The recent surge in nanofluid research has attracted scholars interest worldwide, mostly due to its extensive applications in several industries, espe-cially leading-edge businesses specialising in nano-devices. The utilisation of nanofluids has been extensively explored and documented in a sub-stantial body of research, as referenced in (Goyal & Bhargara, 2014; Boungiorno, 2006; Al-Maman et al., 2019; Muhamad et al., 2023; Amer et al., 2023; Ayub et al., 2016). The rate of heat conduction and transfer in electronic equipment is very sluggish, which significantly hampers the operation of these systems. By incorporating nanoparticles such as copper (Cu) and alumina (Al) into a base fluid, the efficiency of heat transfer can be improved, resulting in optimal performance of the device. For an extensive examination of the nanofluids in various shapes and conditions, refer to the scholarly pub-lications cited in references (Al-Maman et al., 2019; Muhamad et al., 2023; Amer et al., 2023; Goyal & Bhargara, 2014; Kuznetsov & Nield, 2018; Aziz et al., 2012; Dhanal et al., 2015; Ahmed & Elsaid, 2019; Alam et al., 2015). These papers provide valuable perspectives on many subjects pertaining to nanofluid research and its practical implementations. The imperative to consistently pursue enhancement in thermal conductivity is unavoidable, considering that low thermal conductivity poses a significant barrier in the advancement of energy-efficient heat transfer techniques, which are the vital in numerous applications. 

Goyal and Bhargara, (2014) examined velocity slip boundary condition effect on the heat transfer flow of non-Newtonian nanofluids over a stretching sheet. According to the paper, there is a drop in heat and mass transfer rates as the slip parameter increases  Dhanal et al. (2015). Conducted an investigation on the MHD boundary layer flow generated by the stretching and shrinking. According to the authors, viscous dissipation has a beneficial impact on heat and mass transmission, but Brownian motion has a minimal impact. The study conducted by Ali et al. (2021) demonstrated that as the concentration of Cu in water increases, the fluid velocity decreases, but the fluid temperature and concentration increases.  This study examines the influence of a Cu-MHD hybrid nanofluid on heat transfer and flow in a permeable channel. The impacts of heat flux and viscous dissipation are taken into account, using a Newtonian base fluid. They observed that when the volume percentage grows, the velocity of the fluid increases while the temperature lowers. Additionally, the dissipation of viscosity contributes to an increase in the fluids temperature.  In a related development in Goyal and Bhargara, (2018) the boundary layer flow of a nanofluid was addressed using convective boundary conditions with magnetic effects. The Nusselt number was found to decrease when the inclination, buoyancy ratio, Brownian motion, and thermophoresis parameter rise, whereas it increases with an increased Prandtl number. References (Alam et al., 2009; Ali et al., 2013; Chen, 2004; Ali, 2012; Khan et al., 2011) provide in-depth analysis of detail work on inclined planes, whereas references (Aziz & Khan, 2012; Aziz, 2009; Bejan, 2013; Narahari, 2013) address convective boundary problems. 

The study conducted by Ghalambaz, (2014) investi-gated the spontaneous convection of nanofluids on a vertically oriented heated plate.   It was noted that the thickness of the concentration profile is significantly smaller compared to the thickness of the temperature and velocity profiles. Additionally, it was observed that low convective heating leads to increased Brownian motion and affects the temperature profile. Conversely, Gunisetty et al. (2023) conducted research on the movement of nanofluid on a spinning disc, taking into account magnetic and radiative influences. Their primary discoveries the demonstrated a positive correlation between the velocity profile and the Weissenberg number, and an increase in the electric field and radiative parameter resulted in an enhancement of the magnetic field and porosity. Detailed study on porosity is contained in (Ali et al., 2012; Khan et al., 2011; Mohebbi et al., 2020; Olanrewaju et al., 2013). This study draws inspiration from the array of literature discussed above. Many researchers have extensively studied many facets of nanofluids due to its wide-ranging utilisation in the contemporary technological era, where efficient heat transmission has significant impact (Akter et al., 2023).  

Nevertheless, despite the numerous factors taken into account by multiple authors, there has been limited focus on the study of natural convection nanofluid flow over an inclined porous plate subjected to the convective heating and influenced by a magnetic field.  Essentially, this article delves into this novel concept. The mathematical modeling incorporated factors such as the uniform magnetic field, the permeability, and convective boundary conditions, among other considerations. The equations that describe the flow dynamics are derived and then converted into non-dimensional forms using suitable similarity functions. The numerical computations are performed by utilising MAPLE solver. Findings are depicted through graphs and tables, analysed and discussed intuitively

Mathematical Formulation

A two-dimensional, laminar, stable, and income-pressible flow with constant physical properties is considered. The semi-infinite plate is subjected to inclination φ along the vertical axis. The flow direction is tailored towards the horizontal axis, with a constant magnetic field B_0 in the vertical direction. It is assumed that the lower part of the plate is thermally heated by convection through a hot fluid at a temperature T_f  with a heat transfer coefficient h_f. Further, the base fluid and nanoparticles are con-sidered to be in thermal equilibrium. The schematic depiction of the formulation is thus demonstrated. 

The continuity, momentum, energy, and nanoparticle concentration equations in Goyal and Bhargara, (2018) below serve as the governing equations for the flow under the aforementioned presumptions.

∂u/∂x   +  ∂v/∂x=0                     (1)

ρf(u ∂u/∂x+v ∂v/∂x)= μ (∂^2 u)/〖∂y〗^2 - σ_n B_0^2 u-μ/ρk u+[(1-ϕ_∞ )ρf_∞ βg_e (〖T-T〗_∞ )-

 (〖ρ_p-ρf〗_∞ ) g_e (〖ϕ-ϕ〗_∞ ))]Cosφ                                                                                 (2)

u ∂T/∂x+v ∂T/∂y= α_m  〖∂T〗^2/〖∂y〗^2 + τ[D_B  ∂ϕ/∂y  ∂T/∂y  + D_T/T_∞  (∂T/∂y)^2 ]                                              (3)                                          

u ∂ϕ/∂x+v ∂ϕ/∂x= D_B  (∂^2 ϕ)/(∂y^2 )+  D_T/T_∞   (∂^2 T)/(∂y^2 )                                             (4)                                                                                      

Where u and v are the velocity components perpen-dicular to the plate and parallel to it, respectively; B_0 is the magnetic field;  ϕ is the volume fraction of the nanoparticles; β is the  thermal expansion coefficient of the base fluid; φ is the angle of inclination; D_B is the Brownian diffusion coefficient; D_T is the thermo-phortic diffusion coefficient; and T is the dimen-sional temperature. The boundary conditions we propose are;

u = 0, v = 0,  ϕ = ϕ_w,  -k ∂T/∂y=h_f (T_(f-) T)   at y=0                                            (5)

u = 0, v = 0,  ϕ = ϕ_∞,  T = T_∞      as  y→∞                                                         (6)

With u=∂ψ∕∂y and v=-∂ψ∕∂x, equation (1) is identically satisfied.We deployed the similarity variables below  to Equations (2)- (6);

η=y/x Ra_x^(1/4)  ,ψ=α_m Ra_x^(1/4) f(η)  ,      θ(η)=  (T-T_∞)/(T_f-T_∞ )  ,ϕ(η)=   (ϕ-ϕ_∞)/(ϕ_f-ϕ_∞ )                     (7)                   

Where the local Rayleigh number is expressed as 

R_(a_x )=  ((1-ϕ_∞ )βg_e (T_f-T_∞ ) x^3)/v_(α_m )                                                                                      (8)                                                                                         

Using Eqs. (7), and (8) in Eqs. (2)- (6), we have the equations;

 f^+1/〖4P〗_r  (3ff^-2f^(^2 ) )+ (θ-N_r ϕ)Cosφ-(M+K) f^=0   ()                             (9)

θ^+3/4 fθ^+N_b θ^ ϕ^+N_t θ^(^2 )=0                                                                             (10)

ϕ^+N_t/N_b  θ^+  3/4 Lefϕ^=0                                                                                       (11)

The associated boundary conditions are transformed to the following;

f(η)=1 ,   f^ (η)=0 ,    θ^ (η)=-NC[1-θ(η)],ϕ(η)=1     at  η=0                   (12)

f^ (η) = 0, θ(η)=0 ,ϕ(η)=0  as η→∞                                                                 (13) 

In the context of equations (9) - (13), prime represent differentiation with respect to  η, P_r Prandlt number, N_r Buoyancy-ratio parameter, L_e nanoparticle Lewis number N_(c ) convective heating parameter, K the permeability parameter and M magnetic parameter. The parameters are the mathematically represented below Goyal and Bhargara, (2018).

P_r=μ/α_m ,     N_r=(ρ_p-ρ_f )(ϕ_w-ϕ_∞ )/ρ_f(1-ϕ_w )(T_f-T_∞ )  ,   M=(σB_0^2 x^3)/(μ〖Ra〗_x^(1/2) ),   N_c=(hx^(1/4))/k [(vα_m)/((1-ϕ_∞ ) g_e β(T-T_∞ ) )]^(1/4),K=v/ak                                                                                                                   (14) 

It is crucial to remark that the assessment of nano-fluidic device performance necessitates a thorough consideration of heat and mass transmission.  Hence, it is imperative to consider two essential physical parameters pertaining to the rates of heat and mass transport, as indicated by Siddiqa et al. (2014). The Local Nusselt number, denoted as Nux, and the Sherwood number, denoted as Shx, is mathe-matically represented as follows. 

Nu_x=(xq_w)/k(T_f-T_∞ ) ,〖Sh〗_x=(xj_w)/(D_B (ϕ_w-ϕ_∞ ) )                                                                              (15)

In the context of equation (15);

q_w=  heat flux from the plate

j_w = mass flux from the plate 

These terms are mathematically defined as;

q_w= -k(∂T/∂y),j_w=-D_B (∂C/∂y)   at y=0                                                              (16)

With equations (7) and (8), the dimensionless parameters take the form expressed in equation (17) 

Ra_( x)^(1/4) 〖Nu〗_x=〖-θ〗^ (0),Ra_x^(1/4) 〖Sh〗_x=〖-ϕ〗^ (0)                                                                          (17) 

Method of Solution

Equations (9-11), subject to Eqs. (12), and (13) were numerically treated with Maple Solver. The default numerical solution methods employed by the soft-ware for two-point boundary value problems involve the utilisation of a sixth-order Runge-Kutta Fehlberg method in conjunction with shooting techniques. The effectiveness of these strategies has been substan-tiated by scholarly publications Olanrewaju et al. (2013). A comprehensive examination of our find-ings is presented in the next section.

In this section, we provide graphical figures and a comprehensive discussion to the enhance the under-standing of the physical aspects of this work. The obtained results are graphically shown in terms of velocity, temperature, and concentration profiles.  Tables showing the computations of local Nusselt and Sherwood numbers and a diverse set of relevant parameters are shown. As seen in Table 1, the computed value decreases the monotonically for different values of the Prandlt number. This shows a clear correlation with the results articulated in (Bejan, 2013; Kuznetsov and Nield, 2010).  In Table 2, the computation of Nux and Shx numbers under the influence of several pertinent parameters is presented. Results are in close correlation compare to Aziz and Khan, (2012). The dependency of the Nuselt and Sherwood numbers over variations in Pr, Nb, and Nr with other parameters fixed is presented. It is seen that Nusselt and Sherwood numbers increases as Pr increases whereas for fixed Pr, both decreases monotonically as Nb and Nr increases.  The changes in the Magnetic field parameter, thermophoresis parameter, and the Plates inclination angle on Nux and Shx numbers is depicted in Table 4. The performance of heat and transfer decreases as the plates inclination angle and the magnetic field are steadily increased. The corresponding value of Nt are equally represented in Table 4. Fig. 2, 3, and 4 depict the impact of the magnetic field M on the dimensionless velocity, temperature, & concentra-tion, correspondingly. The data presented indicates a decrease in the velocity profile for increased magnetic field strength, while temperature and concentration are both enhanced. This implies that the Lorentz force, which is a type of resistive force, was generated due to the presence of a magnetic field. The force appears to impede the accumulation of momentum thereby decelerating the fluid flow and opposing the influence of M. The thermal energy is generated as a result of the increased force used during the process of pulling the nanofluid. The energy transfer leads to an increase in the temper-ature of the nanofluid. As a result, the presence of the magnetic field causes a reduction in the thickness of the momentum boundary layer and an augment-ation in the thickness of the thermal boundary layer. 

Furthermore, the phenomenon of the nanoparticle diffusion is facilitated by the warming of the boundary layer, resulting in an elevation of the concentration of the fluid, as depicted in Fig. 4. Fig. 5, 6, and 7 depict the effect of the porosity parameter on the fluids dimensionless velocity, temperature, and concentration profiles. Clearly, it is observed that the velocity diminishes as porosity increases. In practice, this means that higher values of porosity are indicative of increased viscous forces, which lead to a dominance of initial forces, which in turn causes the velocity to decrease. However, a gradual increase in porosity value leads to an observable enhancement in both the dimensionless temperature and con-centration (see Fig. 6 and 7). Furthermore, the de-crease in fluid velocity occasioned by an increase in porosity can be explained more by the presence of porous medium. This medium introduces obstruct-tions and resistance to the flow of the fluid. The medium gradually gets more obstructive as the poro-sity parameter rises, resulting in increased resistance, which finally causes a decrease in the fluids velo-city. In Fig. 8, the influence of nanofluid Lewis number on the dimensionless concentration is pre-sented and shows a sharp decline in concentration of the nanoparticles as Le increase.  Practically, for a base fluid of certain thermal diffusivity D_B, a higher Le indicates a lower Brownian diffusion coefficient, resulting in a shorter penetration depth for the concentration boundary layer. The word "thermo-phoresis" describes how particles disperse when there is a temperature gradient present. Fig. 9 and 10 show, respectively, how the thermophoresis para-meter Nt affects the dimensionless temperature and concentration. It goes without saying that altering Nts value causes a rise in the temperature gradient and a strengthening of the force within nanoparticles. This force causes more fluid to the heat up, which likewise boosts the temperature. Enhancing the thermophoresis Nt effect yields a similar outcome for nanoparticle concentration, as shown in Fig.10. Due to constant collisions between nanoparticles and base fluid molecules, Brownian motion is the zigzag movement of nanoparticles inside the base fluid. Fig. 11 and 12 demonstrate the effect of Nt on tem-perature and concentration. The randomness of the nanoparticles is seen to increase rapidly in a chaotic manner as Nb increases. This chaotic behaviour causes more collisions in the system. This increased collision of nanoparticles causes an increase in heat transfer properties and thus increases the temperature of the fluid. Similarly, the concentration of nano-particles along the wall is hampered by a concurrent increase in Nb. As a result, the value of the nano-particle concentration along the wall decreases as the nanoparticles shift from the boundary to the fluid by increasing their random motion. The impact of the convective heating parameter Nc on the velocity is examined in Fig. 13. A rise in Nc increases the fluids velocity, as shown in the figure. The effects of plate inclination angle from vertical for values of 0^0,π/10, π/9, and π/6  on the dimensionless velocity, tem-perature, and concentration are shown in Fig. 14, 15, and 16, respectively. The fluids velocity decreases along with the boundary layer as the plate inclination angle  φ is changed. The alignment of the plate is what creates the effect, through the buoyancy term in equation 2. The value of Cos φ falls in proportion as the value of  φ  grows. This results in the buoyancy effect disappearing as the plates inclination angle φ increases. As a result, the driving force acting on the fluid weakens, lowering its velocity. This obser-vation corresponds well with Alam et al. (2009) in terms of velocity profile. Also, Fig. 15 and 16 clearly reflect how the depletion of the buoyancy effect enhances thermal and nanoparticle diffusion with a variation in the plates inclination φ. Fig. 17, 18, and 19 show the characteristics behaviour of the buoyancy ratio parameter on the dimensionless velocity, temperature, & concentration. As observed in the figures, while increase in the buoyancy para-meter enhances the fluids temperature and concen-tration, it depletes the velocity.

Table 1: Comparison of 〖Nu〗_x of regular fluid for various values of Pr with Le=10,Nb=Nt=〖10〗^(-5),NC=10,φ=k=0.

The following conclusions are interesting drawn from this article. The inclusion of the convective boundary condition, as opposed to the usual con-tinuous heat flux, is a novel idea (see Fig. 13). This phenomenon is relevant in heat exchanger systems where fluid flow through a solid surface affects the solid surfaces conduction. As the permeability of the plate rises, a decrease in nano-fluid temperature is seen. Convective heating, Brownian motion, and thermophoresis parameters, on the other hand, show the opposite pattern. As magnetic field, permeability, and plate inclination values rise, the nanofluid velocity profile decreases. An increase in the Brow-nian motion parameter decreases the nanoparticle concentration, but contrasting observations are noticed in the case of the thermophoresis parameter. As the plate inclination increases, nanoparticle tem-perature and concentration are enhanced, in contrast to the nano-particle velocity. That is the momentum the boundary layer decays. As the magnetic field increases, the thermal and the nanovolume fraction boundary layers grow while the momentum the boundary layer thickness decreases. The Lorentz force, which is created by the magnetic field, slows down fluid motion. Therefore, the heat transmission can be evaluated by the appropriately adjusting M. Cooling devices frequently use this idea. By taking into account various base fluid and nanoparticle combinations, heat and mass transfer rates can be lowered. This idea applies by modifying the heat transfer rates for industries that use inclined plates during production. However, this study focused on steady state nano-fluid flow and the associated parameters of interest. Subsequent studies will expand on this study by examining a time dependent flow of nanofluid over an inclined heated plate with the inclusion of the joule-heating and other con-siderations. 

We, the authors of this article are immensely grateful to the Editor and Reviewers.

The authors declare no conflict of interest.