Central composite design modeling in optimizing heat transfer rate in the dissipative and reactive dynamics of viscoplastic nanomaterials deploying Joule and heat generation aspects

Nomenclature

u , v

velocity components ( m s − 1 )

x , y

Cartesian coordinates ( m )

α

Casson parameter

Ha

magnetic parameter

Gm

solutal buoyancy

Gr

thermal Grashof number

Nr

radiation parameter

Pr

Prandtl number

Nt

thermophoresis parameter

Nb

Brownian motion parameter

J H

Joule heating parameter

Q 0

heat source parameter

Ec

Eckert number

Pr

Prandtl number

Le

Lewis number

Kc

chemical reaction parameter

S

suction parameter

q w

heat flux

q m

mass flux

D T

thermal diffusion

D B

Brownian diffusion

Re x

local Reynolds number

q r

radiative heat flux ( kg s − 3 )

γ

thermal diffusivity

σ s

Stefan–Boltzmann constant ( W m − 2 K − 4 )

A c

mean absorption coefficient ( m − 1 )

T

temperature ( K )

T ∞

ambient temperature ( K )

μ B

dynamic viscosity ( kg m − 1 s − 1 )

c p

specific heat capacity

σ

electrical conductivity

1 Introduction

Due to practical significance in the investigation of combined thermal and solutal transport in hydro-magnetic incompressible fluid across extending/dwindling sheets, numerous researchers have been attracted to investigating the flows through stretching and shrinking sheets. These kinds of flows find applications in the broad context of science and technology, including polymer processing, blood flow problems, glass fiber manufacturing, paper and condensation processes, hot rolling, cooling, etc. The boundary layer flow in elongating sheet investigations was pioneered by Sakiadis [1] who considered the continuous cylindrical surface. Later, Crane [2] scrutinized the fluid transport over a propelling elongating plate. Mabood performed a heat transmission analysis on the hydromagnetic boundary layer nanofluid transport past a nonlinear widening surface, where the flow solutions were obtained numerically. Rama Devi and Reddy [3] discussed the radiative and reactive efforts on hydro-magnetic nanofluid movement via an extending sheet. Consequently, studies of Rama Devi and Reddy, Khan et al., Sadighi et al., and Adel et al. [4,5,6,7] present fascinating findings that showcase the distinct behavior of various physical flow parameters of nanofluids past extending surfaces.

The effects of Ohmic heating and energy dissipation are highly critical in domains of large current density and high-velocity or high-viscosity flows, respectively. Joule heating enhances thermal energy generation and impacts heat transfer structures where thermal transportation is vital for protection, effectiveness, or physical performance. This effect is very important in mechanisms where overheating may result in device failure, inadequacy, or deprivation of materials. Joule heating is significant in industrial metallurgy and material processing, nuclear reactors, biomedical applications, conserving structures and thermal transport, and astrophysical space exploration. Viscous dissipation is more important as internal friction generates significant heat, especially in high-shear regions. Dissipation effects find applications in technology and engineering, for instance, aerospace systems, jet mechanical structures, space dynamics, car engines, etc. With such imperative significance, Swain et al. [8] explained the combined effects of energy dissipation and Ohmic on hydro-magnetic movement and past a continuous surface entrenched in a permeable membrane. Mishra and Kumar [9,10] analyzed the heat efficiency of reactive and dissipative magnetohydrodynamic (MHD) nanofluid transmit past a continuous sheet due to Joule heating. Tarakaramu et al. [11] inspected the effects of dissipative and Ohmic heating and on MHD nanofluid motion over a two-way extending sheet. Some additional related papers on energy dissipation and Joule heating are those of Wahid et al., Jayanthi and Niranjan, Rasool et al., Lund et al., and Ouyang et al. [12,13,14,15,16].

Radiative thermal propagation and heat generation/absorption are crucial phenomena in fluid flow analysis, especially in high-temperature environments where they significantly impact heat transfer. These mechanisms are essential in various industrial applications, including power generation, aerospace engineering, automotive industry, electronics cooling, and chemical processing. Effective thermal management by controlling radiative heat transfer and absorption enhances system durability, energy efficiency, and operational reliability in these fields. Suresh Kumar et al. [17] performed an examination of a radiative MHD Casson nanofluid transport considering heat generation, the Hall effect, and activation energy via numerical solutions. The results of radiative mechanisms and higher-order reactive species on non-Darcy-assorted buoyance-driven MHD nanofluid transport with a variable heat source/sink were presented by Sarkar et al. [18]. Alharbi et al. [19] numerically scrutinized the effects of the radiative process on a bioconvective Maxwell-nanofluid transport in a widening sheet with thermal radiation considering exponential heat source/consumption. Explorations of Swain et al., Prakash et al., Khan et al., and Sekhar et al. [20,21,22,23,24] provided a detailed inquiry on the radiative and heat-generating impact on hydro-magnetic nanofluid transport.

Chemical reaction effects on hydromagnetic nanofluid movement are of astonishing practical implication in engineering and technology fields. Plausible uses of this flow can come across in plentiful industrial procedures, for example, the power systems and chemical mechanisms. The study of reactive effects on MHD nanofluid flow via different shapes has been inspected using various arithmetical techniques. Sademaki et al. [25,26] included the effects of reactive species on hydromagnetic fluid transport and various physical flow parameters employing the finite difference method to find the solutions. Reddy et al. [27] discussed the effects of reactive species on radiative MHD Brinkman-type motion across a wavering permeable plate with a diffusion-thermo effect. Shamshuddin et al. [28] explored the outcomes of reactive species on radiative hydro-magnetic micropolar nanofluid transport incorporating activation energy and the Hall effect past an absorbent extending surface numerically. The studies of Shamshuddin et al., Patil et al., Hussain et al., and Khan et al. [29,30,31,32] are worth mentioning due to their inclusion of reactive processes in the MHD nanofluid transport regime with diverse stream geometries. The thermal and solutal flow study in Casson fluids has significant industrial applications. Shamshuddin et al. [33] studied Casson fluid’s thermal and reactive properties at stagnation point movement past a radiative, reactive sheet. Similarly, Ibrahim et al. [34] discussed how species interactions and thermal sources affected the dissipative magnetized flow of Casson nanofluids over a nonlinear permeable surface. Brewster [35] deliberated thermal radiative characteristics, while Liao [36] scrutinized the “Homotopy Analysis Method” for solving nonlinear system of equations. Nadeem et al. [37] researched the MHD Casson nanofluid movement along convective circumstances.

“Response Surface Methodology (RSM)” is a statistical and arithmetical technique for modeling and analyzing models in which numerous variables influence the response of interest. It optimizes procedures by analyzing the correlations amid input variables and intended output, making it excellent for situations in which multiple factors influence the response. This methodology is very useful for determining optimal conditions and enhancing system performance, making it extensively relevant in engineering, manufacturing, and product design. Mishra et al. [38] presented the sensitivity of improved heat movement in Eyring Powell nanofluid transport, specifically across a convective radiating Riga plate with irregular heat sources and flux constraints. Similarly, Pattnaik et al. [39] used RSM to optimize the heat transfer rate in micro-structured fluid flow via a permeable contracting surface, addressing free convection processes. Ontela et al. [40] expanded the research by exploring thermal flow optimization and sensitivity study in hybridized fluid movement across a radiative sheet, highlighting possible implications in solar-powered charging stations. Farooq et al. [41] and Waqas et al. [42] highlighted enhanced heat transfer capabilities of silver-based hybrid nanofluids in industrial and biomedical applications. Alqawasmi et al. [43] and Mishra [44] investigated flow behavior under nonlinear heat sources and radiative influences, while Mishra et al. [45] employed neural networks to model heat transfer across various geometries. Mishra and Kumar [10] further analyzed MHD nanofluid flow incorporating slip conditions and Joule heating effects.

2 Model description

This exploration extends electrically conducting viscous-incompressible MHD 2D steady state Casson nanofluid assorted convective movement via an angle elevated nonlinear penetrable elongating sheet positioned at the x-axis. The velocity U w = p x n − 1 2 ( p > 0 ) of the surface deviates from the track of the x-axis and along the y -axis with the velocity of v = v w . Moreover, the present transport regime is positive ( y > 0 ) . Additionally, this sheet is regularly executed with the equivalence B ( x ) = B 0 x n − 1 2 by a magnetic strength, which is in the pathway of y -axis, as shown in Figure 1. The matching magnetic field persuades the electrically conducting fluid. It ought to be specified that the persuaded magnetic field approaches zero if the Reynolds number is enormously slight. To extend the wall so that the base remains fixed, whereas two undistinguishable and opposite forces are included along the x-axis path. In this, T w and C w are the measured temperature and concentration of nanoparticles, respectively, and as → ∞ , the temperature and nanoparticle volume fraction ambient values are T ∞ and C ∞ , respectively. Along with the impacts of heat generation and Joule heating, the analysis also takes dissipation and suction into account. Assuming the rheological equation of non-Newtonian Casson fluid as [33]

or

Figure 1

Geometrical model of the considered flow.

In the above equation, μ B is the dynamic viscosity of the Casson fluid, p y is the fluid yield stress, π is the product of deformation rate components, expressed as π = e ij e ij , where e ij is the ( i , j ) th components of the deformation rate, and π c is the critical value of π .

The equations that govern the previously indicated flow system near the oblique stretching surface are expressed as [34]:

They are connected to the following initial and boundary conditions:

where B ( x ) = B 0 x n − 1 2 is an inconstant magnetic field.

The radiative thermal flux q r with the Rosseland approximation was deduced by Brewster [35], expressed as

where σ s and A c denote, respectively, the Stefan–Boltzmann constant and mean absorption coefficient. Letting a slight temperature change between the fluid temperature T and free stream temperature T ∞ ′ is expanded as a linear function about a free transport temperature T ∞ after ignoring the second and higher-order terms in ( T − T ∞ ) produces as a result of the form;

Using equations (6) and (7), equation (3) yields

To transmute the set of partial differential equations (PDEs) (equations (3)–(6)), we apply the following similarity parameters following Ibraham et al. [34]

where ψ , η , f ( η ) , θ ( η ) , and ϕ ( η ) are the stream function, local similarity variable, non-dimensional stream function, non-dimensional temperature, and non-dimensional concentration, respectively, Standard equations of the stream function ψ are defined as u = ∂ ψ ∂ y and v = − ∂ ψ ∂ x .

By the use of equation (9), the PDEs are transformed into the following ordinary differential equations (ODEs):

The transformed initial and borderline settings are given by

where prime denotes d d η ,

The dimensional engineering curiosity connected to the problem is as follows:

Local wall friction coefficient S f = μ B ρ U w 2 1 + 1 α ∂ u ∂ y y = 0

Rate of heat transport NU x = x q w k ( T w − T ∞ )

Rate of mass transport Sh x = x q m D B ( C w − C ∞ ) ,where q w is the heat flux given by q w = − k + 16 σ s T ∞ 3 3 A c ∂ T ∂ y y = 0 and q m is the mass flux given by q m = − D m ∂ C ∂ y y = 0 .

Utilizing the similarity parameters in equation (9), the dimensionless engineering quantities can be reduced as:

where the local Reynolds number is expressed as Re x = U w x υ .

3 Numerical method of solution

Equations (10)–(13) used to simulate this inherently non-linear Casson nano-fluid flow across a nonlinear inclined stretched sheet employing the boundary value problem. Normally, getting closed-form outcomes is rather challenging. To find the right consequences, a numerical procedure that applies the Runge–Kutta technique and the shooting approach is utilized. The Runge–Kutta technique is a well-known family of iterative methods for approximating ODE solutions. Now, the BVP is converted to IVP by converting into first-order ODEs, as described below, along with a flow chart procedure (Figure 2)

Figure 2

Numerical methodology flow chart.

Incorporating equation (16) in equations (10)–(12) leads to first-order ODEs as follows:

Leading boundary conditions

The appropriate guesstimate values are selected, and integration is then carried out. Thus, the appropriate step size Δ η = 0.001 is selected.

4 Discussion and outcomes

To comprehend the physical insight of the modeled problem, the effects of various pertinent parameters fused in the problem on the nanofluid momentum, temperature, and concentration are delineated graphically. Shear stress, Nusselt number, and Sherwood number are computationally solved and exhibited in tabular format. The arithmetical computations have been done by considering the constant values of various incorporated parameters in the flow structure that are well displayed in the graphs.

Figure 3 exhibits the effects of Hartmann number Ha on the fluid velocity. The variation of the factor Ha within the flow domain is presented for the consideration of the standard range described as 0 ≤ Ha ≤ 3 . In general, the numerically assigned value of Ha , Ha = 0 , depicts the non-occurrence of the magnetization on the flow profile, while the nonzero variation portrays the characteristic of the magnetization on the fluid velocity. It is stated that enhancing Ha diminishes fluid velocity. The reason behind this circumstance is the Ha lessens the fluid velocity by strengthening the magnetic damping properties (Lorentz force), which work as an added drag on the fluid regime. The occurrence of the resistivity generated from the Lorentz force is due to the inclusion of applied magnetization in the momentum distribution. This results in lower velocities near the walls presenting a thinner bounding surface thickness leading to shrink nanofluid velocity. This behavior is exhibited in two distinct folds, such as S > 0 leading the case of suction and S < 0 showing the behavior of injection. The comparative analysis displays that suction is more pronounced to attenuate the fluid velocity. Figure 4 displays the influence of a non-linear stretching parameter n on fluid velocity where the behavior is depicted for the particular range of n as 0.5 ≤ n ≤ 2 . It is detected that improving n decelerates the fluid pace. Physically, the nonlinear stretching parameter drops the fluid velocity by snowballing the viscid drag and deepening the boundary stratum, which mutually produce superior opposition to fluid movement near the elongating surface. The outcome inclusively lessens the fluid velocity in the borderline layer domain. Figure 5 showcases the distinction of the Casson parameter α on fluid velocity and to show the characteristic, the numerical variation of the factor is presented as 1 ≤ α ≤ 4 . It noticed that mounting α depresses the fluid velocity. This is because α rises the fluid’s yield stress, which brings resistance toward the flow regime. As α upsurges, the fluid needs a strong force to begin moving, inspissation of the boundary layer, and dipping of the shear rate, hence the velocity drop occurs. The influence of thermal Grashof number ( Gr ) on fluid velocity is showcased in Figure 6, in which the variation of Gr is reported within the standard range of 0 ≤ Gr ≤ 3 . The assignment Gr = 0 clarifies the absence of buoyancy, whereas the effect of buoyancy is deployed for its nonzero variation. Physically, the buoyancy occurs for the density variation between the layers of the fluid, which leads to pressure variation. The observation reveals that the pressure at the bottom layer is dominated by the pressure at the top which shows buoyancy. Here, the positive variation of Gr indicates the cooling of the surface since the temperature transfers from the surface toward the ambient state of low temperature region. It is seen that mounting Gr escalates the fluid momentum. The cause behind this is Gr enumerates the supremacy of buoyancy forces over viscidness in free convective motions. As the Gr upsurges, the buoyancy forces become tougher, generating more dynamic convective currents, hence improving the fluid pace. Figure 7 illustrates the impact of Gm on fluid velocity with a significant variation of the factor as 0.5 ≤ Gm ≤ 2 . In this case, the variation is presented due to the concentration difference, and the positive variation of the factor shows the concentration at the surface is higher than that of the ambient state. It is noted that the fluid speed decreases when Gm is elevated. This is because Gm shrinks the fluid speed when the mass buoyancy forces engendered by concentration gradients resist the heat buoyancy forces, forming an opposition to the fluid stream. In this situation, the interface amid solutal and heat buoyancy effects can diminish the complete generative force for fluid transport, causing high fluid pace. Figure 8 displays the variation of fluid velocity when adjusting the inclination angle β , which is reported as π / 6 ≤ β ≤ π . Mathematically, the cosine value retards with the increase in the inclined angle, and it is observed that increasing β values reduce the fluid momentum. The physics behind is the fact that the inclined angle of a surface disturbs the fluid velocity by fluctuating the gravitational forces, escalating the frictional resistance, and plummeting the buoyancy-driven convection, hence it tends to reduce the fluid pace.

Figure 3

Velocity profile performance toward the magnetic field ( M ) .

Figure 4

Velocity profile performance toward n .

Figure 5

Velocity profile performance toward α .

Figure 6

Velocity profile performance toward Gr .

Figure 7

Velocity profile performance toward Gm .

Figure 8

Velocity profile performance toward β .

Figure 9 presents the effect of radiation Nr on the fluid temperature, where the behavior is projected within the range of 0 ≤ Nr ≤ 3 . Particularly, Nr = 0 signifies the behavior of fluid temperature without the occurrence of radiative heat and the nonzero variation portrays the characteristic of thermal radiation. It realized those mounting values of Nr progresses the magnitude of thermal boundary layers by adding radiant energy to the fluid, which gives a snowballing effect on the fluid temperature. Figures 10 and 11 depict the influence of thermophoresis Nt and Brownian motion parameter Nb on the fluid temperature, respectively. The factor Nt is distributed within the certain range of 0 ≤ Nt ≤ 3 , and Nb is presented as 0.1 ≤ Nb ≤ 0.4 . It is noticed that in both cases the temperature of the fluid increased. This is because both Nt and Nb increase the fluid temperature by enabling the re-dispersion of heat via thermophoresis and augmenting heat transport via Brownian movement. The collective outcomes yield more effective energy distribution, hence improving the fluid temperature. The random motion of the Brownian motion and the thermal gradient impact enhance the heat transport properties which encourage the thermal properties. Figure 12 illustrates the temperature demeanor for the response of Ec , where the factor is reported as 0 ≤ Ec ≤ 0.3 . In general, the numerical consideration of Ec = 0 shows the absence of dissipative heat on the energy transport phenomena, and the nonzero variation indicates the role of dissipative heat affecting the fluid temperature. An increasing Ec boosts the energy of material particles due to high energy input stowed in the boundary stratum as an outcome enhances the temperature of the fluid.

Figure 9

Temperature profile performance toward Nr .

Figure 10

Temperature profile performance toward Nt .

Figure 11

Temperature profile performance toward Nb .

Figure 12

Temperature profile performance toward Ec .

Figure 13 demonstrates the influence of Schmidt number Le on the concentration profile, and the factor Le is projected within the standard range of 1 ≤ Le ≤ 1.3 . Physically, the factor Le described as the relationship between the thermal diffusivity and the Brownian diffusivity and the increasing order of Le shows a significant retardation in the Brownian diffusion. It is spotted that fluid concentration decreases as Le increases. Physically, the expansion of Le conveys higher thermal conductivity compared to mass diffusivity. In this case, the solutes in the fluid diffuse at a very low pace, leading to minimal effectiveness in solute transmission, sharper concentration gradients, and ultimately declining the fluid concentration. Figure 14 shows the variation of Brownian motion parameter Nb in fluid concentration. It is perceived that enhancing the Nb diminishes the fluid concentration. This is because boosting the Nb drops the fluid concentration as it augments the random motion of particles, resulting in superior dispersion and distribution all around the fluid. This condenses the gathering of particles and compresses concentration gradients, subsequently reducing the fluid concentration. Figure 15 displays the variation of thermophoresis on fluid concentration. It is perceived that enhancing the Nt improves the fluid concentration. The physics behind this is, higher Nt heightens the fluid concentration by triggering elements to move from warmer to cooler points, resulting in accretion in cooler points. This effect leads to a deepening boundary stratum and high varying particle dispersion within the fluid, hence increasing the fluid concentration.

Figure 13

Concentration profile performance toward Le .

Figure 14

Concentration profile performance toward Nb .

Figure 15

Concentration profile performance toward Nt .

4.2 RSM

An enhanced and effective design is proposed for analyzing the thermal transport rate, a sophisticated statistical technique, which is presented in this section. To prepare a predictive model, RSM is utilized for the response employing several characterizing factors. In particular, RSM is the assortment of both the arithmetical and statistical methods. Moreover, the major goal of the proposed discussion is to optimize the response which is characterized by several factors considered within their proper range. The proposed technique is employed in describing the role of three distinct factors such as Nb, Nt, and Nr for the response of thermal flow rate, i.e. Nusselt number. The machine simulation process is supported for the ranges of these factors as Nb ∈ [ 0.1 , 0.3 ] , Nt ∈ [ 0.1 , 0.3 ] , and Nr ∈ [ 01 , 0.5 ] , respectively. The main goal of this technique is to find the effective role of the considered three factors that maximize or minimize Nu. Particularly, machine design is developed by utilizing “Central Composite Design” (CCD) and BoxBehnken Design (BBD) in RSM, but for the proposed problem CCD is adopted for the robustness and efficiency. As per the design of the model, three different levels for each of the factors are considered, and these are generally low, medium, and high, where each are coded as −1, 0, and 1, respectively. The range of each of the factor and their levels are presented in Table 5.

Table 5

Range and their levels of factors for Nusselt number

Parameters	Level
	Low (−1)	Medium (0)	High (1)
Nb	0.1	0.2	0.3
Nt	0.1	0.2	0.3
Nr	0.1	0.3	0.5

Table 6 based on the experimental set up for the methodology and then CCD is used to predict a design which contains a total of 20 experimental runts such as factorial, axial, and central points. For each combination of factors presented in Table 6, the response of Nusselt number is recorded. The variation of the parametric values is presented based upon these 20 runs for which the calculation of the Nusselt number is obtained, which is called the observed values of the Nusselt number.

Table 6

Experimental design for heat transfer rate

Runs	Real values			Response	Predicted fits	Predicted standard error fits	Percentile error
	Nb	Nt	Nr	Nusselt number
1	0.1	0.1	0.1	1.0758	1.07274	0.0020217	0.284038
2	0.3	0.1	0.1	0.6593	0.66087	0.0020217	−0.238786
3	0.1	0.3	0.1	0.8083	0.81002	0.0020217	−0.213327
4	0.3	0.3	0.1	0.4654	0.46300	0.0020217	0.514758
5	0.1	0.1	0.5	1.2041	1.20646	0.0020217	−0.196356
6	0.3	0.1	0.5	0.8266	0.82484	0.0020217	0.212398
7	0.1	0.3	0.5	0.9609	0.95929	0.0020217	0.167102
8	0.3	0.3	0.5	0.6395	0.64252	0.0020217	−0.472919
9	0.1	0.2	0.3	1.0123	1.01287	0.0015905	−0.056577
10	0.3	0.2	0.3	0.6490	0.64855	0.0015905	0.068917
11	0.2	0.1	0.3	0.9341	0.93497	0.0015905	−0.093430
12	0.2	0.3	0.3	0.7132	0.71245	0.0015905	0.104777
13	0.2	0.2	0.1	0.7320	0.73415	0.0015905	−0.294088
14	0.2	0.2	0.5	0.8928	0.89077	0.0015905	0.227069
15	0.2	0.2	0.3	0.8185	0.81846	0.0007804	0.005109
16	0.2	0.2	0.3	0.8185	0.81846	0.0007804	0.005109
17	0.2	0.2	0.3	0.8185	0.81846	0.0007804	0.005109
18	0.2	0.2	0.3	0.8185	0.81846	0.0007804	0.005109
19	0.2	0.2	0.3	0.8185	0.81846	0.0007804	0.005109
20	0.2	0.2	0.3	0.8185	0.81846	0.0007804	0.005109

A statistical methodology, in particular analysis of variance (ANOVA) is used to determine the significance of factors involved in RSM. The procedure is conducted for the suitability of the machine criteria of a suitable choice of confidence interval of 95% or the significant level of 5%. For the proposed factors, the quadratic response surface model fit to the data collected from the experiment is expressed as

Nusselt number = 0.818458 − 0.182160 Nb − 0.111260 Nt + 0.078310 Nr + 0.01225 Nb*Nb + 0.00525 Nt*Nt − 0.00600 Nr*Nr + 0.016212 Nb*Nt + 0.007563 Nb*Nr + 0.003888 Nt*Nr .

The simulated F-values, F = MS Regression /MS Residual , were obtained and compared with the critical F-value at a significant level of 0.05 (Table 7). Here, the low p-values ( p < 0.05 ) recommend that the model is significant. It reveals that the design of Nusselt number is dependent upon all the factors involved in it. The adjusted R 2 values in Table 7 indicate a well-fitted model to the data for which the quadratic model is suggested.

Table 7

ANOVA for heat transfer rate

Source	Degree of freedom	Adjusted sum of square	Adjusted mean of square	F-value	P-value	Coefficients
Model	9	0.520441	0.057827	11222.39	0.000	0.818458
Linear	3	0.516935	0.172312	33440.36	0.000
Nb	1	0.331823	0.331823	64396.49	0.000	−0.182160
Nt	1	0.123788	0.123788	24023.39	0.000	−0.111260
Nr	1	0.061325	0.061325	11901.20	0.000	0.078310
Square	3	0.000825	0.000275	53.38	0.000
Nb * Nb	1	0.000413	0.000413	80.15	0.000	0.01225
Nt * Nt	1	0.000076	0.000076	14.74	0.003	0.00525
Nr * Nr	1	0.000099	0.000099	19.18	0.001	−0.00600
Two-way interaction	3	0.002681	0.000894	173.45	0.000
Nb * Nt	1	0.002103	0.002103	408.08	0.000	0.016212
Nb * Nr	1	0.000458	0.000458	88.79	0.000	0.007563
Nt * Nr	1	0.000121	0.000121	23.46	0.001	0.003888
Error	10	0.000052	0.000005
Lack-of-fit	5	0.000052	0.000010
Pure error	5	0.000000	0.000000
Total	19	0.520493

R 2 = 99.99 % ; a djusted R 2 = 99.98 % ; p redicted R 2 = 99.81 % .

Figure 16 shows a contour profile representing the percentile error between the actual versus predicted values. The plot analyses the difference between the actual and predicted values considered in various ranges, using level curves to display the relationship between them. Percentile error quantifies the deviation within the actual and predicted values at particular levels. In the figure, the darker regions indicate higher errors, while lighter regions represent lower errors. The diagonal line signifies perfect predictions. Deviations from this diagonal are referred to as bias, the spread of points around it represents variance, and points far from the diagonal are considered outliers.

Figure 16

Contour profile for percentile error.

In general, residual plots are valuable tools for assessing the goodness of fit in a regression model. It represents the differences between predicted and actual values, and these variations are analyzed through residual plots. Figure 17 shows residual plots in four different levels. The normal probability plot is presented to evaluate whether the residuals follow a normal distribution for the fitted data. Since the residuals are approximately normally distributed, most data points lie close to a straight line, indicating the model’s validity. The residual versus fitted value plot is employed to detect potential issues such as nonlinearity or unequal error variances. The histogram of residuals illustrates their distribution and checks for normality, showing a bell-shaped curve, which further confirms that the residuals are normally distributed. Finally, the plot of residuals against the observation order is used to check for any patterns, ensuring the independence of observations.

Figure 17

Residual profiles for Nusselt number.

Figures 18–23 show the graphical illustration for the Nusselt number utilizing the factors considering the values of these parameters. For each of the factors, the behavior of Nusselt number is depicted through contour and surface maps. These illustrate the Nusselt number and their prediction. In Figures 18 and 19, the effects of the factors Nt and Nt on the heat transfer rate considering the factor Rd at a central level are shown. It is shown that the response of Nusselt number retards the variations of both Nb and Nt. The factor in Figures 20 and 21 shows the role of Nb and Nr in the contour as well as surface plots for the Nusselt number, keeping Nt constant at a central value. There is a significant hike in the heat transfer rate, as depicted with increasing values of thermal radiation, but the Nb retards. In Figures 22 and 23, the contour vis-à-vis surface plots for the heat transfer rate are deployed, considering the variation of Nt and Nr. The factor Nt favors in decelerating the profile, whereas the reverse impact is rendered for the variation of Nr.

Figure 18

Contour plot of Nusselt number for Nb and Nt .

Figure 19

Surface plot of Nusselt number for Nb and Nt .

Figure 20

Contour plot of Nusselt number for Nb and Nr .

Figure 21

Surface plot of Nusselt number for Nb and Nr .

Figure 22

Contour plot of Nusselt number for Nr and Nt .

Figure 23

Surface plot of Nusselt number for Nr and Nt .

5 Conclusion

The current study intended to examine the Joule heating effects on steady reactive and dissipative MHD Casson nanofluid transport past a nonlinear inclined elongating absorbent sheet with heat generation. The model’s leading highly coupled non-linear PDEs have been converted into ODEs using apposite similarity variables and then cracked numerically by the RungeKutta shooting technique. The results are abridged as follows:

• The velocity profile is escalated with expansion of Gr , while a converse trend is detected in enhancing Ha , n , Gm , α , and β .

• The fluid temperature exhibited an upsurge with growing values of Nr , Nt , Ec , and Nb .

• The concentration trend of nanofluid upsurged with an increment in Nt , while Le and Nb exhibited the opposite outcome.

• The wall friction mounts with the growth in Gr and α , but it lessens with Ha , β , S , Gm , and φ .

• The Nusselt number expands with an augmentation in Nr but a converse tendency is detected with an escalation of Ec , Nb , Nt , and Q .

• The Sherwood number escalates on expanding Le , Nb , Nt , and Kc .

The flow model that we have examined is a simple case of geometry. Therefore, it would be interesting to perform in a 3D modeling of the problem, where the assumptions of parabolic shapes are not applicable. However, since this involves a simultaneous solution of a nonlinear system consisting of several PDEs, it requires a large and expensive computation time. Thus, it would be advantageous to develop computer programs based on COMSOL, as they provide fast and reliable solving approaches for such complex configurations.