A semi-analytical solution of the magnetohydrodynamic blood-based ternary hybrid nanofluid flow over a convectively heated bidirectional stretching surface under velocity slip conditions

A*, B*

slip constants

b, c

constants

B₀

magnetic field strength

Bi

thermal Biot number

$Cpf$⁠, $CpThnf$

specific heat of the base and ternary hybrid nanofluids

h_f

heat transfer coefficient

K

porosity factor

K_p

permeability of the porous media

k_f

thermal conductivity of the base fluid

k_Thnf

thermal conductivity of the ternary hybrid nanofluid

q₁

coefficient of the thermal-dependent heat source factor

q₂

coefficient of the space-dependent heat source factor

Q_S

space-dependent heat source factor

Q_T

thermal-dependent heat source factor

T_∞

free-stream temperature

T_f

fluid temperature

T_w

surface temperature

u, v, w

velocity components

x, y, z

coordinate axes

α

slip factor along the x-direction

β

Casson factor

β₁

stretching ratio factor

γ

slip factor along the y-direction

μ_f

dynamic viscosity of the base fluid

μ_Thnf

dynamic viscosity of the ternary hybrid nanofluid

ρ_f, ρ_Thnf

densities of the base and ternary hybrid nanofluids

σ*

Stefan–Boltzmann constant

υ_f

kinematic viscosity of the base fluid

Nanofluid is a type of fluid that incorporates nanometer-sized particles, typically nanoparticles, into a base fluid, such as water, oil, or ethylene glycol. These nanoparticles, often made of materials such as metals, metal oxides, or carbon-based materials, are added to augment the thermal and physical features of the fluid as determined by Choi and Eastman.¹ Nanofluids are known for their improved heat transfer capabilities, making them valuable in various engineering applications, including heat exchangers, cooling systems, and electronic devices.^2,3 The enhanced properties are attributed to the increased surface area and thermal conductivity of the nanoparticles, which contribute to more efficient heat transfer and improved fluid performance compared to traditional fluids. Nanofluids have garnered attention in research and industry for their potential to enhance the efficiency of thermal systems and contribute to advancements in areas such as electronics cooling and renewable energy technologies.⁴ Hybrid nanofluid flow refers to the movement of a fluid that combines nanoparticles with different materials or features within a base fluid. In this context, “hybrid” implies a mixture of two or more types of nanoparticles in the fluid.⁵ The incorporation of multiple types of nanoparticles into the base fluid is aimed at achieving synergistic effects and enhanced properties.⁶ Hybrid nanofluids are often engineered to exhibit improved thermal conductivity, heat transfer efficiency, and other desirable characteristics compared to conventional fluids or single-component nanofluids.⁷ Researchers explore different combinations of nanoparticles to tailor the properties of hybrid nanofluids for specific engineering applications. Recently, a new class of nanofluids has been introduced, termed trihybrid nanofluids, which incorporate three distinct types of nanoparticles within a base fluid.⁸ The purpose of using trihybrid nanofluids is to capitalize on the potential synergistic effects and tailored properties that result from the combination of multiple nanoparticle types. Engineers and researchers are exploring trihybrid nanofluids in applications that demand specific enhancements in thermal conductivity, heat transfer efficiency, or other fluid properties.^9,10 This approach aims to optimize the performance of the nanofluid for particular uses, such as in heat exchangers, cooling systems, or advanced thermal management applications. The study and development of trihybrid nanofluids represent an advanced and nuanced area of nanofluid research within the broader field of fluid dynamics and materials science.

Magnetohydrodynamics (MHD) is a branch of fluid dynamics that investigates the behavior of electrically conducting fluids, incorporating the effects of both electromagnetic fields and fluid motion.¹¹ In the realm of heat transfer, MHD’s influence on convective processes becomes particularly pronounced in systems involving conductive fluids, such as plasmas and liquid metals. Turkyilmazoglu et al.¹² inspected the time-dependent stagnant point MHD fluid flow over a deformable off-centered sheet. The interaction between magnetic fields and these fluids induces complex phenomena that significantly alter heat transfer characteristics. One key aspect is the generation of Lorentz forces, resulting from the interaction between the magnetic field and electric currents within the conductive fluid.¹³ These forces influence fluid flow patterns, introducing unique challenges and opportunities in the context of heat transfer. Turkyilmazoglu¹⁴ studied the MHD fluid flow over a radially elongating and gyrating disk using magnetic field effects in the vertical direction to the flow phenomenon of the system. The Lorentz forces can enhance or impede convective heat transfer, depending on the orientation and strength of the magnetic field. The MHD effects lead to the formation of electric currents and magnetic fields within the fluid, further complicating the heat transfer dynamics. These complex collaborations are crucial in various applications, from designing advanced cooling systems for high-power electronics to optimizing thermal management in fusion reactors and spacecraft. Rahman et al.¹⁵ inspected the time-based MHD nanofluid flow in a three-dimensional system over a slowing gyrating disk using the effects of uniform suction on the fluid flow. In the systems, where efficient heat transfer is paramount, mitigating the MHD effects becomes a critical consideration. Researchers and engineers examine into sophisticated simulations and experiments to unravel the nuanced relationship between magnetic fields, fluid dynamics, and heat transfer. They are aiming to influence the MHD principles for enhanced thermal performance in a range of technological applications, ultimately shaping the future of advanced heat transfer systems. Dey et al.¹⁶ simulated dual solutions for the MHD fluid flow along with thermal and mass transportations over an exponentially elongating and contracting surface placed in a permeable medium. Patel and Patel¹⁷ discussed mass and thermal transportations for the MHD mixed convective fluid flow over a stretched surface with permeability and heat generation/absorption effects.

A porous surface is a material or structure characterized by the presence of interconnected voids or open spaces within its structure. These voids can vary in size, shape, and distribution, creating a network of channels or pores throughout the material. The porous nature of such surfaces provides them with unique physical and thermal properties that have wide-ranging applications in various fields. The porous surface has a profound impact on heat transfer phenomena due to its unique thermal properties.¹⁸ The presence of pores introduces additional pathways for the fluid flow, altering the convective heat transfer characteristics of the surface. The interconnected voids within the porous structure promote increased surface area, facilitating enhanced heat transfer through convective and conductive mechanisms. In addition, porous surfaces have the capability to absorb and store heat within the void spaces, providing a thermal inertia that mitigates temperature fluctuations.¹⁹ This property is particularly valuable in applications such as energy storage systems and thermal management, where porous materials can act as effective heat sinks or reservoirs.²⁰ The complex relationship between the fluid flow, heat conduction, and phase-change phenomena within porous surfaces makes them valuable in a myriad of engineering applications, ranging from heat exchangers and cooling systems to sustainable building materials. Qureshi et al.²¹ mathematically simulated the impacts of nanolayers on the flow of nanofluid over a permeable surface with the injection of SWCNTs. Ashraf et al.²² examined the periodic mixed convective MHD fluid flow over a cone placed in a permeable medium with impacts of varying thermal features. Bejawada et al.²³ discussed the radiative effects of the Casson fluid flow over an inclined sheet with the impacts of the chemical reactivity porous Forchheimer medium. Kalpana and Saleem²⁴ examined the thermal transportation of the MHD dusty stratified fluid flow over an inclined sheet placed in a permeable medium.

Thermal radiation is a form of electromagnetic radiation emitted by the surface of an object due to its temperature. Unlike conduction or convection, which involves the transfer of heat through a material or a fluid medium, thermal radiation does not require a medium and can propagate through a vacuum. A fluid flow coupled with thermal radiation constitutes a complex heat transfer phenomenon with significant implications for various engineering applications.²⁵ In this scenario, a fluid, such as a gas or liquid, undergoes simultaneous convective heat transfer and radiative heat exchange. The fluid flow influences the temperature distribution, and, in turn, the temperature field impacts the radiative heat transfer.²⁶ Convective heat transfer involves the motion of the fluid, driven by factors such as buoyancy or external forces, impacting the overall thermal behavior. Meanwhile, thermal radiation involves the emission, absorption, and transmission of electromagnetic radiation within the fluid.^27,28 The interaction between fluid flow and thermal radiation becomes particularly crucial in high-temperature environments, such as those encountered in combustion processes, aerospace engineering, or industrial furnaces. The presence of radiation alters the temperature profile of the fluid and surrounding surfaces, affecting the heat transfer rates.²⁹ Understanding this coupled phenomenon is essential for optimizing heat transfer systems, as radiation becomes increasingly significant at elevated temperatures where conventional convective heat transfer may be insufficient. Researchers utilize sophisticated computational models and experimental techniques to analyze and predict the complex relationship between fluid flow and thermal radiation, enabling advancements in the design of efficient and sustainable technologies across diverse industries, from power generation to materials processing.³⁰ The integration of fluid flow and thermal radiation considerations allows for a more comprehensive understanding of heat transfer mechanisms, ultimately influencing the development of innovative solutions for enhanced energy efficiency and thermal management in complex engineering systems.³¹ Many such concepts can be seen in Refs. 32–35.

A space-dependent heat source refers to a situation where the generation or absorption of heat within a system varies spatially. In other words, the heat source is not uniform throughout the entire domain but changes in magnitude or location. This can occur in physical systems where different regions or points within the system experience varying rates of heat production or absorption.³⁶ Meanwhile, a thermal-dependent heat source implies that the rate of heat generation or absorption is influenced by the local temperature within the system. In this scenario, the heat source is not constant but varies based on the thermal conditions at a given point or region.³⁷ The heat source may be a function of temperature, and as the temperature changes, so does the rate of heat production or absorption. This kind of dependency is often encountered in systems where temperature-dependent reactions or material properties play a significant role in governing heat transfer processes. The impact of a space-dependent heat source and a thermal-dependent heat source on heat transfer phenomena is profound, influencing the sophisticated dynamics of temperature distribution within a given system.^38,39 In the case of a space-dependent heat source, variations in heat generation or absorption across different spatial locations introduce spatial heterogeneity into the thermal field. This spatial non-uniformity induces complex temperature gradients, altering the flow of heat and creating localized thermal patterns. This phenomenon is crucial in understanding thermal behavior in systems with irregular geometries or heterogeneous material properties. Meanwhile, a thermal-dependent heat source introduces a dynamic aspect to heat generation or absorption, directly linking the heat transfer process to the local temperature conditions.⁴⁰ As temperature changes, the rate of heat production or absorption varies, leading to feedback mechanisms that influence the overall thermal profile of the system. This dependency is particularly significant in situations involving temperature-sensitive reactions or materials, where heat transfer is complicatedly connected to the evolving thermal state. Both space-dependent and thermal-dependent heat sources are essential considerations in fields such as thermal engineering, electronics cooling, and materials processing.⁴¹ Accurately modeling and predicting these heat transfer phenomena require understanding to capture the spatial and temporal variations, enabling the optimization of systems for enhanced efficiency, safety, and performance across diverse applications, from electronic devices to industrial processes and environmental systems.

From the cited literature, it is observed that many investigations have been conducted to discuss thermal transportation using various flow conditions, including heat source, thermal radiations, thermal slip conditions, and MHD effects. However, no investigation has yet been conducted to examine thermal enhancement for the Casson trihybrid nanofluid flow over the dual-directional elongating sheet with the combined effects of the space-dependent heat source and the thermal-dependent heat source, which is the main motivation in this work. This study inspects the thermal transportation of the MHD Casson trihybrid nanofluid flow over a convectively heated dual-directional elongating sheet. The flow is considered as three dimensional passing over the sheet, which is placed in a porous medium. Blood is used as the base fluid, in which the nanoparticles of TiO₂, SiO₂, and Al₂O₃ have been mixed. The effects of thermal radiations and space- and thermal-dependent heat sources are used in energy equations, while magnetic effects are used in momentum equations. Appropriate variables have been used to convert the modeled equations into a dimensionless form, which have then been solved using the homotopy analysis method (HAM). Section I presents the introduction; Sec. II illustrates the problem formulation; the methodology and its convergence and validation are presented in Sec. III; and the results of this work are discussed in Sec. IV, while the main findings are concluded in Sec. V.

The solution’s convergence of the current model is ensured through ℏ-curves such as ℏ_f, ℏ_g, and ℏ_θ using the HAM approach’s 15th-order approximation as depicted in Figs. 2(a)–2(c). The region of convergence for primary velocity as illustrated in Fig. 2(a) is −1.50 ≤ ℏ_f ≤ 0.05; for secondary velocity as illustrated in Fig. 2(b), the region of convergence is −2.50 ≤ ℏ_f ≤ 0.05. From Fig. 2(c), it is noticed that the region of convergence for thermal profiles is −1.0 ≤ ℏ_θ ≤ 1.0.

To confirm that the present analysis is accurate, we have compared our results with the previously established data available in the literature of Yusuf et al.,⁴² Mabood and Das,⁵⁰ and Xu and Lee⁵¹ for different skin friction profiles (⁠$CfxRex1/2$⁠, $CfyRey1/2$⁠) when α and M vary, keeping other factors zero. Table II validates the current results of HAM with the available data for different values of α, keeping all other parameters zero. Table III validates the present results of HAM with the published ones for different values of M, keeping other factors zero. From both these tables, an excellent agreement has been seen among our results and those established previously in the literature that validates our solution.

This work examines the thermal transportation of the MHD Casson trihybrid nanofluid flow over a convectively heated dual-directional elongating sheet. The flow is considered as three dimensional passing over the sheet, which is placed in a porous medium. Blood is used as the base fluid, in which the nanoparticles of TiO₂, SiO₂, and Al₂O₃ have been mixed. The effects of thermal radiations and space- and thermal-dependent heat sources are used in energy equations, while magnetic effects are used in momentum equations. Various emerging factors have been encountered in this work. The impacts of these factors on primary and secondary velocities and thermal distributions will be discussed in Secs. IV A–IV D. Here, we fixed all the governing factors as Pr = 21, Rd = 1.0, β = 0.5, Q_T = 2.0, Q_S = 2.0, Bi = 3.0, K = 1.0, M = 1.0, and φ₁ = φ₂ = 0.02 unless otherwise mentioned in specific figures and tables for both slip and no-slip cases.

The impacts of different factors on $f′η$ are depicted in Figs. 3–5. Figure 2 depicts the impacts of slip factor $α$ along the axis. Here, it is noticed that with growth in $α$⁠, there is a reduction in $f′η$⁠. Actually, slip factor $α$ characterizes the relative motion between the surface and fluid, indicating the extent to which the fluid adheres to or slips along the stretching sheet. As $α$ grows, there is an enhanced slippage of the fluid at the sheet interface. This increased slip effectively reduces the frictional drag, leading to a decrease in $f′η$ for the hybrid nanofluid. The influence of porosity factor $K$ with K = 0.5, 0.7, 0.9 on velocity distribution $f′η$ is depicted in Fig. 4 for the scenarios α = 0 and α = 0.5 by keeping all the other factors fixed. With growth in $K$⁠, there is a reduction in $f′η$ for the fluid flow over the bidirectional stretching sheet for both the scenarios α = 0 and α = 0.5; however, a reduction in $f′η$ is more significant in the scenario α = 0.5. Physically, $K$ represents the fraction of void spaces in the porous medium through which the fluid flows. As $K$ grows, the presence of more void spaces hinders the fluid flow, leading to a reduction in primary velocity. Specifically, when $K$ augments in the scenario α = 0.5, the combination of increased porosity and slip-induced slippage at the sheet interface collectively intensifies the reduction in $f′η$⁠. The enhanced slip effects aggravate the reduction in fluid velocity by allowing greater fluid mobility within the porous structure. Therefore, the simultaneous influence of porosity and slip effects underscores the complex relationship between these factors in shaping the overall fluid dynamics near the surface of bidirectional stretching sheets as depicted in Fig. 4. The influence of magnetic factor $M$ with M = 1.0, 4.0, 8.0 on velocity distribution $f′η$ is depicted in Fig. 5 for the scenarios α = 0 and α = 0.5 by keeping all the other factors fixed. With escalation in $M$⁠, there is a reduction in $f′η$ for the fluid flow over the bidirectional stretching sheet for both the scenarios α = 0 and α = 0.5; however, a reduction in $f′η$ is more significant in the scenario α = 0.5. Actually, $M$ denotes the strength of the applied magnetic field, and as it grows, magnetic forces act against the fluid motion, causing resistance and a consequent decline in primary velocity $f′η$⁠. The added significance of the reduction in $f′η$ for the scenario α = 0.5 can be attributed to the synergistic effects of magnetic forces and slip-induced slippage at the sheet interface. The slip conditions allow the fluid to slide more easily along the stretching sheet, amplifying the overall impact of the magnetic field and intensifying the reduction in fluid velocity. Hence, growth in $M$ results in a reduction in $f′η$ as depicted in Fig. 5, where the reduction in $f′η$ is more obvious for the scenario α = 0.5.

The impacts of different factors on secondary velocity profiles $g′η$ are depicted in Figs. 6–9. Figure 6 depicts the impacts of the slip factor $γ$ along the y-axis. Here, it is noticed that with growth in $γ$⁠, there is a reduction in $g′η$⁠. The slip factor $γ$ along the y-axis represents the relative motion between the fluid and the surface, indicating the degree to which the fluid adheres to slips along the stretching sheet in the y-direction. So, as $γ$ along the y-axis grows, there is an increased slippage of the fluid in the perpendicular direction to the stretching sheet. This augmented slippage diminishes the lateral momentum transfer within the boundary layer, resulting in a reduction in secondary velocity profiles $g′η$⁠. Essentially, the growing slip factor introduces a greater degree of slippage, modifying the fluid–surface interaction and influencing the lateral distribution of velocities, which manifests as a decrease in $g′η$ in the y-direction as depicted in Fig. 6. The influence of the porosity factor $K$ with K = 0.1, 0.2, 0.3 on velocity distribution $g′η$ is illustrated in Fig. 7 for the scenarios γ = 0 and γ = 0.5 by keeping all the other factors fixed. With growth in $K$⁠, there is a reduction in $g′η$ for the fluid flow over the bidirectional stretching sheet for both the scenarios γ = 0 and γ = 0.5; however, a reduction in $g′η$ is more significant in the scenario α = 0.5 along the y-axis. Physically, $K$ represents the fraction of void spaces in the porous medium through which the fluid flows. As $K$ grows, the presence of more void spaces hinders the fluid flow, leading to a reduction in primary velocity. Specifically, when $K$ augments in the scenario γ = 0.5, the combination of increased porosity and slip-induced slippage at the sheet interface collectively intensifies the reduction in $g′η$⁠. The enhanced slip effects aggravate the reduction in fluid velocity by allowing greater fluid mobility within the porous structure. Therefore, the simultaneous influence of porosity and slip effects underscores the complex relationship between these factors in shaping the overall fluid dynamics near the surface of bidirectional stretching sheets as depicted in Fig. 7. The influence of the magnetic factor $M$ with M = 1.0, 4.0, 8.0 on velocity distribution $g′η$ is depicted in Fig. 8 for the scenarios γ = 0 and γ = 0.5 by keeping all the other factors fixed. With escalation in $M$⁠, there is a reduction in $g′η$ for the fluid flow over the bidirectional stretching sheet for both the scenarios γ = 0 and γ = 0.5; however, a reduction in $g′η$ is more significant in the scenario γ = 0.5. Physically, $M$ denotes the strength of the applied magnetic field, and as it grows, magnetic forces act against the fluid motion, resulting in confrontation and a subsequent deterioration in secondary velocity $g′η$⁠. The additional significance of the reduction in $g′η$ for the scenario γ = 0.5 can be endorsed to the synergistic effects of magnetic forces and slip-induced slippage at the sheet interface along the y-axis. The slip conditions allow the fluid to slide more easily along the stretching sheet, amplifying the overall impact of the magnetic field and intensifying the reduction in fluid velocity. Hence, growth in $M$ results in a reduction in $g′η$ as depicted in Fig. 8, where the reduction in $g′η$ is more obvious for the scenario γ = 0.5. The influence of the Casson factor $β$ with β = 0.1, 0.4, 0.8 on velocity distribution $g′η$ is demonstrated in Fig. 9 for the scenarios γ = 0 and γ = 0.5 by keeping all the other factors unchanged. With growth in $β$⁠, there is escalation in $g′η$ for the fluid flow over the bidirectional stretching sheet for both the scenarios γ = 0 and γ = 0.5; however, growth in $g′η$ is more significant in the scenario γ = 0 along the y-axis. Physically, the Casson factor $β$ is associated with the Casson model, which describes non-Newtonian fluids with yield stress. As $β$ increases, the yield stress effect becomes more pronounced, leading to a delayed transition from solid-like to fluid-like behavior. This delayed yielding allows for greater deformation and flow in the fluid, resulting in an augmentation of $g′η$ along the y-axis. For the scenario γ = 0, the increased Casson factor plays a dominant role in shaping the flow dynamics, leading to a more significant enhancement in $g′η$ as portrayed in Fig. 9.

The impacts of different factors on $θη$ are depicted in Figs. 10–13. Figure 10 portrays the impacts of the radiation factor $Rd$ on $θη$⁠. Here, it is noticed that, with growth in $Rd$ there is an escalation in $θη$⁠. Actually, $Rd$ represents the strength of radiative heat transfer in the system. As this factor grows, radiative heat exchange becomes more pronounced, allowing for increased thermal transport within the fluid. The enhanced radiation factor contributes to a more efficient redistribution of thermal energy, leading to an augmentation in the thermal distribution along the bidirectional stretching sheet. This is particularly significant in scenarios where radiation plays a dominant role in heat transfer. The increase in $Rd$ amplifies the impact of radiative heat transfer mechanisms, highlighting the crucial role of radiative effects in shaping the thermal profiles in fluid flow scenarios involving bidirectional stretching sheets. The influence of the thermal-dependent heat source factor $QT$ with Q_T = 0.1, 0.2, 0.3 on thermal distribution $θη$ is illustrated in Fig. 11 for the scenarios Rd = 0 and Rd = 0.5 by keeping all the other factors fixed. With growth in $QT$⁠, there is an escalation in $θη$ for the fluid flow over the bidirectional stretching sheet for both the scenarios Rd = 0 and Rd = 0.5; however, augmentation in $θη$ is more significant in the scenario Rd = 0.5. Actually, $QT$ accounts for the influence of internal heat generation within the fluid. As this factor grows, more heat is generated, leading to an increase in thermal profiles. The added significance of the augmentation in the presence of radiation effects with Rd = 0.5 suggests a synergistic effect between the thermal-dependent heat source and radiative heat transfer. Radiation facilitates efficient heat transfer, allowing the system to better accommodate and distribute the internally generated heat, leading to a more pronounced increase in $θη$⁠. This underscores the combined impact of internal heat generation and radiation in shaping the temperature distribution along bidirectional stretching sheets in fluid flow for both the scenarios, i.e., Rd = 0 and Rd = 0.5. The influence of the space-dependent heat source factor $QS$ with Q_S = 0.1, 0.2, 0.3 on thermal distribution $θη$ is illustrated in Fig. 12 for the scenarios Rd = 0 and Rd = 0.5 by keeping all the other factors fixed. With growth in $QS$⁠, there is an intensification in $θη$ for the fluid flow over the bidirectional stretching sheet for both the scenarios Rd = 0 and Rd = 0.5; however, augmentation in $θη$ is little more significant in the scenario Rd = 0.5. Physically, the space-dependent heat source factor describes the variations in heat generation within the fluid. As this factor increases, the spatial distribution of heat generation becomes more pronounced, leading to an augmentation in $θη$ for the fluid flow over the surface of the dual-directional stretching sheet. The added significance of this augmentation in the presence of radiation effects indicates a cooperative influence between the space-dependent heat source and radiative heat transfer as more augmentation in $θη$ is observed for the scenario Rd = 0.5. The influence of the thermal Biot number $Bi$ with Bi = 0.1, 0.2, 0.3 on thermal distribution $θη$ is illustrated in Fig. 13 for the scenarios Rd = 0 and Rd = 0.5 by keeping all the other factors fixed. With growth in $Bi$⁠, there is an intensification in $θη$ for the fluid flow over the bidirectional stretching sheet for both the scenarios Rd = 0 and Rd = 0.5; however, augmentation in $θη$ is more significant in the scenario Rd = 0.5. Actually, $Bi$ represents the ratio of the internal thermal resistance within the fluid to the external thermal resistance at the fluid–sheet surface interface. As the thermal Biot number increases, the importance of internal thermal resistance becomes more significant. In the presence of radiative effects, the escalation in $θη$ is more pronounced, indicating that radiative heat transfer plays a crucial role in mitigating the external thermal resistance and enhancing the overall thermal transport. The increased thermal Biot number accentuates the contribution of internal thermal resistance, and the synergistic effect of radiative heat transfer amplifies the augmentation in thermal profiles, for the fluid flow over bidirectional stretching sheets in fluid flow scenarios as depicted in Fig. 13.

The impacts of variations in different emerging factors against skin frictions along the x- and y-axes and the Nusselt number are portrayed in Figs. 14–16, respectively. It is noticed from Table IV that with an upsurge in porosity and magnetic factors, the skin friction along the x-axis is escalated in both the scenarios α = 0 and α = 0.5. Similarly, from Table V, it is revealed that with expansion in porosity and magnetic factors, the skin friction along the y-axis has intensified in both the scenarios γ = 0.0 and γ = 0.5. The rise in porosity suggests a greater permeability of the medium, potentially allowing more fluid flow through the material. This increased fluid flow, coupled with the influence of magnetic factors, contributes to heightened skin friction along both axes. The magnetic factor induces changes in the fluid dynamics and alters the interaction between the fluid and the porous medium. However, growth in $CfxRex1/2$ and $CfyRey1/2$ is more significant for the scenarios α = 0.5 and γ = 0.5 as depicted in Tables IV and V, respectively. These phenomena are also portrayed in Figs. 14 and 15, where K = 0.1, 0.2, 0.3, while M = 1.0, 1.5, 2.0, 2.5, 3.0 for the scenarios α = 0 and α = 0.5 and γ = 0.0 and γ = 0.5. From Table VI, it is observed that the Nusselt number augments with escalation in thermal- and space-dependent heat source factors (Q_S, Q_T) and the thermal Biot number (Bi) for both the scenarios Rd = 0.0 and Rd = 0.5. However, augmentation in $NuxRex−1/2$ with an upsurge in Q_S, Q_T, and Bi is more significant for the scenario Rd = 0.5. This phenomenon is also graphically depicted in Fig. 16.

This work inspects the thermal transportation of the MHD Casson trihybrid nanofluid flow over a convectively heated bidirectional elongating sheet. The flow is considered as three dimensional passing over the sheet, which is placed in a porous medium. Blood is used as the base fluid, in which the nanoparticles of TiO₂, SiO₂, and Al₂O₃ have been mixed. The effects of thermal radiations and space-and thermal-dependent heat sources are used in energy equations, while magnetic effects are used in momentum equations. The impacts of different emerging factors on various profiles are discussed in detail. In this work, the following main points have been uncovered:

• Both primary and secondary velocities are weakened with an upsurge in porosity and magnetic factors for all the scenarios α = 0 and α = 0.5 and γ = 0.0 and γ = 0.5; however, this decline is more significant for the scenarios α = 0.5 and γ = 0.5.

• The secondary velocity profiles escalated with an upsurge in the Casson factor, where augmentation is more substantial for the scenario γ = 0.5.

• The thermal transportation is augmented with growth in thermal- and space-dependent heat source factors and thermal Biot number for both the scenarios Rd = 0 and Rd = 0.5; however, escalation in $θη$ is more significant in the scenario Rd = 0.5.

• With an upsurge in the porosity and magnetic factors, the skin frictions $CfxRex1/2$ and $CfyRey1/2$ along the x- and y-axes, respectively, escalated in all the scenarios α = 0 and α = 0.5 and γ = 0.0 and γ = 0.5. However, this growth is more significant for the scenarios α = 0.5 and γ = 0.5.

• The Nusselt number augments with escalation in the thermal- and space-dependent heat source factors and thermal Biot number for both the scenarios Rd = 0.0 and Rd = 0.5. However, this augmentation is more significant for the scenario Rd = 0.5.

• The convergence of the method used in this work is ensured through ℏ-curves.

• The findings of this investigation have been compared with the existing literature, revealing a strong agreement among the present and established results that ensured the validation of the model and method used in this work.

The authors extend their appreciation to the Deanship of ScientificResearch at King Khalid University, Abha, Saudi Arabia, for funding this work through the Research Group Project under Grant Number (RGP.2/505/44).

The authors have no conflicts to disclose.

Showkat Ahmad Lone: Investigation (equal); Methodology (equal); Software (equal); Supervision (equal). Arshad Khan: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Writing – original draft (equal). Zehba Raiza: Conceptualization (equal); Data curation (equal); Writing – original draft (equal). Hussam Alrabaiah: Conceptualization (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal). Sana Shahab: Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). Anwar Saeed: Formal analysis (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal). Ebenezer Bonyah: Writing – review & editing (equal).

The data that support the findings of this study are available within the article.