Generalized diffusion effects on Maxwell nanofluid stagnation point flow over a stretchable sheet with slip conditions and chemical reaction

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 RK-4 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.


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 non-Newtonian 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 two-phase 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 Technical Editor: Cezar Negrao, Ph.D. 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 single-phase 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 rate-type fluids, which calculates stress relaxation so it has become popular. This model also eliminates the complicating behavior of shear-dependent 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.

Mathematical formulation
Let us consider two-dimensional laminar steady heat and mass transfer flow of an electrically conducting Maxwell nanofluid flow passed by a linear stretched surface placed along x-axis and y-axis vertical to the sheet with stagnation point at the origin (as illustrated in Fig. 1). The free stream velocity = e ( ) = and the velocity via which sheet is stretched are = w ( ) = , where a and c are positive constants. The temperature at the surface is conserved at T w and T ∞ far away from the plate; in similar a manner, the nanoparticle volume fractions are C w and C ∞ . An external magnetic field H 0 is applied normal to the sheet.
Under the above assumptions, the required equations are as follows: where is the density, e is the magnetic permeability velocity, is the electrical conductivity, is the Maxwell fluid parameter, and is the kinematic viscosity. Due to hydrostatic and magnetic pressure gradient, the force will be in equilibrium as given by y .
x y Fig. 1 Geometry of the problem The classical form of Fourier's and Fick's laws with the ray of Cattaneo-Christov equations takes the following form: Assume that ∇.q = 0, ∇.J = 0, and for steady state q t = 0, J t = 0, the new equations become: Now in component form, energy and concentration Eqs. (7) and (8) 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 ∞ and T ∞ are the concentration and temperature free streams. Temperature of the sheet is T w = T ∞ + bx, for heated surface b > 0 so T w > T ∞ and for cooled surface b < 0 and T w < T, where b is a constant and D T is known as thermophoresis diffusivity.

and chemical reactive species is
Friction factor coefficient ( C f ) is defined as: Here w denotes the wall shear stress and is given by where Re x = u e x x is local Reynolds number.
three initial guesses to f �� (0) , � (0) and � (0) for approximate solution. Here the step size and convergence criteria are chosen to be 0.001 and 10 −6 (in all cases).

Results and discussion
The main effort of this work is to examine the influence of magnetic field and stagnation point Maxwell nanofluid flow due to a linear stretching surface with slip conditions. The governing differential Eqs. (12)-(15) along with corresponding boundary conditions (16) are solved numerically by implying shooting procedure (Cash and Karp).  Figure 3 depicts the variation of slip parameter k on velocity profile. The influence of slip parameter k significantly enhances the velocity profile. Figure 4 illustrates the variation of fluid relaxation parameter m on velocity profile. It can be analyzed that the velocity of the fluid reduces by enhancing the fluid relaxation parameter m . Figure 5 represents the change in temperature distribution for distinct values of N t . It is found that by enhancing N t , the temperature distribution also increases. Figure 6 depicts the variation of Brownian motion N b on temperature distribution. It can be analyzed that by increasing N b , the mass diffusivity trekked up which leads to enhancement in the temperature and  Figure 7 shows the behavior of Prandtl number Pr on temperature profile. It is found that the temperature profile reduces with rising values of Prandtl Pr . Figure 8 presents the deviation of temperature profile for distinct values of t . It is seen that by enhancing thermal relaxation parameter t , fluid particles require more time to heat the boundary layer region, and as a result temperature profile reduces. Figure 9 displays the effect of relaxation parameter c on concentration distribution. From this figure, it is observed that by increasing the relaxation parameter t , the concentration profile reduces. Figure 10 Figure 11 represents the influence of Lewis number Le on nanoconcentration profile. It is found that the higher values of Lewis number Le lead to reduction in the mass diffusivity, so the concentration profile reduces.  x enhances by enhancing slip parameter k but decreases by increasing the fluid relaxation parameter m . Table 1 shows that the fraction factor rises due to an increase in Hartmann number Ha and fluid relaxation parameter m and opposite behavior is noticed for slip parameter k . The achieved results are in  Table 2. Table 3 is sketched for the comparative investigation between Hsiao x for various values of Ha and k Table 1 Computational results of C f Re  Table 2 Comparison of (f �� (0) + m (f � (0)f �� (0) + f (0)f ��� (0))) with the previous literature when x for large values of slip parameter k but opposing behavior is noticed for fluid relaxation parameter m .