Generalized diffusion effects on Maxwell nanofluid stagnation point flow over a stretchable sheet with slip conditions and chemical reaction
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Abstract
The aim of this article is to investigate the heat and mass diffusion (Cattaneo–Christov model) of the upper convected Maxwell nanomaterials passed by a linear stretched surface (slip surface) near the stagnation point region. Convocational Fourier’s and Fick’s laws are employed to investigate heat and mass diffusion phenomena. Using the similarity transformations, the governing PDEs are rendered into ODEs along with boundary conditions. The boundary value problem is solved numerically using RK4 method along with shooting technique (Cash and Karp). The effects of embedded parameters, namely fluid relaxation parameter, Hartmann number, Brownian moment, thermophoresis parameter, thermal relaxation parameter, Lewis number, chemical reactions concentration relaxation parameter, and slip parameter on velocity, temperature, and concentration distributions, are deliberated through the graphs and discussed numerically. The skin friction coefficient is deliberated numerically, and their numerical values are accessible through graphs and table. The comparison of current article is calculated in the last section, and a good agreement is clear with the existing literature.
Keywords
Maxwell nanofluid Generalized Fourier’s and Fick’s laws Slip conditions Stagnation point Chemical reaction Shooting method1 Introduction
Nanotechnology has important interest in manufacturing, aerospace, and medical industries. The term nanofluid was generated by Choi [1] in 1995, designated as fluids that contain solid nanoparticles having 1–100 nm size dispersed in the base fluids, namely ethylene, water, toluene, oil, etc. Nanoparticles such as coppers, silicone, aluminum, and titanium tend to improve the thermal conductivity and convective heat assignment rate of liquids. Impact of variable viscosity on flow of nonNewtonian material with convective conditions over a porous medium is investigated by Rundora et al. [2]. Babu and Sandeep [3] discussed the numerical solution on MHD nanomaterials over a variable thickness of the surface along with thermophoresis and Brownian motion effects. Haiao [4] presented the numerical solution of magnetohydrodynamic micropolar fluid flow with the addition of nanomaterials toward a stretching sheet with viscous dissipation. Mahdavi et al. [5] illustrated the slip velocity along with multiphase approach of nanofluids. Xun et al. [6] obtained the numerical solution of bioconvection heat flow nanofluid flow over a rotating plate with temperaturebased viscosity. Khan et al. [7] numerically analyzed heat and mass diffusion in Jeffery nanofluid passed by inclined stretching surface. Lebon and Machrafi [8] analyzed the twophase change in Maxwell nanofluid flow along with thermodynamic description. Ansari et al. [9] investigated the comprehensive analysis in order to calculate the relative viscosity of nanofluids. Khan et al. [10] considered the chemical reaction on Carreau–Yasuda nanomaterials over a nonlinear stretching surface.
Magnetohydrodynamic (MHD) flow of heat and mass transfer Maxwell fluid flow over a continuous stretching surface has great significance in several applications in engineering such as melts, aerodynamics extrusion of plastic sheet, geothermal extractions, and purification of molten metals. Numerous researchers have made great interest and evaluated the transport phenomena for magnetohydrodynamic. Zhao et al. [11] solve the differential equations labeling MHD Maxwell fluid in permeable sheet by considering Dufour and Soret impact. Hsiao [12] investigated the combined effects of thermal extraction on MHD Maxwell fluid over stretching surface with viscous dissipation and energy conversion. Ghasemi and Siavashi [13] demonstrated the Cu–water MHD nanofluid in square permeable surface with entropy generation. Nourazar et al. [14] illustrated the heat transfer in flow of singlephase nanofluid toward a stretching cylinder with magnetic field effect. Dogonchi and Ganji [15] addressed the unsteady squeezed MHD nanofluid flow over two parallel plates with solar radiation. Hayat et al. [16] investigated the heat and mass diffusion for stagnation point flow toward a linear stretching surface along with magnetic field. Sayyed et al. [17] investigated the analytical solution of MHD Newtonian fluid flow over a wedge occupied in a permeable sheet. Representative analyses on MHD flow can be seen in Refs. [18, 19, 20].
The Maxwell model is a subclass of ratetype fluids, which calculates stress relaxation so it has become popular. This model also eliminates the complicating behavior of sheardependent viscosity and is thus useful for focusing exclusively on the impact of a fluid’s elasticity on the characteristics of its boundary layer. Nadeem et al. [21] deliberated the numerical study on heat transfer of Maxwell nanofluid flow over a linear stretching sheet. Reddy et al. [22] studied the approximate solution of magnetohydrodynamic Maxwell nanofluid flow over exponentially stretching surface. Liu [23] indicated the 2D flow of frictional Maxwell fluid over a variable thickness. Solution of the differential equations was obtained numerically here by \(L_{1}\) technique. Yang et al. [24] considered the fractional Maxwell fluid through a rectangular microchannel.
Inspired by the above studies, the current study illustrates the MHD Maxwell nanofluid flow over a linearly stretched sheet near the stagnation point and slip boundary conditions. Fourier’s and Fick’s laws are presented in the constitutive relations. The nonlinear ODEs are deduced from the nonlinear PDEs by similarity transaction. The solutions are obtained via shooting method (Cash and Karp). The different involved physical parameters are examined for velocity, concentration, and temperature fields.
2 Mathematical formulation
Here (\(u,\,\;v\)) are the velocity components along the (\(x,\,\;y\)) directions, \(q\), \(J\) are the normal heat and mass flux, respectively, \(\;k_{f}\) represents the thermal conductivity, \(D_{B} \;\) is the Brownian motion, \(\lambda_{T}\), \(\lambda_{C}\) are the relaxation parameters for thermal and concentration, \(\alpha_{f} = \tfrac{{(\rho c)_{s} }}{{(\rho c)_{f} }}\) is the ratio of nanoparticle heat capacity to base fluid thermal capacity, \(\alpha_{f} = \tfrac{{k_{f} }}{{(\rho C_{p} )_{f} }}\) represents thermal diffusion, \(T_{w} (x,\,y)\) is known as temperature at the wall, \(C_{w} (x,\,y)\) is known as concentration at the wall, \(T\;\) and \(C\) are the temperature and concentration of the fluid, respectively, \(C_{p} \;\) is the specific heat, and \(C_{\infty }\) and \(T_{\infty }\) are the concentration and temperature free streams. Temperature of the sheet is \(T_{w} = T_{\infty } + bx,\) for heated surface \(b > 0\) so \(T_{w} > T_{\infty }\) and for cooled surface \(b < 0\) and \(T_{w} < T,\) where \(b\) is a constant and \(D_{T}\) is known as thermophoresis diffusivity.
3 Numerical procedure
Numerical solution of the nonlinear differential Eqs. (13)–(15) along with Neumann boundary conditions (16) is achieved by applying the shooting method with RK4 integration technique for various values of parameters. Let \(y_{1} = f,\,\;y_{2} = f^{{\prime }} ,\,\;y_{3} = f^{{\prime \prime }} ,\,\;y_{4} = \theta ,\,\;y_{5} = \theta^{{\prime }} ,\,\;y_{6} = \phi\) and \(y_{7} = \phi^{{\prime }} .\)
This technique is successfully used to solve the different problems related to boundary layer flows. The boundary conditions \(f^{{\prime }} (0),\,\;\theta (0)\) and \(\phi(0)\) for \(\eta \to \infty\) are converted into finite interval length (here it is \(\eta = 5\)). Insert three initial guesses to \(f^{{\prime \prime }} (0)\), \(\theta^{{\prime }} (0)\) and \(\phi^{{\prime }} (0)\) for approximate solution. Here the step size and convergence criteria are chosen to be \(0.001\) and \(10^{  6}\) (in all cases).
4 Results and discussion
Computational results of \(C_{f} Re_{x}^{{\frac{1}{2}}}\) for different values of \(Ha,\,\;k\;\) and \(\lambda_{m}\) when \(K = A = N_{b} = 0.1,\,\;Le = 1.0,\,\;\delta_{c} = \delta_{t} = 0.0,\,\;N_{t} = 0.5,\) and \(Pr = 1.1\)
Ha  k  \(\lambda_{m}\)  \( C_{f} Re_{x}^{{\frac{1}{2}}}\) 

0.1  0.1  0.9097  
0.2  0.9559  
0.3  0.9999  
0.1  0.9097  
0.2  0.8042  
0.3  0.7225  
0.00  0.8918  
0.05  0.9008  
0.09  0.9097 
Comparison of \((f^{{\prime \prime }} (0) + \lambda_{m} (f^{ \prime } (0)f^{{\prime \prime }} (0)+f(0)f^{{{\prime \prime \prime }}} (0)))\) with the previous literature when \(N_{b} = k = St = 0.1,\,\;\delta t = \delta c = K = A = 0.0,\,\;N_{t} = 0.5\) and \(Le = Pr = 1.0\)
\(\lambda_{m}\)  Ha  [25]  Present results 

0.5  0.0  − 1.68935  − 1.6752 
0.5  0.1  − 1.69715  − 1.6821 
0.5  0.2  − 1.72034  − 1.7000 
0.5  0.3  − 1.75830  − 1.7321 
0.0  0.1  − 1.00499  − 1.0001 
0.1  0.1  − 2.49440  − 2.1898 
Comparison of \( \theta^{{\prime }} (0)\) and \( \phi^{{\prime }} \left( 0 \right)\) with the previous literature when \(\lambda_{m} = Ha = k = St = \delta t = \delta c = K = A = 0.0,\,\;N_{t} = 0.1\) and \(Le = Pr = 10\)
5 Conclusions

Rising the Hartmann number \(\;Ha\) and slip parameter \(k\) leads to decline in velocity profile.

Larger values of fluid relaxation parameter \(\lambda_{m}\), increase the velocity profile.

For increasing values of chemical reaction, \(K,\) Prandtl number \(Pr\), thermal relaxation parameter \(\delta_{t}\), concentration relaxation parameter \(\delta_{c}\) and Lewis number \(Le\) reduce the concentration and temperature profiles.

Temperature profile increases for increasing values of thermophoresis parameter \(N_{t}\) and Brownian motion \(N_{b}\).

Skin friction coefficient reduces \(C_{f} \;Re_{x}^{{\tfrac{1}{2}}}\) for large values of slip parameter \(k\) but opposing behavior is noticed for fluid relaxation parameter \(\lambda_{m}\).
Notes
Acknowledgements
The authors would like to express their gratitude to King Khalid University, Abha 61413, Saudi Arabia, for providing administrative and technical support.
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
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