Combined Effect of Buoyancy Force and Navier Slip on MHD Flow of a Nanofluid over a Convectively Heated Vertical Porous Plate

Abstract

We examine the effect of magnetic field on boundary layer flow of an incompressible electrically conducting water-based nanofluids past a convectively heated vertical porous plate with Navier slip boundary condition. A suitable similarity transformation is employed to reduce the governing partial differential equations into nonlinear ordinary differential equations, which are solved numerically by employing fourth-order Runge-Kutta with a shooting technique. Three different water-based nanofluids containing copper (Cu), aluminium oxide (Al₂O₃), and titanium dioxide (TiO₂) are taken into consideration. Graphical results are presented and discussed quantitatively with respect to the influence of pertinent parameters, such as solid volume fraction of nanoparticles (𝜑), magnetic field parameter (Ha), buoyancy effect (Gr), Eckert number (Ec), suction/injection parameter (𝑓_𝑤), Biot number (Bi), and slip parameter (β), on the dimensionless velocity, temperature, skin friction coefficient, and heat transfer rate.

1. Introduction

Magnetohydrodynamic (MHD) boundary layer flow of an electrically conducting viscous incompressible fluid with a convective surface boundary condition is frequently encountered in many industrial and technological applications such as extrusion of plastics in the manufacture of Rayon and Nylon, the cooling of reactors, purification of crude oil, textile industry, polymer technology, and metallurgy. As a result, the simultaneous occurrence of buoyancy and magnetic field forces on fluid flow has been investigated by many researchers [1–5]. In their investigations, all the authors mentioned above assumed the no-slip boundary conditions. However, more recently, researchers have investigated the flow problem taking slip flow condition at the boundary [6–9].

On the other hand, with the advent of nanofluids, there has been wide usage of recently discovered smart fluid in many industrial and biomedical applications. Nanofluid concept is employed to designate a fluid in which nanometer-sized particles are suspended in conventional heat transfer base fluids to improve their thermal physical properties. Nanoparticles are made from various materials, such as metals (Cu, Ag, Au, Al, and Fe), oxide ceramics (Al₂O₃, CuO, and TiO₂), nitride ceramics (AlN, SiN), carbide ceramics (SiC, tiC), semiconductors, carbon nanotubes, and composite materials such as alloyed nanoparticles or nanoparticle core-polymer shell composites. It is well known that conventional heat transfer fluids, such as oil, water, and ethylene glycol, in general, have poor heat transfer properties compared to those of most solids. Nanofluids have enhanced thermophysical properties such as thermal conductivity, thermal diffusivity, viscosity, and convective heat transfer coefficients compared with those of base fluids like oil or water [10]. Several authors [11–14] have conducted theoretical and experimental investigations to demonstrate that nanofluids distinctly exhibit enhanced heat transfer properties which goes up with increasing volumetric fraction of nanoparticles. Further studies on nanofluids have been currently undertaken by scientists and engineers due to their diverse technical and biomedical applications such as nanofluid coolant: electronics cooling, vehicle cooling, transformer cooling, computers cooling, and electronic devices cooling; medical applications: magnetic drug targeting, cancer therapy, and safer surgery by cooling; process industries; and materials and chemicals: detergency, food and drink, oil and gas, paper and printing, and textiles.

According to Aziz [15], the concept of no-slip condition at the boundary layer is no longer valid for fluid flows in microelectromechanical systems and must be replaced by slip condition. The slip flow model states a proportional relationship between the tangential components of the fluid velocity at the solid surface to the shear stress on the fluid-solid interface [16]. The proportionality is called the slip length, which describes the slipperiness of the surface [7]. Many researchers studied the effect of linear momentum and nonlinear slip on the MHD boundary layer flow with heat/mass transfer of free/forced/combined convection past different geometries [17–20]. In spite of the importance of MHD related studies on boundary layer flow problems, the possibility of fluid exhibiting apparent slip phenomenon on the solid surface has received little attention.

The aim of the present study is to investigate the combined effects of buoyancy, magnetic field, suction, Navier slip, and convective heating on a steady boundary layer flow over a flat surface. In the subsequent sections the boundary layer partial differential equations first transformed into a system of nonlinear ordinary differential equations before being solved numerically using a shooting method together with the fourth-order Runge-Kutta-Fehlberg integration scheme. A graphical representation of the pertinent parameters on the flow field and heat transfer characteristics is displayed and thoroughly discussed. To our best of knowledge, the investigations of the proposed problem are new, and the results have not been published before.

2. Model Formulation

The steady laminar incompressible two-dimensional MHD boundary layer flow of an electrically conducting water-based nanofluid past a convectively heated porous vertical semiinfinite flat plate under the combined effects of buoyancy forces and Navier slip is considered. The nanofluids contain three different types of nanoparticles: Cu, Al₂O₃, and TiO₂. Let the 𝑥-axis be taken along the direction of plate, and let 𝑦-axis be normal to it. The left side of the plate is assumed to be heated by convection from a hot fluid at temperature 𝑇_𝑓, which provides a heat transfer coefficient ℎ_𝑓, while the right surface is subjected to a stream of an electrically conducting cold nanofluid at temperature 𝑇_∞ in the presence of a transverse magnetic field of strength 𝐵₀ applied parallel to the 𝑦-axis, as shown in Figure 1. The induced magnetic field due to the motion of the electrically conducting fluid is negligible. It is also assumed that the external electrical field is zero and that the electric field due to the polarization of charges is negligible (see Table 1).

Table 1 

Thermophysical properties of water and nanoparticles [23, 24].

Figure 1 

Flow configuration and coordinate system.

Assuming a Boussinesq incompressible fluid model, the continuity, momentum, and energy equations describing the flow can be written as 𝜕𝑢+𝜕𝑥𝜕𝑣𝑢𝜕𝑦=0,𝜕𝑢𝜕𝑥+𝑣𝜕𝑢𝜕𝑦=𝑈_∞𝑑𝑈_∞+𝜇𝑑𝑥𝜌^𝑛𝑓_𝑛𝑓𝜕²𝑢𝜕𝑦²+𝛽_𝑛𝑓𝑔𝑇−𝑇_∞−𝜎_𝑛𝑓𝐵²₀𝑢−𝑈_∞𝜌_𝑛𝑓,𝑢𝜕𝑇𝜕𝑥+𝑣𝜕𝑇=𝑘𝜕𝑦𝜌𝑐^𝑛𝑓_𝑝𝑛𝑓𝜕²𝑇𝜕𝑦²+𝜇_𝑛𝑓𝜌𝑐_𝑝𝑛𝑓𝜕𝑢𝜕𝑦²+𝜎_𝑛𝑓𝐵²₀𝜌𝑐_𝑝𝑛𝑓𝑢−𝑈_∞². (1) The boundary conditions at the plate surface and at the free stream may be written as 𝜆𝑢(𝑥,0)=𝜇_𝑓𝜕𝑢()()𝜕𝑦𝑥,0,𝑣𝑥,0=𝑉_𝑤,−𝑘_𝑓𝜕𝑇()𝜕𝑦𝑥,0=ℎ_𝑓𝑇_𝑓(,𝑢()−𝑇𝑥,0)𝑥,∞=𝑈_∞(𝑥),𝑇(𝑥,∞)=𝑇_∞, (2) where (𝑢,𝑣) are the velocity components of the nanofluid in the 𝑥- and 𝑦-directions, respectively, 𝑇 is the nanofluid temperature, 𝑈_∞(𝑥)=𝑎𝑥 is the free stream velocity (which implies that the free stream fluid velocity is increasing with axial distance along the plate surface), 𝑇_∞ is the free stream temperature, 𝑔 is acceleration due to gravity, 𝜆 is the slip coefficient, 𝜇_𝑛𝑓 is dynamic viscosity of the nanofluid, 𝜌_𝑛𝑓 is density of the nanofluid, 𝑘_𝑛𝑓 is thermal conductivity of the nanofluid, 𝜎_𝑛𝑓 is electrical conductivity of the nanofluid, (𝜌𝑐_𝑝)_𝑛𝑓 is heat capacity at constant pressure of the nanofluid, and 𝛽_𝑛𝑓 is volumetric expansion coefficient of the nanofluid which are defined as [21, 22] 𝜇_𝑛𝑓=𝜇_𝑓1−𝜑^2.5,𝜌_𝑛𝑓=1−𝜑𝜌_𝑓+𝜑𝜌_𝑠,𝛽_𝑛𝑓=1−𝜑𝛽_𝑓+𝜑𝛽_𝑠,𝛼_𝑛𝑓=𝑘_𝑛𝑓𝜌𝑐_𝑝𝑛𝑓,𝑘_𝑛𝑓𝑘_𝑓=𝑘_𝑠+2𝑘_𝑓−2𝜑𝑘_𝑓−𝑘_𝑠𝑘_𝑠+2𝑘_𝑓+𝜑𝑘_𝑓−𝑘_𝑠,𝜌𝑐_𝑝𝑛𝑓=1−𝜑𝜌𝑐_𝑝𝑓+𝜑𝜌𝑐_𝑝𝑠,𝜎_𝑛𝑓=1−𝜑𝜎_𝑓+𝜑𝜎_𝑠, (3) where 𝜑 is the nanoparticle volume fraction (𝜑=0 correspond to a regular fluid), 𝜌_𝑓 and 𝜌_𝑠 are the densities of the base fluid and the nanoparticle, respectively, 𝛽_𝑓 and 𝛽_𝑠 are the thermal expansion coefficients of the base fluid and the nanoparticle, respectively, 𝑘_𝑓 and 𝑘_𝑠 are the thermal conductivities of the base fluid and the nanoparticles, respectively, (𝜌𝑐_𝑝)_𝑓 and (𝜌𝑐_𝑝)_𝑠 are the heat capacitance of the base fluid and the nanoparticle, respectively, and 𝜎_𝑠 and 𝜎_𝑓 are the electrical conductivities of the base fluid and the nanofluid, respectively.

In order to simplify the mathematical analysis of the problem, we introduce the following dimensionless variables: 𝑎𝜂=𝜐_𝑓^1/2𝑦,𝜓=𝑎𝜐_𝑓^1/2𝜂,𝜃𝜂=𝑥𝑓𝑇−𝑇_∞𝑇_𝑓−𝑇_∞, (4) where 𝜂 is the similarity variable and 𝜓 is the stream function defined as 𝑢=𝜕𝜓𝜕𝑦,𝑣=−𝜕𝜓.𝜕𝑥 (5) After introducing (5) into (1) and (2), we obtain the following ordinary differential equations: 𝑓^′′′+1−𝜑^2.51−𝜑+𝜑𝜌_𝑠𝜌_𝑓𝑓𝑓^′′−1−𝜑^2.51−𝜑+𝜑𝜌_𝑠𝜌_𝑓𝑓^′2+1−𝜑^2.51−𝜑+𝜑𝜌_𝑠𝜌_𝑓+Gr1−𝜑^2.51−𝜑+𝜑𝜌_𝑠𝜌_𝑓1−𝜑+𝜑𝛽_𝑠𝛽_𝑓𝜃−Ha1−𝜑^2.51−𝜑+𝜑𝜎_𝑠𝜎_𝑓𝑓^′𝜃−1=0,_′′+Pr𝑘_𝑓1−𝜑+𝜑𝜌𝑐_𝑝𝑠/𝜌𝑐_𝑝𝑓𝑘_𝑛𝑓𝑓𝜃^′+PrEc𝑘_𝑓𝑘_𝑛𝑓1−𝜑^2.5𝑓^′′2+HaPrEc𝑘_𝑓𝑘_𝑛𝑓×1−𝜑+𝜑𝜎_𝑠𝜎_𝑓𝑓^′−1²=0. (6) Taking into account the variable plate surface permeability and the hydrodynamic slip boundary functions defined, respectively, as 𝑉_𝑤=−𝑓_𝑤𝑎𝜐_𝑓^1/2,𝜆𝑢(𝑥,0)=𝜇_𝑓𝜕𝑢(),𝜕𝑦𝑥,0 (7) the boundary conditions are 𝑓(0)=𝑓_𝑤,𝑓^′(0)=𝛽𝑓^′′(0),𝜃^′(0)[𝜃(0)],𝑓=Bi−1_′(∞)=1,𝜃(∞)=0, (8) where a prime symbol denotes derivative with respect to 𝜂, 𝑓_𝑤 is a constant with 𝑓_𝑤>0 representing suction rate at the plate surface, 𝑓_𝑤<0 corresponds to injection, 𝑓_𝑤=0 shows an impermeable surface, 𝜆=0 represents highly lubricated surface, and 𝜆=∞ corresponds to a normal surface. The local Reynolds number (Re_𝑥), Grashof number (Gr), Hartmann number (Ha), Prandtl number (Pr), Eckert number (Ec), slip parameter (𝛽), and Biot number (Bi), are defined as Re_𝑥=𝑈_∞𝑥𝜐_𝑓𝛽,Gr=^𝑓𝑔𝑇_𝑓−𝑇_∞𝑈_∞𝑎,𝜎Ha=^𝑓𝐵_𝑜²𝜌_𝑓𝑎𝜐,Pr=^𝑓𝛼_𝑓,𝑈Ec=^2∞𝐶_𝑝𝑓𝑇_𝑓−𝑇_∞𝜇,𝛽=^𝑓𝜆𝑎𝜐_𝑓,ℎBi=^𝑓𝑘_𝑓𝜐_𝑓𝑎. (9) The physical quantities of practical significance in this work are the skin friction coefficient 𝐶_𝑓 and the local Nusselt number Nu, which are expressed as 𝐶_𝑓=𝜏_𝑤𝜌_𝑓𝑈²_∞,Nu=𝑥𝑞_𝑤𝑘_𝑓𝑇_𝑓−𝑇_∞, (10) where 𝜏_𝑤 is the skin friction and 𝑞_𝑤 is the heat flux from the plate which are given by 𝜏_𝑤=𝜇_𝑛𝑓𝜕𝑢𝜕𝑦_𝑦=0,𝑞_𝑤=−𝑘_𝑛𝑓𝜕𝑇𝜕𝑦_𝑦=0. (11) Putting (11) into (10), we obtain Re_𝑥^1/2𝐶_𝑓=11−𝜑^2.5𝑓^′′(0),Re_𝑥^−1/2𝑘Nu=−^𝑛𝑓𝑘_𝑓𝜃^′(0). (12) The set of (6) and together with the boundary conditions (8) are coupled nonlinear boundary value problems which are solved numerically using a shooting algorithm with a Runge-Kutta Fehlberg integration scheme. This method involves transforming (6) and (8) into a set of initial value problems which contain unknown initial values that need to be determined by guessing, after which a fourth order Runge-Kutta iteration scheme is employed to integrate the set of initial valued problems until the given boundary conditions are satisfied. The entire computation procedure is implemented using a program written and carried out using Maple computer language. From the process of numerical computation, the fluid velocity, the temperature, the skin friction coefficient, and the Nusselt number are proportional to 𝑓^′(𝜂), 𝜃(𝜂), 𝑓^′′(𝜂), and 𝜃^′(𝜂), respectively.

3. Results and Discussion

Physically realistic numerical values were assigned to the pertinent parameters in the system in order to gain an insight into the flow structure with respect to velocity, temperature, skin friction, and Nusselt’s number. The results were presented graphically in Figures 2–13, and conclusions are drawn for the flow field. The Prandtl number is kept constant at 6.2 [21]. Ha = 0 corresponds to absence of magnetic field, and 𝜑=0 is regular fluid.

Figure 2 

Velocity profiles for different nanofluids.

Figure 3 

Velocity profiles with increasing Ha, 𝜑, and Ec.

Figure 4 

Velocity profiles with increasing Gr, 𝛽, and 𝑓𝑤.

Figure 5 

Temperature profiles for different nanofluids.

Figure 6 

Temperature profiles with increasing 𝜑, Ha, Bi, and Ec.

Figure 7 

Temperature profiles with increasing 𝛽, Gr, and 𝑓𝑤.

Figure 8 

Local skin friction profiles for different nanofluids.

Figure 9 

Effects of increasing Gr, Ha, and Ec on local skin friction.

Figure 10 

Effects of increasing 𝛽 and 𝑓𝑤 on local skin friction.

Figure 11 

Local Nusselt number for different nanofluids.

Figure 12 

Effects of increasing 𝜑, Bi, and Ha on local Nusselt number.

Figure 13 

Effects of increasing 𝜑, Ha, and Ec on local Nusselt number.

3.1. Dimensionless Velocity Profiles

Figures 2–4 illustrate the effects of various thermophysical parameters on the nanofluids velocity profiles. Generally, it is noted that the fluid velocity increases gradually from zero at the plate surface to the free stream prescribed value far away from the plate satisfying the boundary conditions. Figure 2 shows that the momentum boundary layer thickness for Cu-water nanofluid is smaller than the rest of the nanofluids consequently, Cu-water nanofluid tends to flow closer to the convectively heated plate surface and serves as a better coolant than the other nanofluids. It is observed in Figures 3-4 that an increase in the magnetic field intensity (Ha), nanoparticle volume fraction (𝜑), Eckert number (Ec), Grashof number (Gr), and the suction/injection parameter (𝑓_𝑤) causes an overshoot of the fluid velocity towards the plate surface hence decreasing both the momentum boundary layer thickness and the fluid velocity. From the physics of the problem, an increase in the magnetic field intensity leads to an increase in the Lorentz force which is a retarding force to the transport phenomena. This retarding force can control the nanofluids velocity which is useful in numerous applications such as magneto hydrodynamic power generation and electromagnetic coating of wires and metal. We also note that the fluid velocity at the plate surface increases with an increase in the slip parameter (β). This is in agreement with the fact that higher β implies an increase in the lubrication and slipperiness of the surface.

3.2. Dimensionless Temperature Profiles

Figures 5–7 show the effects of various parameters on the temperature profile. In general, the maximum fluid temperature is achieved at the plate surface due to the convectional heating but decreases exponentially to zero far away from the plate surface satisfying the free stream conditions. As expected, at the plate surface, Cu-water has the highest temperature and a greater thermal boundary layer thickness than the other two nanofluids, as seen in Figure 5. This is in accordance with the earlier observation, since the Cu-water nanofluid is more likely to absorb more heat from the plate surface owing to its close proximity to the hot surface. It is observed from Figure 6, that increasing Ha, φ, Bi, and Ec leads to an increase in both the fluid temperature and the thermal boundary layer thickness. This can be attributed to the additional heating due resistance of fluid flow as a result of the magnetic field, the presence of the nanoparticle, the increased rate at which the heat moves from the hot fluid to the plate and the additional heating as a result of the viscous dissipation.

On the other hand, it is evident that surface slipperiness and suction affect the temperature of the fluid inversely. This is clearly seen from Figure 7, where both temperature and thermal boundary layer decrease as 𝑓_𝑤 and β increase.

3.3. Effects of Parameters Variation on the Skin Friction and Nusselt Number

Figures 8–13 demonstrate the effects of the various pertinent parameters at the plate surface for both the skin friction coefficient and the local Nusselt number (rate of heat transfer). The presence of nanoparticle in the convectional fluid leads to an increase in the skin friction, as seen in Figure 8, where increasing the nanoparticle volume fraction increases the skin friction for the three nanoparticles (Cu, Al₂O₃, and TiO₂) used, with Cu-water exhibiting the highest increment. This is as expected, since Cu-water moves closer to the plate surface leading to an elevation in the velocity gradient at the plate surface. As expected, increasing Ha, Gr, Ec, and 𝑓_𝑤 leads to an increase in the skin friction coefficient, while an increase in β reduces the skin friction coefficient as shown in Figures 9 and 10. There is an increase in the rate of heat transfer with an increase in φ, Bi, and 𝑓_𝑤 as seen in Figures 11-12, with Al₂O₃ exhibiting the highest increment. The converse is seen with increasing Ha as shown in Figure 13.

4. Conclusions

The problem of hydromagnetic boundary layer flow of an incompressible electrically conducting water-based nanofluids past a convectively heated vertical porous plate with Navier slip boundary condition was studied. The governing nonlinear partial differential equations were transformed into a self-similar form and numerically solved using shooting technique with a fourth-order Runge-Kutta-Fehlberg integration scheme, putting into consideration the enhanced electrical conductivity of the convectional base fluid due to the presence of the nanoparticles. Our results showed that the fluid velocity increases, while the local skin friction decreases with the increase in the slip parameter (β), but the reverse is observed with the increase in the magnetic field intensity (Ha), nanoparticle volume fraction (φ), Eckert number (Ec), Grashof number (Gr), and the suction/injection parameter (𝑓_𝑤). Both the temperature and the thermal boundary layer thickness are enhanced by increasing the magnetic field intensity (Ha),nanoparticle volume fraction (φ), Eckert number (Ec), and the intensity of Newtonian heating (Bi), while the cooling effect on the convectively heated plate surface is enhanced by increasing the velocity slip (β) and suction parameter (𝑓_𝑤).

Acknowledgments

The authors would like to thank the Organization for Women in Science for the Developing World (OWSDW), the Swedish International Development Cooperation Agency (SIDA), and the African Union Science and Technology Commission for financial support.