Mixed convection flow of nanofluid with Hall and ion-slip effects using spectral relaxation method

Abstract

In this article, the Hall and ion-slip effects on a mixed convection flow of an electrically conducting nanofluid over a stretching sheet in a permeable medium have been discussed. Using the similarity transformations, the partial differential equations corresponding to the momentum, energy, and concentration equations are transformed to a system of nonlinear ordinary differential equations which are solved numerically using a spectral relaxation method (SRM). The effects of significant parameters on the velocities, temperature, and concentration profiles are analyzed graphically. Moreover, the results of the skin friction coefficients, local Nusselt number, and Sherwood number are determined numerically. The results of the analysis showed that the velocity profile in the flow direction increases with an increase in mixed convection parameter λ, Hall parameter β_h, and ion-slip parameter β_i, and it decreases with an increase in the magnetic parameter M. Furthermore, temperature and concentration profiles decrease as the mixed convection parameter λ and buoyancy ratio Nr increase. It is also observed that the skin friction coefficients, local Nusselt number, and Sherwood number increase with an increase in the Hall parameter β_h, mixed convection parameter λ, and buoyancy ratio Nr.

Introduction

Recently, the study of convection and heat transfer in the presence of nanofluid has obtained significant attention because of its wide applications in science, engineering, and industry. Some of these applications are welding equipment, power generating systems, cooling of nuclear reactor, automobile engines, and heat exchanging in electronics devices. The concept of nanofluid was first commenced by Choi [1] to refer to the fluids with suspended nanoparticles (1–100 nm). The suspended nanoparticles assist to enhance the thermal conductivity of the fluid. The effects of various governing parameters on heat and mass transfer of nanofluid flow through different geometries were studied by different scholars ([2,3,4,5,6,7,8,9,10]). Li et al. [11] numerically modeled electrohydrodynamic nanoparticle flow using control volume-based finite element method. Ramzan et al. [12] analyzed the aqueous-based nanofluid flow containing carbon nanotubes past a vertical cone through a porous medium with entropy generation and thermal radiation. Bilal and Ramzan [13] discussed unsteady two-dimensional flow of mixed convection and nonlinear thermal radiation in the presence of water-based carbon nanotubes over the vertically convected stretched sheet embedded in a Darcy-Forchheimer porous media using Saffman’s proposed model for the suspension of fine dust particles in the nanofluid. Lu et al. [14] scrutinized a thin film flow of a nanofluid consisting of single and multi-walled carbon nanotubes with Cattaneo-Christov heat flux and entropy generation. Suleman et al. [15] studied the impacts of Newtonian heating and homogeneous–heterogeneous reactions on the flow of silver-water nanofluid past a nonlinear stretched cylinder with MHD and nonlinear thermal radiation. Suleman et al. [16] extended the work of Suleman et al. [15] by incorporating viscous dissipation, heat generation/absorption and joule heating effects. Madhu et al. [17] studied the boundary layer flow and heat transfer of a power-law non-Newtonian nanofluid past a nonlinearly stretching sheet. Madhu and Kishan [18] numerically investigated MHD mixed convection flow of heat and mass transfer stagnation-point flow of a non-Newtonian power-law nanofluid towards a stretching surface in the presence of thermal radiation using finite element method. Macha et al. [19] analyzed the MHD boundary layer flow of a viscous, incompressible and electrically conducting non-Newtonian nanofluid obeying power-law model over a nonlinear stretching sheet under the effect of thermal radiation with heat source/sink. Alebraheem and Ramzan [20] studied the heat and mass transfer of Casson nanofluid flow containing gyrotactic microorganisms past a swirling cylinder. Farooq et al. [21] described the static wedge flow of non-viscous Maxwell nanofluid using BVPh2.0 solver. Farooq et al. [22] analyzed the effects of thermophoretic and Brownian motion on MHD non-Newtonian Maxwell fluid with nanomaterials over an exponentially stretching surface using Buongiorno model. Macha et al. [23] investigated the MHD boundary layer flow of Sisko nanofluid flow past a wedge.

Mixed convection flows got a significant consideration because of its numerous technological and industrial applications. Aman and Ishak [24] numerically considered the effects of buoyancy force and convective surface boundary condition on mixed convection flow near a resistant vertical plate using shooting technique. Mahanthesh et al. [25] numerically investigated the influences of buoyancy force, nanoparticles, chemical reaction, and heat source/sink on combined forced and free convection flow of an electrically conducting water-based Cu-nanofluid past a moving/stationary vertical plate using Laplace transform method. The impacts of different governing parameters on mixed convection flow of nanofluid past stretching surfaces were studied by ([26,27,28]). Most recently, Khan and Rasheed [29] scrutinized magnetohydrodynamic (MHD) combined natural and forced convection flow of the Maxwell nanofluid with magnetic field and buoyancy force effects.

The Hall and ion-slip impacts are important under the existence of strong magnetic field because of the significant effect they have on the magnitude and direction of the electric current density and the magnetic force term. Hence, in numerous physical circumstances, it is necessary to incorporate the impact of Hall current and ion-slip terms in the MHD equations. The impacts of Hall and ion-slip parameters on combined forced and free convection flow of an electrically conducting Casson fluid were numerically analyzed using homotopy analysis method by Reddy et al. [30]. Moreover, Reddy et al. [31] analyzed the effects of Hall and ion-slip effects on combined forced and free convection flow of an electrically conducting Newtonian fluid using Adomian decomposition method. The Hall and ion-slip effects on mixed convection flow of an electrically conducting nanofluid through different geometries were discussed by ([32,33,34,35]). Motsa and Shateyi [36] numerically analyzed the effects of Hall and ion-slip currents on magnetomicropolar fluid flow with suction/injection using the spectral linearization method. Moreover, using shooting method, Bilal et al. [37] extended the work of Motsa and Shateyi [36] with the inclusion of nanoparticles neglecting the effect of chemical reaction. The Hall and ion-slip effects on unsteady MHD Couette flow of an electrically conducting fluid has been analytically discussed using Laplace transform method by Ghara et al. [38].

Ali et al. [39] have studied the effect of Hall current on MHD mixed convection flow over a stretched vertical flat plate. However, they did not consider ion-slip and permeability effects. Thus, the objective of the present study is to investigate the effects of Hall and ion-slip parameter on mixed convection flow of nanofluid past a porous stretching sheet. Therefore, the inclusion of the effects of ion-slip and permeability parameters in the presence of mass transfer makes this study a novel one. The governing equations are solved numerically using the SRM. This numerical method is chosen since its accuracy is high and the MATLAB program for the problem is manageable (see: Motsa and Makukula [40], Shateyi and Marewo [41], Shateyi and Marewo [42]). The fluid model considered in this study is solved numerically and hence will present an approximated solution. The other limitation of this flow problem is that the nanoparticle volume fraction is neglected.

Mathematical formulation

The study considered a laminar mixed convection flow of a viscous, incompressible, and electrically conducting nanofluid past a vertical linearly stretching sheet through a permeable medium with velocity V_w(x) in the direction of x-axis. A strong uniform magnetic field M₀ is applied in the way along y-axis as shown in Fig. 1. As a result of this strong magnetic field force, the electrically conducting fluid has the Hall and ion-slip effects which generate a cross-flow in the z-direction, and thus, the flow becomes three dimensional. The induced magnetic field can be ignored in contrast with the applied magnetic field on the presumption that the flow is steady and the magnetic Reynolds number is very small. The mixed convection flow is caused by the buoyancy forces. Further, all the fluid properties are understood to be constant except for density variations in the buoyancy force term. By applying the Boussinesq approximations with the above assumptions, the governing equations of continuity, momentum, energy, and concentration equations for the considered flow problem are ([35, 37, 38]):

$$ \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0 $$

(1)

$$ u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\upupsilon \frac{\partial^2u}{\partial {y}^2}-\frac{\sigma {M}_0^2}{\rho \left({\beta}^2+{\beta}_h^2\right)}\ \left(\beta u+{\beta}_h\ w\right)+\mathrm{g}\left[{\mathrm{B}}_{\mathrm{T}}\left(\mathrm{T}-{\mathrm{T}}_{\infty}\right)+{\mathrm{B}}_{\mathrm{C}}\left(\Phi -{\Phi}_{\infty}\right)\right]-\frac{\upnu}{\kappa }u $$

(2)

$$ u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}=\upupsilon \frac{\partial^2w}{\partial {y}^2}+\frac{\sigma {M}_0^2}{\rho \left({\beta}^2+{\beta}_h^2\right)}\left({\beta}_hu-\beta w\right)-\frac{\upnu}{\kappa }w $$

(3)

$$ u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha \frac{\partial^2T}{\partial {y}^2}+\gamma \left[{D}_B\frac{\partial T}{\partial y}\frac{\mathrm{\partial \Phi }}{\partial y}+\frac{D_T}{T_{\infty }}{\left(\frac{\partial T}{\partial y}\right)}^2\right] $$

(4)

$$ u\frac{\mathrm{\partial \Phi }}{\partial x}+v\frac{\mathrm{\partial \Phi }}{\partial y}={D}_B\frac{\partial^2\Phi}{\partial {y}^2}+\frac{D_T}{T_{\infty }}\frac{\partial^2T}{\partial {y}^2} $$

(5)

Fig. 1

Physical representation of the flow problem

where (u, v, w) are the fluid velocity components along the (x, y, z) coordinates, μ is the dynamic viscosity, ρ is the density, ν is the kinematic viscosity, g is the acceleration due to gravity, T and Φ are the fluid temperature and concentration, respectively, T_∞ and ϕ_∞ are the ambient temperature and concentration, respectively, B_T is the coefficient of thermal expansion, B_C is the solutal coefficient of expansion, σ is the electrical conductivity, M₀ is the constant magnetic field applied normal to the surface, β_h is the Hall parameter, β_i is the ion-slip parameter, β = 1 + β_h β_i is a constant, κ is the permeability of the porous medium, α is the thermal diffusivity, (ρΦ)_p is the heat capacity of the nanoparticle, γ is the ratio of the effective heat capacity of nanoparticle material to the heat capacity of the fluid, D_B the Brownian diffusion coefficient, and D_T the thermophoresis diffusion coefficient.

The corresponding boundary conditions are:

$$ {\displaystyle \begin{array}{c}u={V}_w(x)= Ax,v=w=0,\\ {}T={T}_w(x)={T}_{\infty }+ Bx,\\ {}\Phi ={\Phi}_w(x)={\Phi}_{\infty }+ Cx\kern1em at\ y=0,\\ {}\kern0.50em u\to 0,w\to 0,T\to {T}_{\infty },\\ {}\ \Phi \to {\Phi}_{\infty },\kern1.5em as\ y\to \infty, \end{array}} $$

(6)

where A, C (>0) and B are constants, and T_w(x) and Φ_w(x) are variable surface temperature and concentration, respectively. Moreover, B > 0 denotes heated plate and B < 0 denotes cooled plate.

Let us establish the similarity transformations ([39, 43])

$$ {\displaystyle \begin{array}{cc}\eta ={\left(\frac{A}{\nu}\right)}^{1/2}y,& \psi ={\left( A\nu \right)}^{1/2} xf\left(\eta \right),\\ {}\ w= AxG\left(\eta \right),&\ \theta \left(\eta \right)=\frac{T-{T}_{\infty }}{T_w-{T}_{\infty }},\\ {}\varphi \left(\eta \right)=\frac{\Phi -{\Phi}_{\infty }}{\Phi_w-{\Phi}_{\infty }}\kern0.5em & \end{array}} $$

(7)

to obtain similarity solutions of Eqs. (1)–(5) subject to the boundary conditions (6), where the stream function ψ(x, y) is defined as:

$$ u=\frac{\partial \psi }{\partial y},v=-\frac{\partial \psi }{\partial x}. $$

(8)

Substituting Eqs. (7) into Eqs. (2)–(5), the following ordinary differential equations are obtained:

$$ {f}^{{\prime\prime\prime} }+f{f}^{{\prime\prime} }-{\left({f}^{\prime}\right)}^2-\frac{M}{\left({\beta}^2+{\beta}_h^2\right)}\left(\beta {f}^{\prime }+{\beta}_hG\right)+\lambda \left(\theta + Nr\varphi \right)-K{f}^{\prime }=0 $$

(9)

$$ {G}^{{\prime\prime} }+f{G}^{\prime }-{f}^{\prime }G+\frac{M}{\left({\beta}^2+{\beta}_h^2\right)}\left({\beta}_h{f}^{\prime }-\beta G\right)- KG=0 $$

(10)

$$ {\theta}^{{\prime\prime} }+\mathit{\Pr}\left(f{\theta}^{\prime }-{f}^{\prime}\theta + Nb{\theta}^{\prime }{\varphi}^{\prime }+ Nt{\theta^{\prime}}^2\right)=0 $$

(11)

$$ {\varphi}^{{\prime\prime} }+ PrLe\left(f{\varphi}^{\prime }-f^{\prime}\varphi \right)+\frac{Nt}{Nb}{\theta}^{{\prime\prime} }=0 $$

(12)

with the transformed boundary conditions:

$$ {\displaystyle \begin{array}{c}f(0)=0,\kern0.73em \ {f}^{\hbox{'}}(0)=1,\kern0.73em \ G(0)=0,\\ {}\ \theta (0)=1,\kern0.73em \ \varphi (0)=1,\\ {}{f}^{\hbox{'}}\to 0,G\to 0,\theta \to 0,\\ {}\ \varphi \to 0, as\ \ \ \eta \to \infty, \end{array}} $$

(13)

where $M=\sigma {M}_0^2/\rho A$ is the magnetic parameter, $\lambda ={Gr}_x/{\mathit{\operatorname{Re}}}_x^2$ is the constant mixed convection parameter where Gr_x = gB_T(T_w − T_∞)x³/ν² is the local Grashof number, and Re_x = Ax²/ν is the local Reynolds number, Nr = B_C(Φ_w − Φ_∞)/B_T(T_w − T_∞) is the buoyancy ratio, K = ν/Aκ is the permeability parameter, Pr = ν/α is the Prandtl number, Nb = γD_B(Φ_w − Φ_∞)/ν is the Brownian motion parameter, Nt = γD_T(T_w − T_∞)/νT_∞ is the thermophoresis parameter, Le = α/D_B is the Lewis number, and γ = (ρΦ)_p/(ρΦ)_f is the ratio between the heat capacity of the nanoparticle material and heat capacity of the fluid.

The physical quantities of interest are the skin friction coefficients C_fx and C_fz, the local Nusselt number Nu_x, and the Sherwood number Sh_xwhich are defined as follows:

$$ {C}_{fx}=\frac{2{\tau}_{wx}}{\rho {V}_w^2},\kern0.5em {C}_{fz}=\frac{2{\tau}_{wz}}{\rho {V}_w^2},\kern0.5em {Nu}_x=\frac{x{q}_w}{k\left({T}_W-{T}_{\infty}\right)},\kern0.5em {Sh}_x=\frac{x{\mathrm{j}}_m}{D_B\left({\Phi}_W-{\Phi}_{\infty}\right)} $$

(14)

where τ_wx and τ_wz are the wall shear stresses in the directions of x and z, respectively, q_w is the surface heat flux, and j_m is the surface mass flux, which are given by

$$ {\displaystyle \begin{array}{c}{\tau}_{wx}=\mu \left(\frac{\partial u}{\partial y}\right){\left|{}_{y=0},{\tau}_{wz}=\mu \left(\frac{\partial w}{\partial y}\right)\right|}_{y=0},\\ {}{\left.{q}_w=-k\left(\frac{\partial T}{\partial y}\right)\right|}_{y=0},\\ {}{\left.{\mathrm{j}}_m=-{D}_B\left(\frac{\mathrm{\partial \Phi }}{\partial y}\right)\right|}_{y=0}\end{array}} $$

(15)

From Eqs. (7) and (14), we get

$$ {C}_{fx}=\frac{2}{\sqrt{{\mathit{\operatorname{Re}}}_x}}{f}^{{\prime\prime} }(0),\kern0.5em {C}_{fz}=\frac{2}{\sqrt{{\mathit{\operatorname{Re}}}_x}}{G}^{\prime }(0),\kern0.5em {Nu}_x=-\sqrt{{\mathit{\operatorname{Re}}}_x}{\theta}^{\prime }(0),\kern0.5em {Sh}_x=-\sqrt{{\mathit{\operatorname{Re}}}_x}\varphi^{\prime }(0). $$

(16)

Method of solution

The spectral relaxation method (SRM) is used to solve the system of nonlinear ordinary differential Eqs. (9)–(12) subject to the boundary conditions (13). To implement the SRM, the order of the momentum Eq. (9) is reduced to second order introducing the transformation f ′= H so that f ″= H′ and f ‴= H ″. Hence, Eqs. (9)–(12) and the boundary conditions (13) can be transformed as follows:

$$ f^{\prime }=H $$

(17)

$$ {H}^{{\prime\prime} }+f{H}^{\prime }-{(H)}^2-\frac{M}{\left({\beta}^2+{\beta}_h^2\right)}\left(\beta H+{\beta}_hG\right)+\lambda \left(\theta + Nr\varphi \right)- KH=0 $$

(18)

$$ {G}^{{\prime\prime} }+f{G}^{\prime }- HG+\frac{M}{\left({\beta}^2+{\beta}_h^2\right)}\left({\beta}_hH-\beta G\right)- KG=0 $$

(19)

$$ {\theta}^{{\prime\prime} }+\mathit{\Pr}\left(f{\theta}^{\prime }- H\theta + Nb{\theta}^{\prime }{\varphi}^{\prime }+ Nt{\theta^{\prime}}^2\right)=0 $$

(20)

$$ {\varphi}^{{\prime\prime} }+ PrLe\left(f{\varphi}^{\prime }- H\varphi \right)+\frac{Nt}{Nb}{\theta}^{{\prime\prime} }=0 $$

(21)

subject to the boundary conditions

$$ {\displaystyle \begin{array}{c}f(0)=0,H(0)=1,G(0)=0,\\ {}\theta (0)=1,\kern0.97em \ \varphi (0)=1,\\ {}H\to 0,G\to 0,\theta \to 0,\varphi \to 0\\ {} as\ \ \ \eta \to \infty .\end{array}} $$

(22)

Detail rules of the SRM can be found in ([40,41,42]). The resulting iteration scheme in SRM for Eqs. (17)–(22) can be written as follows:

$$ {f}_{r+1}^{\prime }={H}_r $$

(23)

$$ {H}_{r+1}^{{\prime\prime} }+{f}_{r+1}{H}_{r+1}^{\prime }-{H}_r^2-\frac{M}{\left({\beta}^2+{\beta}_h^2\right)}\left(\beta {H}_{r+1}+{\beta}_h{G}_r\right)+\lambda \left({\theta}_r+ Nr{\varphi}_r\right)-K{H}_{r+1}=0 $$

(24)

$$ {G}_{r+1}^{{\prime\prime} }+{f}_{r+1}{G}_{r+1}^{\prime }-{H}_{r+1}{G}_{r+1}+\frac{M}{\left({\beta}^2+{\beta}_h^2\right)}\left({\beta}_h{H}_{r+1}-\beta {G}_{r+1}\right)-K{G}_{r+1}=0 $$

(25)

$$ {\theta}_{r+1}^{{\prime\prime} }+\Pr \left({f}_{r+1}{\theta}_{r+1}^{\prime }-{H}_{r+1}{\theta}_{r+1}+ Nb{\theta}_{r+1}^{\prime }{\varphi}_r^{\prime }+ Nt{\theta_r^{\prime}}^2\right)=0 $$

(26)

$$ {\upvarphi}_{r+1}^{{\prime\prime} }+ PrLe\left({f}_{r+1}{\upvarphi}_{r+1}^{\prime }-{H}_{r+1}{\upvarphi}_{r+1}\right)+\frac{Nt}{Nb}{\theta}_{r+1}^{{\prime\prime} }=0 $$

(27)

subject to the boundary conditions

$$ {\displaystyle \begin{array}{c}{f}_{r+1}(0)=0,{H}_{r+1}(0)=1,{G}_{r+1}(0)=0,\\ {}\ {\theta}_{r+1}(0)=1,\kern1.58em {\varphi}_{r+1}(0)=1,\\ {}{H}_{r+1}\to 0,\kern0.72em {G}_{r+1}\to 0,{\theta}_{r+1}\to 0,\\ {}{\varphi}_{r+1}\to 0\kern0.50em \ as\ \ \ \eta \to \infty .\end{array}} $$

(28)

In Eqs. (23)–(28), r + 1 and r denote the current and previous iteration, respectively. Implementing Chebyshev differentiation (Trefethen [44]) on iterative scheme defined by Eqs. (23)–(27), we obtain the following discrete form:

$$ {\boldsymbol{B}}_{\mathbf{1}}{\mathbf{f}}_{r+1}={\boldsymbol{R}}_{\mathbf{1}},\kern2em {f}_{r+1}\left({\zeta}_N\right)=0 $$

(29)

$$ {\boldsymbol{B}}_{\mathbf{2}}{\mathbf{H}}_{r+1}={\boldsymbol{R}}_{\mathbf{2}},\kern1.75em {H}_{r+1}\left({\zeta}_N\right)=1,\kern0.75em {H}_{r+1}\left({\zeta}_0\right)=0 $$

(30)

$$ {\boldsymbol{B}}_{\mathbf{3}}{\mathbf{G}}_{\mathrm{r}+1}={\mathbf{R}}_{\mathbf{3}},\kern1.75em {G}_{r+1}\left({\zeta}_N\right)=0,\kern0.75em {G}_{r+1}\left({\zeta}_0\right)=0 $$

(31)

$$ {\boldsymbol{B}}_{\mathbf{4}}{\boldsymbol{\uptheta}}_{\mathrm{r}+1}={\mathbf{R}}_{\mathbf{4}},\kern1.75em {\theta}_{r+1}\left({\zeta}_N\right)=1,{\theta}_{r+1}\left({\zeta}_0\right)=0 $$

(32)

$$ {\boldsymbol{B}}_{\mathbf{5}}\boldsymbol{\upvarphi} \_\mathrm{r}+1={\mathbf{R}}_{\mathbf{5}},\kern1.5em {\varphi}_{r+1}\left({\zeta}_N\right)=1,\kern0.5em {\varphi}_{r+1}\left({\zeta}_0\right)=0 $$

(33)

where

$$ {\boldsymbol{B}}_{\mathbf{1}}=\boldsymbol{D},\kern4.25em \boldsymbol{R}\_\mathbf{1}={H}_r $$

(34)

$$ {\displaystyle \begin{array}{c}{B}_2={D}^2+\operatorname{diag}\left({f}_{r+1}\right)D-\left(\frac{M\beta}{\beta^2+{\beta}_h^2}+K\right)\mathbf{I},\\ {}\mathbf{R}\_2={H}_r^2+\frac{M{\beta}_h}{\left({\beta}^2+{\beta}_h^2\right)}{G}_r-\lambda \left({\theta}_r+ Nr{\varphi}_r\right)\end{array}} $$

(35)

$$ {\displaystyle \begin{array}{l}{B}_3={D}^2+\operatorname{diag}\left({f}_{r+1}\right)D-\operatorname{diag}\left({H}_{r+1}\right)-\left(\frac{M\beta}{\beta^2+{\beta}_h^2}+K\right)\mathbf{I},\\ {}R\_3=-\frac{M{\beta}_h}{\beta^2+{\beta}_h^2}{H}_{r+1}\end{array}} $$

(36)

$$ {\displaystyle \begin{array}{l}{B}_4={D}^2+\mathit{\Pr}\operatorname{diag}\left({f}_{r+1}+ Nb{\varphi}_r^{\hbox{'}}\right)D-\mathit{\Pr}\operatorname{diag}\left({H}_{r+1}\right),\\ {}R\_4=- PrNt{\theta_r^{\hbox{'}}}^2\end{array}} $$

(37)

$$ {\displaystyle \begin{array}{l}{B}_5={D}^2+ Le\Big(\operatorname{diag}\left({f}_{r+1}\right)D- Le\operatorname{diag}\left({H}_{r+1}\right),\\ {}R\_5=-\frac{Nt}{Nb}{\theta}_{r+1}^{\hbox{'}\hbox{'}}\end{array}} $$

(38)

where D = 2D/η_∞, D is the Chebyshev differentiation matrix, and I is the identity matrix of size (N + 1) × (N + 1). The decoupled Eqs. (29)–(33) can be solved independently by choosing a proper initial guesses:

$$ {\displaystyle \begin{array}{c}{f}_0\left(\eta \right)=1-\exp \left(-\eta \right),\\ {}\kern0.5em {G}_0\left(\eta \right)=\eta \exp \left(-\eta \right),\\ {}\kern0.50em {\theta}_0\left(\eta \right)=\varphi \_0\left(\eta \right)=\exp \left(-\eta \right).\end{array}} $$

(39)

Results and discussions

The nonlinear ordinary differential Eqs. (9) to (12) with respect to the boundary conditions (13) are solved numerically using SRM. The numerical solutions are obtained for velocities, temperature, and concentration profiles for different values of governing parameters. The graphical representations of the numerical results are shown in Figs. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22 and 23. Unless otherwise stated, the numerical values for the parameters are taken to be fixed as M = β_h = λ = Nr =  Pr  = Le = 1.0, β_i = 0.1, K = 0.5,   Nb = 0.8, and Nt = 0.2.

Fig. 2

Graph of velocity for various values of M

Fig. 3

Graph of cross flow velocity for various values of M

Fig. 4

Graph of temperature for various values of M

Fig. 5

Graph of concentration for various values of M

Fig. 6

Graph of velocity for various values of β_h

Fig. 7

Graph of cross flow velocity for various values of β_h

Fig. 8

Graph of temperature for various values of β_h

Fig. 9

Graph of concentration for various values of β_h

Fig. 10

Graph of velocity for various values of β_i

Fig. 11

Graph of cross flow velocity for various values of β_i

Fig. 12

Graph of velocity for various values of λ

Fig. 13

Graph of temperature for various values of λ

Fig. 14

Graph of concentration for various values of λ

Fig. 15

Graph of velocity for various values of Nr

Fig. 16

Graph of temperature for various values of Nr

Fig. 17

Graph of concentration for various values of Nr

Fig. 18

Graph of cross flow profiles for various values of K

Fig. 19

Graph of temperature for various values of K

Fig. 20

Graph of concentration for various values of K

Fig. 21

Graph of cross flow velocity for various values of Pr

Fig 22

Graph of velocity for various values of Nt

Fig. 23

Graph of cross flow velocity for various values of Nt

The ranges of parameters used in the figures are 0.1 ≤ M ≤ 2.0, 0.2 ≤ β_h ≤ 1.0, 0.0 ≤ β_i ≤ 1.5, 0.2 ≤ λ ≤ 1.0, 0.0 ≤ Nr ≤ 1.0, 0.0 ≤ K ≤ 1.0, 0.71 ≤  Pr  ≤ 1.5, and 0.0 ≤ Nt ≤ 1.0. For the authentication of the numerical method used, the results were compared with the previously obtained results for various values of parameters and it indicates an excellent accord as shown in Tables 1 and 2.

Table 1 Comparison of −θ ′ (0) for various values of Pr when M = β_h = β_i = λ = Nr = K = Nb = Nt = Le = φ = 0

Table 2 Comparison of −f″(0) and − θ′(0) for different values of M with β_h = 0, 2, λ = Pr  = 1 and β_i = Nr = K = Nb = Nt = Le = φ = 0

From Table 3, it is observed that the skin friction coefficient −f ″ (0) in the x-direction increases with the increasing values of β_h, β_i, λ, Nr, and Nt, and it decreases for increasing values of M and Nb. It is also noticed that the skin friction coefficient G′ (0) in the z-direction increases as M, β_h, λ, Nr, and Nt increase, and it decreases by increasing β_i and Nb. The table also shows that the local Nusselt number −θ′ (0) increases with an increase in β_h, β_i, λ, and Nr, and it decreases with an increase in M, Nb, and Nt. Furthermore, the local Sherwood number −φ′(0) increases as β_h, β_i, λ, Nr, and Nb increase and decreases as M and Nt increase.

Table 3 Numerical values of the skin friction coefficients −f″(0) and G′(0), local Nusselt number −θ′ (0), and local Sherwood number −φ′ (0) for various values of M, β_h, β_i, λ, Nr, Nb, and Nt when K = 0.5,  Pr  = 1, and Le = 1 are fixed

Figures 2, 3, 4 and 5 show the effects of magnetic parameter M on the velocity f ’ (η), cross flow velocity G(η), temperature θ(η), and concentration φ(η) profiles, respectively. The velocity profile f ’ (η) decreases with an increase in the values of M, whereas the cross flow velocity G(η), temperature θ(η) and concentration φ(η) profiles increase as M increases. As M increases, a drag force, called Lorentz force increases. Since this force opposes the flow of nanofluid, velocity in the flow direction decreases. Moreover, since an electrically conducting nanofluid with the strong magnetic field in the direction orthogonal to the flow are considered, an increase in M increases the force in the z-direction which results in an increase in the cross flow velocity profile G(η).

Figures 6, 7, 8 and 9 illustrate the impacts of the Hall parameter β_h on the velocity f ’ (η), cross flow velocity G(η), temperature θ(η), and concentration φ(η) profiles, respectively. It is observed that both the velocity f ’ (η)) and cross flow velocity G(η) profiles increase as β_h increases. But, the temperature and concentration profiles decrease with an increase in β_h as shown in Figs. 8 and 9. This is because the enclosure of Hall parameter decreases the resistive force caused by the magnetic field due to its effect of reducing the effective conductivity. Hence, the velocity component increases as the Hall parameter increases.

The influences of the ion-slip parameter β_i on the velocity f ′ (η) and cross flow velocity G(η) profiles are depicted in Figs. 10 and 11, respectively. The effective conductivity increases with an increase in ion-slip parameter, consecutively damping force decreases on the velocity component in the direction of the flow, and hence, the velocity component increases in the flow direction. Thus, Fig. 10 shows that the velocity profile f ′ (η) increases insignificantly as β_i increases. Conversely, in Fig. 11, a rise in β_i leads to a decline in the cross flow velocity profile G(η).

Figures 12, 13 and 14 disclose the impacts of mixed convection parameter λ on the velocity f′ (η), temperature θ(η), and concentration φ(η) profiles. It is deducted that as λ enhances, the velocity profile f′ (η) rises and conversely, the temperature θ(η) and concentration φ(η) profiles diminish. This is due to the fact that a rise in λ (>0) accelerates the fluid’s flow for g > 0 and hence results in an increase in fluid’s velocity. Furthermore, fluids having larger value of λ have lower thermal diffusivity which reduces the energy capability and the thermal boundary layer thickness.

The effects of the buoyancy ratio Nr on the velocity f ′ (η), temperature θ(η), and the concentration φ(η) profiles are shown in Figs. 15, 16 and 17, respectively. Figure 15 reveals that the velocity profile f ′ (η) increases as Nr increases, while in Figs. 16 and 17, the temperature and concentration φ(η) profiles decrease as Nr increases. The fact is that a rise in Nr decreases fluid’s viscosity. Hence, the result follows.

Figures 18, 19 and 20 demonstrate the effects of the permeability parameter K on the cross flow velocity G(η), temperature θ(η), and concentration φ(η) profiles, respectively. From Fig. 18, it is noticed that as K rises, the cross flow velocity profile diminishes. A rise in resistance against the fluid flow is examined by increasing the thickness of permeable medium which results in a decrease in fluid velocity. In addition, a rise in permeability parameter enhances the thickness of the boundary layer of temperature and concentration. Thus, the temperature and concentration profiles enhance with an increase in K as shown in the Figs. 19 and 20.

Figure 21 divulges the effects of the prandtl number Pr on the cross flow velocity profile G(η). Since fluids having a larger Pr have lower thermal diffusivity, the cross flow velocity profile G(η) decreases as the values of Pr increases.

The influence of the thermophoresis parameter Nt on the velocity f ′ (η) and cross flow velocity G(η) profiles are depicted in Figs. 22 and 23, respectively. As it is observed, both flow velocity and cross flow velocity profiles increase as Nt increases. This is for the reason that the temperature gradient generates a thermophoresis force, and it creates a fast flow away from the stretching sheet. Hence, the velocity profiles increase with an increase in Nt.

Conclusions

A numerical study of a steady mixed convection flow of an electrically conducting nanofluid over a linearly stretching sheet in the presence of Hall and ion-slip effects has been carried out. With the help of similarity transformations the governing partial differential equations for momentum, energy, and concentration equations were reduced to couple nonlinear ordinary differential equations which were then solved numerically using spectral relaxation method. The effects of different governing parameters on the velocities, temperature, and concentration profiles are studied and revealed the following results:

❖ The velocity profile f ′ (η) increases with an increase in β_h, β_i, λ, Nr, and Nt and decreases with an increase in M.

❖ As M, β_h, and Nt increase, the cross flow velocity profile G(η) increases.

❖ As β_i, K, and Pr increase, the cross flow velocity profile G(η) decreases.

❖ The temperature profile θ(η) increases as M and K increase and decreases as β_h, λ, and Nr increase.

❖ The concentration profile φ(η) increases as M and K increase and decreases with an increase in β_h, λ, and Nr.

Nomenclature

A, B, C Constants

x, y, z Cartesian coordinates

u, v, w Velocity components

V_w(x) Velocity of the stretching sheet

M₀ Constant magnetic field strength

g Acceleration due to gravity

T Fluid temperature

T_w Surface temperature

T_∞ Ambient temperature

Φ Concentration of fluid

Φ_w Surface concentration

Φ_∞ Ambient concentration

μ Dynamic viscosity of the fluid

ρ Density of fluid

ν Kinematic viscosity of the fluid

λ Mixed convection parameter

α Thermal diffusivity

σ Electrical conductivity

β_h Hall parameter

β_i Ion-slip parameter

B_T Solutal expansion coefficient

B_C Thermal expansion coefficient

κ Permeability of porous medium

η Dimensionless similarity variable

ψ Stream function

θ Dimensionless temperature

φ Dimensionless concentration

D_B Brownian diffusion coefficient

D_T Thermophoresis diffusion coefficient

M Magnetic parameter

Gr_x Local Grashof number

Re_x Local Reynolds number

Nr Buoyancy ratio

K Permeability parameter

Pr Prandtl number

Le Lewis number

Nb Brownian motion parameter

Nt Thermophoresis parameter

f, G Dimensionless stream functions

(ρΦ)_p Effective heat capacity of a nanoparticle

(ρΦ)_f Heat capacity of the fluid

τ_wx Wall shear stresses in the x-direction

τ_wz Wall shear stresses in the z-direction

C_fx Skin friction coefficient in x−direction

C_fz Skin friction coefficient in z−direction

Nu_x Local Nusselt number

Sh_x Local Sherwood number

q_w Surface heat flux

j_m Surface mass flux

Subscripts

f Fluid

p Nanoparticle

w Condition at the surface

∞ Ambient condition

Superscripts

' Differentiation w. r. t. η