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The present work is focused on the simulation of Casson (nonNewtonian) nanofluid flow over an inclined stretching sheet. The study considers the influence of an imposed magnetic field, heat source/sink, thermal radiation and chemical reaction under the multi slip effects. The study includes the application of wall suction/injection and Navier's firstorder slip to analyse the velocity, temperature, and concentration at the wall. The governing equations have been transformed into nonlinear ordinary differential equations (ODEs) with similarity transformations. By employing the homotopy analysis method (HAM), we have successfully derived the numerical solution for the nonlinear ordinary differential equations (ODEs) and their corresponding boundary conditions. The impact of various parameters on the velocity, temperature, and concentration field has also been demonstrated. Multiple slip flow is utilised in various practical domains like microelectromechanical systems, nanoelectromechanical systems, microorganism flow, and rarefied gas flow, among others.
nanofluid, Casson, MHD, thermal radiation, heat source, multiple slip effects, HAM method
The unique thermal, mechanical, and chemical properties of nanofluids, a mixture of nanoparticles adjourned in a base fluid, make them perfect for enlightening lubrication, increasing thermal conductivity, and improving the performance of a variety of systems. The study of nanofluid dynamics is still in its early stages, with efforts being made to create new nanofluid formulations, increase their stability, forecast their behaviour in various scenarios, and improve their performance in a range of applications. It is true that Choi and Eastman [1] significantly advanced in the field of nanomaterials. Their research concentrated on using nanoparticles to increase a fluid's thermal conductivity. The addition of a small amount of nanoparticles in heat transfer fluids results in the new thermal phenomena of nanofluids (nanoparticlefluid suspensions) reported by Pil Jang and Choi [2]. Meyer et al. [3] investigated computational techniques for studying fluid dynamics, while focusing on numerical approaches to nanofluid flow analysis. Akbari et al. [4] researched on the comparisons between different concepts of fluid flow likely looks at the benefits and distinctions between different models for fluid behaviour prediction. Wong and De Leon [5] engrossed in dispensing an extensive scope of present and anticipated nanofluid applications, stressing their more controllable heat allocation abilities and the unique properties that make nanofluids appropriate for such uses.
Due to of its wideranging applications and implications, nonNewtonian fluid flow requires careful study and modelling in the industrial and engineering sectors. Unlike Newtonian fluids like water or air, nonNewtonian fluids lacked to investigate the direct association between shear tension and shear proportion. Rather, variables such as shear rate, time, or stress can affect their viscosity. For a variety of industries, careful and precise modelling of the behaviour of nonNewtonian fluids is essential. As an illustration: NonNewtonian fluids are frequently found in the production of paints, food products, and polymers. A wide range of food products, including dough, sauces, and creams, behave in ways that are not Newtonian. Siddappa and Abel [6] considered the study of boundary layer flow past a stretching plate in nonNewtonian like viscoelastic fluid flow. Different nonNewtonian nanofluid flows are purposefully designed by Khan [7, 8] for abundant configuration.
Because of their relationship between shear stress and strain, Casson fluids are nonNewtonian due to their sole rheological characteristics. These fluids are appropriate for shear diluting implementation because of their notable shear viscosity and yield stress. In the beginning, Casson [9] introduced the Casson fluid model for silicon dissolution and gel pens. Human blood, concentrated fruit liquids, jellies, honey, soups, and tomato sauces are a few examples of Casson fluids. In a stretching sheet, Dandapat et al. [10] examined about stability of MHD path in a viscoelastic liquid or gas, while Fang et al. [11] investigated the extended Blasius equation. Mamalouka et al. [12] discussed about the difference in solving the second order problems on a fluid that flows over an augmentation. This condition is expected to be mixed convection within the particular field when forced convection and free/natural convection cooperate to facilitate easier heat transfer. The enhanced thermal transference for miscellaneous Casson flow of nanofluid influence by external magnetism over a rotating sheet was investigated by Ali et al. [13]. Intentional aspects of the temperature and molar variation with respect to Arrhenius activation energy are discussed by Alsallami et al. [14].
Magnetohydrodynamic (MHD) flows are driven by curiosity and application: they are recycled to treat cancerous tumours, reduce bleeding from surgical wounds, distribute specific medications using magnetic particles, and detect illness using magnetic resonance imaging (MRI). The study of MHD examines the relationship between magnetic fields and fluid flow, which is a branch of mathematics. The MHD boundary layer flow above a permeable elongation area finds petition claims in the trade of crystal Fiber, paper crafting, plasma education, fuel trades, MHD control producer, boundary regulator in aero mechanics and fear of fissionable containers. Many conceptual as well as creative investigations caused some researchers to argue. Ishak [15] observed (MHD) boundary layer movement brought on by an increasing rapidly elongating sheet with radiation outcome. The viscosity factor of a magnetohydrodynamics (MHD) nanofluid is discussed by Shahid et al. [16] and is dependent on temperature. As a result of its abundant tenders in engineering complications, the influence of an outermost magnetic field on magnetohydrodynamic (MHD) deluge a lengthening area of spreading is highly notable in the field of fluid mechanics. The impact of an outward applied magnetism on Magnetohydrodynamics flow across an elongating membrane was examined by Pavlov [17]. Anderson [18] examined the viscous fluid MHD flow at the end of a stretching sheet. The distribution of heat away from the surface is largely influenced by thermal radiation. It has claims in trade industries like space exploration, missiles, atomic furnaces, planetary vehicles, choppers, satellites, and actions involving high temperatures. More grades regarding the movement of a nanofluid in the occurrence of mass as well as heat were noted in the studies [19, 20]. An elemental response has broad tenders that cover the destruction of crops by freezing, food distribution, newspaper exchange, ventilation, tile work, sunstroke protocols, and river suspensions and petroleum. Seyedi et al. [21] framework is ideal for mathematically analysing the influence of a biological response to direct heat emission on an embracing EyringPowell fluid channel distortion. It is feasible to implement a noslip boundary condition when the fluid particles are near the external and are unable to transfer along with it or after the bond expires coherence. Viscous liquids prevent this marvel from happening because smooth walls sufficient, a firm surface may cause them to slip. In many situations, including wire mesh rough surfaces, greased surfaces, and covered surfaces, slip conditions become important. The change in uses characteristic of nature and modern technology has led to an important amplification in count, heat, and mass transport along with chemical reactions. Fluid flow involves chemical processes such as refrigeration fortifications, mist generation, and ceramic business. The integral transform analysis of mass and heat diffusion of chemically reacting systems with Michaelis–Menten kinetics was prepared by Pinheiro et al. [22].
The mass flow caused by a temperature gradient is indicated by the Soret result. The Soret effect has been incorporated by numerous researchers into their studies to discuss these issues and emphasize their significance. The learning of the boundary layer flow produced by an elongating sheet was primarily done by Crane [23]. He provided a precise fix for the initial issue. As time has passed by, the boundary layer flow over regular and irregular widening of surface areas has piqued the enormous interest of numerous researchers [2427]. The reputation of mutable nanomaterial radius for the nonNewtonian flow of nanofluid promoted through a stretching sheet is discussed by Ali et al. [28]. An elastic limit that is similar to the constant harvest tension in Casson fluid occurs, if the rouleaux entertains similar an elastic solid studied by Fung [29]. The ideal point flow and heat allocation in a Casson fluid flow ended an elongating plate are discussed by Mustafa et al. [30]. Shehzad et al. [31] supplied a thoughtful answer to the stable boundaryed layer flow of a Casson fluid above an absorbent elongating plate. Investigators have been captivated by the potential of exponentially stretching cylinders in a variety of domains, such as polymer dispensation, purification, biomedical tenders, and energy change systems. An investigation into the analogous determination of the flow features and transmission of heat inside a constant Cassonbased nanofluid stirring diagonally crosswise erectly situated cylinder exposed to spreading radial extension was guided by Naseer et al. [32]. Merkin et al. [33] observed the thermal features of the boundaryed sheet flow at a recession fact connecting a volumetric curve that causes exponentially stretched sheet. Malik et al. [34] investigated cautiously the comparable conclusions regarding the flow properties and heat exchange of a stable Cassonbased nanofluid applied inclined to a vertically positioned cylinder subjected to exponential radial stretching sheet. The combination of a magnetic field and a heat source affects on Casson nanofluid flow in a stable boundary layer, as done by Sarojamma and Vendabai [35]. In measuring thermophoretic diffusion properties and Brownian motion, Mustafa et al. [36] examined the merged convective move of magnetonanofluid that was controlled by vertical stretched plate. Awais et al. [37] measured the outcomes of heat source/sink and mass and heat transfer on Casson fluid flow through a vertically exponentially stretched/shrinking sheet using mathematical analysis. Zhai et al. [38] have derived the relation between the mass transfer and heat structures for a variety of disorders and absorbers. Thermal and stratification characteristics of nonNewtonian fluid passes through a porous medium was investigated by Megahed and Abbas [39]. Through the use of FDM, Barik et al. [40] examined the impact of multiple slips effects on the flow of MHD nanofluids over an inclined, radiative, and chemically reactive stretched sheet.
We found that a very few studies have investigated the thermohydrodynamics properties of Casson and nanofluid over an inclined stretching sheet by taking into account all the different phenomena, including variables like magnetic field, linear radiation, an external heat source/sink, and the chemical reaction, from the aforementioned literature survey. It is anticipated that slip exists at the surface in this investigation because nanofluid has been used as the working fluid. To analyze these phenomena, the research employs established similarity variables, converting the primary partial differential equations (PDEs) into a set of interconnected ordinary differential equations (ODEs). These ODEs are numerically resolved by using employing the homotopy analysis method (HAM), This methodology ensures the generation of robust data across various parameters. To validate the computed results, a comparative analysis is performed against existing research data. The research makes a significant contribution to our understanding of nonNewtonian nanofluid flow by elucidating the complex interactions among various elements that govern micro rotational transport in a nonNewtonian nanofluid, as characterized by the Casson nanofluid model.
In the present study, a steady, two‐dimensional, incompressible, laminar flow of nanofluid over a stretching sheet which is inclined at an angle Ω from its vertical axis. The x and y axes are aligned along the inclined surface and normal to the surface as depicted in Figure 1. Stretching has been accomplished by moving the wall with a velocity U_{w}=ax, a>0 along the xaxis. The MHD, thermal, as well as the concentration boundary layers are supposed to develop along the y axis, and grow along the xaxis. The velocity, temperature, and concentration of the nanofluid on the wall are Uw, Tw, and Cw, respectively. The velocity, temperature, and the concentration fields are taken as U_{∞}=bx, T_{∞} and C_{∞} as as y→∞. It is assumed that thermal equilibrium prevailed between the nanoparticles and base fluid with having slip between them. A uniform magnetic field of strength B_{0} has been applied normal to the xaxis, and over the entire fluid domain. In the present study, the effects of the thermal radiation, chemical reaction, and the buoyancy on the velocity, temperature, and concentration profiles in the presence of the multiple slip conditions at the wall are considered under suction/injection. External heat source/sink is also considered in the energy equation. The body force terms due to the thermal and concentration gradients are included in the momentum equation. The properties of Brownian motion and thermophoresis are considered.
Figure 1. Physical model of the flow
The viscoelastic equation of state for an isotropic and flexible flow of Casson fluid is:
$\tau_{i j}=\left[\begin{array}{cl}2\left(\mu_B+\frac{p_y}{\sqrt{2 \pi}}\right) e_{i j}, & \pi>\pi_c \\ 2\left(\mu_B+\frac{p_y}{\sqrt{2 \pi_c}}\right) e_{i j}, & \pi_c>\pi\end{array}\right.$
where, μ_{B} is plastic dynamic viscosity of the nonNewtonian fluid, p_{y} is the yield stress of the fluid, π is the product of the constituent of deformation rate with itself, e_{ij} is the (i,j)th component of the deformation rate and π_{c} is the vital tone of this product based on the nonNewtonian model. For the case of Casson fluid, we measured π>π_{c} and $p_y=\frac{\mu_B \sqrt{2 \pi}}{\beta}$, it is probable to say that the dynamic viscosity $u=\mu_B+\frac{p_y}{\sqrt{2 \pi}}$.
Substituting the value of p_{y} in μ, we get $\mu=\mu_B\left(1+\frac{1}{\beta}\right)$.
In the perspective of the boundary layer approximation, the governing differential equations for continuity, momentum, energy, and the nanofluid concentration are written as follows [40]:
$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$ (1)
$\begin{aligned} & u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}=v\left(1+\frac{1}{\beta}\right) \frac{\partial^2 u}{\partial y^2} +g\left[\beta_T\left(TT_{\infty}\right)+\beta_C\left(CC_{\infty}\right)\right] \cos \Omega +U_{\infty} \frac{d U_{\infty}}{d x}+\frac{\sigma_f B^2(x)}{\rho_f}\left(U_{\infty}u\right)\end{aligned}$ (2)
$\begin{aligned} & u \frac{\partial T}{\partial x}+v \frac{\partial T}{\partial y}=\alpha \frac{\partial^2 T}{\partial y^2} +\tau\left[D_B \frac{\partial C}{\partial y} \frac{\partial T}{\partial y}+\frac{D_T}{T_{\infty}}\left(\frac{\partial T}{\partial y}\right)^2\right]+\frac{Q_0}{(\rho c)_f}\left(TT_{\infty}\right)\frac{1}{(\rho c)} \frac{\partial q_r}{\partial y}\end{aligned}$ (3)
$u \frac{\partial C}{\partial x}+v \frac{\partial C}{\partial y}=D_B \frac{\partial^2 C}{\partial y^2}+\frac{D_T}{T_{\infty}} \frac{\partial^2 T}{\partial y^2}K r\left(CC_{\infty}\right)$. (4)
Succeeding Barik et al. [40], the boundary conditions are considered as:
$\begin{aligned} & u=U_w(x)+\delta_1^*\left(\frac{\partial u}{\partial y}\right), \\ & v=V_w, \quad T=T_w(x)+\delta_2^*\left(\frac{\partial T}{\partial y}\right), \\ & C=C_w(x)+\delta_3^*\left(\frac{\partial C}{\partial y}\right) \text { at } y=0 \\ & u \rightarrow U_{\infty}, \quad T \rightarrow T_{\infty}, \quad C \rightarrow C_{\infty}, \text { as } y \rightarrow \infty\end{aligned}$ (5)
where, $\alpha=\frac{k}{(\rho c)_f}, v=\frac{\mu}{\rho_f}, \tau=\frac{(\rho c)_p}{(\rho c)_f}$.
Succeeding Roseland approximation, the radiative heat flux is:
$q_r=\frac{4 \sigma^*}{3 \kappa^*} \frac{\partial T^4}{\partial y}$ (6)
where, Stefan Boltzmann constant is σ* and κ* is the mean absorption coefficient. Furthermore, we assume that the flow's interior temperature differential is sufficiently vast so that T^{4} is denoted as a linear function of temperature. As a solution, by enlarging T^{4} in Taylor series about T_{∞} and if we disregard terms of higher order, we get:
$T^4 \cong 4 T_{\infty}^3 T3 T_{\infty}^4$, (7)
Using Eqs. (6) and (7), Eq. (3) converts into:
$\begin{aligned} & u \frac{\partial T}{\partial x}+v \frac{\partial T}{\partial y} =\left(\alpha+\frac{16 \sigma^* T_{\infty}^3}{3 \kappa^*(\rho c)_f}\right) \frac{\partial^2 T}{\partial y^2}+\tau\left[D_B \frac{\partial C}{\partial y} \frac{\partial T}{\partial y}+\frac{D_T}{T_{\infty}}\left(\frac{\partial T}{\partial y}\right)^2\right]+\frac{Q_0}{(\rho c)_f}\left(TT_{\infty}\right)\end{aligned}$, (8)
Using following similarity transformations:
$\begin{aligned} & \psi=(a v)^{\frac{1}{2}} x f(\zeta), \theta(\zeta)=\frac{TT_{\infty}}{T_WT_{\infty}}, \\ & \phi(\zeta)=\frac{CC_{\infty}}{C_WC_{\infty}}, \zeta=\sqrt{\frac{a}{v}} y\end{aligned}$ (9)
where, ζ is the similarity variable, ψ is the stream function.
The stream function ψ is formalized in the standard way as:
$u=\frac{\partial \psi}{\partial y}, \quad v=\frac{\partial \psi}{\partial x}$ (10)
substituting Eq. (9) in Eqs. (2), (4) and (8), we obtain:
$\begin{aligned} & \left(1+\frac{1}{\beta}\right) f^{\prime \prime \prime}+f f^{\prime \prime}f^{\prime 2} \\ & +(G r \theta+G c \phi) \cos \Omega+M\left(Af^{\prime}\right)+A^2=0\end{aligned}$, (11)
$\begin{aligned} & \left(1+\frac{4}{3} R\right) \theta^{\prime \prime}+\operatorname{Pr} f \theta^{\prime} \\ & +\operatorname{Pr} N b \phi^{\prime} \theta^{\prime}+\operatorname{Pr} N t \theta^{\prime 2}+\operatorname{Pr} Q \theta=0\end{aligned}$, (12)
$\phi^{\prime \prime}+S c\left(f \phi^{\prime}\gamma \phi\right)+\frac{N t}{N b} \theta^{\prime \prime}=0$. (13)
The boundary conditions are:
$\begin{aligned} & f(0)=S, f^{\prime}(0)=1+\left(1+\frac{1}{\beta}\right) \delta_1 f^{\prime \prime}(0), \\ & \theta(0)=\left(1+\delta_2 \theta^{\prime}(0)\right), \phi(0)=\left(1+\delta_3 \phi^{\prime}(0)\right), \\ & f^{\prime}(\infty) \rightarrow A, \theta(\infty) \rightarrow 0, \phi(\infty) \rightarrow 0\end{aligned}$ (14)
where,
$\begin{aligned} & \beta=\frac{\mu_B \sqrt{2 \pi_c}}{p_y}, M=\frac{\sigma_f B_0^2}{\rho_f a}, G r_x=\frac{g \beta_T\left(T_wT_{\infty}\right) x^3}{v^2}, G r=\frac{G r_x}{R e_x^2}, G c_x=\frac{g \beta_C\left(C_wC_{\infty}\right) x^3}{v^2}, G c=\frac{G c_x}{R e_x^2}, R e_x=\frac{U_w x}{v}=\frac{a x^2}{v}, A=\frac{b}{a}, \operatorname{Pr}=\frac{v}{\alpha}, N b=\frac{\tau D_B\left(C_wC_{\infty}\right)}{v}, \\ & N t=\frac{\tau D_T\left(T_wT_{\infty}\right)}{v T_{\infty}}, R=\frac{4 \sigma^* T_{\infty}^3}{k^* k}, Q=\frac{Q_0}{a(\rho c)_f}, S c=\frac{v}{D_B}, \gamma=\frac{K r}{a}, \delta_1=\delta_1^* \sqrt{\frac{a}{v}}, \delta_2=\delta_2^* \sqrt{\frac{a}{v}}, \delta_3=\delta_3^* \sqrt{\frac{a}{v}}, S=\frac{V_w}{\sqrt{a v}}\end{aligned}$.
2.1 HAM
To express the homotopic results of Eqs. (11) to (14), we gross up the primary deductions and linear operators as follows:
$\begin{gathered}f_0(\zeta)=S+A \zeta+\left(\frac{1A}{1+\delta_1\left(1+\frac{1}{\beta}\right)}\right)\left(1e^{\zeta}\right), \\ \theta_0(\zeta)=\frac{e^{\zeta}}{1+\delta_2}, \phi_0(\zeta)=\frac{e^{\zeta}}{1+\delta_3}, \\ L_f(f)=f^{\prime \prime \prime}f^{\prime}, L_\theta(\theta)=\theta^{\prime \prime}\theta, L_\phi(\phi)=\phi^{\prime \prime}\phi, \\ \text { with } L_f\left(D_1+D_2 e^\zeta+D_3 e^{\zeta}\right)=0, L_\theta\left(D_4 e^\zeta+D_5 e^{\zeta}\right)=0, \\ L_\phi\left(D_6 e^\zeta+D_7 e^{\zeta}\right)=0,\end{gathered}$
where, D_{i} (i=1 to 7) are the arbitrary constants.
We construct the zerothorder deformation equations:
$\begin{aligned} & (1p) L_f\left(f(\zeta ; p)f_0(\zeta)\right) =p \hbar_f N_f[f(\zeta ; p), \theta(\zeta ; p), \phi(\zeta ; p)]\end{aligned}$, (15)
$\begin{aligned} & (1p) L_2\left(\theta(\zeta ; p)\theta_0(\zeta)\right)=p \hbar_\theta N_\theta[f(\zeta ; p), \theta(\zeta ; p), \phi(\zeta ; p)]\end{aligned}$, (16)
$\begin{aligned}& (1p) L_\phi\left(\phi(\zeta ; p)\phi_0(\zeta)\right)=p \hbar_\phi N_\phi[f(\zeta ; p), \theta(\zeta ; p), \phi(\zeta ; p)]\end{aligned}$, (17)
subject to the boundary conditions:
$\begin{gathered}f(0 ; p)=S, f^{\prime}(0 ; p)=\left[1+\delta_1\left(1+\frac{1}{\beta}\right) f^{\prime \prime}(0)\right], \\ f^{\prime}(\infty ; p)=0, \\ \theta(0 ; p)=\left[1+\delta_2 \theta^{\prime}(0)\right], \quad \theta(\infty ; p)=0, \\ \phi(0 ; p)=\left[1+\delta_3 \phi^{\prime}(0)\right], \quad \phi(\infty ; p)=0,\end{gathered}$ (18)
where,
$\begin{aligned} & N_f[f(\zeta ; p), \theta(\zeta ; p), \phi(\zeta ; p)]=\left(1+\frac{1}{\beta}\right) \delta \frac{\partial^3 f(\zeta ; p)}{\partial \zeta^3}+f(\zeta ; p) \frac{\partial^2 f(\zeta ; p)}{\partial \zeta^2} \\ & \left(\frac{\partial f(\zeta ; p)}{\partial \zeta}\right)^2+A^2+M\left(A\frac{\partial f(\zeta ; p)}{\partial \zeta}\right) +(G r \theta(\zeta ; p)+G c \phi(\zeta ; p)) \cos \Omega,\end{aligned}$ (19)
$\begin{aligned} & N_\theta[f(\zeta ; p), \theta(\zeta ; p), \phi(\zeta ; p)] =\frac{1}{\operatorname{Pr}}\left(1+\frac{4}{3} R\right) \frac{\partial^2 \theta(\zeta ; p)}{\partial \zeta^2} \\ & +f(\zeta ; p) \frac{\partial \theta(\zeta ; p)}{\partial \zeta}+N b \frac{\partial \theta(\zeta ; p)}{\partial \zeta} \frac{\partial \phi(\zeta ; p)}{\partial \zeta} +N t\left(\frac{\partial \theta(\zeta ; p)}{\partial \zeta}\right)^2+Q \theta(\zeta ; p),\end{aligned}$ (20)
$\begin{aligned} & N_\phi[f(\zeta ; p), \theta(\zeta ; p), \phi(\zeta ; p)] =\frac{\partial^2 \phi(\zeta ; p)}{\partial \zeta^2}+S c f(\zeta ; p) \frac{\partial \phi(\zeta ; p)}{\partial \zeta}+\frac{N t}{N b} \frac{\partial^2 \theta(\zeta ; p)}{\partial \zeta^2}S c \gamma \phi(\zeta ; p)\end{aligned}$ (21)
where, $p \in[0,1]$ is the embedding parameter, $\hbar_f, \hbar_\theta$ and $\hbar_\phi$ are nonzero auxiliary parameters and N_{f}, N_{θ} and N_{ϕ} are nonlinear operators.
The nthorder deformation equations are follows:
$L_f\left(f_n(\zeta)\chi_n f_{n1}(\zeta)\right)=\hbar_f R_n^f(\zeta)$, (22)
$L_\theta\left(\theta_n(\zeta)\chi_n \theta_{n1}(\zeta)\right)=\hbar_\theta R_n^\theta(\zeta)$ , (23)
$L_\phi\left(\phi_n(\zeta)\chi_n \phi_{n1}(\zeta)\right)=\hbar_\phi R_n^\phi(\zeta)$ , (24)
with the following boundary conditions:
$\begin{aligned} & f_n(0)=0, f_n^{\prime}(0)=\delta_1\left(1+\frac{1}{\beta}\right) f_n^{\prime}(0), f_n^{\prime}(\infty) \rightarrow 0 \\ & \theta_n(0)=\delta_2 \theta_n^{\prime}(0), \quad \theta_n(\infty) \rightarrow 0 ; \phi_n(0)=\delta_3 \phi_n^{\prime}(0), \phi_n(\infty) \rightarrow 0\end{aligned}$ (25)
where,
$\begin{aligned} & R_n^f(\zeta)=\left(1+\frac{1}{\beta}\right) f_{n1}^{\prime \prime \prime}+\sum_{i=0}^{n1} f_{n1i} f_i^{\prime \prime} \\ & \sum_{i=0}^{n1} f_{n1i}^{\prime} f_i^{\prime}+\left(1\chi_n\right)\left(A^2+M\right)M f_{n1}^{\prime} \\ & +\left(G r \theta_{m1}+G c \phi_{m1},\right) \cos \Omega\end{aligned}$ (26)
$\begin{aligned} & R_n^\theta(\zeta)=\frac{1}{\operatorname{Pr}}\left(1+\frac{4 R}{3}\right) \theta_{n1}^{\prime \prime}+\sum_{i=0}^{n1} f_{n1i} \theta_i^{\prime} \\ & +N b \sum_{i=0}^{n1} \theta_{n1i}^{\prime} \phi_i^{\prime}+N t \sum_{i=0}^{n1} \theta_{n1i}^{\prime} \theta_i^{\prime}\operatorname{Pr} Q \theta_{n1}, \end{aligned}$ (27)
$\begin{aligned} & R_n^\phi(\zeta)=\phi_{n1}^{\prime \prime}+S c\left(\sum_{i=0}^{n1} f_{n1i} \phi_i^{\prime}\gamma \phi_{n1}\right)+\frac{N t}{N b} \theta_{n1}^{\prime \prime}, \chi_n= \begin{cases}0, & n \leq 1 \\ 1, & n>1\end{cases} \end{aligned}$ (28)
If we $\operatorname{let} f_n^*(\zeta), \theta_n^*(\zeta)$ and $\phi_n(\zeta)$ as the different results of m^{th} order deformation equations, then the general solution is given by:
$\begin{aligned} & f_n(\zeta)=f_n^*(\zeta)+D_1+D_2 e^\zeta+D_3 e^{\zeta}, \\ & \theta_n(\zeta)=\theta_n^*(\zeta)+D_4 e^\zeta+D_5 e^{\zeta}, \\ & \phi_n(\zeta)=\phi_n^*(\zeta)+D_6 e^\zeta+D_7 e^{\zeta}\end{aligned}$ (29)
where, the integral constants D_{i} (i=1 to 7) using the boundary conditions.
It is effortless to solve the above linear homogeneous equations using MATHEMATICA one after other in the order n=1, 2, ….
2.2 Convergence of HAM
To obtain the appropriate values for the nonzero auxiliary parameters, $\hbar$curves are depicted in Figure 2. From this figure, the auxiliary parameter is given by the supposable interval [1.0,0.0]. The solutions are convergent for whole region of ζ when $\hbar_f=\hbar_\theta=\hbar_\phi=0.72$. Convergence of the method is given in Table 1.
Figure 2. $\hbar$curves for f''(0), θ'(0) and φ'(0) at 15^{th} order approximations
The association between different factors and their consequences on f', θ and ϕ of the fluid flow is explored in detail in this section. Additionally, we elucidate the parameters associations between these profiles and important dimensionless variables, such as the skin friction coefficient, the local Nusselt number and the local Sherwood number. In order to shed light on these connections, we employ a set of informative graphs. As a way to preserve coherence with the numerical results generated in this study, we strictly follow the predetermined values shown in Table 1, unless there are specific deviations indicated in the relevant figures.
Table 1. Convergence of HAM solution for different orders of approximations when β=1.0, M=0.5, Ω=60^{o}, S=R=A=δ_{1}=δ_{2}=δ_{3}=Q=Gr=Gc=0.1, Pr=Sc=1.0, Nb=0.3, Nt=0.2, γ=0.2
Order 
f''(0) 
θ'(0) 
ϕ''(0) 
5 
1.316344 
0.463692 
0.633936 
10 
1.317255 
0.450865 
0.650285 
15 
1.317248 
0.451028 
0.650438 
20 
1.317248 
0.451025 
0.650431 
25 
1.317248 
0.451024 
0.650432 
30 
1.317248 
0.451024 
0.650432 
35 
1.317248 
0.451024 
0.650432 
40 
1.317248 
0.451024 
0.650432 
Figures 35 presented the effects of the Casson fluid parameter β on the velocity, temperature and concentration profiles. It is observed that amplification in β enhances the viscosity of the fluid. Fluid behaves like shearthickening on behalf of an incremental change in β which lessens the fluidity of the fluid and also its wideness of the momentum boundary layer. The velocity, temperature profiles decrease with the increase of Casson fluid parameter, whereas the reverse trend is observed in concentration profile.
Figure 3. Figuration of f'(ζ) for β
Figure 4. Figuration of θ(ζ) for β
Figure 5. Figuration of ϕ(ζ) for β
The influence of magnetic parameter M on the profiles is revealed in Figures 68. f'(ζ) decreases as M's magnitude increases, despite the opposite pattern for concentration and temperature. In actuality, as M increases, the Lorentz force which limits fluid motion increases, which causes the rate of transport to decrease. When the magnetic field was applied over the flow field, the Lorentz force became apparent. This force is strong enough to slow down the fluid's flow and drag it along. As a result, fluid flow velocity decreases as momentum layer thickness increases.
Figure 6. Figuration of f'(ζ) for M
Figure 7. Figuration of θ(ζ) for M
Figure 8. Figuration of ϕ(ζ) for M
The influence of velocity ratio parameter A on the profiles momentum is revealed in Figure 9. The momentum of the fluid increases as velocity ratio parameter A magnitude increases. The effect of velocity ratio parameter A on the evolution of nondimensional temperature and solutal concentration profiles is shown in Figures 10 and 11. Thin thermal and solutal boundary layers form when A values increase because the temperature and fluid concentration decrease asymptotically. An increased amount of heat transfer from the wall to the free stream is encouraged by a higher velocity ratio parameter, which also increases the free stream velocity. Because the free stream velocity increases with increasing velocity ratio parameter A, the fluid concentration likewise decreases.
Figure 9. Figuration of f'(ζ) for A
Figure 10. Figuration of θ(ζ) for A
Figure 11. Figuration of ϕ(ζ) for A
Figure 12. Figuration of f'(ζ) for Ω
Figure 13. Figuration of f'(ζ) for S
Figure 12 demonstrates the decreasing performance of velocity for increased parameter values, specifically angle of inclination Ω. The fluid flow becomes difficult due to Lorentz forces, which lowers the fluid's velocity. The findings in Figure 13 show that increasing suction (S) efficiently lowers the velocity profile in both the gaseous and liquid states.
Figure 14 illustrates the properties of the local Grashof number Gr on the velocity profiles. Here, as Gr values rises, then the momentum of the fluid flow going to be enhanced rapidly. The velocity profiles enhance as the values of modified Grashof parameter Gc rises, as seen in Figure 15.
Figure 14. Figuration of f'(ζ) for Gr
Figure 15. Figuration of f'(ζ) for Gc
The fluid velocity decreases, and the slip velocity rises as the velocity slip constraint δ_{1} growths. This may be the case because, in the event of a slip condition, the stretching sheet's velocity and the stream's velocity differ. This is depicted in Figure 16. The fluid velocity decreases as the slip velocity increases in response to an increase in the velocity slip constraint δ_{1}. This phenomenon happens because the stretching sheet's velocity and the fluid stream's velocity near the sheet are dissimilar under slip conditions.
Figure 17 shows that the temperature drops as the thermal slip constraint δ_{2 }increases. As the thermal slip constraint value rises, the thermal boundary layer's physical width declines even though there is very little heat transfer from the sheet to the fluid. As seen in Figure 18, the influence of the nanoparticle fraction slip constraint δ_{3} on the mass fraction field closely parallels that of δ_{3 }on the temperature field. This similarity stems from the basic impediment to liquid motion caused by slip, which eventually leads to a reduction in net atomic advancement. Consequently, the mass fraction field decreases as a result of decreased molecular development.
Figure 16. Figuration of f'(ζ) for δ_{1}
Figure 17. Figuration of θ(ζ) for δ_{2}
Figure 18. Figuration of ϕ(ζ) for δ_{3}
The connection between temperature and the radiation constraint R is shown in Figure 19. Increased radiative heat energy injection into the system, which increases the temperature, is indicated by higher values of R. Figure 20 shows temperature distributions for a range of Prandtl number Pr values, showing a decline in the temperature profile with increasing Pr. Physically, smaller temperature profiles are produced by increasing Prandtl numbers. Figure 21 illustrates the remarkable impact of Q on θ(ζ). In the instance of air, an increase in the values improves θ. After declining at first, the temperature profile rises away from the wall.
Figure 19. Figuration of θ(ζ) for R
Figure 20. Figuration of θ(ζ) for Pr
Figure 21. Figuration of θ(ζ) for Q
Figures 22 and 23 demonstrate the significance of the Brownian movement parameter Nb affects θ(ζ) and ϕ(ζ). Generally speaking, Brownian movement aids in heating the fluid in the boundary layer and stops particles from depositing on the surface away from the fluid. The temperature rises and the concentration decreases as the amount of Nb in the fluid increases (less than 1). Usually, Brownian motion prevents molecules from depositing away from the liquid surface and heats the liquid inside the boundary layer. Rapid flow at a distance from the extension surface is caused by the thermophoretic force created by the resulting temperature gradient. As a result, as Nb rises, additional fluid is heated away from the surface, raising the temperature of the boundary layer.
Nanoparticles are transported by the fast flow created by the stretching sheet, increasing the width of the boundary layer for mass volume fraction. Figures 24 and 25 provide illustrations of these phenomena. According to reports, the temperature and the concentration of nanoparticles both increase in direct proportion to Nt.
Figure 22. Figuration of θ(ζ) or Nb
Figure 23. Figuration of ϕ(ζ) or Nb
Figure 24. Figuration of θ(ζ) for Nt
Figure 25. Figuration of ϕ(ζ) for Nt
The influence of Schmidt number (Sc) on the concentration profile of ϕ(ζ) is depicted in Figure 26. It has also been observed that the concentration profile decreases as the value increases. In terms of physics, Sc is the ratio of mass diffusivity to momentum diffusivity; an increase in the Schmidt number indicates a decrease in the fluid's mass diffusivity relative to its momentum diffusivity, which implies a decrease in scalar diffusivity and less diffusion and slower concentration changes in the fluid medium.
Figure 26. Figuration of ϕ(ζ) for Sc
Figure 27 illustrates the impact of a chemical reaction parameter γ on ϕ(ζ). It is known that the concentration decreases as the chemical reaction parameter increases.
Figure 28 displays the skinfriction factor on variation of M and δ_{1}. It’s observed that as M and δ_{1} increased the skinfriction coefficient increases.
Figure 27. Figuration of ϕ(ζ) for γ
Figure 28. Figuration of C_{fx} for M and δ_{1}
In Figure 29 it is noted that Nusselt number decreases for increase both the constraints Nt and δ_{2}. Sherwood Number is shown in Figure 30, rise as a result of the fluid's high molecular diffusivity and low heat conductivity. The Nt and δ_{3} variations are displayed. In the boundary layer, the fluid velocity near the wall decreases as Nt and δ_{3 }increases.
Figure 29. Figuration of Nu_{x} for Nt and δ_{2}
Figure 30. Figuration of Sh_{x} for Nt and δ_{3}
Table 2. Comparison of skin friction coefficient for different values of β and A when M=0, δ_{1}=0.0, S=0.0
β 
A 
Oyelakin et al. [41] 
HAM 
1 
0.0 
1.41421 
1.41421 
5 
0.0 
1.09544 
1.09545 
1000 
0.01 
0.99782 
0.99801 
1000 
0.1 
0.96937 
0.96937 
1000 
0.2 
0.91811 
0.91811 
Table 3. Comparison of f''(0) for different values of δ_{1} when M=0, A=0.0, β=1000, S=0.0
δ_{1} 
Ibrahim and Makinde [42] 
Oyelakin et al. [41] 
HAM 
0.0 
1.0000 
1.000000 
1.00000 
0.1 
0.8721 
0.872083 
0.87208 
0.2 
0.7764 
0.776377 
0.77638 
0.5 
0.5912 
0.591195 
0.59121 
1.0 
 
0.430160 
0.43017 
2.0 
0.2840 
0.283979 
0.28397 
3.0 
 
0.214054 
0.21406 
5.0 
0.1448 
0.144714 
0.14484 
10.0 
0.0812 
0.080932 
0.08125 
Table 4. Evaluation of the numerical values of θ'(0) for different values of Pr when Le=10 and in the nonappearance of remaining parameters
Pr 
Rudraswamy et al. [43] 
Gupta et al.[44] 
Mini et al. [45] 
HAM 
0.2 
0.1691 
0.1691382 
0.169124 
0.169118 
0.7 
0.4539 
0.4538682 
0.453917 
0.453853 
2.0 
0.9112 
0.9113432 
0.911358 
0.911341 
To assess the validity and accuracy of the applied numerical scheme, numerical values for skinfriction factor, the heat transfer and mass transfer coefficient for various values parameters and in the absence of different parameters are compared with the available results and the outcome is shown in Tables 24. The results are found in excellent agreement.
Based on numerical studies conducted using a HAM technique, the following results were drawn regarding the MHD properties of a Casson nanofluid flowing over an inclined stretching sheet that was stretched linearly:
(i) The decrease in nanofluid velocity is proportional to the Casson parameter, the magnetic parameter and velocity slip factor. The thermal radiation parameter, the Brownian motion, the heat source parameter, and all contribute to an increase in the nanofluid's temperature.
(ii) The solution boundary layer grows as the Casson parameter, thermophoresis parameter does, but it shrinks as Nb shrinks.
(iii) There is a correlation between the velocity slip parameter and an increase in the heat and mass transfer rates. However, the rate of heat transfer and the rate of mass transfer both decrease as the temperature jump parameter values increase.
(iv) It has been observed that the Nusselt number and the Sherwood number drop when the magnetic parameter increases. Through the process of velocity slip, both the Nusselt number and the Sherwood number fall.
Future scope: It is possible that the current work could be expanded to include the nonNewtonian flow over a nonlinear stretching sheet in our subsequent research activities.
a 
Constant stretching rate 
b 
Free stream rate 
(c)_{f} 
Specific heat at constant pressure 
u,v 
Velocity components in x, y directions 
U_{w} 
Stretching velocity 
U_{∞} 
Free stream velocity 
B_{0} 
Strength of magnetic field 
D_{B} 
Brownian diffusion coefficient 
D_{T} 
Thermophoresis diffusion coefficient 
V_{w} 
Wall injection/suction velocity 
γ 
Chemical reaction parameter 
ζ 
Similarity variable 
$\mu_B$ 
Plastic dynamic viscosity 
$\pi$ 
Product of component deformation 
g 
Gravitational acceleration 
C 
Concentration of the fluid 
C_{w} 
Concentration level of fluid at surface 
C_{∞} 
Ambient concentration 
k_{*} 
Absorption coefficient 
κ 
Thermal conductivity of fluid 
$\hbar_f, \hbar_\theta$ and $\hbar_\phi$ 
Nonzero auxiliary parameters 
$\chi_n$ 
Characteristic function 
D_{i} (i=1 to 7) 
Arbitrary constants 
N_{f}, N_{θ} and N_{ϕ} 
Nonlinear operators 
L_{f}, L_{θ} and L_{ϕ} 
Linear operators 
M 
Magnetic field parameter 
Nt 
Thermophoresis parameter 
Nb 
Brownian motion parameter 
Pr 
Prandtl number 
Ω 
Inclined sheet angle 
σ_{f} 
Electrical conductivity 
ρ_{p} 
Nanoparticles mass density 
(ρc)_{p} 
Heat capacity of the nanoparticles 
σ_{*} 
Stefan Boltzmann constant 
ρ_{f} 
Fluid density 
(ρc)_{f} 
Fluid heat capacity 
Gr 
Local Grashof number due to temperature 
Gc 
Local Grashof number due to concentration 
θ 
Dimensionless temperature 
ϕ 
Dimensionless concentration 
q_{r} 
Radiative heat flux 
A 
Velocity ratio parameter 
Gr_{x} 
Temperature buoyancy parameter 
Gc_{x} 
Concentration buoyancy parameter 
Q_{0} 
Heat generation coefficient 
Kr 
Coefficient of chemical reaction 
P_{y} 
Yield stress 
S 
Suction parameter 
Ge_{x} 
Local Reynolds number 
α 
Thermal diffusivity of the fluid 
β_{c} 
Volumetric coefficient of mass expansion 
β_{T} 
Volumetric coefficient of thermal expansion 
e_{ij} 
(i,j)th component of the deformation 
π_{c} 
Critical value of the product based on nonNewtonian model 
β 
Casson parameter 
R 
Thermal radiation parameter 
Le 
Lewis parameter 
T 
Fluid temperature 
T_{w} 
Convective fluid temperature 
T_{∞} 
Ambient fluid temperature 
q_{w} 
Surface heat flux 
q_{m} 
Surface mass flux 
τ_{w} 
Surface shear stress 
V_{0} 
Initial strength of suction 
C_{fx} 
Coefficient of skin friction 
Nu_{x} 
Local Nusselt number 
Sh_{x} 
Local Sherwood number 
f 
Dimensionless stream function 
f' 
Dimensionless velocity 
Sc 
Schmidt number 
Q 
Heat source parameter 
υ 
Kinematic viscosity 
ζ 
Dimensionless variable 
μ 
Dynamic viscosity 
$\tau=\frac{(\rho c)_p}{(\rho c)_f}$ 
Fraction of Heat Capability of Nanofluid to the Base Fluid 
ψ 
Stream function 
Subscripts 

f 
Fluid 
w 
Wall 
p 
Nanoparticle 
∞ 
Free stream 
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