OPEN ACCESS
In this evaluation, a numerical exploration is inscribed on MHD unsteady heat and mass transfer of Casson nanofluid flow where variable thermal conductivity, radiation and heat absorption are counterfeited. The model is implemented for the nanofluid which annexes the impression of thermophoresis and Brownian motion. The governing models are metamorphosed as a dimensionless silhouette by the renovation rule of mathematics and the procured dimensionless couple of partial differential equations (PDEs) are elucidated by utilizing explicit finite difference method (EFDM). The numerical aggregates are enumerated for the miscellaneous dimensionless parameters on velocity, temperature and concentration profiles along with the skin friction, Nusselt number and Sherwood number with distinct time interims. ForTran programing language is manipulated as the predominated software for executing the numerical values. Also, by wielding the initial boundary conditions U=T=C=0, DX=0.83 and DY=0.50 for Dt=0.0005, convergence criteria are exhibited with Sc³ 0.18 and Le³0.20. At the end, the prevailed culminations are illustrated and perused after the stability convergence test (SCT) by manoeuvring graphics software tecplot9 and streamlines and isotherms are delineated.
casson nanofluid, EFDM, radiation, chemical reaction, MHD, porous medium
Casson fluid is a nonNewtonian shear thinning fluid appending with infinite viscosity at zero shear rates. Pragmatic implementations of Casson fluid comprise in the sector of petroleum drilling, polymer engineering, certain separation processes, manufacturing of foods and paper. In addition, biological, chemical, medical and engineering applications are vitally pivotal affiliated to the Casson fluid. Distinct inquisitors have implemented their factfinding pursuit in this direction enlightening the consequences of numerous parameters on the flow of the Casson fluid. Unsteady Casson nanofluid flow over a stretching sheet have been appraised by Oyelakin et al. [1]. Kataria and Patel [2] have scrutinized the Soret and heat generation sequels on MHD Casson fluid flow. The repercussions on magnetic field in squeezing flow of a Casson fluid have been probed by Ahmed et al. [3]. Furthermore, Mahanta et al. [4] perused the 3D Casson fluid flow. In very recent time, Afikuzzaman et al. [5]; Biswas et al. [6]; Afikuzzaman et al. [7] and Wahiduzzaman et al. [8] have numerically evaluated unsteady MHD Casson fluid flow. Also, Casson nanofluid has been contemplated by abundant scrutinisers such as: Biswas et al. [9]; Khalid et al. [10]; Ali et al. [11] and Ghadikolaei et al. [12].
A nanofluid is a fluid, mingle with nanometersized particles typically made of metals, oxides and carbides. In 1995, the abstraction of nanofluids has been initiated by Chai et al. [13], where the breach of nanaparticles in a common base fluids include water, ethylene glycol and oil has proffered. Nanofluids have some substantial properties that assemble them potentially convenient in many exertions in heat transfer, including microelectronics, fuel cells, and pharmaceutical processes. Ganga et al. [14] delineated the MHD flow of Boungiorno model nanofluid over a vertical plate. In addition, optimal interpretation of nonlinear heat and mass transfer in a twolayer flow with nanoEyring–Powell fluid was assembled by Khan et al. [15]. Denouements of magnetic field and radiation absorption on mixed convective Jeffrey nanofluid flow have been dispatched by Biswas et al. [16]. Explicit numerical study of unsteady hydromagnetic mixed convective nanofluid flow elaborated by Beg et al. [17]. Heat transfer of MHD nanofluid in disparate view have been propounded by multifarious assessors analogous as Dogonchi et al. [18]; Dogonchi et al. [19]; Ahmmed et al. [20]; Leekea et al. [21]; Uddin et al. [22]; Bilal et al. [23] and Arani et al. [24].
In the past several decades, the evaluation of magnetohydrodynamic (MHD) mass transfer flow has procured apocalyptic to profuse scholar’s due to its diverse industrial applications such as the cooling of reactors, power generators, electrostatic precipitation, polymer technology petroleum industry, aerodynamics heating, petroleum industry, the operations of MHD accelerators, fluid droplets sprays, agriculture, geophysics, metrology, petrochemical industry and so on. Heat and mass transfer sequels on MHD natural convective flow was implemented by Murthy et al. [25]. Das et al. [26] have instigated magnetohydrodynamic mixed convective slip flow. Convective heat transfer and MHD viscoelastic nanofluid flow has been actuated by Shit et al. [27]. In contemporary epoch, MHD has become a great factfinding argument of computational fluid dynamics (CFD) which contrived by multitudinous scientists commensurate as Ibrahim [28]; Rani and Kim [29]; Animasaun and Oyem [30] and Lavanya and Ratnam [31].
Homogeneous to MHD deportment, chemical reaction is a stratagem that assists to the transformation of one set of chemical substances to another at a fixed temperature and concentration. Heat and mass transfer on MHD mixed convection axisymmetric chemically reactive flow of Maxwell fluid flow was evoked by Khan et al. [32]. Also, Tripathy et al. [33] have illustrated the chemical reaction effect. Here, authors have renovated their governing equations into a twopoint boundary value predicament manipulating similarity variables and then elucidated numerically by forth order RungeKutta method with shooting modus operandi.
Electromagnetic radiation has anonymous conventional characteristics depending on the frequency and wavelengths of the radiation simulates a vital role in industrial and technological areas such as various space vehicles, satellites, missiles, aircraft, gas turbines and nuclear reactors. Das et al. [34] perused Hall denouements with radiative heat transfer past a porous plate. They disentangled their governing equations by wielding fourth order RungeKutta Fehlberg method with the shooting technique. Thermal radiation culmination was elaborated by Dogonchi by et al. [35]. In addition, combined electrical MHD heat transfer with radiative and viscous dissipation outturns have been unveiled by Hsiao [36]; Biswas et al. [37]; Ahmmed et al. [38] and Biswas et al. [39].
The intention of the contemporaneous interpretation is to evaluate the sequels of MHD unsteady heat and mass transfer of Casson nanofluid flow past an exponentially accelerated plate. The consecutive Casson nanofluid model is reassessed to affix the consequence of thermophoresis and Brownian motion. We have pivoted the culminations of heterogeneous parameters such as of Casson parameter, permeability of porous medium, radiation parameter, Lewis number, thermophoresis parameter, Brownian motion parameter, Soret number, magnetic parameter, Schmidt number, Prandtl number, Heat source parameter, chemical reaction parameter, Grashof number and modified Grashof number. Furthermore, we have evaluated the thermal and momentum boundary layer thickness with isotherms and streamlines analysis. Finally, the present outcomes have been analogized with the foregoing ramifications.
The unsteady two dimensioAccording to Oyelakin et al. [1] and Biswas et al. [9], the rheological equation of Casson fluid flow has been contemplated as:nal free convection Casson nanofluid flow of an electrically conducting viscous incompressible fluid through a vertical porous permeable plate in the manifestation of thermal radiation, thermal diffusion and chemical reaction are enumerated with the xaxis is contemplated along the plate in the precipitously up sloping orientation and the yaxis is stipulated normal to the plate. A uniform magnetic field strength B_{0} is implemented across to the flow direction. It is appraised that the temperature of the wall is T_{w} and concentration at the plate is C_{w} at initially which are constant but at time t>0. The plate is precipitated exponentially with a velocity u=U_{0 }exp(a´t) in its own plane and the temperature and concentration level of the plate are elevated exponentially to T_{w}(=T_{$\infty$}) and C_{w}(=C_{$\infty$}) respectively with time t. The physical configuration and coordinate system of this present model is unveiled in Figure 1.
Figure 1. Physical configuration
According to Oyelakin et al. [1] and Biswas et al. [9], the rheological equation of Casson fluid flow has been contemplated as:
${{\tau }_{ij}}=2\left( {{\mu }_{b}}+\frac{{{P}_{y}}}{\sqrt{2\pi }} \right){{e}_{i\,j}}\,\,\,\,\,\,\text{when}\,\,\pi >{{\pi }_{c}}$ (1)
${{\tau }_{ij}}=2\left( {{\mu }_{b}}+\frac{{{P}_{y}}}{\sqrt{2{{\pi }_{c}}}} \right){{e}_{i\,j}}\,\,\,\,\,\,\text{when}\,\pi <{{\pi }_{c}}$ (2)
where, $\tau$_{ij} is the component of stress tension, π is the product of the component of deformation rate with itself, e_{ij} is (i, j)th component of the deformation rate, π_{c} is critical value based on the nonNewtonian model and P_{y} is comprehended as yield stress of the fluid which is verbalized as:
$P_y=(μ_b √2π)/β$
where, $\mu$_{b} = plastic dynamic viscosity of NonNewtonian fluid. In a case of Casson fluid flow, where $\pi >{{\pi }_{c}}$, therefore
$μ=μ_b+P_y/√2π or, μ=μ_b+((μ_b √2π)/β)/√2π or, υ=μ_b/ρ (1+1/β)$
Under these speculations, the pivotal dimensional governing equations with boundary conditions can be asserted as:
$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$ (3)
$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\upsilon \left( 1+\frac{1}{\beta } \right)\frac{{{\partial }^{2}}u}{\partial {{y}^{2}}}+g{{\beta }_{T}}\left( T{{T}_{\infty }} \right)+g{{\beta }_{C}}\left( C{{C}_{\infty }} \right)\left( 1+\frac{1}{\beta } \right)\frac{\upsilon }{{{k}^{*}}}u\frac{\sigma B_{0}^{2}u}{\rho }$ (4)
$\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{1}{\rho {{C}_{p}}}\frac{\partial }{\partial y}\left( k\frac{\partial T}{\partial y} \right)\frac{1}{\rho {{C}_{p}}}\frac{\partial {{q}_{r}}}{\partial y}+{\tau }'\left\{ {{D}_{B}}\left( \frac{\partial C}{\partial y}\frac{\partial T}{\partial y} \right)+\frac{{{D}_{T}}}{{{T}_{\infty }}}{{\left( \frac{\partial T}{\partial y} \right)}^{2}} \right\}+\frac{Q}{\rho {{C}_{p}}}\left( T{{T}_{\infty }} \right)$ (5)
$\frac{\partial C}{\partial t}+u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}={{D}_{m}}\frac{{{\partial }^{2}}C}{\partial {{y}^{2}}}{{K}_{1}}(C{{C}_{\infty }})+{{D}_{B}}\frac{{{\partial }^{2}}C}{\partial {{y}^{2}}}+\frac{{{D}_{T}}}{{{T}_{\infty }}}\frac{{{\partial }^{2}}T}{\partial {{y}^{2}}}+\frac{{{D}_{m}}{{K}_{T}}}{{{T}_{m}}}\frac{{{\partial }^{2}}T}{\partial {{y}^{2}}}$ (6)
The corresponding boundary conditions are:
$\left. \begin{align} & t\le 0,\,u=0,\,v=0,\,\,T={{T}_{w}},\,C={{C}_{w}}\,\,\,\,\,\,\,\,\,\text{for}\,\,\text{all}\,\,y \\ & t>0,\,u={{U}_{0}}\,{{e}^{\left( {a}'t \right)}},\,v=0,\,T={{T}_{\infty }}+\left( {{T}_{w}}{{T}_{\infty }} \right){{e}^{\left( \frac{U_{0}^{2}}{\upsilon }t \right)\,}} \\ & \,\,\,\,\,\,\,\,\,C={{C}_{\infty }}+\left( {{C}_{w}}{{C}_{\infty }}\, \right)\,{{e}^{\left( \frac{U_{0}^{2}}{\upsilon }t \right)\,}}\,\,\,\,\,\,\,\,\,\,\,\,\text{at}\,\,y=0 \\ & \,\,\,\,\,\,\,\,\,\,u=0,\,\,v=0,\,\,\,T\to {{T}_{\infty }},\,\,C\to {{C}_{\infty }}\,\,\,\,\,\,\text{as}\,\,y\to \infty \\ \end{align} \right\}$ (7)
where, $\tau$'=(rc)p/($\rho$c)f is the ratio of the heat capacity of the nanoparticle material and the heat capacity of the fluid. The dimensionless governing equations have been solicited by appertaining the subsequent dimensionless parameters and variables
$U=\frac{u}{{{U}_{0}}};\,\,V=\frac{v}{{{U}_{0}}};\,\,\,Y=\frac{y{{U}_{0}}}{\upsilon };\,\,X=\frac{x{{U}_{0}}}{\upsilon };\,\tau =\frac{tU_{0}^{2}}{\upsilon };\,$
$T={{T}_{\infty }}+\bar{T}\left( {{T}_{w}}{{T}_{\infty }} \right)\,\,;\,C={{C}_{\infty }}+\bar{C}\left( {{C}_{w}}{{C}_{\infty }} \right);\,$
$\,\Pr =\frac{\upsilon \rho {{C}_{p}}}{k};\,\,Gr=\frac{\upsilon g{{\beta }_{T}}({{T}_{w}}{{T}_{\infty }})}{U_{0}^{3}};\,\,Gm=\frac{\upsilon g{{\beta }_{c}}({{C}_{w}}{{C}_{\infty }})}{U_{0}^{3}}S=\frac{Q\upsilon }{\rho {{C}_{p}}U_{0}^{2}};\,\,M=\frac{\sigma B_{0}^{2}\upsilon }{\rho U_{0}^{2}};\,Le=\frac{\upsilon }{{{D}_{B}}};\,$
$Sc=\frac{\upsilon }{{{D}_{m}}};\,\,Sr=\frac{{{D}_{m}}{{K}_{T}}}{{{T}_{m}}\upsilon }\frac{({{T}_{w}}{{T}_{\infty }})}{({{C}_{w}}{{C}_{\infty }})}\,\gamma =\frac{{{K}_{1}}\upsilon }{U_{0}^{2}};\,Ra=\frac{4{\sigma }'\,T_{\infty }^{3}}{{k}'k};\,Kp=\frac{{{\upsilon }^{2}}}{k*U_{0}^{2}};\,\,\,a=\frac{{{{{a}'}}_{1}}\upsilon }{U_{0}^{2}}\,Nt=\frac{{\tau }'\,{{D}_{T}}({{T}_{w}}{{T}_{\infty }})}{{{T}_{\infty }}\upsilon };\,Nb=\frac{{\tau }'\,{{D}_{B}}({{C}_{w}}{{C}_{\infty }})}{\upsilon }$
Therefore, the dimensionless equations are:
$\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0$ (8)
$\frac{\partial U}{\partial \tau }+U\frac{\partial U}{\partial X}+V\frac{\partial U}{\partial Y}=\left( 1+\frac{1}{\beta } \right)\frac{{{\partial }^{2}}U}{\partial {{Y}^{2}}}+Gr\bar{T}+Gm\bar{C}\left( 1+\frac{1}{\beta } \right)KpUMU$ (9)
$\frac{\partial \bar{T}}{\partial \tau }+U\frac{\partial \bar{T}}{\partial X}+V\frac{\partial \bar{T}}{\partial Y}=\frac{1}{\Pr }\left\{ \frac{\lambda }{1+\lambda \bar{T}}{{\left( \frac{\partial \bar{T}}{\partial Y} \right)}^{2}} \right\}+\frac{1}{\Pr }\left( 1+\frac{4}{3}Ra \right)\frac{{{\partial }^{2}}\bar{T}}{\partial {{Y}^{2}}}+S\,\bar{T}+Nt{{\left( \frac{\partial \bar{T}}{\partial Y} \right)}^{2}}+Nb\frac{\partial \bar{C}}{\partial Y}\frac{\partial \bar{T}}{\partial Y}$ (10)
$\frac{\partial \bar{C}}{\partial \tau }+U\frac{\partial \bar{C}}{\partial X}+V\frac{\partial \bar{C}}{\partial Y}=\frac{1}{Sc}\frac{{{\partial }^{2}}\bar{C}}{\partial {{Y}^{2}}}\gamma \bar{C}+\frac{1}{Le}\frac{{{\partial }^{2}}\bar{C}}{\partial {{Y}^{2}}}+\frac{Nt}{NbLe}\frac{{{\partial }^{2}}\bar{T}}{\partial {{Y}^{2}}}+Sr\frac{{{\partial }^{2}}\bar{T}}{\partial {{Y}^{2}}}$ (11)
The boundary conditions can be elicited as:
$\left. \begin{align} & \tau >0,\,U={{\text{e}}^{(a\tau )}},\,V=0,\,\bar{T}=\text{e}{{\,}^{(\tau )}},\,\bar{C}={{\text{e}}^{(\tau )}}\,\text{at}\,\,Y=0 \\ & \,\,\,\,\,\,\,\,\,\,U=0,\,\,\bar{T}\to 0,\,\bar{C}\to 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{as}\,\,Y\to \infty \\ \end{align} \right\}$ (12)
The skin friction coefficient, Nusselt number and Sherwood number are evolved as:
${{C}_{f}}=\frac{1}{2\sqrt{2}}{{({{G}_{r}})}^{\frac{3}{4}}}{{\left( \frac{\partial U}{\partial Y} \right)}_{Y=0}}$
${{N}_{u}}=\frac{1}{\sqrt{2}}{{({{G}_{r}})}^{\frac{3}{4}}}{{\left( \frac{\partial \bar{T}}{\partial Y} \right)}_{Y=0}}$
${{S}_{h}}=\frac{1}{\sqrt{2}}{{({{G}_{r}})}^{\frac{3}{4}}}{{\left( \frac{\partial \bar{C}}{\partial Y} \right)}_{Y=0}}$
The stream function $\psi$(X,Y) with the velocity components are devised as:
$U=\frac{\partial \psi }{\partial Y},\,V=\frac{\partial \psi }{\partial X}$ (13)
An explicit finite difference scheme has been reconnoitred for deciphering the equations 1417, because of that a rectangular region is ascertained where X axis is pondered along the plate and Yaxis is normal to the plate. Here, plate of height is X_{max}(=125) and Y_{max}(=125). Also, grid spacing are esteemed m=150 and n=300 which are spectacled in Figure 1(a).
Figure 1(a). The finite difference space grid
The constant mash size is wangled ΔX=0.83(0≤x≤125); ΔY=0.50(0≤y≤125) with the smaller timestep, Δ$\tau$=0.0005. Now,
$\frac{{{U}_{i,j}}{{U}_{i1,j}}}{\Delta X}+\frac{{{V}_{i,j}}{{V}_{i,j1}}}{\Delta Y}=0$ (14)
$\frac{{{{{U}'}}_{i,j}}{{U}_{i,j}}}{\Delta \tau }+{{U}_{i,j}}\left( \frac{{{U}_{i,j}}{{U}_{i1,j}}}{\Delta X} \right)+{{V}_{i,j}}\left( \frac{{{U}_{i,j+1}}{{U}_{i,j}}}{\Delta \Upsilon } \right)$
$=\left( 1+\frac{1}{\beta } \right)\left( \frac{{{U}_{i,j+1}}2{{U}_{i,j}}+{{U}_{i,j1}}}{{{\left( \Delta \Upsilon \right)}^{2}}} \right)+Gr{{\bar{T}}_{i,j}}\,+Gm{{\bar{C}}_{i,j}}\left( 1+\frac{1}{\beta } \right)kp{{U}_{i,j}}M{{U}_{i,j}}$ (15)
$\frac{{{{{\bar{T}}'}}_{i,j}}{{{\bar{T}}}_{i,j}}}{\Delta \tau }+{{U}_{i,\,j}}\frac{{{{\bar{T}}}_{i,j}}{{{\bar{T}}}_{i1,j}}}{\Delta X}+{{V}_{i,\,j}}\frac{{{{\bar{T}}}_{i,j+1}}{{{\bar{T}}}_{i,j}}}{\Delta Y}$
$=\frac{1}{\Pr }\left( 1+\frac{4}{3}Ra \right)\frac{{{{\bar{T}}}_{i,j+1}}2{{{\bar{T}}}_{i,j}}+{{{\bar{T}}}_{i,j1}}}{{{\left( \Delta Y \right)}^{2}}}+S\,{{\bar{T}}_{i,j}}\,$
$+\frac{1}{\Pr }\left( \frac{\lambda }{1+\lambda {{{\bar{T}}}_{i,j}}} \right){{\left( \frac{{{{\bar{T}}}_{i,j+1}}{{{\bar{T}}}_{i,j}}}{\Delta \Upsilon } \right)}^{2}}+Nb$
$\left( \frac{{{{\bar{C}}}_{i,j+1}}{{{\bar{C}}}_{i,j}}}{\Delta Y} \right)\left( \frac{{{{\bar{T}}}_{i,j+1}}{{{\bar{T}}}_{i,j}}}{\Delta Y} \right)+Nt{{\left( \frac{{{{\bar{T}}}_{i,j+1}}{{{\bar{T}}}_{i,j}}}{\Delta Y} \right)}^{2}}$ (16)
$\frac{{{{{\bar{C}}'}}_{i,j}}{{C}_{i,j}}}{\Delta \tau }+{{U}_{i,\,j}}\frac{{{{\bar{C}}}_{i,j}}{{{\bar{C}}}_{i1,j}}}{\Delta X}+{{V}_{i,\,j}}\frac{{{{\bar{C}}}_{i,j+1}}{{{\bar{C}}}_{i,j}}}{\Delta Y}$
$=\left( \frac{1}{Sc}+\frac{1}{Le} \right)\frac{{{{\bar{C}}}_{i,j+1}}2{{{\bar{C}}}_{i,j}}+{{{\bar{C}}}_{i,j1}}}{{{\left( \Delta Y \right)}^{2}}}\,\,\gamma {{\bar{C}}_{i,\,j}}+\left( Sr+\frac{{{N}_{t}}}{NbLe} \right)\frac{{{{\bar{T}}}_{i,j+1}}2{{{\bar{T}}}_{i,j}}+{{{\bar{T}}}_{i,j1}}}{{{\left( \Delta Y \right)}^{2}}}$ (17)
The intermeddle boundary periphery in explicit form as:
$\left. \begin{align} & \tau >0,\,\,\,U_{i,\,0}^{n}=\text{e}{{\,}^{(an\Delta \tau )}},\,\,V_{i,\,0}^{n}=0,\,\bar{T}_{i,\,0}^{n}=\text{e}{{\,}^{(n\Delta \tau )}},\,\, \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\bar{C}_{i,\,0}^{n}=\text{e}{{\,}^{(n\Delta \tau )}}\,\,\,\,\,\,\,\,\,\,\text{at}\,\,\text{Y}=\text{0} \\ & \,\,\,\,\,\,\,U_{i,\,L}^{n}=0,\,\,\,\bar{T}_{i,\,L}^{n}\to 0,\,\,\bar{C}_{i,\,L}^{n}\to 0\,\,\text{where}\,\,L\to \infty \\ \end{align} \right\}$ (18)
Here i and j designate to the mesh points with X and Y coordinate respectively and the superscripts n connote a value of time, t=nΔ$\tau$ where, n=0, 1, 2 ….
The exploration persists deficient without the stability and convergence test (SCT) as long as the finite difference scheme has been exerted. For the explicit mesh sizes, the stability criteria of the scheme has been actuated as follows:
The equation (14) has been eradicated because Δ$\tau$ does not transpire in it. The quotidian terms of the Fourier expansion for $U,\,\,\bar{T}\,\text{and}\,\,\bar{C}$ at a logically time $\tau$=0 are all e^{i}^{aX }e^{i}^{bY }aloof form a constant, $i=\sqrt{1}$. At a time $\tau $, these terms become:
$\left. \begin{align} & U:\,\,\,\,\,\,\,\,A(\tau ){{e}^{i\alpha X}}{{e}^{i\beta Y}} \\ & \bar{T}:\,\,\,\,\,\,\,\,C(\tau ){{e}^{i\alpha X}}{{e}^{i\beta Y\,\,\,\,\,}} \\ & \bar{C}:\,\,\,\,\,\,\,\,D(\tau ){{e}^{i\alpha X}}{{e}^{i\beta Y}} \\ \end{align} \right\}$ (19)
After the time step,
$\left. \begin{align} & U:\,\,\,\,\,\,\,\,{A}'(\tau ){{e}^{i\alpha X}}{{e}^{i\beta Y}} \\ & \bar{T}:\,\,\,\,\,\,\,\,{C}'(\tau ){{e}^{i\alpha X}}{{e}^{i\beta Y\,\,\,\,\,}} \\ & \bar{C}:\,\,\,\,\,\,\,\,{D}'(\tau ){{e}^{i\alpha X}}{{e}^{i\beta Y}} \\ \end{align} \right\}$ (20)
By cast away the above values the following palliation equations are erected as:
$\frac{({A}'A){{e}^{i\alpha X}}{{e}^{i\beta Y}}}{\Delta \tau }+\,U\frac{A{{e}^{i\alpha X}}{{e}^{i\beta Y}}(1{{e}^{i\alpha \Delta X}})}{\Delta X}+V\,\frac{A{{e}^{i\alpha X}}{{e}^{i\beta Y}}({{e}^{i\beta \Delta Y}}1)}{\Delta Y}=\,\left( 1+\frac{1}{\beta } \right)$
$\frac{2A{{e}^{i\alpha X}}{{e}^{i\beta Y}}(\cos \beta \Delta Y1)}{{{(\Delta Y)}^{2}}}\,+GrC{{e}^{i\alpha X}}{{e}^{i\beta Y}}+GmDe^{i\alpha X}{{e}^{i\beta Y}}\left( 1+\frac{1}{\beta } \right)KpA{{e}^{i\alpha X}}{{e}^{i\beta Y}}M\,A{{e}^{i\alpha X}}{{e}^{i\beta Y}}$ (21)
$\frac{({C}'C){{e}^{i\alpha X}}{{e}^{i\beta Y}}}{\Delta \tau }+\,U\frac{C{{e}^{i\alpha X}}{{e}^{i\beta Y}}(1{{e}^{i\alpha \Delta X}})}{\Delta X}+V\,\frac{C{{e}^{i\alpha X}}{{e}^{i\beta Y}}({{e}^{i\beta \Delta Y}}1)}{\Delta Y}=\frac{1}{\Pr }\left( 1+\frac{4}{3}Ra \right)$
$+Nb\frac{D{{e}^{i\alpha X}}{{e}^{i\beta Y}}(1{{e}^{i\alpha \Delta X}})}{\Delta X}\frac{C{{e}^{i\alpha X}}{{e}^{i\beta Y}}({{e}^{i\beta \Delta Y}}1)}{\Delta Y}$ (22)
$\frac{({D}'D){{e}^{i\alpha X}}{{e}^{i\beta Y}}}{\Delta \tau }+\,U\frac{D{{e}^{i\alpha X}}{{e}^{i\beta Y}}(1{{e}^{i\alpha \Delta X}})}{\Delta X}+V\,\frac{D{{e}^{i\alpha X}}{{e}^{i\beta Y}}({{e}^{i\beta \Delta Y}}1)}{\Delta Y}=\left( \,\frac{1}{Sc}+\frac{1}{Le} \right)$
$\left[ \frac{2D{{e}^{i\alpha X}}{{e}^{i\beta Y}}(\cos \beta \Delta Y1)}{{{(\Delta Y)}^{2}}} \right]+\left( Sr+\frac{Nt}{NbLe} \right)\left[ \frac{2C{{e}^{i\alpha X}}{{e}^{i\beta Y}}(\cos \beta \Delta Y1)}{{{(\Delta Y)}^{2}}} \right]\gamma \,D{{e}^{i\alpha X}}{{e}^{i\beta Y}}$ (23)
The overhead equations are revealed as
${A}'={{A}_{1}}A+{{A}_{2}}C+{{A}_{3}}D$ (24)
${C}'={{A}_{4}}C$ (25)
${D}'={{A}_{5}}D+{{A}_{6}}C$ (26)
where,
${{A}_{1}}=1+\left( 1+\frac{1}{\beta } \right)\frac{2\Delta \tau }{{{(\Delta Y)}^{2}}}(\cos \beta \Delta Y1)U\frac{\Delta \tau }{\Delta X}(1{{e}^{i\alpha \Delta X}})V\,\frac{\Delta \tau }{\Delta Y}({{e}^{i\beta \Delta Y}}1)\left( 1+\frac{1}{\beta } \right)Kp\,\Delta \tau M\Delta \tau $
${{A}_{2}}=\,Gr\Delta \tau $; ${{A}_{3}}=\,Gm\Delta \tau $
${{A}_{4}}=1+\frac{1}{\Pr }\left( 1+\frac{4}{3}Ra \right)\frac{2\Delta \tau }{{{(\Delta Y)}^{2}}}(\cos \beta \Delta Y1)U\frac{\Delta \tau }{\Delta X}(1{{e}^{i\alpha \Delta X}})V\,\frac{\Delta \tau }{\Delta Y}({{e}^{i\beta \Delta Y}}1)+S\Delta \tau $
$+\frac{1}{\Pr }\left( \frac{\lambda }{1+\lambda \overline{C}} \right)\frac{2\Delta \tau }{{{(\Delta Y)}^{2}}}(\cos \beta \Delta Y1)U\frac{\Delta \tau }{\Delta X}(1{{e}^{i\alpha \Delta X}})V\,\frac{\Delta \tau }{\Delta Y}({{e}^{i\beta \Delta Y}}1)+S\Delta \tau $
${{A}_{5}}=1+\,\left( \,\frac{1}{Sc}+\frac{1}{Le} \right)\frac{2\Delta \tau }{{{(\Delta Y)}^{2}}}(\cos \beta \Delta Y1)U\frac{\Delta \tau }{\Delta X}(1{{e}^{i\alpha \Delta X}})V\,\frac{\Delta \tau }{\Delta Y}({{e}^{i\beta \Delta Y}}1)\gamma \Delta \tau $
${{A}_{6}}=\left( Sr+\frac{Nt}{NbLe} \right)\frac{2\Delta \tau }{{{(\Delta Y)}^{2}}}(\cos \beta \Delta Y1)$
Therefore the matrix form is:
$\,\left[ \begin{matrix} {{A}'} \\ {{C}'} \\ {{D}'} \\ \end{matrix} \right]=\left[ \begin{matrix} {{A}_{1}} & {{A}_{2}} & {{A}_{3}} \\ 0 & {{A}_{4}} & 0 \\ 0 & {{A}_{6}} & {{A}_{5}} \\ \end{matrix} \right]\left[ \begin{matrix} A \\ B \\ C \\ \end{matrix} \right]$ (27)
Therefore, ${\eta }'={T}'\eta$, where:
$\eta '=\left[ \begin{matrix} A' \\ C' \\ D' \\ \end{matrix} \right]$
${T}'=\left[ \begin{matrix} {{A}_{1}} & {{A}_{2}} & {{A}_{3}} \\ 0 & {{A}_{4}} & 0 \\ 0 & {{A}_{6}} & {{A}_{5}} \\ \end{matrix} \right];$ $\eta =\left[ \begin{matrix} A \\ B \\ C \\\end{matrix} \right]$.
This is a fourth order square matrix. So, it is very hard to find out the eigenvalues and Δ$\tau$ is very compact in EFDM, i.e. Δ$\tau\to$ 0. Under this appraisal, A_{2}$\to$0, A_{3}$\to$0 and A_{6}$\to$0.
Therefore,
${T}'=\left[ \begin{matrix} {{A}_{1}} & 0 & 0 \\ 0 & {{A}_{4}} & 0 \\ 0 & 0 & {{A}_{5}} \\\end{matrix} \right];$ (28)
Therefore the eigenvalues are retrieved as l_{1}=A_{1}, l_{2}=A_{4} and l_{3}=A_{5} such that $\left {{A}_{1}} \right\le 1\,\,\,\left {{A}_{4}} \right\le 1\text{and}\,\left {{A}_{5}} \right\le 1\,$ and convinced admissible values are A_{1}=1, A_{4}=1 and A_{5}=1.
So, the stability conditions are:
$(1+1/β) 2Δτ/((ΔY)^2 )+MΔτ/2+U Δτ/ΔX+V Δτ/ΔY+\left( 1+\frac{1}{\beta } \right)\frac{Kp\Delta \tau }{2}\le 1$ (29)
$\frac{1}{\Pr }\left( 1+\frac{4}{3}Ra \right)\frac{2\Delta \tau }{{{(\Delta Y)}^{2}}}+\left( \frac{1}{\Pr }+Nt \right){{\left( \frac{\Delta \tau }{\Delta Y} \right)}^{2}}$
$+U\frac{\Delta \tau }{\Delta X}+\left V \right\,\frac{\Delta \tau }{\Delta Y}\frac{S\,\Delta \tau }{2}\le 1$ (30)
$\left( \frac{1}{Sc}+\frac{1}{Le} \right)\frac{2\Delta \tau }{{{(\Delta Y)}^{2}}}+\frac{\gamma \Delta \tau }{2}+U\frac{\Delta \tau }{\Delta X}+\left V \right\,\frac{\Delta \tau }{\Delta Y}\le 1$ (31)
Exploiting basic agreement, U=V=T=C=0 at t=0 the convergence criteria of the existent problem are retrieved as Sc$\ge$0.18 and Le$\ge$0.20.
The numerical computations have been manipulated for the disparate values of miscellaneous parameters by an explicit finite difference method (EFDM) for the aspiration of scrutinizing the upshots of the contemporary work. The physical stipulation of the contemporaneous model such that velocity, temperature, concentration, skin friction coefficient, Nusselt number, Sherwood number, streamlines and isotherms lines has been procured for distinct parameters which are indicated in Figures 2 to 23. In order to prevail the numerical results, the succeeding values of default parameter are deputed as: M=1.0, Pr=0.71, Sc=0.22, Ra=0.50, $\beta$=0.40, S=0.1, Nb=0.5, Nt =0.8, Le=2.5, Sr=1, Gm=5, $\lambda$=2, Gr=10, Kp=1.0, and $\gamma$=0.50 with time $\tau$=10. These values are critiqued as homogeneous throughout the present study in respective figures.
Figure 2. The illustration of $\beta$ on U
Figure 3. The illustration of Kp on U
The impression of separate values of b and Kp on velocity are elucidated in Figure 2 and Figure 3, respectively. It is proclaimed that velocity are economized 13.84 %, 9.01 %, 20.8 %, and 19.92 % by $\beta$=0.40 to $\beta$=0.80, $\beta$=0.80 to $\beta$=1.00, $\beta$=1.00 to $\beta$=1.50 and $\beta$=1.50 to $\beta$=2.00 at Y=0.4556. Physically, by elaborating the values of b which remises the yield stress suppresses in the fluid velocity. Also, velocity is extenuation due to the prolongation of Kp which ensues in Figure 3. Naturally, permeability of porous medium which propagated a resistive force in the fluid. The numerical evolution of isolated parameters of U are flaunted in the Table 1 and Figures 23.
Table 1. Delineates the curve to curve fluid velocity (Figures. 2, 3) at Y= 0.4556 and $\eta$=10
Values of Parameter 
Numerical Value 
Increase/Decrease in (%) 
$\beta$=0.40 
0.13561 
Initial position 
$\beta$=0.80 
0.11684 
13.84% decrease 
$\beta$=1.00 
0.10632 
9.01% decrease 
$\beta$=1.50 
0.08413 
20.87% decrease 
$\beta$=2.00 
0.06737 
19.92% decrease 
Kp=1.00 
0.13561 
Initial position 
Kp=1.50 
0.10002 
26.24% decrease 
Kp=2.00 
0.07755 
22.47% decrease 
Kp=2.50 
0.06311 
18.62% decrease 
Kp=3.50 
0.04589 
27.29% decrease 
Figure 4. The illustration of Pr $\bar{T}\,$
The impression of Pr, Nb, Ra and S on temperature schemes are found in the Figures 47. It is worth mentioned that temperature allocations are collapsed due to the enhancement of Pr. Instinctively, high Prandtl number have higher viscosity, which constricted the thermal boundary layer depth. Also, the temperature delineations are retrenched considerable with the accrual of Nb because it innovate an unbridled motions in the fluid. Here, temperature depictions are protracted as 35.46 %, 16.02 %, 9.84 %, and 7.06 % for Ra=0.50 to Ra=1.00, Ra=1.00 to Ra=1.50, Ra=1.50 to Ra=2.00 and Ra=2.00 to Ra=2.50 because the thermal boundary layer thickness are prosecuted due to completion of Ra. Further, temperature disbursements are surged due to augmentation of S. The preeminent vindication behind this upshot is that the heat absorption invokes a drop in the kinetic energy as well as thermal energy of the fluid. The numerical disposition of relative parameters are evinced in the Table 2 and Figures 47.
Figure 5. The illustration of Nt on $\bar{T}\,$
Figure 6. The illustration of Ra on $\bar{T}\,$
Figure 7. The illustration of S on $\bar{T}\,$
The denouements of $\gamma$, Sc, Le and Nb on concentration propagations are eliciting in Figures 811. By the offshoot of chemical reaction, concentration is alleviated because the positive values of chemical reaction parameter ($\gamma$>0) presages the sequels of destructive chemical reaction on the concentration field. It is fact that, Sc and Le constricts the concentration. It is true by physically that Schmidt number aggrandizes the density of the fluid as well as concentration buoyancy force in the fluid. Also, from Figure 11, we perceived that concentration disposals abridge to the amplifications of Nb which leads the concentration allocation. Physically, it is intimated that Nt elongates the nanoparticle diffusion. The numerical values of these are dispensed in the Table 3 and Figures 811.
The Figures 12 to 17 epitomize the repercussions of $\beta$, Ra, Gr and Pr on skin friction coefficient (C_{f}), Nusselt number (N_{u}) and Sherwood number(S_{h}) for Casson nanofluid. It is devised that, C_{f} protracts with the accession of $\beta$ and $\gamma$ which is disported in Figure 12 and Figure 13 respectively. Physically, skin friction is aggrandized the rate of velocity of the fluid flow. On the other hand the Nusselt number is a decay function of Ra and Gr but proliferation function of Pr. Generally, convective heat transfer distends with the quadrupling of Nusselt number. Also it is articulated from Figure 17 that, the Sherwood number prolongs with the retrenchment of Grashof number (Gr).
Table 2. Incarnates the curve to curve fluid temperature (Figures 47) at Y= 4.18060 and $\tau $=10.0
Values of Parameter 
Numerical Value 
Increase/Decrease in (%) 
Pr=0.63 
0.11834 
Initial position 
Pr=1.00 
0.10988 
7.14% decrease 
Pr=1.38 
0.09093 
17.24% decrease 
Pr=7.00 
0.05362 
41.03% decrease 
Pr=11.62 
0.05030 
6.19% decrease 
Ra=0.50 
0.11834 
Initial position 
Ra=1.00 
0.18336 
35.46% increase 
Ra=1.50 
0.21834 
16.02% increase 
Ra=2.00 
0.24216 
9.84% increase 
Ra=2.50 
0.26056 
7.06% increase 
Nb=0.50 
0.11834 
Initial position 
Nb=1.00 
0.13127 
9.85% increase 
Nb=1.50 
0.16192 
18.93% increase 
Nb=2.00 
0.19097 
15.31% increase 
Nb=2.50 
0.21830 
12.52% increase 
S=0.10 
0.11834 
Initial position 
S=0.50 
0.16319 
27.48% increase 
S=1.00 
0.24429 
33.19% increase 
S=1.50 
0.36841 
33.79% increase 
S=2.00 
0.56075 
34.33% increase 
Figure 8. The illustration of $\lambda$ on $\bar{C}$
Figure 9. The illustration of Sc on$\bar{C}$
Figure 10. The illustration of Le on $\bar{C}$
Figure 11. The illustration of Nt on $\bar{C}$
Table 3. Emblematizes the curve to curve fluid concentration (Figures 811) at Y= 4.18060 and $\tau$=10.0
Values of parameter 
Numerical value 
Increase/decrease in (%) 
$\gamma$=0.50 
0.24736 
Initial position 
$\gamma$==1.00 
0.19056 
22.96% decrease 
$\gamma$==1.50 
0.14917 
21.72% decrease 
$\gamma$==2.00 
0.11864 
20.47% decrease 
$\gamma$==2.50 
0.09583 
19.22% decrease 
Le=2.50 
0.24736 
Initial position 
Le=3.50 
0.23075 
6.71% decrease 
Le=4.50 
0.22136 
4.06% decrease 
Le=5.50 
0.21532 
2.73% decrease 
Le=6.50 
0.21111 
1.96% decrease 
Sc=0.22 
0.24736 
Initial position 
Sc=0.30 
0.20128 
33.97% decrease 
Sc=0.78 
0.11386 
18.62% decrease 
Sc=0.95 
0.10479 
43.43% decrease 
Sc=5.00 
0.07891 
24.79% decrease 
Nb=0.80 
0.24736 
Initial position 
Nb=1.00 
0.20355 
17.71% decrease 
Nb=1.80 
0.19804 
2.71% decrease 
Nb=2.00 
0.18124 
8.48% decrease 
Nb=2.50 
0.15329 
15.82% decrease 
In this disquisition, the unsteady MHD heat and mass transfer of Casson nanofluid flow in the appearance of variable thermal conductivity, heat absorption and thermal radiation has been propounded. The predicament is evaluated under the influence of strong cross field and consideration has been endorsed to those pragmatic fluids which are altered as liquid metals. The governing model equations are elucidated by utilizing explicit finite difference method (EFDM) and the numerical results are enumerated for the divergent dimensionless parameters. In this observation the Casson parameter and permeability of porous medium abates the velocity profiles. The Nusselt number is plunged by radiation parameter and Grashof number but accumulated by Prandtl number. Also, the skin friction coefficient proliferates with the inflation of Casson parameter and chemical reaction parameter. Further, Brownian motion parameter (Nb), chemical reaction (g), Schmidt number (Sc) and Lewis number (Le) wanes the concentration profiles. But, Brownian motion parameter (Nb) which elevates the temperature profiles. Furthermore, temperature depictions are aggravating due to the amplification of heat source parameter and radiation parameter but dwindling due to the amelioration of Prandtl number.
Table 4. The previous results by Ghadikolaei et al. [12]
Previous Results by Ghadikolaei et al. [12] 

Increased Parameters 
U 
$\bar{T}\,$ 
$\bar{C}\,$ 
Skin fri. 
Nusl. Num. 
Sher. Num. 
Gr 




Inc 

Nb 

Inc 
Dec 



b 
Dec 
Inc 
Inc 



S 

Inc 

Inc 


R 

Inc 


Inc 

g 


Dec 


Inc 
Pr 

Dec 




Le 


Dec 



Table 5. Comparison of the accuracy of the present results with the previous results by Ghadikolaei et al. [12]
Previous Results by Ghadikolaei et al. [12] 

Increased Parameters 
U 
$\bar{T}\,$ 
$\bar{C}\,$ 
Skin fri. 
Nusl. Num. 
Sher. Num. 
Gr 






Nb 

Inc 
Dec 



b 
Dec 





S 

Inc 

Inc 


R 

Inc 


Dec 

g 


Dec 



Pr 

Dec 




Le 


Dec 



Comparison of the accuracy of the present results with the previous results by Ghadikolaei et al. [12] are impersonated in the above Table 4. Here, it is permitted that, all values are same without just only one result is isolated with the works of Ghadikolaei et al. [12]. In this present disquisition, Nusselt number is dwindled by radiation parameter but opposite with the work of Ghadikolaei et al. [12].
x,y 
Cartesian coordinates (m) 
u,v 
Velocity components (m/s) 
$\gamma$ 
Chemical reaction parameter () 
U 
Dimensionless primary velocity (m/s) 
Sc 
Schmidt number () 
Nu 
Nusselt number () 
S 
Heat source parameter () 
Cf 
Skin friction () 
$\rho$ 
Fluid density (kg/m^{3}) 
v 
Kinematic viscosity (m^{2}s^{1}) 
Tm 
Mean fluid temperature (K) 
Cs 
Concentration susceptibility (J/kg·K) 
Sh 
Sherwood number () 
$\bar{T}\,$ 
Dimensionless fluid temperature (K) 
Nt 
Thermophoresis parameter () 
$\lambda$ 
Thermal conductivity variation parameter () 
g 
Acceleration due to gravity (ms^{2}) 
M 
Magnetic parameter (Wbm^{2}) 
bT 
Thermal expansion coefficient () 
Pr 
Prandlt number () 
Gr 
Grashof number () 
Gm 
Modified Grashof number () 
Kp 
Permeability of porous medium () 
b 
Casson parameter () 
Sr 
Soret number () 
T 
Temperature of fluid (K) 
k 
Thermal conductivity (W/m·K) 
Cp 
Specific heat at constant pressure (J/m^{3}·K) 
bc 
Concentration expansion coefficient () 
$\bar{C}\,$ 
Dimensionless fluid concentration () 
Le 
Lewis number () 
Nb 
Brownian motion parameter () 
Dm 
Molecular diffusivity of the species concentration () 
σ 
StefanBoltzmann constant (W/m^{2} K^{4}) 
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