Entropy generation in MHD radiative viscous nanofluid flow over a porous wedge using the bivariate spectral quasi-linearization method

Author’s Accepted Manuscript Entropy generation in MHD radiative viscous nanofluid flow over a porous wedge using the bivariate spectral quasi-linearization method S.P. Goqo, S.D. Oloniiju, H. Mondal, P. Sibanda, S.S. Motsa www.elsevier.com/locate/csite

PII: DOI: Reference:

S2214-157X(18)30278-8 https://doi.org/10.1016/j.csite.2018.10.005 CSITE344

To appear in: Case Studies in Thermal Engineering Received date: 13 September 2018 Revised date: 7 October 2018 Accepted date: 12 October 2018 Cite this article as: S.P. Goqo, S.D. Oloniiju, H. Mondal, P. Sibanda and S.S. Motsa, Entropy generation in MHD radiative viscous nanofluid flow over a porous wedge using the bivariate spectral quasi-linearization method, Case Studies in Thermal Engineering, https://doi.org/10.1016/j.csite.2018.10.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Entropy generation in MHD radiative viscous nanofluid flow over a porous wedge using the bivariate spectral quasi-linearization method S. P. Goqo a, S. D. Oloniiju a School

a,b

, H. Mondala, P. Sibanda a, S. S. Motsa

a,c

of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Scottsville, Pietermaritzburg 3209, South Africa. b DST–NRF

Centre of Excellence in Mathematical and Statistical Sciences (CoE–MaSS).

a,c

University of Eswatini, Department of Mathematics, Swaziland

Abstract We study the viscous nanofluid flow over a non-isothermal wedge with thermal radiation. The entropy due to irreversible processes in the system may degrade the performance of the thermodynamic system. Studying entropy generation in the flow over a porous wedge gives insights into how the system is affected by irreversible processes, and indicate which thermo–physical parameters contribute most to entropy generation in the system. The bivariate spectral quasi-linearization method is used to find the convergent solutions of the model equations. The impact of significant parameters such as the Hartmann number, thermophoreis and Brownian motion parameter on the fluid properties is evaluated and discussed. The Nusselt number, skin friction coefficients and Sherwood number are determined. An analysis of the rate of entropy generation in the flow for various parameters is presented, and among other results, we found that the Reynolds number and thermal radiation contribute significantly to entropy generation. Keywords: Entropy generation; wedge flows; bivariate spectral quasi-linearization method.

1. Introduction In thermal–fluid dynamical systems, entropy generation is a quantitative tool for measuring the irreversibilities that accompany the fluid flow process. The more entropy generated in a system, the higher the scope of irreversible processes occurring in the system. Studies have shown that thermophysical processes such as heat and mass transfer, magnetic effects and viscous dissipation all contribute to the source of the chaos in heat transfer processes. This analysis is important in two–phase flows, optimization of heat exchangers, geothermal energy systems and fuel cells. A large number of studies exist in the literature on entropy generation in a streaming fluid. many such studies follow from the pioneering work of Bejan [1]. Recently, ∗ corresponding

author. Email address: [email protected] (H. Mondala )

Preprint submitted to Case Studies in Thermal Engineering

October 18, 2018

Sithole et al. [2] studied the generation of entropy in magnetohydrodynamic flow of a second grade viscoelastic fluid over a convectively heated stretching sheet with non–linear thermal radiation. Dalir et al. [3] investigated entropy generation in the laminar two–dimensional flow of an incompressible Jeffrey nanofluid over a stretching impermeable isothermal sheet. In Rashidi et al. [4], an analysis of optimization of entropy generation with respect to changes in thermophysical parameters was presented. The analysis was on entropy generation in an unsteady flow of a Newtonian fluid past a malleable rotating disk with time dependent angular velocity. They applied the particle swarm optimization algorithm and an artificial neural network analysis to find the thermophysical parameters that minimizes entropy generation rate in the system. Ibanez and Cuevas [5] investigated energy dissipative process that occur in flow in a microchannel subjected to electromagnetic interactions. They suggested the conditions under which the rate of entropy generation can be minimized in the microchannel flow. The effects of radiation on the free convection in an optically dense viscous incompressible fluid along a heated inclined flat surface in a saturated porous medium was studied by Hossain and Pop [6]. Pal and Mondal [7] studied the non-isothermal wedge with variable temperature dependent viscosity using the fifthorder Runge�–Kutta�–Fehlberg scheme with the shooting technique to solve the conservation equations. The MHD mixed convective heat transfer problem of a second-grade fluid past a wedge with porous medium studied by Hsiao [8]. Using the homotopy analysis method, Rashidi et al. [9] analyzed the mixed convective MHD viscoelastic fluid flow over a porous wedge with thermal radiation. The study of MHD viscous flows is important to technological, industrial and geothermal applications, such as high-temperature plasmas, cooling of nuclear reactors, liquid metal fluids, Magnetohydrodynamic generators and accelerators. Sibanda and Makinde [10] studied the flow over a rotating disk in a porous medium with Ohmic heating and viscous dissipation. Chamkha et al [11] investigated the MHD flow over a permeable stretching sheet having the effects of chemical reaction on mass transfer features. Rashidi et al. [12] studied magnetohydrodynamics boundary layer flow of viscoelastic fluid via a moving surface. Rashidi and Erfani [13] discussed the steady MHD flow on rotating disk with Ohmic heating and viscous dissipation. The problem of heat and mass transfer over a wedge has generated research interest because such flows have a wide range of physical and engineering applications. Flows over a wedge have applications in engine greasing, cooling of hot inclined surfaces and supersonic flows. Considerable emphasis has been put on this kind of problem in the past few decades. One such study is by Watanabe et al. [14] who investigated forced and free mixed convection in flow past a heated and cooled wedge with suction/injection. Liu et al. [15] studied the flow of a viscous incompressible fluid across a wedge using different nanoparticle additives,with a temperature jump and high-order slip boundary. Yih [16] examined the effects of a pressure gradient and viscous dissipation on the wall shear stress, and heat and mass transfer coefficients on flow past a non–isothermal wedge. Devi and Kandasamy [17] investigated the effect of chemical reaction and heat and mass transfer on a flow over a wedge with injection/suction at the wall under electromagnetic interactions. 2

The effects of nanoparticle shape and size on flow over a wedge was studied by Ellahi et al. [18]. The study examined nanoparticles dipped in industrially engineered fluids, and analyzed the changes in entropy generation rates. The effect of nonlinear thermal radiation in Casson nanofluid flow over an inclined porous stretching sheet was studied by Ghadikolaei et al. [19] and Alizadeh et al . [20]. There are many industrial applications in which heat and mass transfer is a consequence of buoyancy effects caused by diffusion of heat and chemical species. The study of such processes is useful for advancing a number of chemical technologies such as in polymer production. Many diffusive operations involve the molecular diffusion of species with a chemical reaction within the boundary layer. The presence of foreign mass in air or water may cause a chemical reaction. During a chemical reaction, heat is also generated or absorbed. A reaction is said to be first-order if the rate of reaction is directly proportional to the chemical concentration. Pal and Mondal [21] studied flow in a magneto-porous medium with thermal radiation and a chemical reaction on mixed convection heat and mass transfer over a stretching sheet. The chemical reaction and thermal radiation on the heat and mass transfer in MHD micropolar flow over a vertical moving porous plate in a porous medium with heat generation was analyzed by Mohamed and Abo-Dahab [22]. Literature indicates that entropy generation under the influence of a chemical reaction on the magnetohydrodynamic viscous nanofluid flow over a porous wedge is yet to be studied. This study aims to bridge this gap. We present a study of entropy generation in an unsteady two–dimensional magnetohydrodynamic flow of an incompressible viscous nanofluid. The transformed partial differential equations describing the mass, momentum, energy and concentration of conservation are solved using the bivariate spectral quasi– linearization method ([23], [24], [25]). The influences of fluid and flow parameters on the entropy generation rate, wall shear stress, heat and mass transfer coefficients, velocity, temperature and concentration profiles is investigated.

2. Mathematical Analysis 2.1. Formulation of the Dynamical Process We consider an unsteady two–dimensional boundary layer flow of an incompressible viscous fluid past a porous wedge. The fluid is under the influence of a uniform magnetic field applied in the y–direction with electrical conductivity σ and constant magnetic strength B0 . The flow is subject to a chemical reaction and viscous dissipation. A uniformity in the size of the nanoparticle is assumed. In view of these assumptions, the continuity, momentum balance, energy and concentration conservation equations are given as [9], [15]:

3

∂u ∂v + =0 ∂x ∂y

(1)

 ( )  2 2  σ B0 ν  ] ∂u ∂u ∂u ∂2 u ∂Ue [ Ω ∗ +u +v = ν 2 + Ue + gβ(T − T ∞ ) + gβ (C − C∞ ) sin −  +  (u − Ue ) ∂t ∂x ∂y ∂x 2 ρ k ∂y  ( )2  ( )2 2   ∂T ∂T ∂ T 1 ∂qr ∂T  ∂C ∂T DT ∂T  ν ∂u +u +v = α 2 + τ DB + − + ∂t ∂x ∂y ∂y ∂y T ∞ ∂y  c p ∂y ρc p ∂y ∂y ∂C ∂C ∂C ∂2 C D B ∂2 T +u +v = DB 2 + − k1 (C − C∞ ), ∂t ∂x ∂y T ∞ ∂y2 ∂y

(2) (3) (4)

where u and v are respectively the x and y components velocity of the nanofluid and Ue is the velocity at the outer edge of the boundary layer, T and C are respectively the temperature and concentration of the nanofluid, while T ∞ and C∞ are the free–stream temperature and concentration. The parameter ν is the kinematic viscosity, g is the acceleration due to gravity, β and β∗ are the thermal and volumetric expansion coefficients, respectively. The Brownian and thermophoresis diffusion coefficients are defined as DB and DT respectively, τ is the effective heat capacity which is the ratio of the heat capacity of the nanoparticle and that of the base fluid. k is the permeability of the porous medium, while α is the thermal conductivity of the nanofluid and k1 is the rate of chemical reaction, such that k1 < 0 signifies a generative reaction, k1 = 0 means there is no reaction and k1 > 0 signifies a destructive reaction, Ω is the total inclination angle of the wedge and qr is the radiative heat flux which is found using Rosseland’s approximation

qr = −

4σ∗ ∂T 4 . 3k∗ ∂y

(5)

Here, k∗ is Rosseland’s mean absorption coefficient and σ∗ is the Stefan–Boltzmann constant. For sufficiently small temperature difference within the flow, ∂qr /∂y is expressed as

3 2 16σ∗ T ∞ ∂ T ∂qr =− . ∗ ∂y 3k ∂y2

(6)

The problem is subject to the following boundary conditions for the velocity, temperature and concentration

at y = 0 : as y → ∞ :

u = 0,

v = 0,

u = Ue ,

T = Tw,

T = T∞,

C = Cw ,

C = C∞ ,

(7)

where T w and Cw are respectively the reference temperature and concentration at the wall. We seek similarity solutions of Equations (1) to (4) by introducing the following transformation variables:

4

√ T − T∞ aνξx f (ξ, η), θ(ξ, η) = , Tw − T∞ √ a η= y, ξ = 1 − e−ε , ε = at. νξ ψ=

ϕ(ξ, η) =

C − C∞ Cw − C∞ (8)

Upon transformation, the system of Equations (1) to (4) become [ ( ) ] η ∂f′ Ω f ′′′ + (1 − ξ) f ′′ + ξ f f ′′ − f ′2 + 1 + (Grt θ + Grc ϕ) sin − (Ha2 + λ)( f ′ − 1) = ξ(1 − ξ) 2 2 ∂ξ η ∂θ 1 + Nr ′′ θ + ξ( f θ′ − f ′ θ) + (1 − ξ)θ′ + (Nbθ′ ϕ′ + Ntθ′2 ) + Ec f ′′2 = ξ(1 − ξ) Pr 2 ∂ξ η Nt ∂ϕ ϕ′′ + Leξ( f ϕ′ − f ′ ϕ) + (1 − ξ)ϕ′ + θ′′ − LeRϕ = Leξ(1 − ξ) , 2 Nb ∂ξ

(9) (10) (11)

and the associated boundary conditions

η=0: η→∞:

f ′ (η, ξ) = 0, f ′ (η, ξ) = 1,

f (η, ξ) = 0, θ(η, ξ) = 0,

θ(η, ξ) = 1,

ϕ(η, ξ) = 1,

ϕ(η, ξ) = 0.

(12)

Here, prime represents differentiation with respect to η. The dimensionless parameters are defined as: gβ(T w − T ∞ ) gβ∗ (Cw − C∞ ) Grt = the temperature Grashof number, Gr = the concentration Grashof number, c a2 x a2 x 2 σB0 ν ν the Hartmann number, λ = the porosity parameter, Pr = the Prandtl number, Nr = Ha2 = ρa ak α 3 16σ∗ T ∞ τDB (Cw − C∞ ) the Brownian motion parameter, Nt = the thermal radiation parameter, Nb = ∗ 3k ρc p ν a2 x2 ν τDT (T w −T ∞ ) the thermophoresis parameter, Ec = the Eckert number, Le = the Lewis number, νT ∞ c p (T w − T ∞ ) DB k1 the chemical reaction parameter. and R = a 2.2. Entropy Generation In thermodynamical systems, disorder occurs in a way that the available energy in the system is conserved. This disorder is due to thermodynamic irreversibilities. In the system considered in this study, these irreversibilities are due to heat transfer, thermal radiation, viscous dissipation, applied magnetic field, and mass transfer. The local entropy generation rate Sgen is calculated based on the second law of thermodynamics. We define this as the sum of all terms contributing to the entropy generation in the system and is given as

Sgen

( ) ( )2 ( )2 ( ) σB20 2 RDB ∂C 2 RDB ∂T ∂C 1 16σ∗ T 3 ∞ ∂T µ ∂u = 2 k+ + + + u + . 3k∗ ∂y T ∞ ∂y T∞ C∞ ∂y T ∞ ∂y ∂y T∞ 5

(13)

The characteristic rate of entropy generation (Sgen )0 is defined as

(Sgen )0 =

k (∆T )2 2 T∞

x2

.

(14)

The entropy generation is the ratio of the local entropy generation rate and characteristics rate of entropy generation and it is defined as

NS (η, ξ) =

Sgen (Sgen )0

NS (η, ξ) = Re(1 + Nr)ξ−1 θ′2 +

BrRe −1 ′′2 BrReHa2 ′2 ReγΣ2 ′2 ReγΣ ′ ′ ξ f + f + ϕ + θϕ, Π Π Π Π2

(15)

where the dimensionless parameters defined as

Re =

ax2 , ν

Br =

µUe2 , k∆T

Π=

∆T , T∞

Σ=

Cw − C∞ , C∞

Ha2 =

σB20 , aρ

γ=

RDBC∞ k

are the Reynold number, Brinkman number, dimensionless temperature difference, dimensionless concentration difference, Hartman number and a constant parameter, respectively.

3. Numerical Solution 3.1. Numerical Scheme The system of partial differential eqs. (9) to (11) is solved using the bivariate spectral quasi–linearization method (BSQLM). The basic principle of the BSQLM is to first linearize the system using the quasi– linearization technique of Bellman and Kalaba [26]. The dependent variables and their respective derivatives from the linearized system are then approximated using the Chebyshev spectral collocation method. Consider a function G(η, ξ) defined in the rectangular physical domain [0, κ∞ ] × [0, ξ f ], such that [0, κ∞ ] is the truncation of the semi–infinite spatial domain and κ∞ ∈ Z+ . The physical domain is transformed into the computational domain [−1, 1] × [−1, 1] using the linear transformation: { {η, ξ} :=

} 1 1 (κ + 1)κ∞ , (ς + 1)ξ f . 2 2

(16)

We use the Chebyshev–Gauss–Lobatto nodes (

πi κi = cos Mη

) Mη , i=0

) M π j ξ ς j = cos , Mξ j=0 (

6

(17)

where Mη and Mξ are the number of collocation points in η and ξ, respectively. The function G(η, ξ) and its derivatives are approximated at the above defined collocation points such that the first order derivatives with respect to η and ξ are defined respectively in terms of the Chebyshev differentiation matrix as dG ∑ = DipG(κ p , ς j ) = DG(κi , ς j ), dη p=0

∑ dG ∑ = d jk G(κi , ςk ) = d jk G(κi , ςk ), dξ k=0 k=0

Mη

Mξ

Mξ

(18)

where D = 2D/κ∞ and d = 2d/ξ f are scaled differentiation matrices. Higher order derivative with respect to η are defined as

G(o) (η, ξ) = D(o)G(κi , ς j ), i = 0, · · · Mη , j = 0, · · · Mξ ,

(19)

where o is the order of the derivative. If we define [ ( ) ] Ω η ∂f′ ′′ ′′ ′2 2 ′ Γ f := f + (1 − ξ) f + ξ f f − f + 1 + (Grt θ + Grc ϕ) sin − (Ha + λ)( f − 1) − ξ(1 − ξ) 2 2 ∂ξ 1 + Nr ′′ η ∂θ Γθ := θ + ξ( f θ′ − f ′ θ) + (1 − ξ)θ′ + (Nbθ′ ϕ′ + Ntθ′2 ) + Ec f ′′2 − ξ(1 − ξ) Pr 2 ∂ξ η Nt ′′ ∂ϕ Γϕ := ϕ′′ + Leξ( f ϕ′ − f ′ ϕ) + (1 − ξ)ϕ′ + θ − LeRϕ − Leξ(1 − ξ) . 2 Nb ∂ξ ′′′

(20) (21) (22)

The bivariate quasi-linearization for Equations (9) to (11) is given as

′′′ ′′ ′ α0r fr+1 + α1r fr+1 + α2r fr+1 + α3r fr+1 + α4r

∂ fr+1 + α5r θr+1 + α6r ϕr+1 = R f ∂ξ

∂θr+1 ′′ ′ + β4r fr+1 + β5r fr+1 + β6r fr+1 + β7r ϕ′r+1 = Rθ ∂ξ ∂ϕr+1 ′ ′′ δ0r ϕ′′r+1 + δ1r ϕ′r+1 + δ2r ϕr+1 + δ3r + δ4r fr+1 + δ5r fr+1 + δ6r θr+1 = Rϕ , ∂ξ ′′ ′ β0r θr+1 + β1r θr+1 + β2r θr+1 + β3r

(23) (24) (25)

subject to the boundary conditions ′ fr+1 (0, ξ) = 0,

fr+1 (0, ξ) = 0,

′ fr+1 (∞, ξ) = 1,

θr+1 (∞, ξ) = 0,

θr+1 (0, ξ) = 1,

ϕr+1 (0, ξ) = 1,

ϕr+1 (∞, ξ) = 0,

(26)

and the terms on the right side of Equations (23) to (25) are defined as

R f = α0r fr′′′ + α1r fr′′ + α2r fr′ + α3r fr + α4r

∂ fr + α5r θr + α6r ϕr − Γ f , ∂ξ

∂θr + β4r fr′′ + β5r fr′ + β6r fr + β7r ϕ′r − Γθ , ∂ξ ∂ϕr Rϕ = δ0r ϕ′′r + δ1r ϕ′r + δ2r ϕr + δ3r + δ4r fr′ + δ5r fr + δ6r θr′′ − Γϕ . ∂ξ Rθ = β0r θr′′ + β1r θr′ + β2r θr + β3r

7

(27) (28) (29)

The linearization coefficients are defined in Section A. Upon applying the Chebyshev spectral collocation to Equations (23) to (25), the numerical scheme becomes   Mξ ∑   [ ] diag(α0r )D3 + diag(α1r )D2 + diag(α2r )D + diag(α3r ) + diag(α4r )D d jk  Fr+1, j + diag(α5r ) Θr+1, j +  [

k=0

] diag(α6r ) Φr+1, j = R f

(30)   Mξ ∑   [ ] 2 2   diag(β4r )D + diag(β5r )D + diag(β6r ) Fr+1, j + diag(β0r )D + diag(β1r )D + diag(β2r ) + diag(β3r ) d jk  Θr+1, j k=0

[ ] + diag(β7r )D Φr+1, j = Rθ [ ] [ ] [ ] diag(δ4r )D + diag(δ5r ) Fr+1, j + diag(β6r )D2 Θr+1, j + diag(δ0r )D2 + diag(δ1r )D + diag(δ2r ) Φr+1, j + diag(δ3r )

Mξ ∑

d jk Φr+1, j = Rϕ ,

(31)

(32)

k=0

and the associated boundary conditions

Mη ∑

D Mη p Fr+1 (κ p , ς j ) = 0,

Fr+1 (κ Mη , ς j ) = 0,

Θr+1 (κ Mη , ς j ) = 1,

D0p Fr+1 (κ p , ς j ) = 1,

Θr+1 (κ0 , ς j ) = 0,

Φr+1 (κ0 , ς j ) = 0,

Φr+1 (κ Mη , ς j ) = 1,

p=0 Mη ∑

(33)

p=0

where r and r + 1 are the previous and current iterations respectively. Fr+1, j , Θr+1, j , Φr+1, j are the solutions f (η, ξ), θ(η, ξ) and ϕ(η, ξ) evaluated at each value of ξ. 3.2. Error and Convergence The numerical solutions obtained are tested for convergence and accuracy using both the convergence error norms and the residual errors. Both errors are defined using the l∞ norm. The convergence error norm is the difference between successive approximations, while the residual error quantifies the extent to which the solutions are approximated. The residual of the solutions f (η, ξ), θ(η, ξ), ϕ(η, ξ) for each level of ξ are defined approximately as

[ ]

Res(q) =

Γq Fr+1, j , Θr+1, j , Φr+1, j

∞

j = 0, 1, · · · , Mξ , q = { f, θ, ϕ} and the convergence error norms are defined as

8

(34)



Eq =

Qr+1, j − Qr, j

∞

(35)

j = 0, 1, · · · , Mξ . We remark here that the initial guesses which approximate the scheme at ξ = 0 are given as

F0,Mξ (η) = η + e−η − 1,

Θ0,Mξ (η) = e−η ,

Φ0,Mξ (η) = e−η .

(36)

Unless varied in the figures or stated otherwise, numerical computations are carried out using the following parameter values Pr = 6.8, Nr = 1, λ = 0.1, Nb = 0.5, Ec = 0.1, Nt = 0.5, Ω = π/3, Grt = 0.6, Grc = 0.6, Le = 0.2, R = 1, Ha = 2, Re = 0.5, Br = 1.5, Π = 10, Σ = 10, ϵ = 0.1 and ξ = 1. These values are assumed from previous studies on flow past a wedge [14, 15]. The residual and the convergence error norms for the solution f (η, ξ) for different values of the thermophoresis parameter are shown in Figure 1. After the third iteration, the errors are seen to be at approximately 10−7 , with the convergence error norms at approximately 10−6 . The residual error and convergence error norms for the solution θ(η, ξ) against iterations for different values of the thermophoresis parameter are presented in Figure 2. The residual errors are seen to be at approximately 10−9 and the convergence error norms are at approximately 10−7 after the second iteration. Figure 3 shows the residual errors and convergence error norms for the solution ϕ(η, ξ) for different values of the thermophoresis parameter. These errors are respectively at approximately 10−10 and 10−7 after the second iteration.

4. Results and Discussion The influence of flow and fluid parameters, namely the Eckert number Ec, the Hartman number Ha, the Brownian diffusion parameter Nb, the thermophoresis parameter Nt, the Reynolds number Re, the Brinkman number Br, the thermal radiation parameter Nr on nanofluid velocity profiles f ′ (η, ξ), nanofluid temperature profiles θ(η, ξ), nanoparticles concentration profiles ϕ(η, ξ), and entropy generation number profiles NS (η, ξ) are investigated. 4.1. Skin friction, Nusselt number and Sherwood number Figures 4 to 6 illustrate the effects of the magnetic, porosity, thermophoresis and Brownian diffusion parameters on the wall stress f ′′ (0, ξ), heat and mass transfer coefficients at the wall −θ′ (0, ξ) & − ϕ′ (0, ξ). Figure 4 shows the effects of the porosity parameter and Hartman number on the skin friction, local Nusselt and local Sherwood numbers. The figure shows both the skin friction and Sherwood number increasing for increasing value of Ha, while the local Nusselt number decrease for increasing value of the Hartman number. 9

On the other hand, both the wall stress and the wall’s mass transfer coefficient increase and the wall’s heat transfer coefficient decreases for increasing value of the porosity parameter. Figure 5 presents the impacts of the thermophoresis and porosity parameters on wall stress, heat and mass transfer coefficient on the wall. The results indicate that as the thermophoresis parameter, increases, the local Nusselt number decreases. However, as the porosity parameter increases, we obtain the results in Figure 4 hold. Figure 6 shows the influence of the Brownian diffusion parameter on the skin friction, local Nusselt number and Sherwood numbers. The result indicates that skin friction and local Nusselt number decrease as mass diffusivity is increased, while the local Sherwood number increases. 4.2. Impacts of thermo–physical parameters variation on flow profiles The effects of thermo-physical parameters on flow velocity profiles, temperature profiles and nanoparticle concentration profiles are shown in Figures 7 to 15. Figures 7 and 8 present the effect of the magnetic parameter on the flow profiles. In Figure 7, the temperature increases with increasing Ha, while the velocity increases with increasing Ha in the vicinity of the wedge. However, as flow advances into the boundary layer region, magnetic effect on the velocity profiles decreases with the Hartman number. Figure 8 shows the effect of Ha on the concentration profiles. At the edge of the wedge (η < 1), the nanoparticle concentration decreases as Ha increases. The effect of the Brownian diffusion parameter on the nanofluid velocity and temperature and nanoparticle concentration are shown in Figures 9 and 10 respectively. A minuscule increase in the Brownian diffusion parameter leads to a slight decrease in the velocity and an increase in the temperature. On the other hand, the nanoparticle concentration diminishes in a large measure for small increase in the Brownian diffusion parameter. A general reason for this variation is because due to random diffusivity, nanoparticles rearrange to form a new structure, thus increasing the thermal conductivity of the nanofluid. Figures 11 and 12 respectively show the effects of the thermophoresis parameter on the nanofluid temperature and the nanoparticle concentration. From Figure 11, we observed that the velocity and temperature increase with increasing thermophoresis parameter. However, Figure 12 shows that close to the wedge, concentration decreases with increasing thermophoresis parameter, before eventually increasing as Nt increases, for flow far from the edge. Viscous dissipation effects on temperature and concentration are illustrated in Figures 13 and 14 respectively. It can be observed that increasing Ec increases the nanofluid temperature. On the contrary, the nanoparticle concentration decreases for increasing Ec. The volume of the nanoparticle diminishes close to the vicinity of the edge of the wedge. 4.3. Entropy generation The effects of various thermo–physical parameters on entropy generation on the flow over a porous wedge are illustrated in Figures 16 to 20. The influence of the viscous effect on heat produced by molecular conduction as in the Brinkman number is shown in Figure 16. There is a direct proportionality between 10

the Brinkman number and entropy generation. Close to the wedge (η < 1), there is a significant increase in the entropy generation number. This is because the Brinkman number has a direct relationship with the product of the fluid’s viscosity and the square of the velocity at the outer edge of the wedge. Viscous effect dominate the flow before this eventually balances out in the region of the boundary layer. Hence, the entropy generation is small for smaller values of Br. Figure 17 shows the effect of viscous heating dissipation; namely the Eckert number on entropy generation. Closer to the edge of the wedge, there is an inverse relationship between Ec and the entropy generation number, with NS (η, ξ) decreasing as Ec increases. However, in the boundary layer region, entropy generation increases with increasing Ec. The same outcome is observed in the effect of viscous dissipation in Figure 13. The magnetic influence on the entropy generation number is illustrated in Figure 18. The magnetic field tends to create a drag force in the flow, thus slowing the fluid motion and decreasing or increasing the temperature and concentration depending on the type of convection in the flow. In this study, we found that an increase in the magnetic parameter increases the entropy generation. This effect is significant near the wedge (η < 1). Figure 19 illustrates the effect of the thermal radiation parameter. An increase in thermal radiation leads to an increase in the thermal boundary layer of the flow as seen in Figure 15. In essence, an increase in thermal radiation leads to an increase in heat transfer resulting in an increase in entropy generation. In Figure 20, the effects of the Reynolds number on entropy generation are shown. With diminished viscous force, the inertial force dominates the flow, resulting in a notable contribution of heat transfer to the entropy generation. There is a proportionality between the Reynolds number and the entropy generation number.

11

100 Nt = 0.1 Nt = 0.1 Nt = 0.3 Nt = 0.3 Nt = 0.5 Nt = 0.5

10-2

10-4

10-6

10-8

10-10

0

2

4

6

8

10

12

14

16

Figure 1: Residual errors ∥Res( f )∥∞ (dash lines) and the convergence error norms E f (solid lines) against iterations for different values of the thermophoresis parameter.

100 Nt = 0.1 Nt = 0.1 Nt = 0.3 Nt = 0.3 Nt = 0.5 Nt = 0.5

10-2

10-4

10-6

10-8

10-10

10-12

0

2

4

6

8

10

12

14

16

Figure 2: Residual errors ∥Res(θ)∥∞ (dash lines) and the convergence error norms Eθ (solid lines) against iterations for different values of the thermophoresis parameter.

12

100 Nt = 0.1 Nt = 0.1 Nt = 0.3 Nt = 0.3 Nt = 0.5 Nt = 0.5

10-2

10-4

10-6

10-8

10-10

10-12

0

2

4

6

8

10

12

14

16

Figure 3: Residual errors ∥Res(ϕ)∥∞ (dash lines) and the convergence error norms Eϕ (solid lines) against iterations for different values of the thermophoresis parameter.

3 = 0.2 = 0.4 = 0.6 = 0.8

2.5 2 1.5 0.52 0.5 0.48 0.46 0.65

0.6

0.55 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Figure 4: Effects of the porosity parameter on skin friction, local Nusselt and Sherwood numbers against Hartman number.

13

Nt = 0.2 Nt = 0.4 Nt = 0.6 Nt = 0.8

2.8 2.7 2.6 2.5 0.8 0.6 0.4 0.2 0.8 0.7 0.6 0.5 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 5: Effects of thermophoresis parameter on skin friction, local Nusselt and Sherwood numbers against the porosity parameter

Nb = 0.2 Nb = 0.4 Nb = 0.6 Nb = 0.8

2.8 2.7 2.6 2.5 0.8 0.6 0.4 0.2 1

0.5

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 6: Effects of Brownian diffusion parameter on skin friction, local Nusselt and Sherwood numbers against the porosity parameter

14

f( , ) 1 1.02 1

0.8

0.98 1.5

2

2.5

0.6 Ha = 0.5 Ha = 1 Ha = 1.5 Ha = 2

0.4

0.2 ( , ) 0 0

0.5

1

1.5

2

2.5

3

3.5

Figure 7: Magnetic effect on the velocity and temperature profiles.

1 Ha = 0.5 Ha = 1 Ha = 1.5 Ha = 2

0.9 0.8 0.7 0.6 0.5 0.4

0.4

0.3

0.38 0.36

0.2

0.34 0.1 2.2

0 0

1

2

2.25

2.3 3

2.35 4

5

6

Figure 8: Magnetic effect on the concentration profile.

15

1 f( , ) Nb = 0.2 Nb = 0.4 Nb = 0.6 Nb = 0.8

0.8

0.6

0.4

0.2 ( , ) 0 0

0.5

1

1.5

2

2.5

3

Figure 9: Effect of Brownian diffusion on the velocity and temperature profiles.

1.8

Nb = 0.2 Nb = 0.4 Nb = 0.6 Nb = 0.8

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

1

2

3

4

5

6

Figure 10: Effect of Brownian diffusion on the concentration profile.

16

1 f( , ) 0.8 Nt = 0.2 Nt = 0.4 Nt = 0.6 Nt = 0.8

0.6

0.4

0.2 ( , ) 0 0

0.5

1

1.5

2

2.5

3

Figure 11: Effect of thermophoresis parameter on velocity and temperature profiles.

1.2 Nt = 0.2 Nt = 0.4 Nt = 0.6 Nt = 0.8

1

0.8

0.6

0.4

0.2

0 0

1

2

3

4

5

6

Figure 12: Effect of thermophoresis parameter on concentration profile.

17

1.2 Ec = 0 Ec = 0.1 Ec = 0.3 Ec = 0.5

1

0.8

0.6

0.4

0.2

0

-0.2 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Figure 13: Effect of viscous dissipation on the temperature profile.

1.2 Ec = 0 Ec = 0.1 Ec = 0.3 Ec = 0.5

1

0.8

0.6

0.52 0.5 0.48 0.46 0.44

0.4

0.2

1.9

1.95

2

0 0

1

2

3

4

5

6

Figure 14: Effect of viscous dissipation on the concentration profile.

18

1 Nr = 0.5 Nr = 0.8 Nr = 1 Nr = 1.3

0.8

0.6

0.4

0.2

0

-0.2 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Figure 15: Effect of thermal radiation on temperature profile.

1.4 Br = 0.5 Br = 1 Br = 1.5 Br = 2

1.2

1

0.8

0.6

0.4

0.2

0 0

1

2

3

4

5

6

7

8

9

10

Figure 16: Influence of the Brinkman number on entropy generation profile.

19

1.3 Ec = 0.05 Ec = 0.1 Ec = 0.2 Ec = 0.3

1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Figure 17: Influence of viscous heating dissipation on entropy generation profile.

1.4 Ha = 0.5 Ha = 1 Ha = 1.5 Ha = 2

1.2

1

0.8

0.6

0.4

0.2

0 0

1

2

3

4

5

6

7

8

9

10

Figure 18: Entropy generation profile for different values of the Hartman number.

20

1.3 Nr = 0.5 Nr = 0.8 Nr = 1 Nr = 1.3

1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0

0.5

1

1.5

2

2.5

3

3.5

4

Figure 19: Effect of thermal radiation on entropy generation profile.

5 Re = 0.5 Re = 1 Re = 1.5 Re = 2

4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0

1

2

3

4

5

6

7

8

9

10

Figure 20: Entropy generation profile for various values of the Reynold number.

5. Conclusion We have presented a study of entropy generation in the flow of a magnetohydrodynamic viscous nanofluid across a permeable wedge. The effects of flow parameters on the skin friction, heat and mass transfer coefficients, fluid properties and entropy generation rate has been analyzed. Among other results, we found that viscous dissipation reduces the nanoparticle concentration in the vicinity of the wall, while it increases the nanofluid temperature. The thermal radiation and viscous dissipation both contribute significantly to entropy generation in the flow. Hence, to reduce irreversibility in the boundary layer region, thermal radiation and viscous dissipation have to be minimized.

21

Conflicts of Interest The authors declare that there are no conflicts of interest regarding this paper.

Acknowledgment This work is based on the research supported wholly by the National Research Foundation of South Africa and the University of KwaZulu-Natal. Appendix A

β4r =

The linearization coefficients are defined as

β5r =

α0r = α1r = α2r = α3r = α4r = α5r = α6r = β0r = β1r = β2r = β3r =

∂Γ f =1 ∂ f ′′′ ∂Γ f η = (1 − ξ) + ξ fr ∂ f ′′ 2 ∂Γ f = −2ξ fr′ − ξ(Ha2 + λ) ∂f′ ∂Γ f = ξ fr′′ ∂f ∂Γ f = −ξ(1 − ξ) ∂ fξ′ ( ) ∂Γ f Ω = ξGrt sin ∂θ 2 ( ) ∂Γ f Ω = ξGrc sin ∂ϕ 2 ∂Γθ 1 + Nr = ∂θ′′ Pr η ∂Γθ = ξ fr + (1 − ξ) + Nbϕ′ + 2Ntθ′ ∂θ′ 2 ∂Γθ = −ξ fr′ ∂θ ∂Γθ = −ξ(1 − ξ) ∂θξ

(37)

β6r =

(38)

β7r =

(39)

δ0r =

(40)

δ1r =

(41)

δ2r =

(42)

δ3r =

(43)

δ4r =

(44)

δ5r =

(45)

δ6r =

(46)

∂Γθ = 2Ec fr′′ ∂ f ′′ ∂Γθ = −ξθr ∂f′ ∂Γθ = ξθr′ ∂f ∂Γθ = Nbθr′ ∂ϕ′ ∂Γϕ =1 ∂ϕ′′ ∂Γϕ η = Leξ fr + (1 − ξ) ∂ϕ′ 2 ∂Γϕ = −Leξ fr′ − LeR ∂ϕ ∂Γϕ = −Leξ(1 − ξ) ∂ϕξ ∂Γϕ = −Leξϕr ∂f′ ∂Γϕ = Leξϕ′r ∂f ∂Γϕ Nt = . ∂θ′′ Nb

(48) (49) (50) (51) (52) (53) (54) (55) (56) (57) (58)

(47)

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