Dynamic Interactions: Non-Integer-Order Heat-Mass Transfer in Magnetohydrodynamic Flow of Non-Newtonian Fluid over Inclined Plates

: The heat and mass transfer phenomenon in the presence of a moving magnetic field has a wide range of applications, spanning from industrial processes to environmental engineering and energy conversion technologies. Understanding these interactions enables the optimization of various processes and the development of innovative technologies. This manuscript is about a non-integer-order heat-mass transfer model for Maxwell fluid over an inclined plate in a porous medium. The MHD flow of non-Newtonian fluid over the plate due to the natural convection of the symmetric temperature field and general motion of the inclined plate is investigated. A magnetic field is applied with a certain angle to the plate, and it can either be fixed in place or move along with the plate as it moves. Our model equations are linear in time, and Laplace transforms form a powerful tool for analyzing and solving linear DEs and systems, while the Stehfest algorithm enables the recovery of original time domain functions from their Laplace transform. Moreover, it offers a powerful framework for handling fractional differential equations and capturing the intricate dynamics of non-Newtonian fluids under the influence of magnetic fields over inclined plates in porous media. So, the Laplace transform method and Stehfest’s numerical inversion algorithm are employed as the analytical approaches in our study for the solution to the model. Several cases for the general motion of the plate and generalized boundary conditions are discussed. A thorough parametric analysis is performed using graphical analysis, and useful conclusions are recorded that help to optimize various processes and the developments of innovative technologies.


Introduction
Non-Newtonian fluids form the wider class of fluids, and these fluids relate shear stress to shear strain in a non-linear relationship.Blood, tooth paste, ketchup, jellies, paint and many others are examples of non-Newtonian fluids.The non-linear rheological properties of non-Newtonian fluids are one of its significant features of importance [1][2][3][4][5].In the literature, we see many models that describe the rheological behavior of non-Newtonian fluids [4].In fact, engineering and industrial problems are non-linear in nature.In comparison, non-Newtonian fluids are more important than viscous and Newtonian fluids due to the fact that they deal with the complex phenomena of real-life problems.Non-Newtonian fluids are employed across various industries to meet specific functional and performance needs.In the oil and gas industry, for instance, they are used as drilling muds in oil wells to effectively suspend and transport drill cuttings to the surface.These fluids exhibit different rheological behaviors depending on the drilling phase.In the pharmaceutical industry, many products have non-Newtonian properties to ensure proper application, absorption and consistency during storage.Blood is another example of a non-Newtonian fluid, displaying shear-thinning behavior.This allows it to flow more easily through narrow capillaries while maintaining higher viscosity at rest to prevent excessive bleeding.Moreover, in latex paints, non-Newtonian behavior helps control flow during application and reduces dripping afterward.Further, non-Newtonian fluids are used in plastic extrusion processes, where their viscosity changes with shear rate can be controlled to achieve desired product properties.
Magnetohydrodynamics (MHD) is the study of electrically conducting fluids, such as plasmas, saltwater and liquid metals, where magnetic and velocity fields interact.This field, also known as hydromagnetics or magneto fluid dynamics, was pioneered by Hannes Alfven.MHD explores how magnetic fields generate currents in conducting fluids, affecting the magnetic field and imposing forces on the fluid.Its applications are diverse, including understanding the ionosphere, Earth's magnetic field generation, the electromagnetic pumping of molten metals, induction furnaces and casting processes.In essence, MHD investigates the complex interplay between magnetic fields, electric currents and fluid motion.MHD has significant implications and applications in various fields, including geophysics, astrophysics and space physics [6,7].A magnetic field may be used to stabilize a turbulent stream to the point where it is compelled to stabilize and revert to a laminar flow; transverse or coplanar magnetic fields may be used to achieve this.In his 1954 work, Stuart [8] studied the effects of coplanar and transverse magnetic fields.A component of velocity which is normal to the lines of force induces a restraining force that is opposite in direction but proportional in magnitude to the applied magnetic field in this case.It also usually minimizes the initial instability.On the other hand, when the magnetic field is perpendicular to the direction of flow, the velocity distribution shifts.At higher magnetic parameter values, this can result in a large increase in laminar run and has a highly positive effect on transition.Additionally, it may increase the instability of fluid flow, leading to turbulence [9][10][11][12][13].
Heat transfer and mass transfer are interconnected in MHD flows due to the interactions between fluid dynamics influenced by Lorentz forces, temperature distribution affected by conductive and convective heat transfer, and species transport driven by diffusion and buoyancy effects from thermal and concentration gradients.Understanding these interactions is essential for accurately predicting the behavior of MHD systems in various practical applications.The effects of the temperature gradient are taken into account while treating convective flow as self-sustaining flow.Numerous theories have been presented in the literature to investigate how heat and mass are transferred in various fluids when they flow and convect [14].Heat transfer occurs naturally from a hotter medium to a colder one, ceasing when thermal equilibrium is achieved.There are three primary modes of heat transfer: (i) conduction, where energy is transferred through the motion of electrons or ions; (ii) convection, which involves the transfer of heat energy through the movement of fluid particles; and (iii) radiation, the transfer of heat through electromagnetic waves.These three mechanisms enable heat transfer, either within a medium or between mediums, until thermal equilibrium is reached.The convective heat transfer phenomenon that naturally arises when a water-based hybrid nano-fluid is in a porous medium with a magnetic field present was examined by Mebarek Oudina [15].Das [16] examined mass and heat transfer effects while accounting for the electrically conducting fluid's spontaneous convective flow across an inclined porous plate.Abo-Dahab [17] considered the Casson nano-fluid flow heat transfer phenomena and examined the issue with the convective boundary conditions and talked about how heat sources and chemical reactions affected the situation.
It is important to keep in mind that most studies on MHD natural convection flows assume an external magnetic field that is fixed to the fluid.However, in studying MHD-free convection flow, Narahari and Debnath [18] looked at two scenarios: a.
The magnetic field is fixed with respect to the fluid (MFFRF).b.The magnetic field is fixed with respect to the plate (MFFRP).Shah et al. [19,20] studied the results for the MFFRF and MFFRP scenarios by changing the chemical reaction parameter and temperature.However, MHD-free convective flows have so many uses in the petroleum industry, nuclear engineering, power generation, space propulsion, solar and stellar structure, radio propagation, electromagnetic pumps, the purification of crude oil and glass fiber, etc. Ahmad [21] investigated free convection, thermal diffusion and mass transfer flow in an electro-conducting viscous MHD fluid with constant density.
Among various non-Newtonian fluids, Maxwell fluids prove invaluable in heat and mass transfer modeling because they adeptly capture the intricate viscoelastic properties exhibited by real-world fluids [22][23][24].Through the integration of time-dependent responses and non-Newtonian flow attributes, Maxwell fluid models facilitate precise forecasts of fluid dynamics, heat transfer and mass transportation across various engineering and scientific domains.
Researchers have recently shown increased interest in studying porous media due to their diverse applications in fields like ceramics, filtration, chromatography, biomechanics, insulation and more.A porous medium is a material with empty spaces between solid particles that allow fluids or gases to pass through.Understanding the complex pore structure is crucial for characterizing permeability and porosity (the amount of void space in the medium).However, mathematically describing fluid flow through porous media is a highly complex task, making it a rarely addressed problem in fluid mechanics literature.Furthermore, porous media play a crucial role in various engineering applications, especially in the study of heat-mass transfer models.The complex network of pores within the media promotes the mixing of fluids, leading to improved heat and mass transfer rates.This mixing helps in distributing heat or mass more uniformly throughout the medium [25,26].
Fractional derivatives have emerged as a powerful tool for describing complex dynamics in various fields, including fluid mechanics.By replacing integer-order derivatives with fractional-order operators (FoOs), ordinary models can be converted into fractional models.FoOs have been successfully used to model the rheological behavior of polymer solutions and melts, capturing their unique characteristics.Moreover, FoOs have been found useful in describing physical problems involving memory effects, such as those with non-Newtonian fluids.Fractional calculus (FC) was developed as a result of the complexity of the artificial and natural phenomena which cannot be solved with the help of classical calculus [27][28][29][30][31][32][33].There were numerous features of natural phenomena that classical calculus was unable to capture.The topic of FC acquired a lot of interest because it deals with complicated non-linear systems.It has been recently noted that it is a useful tool for generalizing physical concepts in several fields, including physics, engineering, mathematics, medicine, biology, finance, signal processing, image processing, material science, environmental science, computer science and robotics [34][35][36][37][38][39][40].For the past thirty years, fractional-order operators (FoOs) have been recognized as more reliable operators for modeling real-world physical phenomenons compared to classical derivatives, and their importance (FoO/FC) has fascinated many researchers to study them more.For dynamic problems, fractional-order models have become very popular these days.The use of fractional calculus to model various physical and engineering problems yields results that are more precise and accurate, aligning closely with experimental findings, compared to the models developed using traditional calculus.For instance, the results obtained for differential and rate-type fluids through fractional modeling have great similarity to the experimental results.Riaz et al. [41] studied the role of non-local and local differential operators in study of the heat-mass transfer for Maxwell fluids and presented a comparative analysis for the model using different FoOs.For further details on the rheological characteristics of fluids using FoOs, we refer to [42][43][44][45].
In the existing literature, the heat-mass transfer models are mostly investigated for Newtonian fluid or MHD-free convection flows for fixed magnetic fields.Moreover, few studies involve non-Newtonian fluids in the regime of fractional calculus modeling, but the choice of FoO is not appropriate.Furthermore, these investigations involve specific boundary conditions.In order to fill this research gap, the main objective of this manuscript is to investigate the influence of heat-mass transfer for a class of non-Newtonian fluids with fractional-order derivatives over an inclined plate in a porous medium.By customizing the parameters involved in the constitutive equation of the non-Newtonian fluid, for example, setting the relaxation parameter equal to zero, the constitutive equation of the Newtonian fluid is recovered.Hence, our model provides a basis for the generalized model approach.Moreover, among various FoOs, the Atangana-Baleanu derivative operator in the sense of Caputo (ABC-FoO) is more suitable for dynamical systems involving temperature effects [41].So, the fractional analogue of the heat-mass transfer model is obtained by employing ABC-FoO.The MHD flow of non-Newtonian fluid over the plate due to the natural convection of the symmetric temperature field and the general motion of an inclined plate is investigated.A magnetic field is applied at an angle to the plate, and it can either be fixed in place or move along with the plate as it inclines.The Laplace transform (LT) method and Stehfest's numerical inversion algorithm are utilized to solve the model with respective initial and boundary conditions.To clarify the direct relevance of previous studies to the current research objectives, the novelty of our current research objectives involves the following: 1.
The investigation of heat-mass transfer for a class of Maxwell fluids (a sub-class of non-Newtonian fluids) with ABC-FoO over an inclined porous medium; 2.
The fluid dynamics are affected by the plate's inclination and the magnetic field's slanted angle.Furthermore, by adjusting the plate's inclination and slanted angle, we may talk about the situations involving horizontal and vertical plates as well as coplanar and transverse magnetic field influences; 3.
The general functions of time determine the general plate motion, concentration and temperature distribution of the plate, and these are taken into consideration in our model; 4.
The LT method and Stehfest's numerical inversion algorithm along with the undetermined coefficient method are utilized to solve the model with respective initial and boundary conditions; 5.
For the validation of our investigation, several results from the literature can be recovered from our general solutions by customizing the parameters and functions in our generalized results.Thus, the issue interconnected to comparable models is entirely resolved; 6.
Several cases for the general motion of the plate and generalized boundary conditions are discussed; 7.
A thorough parametric analysis is performed using graphical analysis, and useful conclusions are recorded that help to optimize various processes and the developments of innovative technologies.

Problem Statement
We will examine the dynamics of a specific type of fluid (viscoelastic, incompressible and electrically conductive Maxwell fluid) caused by the motion of a plate with a varying velocity u 0 g(t).In the xy-plane, the plate is inclined at an angle of 0 ≤ γ ≤ π 2 with respect to the x-axis which we consider as the vertical axis.Additionally, the plate is electrically non-conducting.The plate is exposed to a steady magnetic field with a strength of ⃗ B = (B cos θ, B sin θ) directed at an angle θ.This magnetic field is either attached to plate or to the fluid.Initially, at time t = 0, the plate and fluid are stationary with a uniform temperature and concentration.Suddenly, at time t = 0 + (just after t = 0), the plate begins to move along its own axis.As a result, the temperature and concentration near the plate quickly change to new levels, which are the initial levels plus additional amounts described by the functions f (t) and h(t).These new levels are Θ ∞ + Θ w f (t) for temperature and ϕ ∞ + ϕ w h(t) for concentration, where f (t), g(t) and h(t) are functions that are piecewise continuous and vanish at t = 0.The geometry of the model and our research design is shown in figures 1 and 2 as follows:

Formulation of Proposed Problem
By applying the standard Boussinesq approximation and assuming the external magnetic field dominates the induced field, the mathematical model for Maxwell fluid flow, including shear stress, magneto-free convection and thermal radiation, is described by the following system of PDEs.
S ′ (y, t) represents the shear stress and u(y, t) represents the fluid's velocity.
When the magnetic field lines remain stationary with respect to the fluid, Equation (1) applies.When they are stagnant with respect to the plate, then (1) is substituted by [45][46][47].
The parameter ϵ in the above equation is 1 for MFFRP and 0 for MFFRF, respectively.
Multiplying on both sides of (3) by 1 + λ ′ ∂ ∂t and using (2), we obtain Further equations describing the evolution of the temperature and concentration fields are with constraints Using Rosseland diffusion approximation for an optically thick fluid, In addition, if |Θ − Θ ∞ | << 0, the simplified form of Equation ( 5) is written as follows. where ∞ .By applying the following dimensionless quantities to the following equations, (2), ( 4)-( 9), (11), and removing the star notation, we obtain the following dimensionless PDEs: along with boundary (BC) and initial conditions (ICs) where N = For nomenclature see Table 1.
Permeability of porous medium

The Fractional Analogue of the Model
The fractional analogue of the model is obtained by replacing the ordinary time derivative operator by ABC-FoO in ( 14)-( 16): subject to constraints ( 17)-( 19).In the above relations, ABC D α t denotes the ABC-FoO, defined in [29] as where p (0 < p < 1) is the fractional-order parameter, N(p) is the normalization function and the function N can be any arbitrary function that satisfies the conditions N(0) = N(1) = 1, e.g., it can be N(p) = p!.In the present work, we choose N(p) to be unity.Also, E p is the well-known Mittag-Leffler function [32].Moreover, the LP of the ABC-FoO is given as

Solution of Problem
Our model equations are linear in time, and Laplace transforms form a powerful tool for analyzing and solving linear DEs and systems, while the Stehfest's algorithm [48] enables the recovery of original time domain functions from their Laplace transform.Moreover, it offers a powerful framework for handling fractional differential equations and capturing the intricate dynamics of non-Newtonian fluids under the influence of magnetic fields over inclined plates in porous media.So, the Laplace transform method and Stehfest's numerical inversion algorithm are employed as the analytical approaches in our study for the solution of the model.
Our objective is to determine the velocity and shear stress of the fluid in this specific situation.It would not be feasible without the expressions for temperature and concentration.The method we employ to solve Equations ( 20)-( 22) subject to constraints ( 17)-( 19) is the traditional Laplace transform approach [49] and the solution of second-order ODEs.
The solution of (24) along with BCs (25) is
The solution of (27) along BC (28) is

Solution for Velocity Field
Applying LT (20) and making use of ( 26) and ( 29), we obtain along boundary and initial conditions The solution for the differential equation (30) together with the constraint (31) is where and represents mechanical, thermal and concentration contributions, respectively.

Calculation of Shear Stress
Similarly, the fractional analogue of the shear stress equation is Applying the LT on (36), Substituting (35) into Equation ( 37), we obtain where Expressions ( 26), ( 29), ( 32) and (38) give the dimensionless temperature, concentration, velocity and shear stress in the p-domain.The required results in the t-domain can be obtained by taking the inverse LT of these expressions, which is indeed a tedious job.So, we apply Stehfest's algorithm [48] for Laplace inversion to these expressions.Stehfest's algorithm for Laplace inversion is defined as [48] L where j represents a natural number.

Results and Discussion
A thorough graphical analysis is conducted for dimensionless velocity, concentration and temperature profiles against numerous significant parameters, including C R , N, γ, θ, λ ′ , α, K ′ , S ′ C , P ′ re f f , and we examine their influence on the rheology of fluid.The rationale for selecting specific parameters for analysis in our research is based on several key factors like physical relevance, impact on system behavior, parametric sensitivity analysis and comparative analysis.We choose the boundary conditions in terms of general functions of time, and assigning to these functions' suitable forms, we can determine solutions for any situation with technical relevance of this type.For analysis of temperature profiles, we consider the cases when temperature is the linear function of time, i.e., heating the plate at a constant rate and when it changes exponentially as a function of time, a situation of exposing the plate in a hot medium.Likewise, for concentration profiles, we assume the two scenarios namely when concentration is constant, i.e., chemical equilibrium in a closed system and when it is the linear function of time representing a zero-order reaction in an open system.However, for the analysis of velocity profiles, we consider the cases when concentration and temperature are constant, where the plate moves linearly or as a sinusoidal function of time.To avoid repetition, only important graphs of related factors will be explored and discussed in this section.

Dimensionless Temperature Profiles
In this section, dimensionless temperature profiles are plotted versus χ when f (t) = t and f (t) = e t at different times and different variations of the parameters.

(a) Influence of fractional-order parameter
Figure 3 shows that if we increase the values of non-integer-order parameter α, temperature increases.Basically, increasing α reduces the thermal boundary layer thickness, allowing for a greater temperature gradient and increased heat transfer.

(b) Influence of fractional-order parameter
The variation in the behavior of concentration with varying values of non-integerorder parameter α is illustrated in Figure 6.An increase in α enhances the diffusion profiles, leading to a strengthening of the boundary layer thickness, and as result of this, the concentration profile increases.

(c) Influence of Schmidt number
The Schmidt number S ′ c plays a crucial role in determining the concentration distribution of a fluid in a heat-mass transfer model.Figure 7 illustrates that higher S ′ c numbers can lead to a thicker boundary layer, reducing the concentration gradient and resulting in a more uniform concentration distribution.Additionally, it is noted that the velocity profiles for the MFFRP case are more pronounced as compared to the MFFRF case.

(b) Influence of bouncy force parameter
In Figure 9, we see the effect of bouncy force parameter N on the velocity of the fluid.It has been observed that when N increases, the fluid velocity increases.
Since thermal bouncy force is supported by species diffusion for N > 0, and as a result for rising values of N, the velocity of the fluid increases.
However, for N < 0 species, diffusion resists fluid flow because it opposes thermal bouncy forces.So, for decreasing values of N, the velocity of the fluid decreases.

(c) Influence of angle of inclination of plate
As evident from Figure 10, for increasing values of the angle of inclination of the plate, u(χ, t) decreases.Actually, by increasing the inclination of the plate, the thermal boundary layer thickness also increases, and as a result, the temperature gradient decreases, so velocity decreases.

(d) Influence of porosity parameter
From Figure 11, it is observed that for increasing values of K ′ , the velocity of the fluid decreases because if a medium is more porous, then the fluid finds difficulty flowing due to resistance.So, the velocity decreases.

(e) Influence of Schmidt number
The Schmidt number S ′ c is a dimensionless parameter that represents the ratio of momentum diffusivity to mass diffusivity in a fluid.In the context of heat-mass transfer, Schmidt numbers have a significant influence on the velocity of the fluid (as shown in Figure 12).Higher S ′ c numbers can lead to a thicker boundary layer, reducing the velocity gradient and resulting in a slower fluid velocity.

(f) Influence of slanted angle of magnetic field
It is noticed that the velocity of the fluid decreases as the slanted angle θ ∈ [0, π 2 ] increases (as seen in Figure 13).The strength of the magnetic term M ′ = σB 2 sin θ ρ increases with an increase in θ in the interval [0, π  2 ].This has an impact on electrically conducting fluids by producing the Lorentz force, which tends to slow down fluid motion and reduce fluid velocity.
(g) Influence of fractional-order parameter Figure 14 illustrates that velocity exhibits distinct behavior for varying values of noninteger-order parameter α.Moreover, as α increases, the velocity profiles are elevated due to the thinning of the boundary layer, which in turn enhances fluid flow.It is also observed that velocity profiles are higher in the case of MFFRP as compared to MFFRF.

(i) Influence of relaxation parameter
Figure 16 represents the influence of relaxation time on fluid velocity.It shows that as relaxation time increases, the velocity profile decreases.This is because a longer relaxation time leads to a thicker boundary layer which in turn reduces the fluid's velocity.In other words, as the fluid takes a longer time to return to its equilibrium state, the boundary layer grows and the velocity decreases, and a longer relaxation time allows the fluid to retain its deformation for a longer period, reducing its velocity.The numerical trends of the effects of the parameters are summarized in the following Table 2.

Conclusions
A thorough and detailed study of the non-integer-order heat-mass transfer model of MHD Maxwell fluid over an inclined plate has been carried out here, examining the effects of multiple parameters on the model.For concentration, temperature and velocity, the precise results are obtained satisfying the stated constraints.To find the solution to the model, LTs are used and the effects of various factors on fluid flow as well as various boundary conditions are studied.The effects of the effective Prandtl number, bouncy force parameter, chemical reaction parameter, angle of inclination of the plate, slanted angle of the magnetic field, Maxwell fluid parameter, Schmidt number, non-integer-order parameter and magnetic parameter on the fluid's motion are discussed here.The main results are given below:

•
The velocity of the fluid is higher when the magnetic field is stationary with respect to the plate than when it is stationary with respect to the fluid.

•
As the buoyancy forces ratio parameter N increases, the fluid's velocity also increases.

•
Velocity, temperature as well as concentration profiles increase with the increase in non-integer parameter α.

•
The velocity components related to mechanical, thermal and concentration effects are all significant and must be taken into account.

•
When inclination angle and porosity parameter increase, the fluid's velocity decreases.

•
The fluid's motion is retarded by an increase in the chemical reaction parameter, Schmidt number, the magnetic field's slant angle and the relaxation time parameter.

•
The slant angle of the magnetic field and the strength of the magnetic parameter and inclination of the plate could both be used to control the fluid's motion.

Future Directions
Most theoretical models rely on simplifying assumptions to be solvable, which may not fully capture real-world complexities such as non-uniform material properties or transient effects that could significantly influence the results.The assumption of low magnetic Reynold's number and neglect of viscous dissipation may not be justified in certain situations.Additionally, the Maxwell fluid model assumes specific viscoelastic properties that might not accurately represent the behavior of all non-Newtonian fluids.Real systems often involve further complexities, such as variable properties, irregular geometries, external influences and interactions with other physical phenomena, which are challenging to model comprehensively.Our study focused primarily on developing and validating a theoretical heat-mass transfer model for Maxwell fluids with non-integer-order modeling, not yet validated with experimental data.To ensure consistency and accuracy within the theoretical framework, we validated our results by comparing them with other established theoretical analyses.
In future research, this model could be generalized by considering the heat-mass transfer model for Oldroyd-B or Berger fluid models for different geometries like channels or cylinders, and conducting experiments or obtaining experimental data will confirm the accuracy and applicability of the theoretical model in real-world scenarios.

Figure 2 .
Figure 2. Flow chart of research design

Figure 3 .
Figure 3. Profiles of temperature Θ(χ, t) versus χ at t = 2 and t = 4 for P ′ re f f = 4.5 for different values of α.(b) Influence of effective Prandtl number

Figure 4
Figure 4 shows how the temperature changes in response to different values of the effective Prandtl number.A rise in the effective Prandtl number increases the thermal boundary layer thickness.An increase in thermal boundary layer thickness decreases temperature gradient, so temperature decreases.

Figure 4 .
Figure 4. Profiles of temperature Θ(χ, t) versus χ at t = 2 and t = 4 for α = 0.7 for different values of P ′ re f f .

Figure 5
Figure 5 illustrates how the fluid concentration is affected by the chemical reaction parameter C R .These graphs show that fluid concentration decreases as the chemical reaction parameter C R increases.In reality, it is true because as the chemical reaction

Figure 7 .
Figure 7. Plot of concentration ϕ(χ, t) versus χ at t = 2 and t = 4 for C R = 0.7 for different values of S ′ c .5.3.Dimensionless Velocity Profiles (a) Influence of chemical reaction parameter

Figure 8
Figure8illustrates the impact of the chemical reaction parameter C R on the dimensionless fluid velocity, for various plate motions, while maintaining constant concentration and temperature conditions.From the profiles, we observe that an increase in the values of the chemical reaction parameter decreases the velocity of the fluid.In fact, enhancing the C R reaction parameter leads to a decline in fluid concentration across the entire fluid field, resulting in a reduction in concentration-driven buoyancy effects, and as a result, the velocity profile decreases.
Figure 15 indicates that the velocity of the fluid decreases as the effective Prandtl number P ′ re f f increases, due to the thickening of the thermal boundary layer, which reduces the temperature gradient and subsequently decreases the fluid's velocity.

Table 2 .
Numerical trends of the effects of the parameters.