1. Introduction
During the past few centuries, significant attention has been exhibited in the study of unsteady flows of non-Newtonian fluids. Many materials exclusive of dyes, ketchup, lubricants, mud, certain paints, blood at a low shear rate, and particular care products are non-Newtonian. The Newtonian fluids' flow has a vital role in various applications of engineering, including composite processing, manufacturing of polymer depolarization, boiling, fermentation, bubbles' columns, bubbles' absorption, plastic foam processes, etc. There are various kinds of Non-Newtonian fluids, and their rheological properties are complex compared to the Newtonian. The researchers devoted their attention to studying different non-Newtonian fluid models, including Williamson, Casson, and a few others. The Eyring-Powell fluid model, among these non-Newtonian models, has many advantages but it is complex nature. The mathematical model of an Eyring-Powell fluid defines the characteristics of shear-thinning fluids. Toothpaste, ketchup, and the blood of humans are a few examples of said fluid, but scholars are concerned with examining its thermo-physical properties. A theoretical study of mixed convection MHD Eyring-Powell nanofluid flow over a stretching sheet is examined by Malik et al [1]. The numerical outcomes have been analyzed numerically by using a shooting method. It is reported that the velocity increases for fluid parameters while a drop is observed for the Hartmann number. The influence of thermophoresis and Brownian motion gives high temperatures, while the Prandtl numbers reduce the temperature of the fluid. A comprehensive study related to the Eyring-Powell model is available in the literature [2-5].
The motion of fluid near a stagnated region of a solid body has gained an intense devotion among researchers. The stagnation area of the solid body could be a moving surface or fixed in the liquid. The concept of said mechanism is very common in various applications, including thermal oil recovery, high-speed flows, and thrust bearing. The pioneering effort in this domain is presented by Hiemenz [6]. Later on, Homann and Angew [7] extended the problem based on Hiemenz's work. Nowadays, many scholars carry this out utilizing a different mechanism. The slip impacts on an unsteady stagnation point nanofluid flow towards a stretching sheet is analyzed by Malvandi et al [8].
The behavior of a water-based nanofluid is examined for three various nanoparticles, namely titanium (TiO2) alumina (Al2O3), and copper (Cu). A numerical scheme named Runge-Kutta-Fehlberg is adopted to investigate the dual-natured results of the proposed model. It is reported that a rise in the slip parameter intensifies the rate of heat transfer. Recently, Hayat et al [9] numerically examined the radiation and melting heat transfer impact on the stagnation point flow of carbon-H2O nanofluid. The carbon nanotubes, both single- and multi-walls, are homogeneously isolated in the water. It is reported that velocity is enhanced for a larger ratio of rate constants. The phenomena of stretching sheets are also considered by various authors due to their variety of applications in many technical purposes and engineering, mainly in the polymer and metallurgy industry. For example, plastic strips or gradual cooling of continuous stretching metals, which have several applications in mass production. Crane [10] reported the domain of the stretching surface for the first time. After his idea, many authors investigated this mechanism. Some qualitative studies related to stagnation points and stretching sheets are available in [11-17].
The transport of heat and mass is a vital area of research, and recent developments in nanotechnology have opened another domain of heat transport. The heat transfer phenomena arise in several applications of engineering and science particularly, solar water heating, engine cooling, drag reductions, drilling, jacket water coolant, diesel-electric generators, biomedical, cooling of welding, engine transmission oil, drilling, heating and cooling of buildings, boiler exhaust flue gas recovery, high-power lasers, electronics cooling, thermal storage, transformer and oil cooling, cooling of nuclear systems, refrigeration (chillers, refrigerator, domestic), lubrications, Space, defense, and microwave tubes [18]. The familiar nanofluid models are Buongiorno [19], and Tiwari and Das [20]. Sheikholeslami et al [21] analyzed the impacts of the magnetic field on the unsteady flow of nanofluids and heat transfer. The influence of thermophoresis and Brownian motion has been examined analytically using the differential transformation method. It is noted that the skin friction coefficient has a straight connection with squeeze and Hartmann numbers. One can find some inclusive works related to nanofluids in [22-33, 36].
In account of the above literature, the present motivation is dedicated to analyze the Brownian motion and thermophoresis effects on unsteady stagnation point flow of an Eyring-Powell nanofluid over a stretching sheet. The viable similarity conversion technique reduces the modeled flow equations to an ODE set. The obtained system is tackled numerically via the Galerkin approach. The influence of various involved parameters and physical quantities are carried out numerically and represented graphically. The velocity of the fluid $f^{\prime} \left(\eta \right)$ enhanced with an increase in fluid and magnetic parameters for the case of opposing, but the behavior is reversed for the assisting case. The Brownian motion and thermophoresis parameters cause an increase in temperature for both cases (assisting and opposing). The thermophoresis parameter provides a rise in concentration while a drop is noticed for the Brownian motion parameter. These parameters give an increase in Nusselt numbers, but the assisting case has dominant effects as compared to the opposing case. Similar behavior is noted for Sherwood numbers. Further, a tabular form of comparison is presented to show the reliability of the Galerkin approach with existing literature and a numerical scheme. The method converts the problem under study to a system of algebraic equations and could be solved more efficiently, which is the principal objective of using this method. It is noticed that the suggested approach could be extended to other nonlinear physical problems.
2. Mathematical and geometrical analysis
Suppose that unsteady, viscous and incompressible stagnation point flow of an Eyring-Powell nanofluid and heat transfer in two-dimensions past a stretching sheet. Consider the sheet is stretched with velocity ${u}_{w}=ax/\left(1-ct\right);$ here, $c$ and $a\gt 0$ represent the stretching constant and unsteadiness of the problem, respectively. Assume ${u}_{e}=bx/\left(1-ct\right)$ is the free stream velocity with $b\gt 0$ as the stagnation flow strength. Further, the temperature and concentration at the surface are ${T}_{w}={T}_{\infty }+{T}_{0}x/{\left(1-ct\right)}^{2}$ and ${C}_{w}={C}_{\infty }+{C}_{0}x/{\left(1-ct\right)}^{2}$, respectively with ${T}_{0},{C}_{0}\gt 0.$ The ambient concentration and temperature are represented by ${C}_{\infty }$ and ${T}_{\infty }$, respectively. The magnetic field is applied normal to the surface, with strength $B$ in the direction of the y-axis. According to the above constraints, the continuity, momentum, energy, and mass equations reduce as [1, 16, 17]:
$\begin{eqnarray}\displaystyle \frac{\partial u}{\partial x}+\displaystyle \frac{\partial v}{\partial y}=0,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{\partial u}{\partial t}+u\displaystyle \frac{\partial u}{\partial x}+v\displaystyle \frac{\partial u}{\partial y}=\displaystyle \frac{\partial {u}_{e}}{\partial t}+{u}_{e}\displaystyle \frac{\partial {u}_{e}}{\partial x}+\nu \displaystyle \frac{{\partial }^{2}u}{\partial {y}^{2}}\\ \,+\displaystyle \frac{1}{\beta \gamma {\rho }^{2}}\displaystyle \frac{{\partial }^{2}u}{\partial {y}^{2}}-\displaystyle \frac{1}{2\beta {\gamma }^{3}\rho }{\left(\displaystyle \frac{\partial u}{\partial y}\right)}^{2}\displaystyle \frac{{\partial }^{2}u}{\partial {y}^{2}}\\ \,-\displaystyle \frac{\sigma {B}^{2}}{\rho }\left(u-{u}_{e}\right)\\ \,\pm \,g\left[{\beta }_{T}\left(T-{T}_{\infty }\right)+{\beta }_{C}\left(C-{C}_{\infty }\right)\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{\partial T}{\partial t}+u\displaystyle \frac{\partial T}{\partial x}+v\displaystyle \frac{\partial T}{\partial y}=\displaystyle \frac{k}{\rho {c}_{p}}\displaystyle \frac{{\partial }^{2}T}{\partial {y}^{2}}\\ \,+\tau \left({D}_{B}\displaystyle \frac{\partial C}{\partial y}\displaystyle \frac{\partial T}{\partial y}+\displaystyle \frac{{D}_{T}}{{T}_{\infty }}{\left(\displaystyle \frac{\partial T}{\partial y}\right)}^{2}\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\displaystyle \frac{\partial C}{\partial t}+u\displaystyle \frac{\partial C}{\partial x}+v\displaystyle \frac{\partial C}{\partial y}={D}_{B}\displaystyle \frac{{\partial }^{2}C}{\partial {y}^{2}}+\displaystyle \frac{{D}_{T}}{{T}_{\infty }}\displaystyle \frac{{\partial }^{2}T}{\partial {y}^{2}}.\end{eqnarray}$
The suitable boundary conditions (BCs) linked with equations (1 )-(4 ) are:1 )-(5 ) $\nu $ is denoted as the kinematic viscosity, $\beta $ and $\gamma $ are the Eyring-Powell material liquid parameters, $\rho $ specifies the density, $\sigma $ represents the electrical conductivity, $g$ signifies the gravitational acceleration, and ${\beta }_{T}$ and ${\beta }_{C}$ are the thermal and concentration expansion coefficients. Thermal conductivity is denoted by $k,$ $\tau $ specifies the ratio between effective heat capacity and heat capacity of the base fluid. ${D}_{B},\,{D}_{T},\,{c}_{p},\,{\beta }_{1},\,{\beta }_{2}$ and ${\beta }_{3}$ represent the coefficient of the Brownian motion, coefficient of the thermophoretic diffusion, specific heat, and slip parameters.
$\begin{eqnarray}\left\{\begin{array}{l}\begin{array}{l}y=0;\,u={u}_{w}+{\beta }_{1}\sqrt{1-ct}\nu \displaystyle \frac{\partial u}{\partial y},\,v=0,\\ T={T}_{w}+{\beta }_{2}\sqrt{1-ct}\displaystyle \frac{\partial T}{\partial y},\\ \,C={C}_{w}+{\beta }_{3}\sqrt{1-ct}\displaystyle \frac{\partial C}{\partial y},\end{array}\\ y=\infty ;\,u={u}_{e},T={T}_{\infty },\,C={C}_{\infty }\,\end{array}\right.,\end{eqnarray}$
where velocity components $u$ and $v$ are taken along the $x$- and $y$-axis, respectively. In equations (Consider the following similarity transformations to reduce the differential equations (2 )-(5 ) into the non-dimensional form:
$\begin{eqnarray}\begin{array}{l}u=\displaystyle \frac{bx}{1-ct}f^{\prime} \left(\eta \right),v=-\sqrt{\displaystyle \frac{b\nu }{1-ct}}f\left(\eta \right),\\ \theta =\displaystyle \frac{T-{T}_{\infty }}{{T}_{w}-{T}_{\infty }},\phi =\displaystyle \frac{C-{C}_{\infty }}{{C}_{w}-{C}_{\infty }},\eta =\sqrt{\displaystyle \frac{b}{\left(1-ct\right)\nu }}y.\end{array}\end{eqnarray}$
Using the similarity variables (6 ) into equations (2 )-(5 ), we obtain the following non-dimensional system of differential equations:
$\begin{eqnarray}\begin{array}{l}\left(1+{\rm{\Lambda }}\right)f^{\prime\prime\prime} \left(\eta \right)-{\rm{\Lambda }}{\epsilon }{f}^{{\prime\prime} 2}\left(\eta \right)f^{\prime\prime\prime} \left(\eta \right)-{f}^{{\prime} 2}\left(\eta \right)\\ \,+\left(f\left(\eta \right)-\displaystyle \frac{A}{2}\eta \right)f^{\prime\prime} \left(\eta \right)-\left(M+A\right)\left(f^{\prime} \left(\eta \right)-\,1\right)\\ \,\pm \lambda \left(\theta \left(\eta \right)+R\phi \left(\eta \right)\right)+1=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{1}{{\Pr }}\theta ^{\prime\prime} \left(\eta \right)+f\left(\eta \right)\theta ^{\prime} \left(\eta \right)-f^{\prime} \left(\eta \right)\theta \left(\eta \right)\\ \,-\displaystyle \frac{A}{2}\left(4\theta \left(\eta \right)+\eta \theta ^{\prime} \left(\eta \right)\right)+Nt{\theta }^{{\prime} 2}\left(\eta \right)\\ \,+\,Nb\theta ^{\prime} \left(\eta \right)\phi ^{\prime} \left(\eta \right)=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\phi ^{\prime\prime} \left(\eta \right)+Sc\left(f\left(\eta \right)\phi ^{\prime} \left(\eta \right)-f^{\prime} \left(\eta \right)\phi \left(\eta \right)\right)\\ \,-\displaystyle \frac{ScA}{2}\left(4\phi +\eta \phi ^{\prime} \left(\eta \right)\right)+\displaystyle \frac{Nt}{Nb}\theta ^{\prime\prime} \left(\eta \right)=0.\end{array}\end{eqnarray}$
The BCs take the following form:7 )-(10 ), ${\rm{\Lambda }}$ and ${\epsilon }$ represent the fluid parameter, $A$ is denoted as the steadiness parameter, $M$ specifies the Hartmann number, $\lambda $ is the buoyancy effect due to temperature difference, $\lambda * $ represents the buoyancy effect due to concentration difference, the ratio among $\lambda * $ and $\lambda $ denoted by $R$ and $Pr$ represent the Prandtl number, $Nt$ signifies the thermophoresis parameter, $Nb$ is the Brownian motion parameter, $Sc$ indicates the Schmidt number, ${\lambda }_{f}$ represents the velocity slip parameter, ${\lambda }_{\theta }$ indicates the thermal slip parameter, ${\lambda }_{\phi }$ is the concentration slip parameter, and $S$ indicates the stretching ratio parameter, which is defined as:
$\begin{eqnarray}\left\{\begin{array}{l}\begin{array}{l}f=0,f^{\prime} =S+{\lambda }_{F}f^{\prime\prime} ,\,\theta =1+{\lambda }_{\theta }\theta ^{\prime} ,\\ \,\phi =1+{\lambda }_{\phi }\phi ^{\prime} \,{\rm{as}}\,\eta \to 0,\end{array}\\ f^{\prime} =1,\theta =\phi =0,\,{\rm{as}}\,\eta \to \infty ,\,\end{array}\right.\end{eqnarray}$
where the ± sign characterizes the buoyancy assisting and opposing flow correspondingly when $R\gt 0.$ In equations ( $\begin{eqnarray}\left.\begin{array}{l}{\rm{\Lambda }}=\displaystyle \frac{1}{\beta \mu \gamma },\,{\epsilon }=\displaystyle \frac{{b}^{2}{x}^{3}}{2\nu {\gamma }^{2}},\,A=\displaystyle \frac{c}{b},\,M=\displaystyle \frac{\sigma {B}^{2}}{b\rho },\\ \lambda =\displaystyle \frac{G{r}_{T}}{{{\rm{Re}}}_{x}^{2}},\,G{r}_{T}=\displaystyle \frac{\beta g\left({T}_{w}-{T}_{\infty }\right)}{{\nu }^{2}},\\ {{Re}}_{x}\,=\,\displaystyle \frac{x{u}_{e}}{\nu },\,R=\displaystyle \frac{\lambda * }{\lambda },\,\lambda * =\displaystyle \frac{G{r}_{C}}{{{Re}}_{x}},\,\\ G{r}_{C}=\displaystyle \frac{\beta g\left({T}_{w}-{T}_{\infty }\right)}{{\nu }^{2}},Pr=\displaystyle \frac{\nu }{\alpha },\,Nt=\displaystyle \frac{\tau {D}_{B}\left({T}_{w}-{T}_{\infty }\right)}{\nu {T}_{\infty }}\\ Nb=\displaystyle \frac{\tau {D}_{B}\left({C}_{w}-{C}_{\infty }\right)}{\nu },\,Sc=\displaystyle \frac{\nu }{{D}_{B}},\,{\lambda }_{F}={\beta }_{1}\sqrt{\nu b},\\ {\lambda }_{\theta }={\beta }_{2}\sqrt{\displaystyle \frac{b}{\nu }},\,{\lambda }_{\phi }={\beta }_{3}\sqrt{\displaystyle \frac{b}{\nu }},\,S=\displaystyle \frac{a}{b}\end{array}\right\}.\end{eqnarray}$
The essential physical quantities are the Skin friction coefficient, and local Nusselt and Sherwood numbers, denoted by ${C}_{f},\,N{u}_{x}$ and $S{h}_{x}$, respectively, and defined as [1]:
$\begin{eqnarray}\begin{array}{l}\left.\begin{array}{l}{C}_{f}=\frac{2\mu }{\rho {u}_{e}^{2}}\left(\left(1+\frac{1}{\beta \gamma }\right){\left.\frac{\partial u}{\partial y}\right|}_{y=0}-\frac{1}{6\beta {\gamma }^{3}}{\left.{\left(\frac{\partial u}{\partial y}\right)}^{3}\right|}_{y=0}\right),\\ N{u}_{x}=-\frac{x}{k\left({T}_{w}-{T}_{\infty }\right)}\left(k{\left.\frac{\partial T}{\partial y}\right|}_{y=0}\right),\\ S{h}_{x}=-\frac{x}{{D}_{B}\left({C}_{w}-{C}_{\infty }\right)}\left({D}_{B}{\left.\frac{\partial C}{\partial y}\right|}_{y=0}\right),\end{array}\right\}\,\\ {C}_{f}=\frac{2\mu }{\rho {u}_{e}^{2}}\left(\left(1+\frac{1}{\beta \gamma }\right){\left.\frac{\partial u}{\partial y}\right|}_{y=0}-\frac{1}{6\beta {\gamma }^{3}}{\left.{\left(\frac{\partial u}{\partial y}\right)}^{3}\right|}_{y=0}\right).\end{array}\end{eqnarray}$
Using the non-dimensional form of physical quantities after incorporating the similarity transformations (6 ), we get the following expressions:
$\begin{eqnarray}\begin{array}{l}{{Re}}_{x}^{-1/2}{C}_{f}=\left[\left(1+{\rm{\Lambda }}\right)f^{\prime\prime} \left(0\right)-\displaystyle \frac{{\epsilon }{\rm{\Lambda }}}{3}{f}^{{\prime\prime} 3}\left(0\right)\right],\,\\ {{Re}}_{x}^{-1/2}N{u}_{x}=-\theta ^{\prime} \left(0\right),\,S{h}_{x}=-\phi ^{\prime} \left(0\right),\end{array}\end{eqnarray}$
where ${{Re}}_{x}\,=\,x{u}_{e}/\nu $ are represented by the local Reynolds number.3. Solution procedure
This section is devoted to investigating the numerical solutions of problem (7 )-(10 ) utilizing the Galerkin method, also known as the finite element algorithm [28, 34, 35]. This trial solution is constructed with the help of a basis function chosen from a set of an orthonormal basis. In our method, i.e. the Galerkin method (finite element algorithm), the supposed trial solution necessarily satisfies the associated BCs. To find the residual vectors, we must insert the supposed trial solutions into the differential equation since the supposed trial solutions have constants to be determined. These constants can be obtained by minimizing the weighted residual. To elucidate the procedure, we divided it into the following steps.
Suppose the problem (
$\begin{eqnarray}\begin{array}{l}\left(1+{\rm{\Lambda }}\right)f^{\prime\prime\prime} -{\rm{\Lambda }}{\epsilon }{f}^{{\prime\prime} 2}f^{\prime\prime\prime} -{f}^{{\prime} 2}+\left(f-\displaystyle \frac{A}{2}\eta \right)f^{\prime\prime} \\ \,-\,\left(M+A\right)\left(f^{\prime} -1\right)\pm \lambda \left(\theta +R\phi \right)+1=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{1}{{\Pr }}\theta ^{\prime\prime} +f\theta ^{\prime} -f^{\prime} \theta -\displaystyle \frac{A}{2}\left(4\theta +\eta \theta ^{\prime} \right)+Nt{\theta }^{{\prime} 2}\\ \,+\,Nb\theta ^{\prime} \phi ^{\prime} =0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\phi ^{\prime\prime} +Sc\left(f\phi ^{\prime} -f^{\prime} \phi \right)-\displaystyle \frac{ScA}{2}\left(4\phi +\eta \phi ^{\prime} \right)\\ \,+\,\displaystyle \frac{Nt}{Nb}\theta ^{\prime\prime} =0.\end{array}\end{eqnarray}$
Step 2. The Galerkin method suggests the following trial solutions find the solution to the problem (7 )-(10 )
$\begin{eqnarray}\tilde{f}\left(\eta \right)={\dot{a}}_{0}+{\dot{a}}_{1}\eta +{\dot{a}}_{2}{\eta }^{2}+\ldots +{\dot{a}}_{M}{\eta }^{M}=\displaystyle \sum _{k=0}^{M}{\dot{a}}_{k}{\eta }^{k},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\tilde{\theta }\left(\eta \right)={\ddot{a}}_{0}+{\ddot{a}}_{1}\eta +{\ddot{a}}_{2}{\eta }^{2}+\ldots +{\ddot{a}}_{M}{\eta }^{M}\\ \,=\displaystyle \sum _{k=0}^{M}{\ddot{a}}_{k}{\eta }^{k},\end{array}\end{eqnarray}$
$\begin{eqnarray}\tilde{\phi }\left(\eta \right)={\dddot{a}}_{0}+{\dddot{a}}_{1}\eta +{\dddot{a}}_{2}{\eta }^{2}+\ldots +{\dddot{a}}_{M}{\eta }^{M}=\displaystyle \sum _{k=0}^{M}{\dddot{a}}_{k}{\eta }^{k}.\end{eqnarray}$
In the above $\left\{{\dot{a}}_{i},{\ddot{a}}_{i},{\dddot{a}}_{i}\right\},\,i=0,1,2,\ldots ,M,$ are unknown and need to be determined. M is typically called the order of approximation for well approximation M should be significant. As discussed above the supposed trial solutions for (14 )-(16 ) must satisfy the conditions in (10 ); therefore, we get the following form of trial solutions (17 )-(19 ) after incorporating the conditions from (10 ):
$\begin{eqnarray}\begin{array}{l}\tilde{f}\left(\eta \right)=\left(S-\displaystyle \frac{S-1}{{\lambda }_{f}+{\eta }_{\infty }}{\lambda }_{f}\right)\eta -\displaystyle \frac{1}{2}\displaystyle \frac{S-1}{{\lambda }_{f}+{\eta }_{\infty }}{\eta }^{2}\\ \,+\displaystyle \sum _{k=1}^{M}{\dot{a}}_{k}\left({\eta }^{k+2}-\displaystyle \frac{k+2}{2\left({\lambda }_{f}+{\eta }_{\infty }\right)}{\eta }_{\infty }^{k+1}{\eta }^{2}\right.\\ \,-\left.\displaystyle \frac{k+2}{{\lambda }_{f}+{\eta }_{\infty }}{\lambda }_{f}{\eta }_{\infty }^{k+1}\eta \right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{lll}\tilde{\theta }\left(\eta \right) & = & 1-\displaystyle \frac{{\lambda }_{\theta }}{{\lambda }_{\theta }+{\eta }_{\infty }}-\displaystyle \frac{1}{{\lambda }_{\theta }+{\eta }_{\infty }}\eta +\displaystyle \sum _{k=1}^{M}{\ddot{a}}_{k}\left(\Space{0ex}{3.2ex}{0ex}{\eta }^{k+1}\right.\\ & & -\,\left.\displaystyle \frac{{\lambda }_{\theta }}{{\lambda }_{\theta }+{\eta }_{\infty }}{\eta }_{\infty }^{k+1}-\displaystyle \frac{1}{{\lambda }_{\theta }+{\eta }_{\infty }}\eta {\eta }_{\infty }^{k+1}\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{lll}\tilde{\phi }\left(\eta \right) & = & 1-\displaystyle \frac{{\lambda }_{\phi }}{{\lambda }_{\phi }+{\eta }_{\infty }}-\displaystyle \frac{1}{{\lambda }_{\phi }+{\eta }_{\infty }}\eta +\displaystyle \sum _{k=1}^{M}{\dddot{a}}_{k}\left(\Space{0ex}{3.4ex}{0ex}{\eta }^{k+1}\right.\\ & & -\,\left.\displaystyle \frac{{\lambda }_{\phi }}{{\lambda }_{\phi }+{\eta }_{\infty }}{\eta }_{\infty }^{k+1}-\displaystyle \frac{1}{{\lambda }_{\phi }+{\eta }_{\infty }}\eta {\eta }_{\infty }^{k+1}\right).\end{array}\end{eqnarray}$
We obtained the following residuals ${{\rm{Res}}}_{F},\,{{\rm{Res}}}_{\theta }$ and ${{\rm{Res}}}_{\phi }$ after incorporating the reduced trial solutions (
$\begin{eqnarray}\begin{array}{l}{{\rm{Res}}}_{f}\,=\,\left(1+{\rm{\Lambda }}\right)\tilde{f}^{\prime\prime\prime} -{\rm{\Lambda }}{\epsilon }{\tilde{f}}^{{\prime\prime} 2}\tilde{f}^{\prime\prime\prime} -{\tilde{f}}^{{\prime} 2}\\ \,+\left(\tilde{f}-\displaystyle \frac{A}{2}\eta \right)\tilde{f}^{\prime\prime} -\left(M+A\right)\left(\tilde{f}^{\prime} -1\right)\\ \,\pm \lambda \left(\tilde{\theta }+R\tilde{\phi }\right)+1,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{{\rm{Res}}}_{\theta }\,=\,\displaystyle \frac{1}{{\Pr }}\tilde{\theta }^{\prime\prime} +\tilde{f}\tilde{\theta }^{\prime} -\tilde{f}^{\prime} \tilde{\theta }-\displaystyle \frac{A}{2}\left(4\tilde{\theta }+\eta \tilde{\theta }^{\prime} \right)\\ \,\,+\,Nt{\tilde{\theta }}^{{\prime} 2}+Nb\tilde{\theta }^{\prime} \tilde{\phi }^{\prime} ,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{{\rm{Res}}}_{\phi }\,=\,\tilde{\phi }^{\prime\prime} +Sc\left(\tilde{f}\tilde{\phi }^{\prime} -\tilde{f}^{\prime} \tilde{\phi }\right)-\displaystyle \frac{ScA}{2}\left(4\tilde{\phi }+\eta \tilde{\phi }^{\prime} \right)\\ \,+\,\displaystyle \frac{Nt}{Nb}\tilde{\theta }^{\prime\prime} .\end{array}\end{eqnarray}$
The residual must vanish if the reduced trial solution (20 )-(22 ) is an exact solution.
In this method, it is essential to build a weighted residual error with viable weights and to get the values of the unknown; we need to minimize it, that is
$\begin{eqnarray}\displaystyle \frac{\partial {{E}}_{f}}{\partial {\dot{a}}_{i}}=\displaystyle \int {{\rm{Res}}}_{f}\left(\eta \right){W}_{i}^{f}\left(\eta \right){\rm{d}}\eta =0,\,i=1,2,\ldots ,M,\end{eqnarray}$
$\begin{eqnarray}\displaystyle \frac{\partial {{E}}_{\theta }}{\partial {\ddot{a}}_{i}}=\displaystyle \int {{\rm{Res}}}_{\theta }(\eta ){W}_{i}^{\theta }\left(\eta \right){\rm{d}}\eta =0,\,i=1,2,\ldots ,M,\end{eqnarray}$
$\begin{eqnarray}\displaystyle \frac{\partial {{E}}_{\phi }}{\partial {\dddot{a}}_{i}}=\displaystyle \int {{\rm{Res}}}_{\phi }(\eta ){W}_{i}^{\phi }\left(\eta \right){\rm{d}}\eta =0,\,i=1,2,\ldots ,M.\end{eqnarray}$
In this study, we choose the following weight functions:
$\begin{eqnarray}{W}_{i}^{f}\left(\eta \right)=\displaystyle \frac{\partial \tilde{f}}{\partial {\dot{a}}_{i}},\,{W}_{i}^{\theta }\left(\eta \right)=\displaystyle \frac{\partial \tilde{\theta }}{\partial {\ddot{a}}_{i}},\,{W}_{i}^{\phi }\left(\eta \right)=\displaystyle \frac{\partial \tilde{\phi }}{\partial {\dddot{a}}_{i}}.\end{eqnarray}$
Solve the system of nonlinear algebraic equations achieved from (
4. Results and discussion
This section of the article is dedicated to exploring the physical features of the proposed model, which are explained in the previous section. For this determination, the graphical behavior of dimensionless velocity, temperature, and concentration for different values of the parameters are planned (see figures 1-12). Further, it is essential to mention here that figures 1-12 are plotted for two kinds of flow, entitled as assisting and opposing flow. In figure 1(a), the attitude of velocity for growing values of the Hartman number is explained. Observations show that two diverse types of performance from this figure can be seen. The velocity profile lessens for assisting flow, and the further velocity profile grows for the case of opposing flow.
Figure 1. (a), (b). Behavior of velocity profile $f^{\prime} \left(\eta \right)$ for (a) M and (b) ${\rm{\Lambda }}.$ |
Figure 2. (a), (b). Behavior of velocity profile $f^{\prime} \left(\eta \right)$ for (a) $\lambda $ and (b) $R.$ |
Figure 3. (a), (b). Behavior of temperature profile $\theta \left(\eta \right)$ for (a) ${\lambda }_{\theta }$ and (b) ${\rm{\Lambda }}.$ |
Figure 4. (a), (b). Behavior of temperature profile $\theta \left(\eta \right)$ for (a) $Nt$ and (b) $Nb.$ |
Figure 5. (a), (b). Behavior of concentration profile $\phi \left(\eta \right)$ for (a) ${\lambda }_{\phi }$ and (b) ${\rm{\Lambda }}.$ |
Figure 6. (a), (b). Behavior of concentration profile $\phi \left(\eta \right)$ for (a) $Nt$ and (b) $Nb.$ |
Figure 7. (a), (b). Behavior of skin friction coefficient for (a) M and (b) ${\rm{\Lambda }}.$ |
Figure 8. (a), (b). Behavior of skin friction coefficient for (a) $R$ and (b) ${\lambda }_{f}.$ |
Figure 9. (a), (b). Behavior of Nusselt number coefficient for (a) M and (b) ${\rm{\Lambda }}.$ |
Figure 10. (a), (b). Behavior of Nusselt number coefficient for (a) $Nt$ and (b) $Nb.$ |
Figure 11. (a), (b). Behavior of Schmidt number coefficient for (a) M and (b) ${\rm{\Lambda }}.$ |
Figure 12. (a), (b). Behavior of Schmidt number coefficient for (a) $Nt$ and (b) $Nb.$ |
According to the physical point of view, the incorporation of the magnetic field plays a significant role. It acts like the Lorentz force, and it opposes the fluid particle from its original position. This factor is adjusted in mathematical modeling as a non-dimensional parametric named the Hartman number, and the strong magnetic field increases with the growing values of the Hartman number. This increase in the strength of the magnetic field further opposes the fluid flow for the case of assisting flow. Because of this factor, declining behavior of the velocity assisting the flow can be seen in figure 1(a). In some of the cases, this factor supports the velocity profile for the case of opposing flow. Therefore, increasing the behavior of the velocity for the case of opposing flow is explained in figure 1(a). Similar behavior of the velocity for fluid parameter ${\rm{\Lambda }}$ can be seen in 1(b). In the next figure, 2(a), effects on velocity for different values of the buoyancy parameter are sketched, and it is clear that with the increasing values of $\lambda $ the velocity profile grows for the case of assisting flow.
Reverse consequences of velocity are observed further for opposing flow. In figure 2(a), plotted behavior of the velocity for diverse values of parameter $R$ is explained. Parameter $R$ is the ratio between $\lambda $ (buoyancy effects due to temperature difference) and $\lambda * $ (buoyancy effects due to concentration difference). Therefore it can be observed that $\lambda $ and $\lambda * $ have identical values for the case of $R=1,$ $\lambda $ has dominated effects as compared to $\lambda * $ for the case $R\gt 1$ and $\lambda * $ is dominated compared to $\lambda $ for $R\lt 1.$
Decreasing and increasing behavior of the velocity can be detected for assisting and opposing flows, respectively. Clarifying the attitude of temperature (both supporting and opposing flows) for growing values of the thermal slip parameter is strategized in figure 3(a). The thermal area of the fluid is increasing with the swelling values of the thermal slip parameter. Therefore, the declining effects of temperature can be seen in figure 3(a). This is because we are growing the thermal area, so the temperature has more thermal area to flow. For the case of increasing values of fluid parameter, the reverse attitude of temperature in the case of assisting and opposing flows are plotted in 3(b).
The collision between the fluid particle increase due to the enhancement in the random motion of the molecule and this increment in random motion is because of the growing values of $Nb.$ Due to all these factors, more heat is produced and this further becomes the reason of the system's temperature rise (see 4(a)). Thermophoresis is the procedure in which tiny fluid particles move from a hot area of the surface to a cold area of the surface. This movement of the particles enhances due to the increasing values of $Nt.$ Therefore, the temperature of the system is increasing with the rising values of $Nt$ (see 4(b)). Figure 5(a) is plotted to describe the declining behavior of the concentration profile for rising values of the concentration slip parameter. It is very clear from the modeling that more concentration area has been added due to this parameter. Therefore we can say that particle diffusion has been increased.
The area of concentration profile upsurges due to the growing values of the concentration slip parameter, and hence the concentration profile is decreasing. Observations regarding figure 5(b) demonstrate that the concentration profile with increasing values of the fluid parameter has a conflicting attitude for both the cases of assisting and opposing flow. Rising and lessening attitude of the concentration profile is perceived for the case of increasing values of $Nt$ and $Nb$, respectively (see figures 6(a) and (b)).
The properties of dimensionless velocity, temperature, and concentration at the boundaries of the problem can be witnessed with the help of the study of skin friction coefficient, Nusselt number, and Schmidt number, respectively. For this determination, figures 7-12 are designed at different values of the parameters.
In figures 7(a) and (b), the Hartman number, and fluid parameter have a conflicting performance for both the cases of assisting and opposing flow. On the other hand, the similar behavior of temperature for growing values of the Hartman number and fluid parameter is explained in figures 9(a) and (b). In figure 8(a), the velocity at the boundary is swelling in the case of assisting flow and is reversing for the case of opposing flow. The contradictory effects of velocity can be seen for the case of the velocity slip parameter in figure 8(b). Similar behavior of the Nusselt number for different values of $Nt$ and $Nb$ is elucidated in figures 10(a) and (b). It is perceived from figures 11(a) and (b) that the concentration profile at the boundaries of the modeled problem lessens with growing values of the Hartman number and fluid parameter (for the case of assisting flow) and further reverse behavior can be seen for the case of opposing flow. On the other hand, similar (cumulative) behavior of the Schmidt number is detected for $Nt$ and $Nb$ (see figures 12(a) and(b)). Tables 1 and 2 are constructed to show the efficiency of the proposed method for solving this problem for both the assisting and opposing cases. The obtained results are in excellent agreement with previously published work [16, 17]. Therefore, tables 1 and 2 are evidence that the Galerkin method is an efficient, reliable, and accurate tool to investigate the numerical solution of the problem (7 -10 ) and can be extended for the diversified problem of a nonlinear problem arising in mathematical physics and fluid mechanics.
Table 1. Comparison of the results achieved from the Galerkin method for $f^{\prime\prime} \left(0\right)$ and $\theta ^{\prime} \left(0\right)$ in the case of assisting flow when ${\rm{\Lambda }}=Nt=Nb=M=A={\lambda }_{f}=R={\lambda }_{\theta }=0$ and for various values of the Prandtl number. |
| [16] | [17] | Galerkin approach | ||||
|---|---|---|---|---|---|---|
| Pr | $f^{\prime\prime} \left(0\right)$ $\theta ^{\prime} \left(0\right)$ | $f^{\prime\prime} \left(0\right)$ $\theta ^{\prime} \left(0\right)$ | $f^{\prime\prime} \left(0\right)$ $\theta ^{\prime} \left(0\right)$ | |||
| 0.72 | 0.3645 | 1.0931 | 0.364 49 | 1.093 31 | 0.364 49 | 1.093 11 |
| 6.8 | 0.1804 | 3.2902 | 0.180 42 | 3.289 57 | 0.180 42 | 3.289 57 |
| 20 | 0.1175 | 5.6230 | 0.117 50 | 5.620 14 | 0.117 50 | 5.620 14 |
| 40 | 0.0873 | 7.9463 | 0.087 24 | 7.938 31 | 0.087 24 | 7.938 32 |
| 60 | 0.0729 | 9.7327 | 0.072 84 | 9.718 01 | 0.072 84 | 9.717 99 |
| 80 | 0.0640 | 11.2413 | 0.063 94 | 11.218 75 | 0.063 94 | 11.218 76 |
| 100 | 0.0578 | 12.5726 | 0.057 73 | 12.541 13 | 0.057 73 | 12.541 15 |
Table 2. Comparison of the results achieved from the Galerkin method for $f^{\prime\prime} \left(0\right)$ and $\theta ^{\prime} \left(0\right)$ in the case of opposing flow when ${\rm{\Lambda }}=Nt=Nb=M=A={\lambda }_{f}=R={\lambda }_{\theta }=0$ and for various values of the Prandtl number. |
| [16] | [17] | Galerkin approach | ||||
|---|---|---|---|---|---|---|
| Pr | $f^{\prime\prime} \left(0\right)$ $\theta ^{\prime} \left(0\right)$ | $f^{\prime\prime} \left(0\right)$ $\theta ^{\prime} \left(0\right)$ | $f^{\prime\prime} \left(0\right)$ $\theta ^{\prime} \left(0\right)$ | |||
| 0.72 | −0.3852 | 1.0293 | −0.385 19 | 1.029 25 | −0.385 19 | 1.029 25 |
| 6.8 | −0.1832 | 3.2466 | −0.183 23 | 3.246 09 | −0.183 23 | 3.246 09 |
| 20 | −0.1183 | 5.5923 | −0.118 31 | 5.589 60 | −0.118 31 | 5.589 62 |
| 40 | −0.0876 | 7.9227 | −0.087 58 | 7.914 91 | −0.087 58 | 7.914 90 |
| 60 | −0.0731 | 9.7126 | −0.073 04 | 9.698 18 | −0.073 04 | 9.698 17 |
| 80 | −0.0642 | 11.2235 | −0.064 08 | 11.201 18 | −0.064 08 | 11.201 18 |
| 100 | −0.0579 | 12.5564 | −0.057 83 | 12.525 19 | −0.057 83 | 12.525 17 |
5. Conclusion
The study is devoted to examine an unsteady stagnation point flow of an Eyring-Powell nanofluid over a stretching sheet beside the effects of thermophoresis and Brownian motion. The governing equations are reduced to a system of nonlinear ODEs and tackled numerically by using the Galerkin method. Hence, essential conclusions are stated below:
| • | The velocity of the fluid $f^{\prime} \left(\eta \right)$ enhanced with an increase in fluid and magnetic parameter for the case of opposing flow, but the behavior reversed for the assisting case. The parameters $\lambda $ and $R$ causes an increase in velocity for the assisting case, but the behavior is opposite for the opposing case. |
| • | The Brownian motion and thermophoresis parameters cause an increase in temperature for both cases (assisting and opposing), but the behavior is reverse for ${\lambda }_{\theta }.$ |
| • | The fluid parameter enhances the temperature for the assisting case, but a drop is observed for the opposing case. On the other hand, identical behavior is noticed for the concentration profile. |
| • | The thermophoresis parameter provides a rise in concentration while a drop is noticed for the Brownian motion parameter. |
| • | ${\rm{\Lambda }}$ and $R$ increase the coefficient of skin-friction while ${\lambda }_{f}$ and $M$ cause a decrease in the skin-friction coefficient for the assisting case. The behavior of these parameters is the opposite of the opposing case. |
| • | $M$ and ${\rm{\Lambda }}$ increase the Nusselt number for the opposing case, but the behavior is opposite for the assisting case. The behavior of concentration is similar for $M$ and ${\rm{\Lambda }}.$ |
| • | ${N}_{t}$ and ${N}_{b}$ provide an increase in the Nusselt numbers, but the assisting case has dominant effects compared to the opposing case. Similar behavior is noted for the Sherwood numbers. |
| • | A tabular form of comparison analysis of the outcomes attained via the Galerkin approach and numerical scheme (RK-4) is available to show the credibility of the Galerkin method. The comparison analysis and graphical plots endorse the appropriateness of the Galerkin method. It is concluded that this method could be extended to other problems of a complex nature. |