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MHD slip flow of upper-convected Maxwell nanofluid over a stretching sheet with chemical reaction

Abstract

The present study scrutinizes slip effects and stagnation point flows of upper-convected Maxwell fluid past a stretching sheet. The non-linear ordinary differential equations are obtained from the governing partial differential equations and solved using implicit finite difference method. The impacts of non-dimensional governing parameters such as Brownian motion parameter, velocity ratio, velocity slip parameter, suction/injection parameter, Lewis numbers, upper-convected Maxwell parameter, magnetic field, thermophoresis parameter, chemical reactions parameter, thermal slip parameter, solutal slip parameter, and heat source parameter on the velocity field, heat and mass transfer characteristics are discussed and presented through graphs. The values of local Sherwood number, local Nusselt number, and skin friction coefficient are discussed and presented through tables. The results indicate that when the magnetic field is intensified, it reduces velocity profiles and raises temperature and concentration profiles. Moreover, with an upsurge in velocity slip parameter, the local Nusselt number and local Sherwood number diminish.

Introduction

Nanofluid is a colloidal postponement containing nanoparticles in a base fluid. Nanofluids have enhanced physical properties such as mass diffusivity, thermal diffusivity and conductivity, viscosity, and convective heat transfer coefficients compared to those of base fluids. Nanofluids can be rummage-sale in a plethora of engineering applications extending from the use in the automotive industry to the medical field to use in power plant cooling systems, as well as computers viz. heat transfer applications (in industrial cooling applications as smart fluids, in nuclear reactors, in extraction of geothermal energy sources), automotive applications (as nanofluid coolant and nanofluid in fuel, brake, and other vehicular nanofluids), electronic applications (cooling of microchips, micro scale fluid applications), and biomedical applications (nano drug delivery, cancer therapeutics, cryopreservation, nano cryosurgery), etc. Because of these original properties, nanofluids are important to study. Therefore, more precise researches are presented in this topic in the references [1,2,3,4,5,6].

Because of its comprehensive applications in biomechanics, industry, and engineering, the research on boundary layer flows of non-Newtonian fluids past a stretching surface is very important. Accordingly, Jayachandra and Sandeep [7], Reddy et al. [8], Vijayalakshmi et al. [9], and Tian et al. [10] examined with different parameters on the non-Newtonian fluid flow past various stretching surfaces. The effect of thermal radiations on MHD three-dimensional flow over a stretching sheet was studied by Nasir et al. [11]. In the presence of graphene nanoparticles, Khan et al. [12] investigated the Eyring-Powell slip flow of a nano liquid film.

The upper-convected Maxwell fluid is a type of a viscoelastic or rate type fluid. This model is very important since it predicts the relaxation time effect and it excludes complicated effects of shear-dependent viscosity. Commonly, many researchers have studied upper-convected Maxwell fluid flow. Yu Bai et al. [13], Imran et al. [14], Elsayed Mohamed Abdel Rahman Elbashbeshy et al. [15], Vajravelu et al. [16], Omowaye and Animasaun [17], Alireza Rahbari et al. [18], Gireesha et al. [19], and Meysam Mohamadali [20] scrutinized non-Newtonian Maxwell fluid with different physical conditions such as viscous dissipation, Newtonian heating, homogeneous-heterogeneous chemical reactions, and thermal stratification past different stretching surfaces. Their result shows that as Prandtl number increases, temperature, as well as rate of heat transfer, dwindled.

The research on stagnation point flow of nanofluid over stretching surface has different applications in industries and technology. As a result, Sajid et al. [21], Sirinivasulu et al. [22], and Wubshet [23] investigated MHD stagnation point flow in different non-Newtonian fluids such as Oldroyd-B fluid Casson and upper-convected Maxwell fluid on a stretching sheet with various physical parameters. Similarly, Mageswari and Nirmala [24] scrutinized stagnation point flow on stretching sheet with Newtonian heating. Moreover, Abuzar et al. [25] have examined the effect of radiation and convective boundary condition on oblique stagnation point of non-Newtonian nanofluids over the stretching surface. Furthermore, stagnation point flow of nanofluid due to inclined stretching sheet was studied numerically by Yasin Abdela et al. [26]. The numerical result shows that when velocity ratio parameter, Grashof number, solutal expansion parameter, and angle of inclination velocity increased, the boundary layer thickness increases.

The studies of chemical reaction and slip boundary condition with heat transfer have important application in technology and industry. Accordingly, the slip effect on MHD flow with different model of non-Newtonian nanofluids such as Casson fluid and Jeffrey nanofluid past a stretching sheet with various physical conditions are investigated by Sathies Kumar [27], Raghawendra Mishra [28], Manjula and Jayalakshmi [29], and Mohamed Abd El-Aziz and Ahmed Afify [30]. Their study displays that if velocity ratio, momentum slip, and magnetic parameters increase, then the velocity boundary layer thickness become reduced. Krishnamurthy et al. [31], Mabood et.al [32], and Ibrahim et.al [33] probed the effect of chemical reaction on mass and heat transfer MHD boundary layer flow with viscous dissipation, thermal radiation, mixed convection, etc., past stretching sheet and observed that with an increasing magnetic field, the Nusselt number, skin friction coefficient, and Sherwood number are increased. Madasi krishnaiah et al. [34] studied the effect of slip conditions, viscous dissipation, and chemical reaction on MHD stagnation point flow of nanofluid. They observed that as the values of suction parameter rise, the velocity upsurges, also the temperature and concentration profiles are reduced. Moreover, the reference covering slip effects and chemical reaction are described in references [35,36,37,38,39,40,41,42,43,44,45,46,47].

All the above investigators disregard the effects of nanoparticles with slip effects in the analysis of the problem of MHD stagnation point flow of upper-convected Maxwell fluid with chemical reaction. So, the objective of the present paper is to inspect the effect of nanoparticle and chemical reaction on MHD slip stagnation point flow, the boundary layer flow, and heat and mass transfer of upper-convected Maxwell fluid above a stretching sheet via implicit finite difference method. Therefore, the inclusion of the effect of nanoparticles with chemical reaction and slip effect in upper-convected Maxwell fluids makes this study a novel one.

Mathematical formulation

Let us contemplate time-independent and incompressible MHD slip flow of upper-convected Maxwell (UCM) fluid with chemical reaction along a stretching sheet. It is expected that the free stream velocity Ue(x) and the stretching velocity Uw(x) are of the forms Ue(x) = dx and Uw(x) = cx where c and d are constants. For this, study x-axis is along the sheet and normal to the sheet y-axis is chosen. Over the stretching sheet, the concentration is represented by Cw and temperature is represented by Tw. Moreover, the ambient temperature and the ambient concentration are represented by T and C. Under these assumptions, the governing equations of time-independent and incompressible boundary layer flow nanofluid over stretching sheet are given by:

$$ \frac{\mathrm{\partial u}}{\mathrm{\partial x}}+\frac{\mathrm{\partial v}}{\mathrm{\partial y}}=0 $$
(1)
$$ \mathrm{u}\frac{\mathrm{\partial u}}{\mathrm{\partial x}}+\mathrm{v}\frac{\mathrm{\partial u}}{\mathrm{\partial y}}=\upnu \frac{\partial^2\mathrm{u}}{\partial {\mathrm{y}}^2}-\upxi \left({\mathrm{u}}^2\frac{\partial^2\mathrm{u}}{\partial {\mathrm{x}}^2}+{\mathrm{v}}^2\frac{\partial^2\mathrm{u}}{\partial {\mathrm{y}}^2}+2\mathrm{u}\mathrm{v}\frac{\partial^2\mathrm{u}}{\partial \mathrm{x}\partial \mathrm{y}}\right)+{\mathrm{U}}_{\mathrm{e}}\frac{\partial {\mathrm{u}}_{\mathrm{e}}}{\mathrm{\partial x}}\kern0.5em +\frac{{\upsigma \mathrm{B}}_0^2}{\uprho_{\mathrm{f}}}\left({\mathrm{U}}_{\mathrm{e}}-\mathrm{u}\right) $$
(2)
$$ \mathrm{u}\frac{\mathrm{\partial T}}{\mathrm{\partial x}}+\mathrm{v}\frac{\mathrm{\partial T}}{\mathrm{\partial y}}={\alpha}_m\frac{\partial^2\mathrm{T}}{\partial {\mathrm{y}}^2}+\uptau \left\{{\mathrm{D}}_{\mathrm{B}}\frac{\mathrm{\partial C}}{\mathrm{\partial y}}\frac{\mathrm{\partial T}}{\mathrm{\partial y}}+\frac{{\mathrm{D}}_{\uptau}}{{\mathrm{T}}_{\infty }}{\left(\frac{\mathrm{\partial T}}{\mathrm{\partial y}}\right)}^2\right\}-\frac{1}{{\left(\rho {c}_p\right)}_f}\frac{\partial {q}_r}{\partial y}+\frac{Q_0\left(T-{T}_{\infty}\right)}{{\left(\rho {c}_p\right)}_f} $$
(3)
$$ \mathrm{u}\frac{\mathrm{\partial C}}{\mathrm{\partial x}}+\mathrm{v}\frac{\mathrm{\partial C}}{\mathrm{\partial y}}={\mathrm{D}}_{\mathrm{B}}\frac{\partial^2\mathrm{C}}{\partial {\mathrm{y}}^2}+\frac{{\mathrm{D}}_{\uptau}}{{\mathrm{T}}_{\infty }}\frac{\partial^2\mathrm{T}}{\partial {\mathrm{y}}^2}-{K}_r\left(\mathrm{C}-{C}_{\infty}\right) $$
(4)

The appropriate boundary conditions are:

$$ u={U}_w+\mathrm{A}\frac{\partial u}{\partial y},v=0,T={T}_w+\mathrm{B}\frac{\partial T}{\partial y},C={C}_w+\mathrm{K}\frac{\partial C}{\partial y}, at\;y=0 $$
(5)
$$ u\to {U}_e(x)= dx,v\to 0,T\to {T}_{\infty },C\to {C}_{\infty}\; as\;y\to \infty $$
(6)

where u and v are the velocity components along the x and y directions, respectively, ρf is the density of the base fluid, \( {\alpha}_m=\frac{k}{\rho {c}_f} \) is the thermal diffusivity, ξ is the relaxation time parameter of the fluid, B0 is the strength of the magnetic field, ν is the kinematic viscosity of the fluid, k is the thermal conductivity of the fluid, DB is the Brownian diffusion coefficient, Dτ is the thermophoretic diffusion coefficient, \( \tau =\frac{{\left(\rho c\right)}_p}{{\left(\rho c\right)}_f} \) is the ratio between the effective heat capacity of the nanoparticle material and heat capacity of the fluid, c is the volumetric volume expansion coefficient, and ρ is the density of the particles Fig. 1.

Fig. 1
figure1

Coordinate system and physical model

We can write for the radiation using Rosseland approximation

$$ {q}_r=-\frac{4{\sigma}^{\ast }}{3{k}^{\ast }}\frac{\partial {T}^4}{\partial y} $$
(7)

where σ is the Stefan-Boltzman constant and k is the absorption coefficient; assuming the temperature difference within the flow in such that T4 may be expanded in a Taylor series about T and neglecting higher orders we get \( {T}^4=4T{T}_{\infty}^3-3{T}_{\infty}^4 \).

Hence,

$$ \frac{\partial {q}_r}{\partial y}=-\frac{16{\sigma}^{\ast }{T}_{\infty}^3}{3{k}^{\ast }}\frac{\partial^2T}{\partial {y}^2} $$
(8)

Introducing similarity transformations

$$ \psi =\sqrt{c\nu}f\left(\eta \right),\kern1.25em \theta \left(\eta \right)=\frac{\left(T-{T}_{\infty}\right)}{\left({T}_w-{T}_{\infty}\right)},\kern1.25em \phi \left(\eta \right)=\frac{\left(C-{C}_{\infty}\right)}{\left({C}_w-{C}_{\infty}\right)},\kern0.5em \eta =\sqrt{\frac{c}{\nu }}y $$
(9)

We choose the stream function ψ(x, y) such that

$$ \frac{\partial \psi }{\partial y}=u,\kern0.5em \mathrm{and}\kern0.5em -\frac{\partial \psi }{\partial x}=v\kern0.5em $$
(10)

Using the similarity transformation defined by (9), Eqs. 14 are transformed into the non-dimensional ordinary differential equation form as follows:

$$ {f}^{\prime \prime \prime }+f{f}^{\prime \prime }-{f^{\prime}}^2+{E}^2+M\left(E-{f}^{\prime}\right)+\beta \left(2\mathrm{f}{f}^{\prime }{f}^{\prime \prime }-{f}^2{f}^{\prime \prime \prime}\right)=0 $$
(11)
$$ \left(1+\frac{4}{3R}\right){\theta}^{\prime \prime }+ Prf{\theta}^{\prime }+ PrNb{\phi}^{\prime }{\theta}^{\prime }+ Nt\theta {\prime}^2+ PrQ\theta =0 $$
(12)
$$ {\phi}^{\prime \prime }+ Lef{\phi}^{\prime }+\frac{Nt}{Nb}{\theta}^{\prime \prime }- hLe\phi =0 $$
(13)

The transformed boundary conditions are

$$ f\left(\eta \right)=S,\kern1em {f}^{\prime}\left(\eta \right)=1+\lambda {f}^{\prime \prime}\left(\eta \right),\kern0.75em \theta \left(\eta \right)=1+\delta {\theta}^{\prime \left(\eta \right)},\kern1.25em \phi \left(\eta \right)=1+\gamma {\phi}^{\prime}\left(\eta \right),\mathrm{at}\ \eta =0 $$
(14)
$$ {f}^{\hbox{'}}\left(\eta \right)\to E,\theta \left(\eta \right)\to 0,\phi \left(\eta \right)\to 0, as\ \eta \to \infty $$
(15)

where f is dimensionless velocity, θ is dimensionless temperature, ϕ is dimensionless concentration, and η is the similarity variable. The prime denotes differentiation with respect to η.

The overall governing parameters are defined as the following:

\( S=-{v}_w(x)\sqrt{\frac{2x}{\nu {u}_{\infty }}} \) is the suction-injection parameter, \( E=\frac{d}{c} \) is velocity ratio, β = ξc is Deborah number, \( M=\frac{\sigma {B}_0^2x}{\rho {u}_{\infty }} \)is magnetic field parameter, \( \lambda =\mathrm{A}\sqrt{\frac{c}{\nu }} \)is velocity slip parameter, \( \delta =\mathrm{B}\sqrt{\frac{c}{\nu }} \)is thermal slip parameter, \( \gamma =\mathrm{K}\sqrt{\frac{c}{\nu }} \)is solutal slip parameter, \( \mathrm{R}=\frac{k{k}^{\ast }}{4{\sigma}^{\ast }{T}_{\infty}^3} \)is thermal radiation parameter, k = (ρcp)f, \( \mathit{\Pr}=\frac{\nu }{\alpha } \) is Prandtl number, \( Nb=\frac{{\left(\rho c\right)}_p{D}_B\left({C}_w-{C}_{\infty}\right)}{{\left(\rho c\right)}_f\nu } \) is Brownian motion parameter, \( Nb=\frac{{\left(\rho c\right)}_p{D}_{\tau}\left({T}_w-{T}_{\infty}\right)}{{\left(\rho c\right)}_f\nu {T}_{\infty }} \)is thermophoresis parameter, \( Le=\frac{\nu }{D_B} \)is Lewis number, \( h=\frac{h_r}{c} \)is chemical reaction parameter, and \( Q=\frac{Q_0}{a{\left(\rho {c}_p\right)}_f} \) is heat source parameter.

The skin friction Cf, local Nusselt number Nux, and the Sherwood number Shx are the important physical quantities of interest in this problem which are defined as

$$ {C}_f=\frac{\tau_w}{\rho {u}_w^2},N{u}_x=\frac{x{q}_w}{k\left({T}_f-{T}_{\infty}\right)},S{h}_x=\frac{x{q}_m}{D_B\left({c}_w-{c}_{\infty}\right)} $$

Here \( {\tau}_w=\mu \left(1+\beta \right)\frac{\partial u}{\partial y} \) is the surface shear stress, \( {q}_w=-k{\left(\frac{\partial T}{\partial y}\right)}_{\left(y=0\right)}+{q}_r \) is the surface heat flux, and\( {q}_m=-{D}_B{\left(\frac{\partial C}{\partial y}\right)}_{\left(y=0\right)} \)

Using the similarity transformation in (9) we have the following relations:

\( {C}_f{\mathit{\operatorname{Re}}}_x^{\frac{1}{2}}=f^{\prime\prime }(0), \) \( {Nu}_x{\mathit{\operatorname{Re}}}_x^{-\frac{1}{2}}=-\left(1+\frac{4}{3R}\right)\theta^{\prime }(0), \) \( {Sh}_x{\mathit{\operatorname{Re}}}_x^{-\frac{1}{2}}=-\phi^{\prime }(0) \), where Rex is the local Reynolds number.

Numerical solution

  1. a.

    The Keller-box method

The transfigured ordinary differential Eqs. (11), (12), and (13) subject to the boundary conditions (14) and (15) are solved numerically using an implicit finite difference method (Keller-box) in combination with Newton’s linearization techniques. The key topographies of this method are:

  1. i)

    Unconditionally stable and has a second-order accuracy with arbitrary spacing and attractive extrapolation features.

  2. ii)

    The most reliable and powerful numerical methods for nonlinear boundary layer flows that are generally parabolic in nature.

  3. iii)

    Tolerates very speedy x variations.

The Keller-box scheme comprises four steps:

  1. 1)

    Reducing the order ordinary differential equations into a system of fist order equations.

  2. 2)

    Using central differences write difference equations.

  3. 3)

    Linearizing the resulting algebraic equations by using Newton’s method and writing in matrix-vector form.

  4. 4)

    By using block-tridiagonal elimination method solving the linearized system of equations.

    1. b.

      The finite difference scheme

We write the governing third-order momentum Eq. (11) and second-order energy and concentration Eqs. (12) and (13) in terms of a first-order equations. For this purpose, we introduce new dependent variables u, v, t, θ = s(x, η), ϕ(η) = g(x, η) such that f = u, u = v, s = t, and g = z

Thus Eqs. (11), (12), and (13) can be written as

$$ {v}^{\prime }+ fv-{u}^2+{E}^2+M\left(E-u\right)+\beta \left(2 fuv-{f}^2v^{\prime}\right)=0 $$
(16)
$$ \left(1+\frac{4}{3R}\right){t}^{\prime }+ Prf{t}^{\prime }+ PrNbzt+ Nt{t}^2+ PrQs=0 $$
(17)
$$ z^{\prime }+ Lefz+\frac{Nt}{Nb}t^{\prime }- hLeg=0 $$
(18)

The boundary conditions are

$$ f\left(\eta \right)=S,u\left(\eta \right)=1+\lambda v\left(\eta \right),s\left(\eta \right)=1+\delta t\left(\eta \right),g\left(\eta \right)=1+\gamma z\left(\eta \right), at\;\eta =0 $$
(19)
$$ u\left(\eta \right)\to E,s\left(\eta \right)\to 0,g\left(\eta \right)\to 0, as\kern0.37em \eta \to \infty $$
(20)

where prime denotes the differentiation with respect to η.

We now consider the net rectangle in the x − η plane shown in Fig. 2 and the net points defined as below

$$ {\displaystyle \begin{array}{cccc}{x}^0=0& {x}^i={x}^{i-1}+{k}_i& i=1,2,3,\dots, I& \\ {}{\eta}_0=0& {\eta}_j={\eta}_{j-1}+{h}_j& j=1,2,3,\dots, J& {\eta}_j={\eta}_{\infty}\end{array}} $$
Fig. 2
figure2

Net rectangle for difference approximations

where ki is the ∆x-spacing and hj is the ∆η-spacing. Here, i and j are the sequence of numbers that indicate the coordinate location, not tensor indices or exponents.

Since only first derivatives appear in the governing equations, centered differences and two-point averages can be constructed involving only the four corner nodal values of the “box.” For example, if g represents any of the dependent variables u, v, s, and t then

$$ {\displaystyle \begin{array}{c}{\left[g\right]}_{j-\frac{1}{2}}^i=0.5\left({g}_{j-1}^i+{g}_j^i\right)\\ {}{\left[g\right]}_{j-\frac{1}{2}}^{i-\frac{1}{2}}=0.5\left({\left[g\right]}_{j-\frac{1}{2}}^{i-1}+{\left[g\right]}_{j-\frac{1}{2}}^i\right)\\ {}{\left[\frac{\partial g}{\partial \eta}\right]}_{j-\frac{1}{2}}^{i-\frac{1}{2}}=0.5\left({\left[\frac{\partial g}{\partial \eta}\right]}_{j-\frac{1}{2}}^{i-1}+{\left[\frac{\partial g}{\partial \eta}\right]}_{j-\frac{1}{2}}^i\right)\\ {}{\left[\frac{\partial g}{\partial \eta}\right]}_{j-\frac{1}{2}}^i=0.5\frac{\left({\left[g\right]}_{j-\frac{1}{2}}^i-{\left[g\right]}_{1-\frac{1}{2}}^i\right)}{\left({\eta}_j-{\eta}_{j-1}\right)}\\ {}{\left[\frac{\partial g}{\partial x}\right]}_{j-\frac{1}{2}}^{i-\frac{1}{2}}=0.5\frac{\left({\left[g\right]}_{j-\frac{1}{2}}^i-{\left[g\right]}_{1-\frac{1}{2}}^{i-1}\right)}{\left({x}_i-{x}_{i-1}\right)}\end{array}} $$

Now write the finite difference approximations for first-order ordinary differential equation for the mid-point \( \left({x}^i,{\eta}_{j-\frac{1}{2}}\right) \) of the segment P1P2using centered difference derivatives. This process is called centering about \( \left({x}^i,{\eta}_{j-\frac{1}{2}}\right) \). We get

$$ {\displaystyle \begin{array}{l}{f}_j-{f}_{j-1}-\frac{h_j}{2}\left({u}_j+{u}_{j-1}\right)=0\\ {}{u}_j-{u}_{j-1}-\frac{h_j}{2}\left({\mathrm{v}}_j+{\mathrm{v}}_{j-1}\right)=0\\ {}{s}_j-{s}_{j-1}-\frac{h_j}{2}\left({t}_j+{t}_{j-1}\right)=0\\ {}{g}_j-{g}_{j-1}-\frac{h_j}{2}\left({z}_j+{z}_{j-1}\right)=0\\ {}\left({\mathrm{v}}_j-{\mathrm{v}}_{j-1}\right)+\frac{h_j}{4}\left({f}_j+{f}_{j-1}\right)\left({\mathrm{v}}_j+{\mathrm{v}}_{j-1}\right)-\frac{h_j}{4}{\left({u}_j+{u}_{j-1}\right)}^2+\frac{h_j\beta }{4}\left({f}_j+{f}_{j-1}\right)\left({u}_j+{u}_{j-1}\right)\left({\mathrm{v}}_j+{\mathrm{v}}_{j-1}\right)-\frac{\beta }{4}\left({f}_j+{f}_{j-1}\right)\left({f}_j+{f}_{j-1}\right)\left({\mathrm{v}}_j-{\mathrm{v}}_{j-1}\right)+M{h}_j\left(E-\frac{1}{2}\left({u}_j+{u}_{j-1}\right)\right)+{h}_j{E}^2={P}_{j-\frac{1}{2}}\\ {}\left(1+\frac{4}{3R}\right)\left({t}_j-{t}_{j-1}\right)+\frac{h_j\mathit{\Pr}}{4}\left({f}_j+{f}_{j-1}\right)\left({t}_j+{t}_{j-1}\right)+\frac{h_j PrNb}{4}\left({z}_j+{z}_{j-1}\right)\left({t}_j+{t}_{j-1}\right)+\frac{h_j PrNt}{4}\left({t}_j+{t}_{j-1}\right)\left({t}_j+{t}_{j-1}\right)+\frac{h_j PrQ}{2}\left({s}_j+{s}_{j-1}\right)={S}_{j-\frac{1}{2}}\\ {}\left({z}_j-z\right)+\frac{h_j Le}{4}\left({f}_j+{f}_{j-1}\right)\left({\mathrm{z}}_j+{z}_{j-1}\right)+\frac{Nt}{Nb}\left({t}_j-{t}_{j-1}\right)-\frac{Le{h}_jh}{2}\left({g}_j+{g}_{j-1}\right)={T}_{j-\frac{1}{2}}\end{array}} $$
(21)

where \( {P}_{1-\frac{1}{2}}=-\left({\mathrm{v}}_j-{\mathrm{v}}_{j-1}\right)-{h}_j{\left(f\mathrm{v}\right)}_{j-\frac{1}{2}}+{h}_j{\left({u}^2\right)}_{j-\frac{1}{2}}-2{h}_j{\left( fu\mathrm{v}\right)}_{j-\frac{1}{2}}+\frac{\beta }{4}{\left({f}^2\right)}_{j-\frac{1}{2}}\left({\mathrm{v}}_j-{\mathrm{v}}_{j-1}\right)-M{h}_j\left(E-{(u)}_{j-\frac{1}{2}}\right)-{h}_j{E}^2 \)and

$$ {\displaystyle \begin{array}{l}{S}_{j-\frac{1}{2}}=-\left(1+\frac{4}{3R}\right)\left({t}_j-{t}_{j-1}\right)-{h}_j\mathit{\Pr}{\left(f\mathrm{t}\right)}_{j-\frac{1}{2}}-{h}_j PrNb{(zt)}_{j-\frac{1}{2}}-\beta {h}_j{\left({t}^2\right)}_{j-\frac{1}{2}}-{h}_j PrQ{(s)}_{j-\frac{1}{2}}\\ {}{T}_{j-\frac{1}{2}}=-\left({z}_j-{z}_{j-1}\right)-{h}_j{\left(f\mathrm{z}\right)}_{j-\frac{1}{2}}-\frac{Nt}{Nb}\left({t}_j-{t}_{j-1}\right)+ Le{h}_jh{g}_{j-\frac{1}{2}}\end{array}} $$

We note that \( \kern0.5em {P}_{1-\frac{1}{2}} \), \( {Q}_{j-\frac{1}{2}} \), and \( {S}_{j-\frac{1}{2}} \) involve only known quantities if we assume that the solution is known on

x = xi − 1. In terms of the new dependent variables, the boundary conditions become

$$ {\displaystyle \begin{array}{l}f\left(x,0\right)=S,u\left(x,0\right)=1+\lambda v(0),s\left(x,0\right)=1+\delta t(0),g(0)=1+\gamma z(0)\\ {}u\left(x,\infty \right)=E\ s,\left(x,\infty \right)=0,s\left(x,\infty \right)=0,g\left(x,\infty \right)=0\end{array}} $$
(22)

Equations in Eq. (21) are imposed for j = 1, 2, 3, …, J and the transformed boundary layer thickness ηJ is sufficiently large so that it is beyond the edge of the boundary layer Cebeci and Bradshaw [48]. The boundary condition yields at x = xi are

$$ {f}_0^i=S,{u}_0^i=1+\lambda {v}_0^i,\kern0.5em {s}_0^i=1+\delta {t}_0^i,\kern0.5em {g}_0^i=1+\gamma {z}_0^i,{u}_J^i=E,{s}_J^i=0,{g}_J^i=0 $$
(23)

Newton’s method

Equations in Eq. (21) are nonlinear algebraic equations and therefore have to be linearized before the factorization scheme can be used. Let us write the Newton iterates as follows: For (k + 1)th iterates, we write

$$ {\displaystyle \begin{array}{l}{f}_j^{\left(k+1\right)}={f}_j^{(k)}+\delta {f_j^{(k)}}_{,}\\ {}{u}_j^{\left(k+1\right)}={u}_j^{(k)}+\delta {u}_j^{(k)},\\ {}{\mathrm{v}}_j^{\left(k+1\right)}={\mathrm{v}}_j^{(k)}+\delta {\mathrm{v}}_j^{(k)},\\ {}{t}_j^{\left(k+1\right)}={t}_j^{(k)}+\delta {t}_j^{(k)},\\ {}{s}_j^{\left(k+1\right)}={s}_j^{(k)}+\delta\ {s}_j^{(k)}\\ {}{g}_j^{\left(k+1\right)}={g}_j^{(k)}+\delta\ {g}_j^{(k)}\\ {}{z}_j^{\left(k+1\right)}={z}_j^{(k)}+\delta\ {z}_j^{(k)}\end{array}} $$
(24)

Equation (21) can be written as

$$ {\displaystyle \begin{array}{l}{f}_j+\delta {f}_j-{f}_{j-1}-\delta {f}_{j-1}=\frac{h_j}{2}\left({u}_j+\delta {u}_j+{u}_{j-1}+\delta {u}_{j-1}\right)\\ {}{u}_j+\delta {u}_j-{u}_{j-1}-\delta {u}_{j-1}=\frac{h_j}{2}\left({\mathrm{v}}_j+\delta {\mathrm{v}}_{j,}+{\mathrm{v}}_{j-1}+\delta {\mathrm{v}}_{j-1}\right)\\ {}{g}_j+\delta {g}_j-{g}_{j-1}-\delta {g}_{j-1}=\frac{h_j}{2}\left({z}_j+\delta {z}_j+{z}_{j-1}+\delta {z}_{j-1}\right)\\ {}{s}_j+\delta {s}_j-{s}_{j-1}-\delta {s}_{j-1}=\frac{h_j}{2}\left({t}_j+\delta {t}_j+{t}_{j-1}+\delta {t}_{j-1}\right)\\ {}{\mathrm{v}}_j+\delta {\mathrm{v}}_{\mathrm{j},}-{\mathrm{v}}_{j-1}-\delta {\mathrm{v}}_{j-1}+\frac{h_j}{4}\left({f}_j+\delta {f}_{j\kern0.5em }+{f}_{j-1}+\delta {f}_{j-1\kern0.75em }\right)\left({\mathrm{v}}_j+\delta {\mathrm{v}}_{j,\kern1em }+{\mathrm{v}}_{j-1}+\delta {\mathrm{v}}_{j-1\kern1em }\right)-\frac{h_j}{4}{\left({u}_j+\delta {u}_{j\kern1em }+{u}_{j-1}+\delta {u}_{j-1\kern1em }\right)}^2+\frac{h_j\beta }{4}\left({f}_j+\delta {f}_{j\kern0.5em }+{f}_{j-1}+\delta {f}_{j-1\kern0.5em }\right)\left({u}_j+\delta {u}_{j\kern0.5em }+{u}_{j-1}+\delta {u}_{j-1\kern0.5em }\right)\left({\mathrm{v}}_j+\delta {v}_{j\kern0.5em }+{\mathrm{v}}_{j-1}+\delta {v}_{j-1\kern0.5em }\right)-\frac{\beta }{4}\left({f}_j+\delta {f}_{j\kern0.5em }+{f}_{j-1}+\delta {f}_{j-1\kern0.5em }\right)\left({f}_j+\delta {f}_{j\kern0.5em }+{f}_{j-1}+\delta {f}_{j-1\kern0.5em }\right)\left({\mathrm{v}}_j+\delta {v}_{j\kern0.5em }-{\mathrm{v}}_{j-1}-\delta {v}_{j-1\kern0.5em }\right)+M{h}_j\left(E-\frac{1}{2}\left({u}_j+\delta {u}_{j\kern0.5em }+{u}_{j-1}+\delta {u}_{j-1\kern0.5em }\right)\right)+{h}_j{E}^2={P}_{j-\frac{1}{2}}\\ {}\left(1+\frac{4}{3R}\right)\left({t}_j+\delta {t}_j-{t}_{j-1}-\delta {t}_{j-1}\right)+\frac{h_j\mathit{\Pr}}{4}\left({f}_j+\delta {f}_j+{f}_{j-1}+\delta {f}_{j-1}\right)\left({t}_j+\delta {t}_j+{t}_{j-1}+\delta {t}_{j-1}\right)+\frac{h_j PrNb}{4}\left({z}_j+\delta {z}_j+{z}_{j-1}+\delta {z}_{j-1}\right)\left({t}_j+\delta {t}_j+{t}_{j-1}+\delta {t}_{j-1}\right)+\frac{h_j PrNt}{4}\left({t}_j+\delta {t}_j+{t}_{j-1}+\delta {t}_{j-1}\right)\left({t}_j+\delta {t}_j+{t}_{j-1}+\delta {t}_{j-1}\right)+\frac{h_j PrQ}{2}\left({s}_j+\delta {s}_j+{s}_{j-1}+\delta {s}_{j-1}\right)={Q}_{j-\frac{1}{2}}\\ {}\left({z}_j+\delta {z}_j-{z}_{j-1}-\delta {z}_{j-1}\right)+\frac{h_j Le}{4}\left({f}_j+\delta {f}_j+{f}_{j-1}+\delta {f}_{j-1}\right)\left({z}_j+\delta {z}_j+{z}_{j-1}+\delta {z}_{j-1}\right)+\frac{Nt}{Nb}\left({t}_j+\delta {t}_j-{t}_{j-1}-\delta {t}_{j-1}\right)-\frac{Le{h}_jh}{2}\left({g}_j+\delta {g}_j+{g}_{j-1}+\delta {g}_{j-1}\right)={S}_{j-\frac{1}{2}}\end{array}} $$
(25)

By dropping the quadratic and higher-order terms in \( \delta {f}_j^{(i)},\delta {u}_j^{(i)},\delta {\mathrm{v}}_j^{(i)},\delta {s}_j^{(i)},\delta {g}_j^{(i)},\delta {t}_j^{(i)}, and\ \delta {z}_j^{(i)} \), a linear tridiagonal system of equations will be obtained as follows:

$$ {\displaystyle \begin{array}{l}\delta {f}_j-\delta {f}_{j-1}-\frac{h_j}{2}\left(\delta {u}_j+\delta {u}_{j-1}\right)={\left({r}_1\right)}_{j-\frac{1}{2}}\\ {}\delta {u}_{j\kern0.5em }-\delta {u}_{j-1}-\frac{h_j}{2}\left(\delta {\mathrm{v}}_j+\delta {\mathrm{v}}_{j-1}\right)={\left({r}_2\right)}_{j-\frac{1}{2}}\\ {}\delta {s}_{j\kern0.5em }-\delta {s}_{j-1}-\frac{h_j}{2}\left(\delta {t}_j+\delta {t}_{j-1}\right)={\left({r}_3\right)}_{j-\frac{1}{2}}\kern0.5em \\ {}\delta {g}_j-\delta {g}_{j-1}-\frac{h_j}{2}\left(\delta {z}_j+\delta {z}_{j-1}\right)={\left({r}_4\right)}_{j-\frac{1}{2}}\\ {}\delta {\mathrm{v}}_j-\delta {\mathrm{v}}_{j-1}+\frac{h_j}{2}{\mathrm{v}}_{j-\frac{1}{2}}\left(\delta {f}_{j\kern0.5em }+\delta {f}_{j-1}\right)+\frac{h_j}{2}{f}_{j-\frac{1}{2}}\left(\delta {\mathrm{v}}_j+\delta {\mathrm{v}}_{j-1\kern1em }\right)-{h}_j{u}_{j-\frac{1}{2}}\left(\delta {u}_j+\delta {u}_{j-1}\right)+{h}_j\beta {\left(u\mathrm{v}\right)}_{j-\frac{1}{2}}\left(\delta {f}_{j\kern0.5em }+\delta {f}_{j-1\kern0.5em }\right)+{h}_j\beta {\left(f\mathrm{v}\right)}_{j-\frac{1}{2}}\left(\delta {u}_{j\kern0.5em }+\delta {u}_{j-1\kern0.5em }\right)+{h}_j\beta {(fu)}_{j-\frac{1}{2}}\left(\delta {v}_{j\kern0.5em }+\delta {v}_{j-1\kern0.5em }\right)-\beta {f}_{j-\frac{1}{2}}^2\left(\delta {\mathrm{v}}_j-\delta {\mathrm{v}}_{j-1}\right)-\beta {f}_{j-\frac{1}{2}}\left({\mathrm{v}}_j-{\mathrm{v}}_{j-1}\right)\left(\delta {f}_{j\kern0.5em }+\delta {f}_{j-1\kern0.5em }\right)+M\frac{h_j}{2}\left(\delta {u}_{j\kern0.5em }+\delta {u}_{j-1\kern0.5em }\right)={\left({r}_5\right)}_{j-\frac{1}{2}}\\ {}\left(1+\frac{4}{3R}\right)\left(\delta {t}_j-\delta {t}_{j-1}\right)+\frac{h_j\mathit{\Pr}}{2}{f}_{j-\frac{1}{2}}\left(\delta {t}_j+\delta {t}_{j-1}\right)+\frac{h_j\mathit{\Pr}}{2}{t}_{j-\frac{1}{2}}\left(\delta {f}_j+\delta {f}_{j-1}\right)+\frac{h_j PrNb}{2}{t}_{j-\frac{1}{2}}\left(\delta {z}_j+\delta {z}_{j-1}\right)+\frac{h_j PrNb}{2}{z}_{j-\frac{1}{2}}\left(\delta {t}_j+\delta {t}_{j-1}\right)+{h}_j PrNt{t}_{j-\frac{1}{2}}\left(\delta {t}_j+\delta {t}_{j-1}\right)+\frac{h_j PrQ}{2}\left(\delta {s}_j+\delta {s}_{j-1}\right)={\left({r}_6\right)}_{j-\frac{1}{2}}\\ {}\left(\delta {z}_j-\delta {z}_{j-1}\right)+\frac{h_j Le}{2}{z}_{j-\frac{1}{2}}\left(\delta {f}_j+\delta {f}_{j-1}\right)+\frac{h_j Le}{2}{f}_{j-\frac{1}{2}}\left(\delta {z}_j+\delta {z}_{j-1}\right)+\frac{Nt}{Nb}\left(\delta {t}_j-\delta {t}_{j-1}\right)-\frac{Le{h}_jh}{2}\left(\delta {g}_j+\delta {g}_{j-1}\right)={\left({r}_7\right)}_{j-\frac{1}{2}}\end{array}} $$
(26)

where

$$ {\displaystyle \begin{array}{l}\ {\left({r}_1\right)}_{j-\frac{1}{2}}={f}_{j-1}-{f}_j+\frac{h_j}{2}\left({u}_j+{u}_{j-1}\right)\\ {}{\left({r}_2\right)}_{j-\frac{1}{2}}={u}_{j-1}-{u}_j+\frac{h_j}{2}\left({\mathrm{v}}_j+{\mathrm{v}}_{j-1}\right)\\ {}{\left({r}_3\right)}_{j-\frac{1}{2}}={s}_{j-1}-{s}_j+\frac{h_j}{2}\left({t}_j+{t}_{j-1}\right)\\ {}{\left({r}_4\right)}_{j-\frac{1}{2}}={g}_{j-1}-{g}_j+\frac{h_j}{2}\left({z}_j+{z}_{j-1}\right)\\ {}{\left({r}_5\right)}_{j-\frac{1}{2}}=-\left({\mathrm{v}}_j-{\mathrm{v}}_{j-1}\right)-{h}_j{\left(f\mathrm{v}\right)}_{j-\frac{1}{2}}+{h}_j{u}_{j-\frac{1}{2}}^2-2{h}_j\beta {\left( fu\mathrm{v}\right)}_{j-\frac{1}{2}}+\beta {f}_{j-\frac{1}{2}}^2\left({\mathrm{v}}_j-{\mathrm{v}}_{j-1}\right)-{h}_jM{u}_{j-\frac{1}{2}}-{E}^2{h}_j- ME{h}_j\\ {}{\left({r}_6\right)}_{j-\frac{1}{2}}=-\left(1+\frac{4}{3R}\right)\left({t}_j-{t}_{j-1}\right)-\mathit{\Pr}{h}_j{\left(f\mathrm{t}\right)}_{j-\frac{1}{2}}-\mathit{\Pr}{h}_j Nb{\left(z\mathrm{t}\right)}_{j-\frac{1}{2}}-{h}_j PrNt{t}_{j-\frac{1}{2}}^2-\mathit{\Pr}{h}_jQ{s}_{j-\frac{1}{2}}\\ {}{\left({r}_7\right)}_{j-\frac{1}{2}}=-\left({z}_j-{z}_{j-1}\right)- Le{h}_j{\left(f\mathrm{z}\right)}_{j-\frac{1}{2}}-\frac{Nt}{Nb}\left({\mathrm{t}}_j-{t}_{j-1}\right)- Le{h}_jh{g}_{j-\frac{1}{2}}\end{array}} $$
(27)
$$ {\displaystyle \begin{array}{l}\delta {f}_j-\delta {f}_{j-1}-\frac{h_j}{2}\left(\delta {u}_j+\delta {u}_{j-1}\right)={\left({r}_1\right)}_{j-\frac{1}{2}}\\ {}\delta {u}_j-\delta {u}_{j-1}-\frac{h_j}{2}\left(\delta {\mathrm{v}}_j+\delta {\mathrm{v}}_{j-1}\right)={\left({r}_2\right)}_{j-\frac{1}{2}}\\ {}\delta {s}_j-\delta {s}_{j-1}-\frac{h_j}{2}\left(\delta {t}_j+\delta {t}_{j-1}\right)={\left({r}_3\right)}_{j-\frac{1}{2}}\\ {}\delta {g}_j-\delta {g}_{j-1}-\frac{h_j}{2}\left(\delta {z}_j+\delta {z}_{j-1}\right)={\left({r}_4\right)}_{j-\frac{1}{2}}\\ {}{a}_1\delta {\mathrm{v}}_j+{a}_2\delta {\mathrm{v}}_{j-1}+{a}_3\delta {f}_{j\kern0.5em }+{a}_4\delta {f}_{j-1}+{a}_5\delta {u}_j+{a}_6\delta {u}_{j-1}={\left({r}_5\right)}_{j-\frac{1}{2}}\\ {}{b}_1\delta {t}_j+{b}_2\delta {t}_{j-1}+{b}_3\delta {f}_j+{b}_4\delta {f}_{j-1}+{b}_5\delta {z}_j+{b}_6\delta {z}_{j-1}+{b}_7\delta {s}_j+{b}_8\delta {s}_{j-1}={\left({r}_6\right)}_{j-\frac{1}{2}}\\ {}{c}_1\delta {z}_j+{c}_2\delta {z}_{j-1}+{c}_3\delta {f}_j+{c}_4\delta {f}_{j-1}+{c}_5\delta {t}_j+{c}_6\delta {t}_{j-1}+{c}_7\delta {g}_j+{c}_8\delta {g}_{j-1}={\left({r}_7\right)}_{j-\frac{1}{2}}\end{array}} $$
(28)

Where

$$ {\displaystyle \begin{array}{l}{\left({a}_1\right)}_j=1+\frac{h_j}{2}{f}_{j-\frac{1}{2}}+{h}_j\beta {\left(f\mathrm{u}\right)}_{j-\frac{1}{2}}-\beta {f}_{j-\frac{1}{2}}^2\\ {}{\left({a}_2\right)}_j=-1+\frac{h_j}{2}{f}_{j-\frac{1}{2}}+{h}_j\beta {\left(f\mathrm{u}\right)}_{j-\frac{1}{2}}+\beta {f}_{j-\frac{1}{2}}^2\\ {}{\left({a}_3\right)}_j={\left({a}_4\right)}_j=\frac{h_j}{2}{\mathrm{v}}_{j-\frac{1}{2}}+{h}_j\beta {\left(\mathrm{uv}\right)}_{j-\frac{1}{2}}-\beta {\mathrm{f}}_{j-\frac{1}{2}}\left({\mathrm{v}}_j-{\mathrm{v}}_{j-1}\right)\\ {}{\left({a}_5\right)}_j={\left({a}_6\right)}_j=-\frac{h_j}{2}{u}_{j-\frac{1}{2}}+{h}_j\beta {\left(f\mathrm{v}\right)}_{j-\frac{1}{2}}+\frac{M{h}_j}{2}\\ {}{\left({b}_1\right)}_j=\left(1+\frac{4}{3R}\right)+\frac{h_j\mathit{\Pr}}{2}{f}_{j-\frac{1}{2}}+\frac{h_j PrNb}{2}{z}_{j-\frac{1}{2}}+{h}_j PrNt{t}_{j-\frac{1}{2}}\\ {}{\left({b}_2\right)}_j=-\left(1+\frac{4}{3R}\right)+\frac{h_j\mathit{\Pr}}{2}{f}_{j-\frac{1}{2}}+\frac{h_j PrNb}{2}{z}_{j-\frac{1}{2}}+{h}_j PrNt{t}_{j-\frac{1}{2}}\\ {}{\left({b}_3\right)}_j={\left({b}_4\right)}_j=\frac{h_j\mathit{\Pr}}{2}{\mathrm{t}}_{j-\frac{1}{2}}\\ {}{\left({b}_5\right)}_j={\left({b}_6\right)}_j=\frac{h_j PrNb}{2}{\mathrm{t}}_{j-\frac{1}{2}},{\left({b}_7\right)}_j={\left({b}_8\right)}_j=\frac{h_j PrQ}{2}\\ {}{\left({c}_1\right)}_j=1+\frac{h_j Le}{2}{f}_{j-\frac{1}{2}}\\ {}{\left({c}_2\right)}_j=-1+\frac{h_j Le}{2}{f}_{j-\frac{1}{2}},{\left({c}_3\right)}_j={\left({c}_4\right)}_j=\frac{h_j Le}{2}{f}_{j-\frac{1}{2}}\\ {}{\left({c}_5\right)}_j=\frac{Nt}{Nb},{\left({c}_6\right)}_j=-{\left({c}_5\right)}_j,{\left({c}_7\right)}_j={\left({c}_8\right)}_j=\frac{-{h}_j Le h}{2}\end{array}} $$
(29)

To complete the system (28) we recall the boundary conditions (23) which can be satisfied exactly with no iteration. Therefore, in order to maintain these correct values in all the iterates, we take δf0 = 0,   δu0 = 0,   δs0 = 0, δg0 = 0 δuJ = 0, δsJ = 0, and δgJ = 0.

In a general case, Eq. (31) in vector matrix forms:

$$ \left[A\right]\left[\delta \right]=\left[r\right] $$
(30)

where

$$ A=\left[\begin{array}{cccccccccc}\left[{A}_1\right]& \left[{C}_1\right]& & & & & & & & \\ {}\left[{B}_2\right]& \left[{A}_2\right]& \left[{C}_2\right]& & & & & & & \\ {}& \left[{B}_3\right]& \left[{A}_3\right]& \left[C{}_3\right]& & & & & & \\ {}& & .& .& .& & & & & \\ {}& & & .& .& .& & & & \\ {}& & & & .& .& .& & & \\ {}& & & & & & & & & \\ {}& & & & & & & & & \\ {}& & & & & & & \left[{B}_{J-1}\right]& \left[{A}_{J-1}\right]& \left[{C}_{J-1}\right]\\ {}& & & & & & & & \left[{B}_J\right]& \left[{A}_J\right]\end{array}\right],\delta =\left[\begin{array}{c}\left[{\delta}_1\right]\\ {}\left[{\delta}_2\right]\\ {}.\\ {}.\\ {}.\\ {}\left[{\delta}_{J-1}\right]\\ {}\left[{\delta}_J\right]\end{array}\right],r=\left[\begin{array}{c}\left[{r}_1\right]\\ {}\left[{r}_2\right]\\ {}.\\ {}.\\ {}.\\ {}\left[{r}_{J-1}\right]\\ {}\left[{r}_J\right]\end{array}\right] $$

In Eq. (31), the elements are defined by

$$ {\displaystyle \begin{array}{l}\left[{A}_1\right]=\left[\begin{array}{ccccccc}0& 0& 0& 1& 0& 0& 0\\ {}-\frac{h_1}{2}& 0& 0& 0& -\frac{h_1}{2}& 0& 0\\ {}0& -\frac{h_1}{2}& 0& 0& 0& -\frac{h_1}{2}& 0\\ {}0& 0& -\frac{h_1}{2}& 0& 0& 0& -\frac{h_1}{2}\\ {}{a}_2& 0& 0& {a}_3& {a}_1& 0& 0\\ {}0& {b}_2& {b}_6& {b}_3& 0& {b}_1& {b}_5\\ {}0& {c}_6& {c}_2& {c}_3& 0& {c}_5& {c}_1\end{array}\right]\\ {}\left[{A}_j\right]=\left[\begin{array}{ccccccc}-\frac{h_j}{2}& 0& 0& 1& 0& 0& 0\\ {}-1& 0& 0& 0& -\frac{h_j}{2}& 0& 0\\ {}0& -1& 0& 0& 0& -\frac{h_j}{2}& 0\\ {}0& 0& -1& 0& 0& 0& -\frac{h_j}{2}\\ {}{a}_6& 0& 0& {a}_3& {a}_1& 0& 0\\ {}0& {b}_8& 0& {b}_3& 0& {b}_1& {b}_5\\ {}0& 0& {c}_8& {c}_3& 0& {c}_5& {c}_1\end{array}\right]\kern0.50em \ \ \ 2\le j\le J\\ {}\left[{B}_j\right]=\left[\begin{array}{ccccccc}0& 0& 0& -1& 0& 0& 0\\ {}0& 0& 0& 0& -\frac{h_j}{2}& 0& 0\\ {}0& 0& 0& 0& 0& -\frac{h_j}{2}& 0\\ {}0& 0& 0& 0& 0& 0& -\frac{h_j}{2}\\ {}0& 0& 0& {a}_4& {a}_2& 0& 0\\ {}0& 0& 0& {b}_4& 0& {b}_2& {b}_6\\ {}0& 0& 0& {c}_4& 0& {c}_6& {c}_2\end{array}\right]\kern0.50em \ \ \ 2\le j\le J\\ {}\left[{C}_j\right]=\left[\begin{array}{ccccccc}-\frac{h_j}{2}& 0& 0& 0& 0& 0& 0\\ {}1& 0& 0& 0& 0& 0& 0\\ {}0& 1& 0& 0& 0& 0& 0\\ {}0& 0& 1& 0& 0& 0& 0\\ {}{a}_5& 0& 0& 0& 0& 0& 0\\ {}0& {b}_7& 0& 0& 0& 0& 0\\ {}0& 0& {c}_7& 0& 0& 0& 0\end{array}\right]\ 1\le j\le J\\ {}\left[{\delta}_1\right]=\left[\begin{array}{c}\delta {v}_0\\ {}\delta {t}_0\\ {}\delta {z}_0\\ {}\delta {f}_1\\ {}\delta {v}_1\\ {}\delta {t}_1\\ {}\delta {z}_1\end{array}\right],\left[{\delta}_j\right]=\left[\begin{array}{c}\delta {u}_{j-1}\\ {}\delta {s}_{j-1}\\ {}\delta {g}_{j-1}\\ {}\delta {f}_j\\ {}\delta {v}_j\\ {}\delta {t}_j\\ {}\delta {z}_j\end{array}\right]\;2\le j\le J\;\left[{r}_j\right]=\left[\begin{array}{c}{\left({r}_1\right)}_{j-\frac{1}{2}}\\ {}{\left({r}_2\right)}_{j-\frac{1}{2}}\\ {}{\left({r}_3\right)}_{j-\frac{1}{2}}\\ {}{\left({r}_4\right)}_{j-\frac{1}{2}}\\ {}{\left({r}_5\right)}_{j-\frac{1}{2}}\\ {}{\left({r}_6\right)}_{j-\frac{1}{2}}\\ {}{\left({r}_7\right)}_{j-\frac{1}{2}}\end{array}\right]\;1\le j\le J\end{array}} $$
(31)

To solve Eq. (34), we assume that A is nonsingular and can be factored into

$$ \left[A\right]=\left[L\right]\left[U\right] $$
(32)

where

$$ A=\left[\begin{array}{cccccc}\left[{\alpha}_1\right]& & & & & \\ {}\left[{B}_2\right]& \left[{\alpha}_1\right]& & & & \\ {}& .& .& & & \\ {}& & .& .& & \\ {}& & & .& \left[{\alpha}_{j-1}\right]& \\ {}& & & & \left[{B}_J\right]& \left[{\alpha}_J\right]\end{array}\right] $$

and

$$ U=\left[\begin{array}{cccccc}\left[I\right]& \left[{\Gamma}_1\right]& & & & \\ {}& \left[I\right]& \left[{\Gamma}_2\right]& & & \\ {}& & .& .& & \\ {}& & & .& .& \\ {}& & & & \left[I\right]& \left[{\Gamma}_{j-1}\right]\\ {}& & & & & \left[I\right]\end{array}\right] $$

where [I] is the identity matrix of order 7 and [αi] and [Γi] are 7 × 7 matrices, in which elements are determined by the following equations:

$$ \left[{\alpha}_1\right]=\left[{A}_1\right], $$
(33)
$$ \left[{A}_1\right]\left[{\Gamma}_1\right]=\left[{C}_1\right], $$
(34)
$$ \left[{\alpha}_j\right]=\left[{A}_j\right]-\left[{B}_j\right]\left[{\Gamma}_{j-1}\right],j=2,3,\dots, J $$
(35)
$$ \left[{\alpha}_j\right]\left[{\Gamma}_j\right]=\left[{C}_j\right]\kern0.5em j=2,3,\dots, J-1 $$
(36)

Equation (30) can now be substituted into Eq. (34) and we get

$$ \left[L\right]\left[U\right]\left[\delta \right]=\left[r\right] $$
(37)

If we define

$$ \left[U\right]\left[\delta \right]=\left[W\right] $$
(38)

Equation (37) becomes

$$ \left[L\right]\left[W\right]=\left[r\right] $$
(39)

where

$$ W=\left[\begin{array}{c}\left[{W}_1\right]\\ {}\left[{W}_2\right]\\ {}.\\ {}.\\ {}.\\ {}\left[{W}_{j-1}\right]\\ {}\left[{W}_J\right]\end{array}\right], $$

and the [ Wj] are 7 × 1 column matrices. The element W can be solved from Eq. (39):

$$ \left[{\alpha}_1\right]\left[{W}_1\right]=\left[{r}_1\right] $$
(40)
$$ \left[{\alpha}_j\right]\left[{W}_j\right]=\left[{r}_j\right]-\left[{B}_j\right]\left[{W}_{j-1}\right]\kern0.5em 2\le j\le J $$
(41)

The step in which Γj, αj, and Wj are calculated is usually referred to as the forward sweep. Once the element of W is found, Eq. (38) then gives the solution in the so-called backward sweep, in which the elements are obtained by the following relations:

$$ \left[{\delta}_J\right]=\left[{W}_J\right] $$
(42)
$$ \left[{\delta}_J\right]=\left[{W}_J\right]-\left[{\Gamma}_J\right]\left[{\delta}_{j+1}\right]\kern0.5em 1\le j\le J-1 $$
(43)

These calculations are repeated until some convergence criterion is satisfied and calculations are stopped when

$$ \left|\delta {\mathrm{v}}_0^{(i)}\right|\le {\varepsilon}_1 $$
(44)

where ε1is a small arranged value.

Its exactness and heftiness have been confirmed by different investigators. We have compared our results with investigators Shravani Ittedi et al. [49] to further check on the exactness of our numerical computations and have found to be in an admirable agreement.

Results and discussions

This paper analyzed the effects of slip and chemical reaction on upper-convected Maxwell fluid flow over a stretching sheet. Transfigured governing Eqs. (1113) with the boundary conditions (14) and (15) are coupled non-linear differential equations. Thus, it is impossible to solve directly with the analytical method. Therefore, to solve this coupled non-linear differential equations, we use implicit finite difference (Keller-box) method by MatLabR2013a software. For various values of effective governing parameters such as velocity ratio S, the suction-injection parameter, E velocity ratio, Deborah number β, magnetic field parameter M, velocity slip parameter λ, thermal slip parameter δ, solutal slip parameter γ, thermal radiation parameter M, Prandtl number Pr, Brownian motion parameter Nb, thermophoresis parameter Nt, Lewis number Le, chemical reaction parameter h, and heat source parameter Q, the numerical solutions of velocity, temperature, and concentration are obtained with step size ∆η = 0.1. Unless otherwise the parameters are specified, the value parameters are M = 1.0, β = 0.1, λ = 0.1, δ = 0.1, γ = 0.1,  Pr  = 2.0, Le = 2.0, Nb = 0.1, S = 0.1, Nt = 0.1, R = 0.1, h = 0.1, Q = 0.0, E = 0.1. For this study, the range of parameters is 1 ≤ M ≤ 4, 0 ≤ β ≤ 1, 0 ≤ S ≤ 1.5, 0 ≤ h ≤ 1, 0.1 ≤ E ≤ 1.0, 0.1 ≤ Nb ≤ 1.5, 0.1 ≤ Nt ≤ 1.0, −1.5 ≤ Q ≤ 0.0, 2 ≤ Le ≤ 15, 0 ≤ R ≤ 1.0, 0.0 ≤ λ ≤ 1.5, 0 ≤ δ ≤ 1.5, 0 ≤ γ ≤ 1.0

The comparison of the variation of the skin coefficient f ′  ′ (0) for different values of magnetic field parameter M with respect to another study is presented in Table 1. The values show that our result is in admirable agreement with the results given by researchers Shravani Ittedi et al. [49] in limiting conditions. Moreover, a comparison of heat and mass transfer rate for different values of δ and λ is made in Table 2 to check the accuracy of the numerical solution with Shravani Ittedi et al. [49] and we have found an admirable agreement with him. Therefore, we are assured that for the analysis of our problem, the numerical method is appropriate.

Table 1 Comparison of skin friction coefficient −f ′  ′ (0) for different values of magnetic field parameter M when E = 0, β = 0, Le = 2.0, Pr  = 2.0, and h = 0.1
Table 2 Comparison of Nusselt number −θ ′ (0) and Sherwood number −ϕ ′ (0) for different values of thermal slip parameter δ and concentration slip parameter γ when E = 0, β = 0, Le = 2.0, and Pr  = 2.0

For different values of S, E, λ , and β , the variation of f ′  ′ (0), −θ ′ (0), and −ϕ ′ (0) is given in Table 3. From the table, when the suction-injection parameter S and Deborah number β increases, we see that skin friction coefficient increases but decreases with an increase of velocity ratio E and velocity slip parameter λ. Moreover, the table shows that the local Nusselt number −θ ′ (0) and the local Sherwood number −ϕ ′ (0) of the flow field increases as the values of S and E and decreases with an upsurge of Deborah number β and velocity slip parameter λ.

Table 3 Calculation of skin friction coefficient −f ′  ′ (0), local Nusselt number −θ ′ (0), and local Sherwood number −ϕ ′ (0) for different values of S, E, λ, and β when Nb = 0.1, Nt = 0.1,  Pr  = 2, Le = 2, R = 0.1, δ = 0.1, γ = 0.1, h = 0.1, and Q = 0.1

The sways of magnetic field parameter on flow velocity, temperature, and concentration are displayed through Figs. 3, 4, and 5. From the figures, we have perceived that as magnetic field increased, the velocity of the fluid decreased; in contrast, the temperature and concentration profiles demonstrated the behavior of increasing. These are because of the magnetic field offerings a retarding body force known as Lorentz force performs transverse to the direction of the practical magnetic field. The boundary layer flow and the thickness of the momentum boundary layer are abridged by this body force. Similarly, because of Lorentz force, a fractional resistive force which opposes the fluid motion, it produces heat. Due to this fact the stronger the magnetic field, the thicker the thermal boundary layer and concentration boundary layer. Figures 6, 7, and 8 reveal the characteristic of velocity, temperature, and concentration profiles with respect to the variation in suction parameter S. As the values of S increase the temperature, velocity and concentration profile results diminish. The variations of chemical reaction parameter on concentration profile are executed in Fig. 9. These figures afford a clear insight that increasing the value of chemical reaction, the concentration in the boundary layer falls. This is due to the fact that negative chemical reaction reduces the concentration boundary layer thickness and raises the mass transfer.

Fig. 3
figure3

Velocity graph for various values of M

Fig. 4
figure4

Temperature graph for various values of M

Fig. 5
figure5

Concentration graph for various values of M

Fig. 6
figure6

Velocity graph for various values of S

Fig. 7
figure7

Temperature graph for various values of S

Fig. 8
figure8

Concentration graph for various values of S

Fig. 9
figure9

Concentration graph for various values of h

The influences of velocity ratio parameter on flow velocity and temperature profiles are revealed through Figs. 10 and 11. As the values of velocity ratio upsurge, the boundary layer thickness rises and the flow has boundary layer structure. The graph of velocity is possible when the free steam velocity is less than or equal to the velocity of stretching sheet. That is when velocity ratio is less than or equal to one. But as the value of velocity ratio parameter increases, thermal boundary layer thickness decreases.

Fig. 10
figure10

Velocity graph for various values of E

Fig. 11
figure11

Temperature graph for various values of E

The influence of Brownian motion parameter on concentration and temperature profiles is depicted through Figs. 12 and 13. From the figures, it can be seen that as the values of Brownian motion parameter rises, the thermal boundary layer thickness increases and at the surface, the temperature gradient demises. But we witnessed an opposite result on the concentration profiles and concentration boundary layer thickness as Brownian motion parameter upsurges. Figures 14 and 15 are devoted to demonstrate the impact of thermophoresis parameter on temperature and concentration profiles. From the figures, it is perceived that when thermophoresis parameter rises, there is an improvement of the thermal and concentration boundary layer thickness. Figure 16 is plotted to examine the effect of heat generation/absorption parameter on temperature profiles. From the figure, it can be seen that the temperature profiles increases as heat generation/absorption parameter upsurges. This is because the thermal state of the fluid rises with increasing the heat generation.

Fig. 12.
figure12

Temperature graph for various values of Nb

Fig. 13
figure13

Concentraion graph for various values of Nb

Fig. 14
figure14

Temperature graph for various values of Nt

Fig. 15
figure15

Concentration graph for various values of Nt

Fig. 16
figure16

Temperature graph for various values of Q

Figures 17 and 18 executed that the influence of Deborah number on velocity and temperature profiles. These figures give a clear perception that the velocity boundary thickness tumbles as the value of Deborah number rises, and the thermal boundary layer thickness growths as the values increase. This is because of the fact that as the values upsurge, the force due to the parameter opposes the velocity flow and declines thermal diffusivity in the boundary layer, which encourages the upswing of temperature. The impact of Lewis number on concentration profiles is illustrated through Fig. 19. From the figure, it can be observed that higher values of Lewis number decrease the concentration graph and the concentration boundary layer thickness. The temperature curves for different values of thermal radiation parameter are depicted in Fig. 20. From the graph, it is possible to observe that as the values of thermal radiation parameter upsurge, the temperature graph and the temperature boundary layer thickness are snowballing.

Fig. 17
figure17

Velocity graph for various values of β

Fig. 18
figure18

Temperature graph for various values of β

Fig. 19
figure19

Concentration graph for various value of Le

Fig. 20
figure20

Temperature graph for various values of R

The influences of velocity slip parameter on velocity, temperature, and concentration profiles are portrayed in Figs. 21, 22, and 23. From the graphs, it can be seen that as the value of velocity slip parameter upsurges, the velocity profiles decreases, but the temperature and concentration profiles are snowballing. This is due to the fact that as velocity slip rises, the velocity of fluid declines because pulling of stretching sheet can transmit the fluid. The temperature and concentration profiles are presented in Figs. 24 and 25 for various values of thermal slip parameter. It is observed that as the values of thermal slip parameter increase, both temperature and concentration profiles decline. A decrement of concentration profiles with an increment of concentration slip parameter is executed through Fig. 26. Basically, this is due to the fact that slip retards the motion of the fluid which indicates a decline in concentration profiles.

Fig. 21
figure21

Velocity graph for various values of λ

Fig. 22
figure22

Temperature graph for various values of λ

Fig. 23
figure23

Concentration graph for various values of λ

Fig. 24
figure24

Temperature graph for various values of δ

Fig. 25
figure25

Concentration graph for various values of δ

Fig. 26
figure26

Concentration graph for various values of γ

Conclusions

This study presents MHD slip effect and stagnation point flow of upper-convected Maxwell fluid on a stretching sheet with chemical reaction. Depending on the governing parameters, velocity ratio, suction-injection parameter, Lewis numbers, Deborah number, magnetic field, Brownian motion parameter, thermophoresis parameter, chemical reactions parameter, thermal radiation parameter, velocity slip parameter, thermal slip parameter, solutal slip parameter, and heat source parameter, a similarity solution is obtained. The clarifications of the present study are precised as:

  1. 1.

    The suction-injection parameter on velocity, concentration, and temperature profiles has shown a reduction.

  2. 2.

    When the magnetic field upsurges, it reduces velocity profiles and it upsurge temperature and concentration profiles.

  3. 3.

    A velocity profile is reduced with rising values of the Deborah number, and temperature profile is increased with an increasing value of the Deborah number.

  4. 4.

    Temperature and concentration profiles are intensified with snowballing values of velocity slip parameter but velocity profiles are decreased.

  5. 5.

    By increasing the values of the Brownian motion, chemical reaction, Lewis number, thermal slip parameter, and solutal slip parameter, there is reduction in concentration profiles.

  6. 6.

    The thickness of thermal boundary layer augmented as thermal radiation parameter R upsurges.

  7. 7.

    Both rate of Nusselt number and rate of mass Sherwood number rise with suction-injection parameter S and velocity ratio E, and declines velocity slips parameter and the Deborah number.

Nomenclature

 B0 Strength of magnetic field

Cf Skin friction coefficient

c Volumetric volume expansion coefficient

Cw Uniform concentration over the surface of the sheet

C Ambient concentration

 DB Brownian diffusion coefficient

hf Heat transfer coefficient

 DT Thermophoresis diffusion coefficient

k Thermal conductivity

f Dimensionless velocity stream function

E Velocity ratio

Le Lewis number

h Chemical reaction parameter

M Magnetic parameter

Pr Prandtl number

Nb Brownian motion parameter

S Suction-injection parameter

Nt Thermophoresis parameter

Rex Local Reynolds number

 Nux Local Nusselt number

R Heat source parameter

 Shx Local Sherwood number

Tf Temperature of a hot fluid

Tw Uniform temperature over the surface of the sheet

 T Ambient temperature

T Temperature of the fluid inside the boundary layer

  ue Free steam velocity

(u, v) Velocity component along x- and y-direction

Greeks symbols

 ϕ Dimensionless concentration function at large values of y

 η Dimensionless similarity variable

 β Deborah number

 ν Kinematic viscosity of the fluid

 θ Dimensionless temperature stream function

ρf Density of the fluid

 μ Dynamic viscosity of the fluid

 σ Electrical conductivity

(ρc)f Heat capacity of the fluid

ψ Stream function

(ρc)p Effective heat capacity of a nanoparticle

τ Parameter defined by \( \frac{{\left(\uprho \mathrm{c}\right)}_{\mathrm{p}}}{{\left(\uprho \mathrm{c}\right)}_{\mathrm{f}}} \)

ϕ Dimensionless concentration steam function

αm Thermal diffusivity

 ϕw Dimensionless concentration function at the surface

λ Velocity slip parameter

δ Thermal slip parameter

γ Solutal slip parameter

Subscripts

ξ Relaxation time parameter of the fluid

∞ Condition at the free stream

w Condition at the surface

Availability of data and materials

All the data and materials used in this paper can be freely available.

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The authors would like to express their deep thanks to the referee for his/her careful reading and many valuable suggestions towards the improvement of the paper.

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Ibrahim, W., Negera, M. MHD slip flow of upper-convected Maxwell nanofluid over a stretching sheet with chemical reaction. J Egypt Math Soc 28, 7 (2020). https://doi.org/10.1186/s42787-019-0057-2

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Keywords

  • Upper-convected Maxwell fluid
  • Nanofluid
  • Chemical reaction
  • MHD
  • Slip effects
  • Stagnation point