
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)13471-8
10.1016/j.heliyon.2024.e37440
e37440
Research Article
Sensitivity study on convective heat transfer in a driven cavity with star-shaped obstacle and hybrid nanofluid using response surface methodology
Ziaur R.M. ziaur.math.buet@gmail.com
a⁎
Azad A.K. b
Rahman M.M. a
a Department of Mathematics, Bangladesh University of Engineering and Technology, Dhaka, 1000, Bangladesh
b Department of Natural Sciences, Islamic University of Technology (IUT), Gazipur, 1704, Bangladesh
⁎ Corresponding author. ziaur.math.buet@gmail.com
05 9 2024
15 9 2024
05 9 2024
10 17 e3744012 12 2023
6 6 2024
3 9 2024
© 2024 The Authors
2024
https://creativecommons.org/licenses/by-nc/4.0/ This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
Sensitivity analysis is significant for understanding and measuring the impact of various parameters and input variables on heat transfer phenomena. The main objective of the current work is to examine the sensitivity of a numerical analysis of mixed convection in a lid-driven square cavity with a magnetic field. The cavity also contains a heated, star-shaped obstacle and is filled with a hybrid nanofluid. The sensitivity analysis was conducted employing the statistical response surface methodology (RSM), while the numerical simulations used the Galerkin weighted residual finite element approach to solve the governing PDEs. The study investigates the impacts of four dimensionless factors: Ri, Re, Ha, and ϕ. The numerical observation was made that there exists an upward trend between the average heat transfer rate with Ri, Re, and ϕ, while there exists a downward trend with Ha. Furthermore, the average heat transfer rate increases by almost half (49.54 %) when ϕ increases from 1 % to 10 % and decreases by 5.97 % when the Ha increases from 0 to 60. Finally, the statistical investigation of the current model and testing techniques imply that R2 values for the response function are high (98.72 %), suggesting that this model is appropriate for estimating Nu.

Keywords

Sensitivity analysis
Response surface methodology (RSM)
Hybrid nanofluids
Magnetohydrodynamics (MHD)
Lid-driven cavity
Star-shaped obstacle
==== Body
pmc Nomenclature

	Greek symbol	
B0	magnetic field	α	thermal diffusivity	
g	gravitational acceleration	β	thermal expansion coefficient	
Ha	Hartmann number	θ	temperature (non-dimensional)	
L	length of the enclosure	μ	dynamic viscosity	
Nu	average Nusselt number	υ	kinematic viscosity	
p	pressure	ψ	stream function	
P	dimensionless pressure	ϕ	volume fraction of hybrid nanoparticles	
Pr	Prandtl number	σ	electrical conductivity	
Re	Reynolds number		
Ri	Richardson number	Index	
T	Temperature	bf	base fluid	
u, v	velocity components	c	cold	
U, V	velocity component (non-dimensional)	h	hot	
x, y	x- and y-axes coordinate	hnp	hybrid nanoparticles	
X, Y	cartesian coordinate (non-dimensional)	hnf	hybrid nanofluid	

1 Introduction

Nowadays, there is a strong emphasis on magnetohydrodynamics (MHD) mixed convection, which involves the combined influence of natural and forced convection, in the field of heat transport, which has a diverse range of applications, including solar collectors, electronic devices, heat exchangers, building ventilation, and cooling reactors, among others. In heat transfer, the square geometry with an inner star-shaped obstacle can be utilized in several scenarios to control heat transfer processes. The square geometry with an inner star-shaped obstruction is a flexible solution for heat transfer, having uses in heat exchangers to produce turbulence and improve fluid mixing. It also enhances heat conduction in thermal management applications such as electrical equipment by functioning as a thermal bridge. It helps to create effective cooling solutions for electronic packaging by increasing heat dissipation. Furthermore, its shape optimizes heat transfer rates in solar thermal systems, industrial processes, insulating materials, and thermal energy storage, indicating its broad use across industries. Furthermore, the various geometries of the enclosure, such as triangular, rectangular, square, circular, trapezoidal, and others filled with nanofluid, have a significant influence on convection. Numerous researchers have studied the various types of cavities for the enhancement of heat transmission rate [[1], [2], [3], [4]].

In the last few years, a novel technique for improving thermal conductivity and heat transfer rate has been developed by mixing more than one nanoparticle (<100 nm) of metal, metal oxide, carbon materials, etc., with base fluids such as kerosene, water, engine oil, ethylene glycol, etc. Such type of fluids is known as hybrid nanofluids, which have excellent thermal conductivity properties and can handle the challenge of increasing heat transmission rates. Choi et al. [5] were the first to assert that nanofluid enhances heat transfer rate over pure fluid. He presented his findings at the Argonne National Laboratory in 1995. Mehryan et al. [6] evaluated the influence of Cu-Al2O3/water hybrid nanofluid and Al2O3/water nanofluid on the phenomenon of mixed convection within a square cavity induced by a heated oscillating cylinder. It was revealed that the heat transmission rate through natural convection was significant when using a pure Al2O3/water nanofluid in comparison to a hybrid Cu-Al2O3/water nanofluid. Hirpho et al. [7] employed the Galerkin finite element technique to examine the mixed convection hybrid nanofluid flow in a trapezoidal cavity containing an adiabatic circular cylinder. Their findings observed that as the Ri grew, there was a corresponding rise in the penetration of fluid flow inside the cavity and there was also a decrease in the overall heat transport rate when Ri and volume percentage of hybrid nanoparticles increased. A simulation for the three-dimensional MHD hybrid nanofluid flow inside a 3D cubic enclosure filled with porous material was carried out by Jiang et al. [8]. Two revolving cylinders and an above-wavy wall were presumptive features of the flow area. They discovered that more superb cylinder angular speed values improved heat transport and Nu. Jakeer et al. [9] experimented with the flow of a square cavity containing a magneto Cu-Al2O3/water hybrid nanofluid, including an inner heated square obstruction. It was discovered that, in comparison to the other nanofluids, hybrid nanofluids could transmit heat more quickly. Several studies using hybrid nanofluid are also mentioned [[10], [11], [12], [13], [14], [15]].

In the last few years, mixed convective heat transport inside a cavity with an inner obstruction has been a popular research field. Several researchers have studied inner obstruction inside the cavity. Esfe et al. [16] examined the heat transport within a cavity that was completely filled by a nanofluid, with a focus on mixed convection and there was a rectangular obstacle within the cavity that was heated. They researched that there was a positive connection between the overall Nu and a decrease in Ri across all ranges of solid volume percentage. The finite element approach was employed by Selimefendigil [17] to investigate numerical analysis inside a lid-driven hollow that was fulfilled by a nanofluid consisting of SWCNTs and MWCNTs dispersed in water and including an inner elliptic obstacle. It was found that, the heat transport rate of a SWCNT-water nanofluid with a solid volume fraction of 0.06 exhibited a substantial increase of approximately 120.20 % in comparison to pure water. Karbasifar et al. [18] conducted a study on the mixed convection phenomenon occurring within a lid-driven cavity filled with nanofluid that also contained a centrally positioned heated elliptical cylinder. The findings indicated that increasing the Ri leads to a decrease in average Nu, given a fixed volume fraction due to a raise in Ri reduced the nanofluid velocity. Khan et al. [19] designed a study on thermal management and heat augmentation inside a square cavity, which was then filled with hybrid nanofluids. The experimental setup also included the insertion of a Y-shaped obstruction into the cavity. It was vital to acknowledge that the presence of the Y-shaped obstruction significantly augmented the pace of heat transmission inside the hollow. A double lid-driven square cavity with a solid inner body was studied by Alsabery et al. [20] for conjugate mixed convection. In situations where both the Reynolds and Richardson numbers are high, it was noticed that a large solid body can improve heat transmission. Several studies on convective heat transfer inside a cavity with an inner obstruction are also mentioned [[21], [22], [23], [24]].

Recently, the impacts of magnetic field on combined convection inside a cavity with nanofluid had received outstanding attention in several studies. Several numerical and analytical research have been conducted to better understand the mixed convection phenomenon. In their research, Munawar et al. [25] addressed the phenomenon of MHD mixed convective heat transport in an angled enclosure, including a circular centre heater using a hybrid nanofluid. It was fascinating to observe that the rate of heat transmission was boosted as the Ha increased within the range of 0–10. However, above a Ha of 10, the heat transmission rate experienced a reduction. The numerical investigation conducted by Herouz et al. [26] studied the phenomenon of MHD mixed convection of nano-encapsulated phase change material (NEPCM) inside a hexagonal porous chamber that comes into contact with a square-shaped obstruction. The experimental findings indicated that the heated object exhibited a circular flow pattern when subjected to opposing movements of the two adjacent walls. Rahman et al. [27] looked into the mixed-convective heat transfer inside a square enclosure under the influence of a magnetic field. The enclosure was arranged so that a solid square block, capable of producing heat, was placed at its core. It was determined that the flow behaviour and heat transfer characteristics inside the cavity had a significant dependence on the intensity of the magnetic field. Chamkha et al. [28] examined the impact of a spinning cone on the three-dimensional MHD mixed convection of CNT-water nanofluid in a trapezoidal cavity with porous boundaries. It was observed that using the cone's aspect ratio proved to be a very effective method for enhancing heat transport. This was shown by the significant increase of 95 % in the overall heat transfer rate.

According to the aforementioned literature, it is observed that the researchers have been particularly interested in MHD mixed convection within a lid-driven cavity consisting of a hybrid nanofluid with different types of inner obstructions since it has several applications in various engineering sectors. In recent times, there has been a growing interest in research on sensitivity utilizing response surface methodology (RSM) in heat transfer issues. Numerous studies have been conducted using RSM to examine the sensitivity of convective heat transfer. Shirvan et al. [29] studied the sensitivity using RSM on a mixed convection heat exchanger that was fully filled with nanofluid. A sensitivity analysis of heat transmission within a flat exchanger tube has been conducted by Zheng et al. [30]. Hossain et al. [31] performed sensitivity analysis on MHD mixed convective heat transport in an enclosure using response surface methodology (RSM). Thriveni and Mahanthesh [32] investigated the sensitivity of convective heat transfer between two cylinders in a hybrid nanomaterial employing the statistical response surface methodology (RSM). Recently, Islam et al. [33] researched the sensitivity of MHD mixed convection in the hexagonal heat exchanger using RSM.

Based on the aforementioned literature review, mixed convection analysis with magnetic effect has been studied for the last few years. However, to the best of the authors’ knowledge, no previous studies have been conducted on sensitivity analysis of mixed convection under the influence of a magnetic field inside a double lid-driven square cavity containing a star-shaped obstruction filled with a hybrid nanofluid. As a result, a sensitivity study on numerical assessment for mixed convection in a double lid-driven square cavity with a star-shaped obstacle is performed in this research. The cavity has been filled with a hybrid nanofluid in the presence of a magnetic effect.

The use of a star-shaped obstacle for heat transfer within a square cavity has several benefits: the irregular edges of the star shape induce enhanced turbulence, which facilitates better fluid mixing and improves heat transfer rates; and the increased surface area of the star shape increases heat exchange efficiency by encouraging greater contact between the fluid and the obstacle.

2 Problem formulation

2.1 Physical model description

A two-dimensional physical model, including its boundary conditions and the specified coordinate system filled with the hybrid nanofluid, is shown in Fig. 1. The hybrid nanofluid is composed by of Fe3O4 (50 %) and MWCNT (50 %) nanoparticles with a base fluid (kerosene). L is the height and length of the cavity. A hollow star-shaped block with length L3 is positioned in the centre of the cavity; centre of the star-shaped block is (L2,L2). The hollow star remains at a hot temperature (Th). At the same time, the vertical left and right boundaries stay at a cold temperature (Tc), both of which are moving with constant velocity in opposite directions, and both the horizontal bottom and top boundaries are adiabatic (where, Th > Tc). The current hybrid nanofluid flow model is based on the following assumptions:⁃ The flow is considered as Newtonian steady laminar and incompressible mixed convection flow.

⁃ The nanoparticles (MWCNT and Fe3O4) and base fluid (kerosene) molecules are all in thermal equilibrium, homogenous in size, shape, and no slip.

⁃ There won't be any chemical reaction between the base fluid and the nanoparticles, leading to a thermal equilibrium.

⁃ The fluid's Joule heating impact, nanofluid's viscous dissipation effect, and radiation effect are assumed to be negligible.

⁃ The hybrid nanofluid is considered using the Boussinesq approximation.

⁃ The gravitational effect is considered in the downward Y-direction and a uniform magnetic field (B0) is taken horizontally.

⁃ To simplify and reduce complexity ion slip, and the Hall effect is neglected, as the governing equations become simpler and more tractable, focusing on the primary mechanisms of heat and fluid flow. In addition, these effects have a minimal impact on the overall results [[34], [35], [36], [37]].

Fig. 1 Architectural design with boundary conditions of the problem.

Fig. 1

2.2 Mathematical model

2.2.1 Dimensional governing equations

The dimensional governing equations for two-dimensional steady mixed convective hybrid nanofluids heat transfer models in the presence of a magnetic field may be described as the following continuity equation, momentum equations, and energy equation.(1) ∂u∂x+∂v∂y=0

(2) u∂u∂x+v∂u∂y=−1ρhnf∂p∂x+υhnf(∂2u∂x2+∂2u∂y2)

(3) u∂v∂x+v∂v∂y=−1ρhnf∂p∂y+υhnf(∂2v∂x2+∂2v∂y2)+(ρβ)hnfρhnfg(T−Tc)−σhnfB02ρhnfv

(4) u∂T∂x+v∂T∂y=αhnf(∂2T∂x2+∂2T∂y2)

where, u and v stand for x and y directional velocities, respectively; p, T and Ts stand for fluid pressure, temperature, and solid temperature respectively; β and B0 stand for the coefficient of thermal expansion and magnetic field strength, respectively; and ρ, υ, σ, and α stand for density, kinematic viscosity, electrical conductivity, and thermal diffusivity, respectively. The following are the boundary conditions for this specified problem:(5) Atleftverticalboundary:u=0,v=v0,T=TcAtrightverticalboundary:u=0,v=−v0,T=TcAthorizontalboundaries:u=0,v=0,∂T∂n=0Athollowstar−shappedblock:u=0,v=0,T=Th}

2.2.2 Dimensionless governing equations

To reduce computational expenses and include dimensionless factors, an appropriate scaling approach was used to convert (1), (2), (3), (4) associated with boundary condition (5) into a dimensionless formulation. The dimensionless factors are(6) X=xL,Y=yL,U=uv0,V=vv0,P=pρbfv02,θ=T−TcTh−Tc,Pr=υbfαbf,Gr=gβbf(Th−Tc)L3υbf2,Re=v0Lυbf,Ri=GrRe2,andHa=B0Lσbfρbfυbf}

After applying equation (6), the dimensionless governing equations, namely continuity, momentum, and energy equations are as follows:(7) ∂U∂X+∂V∂Y=0

(8) U∂U∂X+V∂U∂Y=−ρbfρhnf∂P∂X+υhnfυbf1Re(∂2U∂X2+∂2U∂Y2)

(9) U∂V∂X+V∂V∂Y=−ρbfρhnf∂P∂Y+υhnfυbf1Re(∂2V∂X2+∂2V∂Y2)+(ρβ)hnfρhnfβbfRiθ−ρbfρhnfσhnfσbfHa2ReV

(10) U∂θ∂X+V∂θ∂Y=αhnfαbf1RePr(∂2θ∂X2+∂2θ∂Y2)

Then, the dimensionless boundary conditions for this specified problem are(11) Atleftverticalboundary:U=0,V=1,θ=0Atrightverticalboundary:U=0,V=−1,θ=0Athorizontalboundaries:U=0,V=0,∂θ∂N=0Athollowstar−shappedblock:U=0,V=0,θ=1}

2.2.3 Thermo-physical properties and hybrid nanofluids models

The thermophysical properties of both base fluid and hybrid nanoparticles are shown in Table 1 [38,39] and the models of the hybrid nanofluid that are used in (8), (9), (10) will be employed in accordance with the subsequent relationship as shown in Table 2 [40,41].Table 1 Thermophysical properties of base fluid and nanoparticles [38,39].

Table 1Physical properties	Base fluid	Nanoparticles	
Kerosene	Fe3O4	MWCNT	
ρ[kgm−3]	780	5810	2100	
Cp[Jkg−1K−1]	2090	670	711	
k[Wm−1K−1]	0.149	6	3000	
β[K−1]	99 × 10−5	1.3 × 10−5	4.2 × 10−5	
σ[Sm−1]	6 × 10−10	2.5 × 10−4	1.9 × 10−4	
μ[kgm−1s−1]	0.00164	–	–	
Pr	23.004	–	–	

Table 2 Thermophysical properties of hybrid nanofluids [40,41].

Table 2Symbol	Value	Description	
ϕ	ϕFe3O4+ϕMWCNT	volume fraction	
ρhnf	ϕFe3O4ρFe3O4+ϕMWCNTρMWCNT+(1−ϕ)ρbf	effective density	
βhnf	ϕFe3O4βFe3O4+ϕMWCNTβMWCNT+(1−ϕ)βbf	thermal expansion coefficient	
(ρβ)hnf	ϕFe3O4(ρβ)Fe3O4+ϕMWCNT(ρβ)MWCNT+(1−ϕ)(ρβ)bf	buoyancy coefficient	
(ρCp)hnf	ϕFeO4(ρCp)FeO4+ϕMWCNT(ρCp)MWCNT+(1−ϕ)(ρCp)bf	heat capacitance	
μhnf	μbf(123ϕ2+7.3ϕ+1)	dynamic viscosity	
αhnf	khnf/(ρCp)hnf	thermal diffusivity	
khnf	kbf(khnp+2kbf)−2ϕ(kbf−khnp)(khnp+2kbf)+ϕ(kbf−khnp)	thermal conductivity	
khnp	ϕFe3O4kFe3O4+ϕMWCNTkMWCNTϕFe3O4+ϕMWCNT	thermal conductivity of nanoparticles	
σhnf	σbf(σhnp+2σbf)−2ϕ(σbf−σhnp)(σhnp+2σbf)+ϕ(σbf−σhnp)	electric conductivity	
σhnp	ϕFe3O4σFe3O4+ϕMWCNTσMWCNTϕFe3O4+ϕMWCNT	electric conductivity of nanoparticles	

2.2.4 Average Nusselt number and average fluid temperature

The average heat transfer rate of the heated surface is expressed by [42].(12) Nu=−(khnfkbf)1Ls∫0Ls∂θ∂NdS

where, Ls and N represent the length of the heated surface and the unit normal vector on the surface of the hollow star-shaped block, respectively. And the average fluid temperature of the domain is calculated as formula cited from Ref. [42]To construct the visualization of streamlines, the values of stream function ψ(X,Y) has been used which is defined as, in non-dimensional form [43] U=∂ψ∂Y,V=−∂ψ∂X. The positive value of ψ(X,Y) indicates the counterclockwise rotation of the flow, whereas the negative values of ψ(X,Y) indicate the clockwise rotation of the flow of the hybrid nanofluid.

3 Numerical modelling

3.1 Computational technique

The Galerkin weighted residual, based on the finite element technique, was employed to solve the dimensionless governing (7), (8), (9), (10) with the associated initial dimensionless boundary constraints (11). The computational domain is made up of multiple irregular triangular components and the interpolation procedure is used to estimate the dependent variables for each component. The basic governing equation has generated a collection of algebraic equations that reduce the continuum region as discrete triangular patches. The MATLAB implementation of the Newton-Raphson iteration method is employed to generate a set of global nonlinear algebraic equations. These equations may be effectively employed to solve the governing equations. The iteration approach is then applied to solving the derived algebraic equations until the convergence condition of |Γp+1−Γp|<10−5 is fulfilled, p+1 and p indicate two consecutive repeats, whereas Γ(U,V,θ) represents the values obtained during the iteration. The details of the numerical method is available in the published thesis of a co-author of this work, Azad [44]

3.2 Grid independency test

A grid independence test is conducted to ensure the validity of the numerical findings in this research. The selected grid refinement assessment is conducted by focusing on the average Nusselt number for the default values of Ri = 1, Re = 500, Ha = 20, ϕ = 0.03, and Pr = 23.004. Table 3 and Fig. 2 depict the grid sensitivity test for various elements and nodes of the domain on average Nusselt number. The relative error significantly decreases as the element size increases, indicating that satisfactory accuracy in the findings is achieved with element sizes of 34,892 and 44,198. Additionally, it is determined that the Nu for the elements in 34,892 is quite close to the values found for the other higher elements. As a result, from the grid sensitivity test table, the optimum grid is used considering all characteristics is 34892 elements and 19002 nodes for this study. The configuration of the used computer is “Intel (R) Core i5 – 8250U 1.60 GHz, 12 GB RAM.”Table 3 Grid sensitivity test for various number of elements of the domain.

Table 3Number of Elements	Number of Nodes	Nu	Time (s)	Relative Error (%)	
1480	922	4.0518	5	–	
2624	1600	4.3287	6	6.834	
3750	2213	4.4726	7	3.324	
5890	3373	4.5837	8	2.484	
14300	7980	4.8899	10	2.316	
34892	19002	5.0721	20	1.752	
44198	23655	5.0722	27	0.0021	

Fig. 2 Grid sensitivity test for various number of elements.

Fig. 2

3.3 Code verification

To establish the validity of the current research, the model was compared with Khanafer and Aithal [45] in terms of streamlines and isotherms line at default values of Ri = 1, Re = 100, and Pr = 0.7, when there was no cylinder in the cavity. The present results revealed that the streamlines and isothermal lines exhibited similarities to the findings reported in previous results, which is shown in Fig. 3. Also, the model was evaluated in comparison with the work of Moallemi and Jhang [46] in forms of streamline and isothermal line characteristics. This evaluation was conducted using the default values of Ri = 1, Re = 1000, and Pr = 0.1, displayed in Fig. 4. The current findings demonstrate outstanding consistency with the previous results.Fig. 3 Comparison present result with previously published results of Khanafer and Aithal [45].

Fig. 3

Fig. 4 Comparison present result with previously published results of Moallemi and Jhang [46].

Fig. 4

4 Results and discussion

The current study investigated the impact of various dimensionless parameters, specifically the Reynolds numbers (100 ≤ Re ≤ 1000), Richardson numbers (0.01 ≤ Ri ≤ 10), Hartmann numbers (0 ≤ Ha ≤ 60), and hybrid nanoparticles volume fractions (0.01 ≤ ϕ ≤ 0.10), on the phenomenon of mixed convective heat transfer in a square cavity with double lid lid-driven motion. For Ri, using this range, we can learn about the behavior of forced convection (0 <Ri < 1), natural convection (Ri > 1), and mixed convection (Ri = 1). Since we have studied the behavior of laminar flow, that's why we used a lower range of Re. The magnetic parameter (Ha) controlled the flow movement and heat transfer rate inside a cavity. A higher range of the Ha diminishes the heat transfer rate. That's why we kept the lower range of values. Although the experimental works exist for ϕ ≤ 4 %, however, in this study, we used up to 10 % volume fraction for numerical simulation. The numerical results are presented in several visual representations, including temperature distribution (isotherms), flow patterns (streamlines), average Nusselt numbers, and average temperature inside the cavity using the default parameters of Ri = 1, Re = 500, Ha = 20, ϕ = 0.03, and Pr = 23.004 in this study.

4.1 Effect of Reynolds number

The impact of Re on flow movement and temperature distribution inside the cavity, using a range of Re and varying values of the prementioned parameters has been examined. Fig. 5 shows the arrangement of the streamline plots and isothermal plots, which are organized from bottom to top, corresponding to the values of Re = 100, 200, 500, and 1000, respectively.Fig. 5 Effect of Re on (a) streamlines and (b) isotherms at default values of Ri = 1, Ha = 20, ϕ = 0.03, and Pr = 23.004.

Fig. 5

In Fig. 5 (a), for Re = 100, it is seen that two prominent vortices are being formed, one located in the upper left region and the other in the lower right region. Additionally, two minor vortices have also emerged, one positioned at the top and the other at the bottom. The generation of major and minor vortices may be attributed to the impact of a centrally positioned, heat emitting, star-shaped obstruction, as well as two vertically oriented walls driven by a lid. As increasing of Re = 200, the flow strength is also increasing for lower right vortex, while the flow strength for upper left vortex and two minor vortices experiences the same magnitude. Again, for Re = 500 and 1000, the flow strength is increasing for the lower right vortex as well as two minor vortices, while the flow strength for the upper left vortex is decreasing. So, as Re increases, the flow strength of the upper left vortex and two minor vortices exhibits an upward trend, whereas the flow strength of the bottom vortex experiences a downward trend. Hence, at lower Re = 100 and 200, the streamlines exhibit a consistent behavior, however, the chaotic behavior of the streamlines becomes more pronounced as Re increases. Increased Re improves convective heat transfer owing to increased velocity gradients. This complicates the temperature distribution within the cavity. The isothermal lines grow less smooth and more complex. At lower Re, heat transport occurs predominantly by conduction, resulting in more homogeneous and parallel isothermal lines. The isothermal lines may wrap effectively around the obstruction. From Fig. 5 (b), the analysis of isothermal contours reveals that the isothermal lines exhibit little skewness at lower Re = 100 and 200, but higher Re = 500 and 1000 results in more pronounced skewness of the isothermal lines. The skewness of the isothermal lines rises as Re rises, owing to the corresponding increase in fluid velocity. Furthermore, the isothermal lines are more uniform for lower Re compare to higher Re. It may be inferred that higher Re has a greater impact on isothermal lines in comparison to lower Re. The aforementioned reasoning leads to the conclusion that the Re has a significant influence on the flow movement and temperature distribution within the cavity.

Now, Fig. 6 (a) displays the graphical representation of the average heat transfer rate for various Re and ϕ inside the cavity. The analysis of the 2D line graph reveals a positive correlation between the heat transfer rate and both Re and ϕ. It can be attributed to enhanced convective heat transfer due to higher fluid velocities (high Re). This leads to more effective heat transmission from heated surfaces to cold surfaces, which raises the average heat transfer rate. The observation revealed that the minimum heat transport rate occurs at Re = 100 and ϕ = 0.01, while the maximum heat transport rate is observed at Re = 1000 and ϕ = 0.10. Fig. 6 (b) also depicts the average temperature, which exhibits decreasing behaviour with the increasing of Re as well as ϕ.Fig. 6 (a) average Nusselt number and (b) average fluid temperature against Re as well as ϕ at default values of Ri = 1, Ha = 20, and Pr = 23.004.

Fig. 6

4.2 Effect of Richardson number

In Fig. 7 (a), the streamlines have formed two clockwise vortices with equal flow strength around the left as well as right sides of the star-shaped obstruction for Ri = 0.01. This observation suggests that forced convection is the prevailing mechanism in this scenario. The presence of two vortices is seen due to the influence of centrally positioned, warmed star-shaped obstruction and two vertically driven walls. However, when Ri = 1, indicating a state of mixed convection where both free and forced convection are equally dominated and two minor counterclockwise holes with equal flow strength are seen to develop on both the top and bottom sides of the cavity. The increase of Ri = 1 has resulted in an elevation of flow strength in the right vortex, whereas a reduction in flow strength has been seen in the left vortex. The scene exhibits aspects of natural convection, as shown by the guidance of flow by both lids and hot star obstacles. Nevertheless, when the Ri exceeds 1 (specifically, Ri = 5), the primary mode of heat transport transitions to natural convection within the cavity, resulting in amplified flow strength. Three major holes are formed when Ri = 5, where the right hole have a clockwise direction while the left two upper and bottom holes has a counterclockwise direction. As good a similar incidence happens for Ri = 10, however, it has a stronger flow strength than Ri = 5. This observation suggests a significant impact of natural convection. Fig. 7 (b) illustrates that the isothermal plots for Ri = 0.01 exhibit a dominance of forced convection in the flow. The isothermal graphs are seen to be well-aligned with the two opposing lid walls and centrally located hot star. Additionally, it is noted that the isothermal plots exhibit a smooth and evenly dispersed pattern, with no additional forces being apparent except from the movement of the two lids. In the case when Ri = 1, denoting pure mixed convection, the isothermal lines are uniformly spaced from the star obstacle owing to the lack of significant natural convection effects. For a given value Ri = 5, the isothermal plots exhibit noticeable compression, suggesting the presence of induced natural convection forces. Now for Ri = 10, which represents a very dominant natural convection phenomenon, it can be seen that the isothermal plots are being significantly compressed and attempting to expand from the centre. This behaviour is well-documented and is characteristic of natural convection. From the above analysis of the isothermal plots, it is evident that a rise in Ri leads to an augmentation of natural convection. Based on the above discussion, it can be inferred that the Ri significantly influences the flow movement and temperature distribution within the cavity.Fig. 7 Effect of Ri on (a) streamlines and (b) isotherms at default values of Re = 500, Ha = 20, ϕ = 0.03, and Pr = 23.004.

Fig. 7

To summarize, the Ri has a considerable effect on the flow and temperature distribution within a square cavity with a star-shaped hollow obstruction. A low Ri suggests forced convection supremacy, with more regular streamlines and isothermal lines, whereas a high Ri shows natural convection dominance, with more complicated and stratified patterns. The star-shaped obstruction further alters these patterns, increasing complexity and perhaps aiding heat transfer.

The positive correlation between the average Nu and both Ri as well as ϕ inside the cavity shown in Fig. 8 (a) is owning to the increasing dominance of buoyancy-driven natural convection as Ri increases. Enhanced natural convection causes stronger circulatory currents, more efficient fluid mixing, thinner thermal boundary layers, and better interaction with the star-shaped obstruction, all of which contribute to a faster heat transfer rate within the cavity. The minimum average Nu is observed at Ri = 0.01 and ϕ = 0.01, while the greatest average Nu is observed at Ri = 10 and ϕ = 0.10. The average temperature for various Ri and ϕ is also shown by a line graph, which is depicted in Fig. 8 (b). The average temperature increases with Ri, and for lower ϕ = 0.01 and 0.03, it exhibits more curvy behaviour compared to the cases of higher ϕ = 0.05 and 0.10.Fig. 8 (a) average Nusselt number and (b) average fluid temperature against Ri as well as ϕ at default values of Re = 500, Ha = 20, and Pr = 23.004.

Fig. 8

4.3 Effect of magnetic field

The insertion of a magnetic field into a square cavity with a star-shaped hollow obstruction has an effect on the streamlines by damping chaotic fluctuations, altering vortex formations, and changing boundary layer features. The magnetic field creates a Lorentz force that opposes fluid velocity, resulting in smoother, more laminar-like flow patterns and, perhaps, enhanced mixing at certain areas due to the complicated interplay with the obstacle. In Fig. 9 (a), when Ha = 0 (no magnetic effect), the streamlines exhibit the formation of two clockwise holes in the upper left and lower right area, as well as two minor counterclockwise holes in the top and bottom regions inside the cavity. Furthermore, it should be noted that the flow strength of the streamlines reaches its maximum value at Ha = 0. The flow circulation experiences a drop in flow intensity as the rise in Ha = 20. Consequently, the streamlines exhibit the formation of two clockwise and two counterclockwise holes. The hybrid nanofluid's motion inside the cavity experiences a resistive force known as the Lorentz force, which arises from the magnetic effect and has the tendency to decrease the fluid's velocity. The aforementioned procedure is consistently ongoing for Ha = 40. At a Ha = 40, the presence of two minor counterclockwise holes vanishes, resulting in a reduced flow strength of the major holes in comparison to a value of Ha = 20. A comparable occurrence is seen at Ha = 60, but it has less flow strength compared to Ha = 40, resulting in increased resistance to fluid transport. At smaller levels of Ha, the flow exhibits a uniform shape. However, as the Ha grows, the flow becomes disrupted, resulting in a change in its shape. The flow configuration has been squeezed as the Ha rises. As a consequence, the flow intensity experiences a progressive drop as the Ha grows due to the interference of the magnetic field with the flow pattern. For isothermal lines, the magnetic field inhibits convective mixing, resulting in more stratified temperature distributions and steeper thermal gradients near the heated star obstacle. In Fig. 9 (b), the isotherm lines serve as indicators of the heat transport occurring within the cavity. The isothermal lines around a hot star-shaped obstruction represent the maximum temperature, which progressively decreases as it approaches the colder walls. The isotherm lines that are closer together at Ha = 0 indicate higher thermal conductivity and an accelerated rate of heat transmission. The distortion of isotherm lines is greater when Ha = 0, suggesting the absence of magnetic force. When Ha increases from 0 to 20, it is observed that the isotherm lines exhibit less distortion. The non-uniform arrangement of isotherm lines indicates that there are variations in temperature. Furthermore, the isotherm contours exhibit less distortion at Ha = 40. In fact, at Ha = 60, the isotherms are either little distorted or no distorted (i.e., uniform isotherms), so, indicating a more significant influence of convective flow. As a result, the convective type of heat transport eventually transitions to conduction mode, leading to the isotherm lines becoming more regular in shape in the vertical direction. The evenly spaced isotherms indicate an identical distribution of temperature throughout the cavity. It can be inferred that the Ha enormous influences the flow movement and temperature distribution within the cavity.The graphical representation of the average heat transmission rate for various Ha as well as Ri inside the cavity is shown in Fig. 10 (a). The negative correlation between the Nu and Ha is primarily due to the magnetic damping impact caused by the Lorentz force. This effect minimises turbulence, inhibits fluid motion and mixing, thickens boundary layers, and flattens velocity profiles, all of which lead to a reduction in convective heat transfer efficacy. The minimum heat transmission rate occurs at Ri = 0.01 and Ha = 60, while the greatest heat trans rate occurs at Ri = 10 and Ha = 0. In Fig. 10 (b), the average temperature is also shown by a 2D line graph. It exhibits the average temperature rises with Ha and Ri.Fig. 9 Effect of Ha on (a) streamlines and (b) isotherms at default values of Ri = 1, Re = 500, ϕ = 0.03, and Pr = 23.004.

Fig. 9

Fig. 10 (a) average Nusselt number and (b) average fluid temperature against Ha as well as Ri at default values of Re = 500, ϕ = 0.03, and Pr = 23.004.

Fig. 10

4.4 Effect of hybrid nanoparticles volume fractions

The distribution of streamlines is depicted in Fig. 11 (a), showcasing the influences of ϕ. The dispersion of hybrid nanoparticles (consisting of 50 % Fe3O4 and 50 % MWCNT) mixed with a base fluid (kerosene) encounters the resistance within the fluid domain. The addition of hybrid nanoparticles enhances the inertia force of the fluid by augmenting the overall mass of the fluid within the cavity. As a result of the growing inertia forces, there is a minor deceleration in the fluid flow. An additional factor to consider is that the introduction of a greater quantity of nanoparticles results in an increase in the viscosity of the mixture. Furthermore, it is worth noting that the interactions between particles of this mixture are negligible. As the volume fraction increases, the fluid experiences an increase in weight, resulting in a decrease in rotational motion. The efficacy of lowering rotation is shown when ϕ reaches a critical threshold. Consequently, the streamline contours indicate that the introduction of hybrid nanoparticles with varying volume fractions serves to decrease the fluid flow. The streamlines exhibit the presence of two clockwise upper left and lower right vortices, as well as two minor counterclockwise vortices at the top and bottom, across all values of ϕ. When the value of ϕ is low, then the flow pattern of streamlines demonstrates chaotic behavior. However, when the ϕ values grow, there is a noticeable enlargement of the left upper hole and two small holes, leading to an overall increase in flow strength. The impact of ϕ on isothermal lines has been presented in Fig. 11 (b). The isothermal lines around a hot star-shaped obstruction represent the highest attainable temperature, which progressively diminishes as it nears the colder boundaries. Nevertheless, when the value of ϕ rises, the isothermal lines exhibit a gradual displacement towards the cold wall. Consequently, the overall heat transport rate rises with ϕ. The isotherm lines become more regular for more ϕ compared to the less ϕ in shape in the vertical direction. The uniformly spaced isotherms suggest a uniform distribution of temperature within the cavity. As a result, ϕ significantly influences the flow movement and temperature distribution inside the cavity.Fig. 11 Effect of ϕ on (a) streamlines and (b) isotherms at default values of Ri = 1, Re = 500, Ha = 20, and Pr = 23.004.

Fig. 11

In summary, increasing the solid volume fraction of hybrid nanoparticles in a fluid within a square cavity with a star-shaped hollow obstruction causes smoother and more stable streamlines due to higher viscosity and decreased turbulence. Concurrently, improved thermal conductivity results in more uniform and efficient heat distribution, as indicated by smoother and evenly spaced isothermal lines. This combination impact improves the thermal performance and stability of the flow within the cavity.

The graphical representation of the average heat transmission rate for various ϕ and Ha inside the cavity is displayed in Fig. 12 (a). The positive relationship between Nu and ϕ is principally owing to the nanoparticles’ increased thermal conductivity, heat dispersion, convective currents, and specific heat capacity. These characteristics work together to improve the overall heat transfer rate within the cavity, making nanofluids with greater solid volume fractions particularly useful for applications that need efficient thermal management. The average temperature inside the cavity is also depicted in Fig. 12 (b), which exhibits the average temperature rises with ϕ as well as Ha. For Ha = 0 and 20, the average temperature in the cavity increases significantly, while for Ha = 40 and 60, it increases very slightly.Fig. 12 (a) average Nusselt number and (b) average fluid temperature against ϕ as well as Ha at default values of Ri = 1, Re = 500, and Pr = 23.004.

Fig. 12

4.5 Response surface methodology (RSM)

The Response Surface Methodology (RSM) is a very effective approach for simultaneously describing several variables concurrently with little resource allocation, using quantitative data and a well-designed experimental setup. The technique is particularly useful when the response variables are influenced by diverse factors and variables. Furthermore, the analysis included an examination of the interrelation between the aforementioned parameters. Although there exists a range of different RSM models, the 2nd order RSM model which incorporates all linear, square, and interacting variables is often deemed satisfactory for the response estimation [47]. The following formula defines the quadratic polynomial model:(13) y=p0+∑i=14pixi+∑i=14piixi2+∑i=14∑j=14pijxixj

where, y is the response function (output); p0 is the intercept terms; pi is the linear regression coefficient of ith factor; pii is the quadratic regression coefficient of ith factor; pij is the interaction term of ith and jth factors; and xi,xj are the design parameters.

In this particular scenario, the factors Richardson number (Ri), Reynolds number (Re), Hartmann number (Ha), and hybrid nanoparticles volume fraction (ϕ) are employed as independent variables (input factors), while dependent response function (y) is represented by the average Nusselt number (Nu). The objective is to maximize the response of the dependent variable (y) in order to establish a meaningful relationship with the input factors and the response function. In order to accommodate the second-order model, the implementation of a centralized composite design (CCD) was developed in 1992, and was first proposed by Box and Wilson [48]. Recently, the category of design that is most often popular for the purpose of second-order model fitting is quite prevalent. The visual representation of the four factors central composite design (CCD) model for this issue is shown in Fig. 13. The ranges of the input factors, namely, Ri, Re, Ha, and ϕ, are as follows: 0.01 ≤ Ri ≤ 10, 100 ≤ Re ≤ 100, 0 ≤ Ha ≤ 60, and 0.01 ≤ ϕ ≤ 0.10, respectively. The parameters under consideration exhibit three distinct degrees of selection, namely −1, 0, and +1, denoting lowest, medium, and highest levels, respectively. In the framework of CCD, the total number of runs may be calculated by, 2P + 2P + Q. Here, P and Q represent the quantity of key parameters and the number of center points, respectively. Assuming that there are four independent input factors (P = 4) and seven center points of design (Q = 7), the selected design has a total of 31 experimental runs. The design further has 16 cubic points and 8 axial points.Fig. 13 Four factors central composite design (CCD) design for current problem.

Fig. 13

Table 4 highlights a comprehensive overview of the codded (categorized) levels of input variables for CCD. Moreover, Table 5 presents the simulation configurations for both codded and actual data, using the CCD. The statistical evaluation of quadratic polynomial correlation equations, together with the estimation of the coefficients and the assessments of the factors’ impact on variables (represented as codded values), is conducted using analytical software. The adequacy of fit of the experimental results was evaluated by using a designated coefficient of determination and modifying the available resources. This assessment has been made by comparing the obtained p-value with a significant level of 95 %, leading to either the acceptance or rejection of the fitting quality.Table 4 Coded values and design variables for CCD (RSM parameters)

Table 4Factor	Parameter	Ranges	Coded level	
−1 (Low)	0 (Medium)	+1 (High)	
A	Ri	0.01 ≤ Ri ≤ 10	0.01	5.005	10	
B	Re	100 ≤ Re ≤ 1000	100	550	1000	
C	Ha	0 ≤ Ha ≤ 60	0	30	60	
D	ϕ	0.01 ≤ ϕ ≤ 0.10	0.01	0.055	0.10	

Table 5 RSM (CCD based) experimental design

Table 5Run Order	Coded values	Real values	Response	
A	B	C	D	Ri	Re	Ha	ϕ	Nu	
1	0	0	0	0	5.005	550	30	0.055	5.4975	
2	−1	−1	−1	−1	0.01	100	0	0.01	4.2251	
3	1	1	1	−1	10	1000	60	0.01	5.9119	
4	0	0	−1	0	5.005	550	0	0.055	5.7754	
5	0	0	0	0	5.005	550	30	0.055	5.4975	
6	1	−1	1	1	10	100	60	0.10	6.3576	
7	0	0	0	0	5.005	550	30	0.055	5.4975	
8	1	1	1	1	10	1000	60	0.10	7.0484	
9	−1	−1	1	1	0.01	100	60	0.10	6.3575	
10	−1	0	0	0	0.01	550	30	0.055	5.3710	
11	1	−1	−1	−1	10	100	0	0.01	4.2417	
12	0	0	0	0	5.005	550	30	0.055	5.4975	
13	−1	1	−1	−1	0.01	1000	0	0.01	4.8882	
14	1	1	−1	1	10	1000	0	0.10	8.2248	
15	−1	−1	−1	1	0.01	100	0	0.10	6.3706	
16	0	0	0	0	5.005	550	30	0.055	5.4975	
17	0	0	0	1	5.005	550	30	0.10	6.6060	
18	0	0	0	0	5.005	550	30	0.055	5.4975	
19	1	1	−1	−1	10	1000	0	0.01	6.7907	
20	0	1	0	0	5.005	1000	30	0.055	6.3674	
21	−1	1	1	−1	0.01	1000	60	0.01	4.3380	
22	0	−1	0	0	5.005	100	30	0.055	5.1944	
23	1	−1	−1	1	10	100	0	0.10	6.3711	
24	−1	1	1	1	0.01	1000	60	0.10	6.6196	
25	−1	1	−1	1	0.01	1000	0	0.10	7.2503	
26	0	0	0	−1	5.005	550	30	0.01	4.7301	
27	0	0	0	0	5.005	550	30	0.055	5.4975	
28	−1	−1	1	−1	0.01	100	60	0.01	4.2065	
29	1	0	0	0	10	550	30	0.055	5.8062	
30	0	0	1	0	5.005	550	60	0.055	5.2903	
31	1	−1	1	−1	10	100	60	0.01	4.2070	

4.6 Analysis of variance (ANOVA)

Fig. 14 illustrates four separate residual plots that were created subsequent to inputting the data into analytical software to perform an analysis of variance (ANOVA) method. In regression analysis, residual plots are often used to evaluate the adequacy of how well the model fit. According to Fig. 14 (a), the normal probability plots corresponding to the residuals show a satisfactory distribution. Normal probability plots displaying the residual distributions have frequently been utilized to evaluate normality of the observation. The assumption of normality in the residual distribution of values for Nu is made based on the linearity of the relationship. Fig. 14 (b) exhibits a strong connection with the observed values and the results obtained by fitting, as shown by the residual diagrams and fitted values. The residuals frequently show a horizontal distribution centered around the zero line, indicating that the error terms have a similar variance. Each response exhibits a distribution of residuals, with the largest residuals about equal to 0.3. Furthermore, the histogram of the residual is shown in order to determine the presence of skewness or outliers within the dataset. The histogram of the residual has a skewed distribution and does not resemble a symmetrical distribution, as seen in Fig. 14 (c). It is important to acknowledge that none of the residuals exhibit significant deviations from the fundamental random pattern of residuals, suggesting the absence of outliers. In addition, Fig. 14 (d) exhibits the plot of residuals against an order for Nu. It is noticed that the residuals for Nu exhibit random fluctuations around the zero line, indicating a lack of correlation among them. Consequently, the current RSM model exhibits a satisfactory degree of accuracy.Fig. 14 Residual plots for response function (Nu) (a) normal probability plot; (b) residual vs fitted value plot; (c) histogram of residual; and (d) residual vs observation order plot.

Fig. 14

Additionally, Table 6, Table 7, Table 8 provides the outcome of the statistical assessment conducted on a mixed convective system employing RSM. The concept of degrees of freedom (DOF) refers to the highest number of distinct variables in the model, as illustrated in Table 6. The concept of the overall sum of squared (SS) is a statistical technique used to quantify the collective variability resulting from several factors. The adjusted sums of squared (Adj. SS) have a notable significance, as seen by their value of 28.3890. The F-value of 88.40, suggests that the model has statistically significant. The occurrence of a large F-value due to noise is quite unlikely, with a probability of just 0.01 %.Table 6 Analysis of variance (ANOVA) for quadratic model of response function (Nu)

Table 6Source	DOF	SS	Adj. SS	Adj. MS	F-value	p-value	Comment	
Model	14	28.39	28.3890	2.0278	88.40	<0.0001	Significant	
Linear	4	25.1727	25.1757	6.2939	274.39	-		
Ri	1	1.58	1.5798	1.5798	68.87	<0.0001		
Re	1	5.45	5.4536	5.4536	237.75	<0.0001		
Ha	1	0.8027	0.8027	0.8027	34.99	<0.0001		
ϕ	1	17.34	17.3396	17.3396	755.93	<0.0001		
Square	4	0.1543	0.6671	0.1668	7.27	-		
Ri*Ri	1	0.0016	0.0016	0.0016	0.07	0.7970		
Re*Re	1	0.1221	0.1221	0.1221	5.32	0.0348		
Ha*Ha	1	0.0025	0.0025	0.0025	0.11	0.7446		
ϕ*ϕ	1	0.0281	0.0281	0.0281	1.22	0.2848		
Interaction	6	2.5488	2.5462	0.4244	18.50	-		
Ri*Re	1	1.48	1.4774	1.4774	64.41	<0.0001		
Ri*Ha	1	0.0496	0.0496	0.0496	2.16	0.1608		
Ri*ϕ	1	0.2729	0.2729	0.2729	11.90	0.0033		
Re*Ha	1	0.6226	0.6226	0.6226	27.14	<0.0001		
Re*ϕ	1	0.1160	0.1160	0.1160	5.06	0.0390		
Ha*ϕ	1	0.0077	0.0077	0.0077	0.34	0.5700		
Error	16	0.3670	0.3670	0.0229	–	-		
Lack-of-Fit	10	0.3670	0.3670	0.0367	–	-	Insignificant	
Pure Error	6	0.0000	0.0000	0.0000	–	-		
Total	30	28.76	28.7560					
Here, R2= 98.72 %, Adjusted R2 = 97.61 %, Predicted R2 = 89.75 %	

Table 7 Fit summery statistics for response function (Nu)

Table 7Source	SS	DF	MS	F-value	p-value	Comment	
Mean vs Total	1010.98	1	1010.98				
Linear vs Mean	25.18	4	6.29	45.71	<0.0001		
2FI vs Linear	2.55	6	0.4244	8.21	0.0001		
Quadratic vs 2FI	0.6671	4	0.1668	7.27	0.0016	Suggested	
Cubic vs Quadratic	0.3464	8	0.0433	16.84	0.0003	Aliased	
Residual	0.0206	8	0.0026				
Total	1039.74	31	33.54				

Table 8 Model summery statistics for response function (Nu)

Table 8Source	Std. Dev.	R2	Adjusted R2	Predicted R2	Comment	
Linear	0.3711	0.8755	0.8563	0.7867		
2FI	0.2274	0.9640	0.9461	0.8338		
Quadratic	0.1515	0.9872	0.9761	0.8975	Suggested	
Cubic	0.0507	0.9993	0.9973	0.9039	Aliased	

In this particular instance, the p-value serves as a measure of the possibility of whether the null hypothesis holds true for a particular statistical design. The p-value, a numerical measure between 0 and 1, is often used to establish the degree of statistical significance. Typically, three categories of incidence arise based on of p-values. When the p-value is less than 0.01, it is considered to be extremely tiny. In such cases, the null hypothesis is rejected due to the presence of strong evidence, indicating that the observed results are highly statistically significant. Once again, when the p-value falls within the range of 0.01–0.05, it indicates statistical significance. Nevertheless, when the p-value exceeds 0.05, it indicates that there is inadequate proof for rejecting the null hypothesis, since the observed outcome is not deemed statistically significant. Table 6 highlights the significance of the input components in this model, with particular emphasis on those that are not underlined in bold.

Moreover, the analysis of the model demonstrates that the statistical evaluation of the predicted model and subsequent test methods indicates a substantially high R2 value (98.72 %) for the response function (Nu). This finding suggests that the model is appropriate for the computation of Nu values for the response function. Although the adjusted R2 (97.61 %) is lower than the R2 (98.72 %) for response function, it is important to note that the model still demonstrates a considerable reflection of the experimental data. Once again, the predicted R2 of 0.8975 exhibits a satisfactory level of congruence with the adjusted R2 of 0.9761, indicating a discrepancy of less than 0.2. A different approach to crucial parameter to consider is the Lack-of-Fit, which should be minimized in order to ensure the effectiveness of the model. The current model has a Lack-of-Fit value of 0.3670, suggesting that it is appropriate for the given analysis. It is mentioned that the RSM models encompass several orders, such as linear, quadratic, cubic, quadratic vs 2FI, cubic vs 3FI, and so on. Based on the analysis of the important values shown in Table 7, Table 8 and it can be concluded that the ‘Quadratic versus 2FI’ (2-factors interaction) model outperforms all other models. The resulting model generated by the RSM is intended for the analysis of the correlation between the response function (Nu) and the independent input factors (Ri, Re, Ha, and ϕ):(14) y=p0+p1Ri+p2Re+p3Ha+p4ϕ+p11Ri2+p22Re2+p33Ha2+p44ϕ2+p12RiRe+p13RiHa+p14Riϕ+p23ReHa+p24Reϕ+p34Haϕ

where, the coefficients p1, p2, p3, p4, p11, p22, p33, p44, p12, p13, p14, p23, and p24 represent the most accurately fitted regression model's coefficient for the input parameters Ri, Re, Ha, and ϕ in the context of this RSM model. Table 9 represent the predicted coefficients of equation (14) for response function (Nu), that have been obtained in codded units.Table 9 Coded Coefficients for response function (Nu)from RSM

Table 9Term	Coefficient	p-value	
Constant	5.5282	<0.0001	
Ri	0.2963	<0.0001	
Re	0.5504	<0.0001	
Ha	−0.2112	<0.0001	
ϕ	0.9815	<0.0001	
Ri*Ri	0.0246	0.7970	
Re*Re	0.2169	0.0348	
Ha*Ha	−0.0312	0.7446	
ϕ*ϕ	0.1040	0.2848	
Ri*Re	0.3039	<0.0001	
Ri*Ha	−0.0557	0.1608	
Ri*ϕ	−0.1306	0.0033	
Re*Ha	−0.1973	<0.0001	
Re*ϕ	−0.0851	0.0390	
Ha*ϕ	−0.0220	0.5700	

It is worth mentioning that the regression equation has been constructed by selecting only those significant models' variable terms, as their importance warrants their inclusion. The significance of the model's terms is shown by p-values that are less than 0.05. The significant terms of the model in this scenario are Ri, Re, Ha, ϕ, Re2, Ri*Re, Ri*ϕ, Re*Ha, and Re*ϕ. On the other hand, the insignificant terms (for p-value >0.1) are omitted (shown in bold). The terms Ri2, Ha2, ϕ2, Ri*Ha, and Ha*ϕ are deemed statistically irrelevant for response function (Nu) and should be taken out of from the best fitting regression model. The mathematical relation shown above serves as a means to succinctly describe the association between the input parameters (Ri, Re, Ha, and ϕ) and the response function (Nu):(15) Nu=5.5282+0.2963Ri+0.5504Re−0.2112Ha+0.9815ϕ+0.2169Re2+0.3039RiRe−0.1306Riϕ−0.1973ReHa−0.0851Reϕ

4.7 Response surface analysis

Fig. 15, Fig. 16, Fig. 17 provide the visual representation of response surface contour plots obtained via the implementation of response surface analysis. These plots aim to examine the impact of many influential factors, namely Ri, Re, Ha, and ϕ on the response function (Nu), both two-dimensional (2D) and three-dimensional (3D) formats.

The graphical representation in Fig. 15 illustrates the impact of the parameters Ri and Re on Nu, while keeping Ha and ϕ constant. The 2D contour plot in Fig. 15 (a) illustrates that an increase in Ri and Re results in a rise of Nu, while holding Ha = 30, and ϕ = 0.055 constant. The Nu exhibits its highest level when Ri = 10, and Re = 1000, while it reaches its minimum value when Ri = 0.01 and Re = 500. Further, Fig. 15 (b) illustrates a 3D surface plot that facilitates the examination of the impact of Ri and Re on Nu. Both the 2D and 3D surface plot demonstrate same behaviour in relation to the response function. In Fig. 16 (a), the 2D contour plot illustrates that an increase in Ri and a decrease in Ha result in a rise of Nu, while maintaining Re = 550, and ϕ = 0.055 at constant The maximum heat transfer occurs when Ri = 10 and Ha = 0, whereas it strikes minimum when Ri = 0.01 and Ha = 60. Also, Fig. 16 (b) presents a 3D surface plot that illustrates the investigation of the impact of Ri and Ha on Nu. Similarly, Fig. 17 (a) illustrates a 2D contour plot that demonstrates the relationship between increasing Ri and ϕ and the resulting increase in Nu, while maintaining Re at 550, and Ha at 30. Nu responses are highest when Ri = 10 and ϕ = 0.10, whereas its response is minimum when Ri = 0.01 and ϕ = 0.01. Moreover, Fig. 17 (b) displays a 3D surface plot that depicts the investigation of the impact of Ri and ϕ on Nu.Fig. 15 Variations of Nu for factors Ri and Re keeping fixed at Ha = 30 and ϕ = 0.055: (a) 2D contour, (b) 3D plot.

Fig. 15

Fig. 16 Variations of Nu for factors Ri and Ha keeping fixed at Re = 550 and ϕ = 0.055: (a) 2D contour, (b) 3D plot.

Fig. 16

Fig. 17 Variations of Nu for factors Ri and ϕ: keeping fixed at Re = 550 and Ha = 30: (a) 2D contour, (b) 3D plot.

Fig. 17

4.8 Sensitivity analysis

Sensitivity analysis is an essential component of numerical modelling since it enables the assessment of how unpredictability in model inputs has on the resulting model outputs. The determination of the greatest effective parameter may be achieved by ranking the most effective parameters based on their impacts, as shown by the findings of the sensitivity analysis. Sensitivity analysis tests are conducted on computational as well as mathematical models in order to evaluate the impact of uncertainties in parameter values, input variables, and calculations on the model outputs [49]. Based on the outcomes of these tests, the input elements that exhibit the highest degree of effectiveness on the model outputs have been determined. By using the response function's (Nu) derivatives that are partial, it becomes feasible to analytically ascertain the sensitivity of the output factors to certain key factors (Ri, Re, Ha, and ϕ). Consequently, the partial derivatives for equation (15) with respect to the input factors have been determined and are shown as follows:(16) ∂Nu∂Ri=0.2963+0.3039Re−0.1306ϕ

(17) ∂Nu∂Re=0.5504+0.4338Re+0.3039Ri−0.1973Ha−0.0851ϕ

(18) ∂Nu∂Ha=−0.2112−0.1973Re

(19) ∂Nu∂ϕ=0.9815−0.1306Ri−0.0851Re

The sensitivity of response function (Nu) with respect to the input factors Ri, Re, Ha, and ϕ can be computed using (16), (17), (18), (19) as shown in Table 10. The values in concern have been computed via the use of the suggesting mixed convective model. The model incorporates the computation of Ri at codded levels of −1, 0, and +1 (real values 0.01, 5.005, and 10), Re at levels of 0 and + 1 (550 and 1000), Ha at levels of −1 and 0 (0 and 30), and ϕ at levels of −1 (0.01). Moreover, it is crucial to take into account as a positive sensitivity value suggests that an increase in the input factors leads to a corresponding increase in the output parameters. To clarify, it may be stated that average Nusselt number (Nu) exhibits a positive correlation with the Ri, Re, and ϕ. As the values of Ri, Re, and ϕ raised, there is a corresponding rise in the value of Nu. On the contrary, a negative sensitivity value indicates that an increase in the input parameters results in a drop in the output parameter, representing a contrasting behaviour. The overall heat transmission rate is negatively affected by the value of Ha. Therefore, when the value of Ha increases, it is necessary to correspondingly decrease of Nu.Table 10 Sensitivity of input factors to response function (Nu)

Table 10Ri	Re	Ha	ϕ	∂Nu∂Ri	∂Nu∂Re	∂Nu∂Ha	∂Nu∂ϕ	
−1	0	−1	−1	0.4269	0.5289	−0.2112	1.1121	
0	0	−1	−1	0.4269	0.8328	−0.2112	0.9815	
+1	0	−1	−1	0.4269	1.1367	−0.2112	0.8509	
−1	+1	0	−1	0.7308	0.7654	−0.4085	1.0270	
0	+1	0	−1	0.7308	1.0693	−0.4085	0.8964	
+1	+1	0	−1	0.7308	1.3732	−0.4085	0.7658	

The influence of the input factors (Ri, Re, Ha, and ϕ) regarding the response of the function (Nu) may be seen in Fig. 18. The vertical upright bar inside the bar graph indicates positive sensitivity, while the vertical downward bar signifies negative sensitivity in relation to Nu. The length of each bar corresponds to the magnitude of the sensitivity value. In accordance with the sensitivity comparison bar diagram, it can be seen that the Nu are highly sensitive to Ri, Re, and ϕ. Hence, it is essential for researchers to pay more focus towards the Ri, ϕ and Re parameters while developing a mixed convective model aimed at potential heat transfer.Fig. 18 Sensitivity of response function (Nu) at Re = 0, Ha = −1, and ϕ = −1.

Fig. 18

5 Conclusions

A study was conducted to numerically investigate and analyze the sensitivity of the phenomenon of MHD mixed convective heat transport inside a double lid-driven square cavity. The cavity has been filled with a hybrid nanofluid and also contains a centrally positioned heated star-shaped block. Several intriguing factors were discovered during the investigation. The following are a few significant major findings.i. The flow strength increases with Re but the isothermal lines get more distorted as Re rises. The overall heat transport rate within the cavity has a positive correlation with Re.

ii. Interesting behaviour has been observed in various Ri. When the Ri is relatively low, indicating a dominance of forced convection. Also, the isothermal lines exhibit a coherent alignment with the heated star-shaped block and double lids. As the Ri rises, it starts to exhibit the characteristics of natural convection. This is seen in via the emergence of more robust vortices within the flow patterns and the rising skewness of the isothermal lines. The overall heat transport rate has a positive correlation with Ri.

iii. The deformation of streamlines is seen as Ha increases, and the smooth progression of isotherm lines is disrupted with the increase in Ha. The overall heat transmission rate inside the cavity exhibits a declining trend as Ha increase. It decreases by 2.58 %, 5.03 %, and 5.97 %, when the Ha increases from 0 to 20, 40, and 60, respectively at default parameters.

iv. The thermal conductivity of the hybrid nanofluids is shown to rise as ϕ increase. The regularity of isotherms lines and the intensity of flow both also exhibit an increase when ϕ increases. The average heat transfer rate increases by 14.28 %, 31.19 %, and 49.54 %, when ϕ increases from 1 % to 3 %, 5 %, and 10 %, respectively at default parameters.

v. The statistical assessment and testing technique indicate a high R2 values of 98.72 % for response function (Nu), suggesting that this model is appropriate for estimating Nu. Again, the predicted R2 of 0.8975 exhibits a satisfactory level of concordance with the adjusted R2 of 0.9761.

CRediT authorship contribution statement

R.M. Ziaur: Writing – original draft, Methodology, Formal analysis. A.K. Azad: Writing – review & editing, Software, Formal analysis. M.M. Rahman: Supervision, Methodology, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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