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Open Physics

formerly Central European Journal of Physics

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Volume 15, Issue 1

Issues

Volume 13 (2015)

Unsteady mixed convection flow through a permeable stretching flat surface with partial slip effects through MHD nanofluid using spectral relaxation method

Sami M. Ahamed
  • School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Pietermaritzburg-3209, Pietermaritzburg, South Africa
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/ Sabyasachi Mondal
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  • School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Pietermaritzburg-3209, Pietermaritzburg, South Africa
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/ Precious Sibanda
  • School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Pietermaritzburg-3209, Pietermaritzburg, South Africa
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Published Online: 2017-05-20 | DOI: https://doi.org/10.1515/phys-2017-0036

Abstract

An unsteady, laminar, mixed convective stagnation point nanofluid flow through a permeable stretching flat surface using internal heat source or sink and partial slip is investigated. The effects of thermophoresis and Brownian motion parameters are revised on the traditional model of nanofluid for which nanofluid particle volume fraction is passively controlled on the boundary. Spectral relaxation method is applied here to solve the non-dimensional conservation equations. The results show the illustration of the impact of skin friction coefficient, different physical parameters, and the heat transfer rate. The nanofluid motion is enhanced with increase in the value of the internal heat sink or source. On the other hand, the rate of heat transfer on the stretching sheet and the skin friction coefficient are reduced by an increase in internal heat generation. This study further shows that the velocity slip increases with decrease in the rate of heat transfer. The outcome results are benchmarked with previously published results.

keywords: MHD nanofluid; Slip condition; Internal heat source/sink; Spectral relaxation method; Permeable stretching surface

PACS: 02.60.Cb; 02.60.Lj; 02.70.Hm

1 Introduction

Cooling and heating of liquids are significant in many engineering applications and transportation industries. In general, many common heat transfer fluids such as ethylene glycol, oil and water are not efficient heat transfer fluids for their low thermal conductivities and poor physical and chemical characteristics [1]. In contrast, fluids with added nanometer sized metals such as titanium, aluminum, gold, silver, copper and iron or their oxides and composite materials have significantly higher thermal conductivities than these base fluids. The nanoparticles are commonly made by a high-energy-pulsed process from a conductive material. For these reasons, it is prudent to combine base fluids with nanoparticles to achieve both the characteristics of a base fluid and the physical properties of the nanoparticles [2,3]. Suspending nanoparticles affects the base fluid’s homogeneity and the randomness of molecular motion leading to higher conductive rates and better convective heat transfer performances compared to base fluids [47]. There are however mechanisms, which include particle agglomeration, particle shape/surface area, nanoparticle size, temperature and liquid layering on the nanoparticle liquid interface that still need to be fully understood.

Nanofluids have large application in industries, like micro-electro-mechanical or microprocessors electronic systems as well as in the field of biotechnology [8]. Several authors, for instance, [9,10] conducted theoretical and experimental investigations to demonstrate that nanofluids increased the heat transfer properties of base fluid. The numerical solution of the equations that describe the combined effects of thermophoresis and Brownian motion on the nanofluid flow through a flat surface in a saturated porous medium was examined by Nield and Kuznetsov [11]. Kuznetsov and Nield [12] examined a problem on natural convection boundary layer nanofluid flow with showing the effects of thermophoresis and Brownian motion. Buongiorno [13] studied that particle diffusion and thermophoresis play an important role in the flow of a nanofluid. The classical concept of a boundary layer corresponds to a thin region next to a solid surface where viscous forces are important, Blasius [14]. Viscous forces play an essential role in processes such as glass fiber drawing, crystal growing and plastic extrusion. For the case of the Blasius problem, Sakiadis [15] focused on the concept of a boundary layer flow induced by a moving plate in a quiescent ambient fluid. Further studies in this area were made by Schlichting and Gersten [16], Bejan [17], White [18] and Mukhopadhyay [19,20].

The magnetohydrodynamic (MHD) boundary layer flow of a viscous incompressible fluid which is electrically conducted has huge applications in industry and technological sectors, like cooling of nuclear reactors, the extrusion of plastics, textile industry, purification of crude oil, polymer technology, geothermal energy extractions, metallurgy, drug delivery and biological transportation [21, 22]. The application of the magnetic field produces a Lorentz force that increase the mixing processes as an active micro mixing mechanism. The transport system of conductive biological fluids in the micro systems will be benefitted from research [23]. The simultaneous occurrence of buoyancy and magnetic field forces on fluid flow were investigated by many researchers [2428]. In these investigations, many authors have used a no-slip boundary condition. Recently, some researchers have investigated boundary layer flow assuming a slip condition at the boundary [2935].

Stagnation point flow, illustrating the fluid motion near a stagnation region, exists near solid bodies in that fluid flow. Chiam [36] studied fluid flow on a steady stagnation-point through an elastic surface with equal free stream and stretching velocities.

The mail focus of this study is to expand the work of Nandy and Mahapatra [37] to unsteady flow over a permeable stretching flat plate with an internal heat source or sink. For the nanoparticles we have used the boundary condition suggested recently by Kuznetsov and Nield [38] where active control of the nanoparticle volume fraction at the surface is not possible. Appropriate similarity transformations reduce the conservation equations to a set of nonlinear ordinary differential equations. These equations are solved by using a spectral relaxation method, numerically (see Motsa et al. [39]).

2 Problem formulations

Consider an unsteady, viscous, incompressible, laminar nanofluid flow past a stretching flat plate. The flow is restricted to the area y ≥ 0, y and x are the coordinate system normal to the stretching flat plate and along the stretching surface, respectively. The concentration and temperature on the surface are ϕw and Tw, respectively, and away from the surface the values are ϕ and T, respectively. We assume that at a time t ≥ 0, the plate stretching velocity is uw = bx. The free stream velocity is u = ax. Here, b is constant with b > 0 corresponding to a stretching plate and b < 0 to a shrinking plate. The fluid flow is subjected to a uniform magnetic field of strength B0 which is applied in a normal direction to the plane y = 0 and to the heat source/sink. The fluid is electrically conducting with constant properties, except for the density in the buoyancy term in the momentum equation. With these assumptions, the Oberbeck-Boussinesq and the boundary layer approximations, the unsteady momentum, energy and concentration equations can be described as (see Nandy and Mahapatra [37]); ux+vy=0,(1) ρfut+uux+vuyududx=μf2uy2+σB02(uu)+(1ϕ)ρfβ(TT)g(ρpρf)(ϕϕ)g,(2) Tt+uTx+vTy=αf2Ty2+1(ρcp)fq+τDBϕyTy+DTTTy2,(3) ϕt+uϕx+vϕy=DB2ϕy2+DTT2Ty2,(4) where t, u and v represent the time, the fluid velocity components along the x and y axis, respectively, νf is the fluid kinematic viscosity, σ is the fluid electrical conductivity, ρf is the density far from the plate surface, g is the acceleration due to gravity, ρf is the fluid density, T is the fluid temperature, β is the coefficient of thermal expansion, ρp is the density of the particles, (ρ cp)f is fluid heat capacity, ϕ is nanoparticle volume fraction, αf is thermal diffusivity, q‴ is the rate of internal heat sink (< 0) or heat source (>0) coefficient, The ratio of the effective heat capacity of nanoparticle material with the heat capacity of the fluid is τ, DT denotes the thermophoresis diffusion coefficient and DB is the Brownian diffusion coefficient.

The boundary conditions are (see Kuznetsov and Nield [12], Nandy and Mahapatra [37] and Kuznetsov and Nield [38]) v=Vw,u=uw+Luy,aty=0,t0,T=Tw,DBϕy+DTTTy=0aty=0,t0,uu,TT,ϕϕasy,t0,(5) with the initial conditions u0,TT,ϕϕatt<0, where L is a proportionality constant. Vw denotes the surface mass flux. Negative values of Vw imply fluid suction and positive values of Vw imply injection. Also, Vw = 0 corresponds to an impermeable plate.

q‴ is defined according as follows, (see [4042]) q=kuwxνf[A(TwT)eη+B(TT)],(6) where B* and A* are temperature-dependent internal heat source/sink and coefficients of space-dependent, respectively and η is a dimensionless variable defined in Eq. (7). The first term in the Eq. (6) represents the dependence of the internal heat sink or source on the space coordinates while the second term represents its dependence on the temperature. Now, for the case when the case A* < 0 and B* < 0, indicates to internal heat sink while both A* > 0 and B* > 0, indicates to internal heat source.

We introduce non-dimensional variables given by η=bνfξy,ξ=1eτ^,ψ=bνfξxf(η,ξ),θ(η,ξ)=TTTwT,τ^=bt,Φ(η,ξ)=ϕϕϕ,(7) where θ(η,ξ) and Φ(η,ξ) are the non-dimensional temperature and concentration, respectively and the stream function ψ is defined as u=ψyand v=ψx.

Substituting (7) into Eqs. (1) - (4) gives f+12(1ξ)ηf+ξfff2+Mεf+ε2+λ(θNrΦ)=ξ(1ξ)fξ,(8) θ+Pr12(1ξ)ηθ+ξfθ+NbΦθ+Ntθ2+ξAeη+Bθ=Prξ(1ξ)θξ,(9) Φ+Le12(1ξ)ηΦ+ξfΦ+NtNbθ=Leξ(1ξ)Φξ,(10) with boundary conditions f(0,ξ)=S,f(0,ξ)=1+γf(0,ξ),θ(0,ξ)=1,Φ(0,ξ)+NtNbθ(0,ξ)=0,f(,ξ)ε,θ(,ξ)0,Φ(,ξ)0,(11) where the differentiation with respect to η is denoted as prime, S is the dimensionless suction/blowing parameter, γ is the dimensionless slip factor, and the stretching parameter is ε where S=Vwbνfξ,γ=Lbνfξandε=ab.

In Eqs. (8)-(11) the parameters are the magnetic parameter M, the buoyancy parameter λ, Grx is the local Grashof number, Pr is the Prandtl number, Rex is the local Reynolds number, Nt is the thermophoresis parameter, Nr is the buoyancy ratio parameter, Nb is the Brownian motion parameter, and Le is the Lewis number. These parameters are defined as: M=σB02ρfb,λ=GrxRex2,Rex=xuwνf,Nb=τDBϕνf,Grx=(1ϕ)ρfρfgβ(TwT)x3νf2,Nr=(ρpρf)(1ϕ)ρfϕβ(TwT),Pr=νfαf,Nt=τDT(TwT)Tνf,Le=νfDB.(12)

We note that in the case of assisting flow λ > 0, and for opposing flow λ < 0, λ = 0 corresponds to free convection. The skin friction coefficient Cf and the local Nusselt number Nux are defined as Cf=τwρfuw2,Nux=xqwkfTwT,(13) where τw is the shear stress along the stretching surface, qw is the wall heat flux, respectively defined as τw=μfuyy=0,qw=kfTyy=0.(14)

Hence using Eq.(14) we get ξRexCf=f(0,ξ),ξRexNux=θ(0,ξ).(15)

With the revised boundary condition, the Sherwood number which represents the dimensionless mass flux is identically zero (see Kuznetsov and Nield [12], Kuznetsov and Nield [38]).

3 Method of solution

The spectral relaxation method (see [39]) was used to solve the system of non-similar equations (8)(10) with the boundary conditions (11). In the SRM framework, we obtain the iterative scheme fr+1=ur+1(16) ur+1+a1,rur+1+a2,rur+1+a3,r=a4,rur+1ξ,(17) θr+1+b1,rθr+1+b2,rθr+1+b3,r=b4,rθr+1ξ,(18) Φr+1+c1,rΦr+1+c2,r=c3,rΦr+1ξ,(19) with boundary conditions fr+1(0,ξ)=S,ur+1(0,ξ)=1+γf(0,ξ),θr+1(0,ξ)=1,NbΦr+1(0,ξ)+Ntθr+1(0,ξ)=0,ur+1(,ξ)ε,θr+1(,ξ)0,Φr+1(,ξ)0,(20) where a1,r=12(1ξ)η+ξfr,a2,r=ξM,a3,r=ξε2+Mεur2+λ(θrNrΦr),a4,r=ξ(1ξ)(21) b1,r=Pr12(1ξ)η+ξfr+1+NbΦr,b2,r=ξBb3,r=ξAeη+PrNtθr2,b4,r=Prξ(1ξ)(22) c1,r=Le12(1ξ)η+ξfr+1,c2,r=NtNbθr+1,c3,r=Leξ(1ξ).(23)

In Equations (16)(23) the indices r and r + 1 denote the previous and current iteration levels, respectively. Starting from initial approximations denoted by f0, u0, θ0, and Φ0, equations (9)(11) are solved iteratively for fr + 1(η,ξ), ur + 1(η,ξ), θr + 1(η,ξ), and Φr + 1(η,ξ) (r = 0,1,2,…). Equations (8)(10) were discretized using the Chebyshev spectral collocation method in the η direction while the discretization in the ξ direction uses the implicit finite difference method. This leads to a system of linear equations of the form A1fr+1n+1=urn,(24) A2ur+1n+1=B2ur+1n+K2,(25) A3θr+1n+1=B3θr+1n+K3,(26) A4Φr+1n+1=B4Φr+1n+K4,(27) where A1=D,(28) A2=12D2+[a1,rn+12]dD+a2,rn+12Ia4,rn+12ΔξI,B2=12D2+[a1,rn+12]dD+a2,rn+12Ia4,rn+12ΔξI,K2=a3,rn+12,(29) A3=12D2+[b1,rn+12]dD+b2,rn+12Ib4,rn+12ΔξI,B3=12D2+[b1,rn+12]dD+b2,rn+12Ib4,rn+12ΔξI,K3=b3,rn+12,(30) A4=12D2+[c1,rn+12]dDc3,rn+12ΔξI,B4=12D2+[c1,rn+12]dDc3,rn+12ΔξI,K4=c2,rn+12.(31)

Here I is an (N + 1) × (N + 1) identity matrix, and [ . ]d are diagonal matrices of order (N + 1) × (N + 1).

In applying the SRM a computational domain of extent L = 20 was chosen in the η-direction. Through numerical experimentation, this value was found to give accurate results for all the selected physical parameters. Increasing the value of η do not change the results to a significant extent. The number of collocation points is used in the spectral method discretization is Nx = 100 in all cases. The calculations are carried until some desired, tolerance level ε is attained. The tolerance level was defined to be the maximum infinity norm of the difference between the values of the calculated quantities, that is maxfr+1n+1frn+1,θr+1n+1θrn+1,Φr+1n+1Φrn+1<ϵ.

To ensure the accuracy of the results, a sufficiently small step size Δξ was used. The step size was chosen to be small enough such that further reduction in step size did not change the results of the flow properties.

4 Results and discussion

The transformed system of coupled nonlinear ordinary differential equations (8)-(10) including boundary conditions (11) are solved numerically using the spectral relaxation method. The numerical results are presented here to show the velocity, concentration and temperature profiles, the rate of heat transfer and the skin friction coefficient for different physical parameter values. These results are presented both graphically and in tabular form. In the numerical simulations the default parameter values used are, unless otherwise specified; Pr = 6.8, ε = 0.5, λ = 0.5, M = 0.1, Nr = 0.5, A* = 2.0, Nb = 0.5, B* = 1.0, Nt = 0.5, S = 0.1, Le = 10, γ = 0.1 and ξ = 0.5.

To verify the accuracy of our numerical scheme, a comparison of the computed skin friction coefficient is made with earlier results of Anwar et al. [43] in Table 1. Here we observe an excellent agreement validating the accuracy of the current numerical results. The residual error in the numerical simulations against the number of iterations for different values of γ, ε, S and M is shown in Figure 1. These results again confirm that the numerical method used in this study converges.

Residual error for different values of (a) dimensionless slip,(b) stretching or shrinking, (c) suction or injection, (d) magnetic field parameters on velocity profile
Figure 1

Residual error for different values of (a) dimensionless slip,(b) stretching or shrinking, (c) suction or injection, (d) magnetic field parameters on velocity profile

Table 1

Comparison of the reduced skin friction coefficient −f″(0,1) when λ = Nr = γ = 0

Figure 2(a) displays the variation of f′ with respect to η for several values of γ when ε < 1, ε = 1 and ε > 1. Here we note that f′ increases with γ when ε > 1 while the opposite is true when ε < 1. An increase in the slip parameter has the effect of reducing the velocity at the wall. The velocity decreases asymptotically to zero at the edge of the boundary layer. The boundary layer thickness decreases as γ increases. We note that when ε = 1, increasing γ gives no further changes in the velocity profiles because at this stage the external stream velocity becomes equal to the stretching velocity. This causes a frictionless Hiemenz flow [44].

(a) Effects of dimensionless slip factor γ and stretching parameter ε,(b) Effects of buoyancy force λ and buoyancy force ratio Nr parameters, on velocity profiles
Figure 2

(a) Effects of dimensionless slip factor γ and stretching parameter ε,(b) Effects of buoyancy force λ and buoyancy force ratio Nr parameters, on velocity profiles

Figure 2(b) explores the effect of λ and Nr on the velocity profiles. It is seen that the velocity increases as the buoyancy force parameter increases. This is due to the fact that the assisting flow (λ > 0) induces a favorable pressure gradient which enhances the fluid flow in the boundary layer and as a result the momentum boundary layer increases whereas the reverse occurs for the opposing flow when λ < 0.

Figure 3 shows the effects of S and M on f′ with respect to η. Here we note that f′(η, ξ) decreases with increasing values of S and M. This behaviour is due to the fact that M increases the resistive forces on the flat plate which in turn reduces the fluid velocity and hence the motion of the fluid is slowed down. It is known that the wall suction (S > 0) has the tendency to decrease the momentum boundary layer thickness which is the cause of reduction in the velocity. But the the opposite behaviour can be seen for fluid injection (S < 0).

Effects of dimensionless suction/injection S and magnetic M parameters on velocity profiles
Figure 3

Effects of dimensionless suction/injection S and magnetic M parameters on velocity profiles

The internal heat source or sink in the boundary layer has an influence on the temperature fields as illustrated in Figure 4(a). It is clear that increasing A* and B* increases the temperature distribution within the fluid and the thermal boundary layer thickness increases. Figure 4(b) shows the variation of the temperature profiles for various values of the slip and stretching parameters. The temperature profiles increase with increasing γ when ε < 1 but the opposite trend is observed when ε > 1 as the thermal boundary layer thickness decreases.

(a) Effects of space-dependant A* and temperature-dependent B* parameters, (b) Effects of dimensionless slip factor γ and stretching   parameter ε, on temperature profiles
Figure 4

(a) Effects of space-dependant A* and temperature-dependent B* parameters, (b) Effects of dimensionless slip factor γ and stretching parameter ε, on temperature profiles

Figure 5(a) depicts the effect of λ and Nr on the temperature profiles. We note that with the increasing value of buoyancy force parameter decreases the nanofluid temperature. Moreover, positive λ values induce a favorable pressure gradient that enhances the nanofluid temperature in the boundary layer when Nr increases. Consequently, the thermal boundary layer increases while the opposite is observed for negative λ values. Figure 5(b) demonstrates the influence of Le on the temperature distribution within the boundary layer in the presence of magnetic influence, buoyancy force, heat sink/source, thermophoresis/Brownian motion. It is observed that the temperature profiles including the thickness of the temperature boundary layer increases with increases in the values of Le.

(a) Effects of buoyancy force λ and buoyancy force ratio Nr parameters,(b) Effects of Lewis number Le, on temperature profiles
Figure 5

(a) Effects of buoyancy force λ and buoyancy force ratio Nr parameters,(b) Effects of Lewis number Le, on temperature profiles

Figure 6(a) suggests that, in so far as the boundary layer temperature is concerned, thermophoresis plays the same role as the fluid Brownian motion. Thus, the fluid temperature increases with both Nt and Nb in the boundary layer region. The physical reason for this behaviour is that increased random motion of the nanoparticles increases the fluid temperature which enhances the thickness of the thermal boundary layer profiles. Figure 6(b) depicts the response of the temperature profiles to changing suction and magnetic field parameter values. Moreover, as would be expected, we note that the fluid temperature decreases with increase in suction but is higher in the case of injection.

(a) Effects of thermophoresis Nt and Brownian motion Nb parameters, (b) Effects of dimensionless suction/blowing S and magnetic M parameters, on temperature profiles
Figure 6

(a) Effects of thermophoresis Nt and Brownian motion Nb parameters, (b) Effects of dimensionless suction/blowing S and magnetic M parameters, on temperature profiles

The traditional model is revised such that the nanofluid particle volume fraction on the boundary is passively controlled. The effect of this change in boundary conditions can be seen in Figures 7 to 9 for different parameters. The impact of heat sink/source parameters B* and A* on the concentration profiles is presented in Figure 7(a). It can be seen that, away from the boundary, the concentration profiles increase with an increase in the values of A* and B*. The effect of γ and ε on nanoparticle concentration is shown in Figure 7(b). An increase in the value of partial slip parameter value leads to an increase in the nanoparticle concentration when ε < 1 while the opposite trend is observed for ε > 1. In addition, the concentration increases and attains its highest value in the vicinity of the stretching plate near η = 0.9 and then decreases to the zero. Figure 8(a) displays the effect of λ and Nr on the nanoparticle concentration profile distributions. The nanoparticle concentration decreases asymptotically with increasing values of λ. Furthermore, in the case of opposing flow, the nanoparticle concentration decreases with increasing Nr. Figure 8(b) shows the nanoparticle concentration profiles for several values of Le. It is seen that the nanoparticle volume fraction decreases with increases in Le and this manifested through the reduction in the thickness of the concentration boundary layer. Additionally, the nanoparticle concentration profiles decrease asymptotically to zero at the edge of the boundary layer.

(a) Effects of temperature-dependent B* and space-dependent A* parameters, (b) Effects of dimensionless slip γ and stretching ε parameters, on concentration profiles
Figure 7

(a) Effects of temperature-dependent B* and space-dependent A* parameters, (b) Effects of dimensionless slip γ and stretching ε parameters, on concentration profiles

(a) Effects of buoyancy force λ and buoyancy force ratio Nr parameters, (b) Effects of Lewis number Le, on concentration profiles
Figure 8

(a) Effects of buoyancy force λ and buoyancy force ratio Nr parameters, (b) Effects of Lewis number Le, on concentration profiles

(a) Effects of thermophoresis Nt  parameter, (b) Effects of suction/injection S and magnetic M parameters, and (b) Effects of Brownian motion Nb parameters, on concentration profiles
Figure 9

(a) Effects of thermophoresis Nt parameter, (b) Effects of suction/injection S and magnetic M parameters, and (b) Effects of Brownian motion Nb parameters, on concentration profiles

The nanoparticle concentration profiles are presented in Figure 9(a) for various values of the thermophoresis parameter. The results show that the nanoparticle concentration profiles increase with increasing Nt away the boundary layer region. It is interesting to note that the distinctive peaks in the profiles occur in regions adjacent to the surface for higher values of thermophoresis parameter. This means that the nanoparticle concentration profile takes higher value near the plate. Figure 9(b) shows that the nanoparticle concentration profiles increase with increasing M up to a certain value of η, beyond which the opposite trend is observed. It is clear that as S increases the concentration profiles increase to the highest value in the vicinity of the plate and then decreases to zero in the quiescent fluid. As a result, the concentration boundary layer thickness increases close to the plate surface and decreases far from the surface with increasing S. Figure 9(c) shows that the concentration profile for various values of Nb. It is seen that the concentration profiles decrease with increasing Nb. The concentration profiles attain their maximum value near the η = 1.0.

Figure 10(a) shows the variation of the skin friction coefficient with ξ in response to changes in A* and B*. The skin friction coefficient decreases with increases in both A* and B* when ξ increases. This suggests that A* and B* can be useful parameters for reducing the drag coefficient. Variations of the skin friction coefficient as a function of ξ for different values of the slip γ and stretching ε parameters are shown in Figure 10(b). We note that when ε < 1 the skin friction coefficient increases with increasing ξ, and decreases with increasing γ. The highest surface shear stress occurs with the no-slip velocity condition.

(a) Effects of space-dependent A* and temperature-dependent B* parameters, (b) Effects of dimensionless slip γ and stretching ε parameters, on skin friction coefficient
Figure 10

(a) Effects of space-dependent A* and temperature-dependent B* parameters, (b) Effects of dimensionless slip γ and stretching ε parameters, on skin friction coefficient

The variation of skin friction coefficient with λ and Nr is shown in Figure 11(a). We observe that the skin friction coefficient decreases with increasing λ. In the case of opposing flow, the skin friction coefficient increases with ξ and decreases with an increase in Nr. Figure 11(b) shows that the skin friction coefficient decreases with increasing Lewis numbers but increases with increasing ξ.

(a) Effects of buoyancy force λ and buoyancy force ratio Nr parameters, (b) Effects of Lewis number Le on skin friction coefficient
Figure 11

(a) Effects of buoyancy force λ and buoyancy force ratio Nr parameters, (b) Effects of Lewis number Le on skin friction coefficient

Figure 12(a) shows the variation of the skin friction coefficient with different values of Nt and Nb. The skin friction coefficient decreases as Nt and Nb increase but increases with increasing ξ. Figure 12(b) shows the skin friction coefficient with respect to ξ for different values of M and S. The skin friction coefficient increases with increasing M and S. Figures 13 - 14 show the effect of Pr on temperature and concentration profiles with same values of other parameters where dashed line indicates Nb = 0.1 and solid line for Nb = 0.5. The nature of those figures are similar to the Figures 4(a) and 9(b).

(a) Effects of thermophoresis Nt and Brownian motion Nb parameters, (b) Effects of suction/injection S and magnetic M parameters on skin friction coefficient on skin friction coefficient
Figure 12

(a) Effects of thermophoresis Nt and Brownian motion Nb parameters, (b) Effects of suction/injection S and magnetic M parameters on skin friction coefficient on skin friction coefficient

Effect of Pr on temperature profile for with same values of other parameters (dashed line for Nb = 0.1, solid line for Nb = 0.5)
Figure 13

Effect of Pr on temperature profile for with same values of other parameters (dashed line for Nb = 0.1, solid line for Nb = 0.5)

Effect of Pr on concentration profile with same values of other parameters (dashed line for Nb = 0.1, solid line for Nb = 0.5)
Figure 14

Effect of Pr on concentration profile with same values of other parameters (dashed line for Nb = 0.1, solid line for Nb = 0.5)

5 Conclusions

In this work, the unsteady stagnation boundary layer flow of a magnetohydrodynamic nanofluid over a stretching flat plate with velocity slip was investigated. The equations that model the boundary layer equations were solved numerically using the spectral relaxation method. A parametric study was performed to explore the impact of various physical parameters on the flow, and heat and mass transfer characteristics. The internal heat source or sink is shown to enhance the nanofluid motion while reducing the skin friction coefficient and the rate of heat transfer at the stretching surface.

In this study we have shown that increasing the Brownian motion parameter and the Lewis number reduces nanoparticle concentration profiles near the boundary layer region due to increasing mass transfer. The velocity, temperature and concentration profiles and the heat and mass transfer rate on the stretching plate are strongly influenced by the slip parameter. Heat transfer rates decrease with increasing velocity slip parameter values.

Acknowledgement

The authors are grateful to the Claude Leon Foundation and the University of KwaZulu-Natal, South Africa for the financial support.

References

  • [1]

    Choi S.U.S., Nanofluids: from vision to reality through research, J. of Heat Transf, 2009, 131, 1 - 9. Web of ScienceGoogle Scholar

  • [2]

    Choi S.U.S., Enhancing thermal conductivity of fluids with nanoparticles, ASME Fluids Eng. Div., 1995, 231, 99 - 105. Google Scholar

  • [3]

    Wang X., Xu X., Choi S.U.S., Thermal conductivity of nanoparticle-fluid mixture, J. Therm. Phys. Heat Transfer, 1999, 13, 474 - 480. CrossrefGoogle Scholar

  • [4]

    Akbarinia A., Abdolzadeh M., Laur R., Critical investigation of heat transfer enhancement using nanofluids in microchannels with slip and non-slip flow regimes, Appl Therm Eng., 2011, 31, 556 - 65. Web of ScienceCrossrefGoogle Scholar

  • [5]

    Murshed S.M.S., A review of boiling and convective heat transfer with nanofluids, Renewable and Sustainable Energy Reviews, 2011, 15, 2342 - 2354.CrossrefWeb of ScienceGoogle Scholar

  • [6]

    Yu W., France D.M., Routbort J.L., Choi S.U.S., Review and comparison of nanofluid thermal conductivity and heat transfer enhancements, Heat Transfer Engineering, 2008, 29, 432 - 460.Web of ScienceCrossrefGoogle Scholar

  • [7]

    Wong K.V., de Leon O., Applications of nanofluids: current and future, Advances in Mechanical Engineering, 2010, Article ID 519659, 11 pages. Web of ScienceGoogle Scholar

  • [8]

    Kiblinski P., Phillpot S.R., Choi S.U.S., Eastman J.A., Mechanism of heat flow is suspensions of nano-sized particles (nanofluids), Int. J. Heat Mass Transfer, 2002, 45, 855 - 863. CrossrefGoogle Scholar

  • [9]

    Choi S.U.S., Zhang Z.G., Yu W., wood F.E.L., Grulke E.A., Anomalous thermal conductivity enhancement in nanotube suspensions, Applied Physics Letters, 2001, 79, 2252 - 2254. CrossrefGoogle Scholar

  • [10]

    Eastman J.A., Choi S.U.S., Li S., Yu W., Thompson L.J., Anomalously increased effective thermal conductivities of ethylene glycol-based nanofluids containing copper nanoparticles, Applied Physics Letters, 2001, 78, 718 - 720. CrossrefGoogle Scholar

  • [11]

    Nield D.A., Kuznetsov A.V., The Cheng-Minkowycz problem for natural convective boundary-layer flow in a porous medium saturated by a nanofluid, Int. J. Heat Mass Transfer, 2009, 52, 5792 - 5795. Web of ScienceCrossrefGoogle Scholar

  • [12]

    Kuznetsov A.V., Nield D.A., Natural convective boundary-layer flow of a nanofluid past a vertical plate, Int. J. Therm. Sci., 2010, 49, 243 - 247. Web of ScienceCrossrefGoogle Scholar

  • [13]

    Buongiorno J., Convective transport in nanofluids, ASME J. Heat Transfer, 2006, 128, 240 - 250. CrossrefGoogle Scholar

  • [14]

    Blasius H., Grenzschichten in Flü ssigkeiten mit kleiner Reibung, Z. Math. Phys., 1908, 56, 1 - 37. Google Scholar

  • [15]

    Sakiadis B.C., Boundary-layer behaviour on continuous solid surfaces. I. Boundary-layer equations for two-dimensional and axisymmetric flow, AIChE, 1961, 7, 1 - 2. Google Scholar

  • [16]

    Schlichting H., Gersten K., Boundary Layer Theory, Springer Berlin, 2000. Google Scholar

  • [17]

    Bejan A., Entropy Generation Minimization, CRC Boca Raton, 1996. Google Scholar

  • [18]

    White F.M., Viscous Fluid Flow, McGraw-Hill, New York, 2006. Google Scholar

  • [19]

    Mukhopadhyay S., MHD boundary layer flow and heat transfer over an exponentially stretching sheet embedded in a thermally stratified medium, Alex. Eng. J., 2013, 52, 259 - 265. CrossrefGoogle Scholar

  • [20]

    Mukhopadhyay S., Analysis of boundary layer flow over a porous nonlinearly stretching sheet with partial slip at the boundary, Alex. Eng. J., 2013, 52, 607 - 613. Google Scholar

  • [21]

    Capretto L., Cheng W., Hill M., Zhang X., Micromixing within microfluidic devices, Top Curr. Chem., 2011, 304, 27 - 68. CrossrefGoogle Scholar

  • [22]

    Kleinstreuer C., Li J., Koo J., Microfluidics of nano-drug delivery, Int. J. Heat Mass Transfer, 2008, 51, 5590 - 5597. Web of ScienceCrossrefGoogle Scholar

  • [23]

    Yazdi M.H., Abdullah S., Hashim I., Sopian K., Slip MHD liquid flowand heat transfer over non-linear permeable stretching surface with chemical reaction, Int. J. Heat Mass Transfer, 2011, 54, 3214 - 3225. CrossrefGoogle Scholar

  • [24]

    Singh K.R., Cowling T.G., Effect of magnetic field on free convective flow of electrically conducting fluids past a semiinfinite flat plate, Quarterly Journal of Mechanics and AppliedMathematics, 1963, 16, 1 - 15. Google Scholar

  • [25]

    Ahmmed S.F., Sarker A., MHD natural convection flow of viscous incompressible fluid from a vertical flat plate, Journal of Physical Sciences, 2009, 13, 77 - 85.Google Scholar

  • [26]

    Devi S.P.A., Kayalvizhi M., Analytical solution of MHDflow with radiation over a stretching sheet embedded in a porous medium, International Journal of Applied Mathematics and Mechanics, 2010, 6, 82 - 106.Google Scholar

  • [27]

    Rajesh V., Radiation effects on MHD free convection flow near a vertical plate with ramped wall temperature, International Journal of Applied Mathematics and Mechanics, 2010, 6, 60 - 677.Google Scholar

  • [28]

    Reddy M.G., Reddy N.B., Soret and Dufour effects on steady MHD free convection flow past a semi-infinite moving vertical plate in a porous mediumwith viscous dissipation, International Journal of Applied Mathematics and Mechanics, 2010, 6, 1 - 12. Google Scholar

  • [29]

    Devi S.P.A., Raj J.W.S., Thermo-diffusion effects on unsteady hydromagnetic free convection flow with heat and mass transfer past a moving vertical plate with time dependent suction and heat source in a slip flow regime, International Journal of Applied Mathematics and Mechanics, 2010, 7, 20 - 51. Google Scholar

  • [30]

    Makinde O.D., Computational modelling of MHD unsteady flow and heat transfer toward a flat plate with navier slip and newtonian heating, Brazilian Journal of Chemical Engineering, 2012, 29, 159 - 166.CrossrefWeb of ScienceGoogle Scholar

  • [31]

    Mukhopadhyay S., Md Uddin S., Layek G.C., Lie group analysis on MHD boundary layer slip flow past a heated stretching sheet in presence of heat source/sink, International Journal of Applied Mathematics and Mechanic, 2012, 8, 51 - 66.Google Scholar

  • [32]

    Eldesoky I.M.I., Influence of slip condition on peristaltic transport of a compressible Maxwell fluid through porous medium in a tube, International Journal of Applied Mathematics and Mechanics, 2012, 8, 99 - 117.Google Scholar

  • [33]

    El-Shehawy E.F., El-Dabe N.T., ElDesoky I.M.I., Slip effects on the peristaltic flow of a non-Newtonian Maxwellian fluid, Acta Mechanica, 2006, 186, 141 - 159.CrossrefGoogle Scholar

  • [34]

    Eldesoky I.M.I., Kamel M.H., Abumandour R. M., Numerical study of slip effect of unsteady MHD pulsatile flow through porous medium in an artery using generalized differential quadrature method (comparative study), Wor. J. Eng. Techno., 2014, 2,131 - 148.CrossrefGoogle Scholar

  • [35]

    Eldesoky I.M.I., Unsteady MHD pulsatile blood flow through porous medium in stenotic channel with slip at permeable walls subjected to time dependent velocity (injection/suction), Walailak J. Sci. Techno., 2014, 11, 901 - 922. Google Scholar

  • [36]

    Chiam T.C., Stagnation point flow towards a stretching plate, Journal of the Physical Society of Japan., 1994, 63, 2443 - 2444. CrossrefGoogle Scholar

  • [37]

    Nandy S.K., Mahapatra T.R., Effects of slip and heat generation/ absorption on MHD stagnation flow of nanofluid past a stretching/shrinking surface with convective boundary conditions, International Journal of Heat andMass Transfer, 2013, 64, 1091 - 1100. CrossrefGoogle Scholar

  • [38]

    Kuznetsov A., Nield D., The Cheng-Minkowycz problem for natural convective boundry layer flow in a porous medium saturated by a nanofluid: A revised model, Int. J. Heat Mass Trans., 2013, 65, 682 - 685. CrossrefGoogle Scholar

  • [39]

    Motsa S.S., Dlamini P.G., Khumalo M., Spetral relaxation method and spectral quasilinarization method for solving unsteady boundary layer flow problems, Advances inMathematial Physics, 2014, Article ID 341964, 12 pages. Google Scholar

  • [40]

    Cheng P., Minkowycz W.J., Free convection about a vertical flat embedded in a porous mediumwith application to heat transfer from a dike, J. Geophysical Research, 1977, 82, 2040 - 2044. CrossrefGoogle Scholar

  • [41]

    Bataller R.C., Viscoelastic fluid flow and heat transfer over a stretchng sheet under the effects of a non-uniform heat source, viscous dissipatio and thermal radiation, Int. J. HeatMass Tran., 2007, 50, 3152 - 3162.CrossrefGoogle Scholar

  • [42]

    Pal D., Mondal H., Effect of variable viscosity on MHD non-Darcy mixed convective heat transfer over a stretching sheet embedded in a porous medium with non-uniform heat source/sink, Commun. Nonlinear Sci. Numer. Simulat., 2010, 15, 1553 - 1564. CrossrefGoogle Scholar

  • [43]

    Anwar I., Kasim A.R.M., Ismail Z., Salleh M.Z., Shafie S., Chemical reaction and uniform heat generation or absorption effects on MHD stagnation-point flow of a nanofluid over a porous sheet, World Applied Sciences Journal, 2013, 24, 1390 - 1398. Google Scholar

  • [44]

    Sahoo B., Flow and heat transfer of a non-Newtonian fluid past a stretching sheet with partial slip, Communications in Nonlinear Science and Numerical Simulation, 2009, . CrossrefWeb of ScienceGoogle Scholar

About the article

Received: 2016-01-30

Accepted: 2016-11-08

Published Online: 2017-05-20


Citation Information: Open Physics, Volume 15, Issue 1, Pages 323–334, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2017-0036.

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© 2017 Sami M. Ahamed et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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