Improving heat transfer efficiency via optimization and sensitivity assessment in hybrid nanofluid flow with variable magnetism using the Yamada–Ota model

Symbols

r ˆ , z ˆ

coordinates

T ˆ ∞

free stream temperature (K)

T ˆ w ˆ

wall temperature (K)

B ( t )

variable magnetic field

v ˆ 0

suction/injection

λ

stretching/shrinking parameter

β

unsteadiness parameter

ψ

stream function

S

mass flux constant

Q

heat source parameter

k ˆ ∗

mean absorption coefficient (m⁻¹)

σ ˆ ∗

Stefan Boltzmann constant (W m⁻² K⁻⁴)

M

magnetic parameter

Ec

Eckert number

Rd

radiation parameter

μ ˆ

dynamic viscosity (kg m⁻¹ s⁻¹)

ρ ˆ

density (kg m⁻³)

ϕ 1 , ϕ 2

volume fraction for 1st and 2nd nanoparticles

( ρ ˆ c ˆ p ˆ )

specific heat (J m⁻³ K⁻¹)

σ ˆ

electrical conductivity

k ˆ

thermal conductivity (W m⁻¹ K⁻¹)

q r

Radiative heat flux (kg s⁻³)

C f

skin friction

Nu

Nusselt number

C f x

reduced skin friction

Nu x

reduced Nusselt number

q r

radiative heat flux

Re

Reynolds number

Subscript

f

fluid

nf

nanofluid

hnf

hybrid nanofluid

1 Introduction

Hybrid nanofluids, which combine nanoparticles with different base fluids, are the subject of increased research attention. Due to their improved and changed rheological and thermophysical properties, these hybrid systems have a number of advantages over conventional nanofluids. They have consequently attracted a lot of interest in the area of solar energy systems. In recent era, there has been a rising interest in enlightening heat transfer efficiency for various manufacturing applications, vacillating from electronics cooling to thermal management in power generation systems. In this context, the utilization of nanofluids as heat transfer media has gained considerable attention due to their enhanced thermal conductivity properties. Among these nanofluids, the grouping of silver–molybdenum tetra sulphide (Ag–MoS₄) nanoparticles with variable magnetism holds great promise for achieving significant advancements in heat transfer enhancement. However, to fully exploit the potential of such hybrid nanofluids, it is crucial to optimize their flow characteristics and evaluate the sensitivity of key parameters. Managing heat dissipation is vital for maintaining performance and increasing the lifespan of components present in electronic devices. By utilizing hybrid nanofluids, which combine different nanoparticles to leverage their unique thermal properties, the cooling efficiency of heat sinks is significantly improved. This phenomenon is integrated into the cooling systems of high-performance CPUs and other electronic components. In solar thermal systems, hybrid nanofluids are used as the working fluid for the enhanced absorption and transfer of solar energy. In addition, hybrid nanofluidic systems’ behaviour and performance are greatly influenced by variables like nanoparticle concentration and the type of base fluid used, which furthers their distinctive and alluring qualities. Salawu et al. [1] conducted research to evaluate the elasticity and deformation of heat radiation in engine oil using Ag/single-wall carbon nanotube along with MoS₄/multi-wall carbon nanotube-based hybrid structured fluid. The overall focus of the survey was to explore the thermal transportation of hybrid structured nanofluidic in an upward cylindrical surface, with careful consideration given to the thermofluidic properties during the design of the model. The attributes of magnetic hybrid nanomaterials, particularly those suspended in copper and MoS₄ nanoparticles, were examined by Kavya et al. [2]. The variations in thermal conduction, heat production, and the impacts of magneto dynamics were taken into account during these assessments, which were carried out inside of a stretched cylinder. Patel et al. [3] investigated the magnetohydrodynamic (MHD) flows generated by heat transfer in a sheet with an exponential extension or reduction, which contained hybrid nanofluids. These hybrid nanofluids were composed of titanium oxide and silver nanoparticles combined with a base liquid such as H₂O, resulting in hybridized nano-structured fluid. Basit et al. [4] performed in-depth research on hybridized composite, including Au–Ag with base fluid as blood as well as Cu–Fe₃O₄ with base fluid as blood, in two spinning discs. In order to address the Darcy–Forchheimer media, a hybrid nanofluid that is flowing towards the inside of two parallel discs is taken into consideration, along with viscous dissipation association with heat radiation. Hassan et al. [5] directed a survey on the thermal and mass transmission characteristics of two distinct hybridized nano-structured fluids. These hybrid nanofluids were subjected to a magnetized field and Rosseland radiative influences within a spinning cone that was implanted in porosity.

Viscous dissipation is especially significant in viscous fluids, where internal friction is a significant factor. The system’s overall energy balance and thermal behaviour are influenced by the heat dissipated. Understanding and accounting for viscous dissipation is crucial for accurate modelling and analysis in practical applications, such as in industrial processes or the study of fluid flow. Hameed et al. [6] researched heat production and absorption, the two-dimensional flow of a Casson composite nanofluid subjected to a magnetic field, and the dissipation of viscous fluid on a non-linear elongating interface. The improvement of heat transfer performance, which is highly desired in the industrial and technical sectors, was the main goal of this study. Mahesh et al. [7] research focused on determining how radiation affects an MHD pair stress hybrid nanofluid’s ability to move across a porous media with dispersion. This water-based hybrid nanofluid has the efficiency to be implemented in a wide range of technical applications. Its flow is induced by a porous contracting interface with radiative heat as well as viscous dissipation. The movement of characteristics of a nano-structured fluid under an elastic structure was inspected by Riaz et al. [8], specifically focusing on the two-dimensional MHD convection. The study took into account the impact of various factors under slippery restrictions. Additionally, the influence of convective thermal surface along mass flux on the flow characteristics was also considered in the analysis. In order to account for heat radiation association with viscous dissipation, the researchers described the MHD movement of a hybrid nanostructure fluid via a wedge interface. According to Bing Kho et al. [9], growing the volume percentage of titania nanoparticles could improve thermal conductivity. The initial solution additionally showed stability in this flow. Entropy analysis was utilized by Jamshed et al. [10] to inspect the influence of porous media along joule heated on the steam of a second-grade nano-structured fluid towards a horizontally radiating moving flat platform.

A heat source is important in the study of thermodynamics because its primary objective is to accelerate the rate of thermal energy transfer within a fluid. In order to raise the fluid’s temperature, a heat source is added to a system in order to promote and accelerate the flow of thermal energy. Depending on the particular setup and needs of the system, several methods, such as conduction, convection, or radiation, are used to improve thermal energy transmission. A heat sink, conversely, operates with the opposite goal in consideration to lower the temperature of a fluid or item by actively promoting the transfer of heat away from it. A heat sink increases the rate at which thermal energy is passed through it by using materials with high thermal conductivity, thereby successfully establishing a channel for heat to flow away from the target object. Additionally, to enhance the surface area accessible for convective heat transmission, heat sinks frequently have extended surfaces or fin-like structures. In the attendance of a transverse magnetized field, Mishra et al. [11] investigated a continuous stream of an electrically conducted incompressible viscous fluid on a plate in a doubly stratification micropolar fluid. Pattnaik et al. [12] considered the natural steam of an electrically conducted viscoelastic nanocomposite over a widening interface, taking into account the effects of Brownian motion association with thermophoresis on the heat transfer process. Additionally, the study placed significant emphasis on the influence of chemical reactions, which played a crucial role in enhancing the overall understanding of the phenomena being investigated. Baag et al. [13] undertook a study focusing on the magnetized steam of a viscous liquid towards a widening interface. The main objective of the investigation was to examine the effects of thermal buoyancy, significant heat sources or sinks, and the occurrence of a porous media on the fluid flow dynamics. Additionally, special attention was given to the convective heating boundary conditions, as they exert a substantial influence on the alteration of heat transportation features within the system. The investigation conducted by Khan et al. [14] focused on the analysis of radiative mixed convective flow resulting from the movement of a hybridized nanocomposite towards a porous upward cylinder situated within a porous layer. The study placed particular emphasis on studying scenarios where there were uneven distributions of heat sink or heat source conditions. Ishtiaq et al. [15] provided a concise comparative report of stationary point flow in a hybridized nanocomposite using an improved version of two significant models under a widening/shrinking permeability sheet.

Response surface methodology (RSM) is widely used in a wide range of industries. It allows for the optimization of manufacturing processes in industrial settings, resulting in increased productivity, lower costs, and better product quality. RSM supports the creation of efficient therapies, drug formulations, and medical interventions by assisting in the understanding of the relationship between factors and responses in biological and clinical sciences. In order to better understand human behaviour, opinion formation, and decision-making processes, social science researchers use RSM to explore and model complicated interactions between components. Reducing the number of experimental runs necessary to produce accurate and trustworthy data is one of RSM’s primary benefits. RSM enables researchers to forecast and optimize responses within a certain range of factors using statistical models, saving time, resources, and effort. RSM helps in discovering important elements, figuring out their optimal values, and comprehending their relationships by methodically altering and manipulating the factors of interest. The features of three-dimensional flow of a water hybridized nanocomposite towards a porous medium with an expanding or contracting interface were studied by Panda et al. [16]. Understanding the complex interactions between an applied magnetized effect, velocity slippage, and convective heat transportation situations was the main goal of the work. Notably, the observed phenomena were greatly influenced by the presence of an electrically conducting fluid in the system. The use of an ideal technology, notably the response surface approach, to precisely analyse and optimize the heat transfer rate response, was a defining aspect of their work. Mamourian et al. [17] utilized the RSM as well as sensitivity analysis to investigate the effects of MHD and angle of inclination on natural convection heat transfer and entropy production within a square cavity filled with an Al₂O₃–water nanofluid. Their study aimed to understand how these factors influenced heat transportation characteristics and entropy production. Hosseinzadeh et al. [18] created a model of a star-shaped curved porous structure with a spherical cavity that was filled with hybridized nanoparticles. The temperature alteration among the inside cavity and the undulating outside interface, in this configuration, drove the heat flux in the enclosed area. Alhadri et al. [19] inspected the radiation characteristics of a hybrid nanofluid flowing across an extended sheet that used water as its base fluid. They specifically looked at how joule heating and suction affected the properties of heat transfer. Copper along with aluminium oxide nanocomposites was added to the formulation of the nano-structured fluid to improve the fluid’s thermal performance. The researchers applied the existing Darcy–Forchheimer theory to accurately describe the impacts of both inertial and porous medium phenomena. This theory selection made it possible to analyse the system using a model that was both more physically accurate and trustworthy. Rana and Gupta [20] investigated FEM approaches to the quadratic convective and radiated fluxes of an hybridized nanocomposite around a rotary cone with Hall current using RSM. Influential conductivity models such as the Yamada–Ota and Hamilton–Crosser models were studied by Panda et al. [21]. In their study, Chu et al. [22] conducted a radiative thermal analysis using the Keller box numerical method, focusing on four different hybrid nanoparticle varieties that were being affected by an uneven heat source. Alqahtani et al. [23] investigated the thermal analysis of a radiative nanofluid near a cylinder that was expanding and contracting while taking viscous dissipation into consideration. With the impact of bioconvection taken into account, Puneeth et al. [24] study concentrated on a theoretical examination of the thermal characteristics of a Ree–Eyring nanofluid in motion close to a stretching sheet. Some intriguing and recent efforts include investigations [25,26,27,28] that investigated various aspects of nanofluid flow. These works investigated the dynamic behaviour of nanofluids under various circumstances and configurations. Many researchers [29,30,31,32] have conducted considerable research on different types of nanofluid flow over a stretching sheet, taking into account a variety of influential elements. These studies investigated the effects of thermal conductivity, viscosity, and nanoparticle concentration on heat transfer and fluid flow properties. Numerous researchers [33,34,35] have used computational modelling to investigate the parametric influences on fluid flow behaviour. Several researchers [36,37,38] used statistical sensitivity analysis approaches to improve heat transfer in fluid dynamics. These methods involve systematically changing input parameters to assess their impact on heat transfer effectiveness.

2 The theoretical explanation of the contemporary problem

In order to inspect the heat transportation efficiency and the phenomenon of time-dependent, hydromagnetic stagnant point flow in a hybridized nanofluid composed with two nanoparticles, Ag and MoS₄, with the background water fluid, we explore the characteristics of a radiative elongating/contraction permeable interface with a variable velocity profile v ˆ w ˆ ( r ˆ , t ˆ ) . Figure 1 illustrates the geometric configuration of the current problem using cylindrical polar coordinates r ˆ and z ˆ . The stretching/shrinking radiative permeable sheet is positioned at z ˆ = 0 , with the flow occurring to the r ˆ − axis and the z ˆ − axis orthogonal towards the surface. The free stream temperature T ˆ ∞ and the sheet temperature T ˆ w ˆ are assumed to remain constant. Furthermore, a variable magnetic field B ( t ) = B 0 / ( 1 − ct ) 1 / 2 is applied vertically to the axis of the sheet, and the time-dependent heat source is presented as Q 0 ( t ) = Q 0 ⁎ / ( 1 − ct ) .

Figure 1

Schematic configuration of elongating/contracting sheet.

3 Numerical procedure

The provided model, described by Eqs. (2), (3), and (5), is analysed using numerical techniques. This process involves discretizing and differentiating the differential equations, resulting in the appearance of higher-order initial conditions when actual values are absent. These missing initial conditions are determined through a controlled target shooting method. The calculations are carried out in MATLAB using the in-house code bvp5c. bvp5c is a built-in MATLAB function that solves boundary value problems (BVPs) involving ordinary differential equations. In BVPs, the solution must meet conditions at several places, usually at the limits of the interval across which the solution is defined.

The resulting modified first-order ODEs are as follows:

renovated boundary conditions

4 Validation with parametric description

The existing works aims to investigate and improve heat transportation efficiency in a hybrid nanofluid flow. It utilizes numerical methods to analyse the heat transportation features of a hybrid nanofluid composed of Ag–MoS₄ in water with variable magnetism. Table 1 depicts the thermophysical attributes of the nanocomposite as well as the base liquid water. The implementation of the Yamada–Ota model, which is a mathematical model for the thermal conductivity used to describe fluid flow and heat transportation enriches the flow phenomena. However, the descriptions of each of the models for the viscosity, conductivity, etc., are presented through Eqs. 14(a)–(e). The study incorporates various factors that affect heat transfer, such as viscous dissipation and heat source/sink effects. Viscous dissipation refers to the alteration of mechanical energy into thermal energy due to the internal friction of the fluid, while heat source/sink effects account for the presence of external heat sources or sinks in the system. The modelled transformed designed problem is handled numerically using the in-house built-in function bvp5c with the assistance of MATLAB software. The validation of the present outcomes of the Nusselt number in a particular case is obtained with the result of Ishtiaq et al. [15], and the numerical values presented in Table 2 show a good correlation among themselves. Further, the computation is presented by keeping fixed values of the factors as S = 0.3 , β = 0.1 , M = 1 Pr = 6.2 , Rd = 0.3 , ϕ 1 = ϕ 2 = 0.01 , λ = 0.1 , Ec = 0.1 , Q = 0.1 , whereas the development of each constraint deployed the corresponding figure. To optimize the heat transfer efficiency, the researchers employ RSM. RSM is a statistical technique implemented to model and optimize complex systems by exploring the relationship among multiple variables and a response variable (Nusselt number) of interest. Furthermore, sensitivity analysis is also accomplished to determine the relative importance of numerous factors on heat transfer efficiency. This analysis helps identify the key factors that significantly encourage the performance of the system.

Figure 2 characterizes the behaviour of the magnetic constraint on the velocity profile of a hybrid nanofluid flow of Ag–MoS₄, considering variable magnetism in conjunction with suction/injection. The magnetic parameter refers to the strength of the magnetized field applied to the fluid flow. In general, the numerically assigned values such as M = 0 indicates the non-existence of the magnetic field, and the non-zero values, i.e., M ≠ 0 , show the influence of the magnetization of a nanofluid owing to the interaction between the magnetized field and the magnetic nanoparticles suspended in the fluid. For a weak magnetic field, the effect on the velocity profile is minimal, and the flow behaviour is similar to that of a non-magnetic nanofluid. The velocity profile following a typical pattern without significant deviations. As the strength of the magnetized field upsurges, the magnetophoretic force acting on the suspended nanoparticles also increases. This causes the nanoparticles to align with the magnetic field lines, and there is an increase in the magnitude but this resulted in thinning the bounding surface thickness. Similar to suction, a weak and strong magnetic field enhance the velocity profile significantly. However, the comparative analysis shows that suction overrides the fact of injection in retarding the thickness greatly.

Figure 2

Variation of magnetic parameter on the velocity profile.

Figure 3 depicts the consequence of the elongating/contracting parameter on the velocity profile of Ag–MoS₄–water hybrid nanofluid flow that depends on the various conditions such as suction/injection. In general, the shrinking/stretching parameter represents the stretching or compression of a coordinate axis in the governing equations of the fluid flow. This parameter is often introduced to renovate the governing equations into a non-dimensional form, making it easier to analyse the flow characteristics. The effect of the shrinking/stretching parameter on the velocity profile and the variation depends on its value and the specific flow conditions. In particular, when the parameter is negative λ < 0 , it represents a shrinking axis, which compresses the flow and leads to increased velocities. Conversely, a stretching axis λ > 0 expands the flow, potentially resulting in decreased velocities. The association of suction or injection on the velocity profile will depend on the specific boundary restrictions and the behaviour of the fluid near the boundaries. Suction at the boundaries generally tends to draw fluid into the system, altering the flow pattern and potentially affecting the velocity profile. Further, injection introduces fluid into the system, which causes changes in the flow behaviour.

Figure 3

Variation of velocity ratio on the fluid velocity.

Figure 4 illustrates the role of the volume fraction of Ag–MoS₄ nanoparticles on the velocity profile. The volume fraction of nanoparticles in the hybrid nanofluid has a significant influence on the velocity profile. Here, the ϕ 1 is noted for the silver and ϕ 2 denotes the volume fraction of the MoS₄ nanoparticles. Further, the assigned numerical value, i.e., ϕ 1 = 0 = ϕ 2 suggests the behaviour of the pure fluid, and then the non-zero values are assigned for nanofluid and hybrid nanofluid. It is seen that increasing concentration of nanoparticles leads to retards the velocity profile, such as increased viscosity and altered flow patterns. This causes an increase in the surface thickness. In the case of silver nanoparticles, the surface thickness became thinner due to the dominance in the viscosity as well as the density of the particles. The interaction between Ag and MoS₄ nanoparticle interactions leads to the formation of aggregates or clusters, affecting the flow pattern and the velocity profile. The agglomeration or dispersion of nanoparticles is influenced by the presence of surfactants or dispersing agents.

Figure 4

Variation of particle concentration on the fluid velocity.

The significant performance of the Eckert number on the temperature profile of Ag–MoS₄–water-based hybrid nanofluid for the variation of suction/injection is shown in Figure 5. The Eckert number Ec is a dimensionless quantity used in fluid mechanics to characterize the relative importance of thermal effects and kinetic energy effects. It is defined as the ratio of the kinetic energy term to the thermal energy term. In particular, when the Eckert number is low ( Ec ≪ 1 ) , the thermal effects dominate, and the flow behaves in a more heat transfer dominated regime. This means that the temperature changes in the fluid will have a significant influence on the flow behaviour. In this case, the suction/injection of the fluid have a notable impact on the temperature profile. Suction tends to remove fluid and lower the overall temperature, while injection introduces additional fluid and raises the temperature. Moreover, for the higher Eckert number ( Ec ≫ 1 ) , the kinetic energy effects become more significant compared to the thermal effects. In this regime, the flow is more influenced by the fluid velocity rather than temperature changes. In such situations, the suction/injection may have a lesser impact on the temperature profile as related to the velocity profile.

Figure 5

Variation of Eckert number on fluid temperature.

Figure 6 depicts the role of the heat absorption/source on the temperature profile of a hybrid nanofluid flow for the variation of suction/injection. A heat sink is a device or material that absorbs or dissipates heat from a system to maintain or lower its temperature. In the context of a hybrid nanofluid flow, the heat sink is used to remove excess heat generated within the system. A heat sink helps in cooling the nanofluid flow by absorbing the thermal energy. This leads to a reduction in the temperature of the nanofluid as it passes through the heat sink region. The temperature profile will exhibit a decrease in temperature along the heat sink surface. Further, heat source is a component that supplies heat to a system, raising its temperature. In the case of a hybrid nanofluid flow, a heat source can introduce additional thermal energy into the system. The heat source supplies thermal energy to the nanofluid, causing an increase in its temperature. The temperature profile will exhibit a rise in temperature along the heat source surface.

Figure 6

Variation of heat source/sink on fluid temperature.

The influence of thermal radiation on the temperature profile of Ag–MoS₄–water hybrid nanofluid for the variation of suction/injection is presented in Figure 7. Thermal radiation is the transfer of heat energy through electromagnetic waves, and it can play a crucial role in heat transfer phenomena, especially at high temperatures. The nanoparticles in the Ag–MoS₄ nanofluid can absorb and emit thermal radiation that leads to an increase in the temperature of the nanoparticles, while emission results in heat loss. Thermal radiation can transfer heat between different parts of the nanofluid and its surroundings. This radiative heat transfer can influence the overall temperature profile and distribution within the flow.

Figure 7

Variation of radiation on the temperature profile.

Figure 8 exhibits the characteristic of the volume fraction of Ag and MoS₄ nanoparticles on the temperature profile of a hybridized nanocomposite. There are several factors including the size and concentration of the nanocomposites, the thermal conductivity and specific heat capacity of the nanoparticles and base liquid, and the interaction between the nanoparticles and the base fluid participate in the variation of particle concentration as described in the employed models. The tabular result reveals that the thermal conductivity of MoS₄ nanoparticles are higher than that of silver nanoparticles. Therefore, increasing conductivity boosts up the fluid temperature in either of the cases; however, the case of MoS₄ nanoparticles has a better impact than that of silver nanocomposites.

Figure 8

Variation of particle concentration on fluid temperature.

Further, the contribution of several factors on the rate coefficients likely the shear rate and the heat transfer rate is obtained, and the simulated results are depicted in Table 3. The increasing magnetic parameter that produces Lorentz force has a resistive property that reduces the fluid velocity. Here, increasing the magnetic parameter significantly enhances the shear rate. Further, the suction parameter also favours in enhancing the coefficient as well. But, the reverse impact is rendered in the situation of stretching parameter. The particle concentration plays an important role in the parametric behaviour of the flow phenomena. It is observed that increasing particle concentration presents a higher shear rate, whereas the rate of heat transportation retards significantly. Further, the coupling constraint, i.e., the Eckert number for the variation of the dissipative heat along with the additional heat source, also acts as a controlling parameter to restrict the heat transfer rate. However, the radiating heat has a greater impact in enhancing the heat transportation rate.

5 RSM

RSM is a statistical approach employed for modelling and optimizing the correlation between multiple input factors and a response variable. When considering the heat transportation rate, the factors under investigation include particle concentration, thermal radiation, and heat source parameters. By utilizing RSM, it becomes possible to analyse the consequence of these factors on heat transfer rate and identify the optimal conditions that lead to its maximization. The experimental set-up or the design of experiments (DOE) involves establishing the range and levels for each factor under investigation. It is crucial to choose suitable low, middle, and high levels for every factor to encompass the entire operating range. The number of experiments needed depends on the complexity of the response surface and the desired level of accuracy. In this particular design, the range of these three factors is outlined in Table 4 and referred to as factors a, b, and c. 0.1 ≤ ϕ ≤ 0.3 , 0.5 ≤ Rd ≤ 1.0 , and 0.2 ≤ M ≤ 0.6 . A series of experiments are conducted based on the design matrix generated by the DOE. Each experiment consists of a specific combination of particle concentration, thermal radiation, and heat source parameters. The heat transportation rate is measured for each experiment, and the results for the 20 runs, along with their corresponding degrees of freedom, are presented in Table 5. The multivariate classic design for the response related to the specified factor is articulated as

After eliminating insignificant terms, the expression will be

5.1 Model analysis

Model analysis involves evaluating the model’s goodness of fit by examining statistical measures such as the coefficient of determination (R ²), analysis of variance (ANOVA), and lack of fit tests. These measures assess how well the model captures the variation in the response variable. Thus, the accuracy of the model was tested using ANOVA, and the findings are presented in Table 6. The table showcases different testing parameters such as p-value, F-value, errors, and total error for various sources and their degrees of freedom. Typically, for enhanced accuracy, standard outcomes are obtained for F -values > 1 and p -values < 0.05 , considering a significance level of 5% or a confidence level of 95%. ANOVA observations reveal that p-values were obtained for all three components, and a model expressing the response in terms of the Nusselt number and these factors is presented. The results from the table, including adjusted and computed values, indicate that the proposed model is the most suitable choice for the given responses.

Table 6

ANOVA for heat transfer rate

Source	Degree of freedom	Adjusted sum of square	Adjusted mean of square	F-value	p-Value	Coefficients
Model	9	1.58380	0.17598	65060.97	0.000	1.09123
Linear	3	1.56840	0.52280	193285.61	0.000
ϕ	1	1.38816	1.38816	513218.68	0.000	−0.372580
Rd	1	0.06008	0.06008	22211.55	0.000	0.077510
Q	1	0.12017	0.12017	44426.59	0.000	−0.109620
Square	3	0.01187	0.00396	1463.13	0.000
ϕ × ϕ	1	0.00587	0.00587	2168.39	0.000	−0.046182
Rd × Rd	1	0.00001	0.00001	2.39	0.153	−0.001532
Q × Q	1	0.00002	0.00002	6.78	0.026	−0.002582
Interaction	3	0.00352	0.00117	434.18	0.000
ϕ × Rd	1	0.00071	0.00071	262.73	0.000	−0.009425
ϕ × Q	1	0.00259	0.00259	958.29	0.000	−0.018000
Rd × Q	1	0.00022	0.00022	81.52	0.000	−0.005250
Error	10	0.00003	0.00000
Lack-of-fit	5	0.00003	0.00001
Pure error	5	0.00000	0.00000
Total	19	1.58382

Figure 9 is used to analyse the residuals and evaluate the model’s adequacy. Residual plots visually represent the disparities between the predicted and observed Nusselt numbers. The normal probability plot is utilized to assess the assumption of normality for the residuals. If the residuals adhere to a normal distribution, they should roughly align along a straight line. Deviations from this line might suggest non-normality in the residuals. The residual versus predicted plot examines whether the residuals display any systematic trends or heteroscedasticity (unequal variance) across the range of predicted Nusselt numbers. Ideally, the residuals should be evenly dispersed around zero without any noticeable patterns. The presence of non-random patterns or increasing/decreasing spread of residuals with predicted values could indicate a lack of fit or violation of model assumptions. The residuals versus actual values plot investigates whether there is any relationship between the residuals and the actual observed Nusselt numbers. Similar to the residuals versus predicted values plot, a random distribution of residuals around zero is desired. Figure 10 examines the impact of the interplay between particle concentration and thermal radiation on the Nusselt number response using (a) contour plots and (b) surface plots. A contour plot illustrates the Nusselt number on a two-dimensional plane, with particle concentration on one axis and thermal radiation on the other. The contour lines reveal the interaction effect. However, the combination of increased concentration and thermal radiation enhances the heat transportation rate. Conversely, a surface plot offers a three-dimensional depiction of the Nusselt number, incorporating both particle concentration and thermal radiation. The height of the surface represents the Nusselt number, while contour lines indicate regions with equal Nusselt numbers. The irregular surface or unevenly spaced contour lines imply a significant interaction effect.

Figure 9

Residual plots for Nusselt number.

Figure 10

Contour and surface plot for Nusselt number.

Figure 10(c) and (d) depict the relationship among the interaction term of particle concentration and heat source factor and the Nusselt number via contour plots (c) and surface plots (d). The contour plots involve lines that connect points with the same Nusselt number, allowing the observation of trends and patterns. The spacing among contour lines specifies the magnitude of Nusselt number changes, with steeper lines representing more significant profile changes. The surface plot presents the Nusselt number as a continuous interface, providing a detailed understanding of how particle concentration, heat source factor, and the Nusselt number are interconnected. High and low regions on the surface plot indicate areas with higher and lower Nusselt numbers, correspondingly. Figure 10(e) illustrates the contour plot portraying how the Nusselt number fluctuates with the interaction term, thermal radiation, and heat source parameter. The contour lines on the plot indicate different levels or values of the Nusselt number, with each line representing a constant value. In Figure 10(f), a surface plot is presented, which delivers a three-dimensional representation of the relationship among the interaction term, thermal radiation, heat source parameter, and the Nusselt number.

6 Sensitivity analysis

To effectively model and simulate the proposed design, incorporating sensitivity analysis is essential within the process. By employing sensitivity analysis, designers gain insight into how changes in input values affect the model, enabling them to pinpoint the parameters with the greatest influence and make well-informed decisions. Analysts can leverage sensitivity analysis as a valuable tool to assess the impact of specific factors on response variables. In the current study, the sensitivity of heat transport rate has been investigated under well-defined conditions. By utilizing sensitivity analysis, designers can identify the critical parameters that are likely to significantly affect the heat transport rate, empowering them to make robust and informed decisions. Here, the factors are particle concentration, thermal radiation, and the heat source that uses for the response of the Nusselt number. The partial derivative with respect to the effective factors such as ϕ , Rd , Q is characterized as

In Figure 11(a)–(c), we investigate the sensitivity of the Nusselt number to different levels of parameters Rd and Q while keeping parameter ϕ fixed at its medium value. In this analysis, the focus is on assessing the sensitivity of the rate of heat transfer concerning a specific parameter at the medium level of ϕ . By conducting calculations, it becomes apparent that the sensitivity towards parameter Rd exhibits a consistently positive trend. This observation is supported by examining the figures, which reveal that irrespective of the level of parameter Rd , the sensitivity towards Rd remains constant when ϕ is set at its medium level.

Figure 11

Sensitivity plot towards Nusselt number.

7 Conclusive remarks

• The study focuses on improving heat transfer efficiency in a hybrid nanofluid flow of Ag–MoS₄ with applied magnetism.

• The Yamada-Ota model employed considering the effects of viscous dissipation and heat source/sink influences.

• Through optimization and sensitivity assessment, the current study aimed to enhance the understanding of the key parameters affecting heat transfer rate and make informed decisions for optimal system performance.

• By utilizing the Yamada–Ota model, it is beneficial to simulate and analyse the heat transfer characteristics of the hybrid nanofluid.

• The inclusion of applied magnetism allows exploring the influence of external magnetic fields on heat transfer efficiency.

• The present findings revealed the importance of considering viscous dissipation and heat source/sink influences in accurately predicting heat transfer rates within the system.

• Furthermore, sensitivity analysis played a crucial role by systematically varying input parameters and analysing their impact on the heat transfer rate.

This information provided valuable insights to designers, enabling them to prioritize and focus on the parameters that have the greatest influence, ultimately leading to more effective decision-making.

8 Future scope

While this study contributes significant insights into improving heat transfer efficiency in the hybrid nanofluid flow of Ag–MoS₄, there are several areas that warrant further exploration.

• Investigating the influence of different volume fractions of Ag–MoS₄ nanoparticles on heat transfer efficiency could provide deeper insights into the optimal composition for enhanced performance.

• Exploring the effect of varying external magnetic field strengths and orientations on heat transfer characteristics would contribute to a more comprehensive understanding of the system’s behaviour.

• It would be beneficial to extend the study to different geometries and flow regimes, as real-world applications often involve complex systems. This would allow for a more comprehensive evaluation of the proposed design and its potential scalability and applicability in practical scenarios.

• Experimental validation of the model’s predictions would provide a valuable benchmark and ensure the accuracy of the simulations. Conducting experiments under controlled conditions and comparing the results with the model’s outputs would strengthen the reliability and applicability of the findings.