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

S2405-8440(24)13193-3
10.1016/j.heliyon.2024.e37162
e37162
Research Article
Numerical investigation of MHD mixed convection in an octagonal heat exchanger containing hybrid nanofluid
Ali Md. Yousuf yousufmath33@gmail.com
a⁎
Islam Saiful b
Alim M.A. a
Biplob Rejowan Ahmed c
Islam Md. Zohurul d
a Department of Mathematics, Bangladesh University of Engineering and Technology, Dhaka, 1000, Bangladesh
b Department of Mathematics, Bangabandhu Sheikh Mujibur Rahman Science and Technology University, Gopalganj, 8100, Bangladesh
c Department of Electrical and Electronic Engineering, Gono Bishwabidyalay, Dhaka, 1344, Bangladesh
d Department of Mathematics, Jashore University of Science and Technology, Jashore, 7408, Bangladesh
⁎ Corresponding author. yousufmath33@gmail.com
30 8 2024
15 9 2024
30 8 2024
10 17 e3716230 4 2024
2 8 2024
28 8 2024
© 2024 The Authors
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Nowadays, the advancement of heat transmission for the heat exchanger device is an important field of research for many researchers. In this work, a numerical study has been conducted to investigate the thermal performance of a mixed convective flow through the octagonal heat exchanger covered by hybrid nanofluid (Cu-TiO2-H2O). A magnetic field has been introduced inside the cavity to investigate the mixed convective hydrodynamics heat flow characteristics. The nanofluid cores absorb/release energy to manage heat transmission by increasing or decreasing inside the cavity domain as the host fluid and dispersed hybrid nanofluid circulate within the cavity. After transforming the governing equations into a generalized, non-dimensional formulation, the finite element approach is utilized to solve the associated equations. Additionally, response surface methodology is also applied to test the responses of the associated factors. Heat transport was examined in relation to the effects of nanofluids fusion temperature, boundary wall properties, Reynolds number, Hartmann number and nanoparticle volume fractions. The outcomes of this study are analysed by measuring streamline profiles, isotherms, average Nusselt number, velocity profile, and 2D and 3D response surfaces of the computational domain. The underlying flow controlling parameters for instance Reynolds number (10 ≤ Re ≤ 200), Hartmann number (0 ≤ Ha ≤100), and nanoparticle volume fractions (0 ≤ ϕ ≤ 0.1), the influences have been considered. The findings also reveal that the thermal performance is being boosted due to augmentation of Re and ϕ, but reverse behavior is noticed for Ha. Furthermore, the response surfaces obtained from response surfaces methodology express that the Re and ϕ have shown positive influence, and Ha has shown negative influence on Nuav. Utilizing a hybrid nanofluid of Cu-TiO2-H2O increases the heat transfer capacity of water to 25.75 %. Moreover, the findings could guide to design of a mixed convective heat exchanger for industrial purposes.

Keywords

Heat exchanger
Mixed convection
MHD
Response surface
Hybrid nanofluid
==== Body
pmc1 Introduction

The general composition of hybrid nanofluids is a base fluid (such as ethylene glycol, water, or oil) combined with metallic or non-metallic nanoparticles (Cu, Al2O3, Co, Zn, Ag, TiO2, Fe3O4, and so on) to develop the fluid thermal characteristics. The relevance of mixed convection in the realm of heat and mass transfer, together with its numerous uses in science and engineering, has drawn a lot of attention to hybrid nanofluid-filled lid-driven cavities in recent years. To improve heat transmission, recently nanoparticles are involved with base fluid to make a hybrid nanofluid. This can lead to a better thermal performance in mixed convection circumstances which is particularly crucial in real-life applications like electronic cooling systems, heat exchangers, computer processors, cooling of nuclear reactors, solar water heating, solar collectors, cooling of electronic tools, and microelectronics [[1], [2], [3], [4]]. In this sense, Maxwell [5] combined micro or millimeter-sized nanoparticles in convectional fluids to create a new kind of heat transmission fluid. Subsequently, Choi et al. [6] combined nanoscale particles with conventional fluids to create an innovative heat transfer fluid known as nanofluids at the National Argonne Laboratory. Additionally, the better qualities of nanofluids such as their reduced sedimentation and clogging force, improved thermal conductivity can improve the heat transfer efficiency of engineering equipment [[7], [8], [9]].

Moreover, magnetohydrodynamic (MHD) has attracted a lot of curiosity from scientists and engineers because of its vast variety of applicability in geophysical research and engineering, including microelectronic packages, solar technologies, cooling nuclear reactors, and reservoirs of geothermal energy. A complicated interplay between the buoyancy effect, which force occurs in the context of natural, mixed convections related to magnetic field impact, which influences fluid flow and heat transfer processes. Islam et al. [10] studied heat transfer enactment including sensitivity analysis of a mixed convective heat exchanger with MHD in a hexagonal cavity utilizing TiO2-H2O nanofluid. They observed that including TiO2 nanoparticles into water expands heat transfer capability of water up to 17.69 %. A 3D numerical simulation was scrutinized by Hakan et al. [11] to investigate the efficiency of heat transmission employing hybrid nanofluids in a newly built thermal system. In another work, Minea et al. [12] used hybrid nanoparticles to study the parabolic solar system, which shows that at 2 % volume levels, the Cu-MgO nanofluid exhibits a noteworthy spike in average Nusselt number (Nuav) with a rise of 14 % as compared to the normal fluid. Kumar et al. [13] investigated the use of a hybrid nanofluid of a T-shaped cavity for double-diffusive flow in a porous medium. Furthermore, some researches using hybrid nanofluid has been done recently to control the mechanics of heat transfer across multiple types of networking devices [[14], [15], [16], [17], [18], [19], [20]].

Additionally, the current generation of researchers is also fascinated with heat transfer and mixed convection fluid motion in closed cavities, where mixed convection is defined as the combination of natural and forced convection. It has recently drawn a lot of interest in the engineering domains due to the ground-breaking applicability of such concerns as heat exchangers, solar panel collectors, and computer cooling mechanisms [[21], [22], [23]]. In actuality, buoyancy forces within an enclosed space and temperature differential have caused natural convection to occur spontaneously. On the other hand, force convection happens when pressure or force exerts from the outside and enters into the cavity. In other words, these two forms of convection happen together for mixed convective analysis. The mixed convective techniques are influenced by the shape and direction of the cavity. In a cavity, mixed convection usually occurs when there is generated ventilation or a lid velocity on any walls of structure. Scholars conducted a range of investigations on mixed convection using nanofluid. For instance, Yeasmin et al. [24] explored mixed convective heat transfer in a porous lid-driven L-shaped cavity, and concluded that to improve convective heat transfer rate, a greater Darcy number is suggested for inflow conditions. In the context of two spinning cylinders, Garmroodi et al. [25] completed a computational study for MHD and mixed convection in a lid-driven cavity containing a Cu-water nanofluid. The authors demonstrated that mixed convection increases with the circular motion of two circuits, and the intensity of heat transfer falls as Hartmann number (Ha) increases using the multi-phase mixer model and the finite volume technique. Afterward, Ishak et al. [26] explored mixed convection with entropy production in a closed cavity containing a solid cylinder and nanofluid using finite element method (FEM). They declared that the location of solid body and size had a substantial effect on heat transfer and Bejan number within the chamber. In a double-wall lid-driven enclosure with a heat transmission barrier and an electrical field, Ali et al. [27] examined mixed convective Casson fluid motion and identified a large proportion of temperature transfer increase for the proper direction of lid wall. They found that with increasing Casson fluid parameter and Reynolds number, the heat transfer capability boosted up. Later, Ali et al. [28] also conducted another analysis on flow and thermal characteristics in a hollow controlled by a lid and included a revolving flat plate that was exposed to a magnetic field. Their findings guaranteed a 123.03 % improvement in heat transmission while employing 5 % Al2O3 nanoparticles into the nanofluid mixer.

Researchers enhanced hybrid nanofluids, made up of several nanoparticles into a base fluid, for increasing the heat transfer performance much better. The mixture of component nanoparticles in traditional base fluids is connected with the efficacy of hybrid nanofluids. Furthermore, in contrast to mono-nanofluids, hybrid nanofluids also offer more stable, chemically inert, and affordable characteristics. A study about entropy generation for mixed convection in a horizontal channel with a square obstruction containing hybrid nanofluids was completed by Hussain et al. [29]. They anticipated that while magnetic entropy production increases, fluid velocity and heat transport production decrease due to rising magnetic field intensity. In a partially heated square cavity, Akhter et al. [30] applied FEM to investigate mixed convection utilizing hybrid nanofluid that contains also a magnetic field and a spinning cylinder. Mandal et al. [31] used Cu and Al2O3 nanoparticle into water to examine the hydrothermal characteristics of nanofluids in a porous complex enclosure with a uniform magnetic field, where hybrid nanofluids provide a developed heat transfer performance rather than other mono-nanofluids. Table 1 summarizes the literature of prior statistical efforts that depict mixed convection into various confinement types and is relevant to the current investigation.Table 1 Previous investigations on mixed convection related to present study.

Table 1Investigators	Applied method	Type of fluids	Parameters and ranges	Correlation	Applications	Geometry	
Saiful Islam et al. [2]	FEM	Cu-H2O nanofluid	103<Ra<106, 0≤Ha≤30, Pr = 6.83	Nuav=−KnfKbf∫S∂θ∂NdS	a, b, c, d, e, f, p	prismatic heat exchanger	
Abdel-Nour et al. [21]	FEM	Cu-Al2O3 hybrid nanofluid	0≤Ha≤100, 103≤Ra≤106, 10−5≤Da≤10−2	Nuav=1L∫0LNulocaldL	b, e, g, i, k, n, s, t, u	porous square cavity	
Rowsanara et al. [30]	FEM	Ag-Al2O3 hybrid nanofluid	0≤Ha≤50, 0.01≤Ri ≤ 10	Nuav=1Lh∫L/32L/3−khbnfkf∂θ∂XdY	a, c, h, j, m, p, s, u, w	multiple rotating cylinder	
Mandal et al. [31]	TDMA	H2O-Cu-Al2O3 hybrid nanofluid	10 ≤ Re ≤ 300, 0 ≤ Ha ≤70, 0 ≤ ϕ ≤ 0.02, 0.1 ≤ Ri ≤ 100	Nu=kkf1S∫0S(−∂θ∂N|wavywall)dS	a, g, h, m, q, s, t, x, y	porous wavy cavity	
Saha et al. [32]	FEM	H2O-Al2O3 nanofluid	104<Ra<106, 0≤Ha≤50, 0≤ ϕ ≤ 0.05	Nuav=(knfkbf)∫∂θ∂YdX	d, n, o, r, u, w	square wavy cavity	
Zeghbid et al. [33]	FVM	Cu, Ag, Al2O3, TiO2-H2O	0.1≤ Ri ≤ 100, 0≤ ϕ ≤ 1, 103≤Ra≤105	St=∫SgendV	a, c g, j, k, r, t, u, w	square lid driven cavity	
Maneengam et al. [34]	GFEM	Fe3O4-MWCNT-H2O	0 ≤ Ha ≤100 − 500, ≤ Ω ≤ 1000, 10−2 ≤ Da ≤10−5	Nuave=1S2∫0S∫0SNulocdydz	b, h, k, m,o, p, r, u, v, y	3D porous cavity	
Alipour et al. [35]	GFEM, CVFEM	H2O- Al2O3-ethylene glycol	100≤Ra≤1500, 0≤Ha≤40	Nuave=12π∫02πNulocdr	c, d, m, o, p, r, v, w	trapezoidal porous cavity	
*FEM= Finite element method, FVM= Finite volume method, TDMA = Tri-diagonal matrix algorithm, HGF= Higher geometric function, GFEM = Galerkin finite element method.

**(a) heat exchangers; (b) home ventilation; (c) solar collectors; (d) cooling organisms in nuclear reactors; (e) petroleum reservoirs; (f) fire engineering; (g) blood flow control; (h) food processing; (i) paints manufacturing; (j) cancer and tumor treatment; (k) glass fiber manufacturing; (l) shampoo manufacturing; (m) nuclear power plant; (n) solar energy; ((o) cooling of electronic device; (p) cooling of nuclear reactors; (q) geothermal reservoirs; (r) material built-up machinery; (s) gas industries; (t) polymer industries; (u) refrigerator; (v) chemical industry; (w) home ventilation.

As per the previously mentioned literature, it is evident that researchers are highly intrigued by MHD mixed convective heat exchangers because of its innovative application in a wide range of engineering domains, particularly in the design of numerous industrial architects with different geometrical structure. A few research on octagonal enclosures containing different nanoparticles has been conducted recently [36,37], even though separate investigations on diverse closed cavities with varying MHD mixed convection were performed at different stages. The study on the formulation of such heat exchanger devices that can transfer heat energy quickly is very limited [[38], [39], [40]]. The proposed novel heat exchanger device can transfer heat energy between two or more fluids. The prime aim of this underlying model is to illustrate the consequence of employing hybrid nanofluid in a mixed convective octagonal heat exchanger filled with Cu-TiO2-H2O nanofluid, where an exterior uniform magnetic field is present. In order to analyze the enhancement of energy shifting efficiency by the proposed system, we furthermore conducted a statistical technique called response surface methodology (RSM). In this method, the combined impact of significant factors on the rate of heat transfer calculation was assessed for different flow controlling parameters to get the optimal heat transfer. Different 2D and 3D response surfaces are explained physical explanation by the use of RSM. Although multiple numerical studies on octagonal cavities containing nanofluid have been conducted at various times, we use hybrid nanoparticles with MHD in this study, which is one of the novelty of the present study. Furthermore, to the best of the author's knowledge, the analysis of heat transfer behavior using FEM and RSM is the most new aspect of this work. This study uses numerical and statistical approaches to analyze the Reynolds number (Re), Hartmann number (Ha), and nanoparticle volume fraction (ϕ) in a mixed convective octagonal heat exchanger.

2 Mathematical framework

2.1 Physical description

An octagonal-shaped heat exchanger is considered as a fluid domain covered by Titanium oxide (TiO2) and Copper (Cu) solid nanoparticles, and H2O as a common fluid. This mixture is taken 2D steady Newtonian incompressible flow with an external magnetic field. The heat exchanger's length and height are L and H respectively, with two cylindrical pipes (radius 0.1 L) serving as a heater on the left and a cooler on the right side of the domain. The upper straight wall is traveling a lid velocity u0. The physical alignment of this model using hybrid nanofluid is shown in Fig. 1. Here, the right cylindrical pipe taken to be a cooled surface (Tc), while the left pipe considered as a heated surface (Th). The external shielded barriers of the fluid realm are entirely insulated. Moreover, the gravitational force (g) acts in the opposite track of the Y-axis. Additionally, a continuous magnetic field B0 around the container runs from the right to the left. Furthermore, the neighboring media being considered no slip, and TiO2 and Cu nanoparticles are uniform in shape. The thermophysical appearances of the considered nanoparticles and base fluid are shown in Table 2. The fluid flow region is seen and completed using a 2D Cartesian layout, where the left wall is indicated by the Y-direction and the bottom wall is shown by the X-direction.Fig. 1 Physical model of proposed octagonal heat exchanger.

Fig. 1

Table 2 Thermophysical properties of the solid particle and base fluid [31,41].

Table 2Base fluid & nanoparticles	ρ	cp	κ	σ	β	μ	Pr	
H2O	997.1	4179	0.613	5.5 × 10−6	2.1 × 10−4	8.91 × 10−4	6.9	
Cu	8933	385	401	5.96 × 107	1.67 × 10−5	–	–	
TiO2	4250	686.2	8.953	3.5 × 106	0.9 × 10−5	–	–	

2.2 Governing equations

To make a proper mathematical model for this proposed hybrid nanofluid model, the following well-known continuity, momentum, and energy equations are formed with MHD interference [29,32]:(1) ∂u∂x+∂v∂y=0

(2a) ρhnf(u∂u∂x+v∂u∂y)=−∂p∂x+μhnf(∇2u)

(2b) ρhnf(u∂v∂x+v∂v∂y)=−∂p∂y+μhnf(∇2v)+g(ρβ)(T−Tc)hnf−σB02hnfv

(3) (ρcp)hnf(u∂T∂x+v∂T∂y)=κhnf(∂2T∂x2+∂2T∂y2)

There are some essential initial and boundary conditions of the proposed heat exchanger model:(4) atleftcircularedge,u=0,v=0,T=Thatrightcircularedge,u=0,v=0,T=Tcattophorizontalwall,u=u0,v=0,∂T∂y=0andallsurroundingwalls,u=v=0,∂T∂n=0}

where, the vector acting perpendicularly on the heated surface is denoted by n.

2.3 Correlations of hybrid nanofluid

The thermo-physical characteristics of base fluid (H2O) and nanomaterials (Cu and TiO2) are discussed in the current subsection. In actuality, the features of the base fluid and nanomaterials determine the features of the nanofluid. Thus, the regression coefficients in Table 3 are used to compute the appearances of the hybrid nanofluid.Table 3 Correlations of hybrid nanofluid between nanoparticles and base fluid.

Table 3Properties of Hybrid Nanofluid's	Applied correlations	
Concentration of nanoparticles:	:	ϕ=ϕCu+ϕTiO2	
Density of nanofluid:	:	ρhnf=(1−ϕ)ρbf+ϕρsp where ϕρsp=ϕCuρCu+ϕTiO2ρTiO2	
Specific heat capacity:	:	(ρcp)hnf=(1−ϕ)(ρcp)bf+ϕ(ρcp)sp where ϕ(ρcp)sp=ϕCu(ρcp)Cu+ϕTiO2(ρcp)TiO2	
Thermal conductivity	:	khnf=kbf{ksp+2kbf−2ϕ(kbf−ksp)ksp+2kbf+ϕ(kbf−ksp)} where ϕksp=ϕCukCu+ϕTiO2kTiO2	
Thermal diffusivity	:	αhnf=κhnf(ρcp)hnf	
Dynamic viscosity	:	μhnf=μbf(1+2.5ϕ+6.5ϕ2)	
Thermal expansion coefficient	:	(ρβ)hnf=(1−ϕ)(ρβ)bf+ϕ(ρβ)sp where ϕ(ρβ)sp=ϕCu(ρβ)Cu+ϕTiO2(ρβ)TiO2	
Electrical conductivity	:	σhnf=σbf[1+3ϕ(σspσbf−1)(σspσbf+2)−ϕ(σspσbf−1)] where ϕσsp=ϕCuσCu+ϕTiO2σTiO2	

2.4 Dimensional analysis

To transform the governing equations (1), (2a), (2b), (3) into dimensionless form, we define the subsequent non-dimensional variables along with the boundary conditions (4):X=xL,Y=yL,U=uu0,V=υu0,P=pρbfu02andθ=T−TcTh−Tc

The following is a presentation of the non-dimensional versions of the involved dimensional governing equations:(5) ∂U∂X+∂V∂Y=0

(6) U∂U∂X+V∂U∂Y=−(ρbfρhnf)∂P∂X+(μhnfμbf.ρbfρhnf)Pr(∇2U)

(7) U∂V∂X+V∂V∂Y=−(ρbfρhnf)∂P∂Y+(μhnfμbf.ρbfρhnf)Pr(∇2V)+(ρβ)hnfρhnfβbfRiθ−(ρbfρhnf.σhnfσbf)1ReHa2V

(8) U∂θ∂X+V∂θ∂Y=(αhnfαbf)∇2θ

Here, Richardson number (Ri), Prandtl number (Pr) and Hartmann number (Ha) are three significant parameter generated form above equation, respectively as: Ri=gβbf(Th−Tc)υ0, Pr=υbfαbf and Ha=B0Lσbfμbf. Moreover, the dimension free boundary conditions reformed as:(9) onleftcircularedge,U=V=0,θ=1onrightcircularedge,U=V=0,θ=0onthetophorizontalwall,U=U0,V=0,∂θ∂Y=0andonallsurroundingwalls,U=V=0,∂θ∂N=0}

Additionally, from the heated wall, the average Nusselt number (Nuav) is ascertained by:(10) Nuav=−(khnfkbf)∫HS∂θ∂NdS

Moreover, the stream function ψ is defined as: U=∂ψ∂Y,V=−∂ψ∂X. In this case, the positive sign denotes the streamlines' anticlockwise circulation, while the negative sign denotes their clockwise circulation. Therefore, ∂2ψ∂X2+∂2ψ∂Y2=−(∂V∂X−∂U∂Y)=−Ω is the equation of vorticity, and the vorticity vector is Ω.

3 Numerical analysis

3.1 Computational technique

The associated equations are expressed in dimensionless form to avoid dimensional dependencies. Equations (5), (6), (7), (8) related to initial and boundary conditions (9) are computed numerically applying the Galerkin finite element method. To get the finite element equations, the weighted residuals approach, as previously reported by Sienkiewicz and Taylor [36], is applied in equations (5), (6), (7), (8) as:(11) ∫ANα(∂U∂X+∂V∂Y)dA=0

(12) ∫ANα(U∂U∂X+V∂U∂Y)dA=−(ρbfρhnf)∫AHλ(∂P∂X)dA+(μhnfμbf.ρbfρhnf)1Re∫ANa(∂2U∂X2+∂2U∂Y2)dA

(13) ∫ANα(U∂V∂X+V∂V∂Y)dA=−ρbfρhnf∫AHλ(∂P∂Y)dA+(μhnfμbf.ρbfρhnf)1Re∫ANa(∂2V∂X2+∂2V∂Y2)dA+{(ρβ)hnfρhnfβbf}Ri∫ANαθdA−(ρbfρhnf.σhnfσbf)Ha2VRe∫ANαVdA}

(14) ∫ANα(U∂θ∂X+V∂θ∂Y)dA=(αhnfαbf)1RePr∫ANA(∂2θ∂X2+∂2θ∂Y2)dA

Here, Na (a = 1, 2. … …, 6) represents the element's interpolation functions, A is the area of element, and Hλ (λ = l, 2, 3) denotes the interpolation functions for pressure. After that, in equations (12), (13), (14), the Gauss's theorem is involved to create the boundary integral terms related to the heat flux and surface tractions. Then equations (12), (13), (14) converted to:(15) ∫ANα(U∂U∂X+V∂U∂Y)dA+∫AHλ(∂P∂Y)dA+(μhnfρnfνnf)1Re∫A(∂Nα∂X∂U∂X+∂Nα∂Y∂U∂Y)dA=∫S0NαSxdS0}

(16) ∫ANα(U∂V∂X+V∂V∂Y)dA+∫AHλ(∂P∂Y)dA+(μhnfρhnfνbf)1Re∫A(∂Nα∂X∂V∂X+∂Nα∂Y∂V∂Y)dA−{(ρβhnf)ρhnfβbf}Ri∫ANαθdA+(ρbfρhnf.σhnfσbf)Ha2Re∫ANαVdA=∫S0NαSydS0}

(17) ∫ANα(U∂θ∂X+V∂θ∂Y)dA+(αhnfαbf)1RePr∫A(∂Nα∂X∂θ∂X+∂Nα∂Y∂θ∂Y)dA=∫SWNαq1wdSw

Here, Sx and Sy are surface tractions, S0 is outflow barrier, together with the heat flux (qw) that flow into or out of the domain along the boundary wall (Sw). The velocity components U, V, temperature θ, and pressure, P are the essential unknowns for the differential equations that come before. The finite element equations in this study are developed using the six-node triangular element. Only the corner nodes are connected to pressure; the other six nodes are only connected to temperature and velocities. Also, consider:(18) U(X,Y)=NβUβ,V(X,Y)=NβVβ,θ(X,Y)=Nβθβ,P(X,Y)=HλPλ

where,β=1,2,…‥‥,6,andλ=1,2,3. Substituting this into (11) and equations (15), (16), (17), the finite element equations can be written as:(19) RαβxUβ+RαβyVβ=0

(20) RαβγxUβUγ+RαβγyVβUγ+MλμxPμ+(μhnfρhnfνbf)1Re(Sαβxx+Sαβyy)Uβ=Qαu

(21) RαβγxUβVγ+RαβγyVγVγ+MαμyPμ+(μhnfρhnfνf)1Re(Sαβxx+Sαβyy)Vβ−(ρβ)hnfρhnfβbfRiRαβθβ+ρbfσhnfρhnfσbfHa2ReRαβVβ=Qαv}

(22) RαβγxUβθγ+RαβγyVβθγ+αhnfαbf1RePr(Sαβxx+Sαβyy)θβ=Qαθ

where the coefficients concluded the element area and edges are as follows:(23) Rαβx=∫ANαNβ,xdA;Rαβy=∫ANαNβ,ydA;Rαβγx=∫ANαNβNγ,xdA;Rαβγy=∫ANαNβNγ,ydA;Rαβ=∫ANαNβdA;Sαβxx=∫ANα,xNβ,xdA;Sαβyy=∫ANα,yNβ,ydA;Mλμx=∫AHλHμ,xdA;Mαμy=∫AHαHμ,ydA;Qαu=∫S0NαSxdS0;Qαv=∫S0NαSydS0;Qαθ=∫SWNαqwdSw;}

These element matrices are prepared for numerical simulation after a closed-form analysis. The comprehensive computations process is also existing in Refs. [42,43]. The element nodal velocity, temperatures, and pressure components are thus transformed into a set of algebraic equations, with the progressive unknowns expressed as follows:(24) [RuuRuvRuθRupRvuRvvRvθRvpRθuRθvRθθ0RpuRpv00][ΔuΔvΔθΔp]=[FαuFαvFαθFαp]

(25) Ruu=RαβγxUβ+RαβγxUγ+RαβγyVβ+(μhnfρhnfνbf)1Re(Sαβxx+Sαβyy);Ruv=RαβγyUγ;Ruθ=0;Rup=Mαμx;Rvu=RαβγxVγ;Rvp=Rαμy;Rvv=RαβγxUβ+RαβγyVγ+(μhnfρhnfνbf)1Re(Sαβxx+Sαβyy);Rvθ=−(ρβ)hnfρhnfβbfRiRαβ;Rθu=Rαβγxθγ;Rθv=Rαβγyθγ;Rθθ=RαβγxUβ+RαβγyVβ+(αhnfαbf)1RePr(Sαβxx+Sαβyy);Rpu=Mαμx;Rpv=Mαμy;andRθp=Rpθ=Rpp=0;}

After that, the PDE solver in MATLAB has been used to modify the Newton-Raphson iteration approach for solving these nonlinear algebraic equations. Solutions are considered to have converged when the relative error of each variable across successive iterations is less than the convergence criteria ε, meaning that, |ψn+1−ψn|<ε, where n is a number of iteration and ψ=ψ(U,V,θ). The convergence criterion is set to ε=10−5. When comparing FEM to other numerical techniques, there are a number of advantages. One of them is that the equations for each element are determined using FEM. Therefore, it is not a major problem to add new pieces by refining old ones. The approach is helpful for addressing boundary value problems in a range of engineering applications because it uses a set of finite elements to generate computational domains with irregular geometries. A detailed explanation of this method was explained in Ref. [44]. Also, Fig. 2 describes a detailed flowchart of this computational method.Fig. 2 An extensive flow diagram for the computation method.

Fig. 2

3.2 Grid sensitivity analysis

In order to get suitable number of elements for discretization that may be used for FEM, the grid test with the following parameters is explained: ϕ = 0.02, Ha = 50, Re = 100, and Pr = 6.9. Here, the value of Nuav is designated in order to apply the sensitivity test for generating the appropriate meshing of the computational domain. This whole computational domain is divided into five dissimilar quantities of triangular-shaped elements (1992, 2890, 6526, 18958, and 23760). A triangular meshing example is shown in Fig. 3. Additionally, the computed values of Nuav for varied numbers of triangle members in this fluid domain are displayed in Table 4 and Fig. 3. The value of Nuav for 18958 elements is almost the same as the value determined for the following higher component number. Because of this, it is preferred to mesh and finish this fluid model using the 18958 triangular pieces.Fig. 3 Grid sensitivity analysis.

Fig. 3

Table 4 Grid independence check for current study.

Table 4Elements	1992	2890	6526	18958	23760	
Nuav	7.4158	7.4395	7.4890	7.5285	7.5283	

3.3 Code validation

Before initiating the study, the underlying model was verified comprehensively with existing studies conducted by some authors. Firstly, the outcomes (streamlines and isotherms) of Islam et al. [45] is compared, in Fig. 4, to prove the validation of present study. It is clearly visible from Fig. 4 that the pattern of streamlines and isotherm from Islam et al. [45] study and our outcomes are quite similar, which give strong encouragement to validate the present work. The number of two symmetric vortices from Islam et al. [45] study has been produced similar two symmetric vortices, and the patterns of the streamlines has been made the similar trajectories (Fig. 4(a)). It is also observed that the heat energy has been produced in the middle of the streamline contour, which clearly supports the findings from proposed model. Looking at the isotherms contour analysis, the isotherm profiles distribution from Islam et al. [45] study and present study are found to be very similar distribution (Fig. 4(b)). The stratification of the isotherms from present study are fully isolated from the Islam et al. [45] study.Fig. 4 Compareing (a) streamlines, and (b) isotherms of current study (bottom panels) with Islam et al. [45] (top panels).

Fig. 4

In addition to these analyses, we have also formulated another model by inserting two blocks inside the cavity. The code was again validated by optimizing the flow controlling parameters for this model. The hydrodynamic parameters were also taken into account for the model as need as possible by the authors Islam et al. [10]for numerical calculation. The obtained streamlines and isotherms profile were also then compared with the study conducted by Islam et al. [10] investigation. as illustrated in Fig. 5. It is evident that a single vortex has been produced near the top wall of the hexagonal cavity in both of the studies (Islam et al. [10] top and present study bottom) as seen in the streamline analysis (Fig. 5(a)). The obtained isotherm profiles are displayed the thermal stratification from Islam et al. [45] study and the present study. The trajectories of the isotherm curves are showing the similar distribution within the flow domain for the both of the investigation. Overall, the isotherm profiles from Fig. 5(b), our model can correctly reproduce isotherms curves as the isotherm curves obtained by Islam et al. [45] study.Fig. 5 Compareing (a) streamlines, and (b) isotherms of our result (bottom panels) and Islam et al.‘s [10] (top panels).

Fig. 5

Furthermore, we compared the average Nusselt number calculated from Islam et al. [10] findings and the Nusselt number from this current model to assess the accuracy of the present model by computing the heat transfer enhancement. The mean Nusselt number was observed for different Reynolds number to compare the heat transfer enhancement capacity of the present model to validate the model without affecting the solution. As it is seen from Table 5, the calculated Nusselt number (by keeping Ha = 50 and ϕ = 0.5 fixed) is growing with the increasing of Reynolds number that strongly aligned with study conducted by Islam et al. [10]. There are some minor deviations between the results (in Table 5) derived from the current numerical model and the Islam et al. [10] model. The highest variations were noticed to be 1.06 %, which are in line with the acceptable range of the standard error calculation for computational investigation. In this regard, the proposed model may be considered as an optimal model and be able to replicate the height accurate outcomes.Table 5 Result comparision with Islam et al. [10] utilizing Nuav keeping Ha = 50, ϕ = 0.05.

Table 5Re	Islam et al. [10]	Our Findings	Error (%)	
10	2.3517	2.3492	0.1	
105	3.7066	3.6672	1.06	
200	4.1984	4.1852	0.31	

4 Results and discussion

In this work, the significant parameters have been utilized to explore the impacts of MHD mixed convective hybrid nanofluid flow within an octagon-shaped heat exchanger. After validating the model, we run the final simulation to store the data for different aspect of the analysis. The observed results are explained inside the cavity using streamline, isotherm contours, and Nuav for various values of Re, Ha, and ϕ in Fig. 6, Fig. 7, Fig. 8, Fig. 9, Fig. 10, Fig. 11, Fig. 12, Fig. 13, Fig. 14, Fig. 15. Here, Cu-TiO2-H2O hybrid nanofluid occupies the octagonal-shaped cavity, where all nanoparticles are employed in spherical form. Additionally, by few some 2D and 3D response surfaces are explained using RSM to explore how Re, Ha, and ϕ influence the Nuav. The conventional values are taken to complete this simulation, Re = 100, Ha = 10, ϕ = 0.02, Ri = 1, and Pr = 6.9.Fig. 6 Impact of Re on: (a) streamlines, and (b) isotherms.

Fig. 6

Fig. 7 Impact of Re on fluid velocity.

Fig. 7

Fig. 8 Impact of Hartman number on: (a) streamlines, and (b) isotherms.

Fig. 8

Fig. 9 Impact of Ha on fluid velocity.

Fig. 9

Fig. 10 Impact of ϕ on: (a) streamlines, and (b) isotherms.

Fig. 10

Fig. 11 Impact of ϕ on fluid velocity.

Fig. 11

Fig. 12 Importance of addition of ϕ using Nuav for numerous Ha and Re value.

Fig. 12

Fig. 13 Effect on Nuav for Re and Ha: (a) 2D sight; (b) 3D sight.

Fig. 13

Fig. 14 Effect on Nuav for ah and ϕ: (a) 2D sight; (b) 3D sight.

Fig. 14

Fig. 15 Effect on Nuav for Re and ϕ: (a) 2D sight; (b) 3D sight.

Fig. 15

4.1 Impact of Reynolds number

For this mixed convective framework, the streamline, isotherm contours, and Nusselt number in Fig. 6, Fig. 7 reveal the influence of Re. Fig. 6(a) illustrates the way to regulate Re = (10–200) on the fluid's flow pattern through streamline contours assuming Pr = 6.9, Ri = 1, Ha = 10, and ϕ = 0.02. The fluid motion is increased by higher values of Re, which ultimately causes a noticeable enhancement in streamline distribution and flow rotation magnitude. Due to the intensification of Re, the improvement of lid velocity of the octagonal's upper wall produce a single primary vortex (Fig. 6(a)). Threfore, the force convection becomes stronger due to the increase of lid velocity around the obstacle within the cavity. The fluid follows on the upper wall as it transmitted from left to right direction in accordance with the lid velocity. As the magnitude of Re rises, there is an upsurge the fluid velocity from left to right through the top surface, reaching its optimum level at Re = 200. The similar findings have also been observed from Ali et al. study [28]. Conversely, Fig. 6(b) illustrates way isotherm contours regulate fluid temperature by showing how heat transmission within the fluid domain improves by raising the value of Re. There is evidence of spontaneous convection for low values of Re (Re = 10). As a result, from the leftward heated cylinder to the rightward cold one, the isotherm lines change equally. The rising magnitude of Re (Re = 50) causes the lid velocity to increase, and the upper wall's lid velocity causes mixed convection flow within the cavity domain. The stratification of the isotherm lines have been distributed of the most of the cavity domain with the higher value of Reynolds number that might also been observed Akhter et al. [[30], [46]].

In order to measure the impact of Reynolds number on the velocity distribution along the horizontal representative length of the of the cavity, the non-dimensional characteristic velocity (alongside the Y-axis) over the whole domain has been determined and illustrated in Fig. 7. This magnitude of the velocity profile is calculated along the line (0.5, 0) to (0.5, 1). Due to the enhancement of Re, the fluid velocity is increasing near the boundary of the domain (Fig. 7). There is an 83.25 % rise in the heat exchange rate over the prior value. The overall calculated heat transport rate from the heated surface to cooled surface is measured 36.28 % for increasing Reynolds number ranging from Re = 50 to Re = 100. and 22.27 % for Re = 200.

4.2 Impact of Hartman number

Assuming that Pr = 6.9, Ri = 1, Re = 100 and ϕ = 0.02, the streamline and isotherm contours in Fig. 8, Fig. 9 illustrate the effects of the Ha on thermal transit and fluid motion. In reality, Ha illustrates that magnetic force is controlled on this octagonal heat exchanger. As shown in Fig. 8(a), the overall thickness of streamlines is highest when the field of magnetic attraction is not occupied, as well as Ha = 0. Furthermore, fluid flow is maximized if there is no magnetic force (Ha = 0) as opposed to a magnetic field (Ha = 25, 50, 100). Consequently, the isotherm contours will have an identical form in the vertical plane when the convection type of heat transmission progressively yields up to the conduction manner. Because of the magnetism effect, a resistance force known as the Lorentz force is present inside the barrier and has the potential to slow down the mobility of the nanofluid itself. Which is the phenomenon's physical explanation. Hence, the streamlines and the upper lid surfaces are very close together. Additionally, as seen in Fig. 9, the fluid velocity is likewise reduced function of Ha.

This velocity profile is calculated along the line (0.5, 0) to (0.5, 1). Additionally, when Ha grew, Fig. 8(b) isotherm contours depict a minor shift in the temperature field. The heat transmission rate (Nuav) is at its highest when the magnetic effect is dormant (Ha = 0). For Ha = 25, the Nuav is lowered by 25.43 %. When the Ha number is adjusted from 25 to 50, the falling rate is 29.53 %. Furthermore, at the highest magnetic impact value (Ha = 100), this lowered rate is 23.82 %. Similar results have also been noted in the study conducted by Islam et al. [[2], [47]].

4.3 Impact of nanoparticle volume fraction

Fig. 10, Fig. 11 shows the results of ϕ (for 0, 0.025, 0.05, and 0.1) on the streamlines, isotherms, for an octagonal heat exchanger enclosure when Pr = 6.9, Ri = 1, Re = 100, and Ha = 10. Fig. 10(a) shows that the fluid flow pattern is impacted by fluctuations of ϕ. Firstly, observe the base fluid mode (for ϕ = 0) at the equilibrium of the buoyant forces and inertia forces. Consequently, the top lid walls experience a movement effect from the core vortex. Also, the consistent streamlines clearly show the control of mixed convective fluid movements. Furthermore, the vortex in the streamlines widens as the value of ϕ increases. The physical explanation for these specific streamlining shapes is well known. Because due to add of nanoparticles into water, the friction among nanoparticles and water molecules are occurred. As a result, the flid movement reduce slightly, which is clearly explained in Fig. 11. This velocity profile is calculated along the line (0.5, 0) to (0.5, 1). As the value of ϕ grows, the streamline pattern closely resembles pure conduction.

Additionally, a negligible effect of hybrid nanofluids on convection is found upon the adding of hybrid nanoparticles in base fluid within the cavity. On the other hand, the isotherm lines in Fig. 10(b) and the velocity field show that the rate of thermal transit increases as ϕ values rise where the Nuav is also being increased.

4.4 Effect of hybrid nanofluid

Here, Nuav is utilized with the effects of Re and Ha to explain the rate of heat transport for various fluids. The impact of Ha on Nuav for various fluid types, with or without nanoparticles present in the base fluid, is depicted in Fig. 12(a). Clearly, Nuav steadily declines as the magnetic effect (Ha) proliferation's. The Nuav is raised by 11.23 % and 19.76 %, respectively, for TiO2-H2O and Cu-TiO2 nanofluids. Furthermore, the Nuav is quickly flipped for Cu-TiO2-H2O hybrid nanofluid, which is 25.75 % higher than in H2O. Furthermore, as Ra rises, Nuav grows monotonically for pure fluid (H2O), TiO2-H2O nanofluid, Cu-H2O nanofluid, and Cu-TiO2-H2O hybrid nanofluid, as shown in Fig. 12(b). Additionally, the Nuav increases by 3.46 % for TiO2-H2O nanofluid, and 4.93 % for Cu-H2O nanofluid. Also, the rate of Nuav is swiftly flipped for the Cu-TiO2-H2O nanofluid, which is 6.73 % more than H2O.

4.5 Response surface methodology

The response function's impact from the relevant parameters (Ha, Re, and ϕ) Nuav is explained by the statistical RSM method. One of the best methods for modeling multidimensional situations where input elements simultaneously affect the responses that pique attention is response system modeling (RSM) [[48], [49]]. Even though there are numerous RSM models, the second-order RSM model often provides a good enough approximation of the response. According to some, the quadratic RSM model is:(29) y=z0+∑i=13zixi+∑i=13ziixi2+∑i=13zijxixj

Here, the intercept term is z0, the response function (output) denoted by y, the linear term of the ith factor regression coefficient is denoted by zi, the coefficient of quadratic term for ith factor zii, and the collaboration of ith and jth factors are denoted by zij. Furthermore, the pertinent parameters Ha, ϕ, and Re are used as independent factors, while Nuav is considered as the response faction (y). Finding a best-fitted correlation between y and independent factors is the major goal. Here, a second-order RSM model with a CCD basis is used [50]. The range of the independent parameters Re, Ha, and ϕ in this study is and there are 20 runs in this CCD-based RSM model (8 cube, 6 center, and 6 axial points). Table 6 provides information on the codded levels of the parameter for CCD. Table 7 shows the computational run parameters for both coded and actual values centered on the CCD.Table 6 CCD design factors and codded levels.

Table 6Factors	Level	
−1 (lowest)	0 (medium)	1 (highest)	
Re	10	105	200	
Ha	0	50	100	
ϕ	0	0.05	0.1	

Table 7 Levels of input factors and response function.

Table 7Run Order	Codded Values	Real Values	Response	
A: Re	B: Ha	C: ϕ	Re	Ha	ϕ	Nuav	
1	0	0	0	200	100	0	4.2407	
2	−1	0	0	200	100	0.1	26.157	
3	0	−1	0	200	0	0.1	57.397	
4	0	0	0	105	100	0.05	9.1101	
5	0	1	0	10	0	0	3.3431	
6	0	0	0	105	50	0.1	14.282	
7	1	1	−1	200	50	0.05	20.855	
8	0	0	1	10	100	0	2.6246	
9	−1	−1	−1	10	0	0.1	3.8402	
10	−1	1	1	105	0	0.05	20.498	
11	−1	−1	1	200	0	0	11.654	
12	0	0	0	105	50	0.05	10.223	
13	1	−1	1	105	50	0.05	10.223	
14	0	0	−1	105	50	0.05	10.223	
15	−1	1	−1	105	50	0.05	10.223	
16	0	0	0	10	50	0.05	3.0253	
17	1	−1	−1	105	50	0	4.2483	
18	1	0	0	10	100	0.1	3.1759	
19	0	0	0	105	50	0.05	10.223	
20	1	1	1	105	50	0.05	10.223	

Table 8 displays the results of the statistical analysis using RSM. The maximum number of independent terms in this model is represented by the degrees of freedom (DOF). One method for representing the overall variance resulting from several factors is the total sum of squares (SS). The Adj. SS, have a significant value of 316.87. The F-value, which is 31.95 and indicates that noise does not affect the results, indicates that the Nuav model is statistically significant. Furthermore, the p-value, which determines the likelihood that the null hypothesis would be true for a specific statistical model, is a highly significant signal of this statistical investigation. A small p-value (typically ≤0.05) indicates that the predicted outcome is appropriate. It is evident from Table 8 that all input elements are important for this approach. Furthermore, the model statistical analysis and testing methods indicate high values of the R2 (96.64 %) for Nuav, indicating that this model is appropriate for calculating the Nuav response function. Although the model modified R2 (93.6 %) is lower than R2 (96.64 %), the model nevertheless sufficiently explains the experimental data. Another important metric is Lack of Fit, which requires very little to be a suitable model. The general models that RSM created to examine the connection between the effective input parameters (Re, Ha, and ϕ) and the response (Nuav) are as follows:(30) y=z0+z1Re+z2Ha+z3φ+z11Re2+z22Ha2+z33φ2+z12Re.Ha+z13Re.φ+z23Ha.φ}

Table 8 Analysis of variance for Nuav.

Table 8Source	DOF	Adj. SS	F-Value	p-Value	Comment	
Model	9	316.87	31.95	<0.0001	Significant	
Re	1	1087.74	109.67	<0.0001		
Ha	1	264.44	26.66	<0.0004		
ϕ	1	620.02	62.51	<0.0001		
Re2	1	3.75	0.3784	0.5522		
Ha2	1	44.71	4.51	0.0597		
ϕ2	1	6.24	0.6294	0.4460		
Re*Ha	1	173.64	17.51	0.0019		
Re*ϕ	1	554.63	55.92	<0.0001		
Ha*ϕ	1	70.64	7.12	0.0235		
Lack-of-Fit	5	19.84	–	–	Insignificant	
Pure Error	5	0.000	–	–		
Total	19	2951.01	–	–		
**Here, R2 = 96.64 %, Adjusted R2 = 93.61 %.

Here, z0, z1, z2, z3, z11, z22, z33, z12, z13, and z23 are the coefficients of the best fitted regression model for this RSM model. Additionally, Table 9 displays the coded units that are used to compute the projected coefficients of equation (30) for Nuav. To create a suitable regression equation, solely those terms with low p-values (≤0.05) are considered. On the other hand, the terms that are not significant are disregarded (bold indicated). That is, important terms for the response function (Nuav) are Re, Ha, ϕ, Re. Ha, Re. ϕ, and Ha. ϕ. On the other hand, since Re2, ϕ2 and Ha2 have no bearing whatsoever on Nuav, they should not be included in the final best-fitted regression model.Table 9 Regression coefficients for Nuav that are predicted based on RSM.

Table 9Coefficients	z0	z1	z2	z3	z11	z22	z33	z12	z13	z23	
Values	1.2183	0.0439	−0.1017	93.1248	0.00012	0.00161	−602.672	−0.00098	1.7529	−1.1886	
p-values	–	<0.0001	0.0004	<0.0001	0.5522	0.0597	0.4460	0.0019	<0.0001	0.0235	

That is, for Nuav regression equation (30) the terms Re2, Ha2 and ϕ2 have no bearing at all. Therefore, the following mathematical summary may be used to represent the association between Nuav and the components Re, Ha and ϕ:(31) Nuav=1.21830+0.0439Re−0.1017Ha+93.1248φ−0.00098Re.Ha+1.7529Re.φ−1.1886Ha.φ}

Here, equation (31) is the best fitted correlation between the Nuav and the independent factors.

4.6 Response surface analysis

Fig. 13, Fig. 14, Fig. 15 show 2D and 3D contour plots of the RSM-created response surface (Nuav) to explore the effect of included factors on the response function (Nuav). The influence of Re and Ha on Nuav is seen in Fig. 13(a). This 2D contour map makes it evident that, although the other factor ϕ (0.025) is unchanged, the Nuav increases when Re is being raised and Ha is being decreased. The Nuav reaches its maximum at Re = 200 and Ha = 0, and its minimum at Re = 10 and Ha = 100. Moreover, Fig. 13(b) illustrates a 3D surface map that shows both Re and Ha affect Nuav. That is, these two response surfaces display the same behavior which is similar to the outcomes of the applied FEM. Similarly, Fig. 14(a) expresses a 2D graphical depictions that show that with intensification of ϕ and decrease of Ha causing the response function to rise. At that point, the value of Re stays at 100. When ϕ = 0.1 and Ha = 0, the Nuav gets uppermost value, and when ϕ = 0 and Ha = 100, it is at its bottommost. Furthermore, the influence of Ha and ϕ on the response function is shown in a 3D perspective in Fig. 14(b). Similarly, Fig. 15 illustrates Nuav oscillations with Re and ϕ once again. In Fig. 15(a), the Nuav rises when both the values of Re and ϕ are enhanced while holding another component (Ha = 10) constant. The Nuav reaches its maximum at Re = 200 and ϕ = 0.1, and its minimum at Re = 10 and ϕ = 0. Additionally, Fig. 15(b) depicts a different 3D surface plot that shows how Re and ϕ affect Nuav these two response surface plots show a similar action on Nuav.

To prove the validity statistically of the outcomes of the applied FEM, the statistical technique RSM is also used. The outcomes of the FEM are homologous to the RSM. This agreement strongly supports the proposed mixed convective numerical analysis using hybrid nanofluid.

5 Conclusions

The mixed convection properties of hybrid nanofluid (Cu-TiO2-H2O) has been delved in a specially designed octagonal heat exchanger with two heating and cooling source inside cavity domain. The suspended nanofluids in the water-based mixture exhibit the ability to undergo phase transitions, changing between different fluid states as they pass through the enclosing domain. The influence of the most significant parameters Re, Ha, and ϕ on Nuav are analysed using the streamline profiles, isotherm line, and velocity field with 2D and 3D investigation. The accumulation of hybrid nanoparticles significantly changes the heat transfer rate as well. The finite element scheme was applied to integrate the heat transfer underlying the governing equations. The results are rigorously displayed using a range of Nusselt number to provide a thorough assessment of heat transfer and detailed contour plots of field variables, which broaden our understanding of the underlying heat transfer phenomenon from a physical standpoint. The main findings of the study are as follows:1. The mixed convective heat exchanger processes are enhanced by the expansion of Re, whereas the substantial effect of the magnetic field reduces total energy exchange and thermal transport mechanisms.

2. The increasing values of Re from 10 to 50, 100, and 200 have resulted in rising the rate of Nuav by 83.25 %, 36.28 %, and 22.27 %, respectively.

3. The mean Nusselt number, Nuav is negatively impacted by the magnetic field. The decreasing rates of the Nuav are 25.43 %, 29.53 %, and 23.82 %, respectively, with rising the value of Ha from 0 to 25, 50, and 100.

4. The 2D and 3D response surfaces obtained from RSM are geometrically explained by using RSM where the findings are almost homologous to the results of FEM.

5. The size of the nanoparticles of hybrid nanofluid (Cu-TiO2-H2O) can be employed up to 10 % (0.1) and Re = 200 keeping Ha = 0 in order to create an efficient mixed convective octagonal heat exchanger.

Data availability

The study uses a numerical method, and the findings do not include any data.

Funding

This research received no external funding.

CRediT authorship contribution statement

Md. Yousuf Ali: Writing – review & editing, Writing – original draft, Validation, Software, Methodology, Formal analysis, Conceptualization. Saiful Islam: Writing – review & editing, Software, Formal analysis, Conceptualization. M.A. Alim: Writing – review & editing, Supervision, Methodology, Formal analysis. Rejowan Ahmed Biplob: Writing – review & editing, Software, Conceptualization. Md. Zohurul Islam: Writing – review & editing, Validation, Software, Formal analysis, Investigation.

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.

Nomenclature

cp Specific heat at constant pressure (J.kg−1 K−1)

k Thermal conductivity (W.m−1 K−1)

g Gravitational acceleration (m.s−1)

Ri Richardson Number

Nu Nusselt number

Ha Hartmann Number

Pr Prandtl number

P Pressure without dimension

U, V Velocity component without dimension

X, Y Dimensionless Cartesian coordinates

u, v Dimensional velocity component (m.s−1)

x, y Dimensional Cartesian coordinates

TiO2 Titanium Oxide

Cu Copper

L Enclosure length (m)

Greek symbols

α Thermal diffusivity (m2 s−2)

β Coefficient of thermal expansion (K−1)

ϕ Particle concentration

θ Non-dimensional temperature

μ Dynamic viscosity (kg.m−1 s−1)

ν Kinematic viscosity (m2 s−1)

ρ Density (kg.m−3)

Ω Vorticity vector

σ Electric conductivity (Ω−1. m−1)

ψ Stream function

Subscripts

hnf Hybrid nanofluid

bf Base fluid

av Average

Tc Cold surface

Th Heated surface

sp Solid particle

HS Heated surface

Acknowledgement

The authors would like to thank the 10.13039/100007225 Ministry of Science and Technology (MoST) of Bangladesh for its financial support. Also, convey sincere appreciation to the Department of Mathematics at Bangladesh University of Engineering and Technology (BUET) for technical assistance.
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