Heat and Mass Transfer in Magnetohydrodynamics MHD Flow Over a Moving Vertical Plate with Convective

Abstract

Keywords: Vertical plate; similarity solution; magnetic field; thermal radiation; heat and mass transfer; homotopy analysis method (HAM).

1. Introduction

In most of the practical transport processes, the heat transfer is always accompanied by the mass transfer. The study of magnetohydrodynamic (MHD) flow with heat and Mass transfer over a moving surface and the effect of thermal radiation has been of interest due to its wide application scientific and environmental process such as astrophysical flows and geothermal reservoir. As a result of the various application of this problem, it has attracted the attention of many researchers and extensively studied in the literature.

Nomenclatures 𝑔 𝐴𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑑𝑢𝑒 𝑡𝑜 𝑔𝑟𝑎𝑣𝑖𝑡𝑦 𝐷 𝑀𝑎𝑠𝑠 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑣𝑖𝑡𝑦 (𝑥, 𝑦) 𝐶𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 ղ 𝑆𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝜓 𝑆𝑡𝑟𝑒𝑎𝑚 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝜇 𝐷𝑦𝑛𝑎𝑚𝑖𝑐 𝑣𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦

Das (2010) and Seethamahalakshmi et’al. (2011) studied MHD free convection flow and Mass transfer near a moving vertical plate in the presence of thermal radiation. Their results showed that increase in thermal radiation parameter contributes to the decrease in velocity field. Ghara et’al (2012) reported the effect of radiation on MHD free convection flow past an impulsively moving vertical plate with ramped wall temperature, buttressed by Siva et’al (2016). It was reviewed that temperature profile increases the increase in thermal radiation and Eckert number. The radiation effect on the flow past a vertical plate with the mass transfer was examined by Rajput and Kumar (2012). Narahari and Ishaq (2011) reported the radiation effects of free convection flow near a moving vertical plate with Newtonian heating. Das et’al. (2015) and Nepal et’al. (2014) observed the free convection flo w past a vertical plate with heat and mass fluxes in the presence of thermal radiation and concluded that thermal boundary layer thickness increases with increase in radiation parameter. Attention has also been given to the hall effect as Mohammed et’al (2013) observed the heat and Mass transfer in MHD free convection flow over an inclined plate with hall current and reported that there is no effect of the Magnetic parameter and Schmidt number on the temperature field and concentration. However, Gnaneswara (2014) reported the effect of the hall parameter in the temperature is small and the magnetic and hall parameters have opposite effects on the velocity and temperature profiles while studying the effect of thermal radiation, viscous dissipation and hall current effects in the MHD convection flow over a stretched vertical flat plate. Several investigations were performed on porosity in a medium with different conditions. Sandeep et’al (2012), Salem and Rania (2012) studied MHD heat and mass transfer through a porous medium as well as Jhansi et al (2015). Other authors like Idowu et’al (2013), Lakshmi et’al. (2014) and Olubode et’al (2016) investigated MHD flow along a vertical porous plate. Recently, Opiyo and Alfred (2017) studied the effects of Magnetohydrodynamic (MHD) fluid flow on a two-dimension boundary layer flow of a steady free convection heat and Mass transfer in an inclined plane in which the angle of inclination is varied. It was found that the velocity increases with an increase in the thermal and Solutal Grashof numbers. The velocity and concentration of the fluid decrease with an increase in the Schmidt number. The objective of this present investigation is to extend the work in Makinde (2010) to include Heat and Mass transfer in Magnetohydrodynamic (MHD) flow over a moving vertical plate with convective boundary condition in the presence of thermal radiation. The governing equations are solved analytically via Homotopy Analysis Method (HAM), developed by Liao (2003) and effect of different Parameters on fluid flow are considered.

2. Mathematical Formulation

Consider a steady-state two-dimensional boundary layer flow of a stream of cold incompressible electrically conducting fluid along a vertical plate. The surface of the plate is assumed to be heated by convection from a hot fluid at temperature 𝑇𝑓 that produces a heat transfer coefficient ℎ𝑓 . The cold fluid in contact with the surface of the plate generate heat internally at volumetric rate 𝑄0 . A magnetic field 𝐵0 is placed in a transverse direction to the flow. The magnetic Reynolds number is assumed to be small therefore the induced magnetic field is neglected. The joule heating term in energy equation is assumed to be neglected as it really 1032

very small in slow motion free convection flow. 𝑥 − 𝑎𝑥𝑖𝑠 is taken parallel to the plate direction and 𝑦 − 𝑎𝑥𝑖𝑠 normal to it (see fig.1). 𝐶𝑤 is the species concentration while 𝑇∞ and 𝐶∞ represent ambient temperature and concentration respectively. The fluid velocities in 𝑥 and 𝑦 dircetions are denoted by 𝑢 and 𝑣 respectively. The fluid temperature and concentration are respectively taken as 𝑇 and 𝐶.

Figure 1. Flow configuration and coordinate system Under the assumption stated above, boundary layer approximation and usual Boussinesq's approximation, the governing equations the present problem can be expressed as 𝜕𝑢 𝜕𝑥

with the following boundary conditions 𝑈(𝑥, 0) = 𝑈0 , 𝑉(𝑥, 0) = 0, −𝑘

where 𝜆 denotes the power index of the concentration and 𝑘 is the thermal conductivity coefficient. The radiative heat flux by Roseland is adopted and expressed as 𝑞𝑟 =

where 𝜎 is the Sterfan-Boltzmann constant and 𝐾 ∗ is the mean of absorption coefficient. It is

assumed that the temperature differences within the flow are such that the term 𝑇 4 can be expressed as a linear function of temperature by expanding 𝑇 4 in a Taylor series about 𝑇∞ as; 𝑇 4 = 𝑇∞4 𝑇 + 4𝑇∞3 (𝑇 − 𝑇∞ ) − 6𝑇∞2 (𝑇 − 𝑇∞ )2 + . . .

and neglecting higher order terms beyond the first degree in (𝑇 − 𝑇∞ ) gives 𝑇 4 ≈ 4𝑇∞3 𝑇 − 3𝑇∞4

The substitution of equations (6) and (8) in equation (3) gives a modified equation of the form 𝑢

Following Makinde (2010) and Mohammed et’.al (2015), the continuity equation (1) is satisfied automatically by invoking the stream function defined by 𝑢=

and obtained similarity equations of the problem by introducing the following similarity transformation 𝑈

Where ղ is an independent similarity variable, 𝜃(ղ) and ∅(ղ) are dimensionaless temperature and concentration respectively, 𝑈0 is the velocity of the plate. Apply equation equations (10) and (11) into equations (2), modified equation (9) and equation (4), we have 1

Which agreed with Makinde (2010), Rout et’al (2013), Lakshmi et’al.(2014), Hemalatha and Bhaskar (2015) where the prime symbol represents differentiation with respect to ղ and 𝐻𝑎 = 𝑃𝑟 =

For 𝐻𝑎 is the local magnetic field parameter, 𝐺𝑟 is the local thermal Grashof number, 𝐺𝑐 is the Solutal Grashof number, 𝐵𝑖 is the local convective heat transfer parameter, 𝑃𝑟 is the Prandtl number, 𝑆𝑐 is the Schmidt number, 𝑄 is the heat source, 𝐸𝑐 is the Eckert number and 𝑅𝑎 is the Radiation parameter. The corresponding boundary conditions are as follows 𝑓(0) = 0, 𝑓 ′ (0) = 1, 𝜃 ′ (0) = 𝐵𝑖[𝜃(0) − 1], ∅(0) = 1 (16) 𝑓 ′ (∞) = 0, 𝜃(∞) = 0, ∅(∞) = 0

The local parameters 𝐵𝑖, 𝐻𝑎, 𝐺𝑟, Q and 𝐺𝑐 in (12-14) denotes the function of 𝑥. In an attempt to have similarity solution, we assume the following parameters ℎ𝑓 =

Where 𝑝, 𝑞, 𝑟, 𝑥, and 𝑡 are constant under the appropriate dimension. The coupled equations (12-14) subject to the boundary conditions of equations (16) and (17) are solved analytically by Homotopy Analysis Method as shown in (3.0) below. For the purpose of Engineering application, we compute the local skin friction coefficient, the Local Nusselt number, the Local Sherwood number and the plate surface temperature are considered interms of 𝑓 ′′ (0), −𝜃 ′ (0), −∅′ (0) and 𝜃(0) respectively and the results obtained are presented in the tabular form.

3. Homotopy Analysis Method

The set of coupled Non-linear differential equations are usually inevitable and has become a culture in mathematical modeling. They are solved by a different method, among which are;

Adomian Decomposition, Variation Iteration Method and so on. Homotopy Analysis Method (HAM), discovered by Liao (2003) was preferred over another method due to its efficiency in solving both Linear and non-linear differential equation particularly at infinite domain. We consider the differential equations 𝐿[𝑓(ղ)] = 0

where 𝐿 and 𝑁 are called Linear and non-linear function respectively (for Algebra Equation) or Linear and Non-linear operator respectively (for differential Equations), ղ represents an independent variable, while 𝑓(ղ) is the solution of (19). Let 𝑓0 (ղ) be initial guess for 𝑓(ղ) and ℏ ≠0, 𝐻(ղ) ≠ 0 denote the auxillary parameter, auxiliary function respectively, we then construct a family equation of the form (1 − 𝑟)𝐿[𝑓(ղ; 𝑟) − 𝑓0 (ղ)] = 𝑟ℏ𝐻(ղ)𝑁[𝑓(ղ; 𝑟)]

where 𝑟 ∈ [0,1] is called an embedding parameter. When, 𝑟 = 0, we have 𝑓(ղ; 0) = 𝑓0 (ղ) 𝑎𝑛𝑑 𝑁[𝑓(ղ; 1)] = 0 𝑏𝑢𝑡 ℏ𝐻(ղ) ≠ 0 for, 𝑟 = 1

Hence, in respect to the boundary conditions (16) and (17), 𝑓(ղ), 𝜃(ղ) and ∅(ղ) Can be expressed by the set of base functions { ղ𝑗 exp(−𝑛𝑗) | 𝑗 ≥ 0, 𝑛 ≥ 0 }

in the following form ∞ 𝑘 ∞ 𝑘 ∞ 𝑘 𝑘 𝑓(ղ) = ∑∞ 𝑛=0 ∑𝑘=0 𝑎𝑛,𝑘 ղ exp(−𝑛𝑗), 𝜃(ղ) = ∑𝑛=0 ∑𝑘=0 𝑏𝑛,𝑘 ղ exp(−𝑛𝑗) 𝑎𝑛𝑑 ∅(ղ) = ∞ 𝑘 ∞ 𝑘 ∑𝑛=0 ∑𝑘=0 𝑐𝑛,𝑘 ղ exp(−𝑛𝑗)

𝑘 𝑘 𝑘 where 𝑎𝑛,𝑘 , 𝑏𝑛,𝑘 and 𝑐𝑛,𝑘 are coefficients. However, as long as such a set of base functions is

determined, the auxiliary function 𝐻(ղ), initial approximation 𝑓0 (ղ), 𝜃0 (ղ), ∅0 (ղ), and the auxiliary linear operators 𝐿𝑓 , 𝐿𝜃 , and 𝐿∅ must be chosen in such a way that solution of the corresponding high-order deformation exist (see Farooq et’al (2015) and Olubode et’al (2016)). The point raised above is essential in the framework of homotopy Analysis Method as its provide us with a basic rule called the rule of solution expression for 𝑓(ղ), 𝜃(ղ) and ∅(ղ). In accordance with the rule of solution and boundary conditions (16) – (17), we choose the initial guess 𝑓0 (ղ) = 1 − 𝑒𝑥𝑝(−ղ),

as the initial linear approximations of 𝑓(ղ), 𝜃(ղ) and ∅(ղ). The auxiliary linear operations 𝐿𝑓 , 𝐿𝜃 , and 𝐿∅ are; 𝐿𝑓 [𝑓(ղ; 𝑟)] =

agreed with the following properties 𝐿𝑓 [𝐶1 + 𝐶2 𝑒𝑥𝑝(ղ) + 𝐶3 𝑒𝑥𝑝(−ղ)] = 0, 𝐿𝜃 [𝐶4 + 𝐶5 𝑒𝑥𝑝(−ղ)] = 0 𝑎𝑛𝑑 𝐿∅ [𝐶6 + 𝐶7 𝑒𝑥𝑝(−ղ)] = 0

4. Validation Of The Study

Table 1. Comparison of the present result with Makinde (2010) Makinde (2010)

Here, we first ensure the successful implementation of the numerical result by comparing it with the previous work done. So, these present results are compared to those obtained by Makinde (2010) for the local skin-friction, Nusselt Number, Sherwood number and plate surface temperature by setting 𝑄 = 0, 𝑅𝑎 = 0, 𝐸𝑐 = 0. The results strongly agreed with each other (see Table 1).

5. Discussion Of Results

In order to get a physical understanding of the present problem, equation (12)-(14) with the boundary conditions (16) and (17) have been solved using Homotopy Analysis Method (HAM) at 20th –order to meet the far field boundary condition at infinite domain. The resulting effects of various parameters embedded in the flow system such as; Magnetic Parameter (𝐻𝑎), Thermal Grashof Number (𝐺𝑟), Solutal Grashof Number (𝐺𝑐), Prandtl Number (𝑃𝑟. ), Schmidt Number (𝑆𝑐), Local Heat transfer parameter (𝐵𝑖), Heat Source Parameter (𝑄), Eckert number (𝐸𝑐), and Radiation Parameter (𝑅𝑎) on Velocity profile, Temperature profile, Concentration profile, Local Skin-friction, Local Nusselt Number, plate surface temperature and Sherwood number were presented in graphically and numerically. During the numerical computation, the Prandtl number was considered to be 0.72 which correspond to air and it is mostly encountered fluid in nature and commonly used in engineering.

The positive values of thermal Grashof number and Solutal Grashof number which are collectively referred to as buoyancy parameter correspond to the greater cooling of the surface and shows that the concentration at the plate surface is higher than the free stream concentration respectively. The cooling surface such as nuclear reactors is frequently encountered in engineering and industry. The values of Schmidt number 𝑆𝑐 for diffusing chemical species in air were chosen to be 𝑆𝑐 = 0.24 (𝐻2 ), 0.62 (𝐻2 𝑂), 𝑆𝑐 = 0.78 (𝑁𝐻3 ) and 𝑆𝑐 = 2.62 (𝐶9 𝐻12 ). Other parameters were discussed by holding 𝐻𝑎 = 𝐺𝑟 = 𝐺𝑐 = 𝐵𝑖 = 𝐸𝑐 = 0.1, 𝑆𝑐 = 0.62, 𝑃𝑟 = 0.72, 𝑄 = 0.01, 𝑅𝑎 = 0.7 constant for each varying parameter.

Figure 7. Concentration profile for 𝐻𝑎 Figure 5, 6 and 7 reveals the variation effects of Magnetic Parameter 𝐻𝑎, on velocity, temperature and concentration profile respectively. It is obvious from the figure 5 as expected that the velocity distribution across the boundary layer decreases with the increase in 𝐻𝑎. This obvious decrease is true due to the fact that increase in Magnetic field brings about an opposing force to the flow called Lorentz force which has tendency to resist motion of fluid and decrease the momentum boundary layer. However, increase in 𝐻𝑎 as well results in frictional heating and increase the fluid temperature, magnitude of the local skin-friction, plate surface temperature and the concentration of the fluid while the Nusselt and Sherwood numbers decrease (See Fig.(6-7) and Table 2). Note that the thickness of the thermal and concentration boundary layer improve as the fluid temperature and its concentration increase.

Figure 13. Concentration profile for 𝐺𝑐 Figures (8-13) illustrate the influence of thermal Grashof and Solutal Grashof numbers on velocity, temperature, and concentration profiles. The thermal Grashof number (𝐺𝑟) signifies the relative importance of buoyancy force to the viscous hydrodynamic force within the boundary layer while the solutal Grashof number (𝐺𝑐) defines the ratio of the species buoyancy force to the viscous hydrodynamic force. It can be seen from the figures 8 and 11, that increase in (𝐺𝑟,𝐺𝑐) gives rise to the fluid velocity within the boundary layer and suddenly fall monotonically to the free stream zero value far away from the plate surface agreeing with the far field boundary conditions which inturns increases the thickness of momentum boundary layer (See fig.8 and fig.11). It is interesting to note that the positive values of (𝐺𝑟,𝐺𝑐) correspond to the cooling of the plate as the fluid Temperature, Plate Surface Temperature and the fluid concentration decrease which inturns deteriorate the thickness of thermal and concentration boundary layers as shown in fig.(9-10), fig.(12-13). The Local Skin Friction, Nusselt number and Sherwood number increase with the increase in (𝐺𝑟,𝐺𝑐) (see table 2).

Figures (14-16), depicts the influence of Prandtl number 𝑃𝑟 on the velocity, temperature and concentration profiles respectively. Prandtl number 𝑃𝑟 is a dimensionaless number, approximating the ratio of momentum diffusivity to thermal diffusivity. Increase in 𝑃𝑟 as a result of low thermal diffusivity results in an increase in magnitude of local skin-friction, Nusselt number with a reverse phenomenon on Plate Surface Temperature and Sherwood number as shown in table 2. It can be seen from the figure that increase in 𝑃𝑟 leads to a fall in velocity field and rapid decrease in the thermal boundary layer thickness which inturns lowers the average temperature across the boundary layer. The main reason is that, the smaller values of 𝑃𝑟 are equivalent to increase in the thermal conductivity. This however, enable the heat to diffuse away from the heated surface more rapidly than the higher values. Table 2. Numerical values of the Skin-friction coefficient, Local Nusselt number, Local Sherwood number and plate surface temperature 𝐻𝑎 𝐺𝑟

6. Conclusion

This study has vast application in industries and engineering disciplines in understanding the dynamic flowing phenomenon which is a major language in science and technology such as cooling of nuclear reactors, cooling of electronic components and enhanced oil recovery e.t.c. In this present investigation, an analysis is made to study heat and mass transfer in hydromagnetic boundary layer flow over a moving vertical plate with convective boundary condition in the presence of thermal radiation. The resulting partial differential equations which describe the problem are transformed to dimensionless equations using Similarity method with the corresponding dimensionless variables. We then solve the equations by Homotopy Analysis Method and the results are discussed through graphs and tables for different values of embedding parameters and the following conclusion are drawn from the results obtained.  Cooling problem is guaranteed with the positive values of (𝐺𝑟, 𝐺𝑐) which is often encountered in engineering application such as cooling of electronic component and nuclear reactors.  The Nusselt number increased as the values of Prandtl number, radiation parameter, and convective heat parameter increase but decrease with the increase in Schmidt number and viscous dissipation  The momentum boundary layer thickness decrease while the thermal and concentration boundary layers thickness increase on the increase in the magnetic parameter.  An increase in viscous dissipation parameter enhanced the velocity and temperature profiles with a reverse phenomenon on concentration profile.  Higher values of radiation parameter 𝑅𝑎, pioneer the dominance of conduction over radiation and consequently depressed the thermal boundary layer thickness.

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