Multiple Slips Boundary Layer Flow of Chemically MHD Casson Nanofluid via Homotopic Procedure with Heat Source and Radiation Impacts over an Exponentially Inclined Stretched Surface

With varying shear, non-Newtonian fluids' viscosity varies rates and pressures because they don’t adhere newtons law on viscosity. Numerous sectors can benefit from studying non-Newtonian fluids. It is well-known that fluids whose behaviour is not in accordance with Newton's laws include things like petroleum, nuclear waste disposal fluids, hydrology of groundwater, systems for cooling transpiration, almond juice, natural honey & other lotions and edible oils required for the body. A threshold stress property is displayed by Casson fluid, of non-Newtonian fluids category. Human blood is a complex fluid that can be classed as a Casson fluid since it contains blood cells, has heat transmission capabilities, and contains complex components like fibrinogen, synovial fluid, and rouleaux. Casson first proposed the rheological model, which is another name for the Casson model [1]. Using thermal radiation and chemical processes as variables, Reddy [2] performed theoretical research on the multi-harmonic flow of connected boundary layers through a porous medium of a Casson fluid over a stretched surface with an exponential inclination. A fluid of Casson model within a tube flow of blood was developed by Walawender et al. [3], and the basic assets of Casson fluid in pipes within a similar kind of absorbent channel were studied by Dash et al. [4]. Researchers Mukhopadhyay et al. [5] looked into the time-dependent flow on a stretchy surface of Casson fluid with double dimension. Casson fluid`s boundary layer flow (BL) of thermal energy transfer towards a stretching top surface had been studied by Pramanik [6], who took blowing and suction effects into account. Rao et al. [7] applied similar transformation techniques of triple-dimensional flow of Casson motion upon a stretched membrane with various parameters.

Heat transfer fluids like engine oils, ethylene glycol and water, known for their low thermal conductivity are termed base fluids. The thermal energy exchange capabilities of the base fluid are substantially enhanced by adding nanoparticles to the base fluid to form nanofluids. Nanoparticles are nano-sized particles (1-100 nm) which when added to the base fluid exhibit improve the nanofluid's thermal energy transfer properties. Choi [8] was the first to investigate thermal behaviour of nanofluids and the study revealed that nanofluids enhance the heat transfer rate. Reddy and Ramasekhar [9] investigated enhancement of heat transfer and operational performance using nanofluid in engine oil. Dzulkifli et al. [10] investigated heat transfer and unsteady flow in a Cu-water nanofluid across a stretching sheet. They discovered that as the unsteadiness parameter and nanoparticle volume fraction increase, the skin friction coefficient decreases.

An integrated fluid system that exhibits properties of nanofluids, magneto-hydrodynamics (MHD), and non-Newtonian fluids (Casson) is called a magneto-Casson nanofluid. Consisting of a base fluid—which may or may not be conducting—with suspended nanoparticles, behaving Casson as a fluid, and being affected by an applied magnetic field. These fluids have potential in cutting-edge heat transfer applications that make use of magnetic fields, such as non-Newtonian fluidic systems for magnetically enhanced heat transfer, magnetic hyperthermia to treat cancer, and magnetic drug targeting.

Electromagnetic waves may travel in any direction and are known as radiation. The radiation that an object emits determines its classification. Ionizing and non-ionizing radiations are the two main categories of radiation. A number of fields rely heavily on radiation, including engineering, astronomy, and climate science. To design solar panels, study the distribution of energy on Earth, and assess heat transmission in industrial processes, one must have a thorough understanding of radiation.

The Casson nanofluid's magnetohydrodynamic (MHD) flow on a surface that is not stationary and is shrinking is examined by Lund et al. [11]. Shah et al. [12] derived the magneto hydrodynamic flow of Casson nanofluid time-fractional method by applying the generalized Fourier and Fick’s laws over a slanted channel. The magneto hydro dynamic Couple stress Casson flow of a hybrid nanofluid was explored by Rehman et al. [13]. Due to its significant uses in modern engineering devices, specifically by improving heat management and efficacy in several industrial processes, the stress of the MHD Couple Rehman et al. [13] looked into the Casson flow of a hybrid nanofluid. Panigrahi et al. [14] suggested an analysis to an inclined magnetic field affects Casson nanofluid flowing across an extending spread sheet that's embedded into a porous matrix with saturation. Irshad et al. [15] studied a challenging numerical problem involving the magnetohydrodynamic Casson nanofluid flow in a porous medium across a stretched membrane. They used the Keller box technique, which relies on the finite difference method (FDM), to solve this issue.

The flow of nanofluids towards a heated permeable sheet with extensions and compressions in the MHD stagnation area, where fluid velocity tends to be zero, was investigated by Chaudhary and Kanika [16]. They investigated the heat production and absorption, as well as thermal radiation, influenced the flow of nanofluids. An investigation was conducted by Reddy et al. [17] to determine the impact that thermal radiation has on the magneto hydrodynamic boundary layer flow of a non-Newtonian nanofluid as it travels across a stretched surface that contains porous media.

Nanofluid flow across a stretching sheet in a porous media was the subject of an investigation by Pal and Mandal [18], who looked into the ways in which thermal radiation and chemical reactions influence mass transfer and mixed convection heat. The thermal and stratification properties of non-Newtonian fluids that are flowing across porous media were explored by Megahed and Abbas [19]. The FDM utilized by Barik et al. [20] in order to explore the effects of repeated slips on the flow of MHD nanofluids across a stretched sheet that was both chemically reactive and radiative. The sheet was also inclined.

The impact of dual stratification on the stagnation point had been investigated by Sadaf et al. [21]. They considered the radiative Riga plate flow of Walters' B nanofluid. Ramana et al. [22] provided an explanation of the continuous magnetohydrodynamic stagnation point fluid flow for a fluid of Casson across a stretched sheet with a source of heat and a first-order reaction of a chemical by including multiple slip boundary conditions. According to their observations, the thermal profile and chemical reactivity decreased when the heat source and chemical reaction parameters rose. In order to study the problem of stagnating location (where fluid velocity tends to zero) flow of MHD methanol-based nanofluid through convectively heated stretched sheets while keeping in mind the influence of heat radio activity and generation via a porous medium, Nandi et al. [23] used statistical and numerical perspectives.

Particles in fluids, particularly those at the microscopic level, are controlled by two separate systems: thermophoretic diffusion and Brownian motion. Particles in a liquid or gas can be described by the random and unexpected motion that occurs when they collide with the molecules of the fluid, a process known as Brownian motion. When there is a change in temperature throughout a fluid, this process is called thermophoretic diffusion. When there is a difference in temperature between the fluid and the particles move from hotter to colder areas by a process called thermophoresis. Particle displacement in fluids is a common feature of both Brownian motion and thermophoretic diffusion, yet these two phenomena have different origins. Thermophoretic diffusion happens when particles interact with the temperature differential in the fluid, as opposed to Brownian motion, which is primarily produced by random collisions with molecules in the fluid. Microfluidic device creation, colloidal system analysis, and atmospheric dynamics understanding are just a few of the scientific and technological fields that rely on these two phenomena.

In order to investigate the effects of Brownian motion, thermophoretic diffusion, and thermal advancement across an incline on MHD micropolar nanofluid, Thabet et al. [24] utilized computer simulations to investigate the interaction between these three phenomena. In their study, Shahzad et al. [25] investigated the Darcy-Forchheimer flow of bioconvective micropolar nanofluid between two discs with Cattaneo-Christov heat flux. They also investigated the relationship between thermophoretic diffusion and Brownian motion. Hazarika and Ahmed [26] conducted research on thermophoresis and Brownian motion in micro-polar nanofluids by employing the bvp4c numerical approach. Irfan [27] conducted a study in which he explored the effects of diffusion through thermophoresis and Brownian motion. The study involved Carreau nanofluids in a non-linear convectional mixed flow of nanofluids with varying parameters. Shah et al. [28] observed diffusion effects of thermophoretics and Brown's law in their research on the MHD dynamics with Higher-Convection flow of Maxwell nanofluids down a typically inclined surface. This research was conducted in addition to the MHD dynamics. Recently, literature in this field has drawn on thermophoretic transmission and Brownian law studies [29-34].

A fluid's movement across a surface area that is sloping or inclined in relation to a straight line is referred to as fluid movement across a surface with inclination. Gravitational forces, fluid viscosity, and gradient are some of the variables that might affect this flow was studied by Pera and Gebhart [35]. The value-based study of MHD free convective layer of limitations motion of Casson substance brought on by a porous surface that is angled vertically, as well as the consequences of thermal energy and chemical changes, was reported by Raju [36]. Rafique et al. [37] used the Soret and Dufour effect to study the Casson Nanofluid limit layer's movement on a sloped elongated surface area. Tawade et al. [38] used the Runge-Kutta 4th-order with the shooting technique for resolving ordinary differential calculations that are non-linear for nanofluid movement that was smooth and unsteady past stretching surfaces in double dimensions. Additional issues about fluid flow beneath a stretchy sheet with varying thicknesses under various circumstances can be found in the literature by Pal et al. [39] and Abbas and Megahed [40].

Mathematica is a sophisticated numerical tool that is known for its efficiency and accuracy in solving nonlinear boundary value problems. This research makes use of the Homotopy Analysis Method (HAM) technique with the assistance of the BVPh2.0 package in Mathematica with the intention of achieving similar results. An efficient framework for numerical calculations in fluid dynamics is provided by HAM, which is characterized by its compact support and qualities that pertain to multiple solutions. Due to the fact that the HAM method incorporates the benefits of various strategies that are designed to handle complicated systems in an effective manner, it is particularly well-suited for the challenge at hand. The purpose of this study is to evaluate the influence that various parameters, including the intensity of the magnetic field, the features of the Casson fluid, the Prandtl quantity, and the velocity ratio, as well as the impacts of multi slip, have on the distributions of velocity, concentration, and temperature. In addition, important technical metrics like as the Sherwood number, the Nusselt number, and the skin friction coefficient are evaluated using this process. The findings provide important insights into the interaction between magnetic fields, non-Newtonian fluid dynamics, and heat transfer. These discoveries have substantial implications for the optimization of industrial processes and the improvement of engineering systems.

Take note of a stable, two-dimensional Casson flow of nanofluids over a substantially elongated sheet that is disposed at an inclined angle $\Omega$ with its perpendicular axis. The x‐ and y‐ As shown, the axes are normal to the surface and aligned along the inclined surface in Figure 1. Here, elongating and liberated fluid Acceleration of streams are represented as and $U_{\infty}(x)=b e^{\frac{x}{L}}$ correspond, where a and b are any constants, x is the value of regulated together the surface that is being stretched and L is the sheet length. The temperature near and distant from surface are denoted as T_w and T_∞, respectively, whereas concentration C_w and C_∞ are also respectively. A Horizontal electromagnetic nonuniformity induction of strength $B(x)=B_0 e^{\frac{x}{2 L}}$ is applied parallel to y axis, where B₀ is the uniform magnetic induction power. It is assumed that the magnetic field generated by the fluctuating electrical current is negligible. Furthermore, it is assumed that there is no external electric field and that the charge polarization-induced electrical field is insignificant. In Figure 1, the flow is described. Brownian motion and thermophoresis properties are considered. Chemical responses, the vacuum, injection, thermal energy, and multi-slip effects are also discussed.

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Figure 1. An actual physical manifestation of the flow

The rheological computation of state is presented as follows within the framework of a Casson fluid flow that is isotropic and incompressible:

$\tau_{i j}=\left[\begin{array}{ll}2\left(\mu_B+\frac{p_y}{\sqrt{2 \pi}}\right) e_{i j}, & \pi>\pi_c \\ 2\left(\mu_B+\frac{p_y}{\sqrt{2 \pi_c}}\right) e_{i j}, & \pi_c>\pi\end{array}\right.$

here, μ_B is the non-Newtonian fluid's material elasticity. p_y is the fluid's stress due to yield, π is the item of the component of distortion rate in relation to itself, e_ij is the (i,j)th part of the amount of distortion and π_c is a critical value of this item based on the model that is not Newtonian. Regarding the particular situation of Casson fluid, we considered π>π_c and $p_y=\frac{\mu_B \sqrt{2 \pi}}{\beta}$, it is possible in other words, the fluctuating of the consistency $\mu=\mu_B+\frac{p_y}{\sqrt{2 \pi}}$.

Changing the amount of p_y in μ, we get:

$\mu=\mu_B\left(1+\frac{1}{\beta}\right)$

For this investigation, the process of mathematical equations is provided by [41]:

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$,                          (1)

$\begin{aligned} & u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}=v\left(1+\frac{1}{\beta}\right) \frac{\partial^2 u}{\partial y^2} +g\left[\beta_T\left(T-T_{\infty}\right)+\beta_C\left(C-C_{\infty}\right)\right] \cos \Omega +U_{\infty} \frac{d U_{\infty}}{d x}+\frac{\sigma_f B^2(x)}{\rho_f}\left(U_{\infty}-u\right),\end{aligned}$                        (2)

$\begin{aligned} & u \frac{\partial T}{\partial x}+v \frac{\partial T}{\partial y}=\alpha \frac{\partial^2 T}{\partial y^2}+\tau\left[D_B \frac{\partial C}{\partial y} \frac{\partial T}{\partial y}+\frac{D_T}{T_{\infty}}\left(\frac{\partial T}{\partial y}\right)^2\right] \\ & +\frac{Q_0}{(\rho c)_f}\left(T-T_{\infty}\right)-\frac{1}{(\rho c)_f} \frac{\partial q_r}{\partial y}\end{aligned}$                        (3)

$u \frac{\partial C}{\partial x}+v \frac{\partial C}{\partial y}=D_B \frac{\partial^2 C}{\partial y^2}+\frac{D_T}{T_{\infty}} \frac{\partial^2 T}{\partial y^2}-K r\left(C-C_{\infty}\right)$                         (4)

Following Rao [41], the circumstances surrounding limits are considered as:

$\left.\begin{array}{l}u=U_w(x)+\delta_1^*\left(1+\frac{1}{\beta}\right)\left(\frac{\partial u}{\partial y}\right), \quad v=V_w, \\ T=T_w(x)+\delta_2^*\left(\frac{\partial T}{\partial y}\right), C=C_w(x)+\delta_3^*\left(\frac{\partial C}{\partial y}\right)\end{array}\right\}$ at $y=0\left.\begin{array}{rl}u\rightarrow&U_{\infty}, T \rightarrow T_{\infty}+T_0 e^{x /(2 L)}, \\ & C \rightarrow C_{\infty}+C_0 e^{x /(2 L)}\end{array}\right\}$ as $y \rightarrow \infty$                             (5)

where,

$\alpha=\frac{k}{(\rho c)_f}, v=\frac{\mu}{\rho_f}, \tau=\frac{(\rho c)_p}{(\rho c)_f}, V_w(x)=V_0 e^{x /(2 L)}$

Following the thermal flux, using the Roseland estimation, is

$q_r=-\frac{4 \sigma^*}{3 \kappa^*} \frac{\partial T^4}{\partial y}$                       (6)

where, σ^* is Stefan Boltzmann constant and average value of consumption is $\kappa^*$. Furthermore, we assume that the internal flow thermal difference is sufficiently wide, so that T⁴ is shown as an inverse function of temperature. Therefore, by enlarging T⁴ in Taylor's sequence regarding T_∞ and if we neglect higher order words, we get:

$T^4 \cong 4 T_{\infty}^3 T-3 T_{\infty}^4$,                         (7)

Making use of Eqs. (6) and (7) the Eq. (3) converts into:

$\begin{aligned} & u \frac{\partial T}{\partial x}+v \frac{\partial T}{\partial y}=\left(\alpha+\frac{16 \sigma^* T_{\infty}^3}{3 \kappa^*(\rho c)_f}\right) \frac{\partial^2 T}{\partial y^2} +\tau\left[D_B \frac{\partial C}{\partial y} \frac{\partial T}{\partial y}+\frac{D_T}{T_{\infty}}\left(\frac{\partial T}{\partial y}\right)^2\right]+\frac{Q_0}{(\rho c)_f}\left(T-T_{\infty}\right)\end{aligned}$,                          (8)

The non-linear partial differential equations are reduced to non-linear ordinary differential equations by applying the following similarity transformations.

$\left.\begin{array}{c}\zeta=\left(\frac{a}{2 v L}\right)^{1 / 2} e^{x / 2 L} y, u=a e^{x / L} f^{\prime}(\zeta), \\ v=-\sqrt{\frac{v a}{2 L}} e^{x / 2 L}\left(f(\zeta)+\zeta f^{\prime}(\zeta)\right), \\ T=T_w=T_{\infty}+T_0 e^{x /(2 L)} \theta(\zeta), \\ C=C_w=C_{\infty}+C_0 e^{x /(2 L)} \theta(\zeta) .\end{array}\right\}$                          (9)

where, $\zeta$ is the similarity variable.

Substituting Eq. (9) in Eqs. (2), (4), and (8), we obtain:

$\begin{aligned} & \left(1+\frac{1}{\beta}\right) f^{\prime \prime \prime}+f f^{\prime \prime}-2 f^{\prime 2}+2 A^2+M\left(A-f^{\prime}\right)  +(G r \theta+G c \phi) \cos \Omega=0,\end{aligned}$                     (10)

$\begin{aligned} & \left(1+\frac{4}{3} R\right) \theta^{\prime \prime}+\operatorname{Pr}\left(f \theta^{\prime}-f^{\prime} \theta\right)+\operatorname{Pr} N b \phi^{\prime} \theta^{\prime} +\operatorname{Pr} N t \theta^{\prime 2}+\operatorname{Pr} Q \theta=0\end{aligned}$,                      (11)

$\phi^{\prime \prime}+S c\left(f \phi^{\prime}-f^{\prime} \phi-\gamma \phi\right)+\frac{N t}{N b} \theta^{\prime \prime}=0$.                     (12)

The limited conditions are:

$\begin{aligned} & f(0)=S, f^{\prime}(0)=1+\delta_1\left(1+\frac{1}{\beta}\right) f^{\prime \prime}(0), \\ & \theta(0)=\left(1+\delta_2 \theta^{\prime}(0)\right), \phi(0)=\left(1+\delta_3 \phi^{\prime}(0)\right), \\ & f^{\prime}(\infty) \rightarrow A, \theta(\infty) \rightarrow 0, \phi(\infty) \rightarrow 0,\end{aligned}$                         (13)

where,

$\beta=\frac{\mu_B \sqrt{2 \pi_c}}{p_y}, M=\frac{2 L \sigma_f B_0^2}{\rho_f a}, G r=\frac{G r_x}{\mathrm{Re}_x^2}, A=\frac{b}{a}$,

$G r_x=\frac{2 L g \beta_T\left(T_w-T_{\infty}\right) x^2}{a^2}, G r=\frac{G r_x}{\operatorname{Re}_x^2}, S c=\frac{v}{D_B}$,

$G c_x=\frac{g 2 L \beta_C\left(C_w-C_{\infty}\right) x^2}{a^2}, G c=\frac{G c_x}{\mathrm{Re}_x^2}, \operatorname{Pr}=\frac{v}{\alpha}$,

$N b=\frac{\tau D_B\left(C_w-C_{\infty}\right)}{v}, N t=\frac{\tau D_T\left(T_w-T_{\infty}\right)}{v T_{\infty}}$,

$\gamma=\frac{2 L k_0}{U_0}, R=\frac{4 \sigma^* T_{\infty}^3}{k^* k}, \delta_1=\delta_1^* \sqrt{\frac{a}{2 L v}}$,

$\delta_2=\delta_2^* \sqrt{\frac{a}{2 v L}}, \delta_3=\delta_3^* \sqrt{\frac{a}{2 L v}}, S=-\frac{V_w}{\sqrt{a v}}$,

$Q=\frac{Q_0}{a(\rho c)_f}, \mathrm{Re}_x=\frac{a x e^{x / L}}{v}$.

The primary physical factors that are relevant to this topic right now are ${Sh}_x\left(=\frac{x q_m}{k\left(C_w-C_{\infty}\right)}\right)$ is the local Sherwood number, $N u_x\left(=\frac{x q_w}{k_{\infty}\left(T_w-T_{\infty}\right)}\right)$, is the local Nusselt number, $C_f\left(=\frac{\tau_w}{\rho U_w^2}\right)$, is the local Skin-friction component.

Here, wall-shear stress $\tau_w=\mu_B\left(1+\frac{1}{\beta}\right)\left(\frac{\partial u}{\partial y}\right)_{y=0}$, wall heat flux $q_w=\left(-\left(k_{\infty}+\frac{16 \sigma^* T_{\infty}^3}{3 k^*}\right)\left(\frac{\partial T}{\partial y}\right)\right)_{y=0}$, mass flux $q_m=-\left(D_B\left(\frac{\partial C}{\partial y}\right)\right)_{y=0}$.

With the modifications described above, we have:

$\begin{aligned} & \operatorname{Re}_x^{-1 / 2} S h_x=-\phi^{\prime}(0), \\ & \operatorname{Re}_x^{-1 / 2} N u_x=-\left(1+\frac{4}{3} R\right) \theta^{\prime}(0), \\ & \operatorname{Re}_x^{1 / 2} C_f=\left(\left(1+\frac{1}{\beta}\right) f^{\prime \prime}(0)\right) .\end{aligned}$

Boundary conditions have been addressed by taking advantage of the modified equations that have been exposed to HAM. Graphs are plotted for different profiles for varying values of the controlling constraints. To determine the validity of our effort, a comparison with historical trends was conducted, and Table 2 clearly shows the enormous agreement we obtained. While most of the figures are generated by varying a specific parameter over a range of values, some remain fixed at a single value throughout the entire simulation, such as $\beta$=1.0, M=0.5, A=0.5, $\Omega=60^{\circ}$, S=0.5, Gr=0.5, Gc=0.5, $\delta_1=\delta_2=\delta_3=0.3$, R=0.5, Pr=0.71, Q=0.5, Nb=0.5, Nt=0.5, Sc=0.6, and $\gamma=0.5$.

Table 2. Comparison of $-\theta^{\prime}(0)$ for various values of $M, \operatorname{Pr}$ and $R$ in the absence of remaining constraints

The way the momentum is affected by the Casson fluid parameter β, thermal & solutal contours were revealed in Figures 4 to 6. Amplification in β has been found to decline the fluid's viscosity. As a change in β causes the fluid to behave like shear-thickening, which diminishes the fluid's fluidity and the width of its momentum barrier layer. While the velocity contour displays the opposite tendency, the thermal, solutal contours grow as the Casson fluid parameter increases.

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Figure 4. Figuration of $f^{\prime}(\zeta)$ for β

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Figure 5. Figuration of $\theta(\zeta)$ for β

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Figure 6. Figuration of $\phi(\zeta)$ for β

Figures 7 to 9 validate that the magnetic parameter M affects the profiles. Although the trend for concentration and temperature is the reverse, $f^{\prime}(\zeta)$ falls as M's magnitude upsurges. The rate of transfer actually declines as M increases as a result of which it upsurges the Lorentz force that limits the flow of fluids. The Lorentz force was visible when the flow field was going through a magnetic field. This force is powerful enough to drag the fluid along and slow its movement. Consequently, the fluid flow momentum decreases as the velocity layer thickness increases.

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Figure 7. Figuration of $f^{\prime}(\zeta)$ for M

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Figure 8. Figuration of $\theta(\zeta)$ for M

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Figure 9. Figuration of $\phi(\zeta)$ for M

Figure 10 exemplifies that the momentum ratio significator A affects the momentum of the profiles. The fluid's momentum gains as the momentum ratio significator the magnitude upsurges. Figures 11 and 12 display the momentum ratio significator. A affects the development of non-dimensional thermal and solutal contours. As A surge, the thermal and solutal of the fluid asymptotically decline, forming thin temperature and concentration boundary layers. A greater momentum ratio signifies more. The momentum of the free stream is increased by heat transfer from the wall. The concentration of the fluid upsurges in tandem with the free stream velocity as momentum ratio significator A upsurges.

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Figure 10. Figuration of $f^{\prime}(\zeta)$ for A

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Figure 11. Figuration of $\theta(\zeta)$ for A

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Figure 12. Figuration of $\phi(\zeta)$ for A

The impact of an inclined angle on the momentum contour is illustrated in Figure 13. As the increases, the momentum contour decreases. This is explained by the fact that the buoyant force caused by thermal diffusion is reduced by a factor of $\cos \Omega$. As higher values suction (S) effectively reduce the momentum contour both in fluid and gaseous forms, as shown in Figure 14. The features of the local Grashof number are Gr on the momentum contours are depicted in Figure 15. In this case, the fluid flow velocity will increase quickly as Gr values rise.

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Figure 13. Figuration of $f^{\prime}(\zeta)$ for $\Omega$

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Figure 14. Figuration of $f^{\prime}(\zeta)$ for S

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Figure 15. Figuration of $f^{\prime}(\zeta)$ for Gr

Figure 16 demonstrates that the velocity contours improve when the adjusted Grashof parameter Gc values upsurge. Figure 17 depicts the restriction on velocity slip δ1 upsurges, the fluid speed falls and the slip speed upsurges. This could be because the velocity of the stream and the stretched sheet are different in the case of a slip scenario. An upsurge in the velocity slip limitation δ1 causes the fluid velocity to drop as the slip velocity increases. This phenomenon occurs when, in slip conditions, the velocity of the fluid stream close to the stretching sheet differs from that of the document for extending. As the thermal slip limitation δ2 rises, Figure 18 demonstrates that the temperature declines. Although there is very little heat and the physical width of the temperature boundary layer decreases as the thermal slip constraint value increases during the transfer from the sheet to the temperature boundary layer.

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Figure 16. Figuration of $f^{\prime}(\zeta)$ for Gc

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Figure 17. Figuration of $f^{\prime}(\zeta)$ for δ1

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Figure 18. Figuration of $\theta(\zeta)$ for δ2

Figure 19 shows the connection between thermal radiation and temperature limitation R. Higher values of R indicate a greater infusion of heat energy that radiates into the system, raising the temperature. Figure 20 displays the temperature contours change for different Prandtl values. The Prandtl number is found to be upsurges, the boundary layer's thermal declines. The extraordinary effect of Q on θ(ζ) is revealed in Figure 21. An increase in the values improves θ in the case of air. The thermal contour initially declines before rising away from the wall.

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Figure 19. Figuration of $\theta(\zeta)$ for R

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Figure 20. Figuration of $\theta(\zeta)$ for ${Pr}$

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Figure 21. Figuration of $\theta(\zeta)$ for Q

Figure 22 illustrates the significance of the Brownian movement significator in a clear and concise manner. Nb may have an effect on θ(ζ). In general, Brownian mobility helps to regulate the temperature of the fluid in the boundary layer by preventing materials from accumulating on the surface that is located at a distance from the fluid. The temperature rises when there is a greater concentration of Nb in the fluid (one that is less than 1). Most of the time, Brownian motion is responsible for preventing molecules from depositing away from the surface of the liquid and warming the liquid that is contained within the boundary layer. At some distance from the extension surface, rapid flow is brought about by the thermophoretic force that is produced by the temperature gradient that is the result of the extension. As Nb increases, a greater quantity of fluid is heated away from the surface, which results in an increase in the thermal properties of the boundary layer. However, concentration profiles exhibit the reverse pattern, as illustrated in Figure 23.

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Figure 22. Figuration of $\theta(\zeta)$ for Nb

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Figure 23. Figuration of $\phi(\zeta)$ for Nb

The stretched sheet creates a rapid flow that carries nanoparticles, widening the boundary layer for the mass volume fraction. These phenomena are demonstrated in Figures 24 and 25. It has been reported that Nt directly correlates with both the thermal and the solutal of nanoparticles.

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Figure 24. Figuration of $\theta(\zeta)$ for Nt

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Figure 25. Figuration of $\phi(\zeta)$ for Nt

Figure 26 portrays the Schmidt number (Sc) affects the ϕ(ζ) concentration contour. Additionally, it has been noted that when the value growths, the concentration contour falls. A lower mass diffusivity relative to velocity diffusivity is indicated by a higher Schmidt number, which implies a decrease in scalar diffusivity, less diffusion, and slower solutal changes in the fluid medium. The ratio of mass diffusivity to momentum diffusivity is known as Sc in physics. The effect of a chemical reaction significator γ on ϕ(ζ) is portrayed in Figure 27. It is well known that when the chemical reaction significance increases, the solutal falls. Figure 28 exemplifies the nanoparticle fraction slip limitation δ₃ has a similar effect on the mass fraction field as it does on the temperature field. This resemblance results from slip's fundamental obstruction to liquid motion, which ultimately lowers net atomic advancement. Consequently, reduced molecular development leads to a decline in the mass fraction field.

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Figure 26. Figuration of $\phi(\zeta)$ for Sc

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Figure 27. Figuration of $\theta(\zeta)$ for γ

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Figure 28. Figuration of $\phi(\zeta)$ for δ₃

Figure 29 exhibits the skin-friction factor on variation $\beta$ and M. It’s observed that as $\beta$ and M heightened the skin-friction factor declines. Figure 30 demonstrated that the Nusselt number on variations of $\beta$ and M. It is observed that as $\beta$ and M increased the Nusselt number declines. Figure 31 depicted that the local Sherwood number on variations of $\beta$ and M. It is noticed that as $\beta$ and M. increased the local Sherwood number declines.

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Figure 29. Figuration of $C_{f x}$ for $\beta$ and M

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Figure 30. Figuration of $N u_x$ for $\beta$ and M

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Figure 31. Figuration of $S h_x$ for $\beta$ and M