Analysis of MHD Casson Nanofluid Flow over a Nonlinearly Stretching Surface with Joule Heating, Radiation and Suction Effects

Many industries require energy-efficient heat transfer of fluids. The only limitation is that thermal conductivity must be low in developing such fluids. So to meet the requirements of industries, a new type of fluid is generated by suspending manometer-sized particles in base fluid [1-5]. Non-Newtonian boundary-layer flows have been the subject of extensive research for several decades due to their wide range of applications in areas such as radial diffusers, thermal oil recovery, and drag reducing agents. Unlike Newtonian fluids, which can be described by a single equation, non-Newtonian fluids exhibit diverse characteristics that cannot be captured by a single model. As a result, various models have been developed to represent the behavior of these fluids. One such model is the Casson fluid model, which has been found to provide more accurate rheological data for many materials compared to other non-Newtonian fluids. Li and Xuan [6] experimented with and investigated nanoparticles' suspension, strangely escalating base fluid's heat exchange behaviour. The heat transmission attribute of a nanofluid escalates by the nanoparticles. Kang et al. [7] clarified the procedure of enhancing of heat conductivity of fluid by using nanoparticles. Jang and Choi [8] discussed various parameters of nanoparticles like volume fraction, Nano unit dimension, and temperature that will describe the heat conductivity of the fluid. Yu et al. [9] reviewed the literature and explained different nanofluids' heat transfer performance. Later, Anoop et al. [10] investigated particle size in the progression of thermal conductivity. They concluded that the mean heat exchange coefficient estimations were augmented with an escalation in the concentration of particle and stream rate.

Casson fluid shows yield stress, and also Casson-type fluid is a pseudoplastic fluid with an extreme thickness at zero shear rates and an elastic limit with no underneath flow. If a shear load is less than the elastic limit operated on the liquid, it performs as a rigid, but if it is more than the elastic limit operated, it initiates progress. Also, Casson nanofluid is particularly useful in cooling of electronic appliances, efficient heat exchanger, drug delivery systems in medical field, processing of food products. Due to these applications Casson nano fluid flow analysis is considered in this study. Dash et al. [11] analysed Casson-type liquid flowing features inside an absorbent medium and noted that flow radius relies on the elastic limit and permeability of the absorbent medium. Hady et al. [12] considered a nanofluid's heat and flow features over a nonlinear extending sheet. They observed that on increasing the radiation parameter, the nonlinear sheet's stretching parameter 'n' causes a drop in the temperature of nanofluid but grows the heat exchange rates. Nadeem et al. [13] examine Casson-type fluid flow across a sheet, which exponentially shrinks and shows an increment in momentum boundary layer thickness. Mukhopadhyay et al. [14] examine a time-dependent Casson-type fluid stream upon an elongating surface and conclude that greater Casson parameter observations enhance the temperature distribution. Swati [15] discussed Casson fluid movement on the non-linear stretching sheet. The nonlinear stretching parameter will increase the skin friction coefficient. Shehzad et al. [16] studied a Casson-type fluid stream with suction and chemical reaction and found that intensification in the chemical reaction parameter will drop the concentration of the liquid. Afterwards, Mukhopadhyay et al. [17] pondered the Casson-type liquid flowing upon a symmetric wedge. They noticed that the heat conductivity rises with increasing Casson parameter. Pramanik [18] examines Casson fluid movement via extending surface through a porous medium and radiation and detected that increasing the Casson parameter will enhance thermal boundary layer thickness. Sheikholeslami et al. [19] considered the radiation influence on the nanofluid flow and determined that heat transmission grows with the rise in radiation. Haq et al. [20] assumed the stagnant nanofluid movement upon an extending sheet and noticed that the radiation parameter enhances temperature distribution. Hussain et al. [21] scrutinized the Casson-type nanofluid flow within the boundary layer with viscous scatteredness. They concluded that the temperature is boosted by the progressive values of thermophoresis and Brownian motion parameters. Hayat et al. [22] analysed a Casson fluid stream with nanoparticles and observed adaptable heat conductivity enhances temperature. Ibrahim and Makinde [23] studied Casson-type nanofluid movement upon an extending sheet through convective conditions and observed that the drag coefficient escalates with an upsurge in the Casson parameter. Ullah et al. [24] deliberated Casson nanofluid stream over a nonlinear extending sheet with thermal radiation using the Keller box method and concludes increasing values of nonlinear parameter diminishes the velocity of the fluid. Kataria and Patel [25] examined the Casson-type fluid flow along with chemically reactive species and thermal radiative via an upright plate and observed that temperature decays with the growth in the Prandtl quantity. Qing et al. [26] deliberate Casson-type nanofluid movement upon a stretchable surface with a permeable medium and realized that the rapidity outline diminishes for the magnetic parameter. Zaib et al. [27] scrutinized the Casson-type nanofluid upon a plate dipped in a permeable material with activation energy and chemically reactive flow. They concluded that the thickness of the peripheral concentration layer intensifies because of activation energy. After that Naqvi et al. [28] debated the Casson-type nano liquid flowing upon an absorbent extending cylinder, and improved temperature is observed for growing estimations of magnetic and the Casson parameters.

The term joule heating refers to the increase in temperature of a liquid caused by its resistance to electric current in electro kinetic flow. Electric energy derived from a flow of electric current is converted into heat when the current passes through a solid or liquid with finite conductivity. The degree of heat experienced by the conductor material is evidenced by an increase in its temperature. Following the principle of energy conservation, Joule heating can be considered a transformation between electrical energy and thermal energy. Hossain [29] studied free convection flow and observed that the presence of dissipation reduces skin friction and heat conductivity. Later, Chen [30] measured the Joule heating impression upon the fluid flow across an extending surface and noticed that the Joule heating decreases the skin friction coefficient. Afterwards, Hayat and Qasim [31] measured the effect of joule heating along with radiation upon the Maxwell fluid flow using the HAM method. They concluded that the concentration field decreases by increasing the Schmidt number. Bachok et al. [32] investigated the nanofluid flow within the boundary layer and observed that the heat transmission boosts with low estimations of the Prandtl quantity. Chamkha et al. [33] probed the influence of Joule heating, radiative, and chemically reactive species upon free convective fluid flow. They detected that boosting the suction/injection variable reduces drag force. Hayat et al. [34] researched the influence of Joule heating over a radiative surface. They noticed that the rapidity diminishes and skin friction growth for the magnetic parameter. Kempannagari et al. [35] explored the Joule heating effect on the exponentially stretched curved surface. They conclude progressive observations of Eckert number outcomes in growth in temperature. In the analysis of Wakif et al. [36] with the rising observations of thermophoresis enhances the temperature distribution. Later Wakif et al. [37] reported in their study that the stability of the hybrid nanofuid will be significantly enhanced by decreasing the spherical size of nanoparticles. Ganesh and Sridhar [38] measured the influence of the chemically reactive Casson-type nanofluid stream above a moving plate by adaptable thickness. Then, the velocity outline declines for the magnetic, Casson, and permeability parameters. BalajiPrakash et al. [39] explored radiative and heat source impacts on vertical porous plates. They concluded that temperature decreases with the increments in the heat source and Prandtl number. Ganesh and Sridhar [40] examined Casson nanofluid flow across Darcy Forchhiemer porous medium and concluded for incremental values of Prandtl number, both thermal and concentration sketches display a falling tendency. Abo-Dahab et al. [41] analysed Casson type nanofluid flow upon a sheet that is nonlinearly stretched through a porous media. They realized that the temperature diminishes with escalating suction/injection parameter amounts. It is observed in the report of Rasool and Wakif [42] velocity field and the associated boundary layer show incremental trend for enhanced observations of modified Hartman factor. Wakif et al. [43] determined that the Nusselt number, which is proportional to the heat transfer rate, can be reduced by an increase in the Lorentz force. Wakif et al. [44] analysed the stability of nanofluid using Wakif-Galerkin technique and concluded that onset of convection is delayed due to the presence of nanoparticles in the fluid. Gumber et al. [45] demonstrated the buoyancy-driven in the micropolar hybrid nanofluid flow upon an upright surface, considering the suction/injection, magneto force, and thermal radiative effect. They have shown that the injection effect is recommended for better heat transfer. Using the Keller-Box technique, Ganesh et al. [46] computationally investigated the forced convection heat transfer of a magneto-Casson-type liquid in a nonlinear porous medium. Moreover, the mixed convection of Casson liquid has been thoroughly investigated by Sridhar et al. [47]. For incremental observations of Brownian motion argument an enhancing pattern on the concentration of nanoparticles nearer to elongating sheet is noted for Wakif et al. [48], Later Wakif et al. [49] remarked that the flow speed of nanofluid reduces expressively with the upsurge in the magnetic constraint and also Substantial improvement in the temperature distribution Wakif and Shahb [50] noted in his study that the surface drag force factor can be reduced by reducing the magnetic constraint.

Ghadikolaei and Gholinia [51] discussed Hydrogen specifications on Cu hybrid nanoparticles and mentioned that the shape nanoparticle has a major effect on the growing Nusselt number. Velocity decreases corresponding to greater observations of the Hartman number mentioned by Gulzar et al. [52]. The remarkable improvement in thermal conductivity resulting from the strong hydrogen bonding of hybrid nanofluids has been extensively discussed by Ghadikolaei and Gholina [53]. In their study, they explore the fascinating effects of hydrogen bonding on thermal conductivity, highlighting the significant advancements that can be achieved in this field. Ghadikolaei [54] further provides a comprehensive review of methods for cooling solar PV cells and emphasizes the efficient utilization of renewable resources to mitigate pollution. The effectiveness of cooling is found to be closely correlated with the volume ratio of nanoparticles, as elucidated by Ghadikolaei [55]. In a study conducted by Akbari et al. [56], the non-Fourier heat conduction over a three-dimensional hollow sphere is analyzed using the Cattaneo-Vernotte model. The researchers shed light on the intricacies of heat conduction in this unique geometry, contributing to our understanding of heat transfer phenomena. Siahchehrehghadikolaei et al. [57] propose a model for efficient CPU cooling and suggest that silver heat sinks outperform copper and nickel heat sinks due to their superior heat conductivity. They highlight the potential of silver heat sinks in improving CPU cooling efficiency. Faghiri et al. [58] investigate blood flow through a circular tube and observe that the Nusselt number for shear-thinning blood flow is significantly higher compared to Newtonian fluid flow. This finding has important implications for understanding blood flow dynamics and its implications on heat transfer processes. Rostami et al. [59] investigated dusty hybrid nanofluid flow utilizing RBF technique and observed that for growing observations of the Eckert number temperature of the nanofluid enhances. Attar et al. [60] investigated a novel model to determine the fractional differential equations. Fallah Najafabadi et al. [61] studied nanofluid flow with the RBF method and concludes increasing the Brownian motion parameter enhances the temperature of the fluid. Zangooee et al. [62] studied hybrid nanofluid flow using 5th order RK method, progressive values of thermal slip, and coefficient of heat transfer reduces, Ghadikolaei et al. [63] proposed a model of CGNP/H2O nanofluid flow model for CPU cooling. Hosseinzadeh et al. [64] investigated non-Newtonian fluid flow over a curved stretching surface and noted that Enhancement in heat transfer rate is observed for greater values of Brownian diffusion and thermophoresis.

When comparing the flow of Newtonian nanofluids to Casson nanofluids, it becomes evident that fluid flow over stretched sheets offers enhanced cooling and reduced friction. This phenomenon finds practical applications in various industries, such as aerodynamic extrusion involves the shaping and forming of polymer sheets using the forces and principles of aerodynamics, hot rolling processes, metallic plates cooling, condensation processes with liquid films, and glass-fiber production. Recognizing the significance of fluid flow over stretching sheets, this study investigates the effects of radiation, Joule heating, chemical reactions, suction/injection, and magnetic fields on Casson-type nanofluid flow over a nonlinearly stretched sheet. By analyzing the influence of these parameters, the study aims to gain insights into the behavior and characteristics of Casson nanofluid flow over a non-linear stretching sheet. The findings are presented through graphs that illustrate the impact of MHD (magnetohydrodynamics), Joule heating, radiation, and chemical reaction on the flow dynamics of Casson nanofluids. This research contributes to our understanding of the complex interplay between various factors in nanofluid flow over stretched sheets and provides valuable insights for practical applications in different industries. The validation of the numerical method is observed by calculating local parameters and comparison with previous literature.

The non-linear ODE mentioned in Eqs. (8)-(10) was further solved using the Keller box method.

1. First, we convert a set of non-linear ODEs to ODEs of the 1^st order.

2. Next, using Newton's scheme, we use finite differences to linearize the system of equations.

3.Furthermore, we obtain a system of linear equations represented in matrix notation.

4. Atlast, the Block tri-diagonal elimination procedure has been applied to Simplify the system. It is solved using MATLAB.

Introducing,

$f^{\prime}=p, p^{\prime}=q, g=\theta, g^{\prime}=t, s=\phi, s^{\prime}=n$

Eqs. (8)-(10) converted to:

$\left(1+\frac{1}{\beta}\right) q^{\prime}+\left(\frac{N+1}{2}\right) f q-K p-M p-N p^2=0$         (14)

$\begin{gathered}\left(1+\frac{4}{3} R d\right) t^{\prime}+\frac{N+1}{2} \operatorname{Pr} f t+\operatorname{Pr} \lambda g+\operatorname{Pr} N b t n+\operatorname{Pr} N t t^2+\left(1+\frac{1}{\beta}\right) \operatorname{Pr} E c q^2+\operatorname{Pr} J p^2=0\end{gathered}$         (15)

$n^{\prime}+\frac{N+1}{2} S c f n+\frac{N t}{N b} t^{\prime}-S c \gamma s=0$      (16)

Introducing finite differences,

$\left.\begin{array}{c}\frac{1}{h_j}\left(-f_{j-1}+f_j\right)=\frac{1}{2}\left(p_{j-1}+p_j\right), \\ \frac{1}{h_j}\left(-p_{j-1}+p_j\right)=\frac{1}{2}\left(q_{j-1}+q_j\right), \\ \frac{1}{h_j}\left(-g_{j-1}+g_j\right)=\frac{1}{2}\left(t_{j-1}+t_j\right), \\ \frac{1}{h_j}\left(-s_{j-1}+s_j\right)=\frac{1}{2}\left(n_{j-1}+n_j\right), \\ \left(1+\frac{1}{\beta}\right)\left(\frac{-q_{j-1}+q_j}{h_j}\right)+\left(\frac{N+1}{2}\right)\left(\frac{f_{j-1}+f_j}{2}\right)\left(\frac{q_{j-1}+q_j}{2}\right)-K\left(\frac{p_{j-1}+p_j}{2}\right) \\ -M\left(\frac{p_{j-1}+p_j}{2}\right)-N\left(\frac{p_{j-1}+p_j}{2}\right)^2=0, \\ \left.+\frac{t_j-t_{j-1}}{3+4 R r}\right)+3 \frac{\operatorname{Pr}^2}{3+4 R d}\left(\frac{g_j+1}{2}\right)\left(\frac{f_j+f_{j-1}}{2}\right)\left(\frac{t_j+t_{j-1}}{2}\right) \frac{3 \operatorname{Pr} N b}{3+4 R d}\left(\frac{t_j+t_{j-1}}{2}\right)\left(\frac{n_j+n_{j-1}}{2}\right) \frac{3 P r N t}{3+4 R d}\left(\frac{t_j+t_{j-1}}{2}\right)^2 \\ +\frac{3 P r E c}{3+4 R d}\left(\frac{\beta+1}{\beta}\right)\left(\frac{q_j+q_{j-1}}{2}\right)^2+\frac{3 P r J}{3+4 R d}\left(\frac{p_j+p_{j-1}}{2}\right)^2=0, \\ \frac{n_j-n_{j-1}}{h_j}+\left(\frac{N+1}{2}\right) S c\left(\frac{f_j+f_{j-1}}{2}\right)\left(\frac{n_j+n_{j-1}}{2}\right)+\frac{N t}{N b}\left(\frac{t_j-t_{j-1}}{h_j}\right) \\ -S c \gamma\left(\frac{s_j+s_{j-1}}{2}\right)=0 .\end{array}\right\}$        (17)

To linearize the equations in (17) using Newton's method, we consider:

$\left.\begin{array}{rl}f_j^{k+1} & =f_j^k+\delta f_j^k, \\ p_j^{k+1} & =p_j^k+\delta p_j^k, \\ q_j^{k+1} & =q_j^k+\delta q_j^k, \\ g_j^{k+1} & =g_j^k+\delta g_j^k, \\ t_j^{k+1} & =t_j^k+\delta t_j^k, \\ s_j^{k+1} & =s_j^k+\delta s_j^k, \\ n_j^{k+1} & =n_j^k+\delta n_j^k .\end{array}\right\}$        (18)

We make the given below usage of linearly formulas:

$\left.\begin{array}{c}\delta f_j-\delta f_{j-1}-\frac{h_j}{2}\left(\delta p_j\right)-\frac{h_j}{2}\left(\delta p_{j-1}\right)=r_{1, j}, \\ \delta p_j-\delta p_{j-1}-\frac{h_j}{2}\left(\delta q_j\right)-\frac{h_j}{2}\left(\delta q_{j-1}\right)=r_{2, j}, \\ \delta g_j-\delta g_{j-1}-\frac{h_j}{2}\left(\delta t_j\right)-\frac{h_j}{2}\left(\delta t_{j-1}\right)=r_{3, j}, \\ \delta s_j-\delta s_{j-1}-\frac{h_j}{2}\left(\delta n_j\right)-\frac{h_j}{2}\left(\delta n_{j-1}\right)=r_{4, j}, \\ a_{1, j} \delta q_j+a_{2, j} \delta q_{j-1}+a_{3, j} \delta f_j+a_{4, j} \delta f_{j-1} \\ +a_{5, j} \delta p_j+a_{6, j} \delta p_{j-1}=r_{5, j}, \\ b_{1, j} \delta t_j+b_{2, j} \delta t_{j-1}+b_{3, j} \delta f_j+b_{4, j} \delta f_{j-1} \\ +b_{5, j} \delta g_j+b_{6, j} \delta g_{j-1}+b_{7, j} \delta p_j \\ b_{8, j} \delta p_{j-1}+b_{9, j} \delta n_j+b_{10, j} \delta n_{j-1} \\ +b_{11, j} \delta q_j+b_{12, j} \delta q_{j-1}=r_{6, j},\\ c_{1, j} \delta n_j+c_{2, j} \delta n_{j-1}+c_{3, j} \delta f_j+c_{4, j} \delta f_{j-1}+ \\ c_{5, j} \delta t_j+c_{6, j} \delta t_{j-1}+c_{7, j} \delta s_j+c_{8, j} \delta s_{j-1}=r_{7, j},\end{array}\right\}$           (19)

herein

$\left.\begin{array}{c}a_{1, j}=1+\frac{\beta h_j}{8(\beta+1)}(N+1)\left(f_j+f_{j-1}\right), \\ a_{2, j}=a_{1, j}-2.0, \\ a_{3, j}=\frac{\beta h_j}{8(\beta+1)}(N+1)\left(q_j+q_{j-1}\right), \\ a_{4, j}=a_{3, j}, \\ a_{5, j}=\frac{-K \beta h_j}{2(\beta+1)}-\frac{M \beta h_j}{2(\beta+1)}-\frac{N \beta h_j}{2(\beta+1)}\left(p_j+p_{j-1}\right), \\ a_{6, j}=a_{5, j},\end{array}\right\}$        (20)

$\left.\begin{array}{c}b_{1, j}=1+\frac{3 \operatorname{Pr} h_j(N+1)}{8(3+4 R d)}\left(f_j+f_{j-1}\right)+\frac{3 \operatorname{Pr} N b h_j}{4(3+4 R d)}\left(n_j+n_{j-1}\right) \\ +\frac{3 \operatorname{Pr} N t h_j}{2(3+4 R d)}\left(t_j+t_{j-1}\right) \\ b_{2, j}=b_{1, j}-2.0, \\ b_{3, j}=\frac{3 \operatorname{Pr} h_j(N+1)}{8(3+4 R d)}\left(t_j+t_{j-1}\right), \\ b_{4, j}=b_{3, j}, \\ b_{5, j}=\frac{3 \lambda \operatorname{Pr} h_j}{2(3+4 R d)}, \\ b_{6, j}=b_{5, j}, \\ b_{7, j}=\frac{3 \operatorname{Pr} J h_j}{2(3+4 R d)}\left(p_j\right)+\frac{3 \operatorname{Pr} h_j}{2(3+4 R d)}\left(p_{j-1}\right) \\ b_{8, j}=b_{7, j}, \\ b_{9, j}=\frac{3 \operatorname{Pr} N b h_j}{4(3+4 R d)}\left(t_j\right)+\frac{3 \operatorname{Pr} N b h_j}{4(3+4 R d)}\left(t_{j-1}\right), \\ b_{10, j}=b_{9, j}, \\ b_{11, j}=\frac{3 \operatorname{Pr} E c h_j(\beta+1)}{2 \beta(3+4 R d)}\left(q_j+q_{j-1}\right), \\ b_{12, j}=b_{11, j},\end{array}\right\}$        (21)

$\left.\begin{array}{rl}c_{1, j} & =1+\frac{S c h_j(N+1)}{8}\left(f_j+f_{j-1}\right), \\ c_{2, j} & =c_{1, j}-2.0, \\ c_{3, j} & =\frac{S c h_j}{8}(N+1)\left(n_j+n_{j-1}\right), \\ c_{4, j} & =c_{3, j}, \\ c_{5, j} & =\frac{N t}{N b}, \\ c_{6, j} & =-\frac{N t}{N b}, \\ c_{7, j} & =-\frac{S c \gamma h_j}{2} ,\\ c_{8, j} & =c_{7, j},\end{array}\right\}$        (22)

$\left.\begin{array}{c}r_{1, j}=\frac{1}{2}\left(2\left(f_{j-1}-f_j\right)+h_j\left(p_j+p_{j-1}\right)\right), \\ r_{2, j}=\frac{1}{2}\left(2\left(p_{j-1}-p_j\right)+h_j\left(q_j+q_{j-1}\right)\right), \\ r_{3, j}=\frac{1}{2}\left(2\left(g_{j-1}-g_j\right)++h_j\left(t_j+t_{j-1}\right)\right), \\ r_{4, j}=\frac{1}{2}\left(2\left(s_{j-1}-s_j\right)++h_j\left(n_j+n_{j-1}\right)\right), \\ r_{5_{i j}}=q_{j-1}-q_j-\frac{\beta h_j}{8(\beta+1)}(N+1)\left(f_j+f_{j-1}\right)\left(q_j+q_{j-1}\right) \\ +\frac{K \beta h_j}{2(\beta+1)}\left(p_j+p_{j-1}\right)+\frac{M \beta h_j}{2(\beta+1)}\left(p_j+p_{j-1}\right) \\ +\frac{N \beta h_j}{4(\beta+1)}\left(p_j+p_{j-1}\right)^2, \\ r_{6, j}=-t_j+t_{j-1}-\frac{3 P r h_j(N+1)}{8(3+4 R d)}\left(f_{j-1}+f_j\right)\left(t_{j-1}+t_j\right) \\ \frac{3 \lambda P r h_j}{2(3+4 R d)}\left(g_{j-1}+g_j\right)-\frac{3 \operatorname{Pr} N b h_j}{4(3+4 R d)}\left(t_{j-1}+t_j\right)\left(n_{j-1}+n_j\right) \\ -\frac{3 \operatorname{Pr} N t h_j}{4(3+4 R d)}\left(t_{j-1}+t_j\right)^2-\frac{3 \operatorname{Pr} E c h_j(\beta+1)}{4 \beta(3+4 R d)}\left(q_{j-1}+q_j\right)^2+ \\ -\frac{3 \operatorname{Pr} J h_j}{4(3+4 R d)}\left(p_{j-1}+p_j\right)^2, \\ r_{7, j}=n_{j-1}-n_j-\frac{S c h_j}{8}(N+1)\left(f_j+f_{j-1}\right)\left(n_j+n_{j-1}\right) \\ -\frac{N t}{n b}\left(t_j-t_{j-1}\right)+\frac{S c \gamma h_j}{2}\left(s_j+s_{j-1}\right) .\end{array}\right\}$       (23)

The linear Eq. (17) resultant system is tackled by the Block elimination technique.

The system's equations in matrix form are:

$\left.\begin{array}{c}{\left[A_1\right]\left[\delta_1\right]+\left[C_1\right]\left[\delta_2\right]=\left[r_1\right],} \\ {\left[B_1\right]\left[\delta_1\right]+\left[A_2\right]\left[\delta_2\right]+\left[C_2\right]\left[\delta_3\right]=\left[r_2\right],} \\ {\left[B_{j-1}\right]\left[\delta_1\right]+\left[A_{j-1}\right]\left[\delta_2\right]+\left[C_{j-1}\right]\left[\delta_3\right]=\left[r_{j-1}\right],} \\ {\left[B_j\right]\left[\delta_{j-1}\right]+\left[A_j\right]\left[\delta_j\right]=\left[r_j\right],}\end{array}\right\}$         (24)

where,

$\begin{gathered}{A_1=\left[\begin{array}{ccccccc}0 & 0 & 0 & 1 & 0 & 0 & 0 \\ -0.5 h_j & 0 & 0 & 0 & -0.5 h_j & 0 & 0 \\ 0 & -0.5 h_j & 0 & 0 & 0 & -0.5 h_j & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 & -0.5 h_j \\ \left(a_2\right)_1 & 0 & 0 & \left(a_3\right)_1 & \left(a_1\right)_1 & 0 & 0 \\ \left(b_{12}\right)_1 & \left(b_2\right)_1 & 0 & \left(b_3\right)_1 & \left(b_{11}\right)_1 & \left(b_1\right)_1 & \left(b_9\right)_1 \\ 0 & \left(c_6\right)_1 & \left(c_8\right)_1 & \left(c_3\right)_1 & 0 & \left(c_5\right)_1 & \left(c_1\right)_1\end{array}\right]}, \\ A_j=\left[\begin{array}{ccccccc}-0.5 h_j & 0 & 0 & 1 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & -0.5 h_j & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & -0.5 h_j & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 & -0.5 h_j \\ \left(a_6\right)_j & 0 & 0 & \left(a_3\right)_j & \left(a_1\right)_j & 0 & 0 \\ \left(b_8\right)_j & \left(b_6\right)_j & 0 & \left(b_3\right)_j & \left(b_{11}\right)_j & \left(b_1\right)_j & \left(b_9\right)_j \\ 0 & 0 & \left(c_8\right)_j & \left(c_3\right)_j & 0 & \left(c_5\right)_j & \left(c_1\right)_j\end{array}\right]\end{gathered}$

where, $2 \leq j \leq J$,

$B_j=\left[\begin{array}{ccccccc}0 & 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -0.5 h_j & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -0.5 h_j & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -0.5 h_j \\ 0 & 0 & 0 & \left(a_4\right)_j & \left(a_2\right)_j & 0 & 0 \\ 0 & 0 & 0 & \left(b_4\right)_j & \left(b_{12}\right)_j & \left(b_2\right)_j & \left(b_{10}\right)_j \\ 0 & 0 & 0 & \left(c_4\right)_j & 0 & \left(c_6\right)_j & \left(c_2\right)_j\end{array}\right]$,

where, $1 \leq j \leq J-1$,

$C_j=\left[\begin{array}{ccccccc}-0.5 h_j & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ \left(a_5\right)_j & 0 & 0 & 0 & 0 & 0 & 0 \\ \left(b_7\right)_j & \left(b_5\right)_j & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \left(c_7\right)_j & 0 & 0 & 0 & 0\end{array}\right]$

where, $2 \leq j \leq J$.

The LU decomposition method can work out the system's subsequent block diagonal form. The calculations are executed until the desired convergence criterion is met, and the calculations are terminated when the value of $\left|\delta g_0^{(i)}\right|<\varepsilon$, which is very small, reaches the recommended threshold.

The present study shows MHD Casson-type nanofluid movement via a nonlinearly stretchable plate along with suction/injection. Also, Joule heating, and radiation influences are considered. Fluid flow is across a porous material. The chemical reaction is also taken into consideration. Flow due to a nonlinearly stretching sheet has potential applications in various fields of science, such as metallurgy and chemical engineering processes with industrial applications which include glass fibre, paper production, hot rolling, metal spinning, wire drawing, etc.

It is determined that:

• The liquid flowing rapidity is reduced in the presence of the absorbent medium, suction/injection, and non-linear stretching parameter.
• The fluid temperature is enhanced in case of progressive observations of Casson, radiation, Joule heating, and heat source parameters. The opposite nature is detected in the nonlinearly stretching parameters.
• The fluid concentration is increased for the Casson factor and decayed for heat source variable.
• Fluid viscosity and thermal radiation, chemical reaction, porosity and Joule heating have significant impact on the Casson nanofluid flow.
• The remarkable phenomenon of Joule heating effect finds its applications in diverse industries such as food processing, materials synthesis, medicine, pharmacology, and chemical industries. By harnessing the power of Joule heating, we can ensure the maintenance of high-quality food products and facilitate the synthesis and processing of various materials.
• In future this work can be carried out for new types of flow flat forms, adding few more effects and different types of Nano fluids. Also, this study can be extended for hybrid Nano fluids.