microelectronics and micro-electro-mechanical systems is observed. Numerical investigations^12–16 undertaken on the thermal efficiency of a nanofluid moving over a Riga plate showed that an elevation in the dissipation factor results in escalated temperatures of the nanoliquid and a growth in the energy transmission. The process of suction has garnered significant interest due to its effectiveness in extracting reactants during chemical reactions. Numerous numerical analyses have explored the influence of this mechanism on fluid flow and thermal performance. Khatun et al. ¹⁷ demonstrated that an increased suction coefficient not only reduces the flow rate but also enhances shear stress. A decline in the temperature and enhancement in surface energy transport are observed with the higher of the suction factor, according to Zaydan et al.’s study. ¹⁸ The numerical analysis conducted by Shah et al. ¹⁹ revealed that enhancements in the suction factor correspond to an increase in drag forces and a reduction in Nusselt number values.

For decades, nanomaterial fluid enhancement technology has been recognized as one of the most effective techniques for enhancing energy transfer efficiency in several fields of application.^20,21 Metallic or ceramic nanomaterials were widely utilized in the synthesis of the mono-nanoliquid. The application of metallic nanomaterials greatly enhances the original liquid’s thermal conductivity but tends to reduce its stability, whereas the use of ceramic nanomaterials enhances stability at the expense of reduced thermal conductivity.^22–24 To tackle this issue, the fabrication of hybrid nanofluids that combined the original fluid with two nanomaterials was initiated.^25–31 In recent years, a novel class of highly effective nanofluids known as ‘trihybrid’ nanofluids has been discovered and thoroughly investigated. The tri-hybrid nanofluid consists of a host liquid and a combination of three nanoparticles, exhibiting distinctive flow characteristics and near-perfect heat transfer. ³² Computational investigations have addressed the thermal performance of the trihybrid nanofluid, with studies^33–38 revealing that tri-hybrid nanoliquids demonstrate enhanced energy transfer rates, velocity, and temperature compared to their hybrid and single nanoliquid counterparts. On the other hand, nanoparticles’ surface area and geometric configuration, which characterize the shape factor, influence both the thermal boundary layer thickness and the convective energy transfer coefficient.^39,40 Numerically, Khashi’ie et al. ⁴¹ examined the impact of spherical, brick, and blade shapes on the thermal properties of a hybrid nanoliquid moving past an electromagnetohydrodynamic Riga plate. Research findings indicated that blade nanosolids exhibit the most rapid rate of energy transfer, whereas sphere nanosolids demonstrate the smallest rate. According to Ahmed et al.’s study ⁴² on the mass and energy transfer of nanoliquid across a Riga plate, taking into account different nanomaterial shapes, spherical nanomaterials generate the highest drag forces, whereas platelet nanomaterials generate the lowest.

The thermal behavior of nanofluids as its flow over Riga surface using Casson mathematical model has been investigated in prior research, including investigating the impacts of the chemical reaction on nanoliquids over a porous Riga plate, ⁴³ investigating the energy transport and entropy generation optimization of EMHD radiative nanoliquid on a melting Riga plate, ⁴⁴ analyzing the characteristics of energy transmission and entropy generation on MHD second-grade nanoliquid while taking the radiation factor into account, ⁴⁵ computationally and statistically analyzing the flow of hybrid nanoliquid across a perforated Riga surface, considering the effects of activation energy and radiation, ⁴⁶ studying the thermal efficiency of hybrid nanoliquid over a Riga plate, considering suction/injection and thermal slip factor, ⁴⁷ and examining the thermal behavior of electro-magneto nanoliquid with variable viscosity on a Riga surface. ⁴⁸ This study presents a comprehensive computational and statistical exploration of energy transfer in an electromagnetohydrodynamic (EMHD) Casson tri-hybrid nanoliquid flow over a radiative vertical Riga surface. Uniquely integrating the combined effects of nanoparticle morphology (shape factor), suction, and viscous dissipation under mixed convection. To the best of our knowledge, this study offers the first cohesive framework for tackling these related issues, allowing for precise energy transfer prediction in EMHD systems.

Thus, the objectives of this study are to:

1. Create a thorough computational model that integrates shape factor, suction, and viscous dissipation for EMHD Casson tri-hybrid nanofluid flow on a vertical radiative Riga surface.

2. Providing a numerical solution to the mathematical model created using the technique of hybrid spectral.

3. Quantify the effects of each of these parameters separately and in combination on thermal performance.

4. Create predictive regression models for skin friction and heat transfer.

Mathematical representation

Consider an EMHD combined convection flow of a dissipative Casson tri-hybrid nanoliquid past a radiative vertical Riga surface, as illustrated in Figure 1. $\stackrel{\vee }{v}\mathrm{and}\stackrel{\vee }{w}$ are the velocity components along $\stackrel{\vee }{x}\mathrm{and}\stackrel{\vee }{y}$ respectively. The $\stackrel{\vee }{x}$ -axis runs parallel to the plate’s surface, the $\stackrel{\vee }{y}$ -axis extends perpendicular to it, and the plate is situated at $\stackrel{\vee }{y}$  = 0. It is assumed that the free stream velocity is ${\stackrel{\vee }{V}}_e\left(x\right)=\frac{a\stackrel{\vee }{x}}{L}$ , $a>0.$ The plate’s temperature $T_w(x)=T_{\infty }+T_0\left(\frac{\stackrel{\vee }{x}}{L}\right)$ , where the temperature of the surrounding fluid is $T_{\infty },L$ denotes the Riga surface’s characteristic length and $T_0$ is the characteristic temperature. If $T_w$  >  $T_{\infty }$ is assumed, a heated surface (assisting flow) can be observed, while a cooled surface (resisting flow) is indicated if $T_w$  <  $T_{\infty }$ .

Figure 1.

Physical configuration.

In light of the stated assumption, the simulation model is^49–51:

$$\frac{\partial \stackrel{\vee }{v}}{\partial \stackrel{\vee }{x}}+\frac{\partial \stackrel{\vee }{w}}{\partial \stackrel{\vee }{y}}=0,$$

(1)

$$\begin{array}{c}\stackrel{\vee }{v}\frac{\partial \stackrel{\vee }{v}}{\partial \stackrel{\vee }{x}}+\stackrel{\vee }{w}\frac{\partial \stackrel{\vee }{v}}{\partial \stackrel{\vee }{y}}={\stackrel{\vee }{v}}_e\frac{d{\stackrel{\vee }{v}}_e}{d\stackrel{\vee }{x}}+\frac{{\mu }_{\mathrm{thnl}}}{{\rho }_{\mathrm{thnl}}}(1+\frac{1}{\beta })\frac{{\partial }^2\stackrel{\vee }{v}}{\partial {\stackrel{\vee }{y}}^2}\hfill \\ +(\rho \beta {)}_{\mathrm{thnl}}g(T-T_{\infty })+\frac{\pi M_0{J}_0{e}^{\left(\frac{-\pi y}{p}\right)}}{8{\rho }_{\mathrm{thnl}}},\hfill \end{array}$$

(2)

$$\begin{array}{c}\stackrel{\vee }{v}\frac{\partial T}{\partial \stackrel{\vee }{x}}+\stackrel{\vee }{w}\frac{\partial T}{\partial \stackrel{\vee }{y}}={\alpha }_{\mathrm{thnl}}\frac{{\partial }^{2}T}{\partial {\stackrel{\vee }{y}}^2}-\frac{1}{{\left(\rho \mathrm{cp}\right)}_{\mathrm{thnl}}}\frac{\partial q_r}{\partial \stackrel{\vee }{y}}\hfill \\ +\frac{{\mu }_{\mathrm{thhl}}}{{\left(\rho \mathrm{cp}\right)}_{\mathrm{thnl}}}(1+\frac{1}{\beta }){\left(\frac{\partial \stackrel{\vee }{v}}{\partial \stackrel{\vee }{y}}\right)}^2.\hfill \end{array}$$

(3)

where $\beta$ is the Casson factor, $\rho$ is the density, $\mu$ is the dynamic viscosity, $(\rho \mathrm{cp})$ is heat capacity, $(\rho \beta )$ is the thermal expansion, $p$ signifies the magnets and electrodes widths’, $M_0$ reflects the magnetization intensity of the permanent magnets, and $J_0$ indicates the current density that is applied to the electrodes. The subscript $\mathrm{thnl}$ indicates the trihybrid nanoliquid. ${\alpha }_{\mathrm{thnl}}=\frac{k_{\mathrm{thhl}}}{{\left(\rho \mathrm{cp}\right)}_{\mathrm{thnl}}}$ is the thermal diffusivity of tri-hybrid nanoliquid, $q_r=-\frac{4\sigma }{3k^{*}}{\left(\frac{\partial T^4}{\partial \stackrel{\vee }{y}}\right)}_{\stackrel{\vee }{y}=0}$ is the radiative heat flow, and $T^4\cong 4T_{\infty }^{3}T-3T_{\infty }^4.\sigma ,\mathrm{and}{k}^{*}$ are the Stefan–Boltzmann and mean absorption coefficients, respectively.

Incorporating the effects of suction, the boundary conditions can be stated as:

$$\begin{array}{c}\stackrel{\vee }{v}=0,\stackrel{\vee }{w}={\stackrel{\vee }{w}}_m,T=T_w\mathrm{at}\stackrel{\vee }{y}=0,\stackrel{\vee }{v}\to {\stackrel{\vee }{v}}_e,\hfill \\ T\to T_{\infty }\mathrm{as}\stackrel{\vee }{y}\to \infty .\hfill \end{array}$$

(4)

where ${\stackrel{\vee }{w}}_m=-\sqrt{\frac{{\stackrel{\vee }{v}}_e{v}_l}{x}}S$ ,and $S$ is suction factor.

Through the succeeding mathematical formulas, the shape-dependent thermophysical characteristics can be articulated^40,52–55:

Thermal conductivity and viscosity of single-compound nanoliquid are:

$$\begin{array}{c}\frac{k_{\mathrm{nl}}}{k_l}=\frac{k_s+(m-1)k_l-(m-1){\chi }_s(k_l-k_s)}{k_s+(m-1)k_l+{\chi }_s(k_l-k_s)},\hfill \\ \frac{{\mu }_{\mathrm{nl}}}{{\mu }_l}=1+{\chi }_s(D_1+D_2{\chi }_s)\hfill \end{array}$$

(5)

here, $k$ is the thermal conductivity, $m$ stands for the empirical shape parameter. Coefficients of viscosity are denoted by $D_1\mathrm{and}{D}_2$ . The subscripts $s,l\mathrm{and}\mathrm{nl}$ indicate the nanosolid, original liquid, and mono-nanoliquid. The viscosity coefficients and shape parameters of the nanosolids used in this investigation are detailed in Table 1.

Table 1.

Viscosity’s coefficients and shape parameter.^52,53

Nanomaterials’ shape	Viscosity’s coefficients		Shape parameter ( $m)$
	$D_1$	$D_2$
Blades	14.6	123.3	8.6
Platelets	37.1	612.6	5.7
Cylinders	13.5	904.4	4.9
Bricks	1.9	471.4	3.7
Spheres	2.5	6.2	3

Thermophysical characteristics of trihybrid nanoliquid are ⁵⁵ :

$$\begin{array}{c}\frac{k_{\mathrm{thnl}}}{k_l}=\frac{k_{\mathrm{nl}1}{\chi }_1+k_{\mathrm{nl}2}{\chi }_2+k_{\mathrm{nl}3}{\chi }_3}{\chi k_l},\hfill \\ \frac{{\mu }_{\mathrm{thnl}}}{{\mu }_l}=\frac{{\mu }_{\mathrm{nl}1}{\chi }_1+{\mu }_{\mathrm{nl}2}{\chi }_2+{\mu }_{\mathrm{nl}3}{\chi }_3}{\chi {\mu }_l},\hfill \\ {\rho }_{\mathrm{thnl}}={\chi }_1{\rho }_1+{\chi }_2{\rho }_2+{\chi }_3{\rho }_3+[(1-\chi ){\rho }_l],\hfill \\ (\rho c_p{)}_{\mathrm{thnl}}={\chi }_1(\rho c_p{)}_1+{\chi }_2(\rho c_p{)}_2+{\chi }_3(\rho c_p{)}_3\hfill \\ +[(1-\chi )(\rho c_p{)}_l],(\rho \beta {)}_{\mathrm{thnl}}={\chi }_1(\rho \beta {)}_1+{\chi }_2(\rho \beta {)}_2\hfill \\ +{\chi }_3(\rho \beta {)}_3+[(1-\chi )(\rho \beta {)}_l].\hfill \end{array}$$

(6)

In this context, $\chi ={\chi }_1+{\chi }_2+{\chi }_3.$ The subscripts 1 and 2 refer to Al₂O₃ and SiO₂, respectively, while subscript 3 represents TiO₂. Introducing the similarity transformations ⁵⁰ :

$$\begin{array}{c}\eta =\stackrel{\vee }{y}\sqrt{\frac{a}{L{v}_l}},\stackrel{\vee }{v}=\frac{a\stackrel{\vee }{x}}{L}{f}^{\text{'}}\left(\eta \right),\hfill \\ \stackrel{\vee }{w}=\sqrt{\frac{a{v}_f}{L}}f\left(\eta \right),T-T_{\infty }=(T_w-T_{\infty })\theta .\hfill \end{array}$$

(7)

Using similarity transformations (7) yields the following reduced mathematical model:

$$\begin{array}{c}\frac{{\rho }_l}{{\rho }_{\mathrm{thnl}}}\frac{{\mu }_{\mathrm{thnl}}}{{\mu }_l}(1+\frac{1}{\beta })f″\text{'}(\eta )-{f^{\text{'}}}^2\left(\eta \right)+f\left(\eta \right)f^{″}\left(\eta \right)\hfill \\ +\frac{{(\rho \beta )}_{\mathrm{thnl}}}{{(\rho \beta )}_l}\lambda \theta \left(\eta \right)+\frac{{\rho }_f}{{\rho }_{\mathrm{thnl}}}H{e}^{-d\eta }+1=0,\hfill \end{array}$$

(8)

$$\begin{array}{c}\frac{1}{\mathrm{Pr}}\frac{{(\rho c_p)}_l}{{\left(\rho \mathrm{cp}\right)}_{\mathrm{thnl}}}(\frac{k_{\mathrm{thnf}}}{k_f}+\frac{4}{3}R){\theta }^{″}\left(\eta \right)-f^{\text{'}}\theta \left(\eta \right)+f{\theta }^{\text{'}}\left(\eta \right)\hfill \\ +\frac{{\mu }_{\mathrm{thhl}}}{{\mu }_l}\frac{{\left(\rho \mathrm{cp}\right)}_l}{{\left(\rho \mathrm{cp}\right)}_{\mathrm{thnl}}}\mathrm{Ec}(1+\frac{1}{\beta })(f{\text{'}}^{\text{'}}\left(\eta \right){)}^2=0.\hfill \end{array}$$

(9)

Subject to:

$$\begin{array}{c}{f}^{\prime }(\eta )=0,f(\eta )=S,\theta (\eta )=1\mathrm{at}\eta =0,\hfill \\ f^{\prime }\left(\eta \right)=1,\theta \left(\eta \right)=0\mathrm{as}\gamma \to \infty .\hfill \end{array}$$

(10)

here $\lambda =\frac{\mathrm{Gr}}{R{e}^2}$ is the combined convection factor, $\mathrm{Gr}=\frac{g{(\rho \beta )}_l(T_w-T_{\infty })x^3}{{v_l}^2}$ is the Grashof number,

$\mathrm{Re}=\frac{a{x}^2}{v_{l}L}$ is the Reynolds number, $H=\frac{\pi M_0{J}_{0}x}{8{V_e}^2{\rho }_l}$ is the EMHD factor, $d=\frac{\pi }{p}{\left(\frac{v_{l}L}{a}\right)}^{1/2}$ is a dimensionless factor, $\mathrm{Pr}=\frac{v_f}{{\alpha }_f}$ is the Prandtl number, $R=\frac{4\sigma T_{\infty }^3}{k^{*}{k}_l}$ is the radiation factor, and $\mathrm{Ec}=\frac{V_e^2{\rho }_f}{(T_w-T_{\infty }){\left(\rho \mathrm{cp}\right)}_f}$ is the Eckert number.

The following formulas represent the Nusselt number $\mathrm{Nu}$ and skin friction $C_f$ , which are crucial for forecasting energy transport and drag force in fluid flow applications:

$$\mathrm{Nu}=\left(\frac{x{q}_w}{k_l(T_w-T_{\infty })}\right),C_f=\frac{{\tau }_w}{V_e^2{\rho }_l}.$$

(11)

Where ${\tau }_w={\mu }_{\mathrm{nthf}}(1+\frac{1}{\beta }){\left(\frac{\partial \stackrel{\vee }{v}}{\partial \stackrel{\vee }{y}}\right)}_{\stackrel{\vee }{y}=0},q_w=-(k_{\mathrm{thnf}}{\left(\frac{\partial T}{\partial \stackrel{\vee }{y}}\right)}_{\stackrel{\vee }{y}=0}-{(q_r)}_{\stackrel{\vee }{y}=0})$ .

Applying similarity transformations (7) yields the following reduced form of $\mathrm{Nu}$ and $C_f$ :

$$\begin{array}{c}R{e}^{-1/2}\mathrm{Nu}=-(\frac{k_{\mathrm{thnl}}}{k_l}+\frac{4}{3}R){\theta }^{\text{'}}\left(0\right),\hfill \\ R{e}^{1/2}{C}_f=\frac{{\mu }_{\mathrm{nthl}}}{{\mu }_l}(1+\frac{1}{\beta })f^{″}\left(0\right).\hfill \end{array}$$

(12)

here, $\mathrm{Re}=\frac{\mathrm{xVe}}{v_l}$ represents the Reynold number.

Computational approaches and results’ validation

The hybrid spectral numerical technique is an advanced computational method that integrates spectral methods with linearization approaches to effectively solve complex nonlinear differential equations.^55–57 By employing this technique, the governing model is solved numerically, with computations generated through MATLAB, achieving an accuracy level of up to 10⁻⁶. The results obtained demonstrate significant compatibility with previous findings, as highlighted by the numerical results presented in Tables 2 and 3.

Table 2.

Comparison of $R{e}^{1/2}{C}_f$ values in the absence of the effects of $\chi ,\beta ,H,\mathrm{Ec},R,\mathrm{and}S.$

$\mathrm{Pr}$	$R{e}^{1/2}{C}_f\mathrm{at}\lambda =1$			$R{e}^{1/2}{C}_f\mathrm{at}\lambda =-1$
	Present results	Roşca et al. 49	Ramachandran et al. 58	Present results	Roşca et al. 49	Ramachandran et al. 58
$0.7$	1.7063	1.7063	1.7063	$0.6917$	$0.6917$	$0.6917$
$7$	1.5179	1.5179	1.5179	$0.9235$	$0.9235$	$0.9235$
$20$	1.4485	1.4485	1.4485	1.0031	$1.0031$	$1.0031$
$40$	1.4101	1.4101	1.4101	$1.0459$	$1.0459$	$1.0459$
$60$	1.3903	1.3903	1.3903	$1.0677$	$1.0677$	$1.0677$
$80$	1.3774	1.3774	1.3774	$1.0817$	$1.0817$	$1.0817$
$100$	1.3680	1.3680	1.3680	$1.0918$	$1.0918$	$1.0918$

Table 3.

Comparison of $R{e}^{-1/2}\mathrm{Nu}$ values in the absence of the effects of $\chi ,\beta ,H,\mathrm{Ec},R,\mathrm{and}S.$

$\mathrm{Pr}$	$R{e}^{-1/2}\mathrm{Nu}\mathrm{at}\lambda =1$			$R{e}^{-1/2}\mathrm{Nu}\mathrm{at}\lambda =-1$
	Present results	Roşca et al. 49	Ramachandran et al. 58	Present results	Roşca et al. 49	Ramachandran et al. 58
$0.7$	0.7641	0.7641	0.7641	$0.6332$	$0.6332$	$0.6332$
$7$	1.7224	1.7224	1.7224	$1.5460$	$1.5460$	$1.5460$
$20$	2.4576	2.4576	2.4576	$2.2683$	$2.2683$	$2.2683$
$40$	3.1011	3.1011	3.1011	2.9054	$2.9054$	$2.9054$
$60$	3.5514	3.5514	3.5514	$3.3527$	$3.3527$	$3.3527$
$80$	3.9095	3.9095	3.9095	$3.7089$	$3.7089$	$3.7089$
$100$	4.2116	4.2116	4.2116	$4.0097$	4.0097	$4.0097$

Graphical results and discussion

This section presents analyses of the impact of key factors on temperature and velocity distributions, specifically the nanosolid volume fraction and its shape, the Eckert number, the suction factor, the combined convection factor, the Casson factor, and the modified Hartmann number. Additionally, it includes an assessment of heat transfer rates and drag forces by analyzing the numerical outcomes for both the Nusselt number and the skin friction coefficient. The original fluid is a mix of 50% ethylene glycol and 50% water (H₂O-EG), and the nanosolids used to create the trihybrid nanosolid are aluminum oxide (Al₂O₃), silicon dioxide (SiO₂), and titanium dioxide (TiO₂). Their thermo-physical features shown in Table 4 were employed to obtain the graphical and tabular results.

Table 4.

Thermophysical characteristics of the original liquid and nanoparticles.^59–61

Property	H2O-EG 50%-50%	Al2O3.0.45	TiO20.250	SiO20.30
$\left(\rho \mathrm{cp}\right)$ (J/kg K)	3288	773	686	765
$(\rho \beta )$  × 10−5(K−1)	0.00341	0.85	0.90	0.056
ρ (kg/m3)	1056	3970	5200	6310
k (W/m K)	0.425	40	8.95	32.9
Pr	29.86	…	…	…

The effect of the shapes of blades, platelets, cylinders, bricks, and spheres on velocity, temperature, skin friction, and Nusselt number is investigated through Figures 2 to 5. It was found that spherical nanoparticles achieve the maximum velocity for the initial fluid, outperforming the other shapes, while platelet nanoparticles provide the lowest velocity. This occurs because the smooth surfaces of spherical particles minimize drag forces, enabling streamlined flow, whereas platelet nanoparticles’ irregular edges increase drag resistance. Platelet nanoparticles yield the highest temperatures, while spherical nanoparticles give the lowest temperatures due to their efficient thermal diffusion versus platelet particles’ disruptive heat transfer pathways. Compared to spherical nanoparticles, cylindrical, brick, platelet, or blade nanoparticles reduce the Nusselt number by 0.23%–5.3% by disrupting thermal boundary layer development. Non-spherical nanoparticles increase skin friction by 1.6%–14% through enhanced viscous dissipation at sharp edges. Figures 6 and 7 show the impact of the mixed convection parameter λ on both velocity and temperature of the tri-hybrid nanofluid in assisting and opposing flow scenarios. Increased buoyancy forces from higher λ values intensify fluid motion, enhancing tri-hybrid nanofluid velocity. Concurrently, heightened forced convection improves advective heat removal, reducing temperature. Figures 8 and 9 illustrate the substantial impact of the Casson parameter β on the velocity and temperature of the tri-hybrid nanoliquid. Higher β elevates apparent viscosity, shrinking the momentum boundary layer, restricting fluid motion (reducing velocity), and impeding energy transport (diminishing temperature). Figures 10 and 11 manifest the potency of the radiation factor in velocity and temperature distributions. An increase in the radiation factor leads to increased trihybrid nanofluid velocity and elevated temperature. In a real sense, a higher radiation factor typically leads to the release of more external energy into the trihybrid nanoliquid, which can elevate the overall temperature and increase velocity profiles within the trihybrid nanoliquid. Figures 12 and 13 illustrate the velocity and temperature distributions influenced by the Eckert number. According to the figures, there is a direct relationship between the Eckert number on the one hand and both the velocity and temperature profiles on the other. The physical mechanisms that transform kinetic energy into thermal energy provide an explanation for this relationship. Work done against the strains of the viscous fluid converts kinetic energy into internal energy as the Eckert number raises. Consequently, the trihybrid nanofluid’s temperature and velocity are both improved when the Eckert number escalates. Figures 14 and 15 introduce the influence of the suction parameter on velocity and temperature profiles. The ascending values of the suction factor contribute to the decrease in velocity and temperature of the trihybrid nanoliquid. A growing suction factor indicates that more trihybrid nanoliquid is sucked away from the surface, thereby tending to slow down the trihybrid nanoliquid at the surface. In addition, the temperature of the trihybrid nanoliquid at the boundary layer decreases as the trihybrid nanoliquid is sucked away since cooler fluid from the bulk of the flow replaces the warmer fluid close to the surface. Figures 16 and 17 expose the behavior of velocity and temperature under EMHD parameter H influence. Elevated H strengthens Lorentz forces, accelerating fluid via electromagnetic body forces (increasing velocity) while enhancing convective cooling (marginally reducing temperature). Figures 18 and 19 show the influence of the volume fraction parameter on velocity and temperature profiles. The incorporation of nanoparticles at low volume fractions can enhance heat transfer and improve the thermal conductivity of the original liquid without significantly affecting viscosity, optimizing the thermal transportation effectiveness of the host liquid. As a result, the velocity will increase while the temperature will decrease.

Figure 2.

Impact of $m$ on velocity.

Figure 3.

Impact of $m$ on temperature.

Figure 4.

Impact of $m$ on $R{e}^{-1/2}\mathrm{Nu}$ .

Figure 5.

Impact of $m$ on $R{e}^{1/2}{C}_f$ .

Figure 6.

Impact of $\lambda$ on velocity

Figure 7.

Impact of $\lambda$ on temperature.

Figure 8.

Impact of β on velocity.

Figure 9.

Impact of $\beta$ on temperature.

Figure 10.

Impact of $R$ on velocity.

Figure 11.

Impact of $R$ on temperature.

Figure 12.

Impact of $\mathrm{Ec}$ on velocity.

Figure 13.

Impact of $\mathrm{Ec}$ on temperature.

Figure 14.

Impact of $S$ on velocity.

Figure 15.

Impact of $S$ on temperature.

Figure 16.

Impact of $H$ on velocity.

Figure 17.

Impact of $H$ on temperature.

Figure 18.

Impact of χ on velocity.

Figure 19.

Impact of χ on temperature.

Numerical results of the variation of $R{e}^{-1/2}\mathrm{Nu}$ and $R{e}^{1/2}{C}_f$ under the effect of key factors are listed in Table 5.

Table 5.

Variation in $R{e}^{-1/2}\mathrm{Nu}$ and $R{e}^{1/2}{C}_f$ when a single parameter is varied, keeping other parameters fixed $\lambda =1,\chi =0.03,\beta =0.4,R=0.2,\mathrm{Ec}=0.01,H=0.4,S=0.02,\mathrm{\&}d=0.5$ .

$\lambda$	$\chi$	$\beta$	$R$	$\mathrm{Ec}$	H	$S$	$R{e}^{-1/2}\mathrm{Nu}$	$R{e}^{1/2}{C}_f$
−1							2.4413	0.7143
−0.5							2.7396	1.8762
0.5							3.0505	3.7051
1							3.1476	4.5090
	0.009						3.0115	3.3953
	0.018						3.0741	3.8640
	0.027						3.1302	4.3455
	0.036						3.1803	4.8406
		0.2					2.8294	5.5583
		0.4					3.1476	4.5090
		0.6					3.3200	4.0629
		0.8					3.4319	3.8082
			0.2				3.1476	4.5090
			0.4				3.5529	4.6011
			0.6				3.9380	4.6815
			0.8				4.3066	4.7531
				0.01			3.1476	4.5090
				0.02			2.8702	4.5792
				0.03			2.5774	4.6506
				0.04			2.2687	4.7230
					0.4		3.1476	4.509
					1		3.2067	4.9852
					1.6		3.2528	5.4573
					2.2		3.2875	5.9251
						0.01	2.9928	4.5398
						0.02	3.1476	4.5090
						0.03	3.3075	4.4788
						0.04	3.4726	4.4491

Multiple linear regression analysis

Statistical multiple linear regression is a technique frequently applied in many fields to model the relationship between a dependent variable (the response) and multiple independent variables (the predictors) through a linear equation. In heat transfer parametric analysis, it is an effective method for analyzing the relationship between thermal and flow fields and various key factors affecting them as well as determining the extent of their contributions.^62,63 Using data in Table 5, the descriptive statistics of the predictor factors are presented in Table 6.

Table 6.

Descriptive statistics of the predictor factors.

Factor	$\overline{x}$	$s$
$\lambda$	0.8571	0.4686
$\chi$	0.0289	0.0047
$\beta$	0.4143	0.0932
$R$	0.2429	0.1372
$\mathrm{Ec}$	0.0121	0.0067
$H$	0.5286	0.4117
$S$	0.0207	0.0047

where $\overline{x},\mathrm{and}s$ represent the sample mean and standard deviation of the values of each factor in Table 5. Predictors (λ, χ, β, R, Ec, H, S) are standardized (z-Scored) to enhance model interpretability and enable direct coefficient comparison. The standardized value of predictor F for observation i is^62,64:

$$F_{\mathrm{is}}=\frac{F_i-{\overline{x}}_F}{s_F}.$$

(13)

where $F_i$ is the real value of predictor, and ${\overline{x}}_F,s_F$ are the factor-specific mean and standard deviation. The relationship between dimensionless $R{e}^{-1/2}\mathrm{Nu},$ and $R{e}^{1/2}{C}_f,$ and the predictor factors are modeled as^62,64:

$$\begin{array}{c}{\left(R{e}^{-\frac{1}{2}}\mathrm{Nu}\right)}_{\mathrm{ir}}=a_0+a_1{\lambda }_{\mathrm{is}}+a_2{\chi }_{\mathrm{is}}+a_3{\beta }_{\mathrm{is}}+a_4{R}_{\mathrm{is}}\hfill \\ +a_{5}E{c}_{\mathrm{is}}+a_6{H}_{\mathrm{is}}+a_7{S}_{\mathrm{is}}+{\epsilon }_{i0},(R{e}^{1/2}{C}_f{)}_{\mathrm{ir}}\hfill \\ \multicolumn{1}{c}{=b_0+b_1{\lambda }_{\mathrm{is}}+b_2{\chi }_{\mathrm{is}}+b_3{\beta }_{\mathrm{is}}+b_4{R}_{\mathrm{is}}+b_{5}E{c}_{\mathrm{is}}}\\ +b_6{H}_{\mathrm{is}}+b_7{S}_{\mathrm{is}}+{\epsilon }_{i1}.\hfill \end{array}$$

(14)

where ${\left(R{e}^{-\frac{1}{2}}\mathrm{Nu}\right)}_{\mathrm{ir}}\mathrm{and}(R{e}^{\frac{1}{2}}{C}_f{)}_{\mathrm{ir}}$ are regression estimates for observation i, $a_0\mathrm{and}{b}_0$ are intercepts, $a_i\mathrm{and}{b}_i$ are regression coefficients, and ${\epsilon }_{i0},{\epsilon }_{i1}$ are observation-specific error terms. Regression analyses were performed using SPSS software with standardized predictors and Table 5 data. Results are shown in Tables 7 and 8.

Table 7.

Regression analysis of $R{e}^{-1/2}\mathrm{Nu}$ .

Regression statistics
Multiple R	0.9956
R-square	0.9913
Adjusted R square	0.9882
Standard error	0.0439
Observations	28
ANOVA
Source	df	SS	MS	F	Significance F
Regression	7	4.3695	0.6242	324.4994	3.5666E-19
Residual	20	0.0385	0.0019
Total	27	4.4080
Standardized factor	Coefficients	Standard error	t Stat	p-Value	Lower 95%	Upper 95%
Intercept	3.1487	0.0083	379.8515	4.6073E-40	3.1314	3.1660
${\lambda }_{\mathrm{st}}$	0.1475	0.0087	16.9410	2.5014E-13	0.1294	0.1657
${\chi }_{\mathrm{st}}$	0.0281	0.0086	3.2678	3.8502E-03	0.0102	0.0461
${\beta }_{\mathrm{st}}$	0.0835	0.0085	9.7888	4.5245E-09	0.0657	0.1013
$R_{\mathrm{st}}$	0.2701	0.0087	31.0220	2.1759E-18	0.2520	0.2883
$E{c}_{\mathrm{st}}$	−0.1922	0.0085	−22.5813	1.0515E-15	−0.2100	−0.1745
$H_{\mathrm{st}}$	0.0360	0.0087	4.1356	5.1237E-04	0.0178	0.0542
$S_{\mathrm{st}}$	0.0767	0.0086	8.9243	2.0685E-08	0.0587	0.0946

Table 8.

Regression analysis of $R{e}^{1/2}{C}_f$ .

Regression statistics
Multiple R	0.9934
R-square	0.9868
Adjusted R square	0.9822
Standard error	0.1357
Observations	28
ANOVA
Source	df	SS	MS	F	Significance F
Regression	7	27.5318	3.9331	213.5744	2.21208E-17
Residual	20	0.3683	0.0184
Total	27	27.9001
Standardized factor	Coefficients	Standard error	t Stat	p-Value	Lower 95%	Upper 95%
Intercept	4.3259	0.0256	168.6639	5.1635E-33	4.2724	4.3794
${\lambda }_{\mathrm{st}}$	0.8812	0.0269	32.7084	7.6972E-19	0.8250	0.9374
${\chi }_{\mathrm{st}}$	0.2659	0.0266	9.9875	3.2313E-09	0.2104	0.3215
${\beta }_{\mathrm{st}}$	−0.2355	0.0264	−8.9222	2.0762E-08	−0.2905	−0.1804
$R_{\mathrm{st}}$	0.0379	0.0269	1.4060	1.7506E-01	−0.0183	0.0941
$E{c}_{\mathrm{st}}$	0.0287	0.0263	1.0895	2.8890E-01	−0.0262	0.0836
$H_{\mathrm{st}}$	0.3053	0.0269	11.3293	3.7302E-10	0.2491	0.3615
$S_{\mathrm{st}}$	−0.0245	0.0266	−0.9217	3.6765E-01	−0.0799	0.0309

In Table 7, the coefficient of determination (R-Square) value indicates that variations in the key factors collectively contribute to 99% of the total changes seen in $R{e}^{-1/2}\mathrm{Nu}$ . All of the predictor p-values fall below the 0.05 level, and the small value of significance F (3.5666E-19) shows that R-Square and the independent factors are statistically significant. A standard error of 0.0439 indicates that the model fits closely to the data, and the adjusted R-square of 0.9882 supports the model’s reliability. The predictors λ, H, β, and S exhibit positive influences on the response $R{e}^{-1/2}\mathrm{Nu}$ . Conversely, Ec exhibits a notable negative effect, with a coefficient of −0.1922. For R and $\chi$ , they have the major and minor relative positive effects, respectively, with coefficients of 0.0281 and 0.2701. In Table 8, the coefficient of determination shows that the key factors are responsible for about 99% of $R{e}^{1/2}{C}_f$ variation. The small value of significance F (2.22033E-17) shows that the coefficient of determination is statistically significant. Both the adjusted R-squared value of 0.9868 and the standard error of 0.9822 show that the model fits the data closely, indicating dependability. The significant statistical predictors λ, H, and χ exhibit positive influences on the response $R{e}^{1/2}{C}_f$ , while $\beta$ exhibits a negative effect. The non-significant statistical predictors R and Ec positively contribute, while S possesses a negative non-significant statistical effect.

Based on Tables 7 and 8, the linear models incorporating multiple predictors of $R{e}^{-1/2}\mathrm{Nu}$ and $R{e}^{1/2}{C}_f$ are:

$$\begin{array}{c}(R{e}^{-1/2}\mathrm{Nu}{)}_{\mathrm{ir}}=3.1487+0.1475{\lambda }_{\mathrm{is}}+0.0281{\chi }_{\mathrm{is}}\hfill \\ +0.0835{\beta }_{\mathrm{is}}+0.2701R_{\mathrm{is}}-0.1922E{c}_{\mathrm{is}}\hfill \\ +0.0360H_{\mathrm{is}}+0.0767S_{\mathrm{is}},(R{e}^{1/2}{C}_f{)}_{\mathrm{ir}}=4.3259\hfill \\ +0.8812{\lambda }_{\mathrm{is}}+0.2659{\chi }_{\mathrm{is}}-0.2355{\beta }_{\mathrm{is}}+0.0379R_{\mathrm{is}}\hfill \\ +0.0287E{c}_{\mathrm{is}}+0.3053H_{\mathrm{is}}-0.0245S_{\mathrm{is}}.\hfill \end{array}$$

(15)

Figures 20 and 21 illustrate the numerical solution versus the regression estimation of $R{e}^{-1/2}\mathrm{Nu}$ and $R{e}^{1/2}{C}_f$ . Strong alignment refers to the effectiveness of models in capturing underlying relationships within the data.

Figure 20.

Numerical solution versus regression estimation of $R{e}^{-1/2}\mathrm{Nu}$ .

Figure 21.

Numerical solution versus regression estimation of $R{e}^{1/2}{C}_f$ .

Conclusion

Numerical simulation and statistical regression analyzed an EMHD Casson tri-hybrid nanoliquid flow over a radiative vertical Riga plate, incorporating viscous dissipation, suction, and nanoparticle shape effects. Key findings are:

1. Spherical nanoparticles maximize velocity and minimize temperature; platelet nanoparticles minimize velocity and maximize temperature.

2. Non-spherical nanoparticles reduce the Nusselt number (0.23%–5.3%) and increase skin friction (1.6%–14%) compared to spherical nanoparticles.

3. Velocity falls with an increase in the suction factor, but it increases with other factors examined.

4. Escalating the values of $R,\mathrm{or}\mathrm{Ec}$ elevates temperature, whereas enlarging $\lambda ,\chi ,\beta ,H,\mathrm{or}S$ reduces temperature.

5. $R{e}^{-1/2}\mathrm{Nu}$ is inversely related to $\mathrm{Ec}$ , while it has a positive relationship with $\lambda ,\chi ,\beta ,R,H,\mathrm{or}S.$

6. $R{e}^{1/2}{C}_f$ is a decreasing function of $\beta$ , or $S$ but it is an increasing function of $\lambda ,\chi ,R,\mathrm{Ec},\mathrm{or}H$ .

7. Multiple regression indicates that thermal radiation is the most positive contributor in boosting energy transfer rates, while the mixed convection parameter is the primary driver of skin friction augmentation.

In addition to its utility in academic research, the numerical data and multiple linear regression models from this study may be employed to optimize thermal systems such as cooling devices and heat exchangers. Future studies could concentrate on tri-hybrid nanofluids, utilizing different mathematical models. Furthermore, analyzing flow behavior over complex geometries, optimizing nanoparticle form factors, and undertaking stability analyses would yield useful insights.