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The boundary layer phenomena for Siskonano fluid flow is being observed with the effect of MHD and thermal radiation on a nonlinear stretched surface. For developing a fundamental flow model, a boundary layer approximation is done, which represents time subservient momentum, concentration and energy equations. By taking the assistance of Compaq Visual Fortran, the fundamental equations are analysed by imposing a finite difference scheme explicitly. A stability and convergence study is also exhibited, and the ongoing investigation is found converged for Lewis number, Le≥0.161 and Prandtl number, Pr≥0.668. The impression of Sisko fluid parameter (A1, A2) along with diversified appropriate parameters is depicted in various flow fields. However, the developed visualisation of fluid flow is also depicted through streamlines and isotherms.
stability and convergence analysis, Sisko nanofluid, higher order chemical reaction, porous plate, MHD, thermal radiation
Sisko fluid is a significant type of nonNewtonian fluid in the field of oil engineering, cement slurries, waterborne coating, drilling fluids, blood flow, mud and paint. Dominant Physical properties of this fluid are it gives higher viscosities at minimum share rates and lower viscosities at higher share rates, which is proposed by Sisko [1]. Then Na and Hansen obtained a theoretical solution by using geometry and powerlaw model for flowing Sisko fluid between two parallel circular disks [2]. They revealed radial distance almost linearly proportional to the pressure. By applying Homotopy Analysis Method (HAM) on Sisko fluid was showed an inverse relation between speed of the vertical belt and the nonNewtonian effect conducted by Nemati et al. [3]. Khan and Shahzad [4] have done numerical investigation to understand the flow attitude of Sisko fluid on stretched surface by imposing HAM. They concluded that the effect of ascents grade of powerlaw index is to descent the velocity as well as boundary layer thickness. Similar investigation analysed by Munir et al. in combination with nonintegral and integral data of powerlaw index by applying shooting technique [5]. Sisko fluid was analysed by Hayat et al. in consider with impact of hall and heat transfer [6].
Additionally, Khan et al. have done a work on Sisko fluid regarding radially stretching sheet with MHD and without MHD effect [78]. Khan et al. addressed the flow of Sisko fluid along with forced convective heat transfer on a stretched cylinder [9]. Moosavi et al. considered the variational iteration method (VIM) for Sisko fluid flow to scrutinize the fluid behavior through a moving belt and also in a collector [10]. Khan et al. and Malik et al. demonstrated CattaneoChristov heat flux and stagnation point flow of Sisko fluid by using HAM method [11, 12]. Abbasbandy et al. studied OldroydB fluid with MHD effect by applying HAM and Kellerbox method [13]. They found good uniformity between series and numerical solution in case of skin friction. Rashidi et al. devoted to a study on entropy generation by using HAM where fluid flow considered over stretching rotating disk [14]. Many researchers addressed heat conducting phenomenon of Sisko fluid very recently [15, 16]. Choi introduced a new idea to enhance the thermal conductivity of fluid mixing with nanoparticles (Cu, Al) [17]. Then the researchers focused on an added nanoscale particle in fluid for industrial and engineering application. Therefore, in asymmetric channel Akbar depicted the attitude of Sisko nanofluid by employing 4^{th} and 5^{th} order RungeKuttaFehlberg scheme [18]. Khan et al. described how the flow of Sisko nanofluid behaved on a nonlinearly stretched flat plate [19]. They observed a monotonically increasing pattern for thermophoresis and Brownian parameters. However, for the powerlaw index, Prandtl number and material parameter inverse phenomena were observed. The flow attitude of 3D MHD Sisko nano and ferrofluids were examined by Raju and Sandeep on a bidirectional stretched surface [20]. For steady Sisko nano fluid flow, Ramanaiah et al. showed temperature profile develops for improving value of thermophoresis, Brownian and thermal radiation parameters on a nonlinear stretched surface [21]. Mahmood et al. conducted a research by focusing on similar parameter with combined effect of MHD and radiation [22]. It was depicted that thermal boundary layers developed for increasing thermophoretic parameter, but opposite phenomena was observed for Brownian parameter. The impressions of nonlinear chemical reaction along with thermal radiation were examined by Prasannakumara et al. over a nonlinear stretching sheet [23]. They deduced an incremented character of chemical reaction parameter, thermophoresis parameter with concentration profile.
Porous medium has been extensively used in practical engineering application, i.e., oil production, cooling of nuclear reactors, solar collector, ventilation procedure, electronic cooling. Hayat et al. devoted the study about a fluid of 4^{th} grade for unsteady flow on porous plate [24]. For nonNewtonian nanofluid flow, the influence of mixed convection was depicted by author Rashad et al. in a porous medium [25]. Rashidi et al. reported about the fluid flow over a rotating porous plate to reveal the entropy generation [26]. They found radial outflow decrease by the impact of magnetic field. Raju et al. obtained the numerical solution of nanofluid by the effect of radiation and Soret in a porous medium [27]. Using Von Karman Method by Rashidi et al. conducted research of fluid flow on porous plate with different conditions [28]. Pandey and Kumar considered the impact of natural convection with thermal radiation for nano fluid which was flowing from a stretched cylinder [29]. Arifuzzaman et al. reported about the viscoelastic nanofluid flowing from stretched surface by imposing explicit scheme [30]. Be´g et al. investigated nanofluid flow mixed with convective boundary layer on adjacent nonDarcian porous medium on exponentially stretched sheet by imposing explicit finite scheme [31]. By applying shooting with RK method, Khan et al. examined the flow character of nanofluid past a linearly stretchedsurface [32]. They found Williamson and radiation parameter accent by the increase of nanoparticle thermal properties. However, the latest review of nanofluid flow was discussed by Kasaeian et al. in porous media, which indicated significant surface contact area between porous structure and working liquid render huge heat transfer [33]. Biswas et al. examined the character of hydromagnetic nanofluid explicitly on a perpendicular stretched sheet/plate with radiation absorption [34]. Arifuzzaman et al. and Rabbi et al. analysed the character of different fluids flow by using EFDM over porous plate [3537, 3942]. A similar method is applied by Biswas et al. to observe the 2D transient optically dense Grey nanofluid impact of periodic magnetic field [38].
To author’s best idea, the following specific objectives of this numerical investigation have remained undone, and the objectives are:
Time subservient hydromagnetic Sisko nanofluid flow resulting from the stretched surface with the impression of heat source, radiation absorption along with nonlinear chemical reaction have been studied. Here, yaxis is considered as the fluid flow direction. U_{0}=cx^{s} is the power law velocity. Here, s>0 stands for stretching rate. Here T_{w} and C_{w} are the fluid temperatures and concentration close to the surface whereas, T_{∞} and C_{∞} exhibit the same phenomena outside the boundary layer. B_{y}=B_{0} is the magnetic field assumed towards the flow region (Figure 1).
Figure 1. Flow pattern of Sisko nanofluid
Continuity equation,
$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$ (1)
Momentum equation,
$\begin{align} & \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\frac{a}{\rho }\frac{{{\partial }^{2}}u}{\partial {{y}^{2}}}\frac{b}{\rho }\frac{\partial }{\partial y}{{\left( \frac{\partial u}{\partial y} \right)}^{n}} \\ & \frac{\sigma B_{0}^{2}}{\rho }u+g\beta (T{{T}_{\infty }})+g{{\beta }^{*}}(C{{C}_{\infty }})\frac{\upsilon }{k}u \\\end{align}$ (2)
Energy equation,
$\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha \left( \frac{{{\partial }^{2}}T}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}T}{\partial {{y}^{2}}} \right)+\frac{\upsilon }{{{c}_{p}}}{{\left( \frac{\partial u}{\partial y} \right)}^{2}}\\+\frac{{{Q}_{0}}}{\rho \,{{c}_{p}}}(T{{T}_{\infty }})+\frac{Q_{1}^{*}}{\rho \,{{c}_{p}}}(C{{C}_{\infty }})\frac{1}{\rho {{c}_{p}}}\frac{\partial {{q}_{r}}}{\partial y}\\+\frac{{{D}_{m}}{{\kappa }_{T}}}{{{c}_{s}}\,{{c}_{p}}}\frac{{{\partial }^{2}}C}{\partial {{y}^{2}}}+\Gamma \left\{ \begin{align} & {{D}_{B}}\left( \frac{\partial T}{\partial x}\frac{\partial C}{\partial x}+\frac{\partial T}{\partial y}\frac{\partial C}{\partial y} \right) \\ & +\frac{{{D}_{T}}}{{{T}_{\infty }}}\left[ {{\left( \frac{\partial T}{\partial x} \right)}^{2}}+{{\left( \frac{\partial T}{\partial y} \right)}^{2}} \right] \\\end{align} \right\}$ (3)
Concentration equation,
$\begin{align} & \frac{\partial C}{\partial t}+u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}={{D}_{B}}\left( \frac{{{\partial }^{2}}C}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}C}{\partial {{y}^{2}}} \right) \\ & +\frac{{{D}_{T}}}{{{T}_{\infty }}}\left( \frac{{{\partial }^{2}}T}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}T}{\partial {{y}^{2}}} \right){{K}_{c}}{{(C{{C}_{\infty }})}^{P}} \\\end{align}$ (4)
with boundary conditions,
$u={{U}_{0}}=c{{x}^{s}},\,T={{T}_{w}},\,\,C={{C}_{w}}\,\,\,\,\text{at}\,\,\,\,\,\,y=0$
$u\to 0,\,T\to {{T}_{\infty }},\,C\to {{C}_{\infty }}\,\,\text{at}\,\,\,\,\,\,y\to \infty \,$
Here, Q_{0} is the heat source, Q_{1}^{*} denotes the radiation absorption, K_{c} species chemical reaction and p is the order. The Rosseland approximation is exhibited as, ${{q}_{r}}=\frac{4{{\sigma }_{s}}}{3{{k}_{e}}}\frac{\partial {{T}^{4}}}{\partial y}$. Then the equation (3) becomes,
For solving the fundamental equations (1)(5) the dimensionless quantities are adopted as,
$\begin{align} & X=\frac{x{{U}_{0}}}{\upsilon },Y=\frac{y{{U}_{0}}}{\upsilon },\,U=\frac{u}{{{U}_{0}}},V=\frac{v}{{{U}_{0}}},\,\tau =\frac{tU_{0}^{2}}{\upsilon },\,\, \\ & \theta =\frac{T{{T}_{\infty }}}{{{T}_{w}}{{T}_{\infty }}},\,\phi =\frac{C{{C}_{\infty }}}{{{C}_{w}}{{C}_{\infty }}}. \\\end{align}$
Hence nondimensional forms are obtained as,
Continuity equation,
$\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0$ (6)
Momentum equation,
$\begin{align} & \frac{\partial U}{\partial \tau }+U\frac{\partial U}{\partial X}+V\frac{\partial U}{\partial Y}=\frac{{{\partial }^{2}}U}{\partial {{Y}^{2}}}\left[ {{A}_{1}}{{A}_{2}}n{{(\frac{\partial U}{\partial Y})}^{n1}} \right] \\ & +{{G}_{r}}\theta +{{G}_{c}}\phi MU\frac{U}{{{D}_{a}}} \\ \end{align}$ (7)
Energy equation,
$\begin{align} & \frac{\partial \theta }{\partial \tau }+U\frac{\partial \theta }{\partial X}+V\frac{\partial \theta }{\partial Y}=\frac{1}{{{P}_{r}}}\left( 1+\frac{16R}{3} \right)\frac{{{\partial }^{2}}\theta }{\partial {{Y}^{2}}}+\frac{1}{{{P}_{r}}}\frac{{{\partial }^{2}}\theta }{\partial {{X}^{2}}} \\ & +Q\theta +{{Q}_{1}}\phi +{{E}_{c}}{{\left( \frac{\partial U}{\partial Y} \right)}^{2}}+{{D}_{u}}\frac{{{\partial }^{2}}\phi }{\partial {{Y}^{2}}} \\ & +{{N}_{b}}\left( \frac{\partial \theta }{\partial X}\frac{\partial \phi }{\partial X}+\frac{\partial \theta }{\partial Y}\frac{\partial \phi }{\partial Y} \right)+{{N}_{t}}\left[ {{\left( \frac{\partial \phi }{\partial X} \right)}^{2}}+{{\left( \frac{\partial \theta }{\partial X} \right)}^{2}} \right] \\\end{align}$ (8)
Concentration equation,
$\begin{align} & \frac{\partial \phi }{\partial \tau }+U\frac{\partial \phi }{\partial X}+V\frac{\partial \phi }{\partial Y}=\frac{1}{{{L}_{e}}}\left( \frac{{{\partial }^{2}}\phi }{\partial {{X}^{2}}}+\frac{{{\partial }^{2}}\phi }{\partial {{Y}^{2}}} \right) \\ & +\frac{{{N}_{t}}}{{{N}_{b}}{{L}_{e}}}\left( \frac{{{\partial }^{2}}\theta }{\partial {{X}^{2}}}+\frac{{{\partial }^{2}}\theta }{\partial {{Y}^{2}}} \right){{K}_{c}}{{\phi }^{P}} \\\end{align}$ (9)
With the conditions,
$\begin{align} & U=1,\,\,\,\theta =1,\,\,\varphi =1\,\,\,\,\,\text{at}\,\,\,\,\,\,y=0 \\ & U=0,\,\,\theta =0,\,\varphi =0\,\,\,\,\,\text{at}\,\,\,\,\,y\to \infty \, \\\end{align}$
where, Magnetic parameter, $M\text{=}\frac{\sigma '{{B}_{o}}^{2}v}{\rho{{U}_{o}}^{2}}$, mass Grashof number, ${{G}_{c}}=\frac{g\beta *({{c}_{w}}{{c}_{\infty }})v}{U_{_{o}}^{2}}$, Grashof number, ${{G}_{r}}=\frac{g\beta ({{T}_{w}}{{T}_{\infty }})}{U_{0}^{3}}$, Darcy number, ${{D}_{a}}=\frac{k'U_{0}^{2}}{{{v}^{2}}}$, Prandtl number, ${{\operatorname{P}}_{r}}=\frac{\rho {{c}_{p}}v}{k}$, Eckert number, ${{E}_{\text{c}}}=\frac{U_{0}^{2}}{{{c}_{p}}({{T}_{w}}{{T}_{\infty }})}$, radiation parameter, $R=\frac{\sigma T_{\infty }^{3}}{{{k}_{e}}\kappa }$, heat source parameter, $Q=\frac{{{Q}_{o}}v}{U_{\text{o}}^{2}\rho {{c}_{p}}}$, radiation absorption parameter, ${{Q}_{1}}=\frac{Q_{1}^{*}v}{U_{\text{o}}^{2}\rho {{c}_{p}}}\frac{({{C}_{w}}{{C}_{\infty }})}{({{T}_{w}}{{T}_{\infty }})}$, Dufour number, ${{D}_{u}}=\frac{{{D}_{m}}kT}{{{c}_{s}}{{c}_{p}}v}\frac{({{C}_{w}}{{C}_{\infty }})}{({{T}_{w}}{{T}_{\infty }})}$, Lewis number, $L_e=\frac{v}{D_m}$, Sisko fluid parameter, $A_1=\frac{a}{ρυ}$ and ${{A}_{\text{2}}}\text{=}\frac{bU_{0}^{2n2}}{\rho {{v}^{n}}}$, Brownian parameter, ${{N}_{b}}\text{=}\frac{r{{D}_{B}}({{C}_{w}}{{C}_{\infty }})}{v}$, thermophoresis parameter ${{N}_{t}}\text{=}\frac{r{{D}_{T}}}{{{T}_{\infty }}v}({{T}_{w}}{{T}_{\infty }})$, chemical reaction, ${{K}_{c}}\text{=}\frac{v{{K}_{c}}({{C}_{w}}{{C}_{\infty }})}{U_{0}^{2}}$ and Order of chemical reaction = P.
Here, equation (6) is satisfied by stream function ψ and associated with velocity component as, $U=\frac{∂ψ}{∂Y}$ and $V=\frac{∂ψ}{∂X}$.
The impression of different parameters on local and average shear stress is being computed from the velocity. Here, Local and average shear stresses are $\tau A=\mu \int{{{(\frac{\partial u}{\partial y})}_{y=0}}}dx$ respectively which are proportionate to ${{(\frac{\partial u}{\partial y})}_{y=0}}$ and $\int _{0}^{100}{{(\frac{\partial u}{\partial y})}_{y=0}}dX$ respectively. Now, the influence of different parameters has been examined from the temperature field on local as well as average heat transfer coefficient. In this case, the local and average Nusselt number, ${{N}_{uL}}=\mu {{(\frac{\partial T}{\partial y})}_{y=0}}$ and ${{N}_{uA}}=\mu {{(\frac{\partial T}{\partial y})}_{y=0}}dx$ are also proportionate to ${{(\frac{\partial \theta }{\partial Y})}_{y=0}}$ and $\int_{0}^{100}{{{(\frac{\partial \theta }{\partial Y})}_{y=0}}}$. However, from concentric field, the average and local mass transfer have been analysed with the impact of diversified parameters such that average ${{S}_{hA}}=\mu \int{{{(\frac{\partial C}{\partial y})}_{y=0}}}$ and local ${{S}_{hL}}=\mu {{(\frac{\partial C}{\partial y})}_{y=0}}$ mass transfer rate proportionate $\int_{0}^{100}{{{(\frac{\partial \varphi }{\partial Y})}_{Y=0}}}dX$ and ${{(\frac{\partial \varphi }{\partial Y})}_{Y=0}}$ respectively.
Equations (6)(9) are being solved by imposing explicit finite scheme within the given boundary criterion. A rectangular shape flow region is chosen in which the grid lines are distributed parallel to x and yaxes (Figure 2). For the existing problem, it is adopted as Y_{max }= 20, which represents the length of the plate. It changes from o to 20 as Y→∞. However, the grid spaces are also considered as, m = 100 and n = 200 respectively and Δτ = 0.005. Now, we adopt the following equations by employing explicit finite scheme.
Figure 2. Illustration of grid spacing
$\frac{{{U}_{i,j}}{{U}_{i,j1}}}{\Delta X}+\frac{{{V}_{i,j+1}}{{V}_{i,j}}}{\Delta Y}=0$ (10)
$\begin{align} & \frac{U_{i,j}^{'}{{U}_{i,j}}}{\Delta \tau }+{{U}_{i,j}}\frac{{{U}_{i,j}}{{U}_{i1,j}}}{\Delta X}+{{V}_{i,j}}\frac{{{U}_{i,j+1}}{{U}_{i,j}}}{\Delta Y}={{G}_{r}}{{\theta }_{i,j}} \\ & +{{G}_{c}}{{\phi }_{i,j}}\,(M+\frac{1}{{{D}_{a}}}){{U}_{i,j}}+\frac{{{U}_{i,j+1}}2{{U}_{i,j}}+{{U}_{i,j1}}}{{{(\Delta Y)}^{2}}} \\ & \left\{ {{A}_{1}}{{A}_{2}}n{{\left( \frac{{{U}_{i,j+1}}{{U}_{i,j}}}{\Delta Y} \right)}^{n1}} \right\} \\\end{align}$ (11)
$\begin{align} & \frac{\theta _{i,j}^{'}{{\theta }_{i,j}}}{\Delta \tau }+{{U}_{i,j}}\frac{{{\theta }_{i,j}}{{\theta }_{i1,j}}}{\Delta X}+{{V}_{i,j}}\frac{{{\theta }_{i,j+1}}{{\theta }_{i,j}}}{\Delta Y}=Q{{\theta }_{i,j}} \\ & +\frac{1}{{{P}_{r}}}(1+\frac{16}{3}R)\frac{{{\theta }_{i,j+1}}2{{\theta }_{i,j}}+{{\theta }_{i,j1}}}{{{(\Delta Y)}^{2}}}+{{Q}_{1}}{{\phi }_{i,j}} \\ & +\frac{1}{{{P}_{r}}}\left( \frac{{{\theta }_{i+1,j}}2{{\theta }_{i,j}}+{{\theta }_{i1,j}}}{{{(\Delta X)}^{2}}} \right)+{{D}_{u}}\frac{{{\phi }_{i,j+1}}2{{\phi }_{i,j}}+{{\phi }_{i,j1}}}{{{(\Delta Y)}^{2}}} \\ & +{{E}_{c}}{{\left( \frac{{{U}_{i,j+1}}{{U}_{i,j}}}{\Delta Y} \right)}^{2}}+{{N}_{t}}\left\{ {{\left( \frac{{{\phi }_{i,j}}{{\phi }_{i1,j}}}{\Delta X} \right)}^{2}}+{{\left( \frac{{{\theta }_{i,j+1}}{{\theta }_{i,j}}}{\Delta Y} \right)}^{2}} \right\} \\ & +{{N}_{b}}\left( \frac{{{\theta }_{i,j}}{{\theta }_{i1,j}}}{\Delta X}.\frac{{{\varphi }_{i,j}}{{\varphi }_{i1,j}}}{\Delta X}+\frac{{{\theta }_{i,j+1}}{{\theta }_{i,j}}}{\Delta Y}.\frac{{{\varphi }_{i,j+1}}{{\varphi }_{i,j}}}{\Delta Y} \right) \\\end{align}$ (12)
With boundary conditions,
$U_{i,0}^{n}=1,\,\,\theta _{i,0}^{n}=1,\,\,\phi _{i,0}^{n}=1\\U_{i,L}^{n}=0,\,\,\theta _{i,L}^{n}=0,\,\,\phi _{i,L}^{n}=0$ where, $L\to \infty $
Here, i=j= grid points along X and Y axes and $\tau =n\Delta \tau $, where, n= positive number.
Due to the implementation of explicit finite scheme the ongoing investigation demands the study of stability and convergence test. It won’t be necessary to use equation (6) because Δτ doesn’t appear on it. At an arbitrary time the Fourier transformation gives the following equations.
$\left. \begin{align} & U:\psi (\tau ){{e}^{i\alpha X}}{{e}^{i\beta Y}} \\ & \theta :\theta (\tau ){{e}^{i\alpha X}}{{e}^{i\beta Y}} \\ & \varphi :\varphi (\tau ){{e}^{i\alpha X}}{{e}^{i\beta Y}} \\\end{align} \right\}$ (14)
And after a time step we adopt,
$\left. \begin{align} & U:{{\psi }^{'}}(\tau ){{e}^{i\alpha X}}{{e}^{i\beta Y}} \\ & \theta :{{\theta }^{'}}(\tau ){{e}^{i\alpha X}}{{e}^{i\beta Y}} \\ & \varphi :{{\varphi }^{'}}(\tau ){{e}^{i\alpha X}}{{e}^{i\beta Y}} \\\end{align} \right\}$ (15)
Substituting Equation (14) and (15) to Equations (11)(13) we attain,
$\begin{align} & {{\psi }^{'}}=\psi +\Delta \tau [{{G}_{r}}\theta +{{G}_{c}}\varphi (M+\frac{1}{{{D}_{a}}})\psi +\frac{2(\cos \beta \Delta Y1)}{{{(\Delta Y)}^{2}}}\psi \\ & \frac{U(1{{e}^{i\alpha \Delta X}})}{\Delta X}\psi \frac{V({{e}^{i\beta \Delta Y}}1)}{\Delta Y}\psi \\ & \frac{2\psi (\cos \beta \Delta Y1)}{{{(\Delta Y)}^{2}}}\left\{ {{A}_{1}}{{A}_{2}}n\psi {{\left( \frac{({{e}^{i\beta \Delta Y}}1)}{\Delta Y} \right)}^{n1}} \right\}] \\\end{align}\\herefore {\psi }'={{A}_{1}}\psi +{{A}_{2}}\theta +{{A}_{3}}\varphi$ (16)
where, ${{A}_{2}}=\Delta \tau {{G}_{r}}$ and ${{A}_{3}}=\Delta \tau {{G}_{c}}$ and
$\begin{align} & {{A}_{1}}=1+\Delta \tau \frac{2(\cos \beta \Delta Y1)}{{{(\Delta Y)}^{2}}}\left\{ {{A}_{1}}{{A}_{2}}n{{\left( \frac{({{e}^{i\beta \Delta Y}}1)}{\Delta Y} \right)}^{n1}} \right\} \\ & \Delta \tau (M+\frac{1}{{{D}_{a}}})\frac{U\Delta \tau (1{{e}^{i\alpha \Delta X}})}{\Delta X}\frac{V\Delta \tau ({{e}^{i\beta \Delta Y}}1)}{\Delta Y} \\\end{align}$
For temperature equation,
where,
\[{{A}_{5}}={{D}_{u}}\frac{\Delta \tau 2(\cos \beta \Delta Y1)}{{{(\Delta Y)}^{2}}}+{{Q}_{1}}\Delta \tau +\Delta \tau {{N}_{t}}\left\{ C{{\left( \frac{(1{{e}^{i\alpha \Delta X}})}{\Delta X} \right)}^{2}} \right\}\]
And
For the concentration equation,
where, $\begin{align} & {{A}_{6}}=1+\frac{\Delta \tau }{{{L}_{e}}}\left\{ \frac{2(\cos \beta \Delta X1)}{{{(\Delta X)}^{2}}}+\frac{2(\cos \beta \Delta Y1)}{{{(\Delta Y)}^{2}}} \right\}\Delta \tau {{K}_{r}} \\ & \frac{U(1{{e}^{i\alpha \Delta X}})}{\Delta X}\frac{V({{e}^{i\beta \Delta Y}}1)}{\Delta Y} \\\end{align}$ and ${{A}_{7}}=\frac{{{N}_{t}}}{{{N}_{b}}}\left\{ \frac{2(\cos \beta \Delta X1)}{{{(\Delta X)}^{2}}}+\frac{2(\cos \beta \Delta Y1)}{{{(\Delta Y)}^{2}}} \right\}$
Equation (16)(18) can be expressed in matrix notation,
$\left[ \begin{matrix} \psi ' \\ \theta ' \\ \varphi ' \\\end{matrix} \right]=\left[ \begin{matrix} {{A}_{1}} & {{A}_{2}} & {{A}_{3}} \\ 0 & {{A}_{4}} & {{A}_{5}} \\ 0 & {{A}_{7}} & {{A}_{6}} \\\end{matrix} \right]\left[ \begin{matrix} \psi \\ \theta \\ \varphi \\\end{matrix} \right]$ i.e. $\eta '=T'\eta $
Diversified data Δτ→0 makes the investigation critical. Hence, for Δτ→0 we adopt, A_{2}→0,A_{3}→0,A_{5}→0 and A_{7}→0.
$\therefore T'=\left[ \begin{matrix} {{A}_{1}} & 0 & 0 \\ 0 & {{A}_{4}} & 0 \\ 0 & 0 & {{A}_{6}} \\\end{matrix} \right]$
So, the Eigenvalues are attained as A_{1}=λ_{1},A_{4}=λ_{2} and A_{6}=λ_{3} which satisfies, A_{1} ≤1,A_{4} ≤1 and A_{6}≤1
Now taking,
\[\begin{align} & {{a}_{1}}=\Delta \tau ,\,{{b}_{1}}=U\frac{\Delta \tau }{\Delta X},\,{{c}_{1}}=\left V \right\frac{\Delta \tau }{\Delta Y}\,\,,{{d}_{1}}=2\frac{\Delta \tau }{{{\left( \Delta X \right)}^{2}}}\text{,}{{\text{e}}_{1}}=2\frac{\Delta \tau }{{{\left( \Delta Y \right)}^{2}}} \\ & \alpha \Delta Y=m\pi ,\alpha \Delta X=n\pi ,U=\,\text{positive}\,\,\text{and}\,\,V=\,\text{negative}. \\\end{align}\]
Keeping in mind the aboveconsidered things the stability criterion of this investigation can be achieved after simplification as,
\[\begin{align} & U\frac{\Delta \tau }{\Delta X}+V\frac{\Delta \tau }{\Delta Y}+\frac{2}{{{P}_{r}}}(1+\frac{16}{3}R)\frac{\Delta \tau }{{{\left( \Delta Y \right)}^{2}}}+\frac{2}{{{P}_{r}}}\frac{\Delta \tau }{{{\left( \Delta X \right)}^{2}}}+\frac{\Delta \tau \,Q}{2} \\ & +4{{N}_{b}}C\left( \frac{\Delta \tau }{{{\left( \Delta X \right)}^{2}}}+\frac{\Delta \tau }{{{\left( \Delta Y \right)}^{2}}} \right)+2{{N}_{t}}T\frac{\Delta \tau }{{{\left( \Delta Y \right)}^{2}}}\le 1 \\\end{align}\]
and $U\frac{\Delta \tau }{\Delta X}+V\frac{\Delta \tau }{\Delta X}+\frac{2}{{{L}_{e}}}\left( \frac{\Delta \tau }{{{\left( \Delta X \right)}^{2}}}+\frac{\Delta \tau }{{{\left( \Delta Y \right)}^{2}}} \right)+\frac{\Delta \tau \,{{K}_{c}}}{2}\le 1$
For U=V=T=C=0,Δτ=0.005,ΔY=0.25 and ΔX=0.20, the existing problem converged at L_{e}≥0.161 and P_{r}≥0.668.
The flow character of hydromagnetic naturally convective Sisko fluid through a perpendicular nonlinear stretching sheet with the appearance of nanoparticles is being studied numerically. The numerical computation for different flow fields is depicted by FORTRAN. The timeindependent resolutions have executed up to nondimensional time $\tau =30$. Graphical results are analysed using physical parameters, such as G_{r}=8.00, D_{a}=1.60, G_{m}=2.00, M=1.20, A_{1}=1.20, A_{2}=0.60, L=1, P_{r}=2, D_{u}=0.03, R=0.30, E_{c}=0.002, Q=1.40, Q_{1}=0.06, L_{e}=8.00, N_{t}=0.1, N_{b}=0.1 and K_{c}=0.50.
Figure 3. Variation of concentration profiles due to Le
Figure 3 depicts the impact of Lewis number, L_{e} on concentric fields. By escalating the data of L_{e} indicate the decline of concentric fields with initial thin line spread out by the difference of large value from value of Y=1.1, where this behaviour is attributed to the effect Le (inversely proportional to D_{B}) as L_{e} rises when D_{B} decreased. Figure 4 demonstrates the impact Lewis number on Sherwood number where the plot is shown downward pattern of S_{h }concerning Lewis number. The impression of Dufour number, D_{u}, is drafted on temperature profiles in Figure 5.
Figure 4. Variation of Sherwood number due to L_{e}
Figure 5. Variation of temperature profiles due to D_{u}
It can be perceived that for the growing values of D_{u} noticed the increase of temperature distribution from initial to the point of Y=1.6. After this point, the decremented pattern is shown to the end with insignificant differences. The upshots of thermophoresis, N_{t}, and Brownian, N_{b}, parameters on temperature fields are established in Figures 6 and 7. Furthermore, the numerical change of N_{b} and N_{t} concerning temperature by percentage is shown in Table 2 were small change occurred in case of N_{b} instead of significant change in N_{t}.
The fundamental causes behind are thermophoresis parameter directly proportional to temperature difference and Brownian motion increase by the effect of incremented concentration. The contrasts of Prandtl number, P_{r}, are described in Figure 8. It is witnessed that initially enlarged data of temperature profile become reverse at the point near about Y=2.
Table 1. Comparison of skin friction coefficient with Prasanna kumara et al. [23] when, Nr=R=1.5, θ_{w}=1.2, N_{b}=N_{t}=0.45, L_{e}=10, , γ=K_c=0.1, P_{r}=6.2
Linear Stretching Sheet (n=1) 
Nonlinear Stretching Sheet (n=3) 

A, A_{1}, A_{2} 
Q 
S 
Skin Friction [23] 
Present Study 
Skin Friction [23] 
Present Study 
0.5 
0.3 
0.5 
1.1583 
1.1623 
0.9486 
0.9684 
1 
0.3 
0.5 
1.3417 
1.1418 
1.1259 
1.1339 
2 
0.3 
0.5 
1.6582 
1.1695 
1.4526 
1.1506 
0.5 
0 
0.5 
0.9530 
0.9936 
0.7567 
0.8019 
0.5 
0.3 
0.5 
1.1583 
1.1590 
0.9486 
0.9713 
0.5 
0.6 
0.5 
1.3360 
1.3452 
1.1248 
1.1302 
0.5 
0.3 
0 
0.8559 
0.8831 
0.5365 
0.6348 
0.5 
0.3 
0.5 
1.1583 
1.1632 
0.9486 
0.9684 
Table 2. Variation of Curve for diversified data of N_{b} and N_{t} in Figure 6 and Figure 7 at Y=4
N_{b} 
$\theta$ 
Increase 
N_{t} 
θ 
Increase 
1.10 
0.26876 

0.20 
0.36071 

1.50 
0.26881 
0.005 % 
0.25 
0.38820 
2.749 % 
1.90 
0.26889 
0.008 % 
0.30 
0.41705 
2.885 % 
2.40 
0.26898 
0.009 % 
0.40 
0.47878 
6.173 % 
Figure 6. Variation of temperature profiles due to N_{b}
Then it is established the reduction of temperature distribution by the impact of incremented P_{r}. Because of high Prandtl number creates low thermal conductivity. To explore the character of Q (heat source), on temperature profiles, Figure 9 is plotted. It is asserted that initially temperature profile incremented with rising of Q but finally decreased. Before the value of Y=2.4, it shows incremented with difference of Q=0.20 and Q=0.40 about 119.156% at Y=2.00 but finally decreased by 1.014% at Y=6.00. Figure 10 displays the impression of Q_{1} (radiation absorption), upon temperature profiles for diversified rising values of Q_{1}. It is seen that temperature of Sisko nanofluid drops as Q_{1 }upsurge.
Figure 7. Variation of temperature profiles due to N_{t}
_{}
Figure 8. Variation of temperature profiles due to P_{r}
_{}
Figure 9. Variation of temperature profiles due to Q
Figure 10. Variation of temperature profiles due to Q_{1}
Figure 11 depicts chemical reaction impact on concentration profiles. On the evidence of these figures, it is investigated that the values of K_{c} are raised, the concentration profile depresses. This occurs because the incremented of K_{c} enhance the chemical reaction and consequently the concentration profile reduces.
Furthermore, the velocity distribution for various data of Sisko fluid parameters A_{1 }andA_{2 }is organised in Figures 12 and 13. Here, the velocity distribution decremented with growing value only for A_{1} but reversed for A_{2}. Both figures exhibit their change in about 50% before twisting at a point nearly Y=1.8. However, the Darcy number, D_{a}, impact on velocity profile is sketched in Figure 14. The fluid velocity is found increasing with increased D_{a}.
Figure 11. Variation of concentration profiles due to K_{r}
Figure 12. Variation of velocity profiles due to A_{1}
_{}
Figure 13. Variation of velocity profiles due to A_{2}
Table 3. Variation of Curve for different value of A_{1} and A_{2} in Figure 12 and Figure 13 at Y=3
A_{1} 
U 
Increase 
A_{2} 
U 
Decrease 
1.00 
4.65671 

0.60 
5.17792 

1.20 
5.17792 
52.121 % 
0.80 
4.65671 
52.121 % 
1.40 
5.65896 
48.104 % 
1.00 
4.11640 
54.031 % 
1.60 
6.09619 
43.723 % 
1.20 
3.66987 
44.653 % 
The change of percentage for Y=3, from range of value D_{a}= 2.00 to 2.60 is described numerically at Table 4. Finally, the change becomes indistinct for their similar value at far away from plate. The variation of radiation parameter, R, is displayed in Figure 15. It is anticipated that fluid temperature enhances with ascending values of R. It is claimed that the radiation parameter is related with third power of the temperature. Initially, downward pattern changes after Y=1.6 until the end of plate which detects numerically at point Y=4 in Table 5. From Figure 16, it is examined that an increment in R_{a} initially created reduction pattern but finally incremented with Nusselt number.
Table 4. Variation of Curve for different value of D_{a} in Figure 14 at Y=3
D_{a} 
U 
Increment in Percentage 
2.00 
1.80661 

2.20 
1.84366 
3.705 % 
2.40 
1.87461 
3.095 % 
2.60 
1.90071 
2.610 % 
Figure 14. Variation of velocity profiles due to D_{a}
Figure 15. Variation of temperature profiles due to R
Figure 16. Variation of Nusselt number due to R
Table 5. Variation of curve for different R in Figure 15 and Figure 16 at Y=4
Figure 15 
Figure 16 

R 
θ 
Increase 
R 
N_{u} 
Decrease 
0.00 
0.06244 

0.00 
0.17853 

0.10 
0.13794 
7.550 % 
0.10 
0.15646 
2.207 % 
0.20 
0.20878 
7.084 % 
0.20 
0.14102 
1.544 % 
0.30 
0.26921 
6.043 % 
0.30 
0.12945 
1.157 % 
In case of Nusselt number change of percentage from the value of R=0.00 to R=0.30 is displayed in Table 5 which continue at the point of Y=6.2. Furthermore, Figure 17, we elucidated the dleomination of magnetic parameter, M, on velocity. Here we detect a rising value of M indicates the descending value of velocity profile.
Figure 17. Variation of velocity profiles due to M
The fact behind this, the appearance of magnetic field develops Lorentz force. This force retracts fluid flows. The decremented pattern becomes more distinct by numerically (Table 6). That’s shown differences between a line to another line in Figure 17 with respect considering grid by percentage. The impact of M on skin friction is given in Figure 18. It is evident from this Figure, the value of M enhances by decreasing skin friction. In table 6 represents the change in the percentage of M from 1.90 to 2.50.
Table 6. Variation of curve for different M in Figure 17 and Figure 18 at Y = 3
Figure 17 
Figure 18 

M 
U 
Increase 
M 
U 
Decrease 
1.90 
1.00763 

1.90 
0.60374 

2.10 
0.91484 
7.550 % 
2.10 
0.56139 
2.207 % 
2.30 
0.83544 
7.084 % 
2.30 
0.52236 
1.544 % 
2.50 
0.76786 
6.043 % 
2.50 
0.48618 
1.157 % 
Figure 18. Variation of Skin friction due to M
It indicates the apparent features of different M from Y = 1.1 to the end of the plate. The analysis of isotherms and streamlines are depicted in Figures 19 to 21 for exhibiting the advanced visualisation of fluid fields. In Figures. 19 and 20, isotherms for different M with different view (line and flood) is portrayed, which indicate thermal boundary layers increase due to with developing magnetic parameter, M. To reveal the distinct visualisation of fluid flow, the streamlined flow is illustrated in Figures 21 and 22 for increasing values of M. It can be perceived by sketching tangent on the velocity direction of fluids. Here it is experienced that, momentum boundary layers get suppress for increasing magnetic parameter.
Figure 19. Isotherms view for M=1.20 (red solid line) and M=2.00 (green solid line)
Figure 20. Isotherms flood view for different M
Figure 21. Streamline line view for M=1.20 (red solid line) and M=2.00 (green solid line)
Figure 22. Streamline flood view for different M
Computational modelling of Sisko fluid with nanoparticle moving to a porous stretching sheet with nonlinear chemical reaction is being analysed. The following results are noticed after conducting the complete study:
A, A_{1}, A_{2} 
Sisko fluid parameter () 

B_{○} 
magnetic component, (Wb m^{2}) 

C_{f} 
skinfriction, () 

Cp 
specific heat at constant pressure, (J kg^{1} K^{1}) 

D_{a} 
Darcy number, () 

D_{B} 
The Brownian diffusion coefficient, () 

D_{u} 
Dufour number, () 

E_{c} 
Eckert number, () 

G_{r} 
Grashof number, () 

G_{c} 
modified Grashof number, () 

K^{/} 
the permeability of the porous medium, () 

k_{e} 
mean absorption coefficient 

K_{r} 
chemical reaction parameter, () 

Le 
Lewis number, () 

N_{b} 
The Brownian parameter, () 

Nt 
thermophoresis parameter, () 

N_{u} 
local Nusselt number, () 

Pr 
Prandtl number, () 

Q 
heat source parameter, () 

q_{r} 
unidirectional radiative heat flux, (kg m^{2}) 

Q_{1} 
radiation absorption, () 

R 
radiation parameter () 

S_{h} 
Sherwood number, () 

T 
Fluid temperature, (K) 

Tw 
The temperature at the plate surface, (K) 

T_{∞} 
ambient temperature as y tends to infinity, (K) 

U_{○} 
uniform velocity 

u, v 
velocity components 

x, y 
Cartesian coordinates 

Greek symbols 

β 
thermal expansion coefficient 

β^{*} 
concentration expansion coefficient 

κ 
thermal conductivity, (Wm^{1} K^{1}) 

μ 
dynamic viscosities 

ν 
kinematic viscosity, (m^{2} s^{1}) 

ρ 
the density of the fluid, (kg m^{3}) 

σ' 
electric conductivity 

σ_{s} 
StefanBoltzmann constant, 5.6697 × 10^{8} (W/m^{2}K^{4}) 
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