MHD Forced Convection Laminar Boundary Layer Flow of Alumina-Water Nanofluid over a Moving Permeable Flat Plate with Convective Surface Boundary Condition

Abstract

This paper analyses a two-dimensional steady forced convection boundary layer viscous incompressible flow of alumina-water nanofluid over a moving permeable vertical flat plate under the effect of a magnetic field normal to the plate. Thermal convective surface boundary condition is applied. The nanofluid formulated in the present study is water dispersed with various volumetric fractions of the alumina (Al₂O₃) nanoparticles. The plate velocity and the free stream velocities are considered to be proportional to 𝑥^𝑛, while the magnetic field and suction velocities are taken to be proportional to (𝑥)^(𝑛−1)/2. The similarity solution of the governing problem is obtained. Numerical studies are presented to show the effect of the nanoparticle volume fraction 𝜙, the convective heat transfer parameter 𝑏, the power law exponent 𝑛, the wall velocity parameter 𝐴, and the suction parameter f_w on the velocity, temperature, skin-friction coefficient, and Nusselt number.

1. Introduction

It is known that nanofluids can tremendously enhance the heat transfer characteristics of the original (base) fluid. Thus, nanofluids have many applications in industry such as coolants, lubricants, heat exchangers, and microchannel heat sinks. Therefore, numerous methods have been taken to improve the thermal conductivity of these fluids by suspending nano-/microsized particle materials in liquids. Several recent studies on the modeling of free/mixed convection heat transfer in nanofluids have been published: Congedo et al. [1], Ghasemi and Aminossadati [2], and Ho et al. [3, 4]. These studies have used traditional finite difference and finite volume techniques with the tremendous call on computational resources that these techniques necessitate. Hamad et al. [5] used the application of a one-parameter group to present similarity reductions for problems of magnetic field effects on free-convection flow of a nanofluid past a semi-infinite vertical flat plate. Hamad and Pop [6] studied theoretically the similarity solution of the steady boundary layer flow near the stagnation-point flow on a permeable stretching sheet in a porous medium saturated with a nanofluid and in the presence of internal heat generation/absorption. Hamad [7] found the analytical solutions of convective flow and heat transfer of an incompressible viscous nanofluid past a semi-infinite vertical stretching sheet in the presence of magnetic field. Abu-Nada and Chamkha [8] investigated the natural convection heat transfer characteristics in a differentially heated enclosure filled with a CuO-EG-water nanofluid for different published variable thermal conductivity and variable viscosity models. Teng et al. [9] studied the effect of particle size, temperature, and weight fraction on the thermal conductivity ratio of Al₂O₃-water nanofluid. Ho et al. [10] presented an experimental study for natural convection heat transfer of a nanofluid in vertical square enclosure of different sizes. Bachok et al. [11] studied the steady boundary-layer flow of a nanofluid past a moving semi-infinite flat plate in a uniform free stream. Hady et al. [12] studied the natural convection boundary-layer flow over a downward-pointing vertical cone in a porous medium saturated with a non-Newtonian nanofluid in the presence of heat generation or absorption. The numerical modeling of the conjugate heat transfer and fluid flow of Al₂O₃/water nanofluid through the microchannel heat sink was presented by Lelea [13]. In this paper, the laminar flow regime was considered along with viscous dissipation effect. Also, the microchannel heat sink with square microchannels and Dh = 50 μm was considered. Nandy and Mahapatra [14] analyzed the effects of velocity slip and heat generation/absorption on magnetohydrodynamic (MHD) stagnation-point flow and heat transfer over a stretching/shrinking surface, with convective boundary conditions, in the presence of nanoparticle fractions. They considered also the effects of Brownian motion and thermophoresis. Numerical study for the effects of thermal radiation and viscous dissipation on hydromagnetic mixed convection flow over a nonlinear stretching and shrinking sheets in nanofluids was presented by Pal et al. [15]. Rohni et al. [16] theoretically investigated the steady two-dimensional boundary-layer flow past a moving semi-infinite flat plate in a water-based nanofluid containing three different types of nanoparticles: copper (cuprum), alumina (aluminium oxide), and titania (titanium dioxide). For more information, see also Das et al. [17], Wang and Mujumdar [18–20], and Kakaç and Pramuanjaroenkij [21].

Heat transfer due to forced convection occurs in many areas. Examples include chemical process, biochemical process, and engineering. Its study is important due to the need to improve the efficiency of the heat transfer. Ali [22] studied the effect of temperature dependent viscosity on laminar mixed convection boundary layer flow and heat transfer on a continuously moving vertical isothermal surface and obtained local similarity solutions. According to previous studies, the forced convection flow over moving plate has many practical engineering applications such as liquid films in condensation process and in aerodynamics. Similarity solutions about moving plate were investigated by many authors. Among them, Ishak et al. [23] studied the boundary layer flow on a moving permeable plate parallel to a moving stream and concluded that dual solutions exist if the plate and the free stream move in the opposite directions. Magyari [24] studied the moving plate thermometer with uniform velocity. His approach was based on a series solution of the problem using the Merkin transformation method. Hoernel [25] investigated similarity solution for steady laminar boundary layer equations governing MHD flow near forward stagnation point of two dimensional moving axisymmetric bodies. Recently, Pal and Mondal [26] studied MHD forced convection and heat transfer of an incompressible Newtonian electrically conducting and heat generating/absorbing fluid having temperature-dependent viscosity over a nonisothermal permeable wedge in the presence of thermal radiation. In all of the above studies, convectional thermal boundary conditions (either prescribed heat flux or prescribed surface temperature) were used.

According to previous scholars, the MHD flow has received the attention of many researchers due to its engineering applications. In metallurgy, for example, some processes involve the cooling of many continuous strips by drawing them through an electrically conducting fluid subject to a magnetic field (Kandasamy and Muhaimin [27]). This allows the rate of cooling to be controlled and final product with the desired characteristics to be obtained. Another important application of hydromagnetic flow in metallurgy is in the purification of molten metals from nonmetallic inclusions through the application of a magnetic field. Research has also been carried out by previous researchers on the flow and heat transfer effects of electrically conducting fluids such as liquid metals, water mixed with a little acid and other equivalent substances in the presence of a magnetic field. The studies have involved different geometries and different boundary conditions. Herdricha et al. [28] studied MHD flow control for plasma technology applications. They identified potential applications for magnetically controlled plasmas in the fields of space technology as well as in plasma technology. Seddeek et al. [29] investigated the similarity solution in MHD flow and heat transfer over a wedge taking into account variable viscosity and thermal conductivities. Alam et al. [30] investigated the effects of variable suction and thermophoresis on steady MHD combined free-forced convective heat and mass transfer flow over a semi-infinite permeable inclined plate in the presence of thermal radiation. Aydin and Kaya [31] studied MHD mixed convection of a viscous dissipating fluid about a permeable vertical flat plate and found that the value of Richardson number determines the effect of the magnetic parameter on the momentum and heat transfer. Recently, Rahman and Salahuddin [32] investigated the effects of variable electric conductivity and viscosity on hydromagnetic heat and mass transfer flow along a radiated isothermal inclined permeable surface in a stationary fluid which includes internal heat generation.

The aim of the present study is to analyze steady, two-dimension, laminar MHD forced convection flow of alumina-water nanofluid on moving permeable flat plate with convective surface boundary condition. The similarity representation of the partial differential equations governing this problem is obtained. The motivation of this study is to explore the influence of nanofluid parameters with the effect of suction, convective heat transfer, and velocity parameter. The variation distribution of the shear stress and heat transfer rates (the physical quantities of interest in this study) with the parameters governing the problem are presented.

2. Formulation of the Problem

Consider a two-dimensional forced convective flow of water-based alumina nanoparticles over a moving permeable flat plate. A rectangular coordinate frame is selected in which the 𝑥-axis is vertical and 𝑦-axis is horizontal. A magnetic field of variable strength 𝐵(𝑥) is applied perpendicular to the plate. The temperature of the plate on the top is held at a uniform temperature 𝑇_𝑤 which is assumed greater than the ambient temperature 𝑇_∞. It will be further assumed that the bottom surface of the plate is heated by convection from a hot fluid of temperature 𝑇_𝑓, and this generates a heat transfer coefficient ℎ_𝑓 (see Aziz [33]). For the thermophysical properties of the alumina, see Oztop and Abu-Nada [34]. The flow configuration and the coordinate system are shown in Figure 1. Kinematic viscosity and thermal diffusivity are assumed constants. Particle coagulation, the magnetic Reynolds number, the electric field owing to the polarization of charges, viscous dissipation term, and Hall effects are assumed to be negligible. The field variables of this study are the velocity 𝑉=(𝑢,𝑣) and the temperature𝑇. Under the above assumptions, the boundary layer equations can be written in dimensional form as (see Schlichting and Gersten [35], Kays and Crawford [36]) 𝜕𝑢+𝜕𝑥𝜕𝑣𝜌𝜕𝑦=0,_nf𝑢𝜕𝑢𝜕𝑥+𝑣𝜕𝑢𝜕𝑦=−𝜕𝑝𝜕𝑥+𝜇_nf𝜕²𝑢𝜕𝑦²−𝜎𝐵²(𝑥)𝑢𝑢,𝜕𝑇𝜕𝑥+𝑣𝜕𝑇𝜕𝑦=𝛼_nf𝜕²𝑇𝜕𝑦². (1)(2)(3) The corresponding boundary conditions are 𝑢=𝑢_𝑤(𝑥),𝑣=−𝑣_𝑤(𝑥),−𝜅_nf𝜕𝑇𝜕𝑦=ℎ_𝑓𝑇_𝑓−𝑇_𝑤at𝑦=0,𝑢⟶𝑈_𝑒(𝑥),𝑇⟶𝑇_∞,𝐶⟶𝐶_∞as𝑦⟶∞. (4)

Figure 1 

Flow configuration and coordinate system.

Here, 𝑝 is the pressure, 𝜎 is the electric conductivity, 𝑈_𝑒(𝑥) is potential velocity, 𝑢_𝑤(𝑥) is the velocity of the moving plate, 𝑣_𝑤(𝑥) is the velocity normal to the plate with 𝑣_𝑤(𝑥)>0 for injection (blowing) and 𝑣_𝑤(𝑥)<0 for suction, and 𝑣_𝑤(𝑥)=0 corresponds to an impermeable sheet. As usual 𝜇_nf is the nanofluid viscosity, 𝜌_nf is the nanofluid density, 𝛼_nf is the nanofluid thermal diffusivity and, 𝜅_nf is the nanofluid thermal conductivity, which are given by 𝜇_nf=𝜇_𝑓1−𝜙^2.5,𝛼_nf=𝜅_nf𝜌𝐶_𝑝nf,𝜌𝐶_𝑝nf=1−𝜙𝜌𝐶_𝑝𝑓+𝜙𝜌𝐶_𝑝𝑠,𝜌_nf=1−𝜙𝜌_𝑓+𝜙𝜌_𝑠,𝜅_nf=𝜅_𝑓𝜅_𝑠+2𝜅_𝑓−2𝜙𝜅_𝑓−𝜅_𝑠𝜅_𝑠+2𝜅_𝑓+𝜙𝜅_𝑓−𝜅_𝑠, (5) where 𝜙 is the nanoparticle volume fraction, (𝜌𝐶_𝑝)_nf is the nanofluid heat capacity, 𝜅_𝑓 and 𝜅_𝑠 are thermal conductivities of the fluid and of the solid fractions, respectively, and 𝜌_𝑓 and 𝜌_𝑠 are the densities of the fluid and of the solid fractions, respectively.

In the free stream, 𝑢=𝑈_𝑒(𝑥), and so (2) reduces to 𝑈_𝑒𝑑𝑈_𝑒1𝑑𝑥=−𝜌_nf𝜕𝑝−𝜕𝑥𝜎𝐵²(𝑥)𝜌_nf𝑈_𝑒(𝑥). (6) Eliminating the pressure gradient term 𝜕𝑝/𝜕𝑥 between (2) and (6), we get 𝑢𝜕𝑢𝜕𝑥+𝑣𝜕𝑢𝜕𝑦=𝑈_𝑒𝑑𝑈_𝑒+1𝑑𝑥𝜌_nf𝜇_nf𝜕²𝑢𝜕𝑦²−𝜎𝐵²(𝑥)𝑢−𝑈_𝑒. (7)By introducing stream function 𝜓, such that𝑢=𝜕𝜓𝜕𝑦,𝑣=−𝜕𝜓.𝜕𝑥 (8) We also introduce the following dimensionless transformations: 𝑦𝜂=𝑥√𝑛+1Re_𝑥^1/2𝑣,𝜓=^𝑓√𝑛+1Re_𝑥^1/2𝑓𝜂,𝜃𝜂=𝑇−𝑇_∞𝑇_𝑓−𝑇_∞,Re_𝑥=𝑈_∞𝑥^𝑛+1𝑣_𝑓,𝐵(𝑥)=𝐵₀𝑥^(𝑛−1)/2,𝑢_𝑤(𝑥)=𝑢₀𝑥^𝑛,𝑣_𝑤(𝑥)=𝑣₀𝑥^(𝑛−1)/2,𝑈_𝑒(𝑥)=𝑈_∞𝑥^𝑛, (9) where 𝐵₀,𝑢₀,𝑣₀, and𝑢_∞ are constants and where Re_𝑥 is the Reynolds number based on position 𝑥 along flow direction (see Hoernel [25]). Here, 𝑛 (not necessarily an integer) is the power law exponent, 𝑢₀>0 means that the plate is moving along the positive direction of 𝑥-axis, and 𝑢₀<0 means that the plate is moving in the negative direction of 𝑥-axis. In our analysis, we choose𝑢₀>0. The quantity 𝑣₀ represents suction or injection, 𝑣₀>0 represents suction, and 𝑣₀<0 represents injection.

It is noticed that the continuity equation (1) is satisfied automatically and (3) and (7) yield 𝑛+11−𝜙^2.5𝑓^′′′+𝜌1−𝜙+^𝑠𝜌_𝑓𝜙×𝑛+12𝑓𝑓^′′−𝑛𝑓^′2+𝑛+𝑀𝑓^′1−1=0,𝜅Pr𝜅^nf_𝑓𝜃^′′+121−𝜙+𝜌𝐶_𝑝𝑠𝜌𝐶_𝑝𝑓𝜙𝑓𝜃^′=0, (10)(11) where Pr=(𝜇𝑐_𝑝)_𝑓/𝜅_𝑓 is the Prandtl number and 𝑀=𝜎𝐵²₀/(𝜌_𝑓𝑢_∞) is the magnetic parameter. Here, we assume that 𝑇_𝑓>𝑇_𝑤>𝑇_∞.

The corresponding boundary conditions in (4) become 𝑓^′(0)=𝐴,𝑓(0)=𝑓_𝑤,𝜃^′(0)𝜅=−𝑏^𝑓𝜅_nf[1−𝜃(0)],𝑓^′(∞)=1,𝜃(∞)=0, (12) where 𝐴=𝑢₀/𝑈_∞ is a constant and is considered as a velocity parameter and 𝑓_𝑤=−2𝑣₀/𝑣𝑈_∞(𝑛+1) is the suction parameter. The parameter 𝐴>0 means that the plate is moving in the same direction as the free stream velocity, 𝐴<0 means that the plate is moving in the opposite direction of the free steam, and 𝐴=0 stands for static plate.

The parameter 𝑏=(ℎ_𝑓/𝜅)𝑣𝑥/𝑈_𝑒(𝑛+1) is the convective heat transfer parameter, which should be constant. To fulfill this criterion, the coefficient of heat transfer ℎ_𝑓 must be directly proportional to 1/√𝑥; that is, ℎ_𝑓=𝑐₁/√𝑥.

2.1. Particular Cases

(i)Here, we should mention that if 𝑛=0(the full stream velocity is constant),𝑓_𝑤=0 (the plate is impermeable), 𝑢_𝑤=0 (the plate is stationary), and𝑀=0 (absence of magnetic field), our system (10)–(12) reduces to that in Aziz [33].(ii)If𝑓_𝑤≠0 (the plate is permeable), 𝑢_𝑤=0, and𝑀=0, our system (10)–(12) reduces to that in Ishak [37].

The parameters of engineering interest for the problem are the skin friction 𝐶_𝑓 and the local Nusselt number Nu, respectively. These physical quantities can be derived from the following definitions: 𝐶_𝑓=𝜇_nf𝑢²_𝑤𝜌_nf𝜕𝑢𝜕𝑦_𝑦=0,Nu=−𝑥𝑇_𝑓−𝑇_∞𝜅_nf𝜅_𝑓𝜕𝑇𝜕𝑦_𝑦=0. (13) By substituting from (8) and (12) into (13), we get Re_𝑥^1/2𝐶_𝑓=√𝑛+1𝐴²1−𝜙^2.5𝑓^′′(0),Re^−1/2Nu=−√𝜅𝑛+1^nf𝜅_𝑓𝜃^′(0). (14)

3. Results and Discussion

The transformed system of nonlinear ordinary differential equations (10) and (11) together with boundary conditions (12) containing various parameters, namely, the power law parameter 𝑛, magnetic parameter 𝑀, the Prandtl number Pr, the nanoparticles volume fraction 𝜙 and convective heat transfer parameter 𝑏, suction parameter 𝑓_𝑤and the velocity parameter 𝐴. Equations (10) and (11) together with boundary conditions given in (12) are solved numerically by an implicit finite difference method. To exhibit our results obtained by numerical computations, we plot the graphs given in Figures 2 to 10, respectively.

Figure 2 

𝑏=0.5,𝑓𝑤=0.5,Pr=6.2,𝑀=0.5, and𝐴=0.5.

To exhibit the effects of various values of the power law exponent 𝑛 on velocity 𝐹^′(𝜂) and temperature 𝜃(𝜂) for two values of the nanoparticles volume fraction 𝜙, which are 𝜙=0 (pure water) and 𝜙≠0 (=0.1) (Al₂O₃-water), keeping the other parameters fixed (Pr=6.2, 𝑀=0.5, 𝑏=0.5,𝑓_𝑤=0.5, and 𝐴=0.5), we plot Figures 2 and 3, respectively. It is clear from Figure 2 that the increased power law exponent increases the velocity. Also, it is clear that the presence of alumina nanoparticles reduces the momentum boundary layer thickness and increases the velocity. However, from Figure 3, it is seen that the increased power law exponent decreases the temperature and the existence of alumina nanoparticles increases the thermal boundary layer thickness. This is because the thermal conductivity of alumina is higher than that of water.

Figure 3 

𝑏=0.5,𝑓𝑤=0.5,Pr=6.2,𝑀=0.5, and𝐴=0.5.

Figure 4 displays the influence of various values of the convective heat transfer parameter 𝑏 and the nanoparticles volume fractions 𝜙on temperature distribution. We have noted that the thermal boundary layer thickness increases when 𝑛 or 𝜙increases. This agrees with the physical behavior that, when the convective heat transfer increases, the hot fluid side convection resistance decreases, and, consequently, the surface temperature increases (see Ishak [37]); because of this, the thermal boundary layer thickness increases, and because also alumina has high thermal conductivity, then the thickness increases more for alumina-water nanofluid.

Figure 4 

𝑛=2,𝑓𝑤=0.5,Pr=6.2,𝑀=0.5, and𝐴=0.5.

Figure 5 shows the effect of the variation of suction parameter𝑓_𝑤on temperature for two cases of convective heat transfer parameter 𝑏, first when 𝑏=0.5 (convective boundary condition) and second when 𝑏=∞ (neglect convective condition, i.e., back to normal condition 𝜃(0)=1). All other parameters are nonzero. It can be seen that the thickness of the thermal boundary layer reduces with increasing values of𝑓_𝑤for the two cases of 𝑏.

Figure 5 

𝑛=2,𝜙=0.1,Pr=6.2,𝑀=0.5, and𝐴=0.5.

Effects of the velocity parameter 𝐴 on the velocity and temperature are shown in Figures 6 and 7, respectively. We have observed from Figure 6 that the velocity increases when 𝐴 increases. We have noticed also that for each 𝐴<1 (𝑢₀<𝑈_∞), the velocity increases till it becomes 1 and, for each 𝐴>1 (𝑢₀>𝑈_∞) the velocity decreases till it becomes 1 while, from Figure 7, it is noted that the temperature decreases when 𝐴 increases and the wall temperature also decreases when 𝐴 increases.

Figure 6 

𝑏=0.1,𝑓𝑤=0.1,Pr=6.2,𝑀=0.5,𝑛=2, and𝜙=0.1.

Figure 7 

𝑏=0.1,𝑓𝑤=0.1,Pr=6.2,𝑀=0.5,𝑛=2, and𝜙=0.1.

Figure 8 shows the effects of 𝑏 on the shear stress versus the parameter 𝑛 for pure water and alumina-water nanofluid (𝜙≠0(=0.1)). It is clear that the value of shear stress without considering the convictive boundary condition (𝑏=∞) is greater than when considering the convective boundary condition for both 𝜙=0and 𝜙≠0, also for the nanofluid the shear stress is greater more than for pure water. It is also noticed that increasing parameter 𝑛leads to increasing the shear stress, as well as at each value of 𝑛, the shear stress be higher for higher value of 𝑏both 𝜙=0 and 𝜙≠0.

Figure 8 

Pr=6.2,𝑀=0.5,𝐴=0.5, and𝑓𝑤=0.1.

Figures 9 and 10 show the effect of suction parameter 𝑓_𝑤and the velocity parameter 𝐴 on the shear stress and rate of heat transfer versus 𝑛, respectively. From Figure 9, we have seen that the shear stress increases when 𝑓_𝑤increases. For 𝐴<1 as in Figure 9(a), we have noted that the shear stress decreases when 𝐴 increases, but, for 𝐴>1 as in Figure 9(b), the shear stress increases with the increased 𝐴. However, from Figures 10(a) and 10(b), it is observed that the rate of heat transfer increases with the increasing value of suction parameter 𝑓_𝑤or the increasing value of the power law parameter 𝑛.

(a) A < 1

(b) A > 1

Figure 9 

Pr=6.2,𝑀=0.5,𝑏=0.5, and𝜙=0.1.

(a) A < 1

(b) A > 1

Figure 10 

Pr=6.2,𝑀=0.5,𝑏=0.5, and𝜙=0.1.

4. Conclusion

In this paper, we have studied the problem of heat transfer in MHD forced convection flow of alumina-water nanofluid over moving permeable vertical flat plate with convective surface boundary condition. New similarity transformations have been presented and solved numerically. It was found that alumina nanoparticles when suspended in a fluid are capable of increasing the heat transfer capability of the base fluid. The shear stress as well as heat transfer rate increases with nanoparticle volume fraction. Thermal boundary layer thickness as well as heat transfer rate increases when the convective heat transfer parameter 𝑏 increases. Thermal boundary layer thickness decreases while both shear stress and heat transfer rate increase with the increase of suction parameter. Temperature and wall temperature decrease while velocity and heat transfer rate increase when the velocity parameter 𝐴 increases.