A theoretical investigation is carried out to examine the effects of volume fraction of nanoparticles, suction/injection, and convective heat and mass transfer parameters on MHD stagnation point flow of water-based nanofluids (Cu and Ag). The governing partial differential equations for the fluid flow, temperature, and concentration are reduced to a system of nonlinear ordinary differential equations. The derived similarity equations and corresponding boundary conditions are solved numerically using Runge-Kutta Fehlberg fourth-fifth order method. To exhibit the effect of the controlling parameters on the dimensionless velocity, temperature, nanoparticle volume fraction, skin friction factor, and local Nusselt and local Sherwood numbers, numerical results are presented in graphical and tabular forms. It is found that the friction factor and heat and mass transfer rates increase with magnetic field and suction/injection parameters.
1. Introduction
In recent years, the requirements of modern technology have encouraged interest in fluid flow studies which involve interaction of several phenomena. One such study is stagnation point flow over a permeable surface which plays an important role in many engineering problems including petroleum industries, ground water flows, extrusion of a polymer sheet from a dye, and boundary layer control. The study of a stagnation point flow towards a solid surface in moving fluid is traced back to Hiemenz in 1911. He analyzed two-dimensional stagnation point flow on stationary plate using a similarity transformation. Consequently, many investigators have extended that idea to different aspects of the stagnation point flow problem. For instance, Chamkha [1] investigated the mixed convection MHD flow near the stagnation point of a vertical semi-infinite surface. Ramachandran et al. [2] studied the dual solution analysis for different range of the buoyancy parameter. Devi et al. [3] extended the work of [2] and obtained a similarity solution for unsteady case. Grosan and Pop [4] studied axisymmetric mixed convection nanofluid flow past a vertical cylinder, Shateyi and Makinde [5] investigated MHD stagnation point flow towards a radially stretching convectively heated disk, Ibrahim et al. [6] examined MHD stagnation point flow and heat transfer due to nanofluid towards a stretching sheet, and Mustafa et al. [7] reported stagnation point flow of a nanofluid towards a stretching sheet.
In modern metallurgy and metal-working process, the magnetohydrodynamic (MHD) flow of an electrically conducting fluid towards a stretching surface is significant. These processes include the fusing of metals with electrical furnace by using magnetic field and cooling the inner first wall of the nuclear reactor containment vessel where the hot plasma is isolated from the wall by applying a magnetic field. Ishak et al. [8] studied MHD mixed convection flow near the stagnation point on a vertical permeable surface. Ishak et al. [9] numerically investigated the MHD stagnation point flow over a stretching sheet by using Keller box method and observed that velocity at a point increases with strong magnetic field when the free stream velocity is greater than the stretching velocity. Furthermore, Mahapatra et al. [10] investigated MHD stagnation point flow of a non-Newtonian fluid towards a stretching sheet and observed that, for a given magnetic parameter, the skin friction increases with an increase in power law index. Some more related studies to stagnation point flow can be found in [11–18].
Choi and Eastman [19] was the first who introduced the theory of nanofluid. Since then it has been an active field of research for about two decades. Nanofluid is a single phase mixture of suspended nanometer sized solid particles and fibers in conventional base fluids. Water, ethylene glycol mixture, toluene, and so forth are commonly used as base fluids. Wang and Mujumdar [20] found that the addition of very small amount of nanoparticles to conventional heat transfer liquids enhanced the thermal conductivity of the fluid up to approximately two times. Nanofluids offer many diverse advantages in industrial applications such as microelectronics, biomedicine, transportation, fuel cell, and nuclear reactors [21]. The literature is comprehensively occupied by nanofluids studies [22–29].
In this study, our main objective is to analyze the effect of volume fraction of nanoparticles on MHD stagnation point flow towards a moving surface with convective heat and mass transfer parameters. The governing boundary layer equations have been transformed to a two-point boundary value problem using similarity variables. These similarity equations were solved numerically using Runge-Kutta Fehlberg fourth-fifth order method. The effects of governing parameters on the dimensionless velocity, temperature, and particle concentration as well as on local skin friction, and Nusselt and Sherwood numbers have been investigated.
2. Mathematical Formulation
We considered the two-dimensional, steady, and laminar boundary layer flow of water-based nanofluids containing two types of nanoparticles, Cu and Ag, over a moving flat plate. A uniform magnetic field of strength B0 was applied in a direction normal to plane y = 0. We assumed that the plate was heated by a fluid with heat transfer coefficient hf and temperature Tf. The concentration of the fluid at the plate surface was assumed to be Cw while the uniform temperature and concentration far away from the plate were taken as T∞ and C∞, respectively. Under these assumptions, the steady boundary layer equations governing the flow, heat, and mass transfer are(1)∂u∂x+∂v∂y=0,(2)u∂u∂x+v∂u∂y=U∞dU∞dx+νnf∂2u∂y2+σnfB02ρnfU∞-u,(3)u∂T∂x+v∂T∂y=αnf∂2T∂y2,(4)u∂C∂x+v∂C∂y=D∂2C∂y2.The boundary conditions for the velocity components, temperature, and nanoparticle fraction are defined as(5)y=0:u=0,v=-v0,-knf∂T∂y=hfTf-T,-Dm∂C∂y=hmCf-C,y⟶∞:u=U∞=ax,T=T∞,C=C∞.Here, u and v are the velocity components along the axes x and y, respectively. U∞ is the free stream velocity, D is species diffusivity, σ is electrical conductivity, C is concentration of the nanofluid, and νnf,ρnf,αnf are the kinematic viscosity, density, and thermal diffusivity of the nanofluid, respectively, which are expressed as follows [7, 29]:(6)ρnf=1-ϕρf+ϕρs,μnf=μf1-ϕ2.5,αnf=knfρcpnf,ρcpnf=1-ϕρcpf+ϕρcps,νf=μfρf,knf=kfks+2kf-2ϕkf-ksks+2kf+2ϕkf-ks.We introduce the following transformations:(7)ψ=U∞xνffη,η=yU∞xνf,θη=T-T∞Tf-T∞,hη=C-C∞Cw-C∞,where hη is the dimensionless concentration and ψ represents the stream function and is defined as u=∂ψ/∂y, v=-∂ψ/∂x, so that (1) is satisfied identically. The governing equations are reduced by using (7) as follows:(8)11-ϕ2.51-ϕ+ϕρs/ρff′′′+ff′′-f′2+11-ϕ+ϕρs/ρfM1-f′+1=0,1Prknf/kf1-ϕ+ϕρcps/ρcpfθ′′+fθ′=0,h′′+Scfh′=0.The transformed boundary conditions are(9)f0=fw,f′0=0,θ′0=-kfknfNc1-θ0,h′0=-Nd1-h0,f′∞=1,θ∞=0,h∞=0,where primes denote differentiation with respect to η and the four parameters are defined as(10)fw=v0avf,M=σB02ρfD,Pr=νfαf,Sc=νfD,Nc=hfkfνfa,Nd=hmDmνfa.Here, fw is suction (fw>0) and injection (fw<0) parameter, M is magnetic parameter, Pr is Prandtl number, Sc is Schmidt number, Nc is convective heat transfer parameter, and Nd is convective mass transfer parameter.
The quantities of practical interest, in this study, are the local skin friction coefficient and local Nusselt and Sherwood numbers which are defined as(11)Cf=τwρfU∞2,Nux=-xqwkTf-T∞,Shx=xqmDCw-C∞,where τw is the shear stress at surface of the wall and qw and qm are the wall heat and mass fluxes, respectively. Using (7), we obtain(12)RexCfx=11-ϕ2.5f′′0,NuxRex=-knfkfθ′0,ShxRex=-h′0,where Rex=U∞x/νf is the local Reynolds number.
3. Numerical Solution
Using similarity transformation, the governing equations of the problem are reduced to a system of nonlinear, coupled ordinary differential equations (8) which are solved numerically by Runge-Kutta-Fehlberg fourth-fifth order method for different values of parameters such as magnetic parameter, suction/injection parameter, nanoparticles volume fraction, and convective heat and mass transfer parameter for water-based nanofluids. The effects of these parameters on the dimensionless velocity, temperature, and concentration as well as skin friction and the rate of heat and mass transfer are investigated. Since the physical domain of considered problem is unbounded, whereas the computational domain has to be finite, in all our computations we have used the value of ηmax=10 to ensure that all numerical solutions approach the asymptotic values correctly for all values of physical quantities considered in this study. For these numerical computations, the step size and convergence criteria are chosen to be 0.001 and 10^{−6}, respectively.
The physical properties of the fluid, water, and the nanoparticles are given in Table 1. To validate the obtained numerical solution, comparison has been made with previously published data from the literature for skin friction in Table 2 and they are found to be in a favorable agreement. Simultaneous effects of different parameters on friction factor and Nusselt and Sherwood numbers are presented in Table 3, while other parameter values have been kept preset. From Table 3, it is clear that all physical quantities of interest, that is, skin friction and Nusselt and Sherwood numbers, are the increasing functions of M and fw.
Thermophysical properties of water and nanoparticles [21].
Fluids
ρ (kg/m^{3})
cp (J/kg K)
k (W/m K)
β×105 (K^{−1})
Pure water
997.1
4179
0.613
21
Copper (Cu)
8933
385
401
1.67
Silver (Ag)
10500
235
429
1.89
Alumina (Al_{2}O_{3})
3970
765
40
0.85
Titanium oxide (TiO_{2})
4250
686.2
9
0.9
Comparison of skin friction for different values of ϕ at M=fw=0.
ϕ
Grosan and Pop [4]
Rashidi et al. [21]
Present results
0.0
1.232587
1.232745
1.232587
0.1
1.884324
—
1.884324
0.2
2.622743
2.622880
2.622743
Values related to the reduced skin friction coefficient and reduced Nusselt and Sherwood numbers for different values of the governing parameters (for Cu-water).
ϕ
Nc
Nd
fw
↓M,Sc→
f′′(0)
-θ′(0)
-ϕ′(0)
5
10
5
10
5
10
0.0
0.1
∞
0
0
1.23259
1.23259
0.09186
0.09186
1.04343
1.33879
0.2
1.44798
1.44798
0.07010
0.07010
1.08967
1.40160
0.0
1.50135
1.50135
0.05405
0.05405
1.10014
1.41591
0.1
1
1.58533
1.58533
0.09229
0.09229
1.10497
1.42486
0.2
1.69107
1.69107
0.07026
0.07026
1.12713
1.45457
0.1
1.67995
1.67995
0.05411
0.05411
1.12711
1.45415
0.5
5
2.43392
2.43392
0.28468
0.28468
1.21832
1.58504
1
2.43392
2.43392
0.45848
0.45848
1.21832
1.58504
10
2.43392
2.43392
1.01768
1.01768
1.21832
1.58504
100
2.43392
2.43392
1.15905
1.15905
1.21832
1.58504
1000
2.43392
2.43392
1.17538
1.17538
1.21832
1.58504
2
1
0.3
2
2.14014
2.14014
0.87883
0.87883
0.68792
0.78676
2
2.14014
2.14014
0.87883
0.87883
1.04862
1.29700
3
2.14014
2.14014
0.87883
0.87883
1.27069
1.65463
0.2
3
5
−1.0
10
2.14294
2.14294
0.03866
0.03866
0.01250
0.00025
−0.5
2.45791
2.45791
0.26187
0.26187
0.22729
0.08675
0.5
3.23291
3.23291
0.98643
0.98643
1.91318
2.62646
1.0
3.68986
3.68986
1.18305
1.18305
2.58038
3.35962
4. Results and Discussion
Numerical computations are carried out for several sets of values of the governing parameters using the Runge-Kutta-Fehlberg fourth-fifth order method. In order to illustrate the salient features of the model, the numerical results are presented in Figures 1–9. In this study we considered Pr=6.2. Figures 1(a) and 1(b) are presented to show the influence of the volume fraction of nanoparticles on the dimensionless velocity. It is observed that the momentum boundary layer thickness is smaller for Cu-water as compared to Ag-water. In the absence of magnetic field, the dimensionless velocity, inside the boundary layer, is smaller and increases with magnetic field. It is also observed that the magnetic field reduces the boundary layer thickness in both cases. This decrease is because of Lorentz force which resists the transport phenomena. From Figures 2(a) and 2(b), it is evident that the nanofluid velocity profiles width decreases (becomes narrow) with the increasing value of nanoparticle volume fraction ϕ, and vice versa. Increasing the volume fraction of nanoparticle, on the other hand, increases the thermal conductivity of the nanofluid, and we therefore observe that the thermal boundary layer thickness becomes thicker.
Effects of M and ϕ on dimensionless velocity.
Effect of ϕ on dimensionless velocity and temperature.
Effects of fwandϕ on dimensionless velocity.
Effects of Mandϕ on dimensionless temperature.
Effects of Mandϕ on dimensionless concentration.
Variation of skin friction with fwandϕ.
Variation of skin friction with volume fraction of nanoparticles.
Variation of dimensionless heat transfer rate with Nc,M,andϕ.
Variation of dimensionless mass transfer rate with Nd, M, and ϕ.
Figures 3(a) and 3(b) illustrate the effect of the volume fraction of nanoparticles in the absence/presence of suction parameter on the dimensionless velocity. It is noticed that the hydrodynamic boundary layer thickness is smaller for Cu-water as compared to Ag-water. In the absence of suction, the dimensionless velocity, inside the boundary layer, is smaller and increases with suction. It is also evident that the increasing value of suction reduces the boundary layer thickness in both the cases.
Figures 4(a) and 4(b) are drawn to analyze the influence of volume fraction of nanoparticles on the dimensionless temperature in both hydrodynamic and hydromagnetic flows of water-based nanofluids. It is observed that the dimensionless temperature at the wall is higher for hydrodynamic flows and lowers with an increase in the volume fraction of nanoparticles in both cases. This is due to the increase in thermal conductivity with an increase in the volume fraction of nanoparticles. It is also noticed from Figures 4(a) and 4(b) that the wall temperature decreases with an increase in the magnetic field. It is due to the fact that transverse magnetic field creates a drag (Lorentz force) which resists the flow and decreases the wall temperature.
Figures 5(a) and 5(b) are presented to show the effect of magnetic parameter and volume fraction of nanoparticles (Cu and Ag) on the dimensionless concentration. It is observed that in the absence of nanoparticles and the magnetic parameter, the dimensionless concentration is higher within the boundary layer and decreases with an increase in the magnetic field. However, no appreciable effect of both parameters on the dimensionless concentration could be observed.
The variation of skin friction with magnetic parameter, suction/injection parameter, and volume fraction of nanoparticles is presented in Figures 6 and 7 for both nanofluids. As expected, for conventional fluid (ϕ=0), there is no change in the skin friction of both nanofluids. However, as the volume fraction of nanoparticles increases, the skin friction also increases. This is due to increase in density of nanofluids with the volume fraction of nanoparticles. Due to higher density of Cu, the skin friction is found to be higher for Cu-water nanofluids. A monotonic increase is noticed in skin friction with suction/injection parameter. It is also important to note that the magnetic field increases the skin friction due to an additional force which opposes the velocity of nanofluids.
Figure 8 presents the variation of Nusselt number with nanoparticle volume fraction and magnetic and convection parameters. It is evident that, for both nanofluids, the local Nusselt number or the local heat transfer coefficient increases as the solid volume fraction increases. This increase in local heat transfer coefficient is the result of the increase in thermal conductivity of nanofluid caused by increase in the population of high thermal conductivity nanoparticles. Since the magnetic field reduces the surface temperature (see Figure 4), the local Nusselt number increases with magnetic field. Also the convective parameter helps in enhancing the heat transfer from the surface. This can be observed in Figure 8. It is important to note that when Nc→∞, the convective boundary condition is reduced to an isothermal boundary condition.
Finally, Figure 9 shows the effects of magnetic parameter, convective mass transfer parameter, and Schmidt number for both Cu and Ag nanofluids on Sherwood number. It is seen from the figure that the mass transfer rate increases with an increase in magnetic parameter and Schmidt number. Further, it is found that the mass transfer rate is enhanced with convective mass transfer parameter.
5. Conclusion
The effect of volume fraction of nanoparticles, suction/injection, convective heat, and mass transfer for MHD stagnation point of water-based nanofluids over a moving plate is investigated numerically. The governing equations are converted into ordinary differential equations by using appropriate similarity transformations. The similarity equations are then solved by Runge-Kutta Fehlberg fourth-fifth order method. The obtained results are displayed graphically to illustrate the effect of the different physical parameters on the dimensionless velocity, temperature, and concentration as well as the local skin friction and the local Nusselt and Sherwood numbers. The main findings of the study are as follows:
The hydrodynamic boundary layer decreases with magnetic and suction parameters.
The skin friction factor monotonically increases with suction parameter.
The local Nusselt and Sherwood numbers are the increasing functions of magnetic and suction parameters.
Competing Interests
The authors declare that they do not have any competing interests in their submitted paper.
ChamkhaA. J.Hydromagnetic mixed convection stagnation flow with suction and blowingRamachandranN.ChenT. S.ArmalyB. F.Mixed convection in stagnation flows adjacent to vertical surfacesDeviC. D. C.TakharH. S.NathG.Unsteady mixed convection flow in stagnation region adjacent to a vertical surfaceGrosanT.PopI.Axisymmetric mixed convection boundary layer flow past a vertical cylinder in a nanofluidShateyiS.MakindeO. D.Hydromagnetic stagnation-point flow towards a radially stretching convectively heated diskIbrahimW.ShankarB.NandeppanavarM. M.MHD stagnation point flow and heat transfer due to nanofluid towards a stretching sheetMustafaM.HayatT.PopI.AsgharS.ObaidatS.Stagnation-point flow of a nanofluid towards a stretching sheetIshakA.BachokN.NazarR.PopI.Mixed convection stagnation point flow near the stagnation-point on a vertical permeable surfaceIshakA.JafarK.NazarR.PopI.MHD stagnation point flow towards a stretching sheetMahapatraT. R.NandyS. K.GuptaA. S.Magnetohydrodynamic stagnation-point flow of a power-law fluid towards a stretching surfaceMaboodF.KhanW. A.IsmailA. I. M.Approximate analytical solution of stagnation point flow and heat transfer over an exponential stretching sheet with convective boundary conditionKhaliliS.DinarvandS.HosseiniR.TamimH.PopI.Unsteady MHD flow and heat transfer near stagnation point over a stretching/shrinking sheet in porous medium filled with a nanofluidNandyS. K.Analytical solution of MHD stagnation-point flow and heat transfer of casson fluid over a stretching sheet with partial slipRashidiM. M.FreidoonimehrN.Analysis of entropy generation in MHD stagnation-point flow in porous media with heat transferHayatT.AnwarM. S.FarooqM.AlsaediA.MHD stagnation point flow of second grade fluid over a stretching cylinder with heat and mass transferAliF. M.NazarR.ArifinN. M.PopI.MHD stagnation-point flow and heat transfer towards stretching sheet with induced magnetic fieldMaboodF.KhanW. A.Approximate analytic solutions for influence of heat transfer on MHD stagnation point flow in porous mediumRashidiM. M.ErfaniE.A new analytical study of MHD stagnation-point flow in porous media with heat transferChoiS. U. S.EastmanJ. A.Enhancing thermal conductivity of fluids with nanoparticlesWangX.-Q.MujumdarA. S.A review on nanofluids—part I: theoretical and numerical investigationsRashidiM. M.BegO. A.Freidooni MehrN.HosseiniA.GorlaR. S.Homotopy simulation of axisymmetric laminar mixed convection nanofluid boundary layer flow over a vertical cylinderEbaidA.Al MutairiF.KhaledS. M.Effect of velocity slip boundary condition on the flow and heat transfer of Cu-water and TiO_{2}-water nanofluids in the presence of a magnetic fieldSheikholeslamiM.AbelmanS.GanjiD. D.Numerical simulation of MHD nanofluid flow and heat transfer considering viscous dissipationNoghrehabadiA.GhalambazM.GhanbarzadehA.Comparing thermal enhancement of Ag-water and SiO_{2}-water nanofluids over an isothermal stretching sheet with suction or injectionAbdEl-GaiedS. M.HamadM. A.MHD forced convection laminar boundary layer flow of alumina-water nanofluid over a moving permeable flat plate with convective surface boundary conditionShateyiS.PrakashJ.A new numerical approach for MHD laminar boundary layer flow and heat transfer of nanofluids over a moving surface in the presence of thermal radiationHamadM. A. A.FerdowsM.Similarity solution of boundary layer stagnation-point flow towards a heated porous stretching sheet saturated with a nanofluid with heat absorption/generation and suction/blowing: a Lie group analysisMakindeO. D.AzizA.Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary conditionKameswaranP. K.NarayanaM.SibandaP.MurthyP. V. S. N.Hydromagnetic nanofluid flow due to a stretching or shrinking sheet with viscous dissipation and chemical reaction effects