The problem of unsteady magnetohydrodynamic convective flow with radiation and chemical reaction past a flat porous plate moving through a binary mixture in an optically thin environment is considered. The governing boundary layer equations are converted to nonlinear ordinary differential equations by similarity transformation and then solved numerically by MATLAB “bvp4c” routine. The velocity, temperature, and concentration profiles are presented graphically for various values of the material parameters. Also a numerical data for the local skin friction coefficient, the local Nusselt number, and local Sherwood number is presented in tabular forms.
1. Introduction
Combined heat and mass transfer problems with chemical reaction are of importance in many processes and have, therefore, received a considerable amount of attention in research. In processes such as drying, evaporation at the surface of a water body, energy transfer in a wet cooling tower, and the flow in a desert cooler, heat and the mass transfer occur simultaneously. Study towards boundary layer flow of a binary mixture of fluids is very important in view of its application in various branches of engineering and technology. A familiar example is an emulsion which is the dispersion of one fluid within another fluid. Typical emulsions are oil dispersed within water or water within oil. Another example where the mixture of fluids plays an important role is in multigrade oils. Some polymeric type fluids are added to the base oil so as to enhance the lubrication properties of mineral oil [1]. Moreover through chemical reaction, all industrial chemical processes are designed to transform cheaper raw materials to high value products. Naturally these transformations occur in reactors. Fluid dynamics plays a pivotal role in establishing relationship between the reactor hardware and reactor performance. Unsteady free convection boundary layer flows with heat and mass transfer encounter an important criterion of species chemical reaction with finite Arrhenius activation energy defined by Makinde [2]. This phenomenon is useful in the areas such as geothermal or oil reservoir engineering where more numbers of experimental works happen. It is very important for theoretical works to predict the effects of the activation energy in these flows. But very few theoretical works are available in the literature as the chemical reaction process involved in the systems are quite complex. This may be simplified by restricting the reaction to binary type chemical reaction. The thermomechanical balance equations for a mixture of general materials were first formulated by Truesdell [3]. Thereafter, Mills [4], Beevers, and Craine [5] have obtained some exact solutions for the boundary layer flow of a binary mixture of incompressible Newtonian fluids. A particular contribution towards the context of binary mixture theory was done by Al-Sharif et al. [6] and Wang et al. [7]. Recently Kandasamy et al. [8] studied the combined effects of chemical reaction, heat, and mass transfer along a wedge with heat source and concentration in presence of suction or injection. Their result shows that the flow field is influenced appreciably by chemical reaction, heat source, and suction or injection at the wall of the wedge. El-Hakiem [9] studied the unsteady MHD oscillatory flow on free convection-radiation through a porous medium with a vertical infinite surface that absorbs the fluid with a constant velocity. Raptis et al. [10] studied the effect of radiation on two-dimensional steady MHD optically thin gray gas flow along infinite vertical plate taking the induced magnetic field into account. Israel-Cookey et al. [11] discussed the influence of viscous dissipation and radiation on unsteady MHD free convection flow past on infinite heated vertical plate in a porous medium with time-dependent suction. Abd El-Naby et al. [12] employed an implicit finite-difference method to study the effect of radiation on MHD unsteady free convection flow past a semi-infinite vertical porous plate without viscous dissipation. Singh and Dikshift [13] investigated the hydromagntic flow past a continuously moving semi-infinite plate at large suction. Takhar et al. [14] observed the radiation effects on MHD free convection flow past a semi-infinite vertical plate. Kim [15] studied unsteady MHD convective heat transfer past a semi-infinite vertical porous moving plate. He found that an increase in the Prandtl number and magnetic field intensity decreases the fluid velocity. Vajravelu and Hadjinicolaou [16] studied the heat transfer characteristics in the laminar boundary layer of a viscous fluid over a stretching sheet with viscous dissipation or frictional heating and internal heat generation. Alam et al. [17] studied the problem of free convection heat and mass transfer flow past an inclined semi-infinite heated surface of an electrically conducting and steady viscous incompressible fluid in the presence of a magnetic field and heat generation. Chamkha [18] investigated unsteady convective heat and mass transfer past a semi-infinite porous moving plate with heat absorption. Hady et al. [19] studied the problem of free convection flow along a vertical wavy surface embedded in an electrically conducting fluid saturated porous media in the presence of internal heat generation or absorption. More recently Makinde and Olanrewaju [20] have studied the effects of chemical reaction and radiative heat transfer for an unsteady convection flow past porous plate moving through a binary mixture.
In this paper, we have considered the Arrhenius kinetics and thermal radiation in an unsteady MHD convective flow over a moving plate through a binary mixture with suction or injection at the plate surface. The governing equations are converted to ordinary differential equations by applying the similarity transformation. The numerical solution of the similarity equations are then obtained through the MATLAB “bvp4c” routine. The analysis of the results obtained shows that the flow field is influenced by the presence of magnetic field parameter, thermal radiation, chemical reaction, buoyancy force, and suction/injection parameter at surface of the plate. The profiles of velocity, temperature, and concentration are represented graphically. The skin friction, heat transfer, and mass transfer are displayed in tabular forms for different parameters.
2. Mathematical Formulation
Consider the unsteady one-dimensional hydromagnetic convective flow with chemical reaction and radiative heat transfer past a vertical porous plate moving through a binary mixture (see Figure 1). Assume that the fluid is electrically conducting and the boundary wall to be of infinite extended so that all quantities are homogeneous in x and hence all derivatives with respect to x are omitted. Further assume the fluid is optically thin with absorption coefficient α≪1. Let the x-axis be directed in upward direction along the plate and the y-axis is normal to the plate. Let u and v be the velocity components along the x- and y-axes, respectively. A magnetic field B0 of uniform strength is applied transversely to the direction of the flow. Since the fluid pressure is constant, it is assumed that induced magnetic field is small in comparison to the applied magnetic field; therefore, it is neglected.
Flow configuration and coordinate system.
Under these assumptions the momentum, energy, and chemical species concentration balance equations which govern the flow may be written as follows:
(1)∂v∂y=0,(2)∂u∂t+v∂u∂y=υ∂2u∂y2+gβT(T-T∞)+gβC(C-C∞)-σρB02u,(3)ρCp(∂T∂t+v∂T∂y)=k∂2T∂y2+Q-4σα2T4,(4)∂C∂t+v∂C∂y=Df∂2C∂y2-RA,
where Q=(-ΔH)RA is the heat of chemical reaction and is called the activation enthalpy and
(5)RA=kre-EA/RGTCn
is the Arrhenius type of the nth order irreversible reaction, kr is the chemical reaction rate, RG is the universal gas constant, and EA is the activation energy parameter. The boundary conditions of the above problem are assumed to be
(6)u(y,0)=0,T(y,0)=Tw,C(y,0)=Cw,u(0,t)=U0,T(0,t)=Tw,C(0,t)=Cw,t>0,u⟶0,T⟶T∞,C⟶C∞asy⟶∞,t>0,
where U0 is the plate characteristic velocity. From the equation of continuity (1), it can be noted that v is either constant or a function of time. Following Makinde [2], we take
(7)v=-c(υt)1/2,
where c>0 is the suction parameter and c<0 is the injection parameter.
We introduce the dimensionless quantities and parameters
(8)u=U0F(η),(θ,θw)=(T,Tw)T∞,(ϕ,ϕw)=(C,Cw)C∞,Gr=4tgβTT∞U0,Gc=4tgβCC∞U0,Pr=υλ,λ=kρCp,Sc=υDf,γ=EARGT∞,η=y2υt,h=(-ΔH)C∞ρCpT∞,Ra=16σα2tT∞3ρCp,Da=4tk0C∞n-1,k0=kre-EA/RGT∞,M=4tσB02ρ.
With (8) equations in (2), (3), and (4) become
(9)F′′+2(η+c)F′-MF=-Gr(θ-1)-Gc(ϕ-1),1Prθ′′+2(η+c)θ′=-hϕnexp{γ(1-1θ)}+Raθ4,1Scϕ′′+2(η+c)ϕ′=Daϕnexp{γ(1-1θ)},
with boundary conditions
(10)F(0)=1,θ(0)=θw,ϕ(0)=ϕw,F(∞)=0,θ(∞)=1,ϕ(∞)=1,
where Da is the Damköhler number, Ra is the radiation parameter, γ is the activation energy parameter, Gr is the thermal Grashof number, Gc is the solutal Grashof number, Kr is the chemical reaction rate, Pr is the Prandtl number, and M is the magnetic field parameter.
The wall skin-friction
(11)τw=-μ(∂u∂y)y=0=-12μU0υtF′(0).
Hence, the skin-friction coefficient
(12)Cf=2τwρU02=-μρU0υtF′(0)=-1ReF′(0)∝-F′(0),
where Re=ρU0υt/μ is the Reynolds number.
At the wall, the heat flux (qw) and the mass flux (mw) are given by
(13)qw=-k(∂T∂y)y=0,mw=-Df(∂C∂y)y=0.
The Nusselt number (Nu) and Sherwood number (Sh) are defined as
(14)Nu=qwυtk(Tw-T∞)=-12θ′(0)∝-θ′(0),(15)Sh=mwυtDf(Cw-C∞)=-12ϕ′(0)∝-ϕ′(0),
where υt is characteristic length.
The coefficients presented in (12) and (14) are obtained from the procedure of the numerical computations and are sorted for different parameters given in Tables 1–8.
Comparison values of F′(0),θ′(0), and ϕ′(0) for different Da (c=Ra=Gr=Gc=γ=θw=ϕw=0.1,Sc=0.22,h=n=1).
Da
F′(0)
θ′(0)
ϕ′(0)
Makinde and Olanrewaju [20]
Present
Makinde and Olanrewaju [20]
Present
Makinde and Olanrewaju [20]
Present
0.1
-1.363220887
-1.363874729
0.940099503
0.939692306
0.45299820
0.450546865
0.2
-1.362524805
-1.363259605
1.017519463
1.019231925
0.40825426
0.403754294
0.3
-1.362049836
-1.362888018
1.085715626
1.088738943
0.36731514
0.361127499
Comparison values of F′(0),θ′(0), and ϕ′(0) for different Sc (c=Ra=Da=Gr=Gc=γ=θw=ϕw=0.1,h=n=1).
Sc
F′(0)
θ′(0)
ϕ′(0)
Makinde and Olanrewaju [20]
Present
Makinde and Olanrewaju [20]
Present
Makinde and Olanrewaju [20]
Present
0.22
-1.363220887
-1.363874729
0.940099503
0.939692306
0.45299820
0.450546865
0.62
-1.347622867
-1.348351348
0.950736367
0.950277754
0.76629786
0.762196009
0.78
-1.344659445
-1.345402697
0.952741674
0.952272964
0.86343256
0.858815122
Comparison values of F′(0),θ′(0), and ϕ′(0) for different c(Da=Ra=Gr=Gc=γ=θw=ϕw=0.1,Sc=0.22,h=n=1).
c
F′(0)
θ′(0)
ϕ′(0)
Makinde and Olanrewaju [20]
Present
Makinde and Olanrewaju [20]
Present
Makinde and Olanrewaju [20]
Present
0.1
-1.363220887
-1.363874729
0.940099503
0.939692306
0.45299820
0.450546865
1.0
-2.737972223
-2.739021858
1.796428656
1.795804409
0.68928294
0.685864239
-.1
-1.108018112
-1.108575893
0.779055070
0.778698841
0.40619537
0.403951603
-1
-0.311091119
-0.311271144
0.244583695
0.244436402
0.22690545
0.225521578
Comparison values of F′(0),θ′(0), and ϕ′(0) for different Ra (c=Da=Gr=Gc=γ=θw=ϕw=0.1,Sc=0.22,h=n=1).
Ra
F′(0)
θ′(0)
ϕ′(0)
Makinde and Olanrewaju [20]
Present
Makinde and Olanrewaju [20]
Present
Makinde and Olanrewaju [20]
Present
0.1
-1.363220887
-1.363874729
0.940099503
0.939692306
0.45299820
0.450546865
0.2
-1.370176013
-1.371318069
0.867590872
0.864748108
0.45322418
0.450790024
0.3
-1.375748994
-1.377248017
0.812511169
0.808288321
0.45342377
0.451004257
Comparison values of F′(0),θ′(0), and ϕ′(0) for different n(c=Da=Ra=Gr=Gc=γ=θw=ϕw=0.1,Sc=0.22,h=1).
n
F′(0)
θ′(0)
ϕ′(0)
Makinde and Olanrewaju [20]
Present
Makinde and Olanrewaju [20]
Present
Makinde and Olanrewaju [20]
Present
1
-1.363220887
-1.363874729
0.940099503
0.939692306
0.45299820
0.450546865
3
-1.363537283
-1.364193050
0.914076370
0.913256158
0.46481660
0.462643274
5
-1.363673665
-1.364329690
0.903949316
0.902860087
0.46999599
0.468013723
Comparison values of F′(0),θ′(0), and ϕ′(0) for different Gr (c=Da=Ra=Gc=γ=θw=ϕw=0.1, Sc=0.22,h=n=1).
Gr
F′(0)
θ′(0)
ϕ′(0)
Makinde and Olanrewaju [20]
Present
Makinde and Olanrewaju [20]
Present
Makinde and Olanrewaju [20]
Present
0.1
-1.363220887
-1.363874729
0.940099503
0.939692306
0.45299820
0.450546865
1.0
-1.716737895
-1.717731585
0.940099503
0.939692306
0.45299820
0.450546865
5.0
-3.287924599
-3.290428721
0.940099503
0.939692306
0.45299820
0.450546865
Comparison values of F′(0),θ′(0), and ϕ′(0) for different Gc (c=Da=Ra=Gr=γ=θw=ϕw=0.1, Sc=0.22,h=n=1).
Gc
F′(0)
θ′(0)
ϕ′(0)
Makinde and Olanrewaju [20]
Present
Makinde and Olanrewaju [20]
Present
Makinde and Olanrewaju [20]
Present
0.1
-1.363220887
-1.363874729
0.940099503
0.939692306
0.45299820
0.450546865
1.0
-1.950303852
-1.956502423
0.940099503
0.939692306
0.45299820
0.450546865
5.0
-4.559561477
-4.590403287
0.940099503
0.939692306
0.45299820
0.450546865
Values of F′(0) for different M(Da=Ra=Gr=Gc=γ=θw=ϕw=0.1,Sc=0.22,h=n=1).
M
F′(0) (suction, c=0.1)
F′(0)(injection, c=-0.1)
0
-1.363874729
-1.110534369
1
-1.689339475
-1.451374690
2
-1.966674451
-1.737579643
5
-2.635097137
-2.418368678
10
-3.475594470
-3.265944783
3. Results and Discussion
To get a clear insight of the physical problem, we have assigned various numerical values to the parameters that are incorporated in the problem with which one can discuss the profiles of velocity, temperature, and concentration. In order to compare the present results with previous work [20], we have taken the values of Schmidt number (Sc) for hydrogen 0.22, water vapour 0.62, ammonia 0.78, and propyl benzene 2.62 at temperature 25°C and one atmospheric pressure. The value of Prandtl number is chosen to be Pr=0.71 which represents air at temperature 25°C and one atmospheric pressure. Moreover the focus is made towards the positive values of the buoyancy parameters; that is, Grashof number Gr>0, corresponds to the cooling problem, and solutal Grashof number Gc>0 indicates that the concentration in the free stream region is less than the concentration at the boundary surface. It is worthy to note from Table 1 that increase in chemical reaction parameter (Da) enhances the heat transfer rate and reduces both skin friction and mass transfer rate. From Table 2, it is seen that the increase in the values of Sc leads to enhance both heat, and mass transfer rates leaving a decrement in the values of skin friction. We note from Table 3 that, in the case of wall suction, increase in the value of c shows an increment in skin friction, heat, and mass transfer rates whereas, in case of injection, increment in c shows a decrement in the respective fluid properties. From Tables 4 and 5 it is seen that the effect of increasing values of Ra and n is to increase both mass transfer rate and skin friction and to decrease the heat transfer rate. From Tables 6 and 7 it is observed that increase in buoyancy parameters enhance the skin friction at the moving plate surface. Also from Table 8 it is observed that skin friction increases with the increase in magnetic field parameter M.
Figures 2–5 display the velocity profiles for different material parameters. Figure 2 illustrates the effect of magnetic field parameter on velocity profiles when the other parameters are fixed. The magnetic field within the boundary layer has produced a resistive type force known as Lorentz force. Due to this force, retardation in the fluid motion along surface is observed. Therefore, it is clear from the same figure that the momentum boundary layer thickness decreases with the increase of magnetic field parameter. Figures 3 and 4 illustrate the effect of buoyancy forces on the horizontal velocity component in the momentum boundary layer. In presence and absence of magnetic field parameter, fluid velocity is highest at the moving plate surface and decreases to free stream zero velocity far away from the plate satisfying the boundary conditions. Also it is observed that in presence of uniform suction at the plate surface, increase of buoyancy forces lead to retardation in the flow and thereby giving rise to a decrease in the velocity profiles; that is, momentum boundary layer thickness decreases with an increase of buoyancy forces. It is observed that a reverse flow occurs within the boundary layer as the intensity of buoyancy forces increases. It is worthy to note that the reverse flow is less in hydromagnetic fluids. Figure 5 depicts the effect of wall suction and injection on the horizontal velocity in momentum boundary layer. It is observed that the momentum boundary layer thickness decreases with increasing the wall suction (c>0) and increases with increasing wall injection (c<0). The same trend is noticed in the magnetohydrodynamic flows.
Effect of magnetic field parameter on velocity profiles when (Da=c=Ra=Gr=Gc=γ=θw=ϕw=0.1,Sc=0.62,h=n=1).
Effect of thermal Grashof number on velocity profiles when (Da=c=Ra=Gc=γ=θw=ϕw=0.1,Sc=0.62,h=n=1).
Effect of solutal Grashof number on velocity profiles when (Da=c=Ra=Gr=γ=θw=ϕw=0.1,Sc=0.62,h=n=1).
Effect of suction/injection parameter on velocity profiles when (Da=Ra=Gr=Gc=γ=θw=ϕw=0.1,Sc=0.62,h=n=1).
The effects of various material parameters on the fluid temperature are illustrated in Figures 6–10. It is seen from Figure 6, the variation of temperature profile against similarity variable η for varying values of wall temperature parameter under uniform magnetic field and uniform suction. The temperature increases towards the free stream temperature whenever the surface temperature is lower than the free stream temperature. The temperature decreases toward the free stream temperature whenever the plate temperature is higher than the free stream temperature. Figure 7 shows the effect of chemical reaction rate on the fluid temperature profile within the boundary layer when the wall temperature is lower than the free stream temperature in the presence of uniform suction. It is observed that increasing value of the Damköhler number Da enhances the fluid temperature. This is because of internal heat generation in the fluid due to Arrhenius kinetics. Figure 8 depicts the effect of radiation parameter Ra on temperature profiles in uniform magnetic field. It is noticed from this figure that the fluid temperature starts from a minimum value at the moving plate surface and then increases till it reaches the free stream temperature value at the of the boundary layer for all the values of radiation parameter. It is also observed that temperature of the fluid decreases with increase in the value of Ra. Figure 9 represents the effect of suction and injection parameter on the fluid temperature for different values of magnetic field parameter. It is noticed that the effect of magnetic field parameter on temperature profiles is insignificant. The fluid temperature increases with increase of suction and decreases with increase of injection. The effect of reaction order parameter n, on the temperature profile is represented in Figure 10. The fluid temperature decreases with increasing order of chemical reaction. Figures 11–14 illustrate the effects of various parameters on concentration profiles. The effect of chemical reaction rate parameter Da on concentration profiles in presence and absence of magnetic field parameter is shown in Figure 11. It is noticed that magnetic field parameter does not affect the concentration profiles. An increase in chemical reaction rate causes a decrease in the concentration of the chemical species in the boundary layer supporting the fact that chemical reaction rate reduces the local concentration. Figure 12 shows the effect of Schmidt number Sc on concentration profiles. From this figure we observed that chemical species concentration within the boundary layer increases with an increase in Sc. From Figure 13 it is noted that increasing the order of chemical reaction enhances the species concentration within the boundary layer. Figure 14 shows the effect of suction and injection parameter on the concentration profiles. The species concentration is higher for suction and lower for injection. Figure 15 shows the variation of concentration profiles against similarity variable η for varying values of wall concentration parameter under uniform magnetic field and uniform suction. The concentration increases towards the free stream concentration whenever the species concentration at surface is lower than the free stream concentration. The reverse trend is observed when surface concentration is higher than the free stream concentration.
Effect of wall temperature on temperature profiles when (M=2,Da=c=Ra=Gr=Gc=γ=ϕw=0.1,Sc=0.62,h=n=1).
Effect of chemical reaction on temperature profiles when (c=Ra=Gr=Gc=γ=θw=ϕw=0.1,Sc=0.62,h=n=1).
Effect of radiation on temperature profiles when (M=2,c=Da=Gr=Gc=γ=θw=ϕw=0.1,Sc=0.62,h=n=1).
Effect of suction/injection parameter on temperature profiles when (Da=Ra=Gr=Gc=γ=θw=ϕw=0.1,Sc=0.62,h=n=1).
Effect of order of reaction parameter on temperature profiles when (M=2,Da=c=Ra=Gr=Gc=γ=θw=ϕw=0.1,Sc=0.62,h=1).
Effect of chemical reaction on concentration profiles when (c=Ra=Gr=Gc=γ=θw=ϕw=0.1,Sc=0.62,h=n=1).
Effect of Schmidt number on concentration profiles when (M=2,Da=c=Ra=Gr=Gc=γ=θw=ϕw=0.1,h=n=1).
Effect of order of reaction on concentration profiles when (M=2,Da=c=Ra=Gr=Gc=γ=θw=ϕw=0.1,Sc=0.62,h=1).
Effect of suction/injection on concentration profiles when (M=2,Da=Ra=Gr=Gc=γ=θw=ϕw=0.1,Sc=0.62,h=n=1).
Effect of wall concentration on concentration profiles when (M=2,Da=c=Ra=Gr=Gc=γ=θw=0.1,Sc=0.62,h=n=1).
4. Conclusion
The effects of magnetism, thermal radiation, suction/injection, buoyancy forces, nth order Arrhenius chemical reaction, and Damköhler number on unsteady convection of viscous incompressible fluid past a vertical porous plate are studied. A set of nonlinear coupled differential equations governing the fluid velocity, temperature, and chemical species concentration are solved numerically for various material parameters. Results for the velocity, temperature, and concentration are presented and discussed graphically. In the present study we noticed that fluid velocity within the boundary layer decreases with the increasing values of magnetic field parameter and buoyancy forces. Also it is observed that within the boundary layer fluid velocity decreases with increasing values of wall suction and increases with wall injection. The surface temperature decreases in presence of radiation and increases with increasing rate of exothermic chemical reaction Da and reaction order n. Also it is observed that the chemical species concentration within the boundary layer decreases with increasing values of Da and wall injection. We also noticed that the skin friction increases with increase in magnetic field parameter under uniform suction and injection.
List of Symbols(x,y):
Cartesian coordinates
(u,v):
Velocity components along x, y directions, respectively
t:
Time
g:
Acceleration due to gravity
T:
Temperature of the fluid
C:
Concentration of the fluid
Tw:
Surface temperature
Cw:
Surface concentration
T∞:
Free stream temperature
C∞:
Free stream concentration
Q:
Activation enthalpy
Df:
Diffusion coefficient
c:
Suction/injection parameter
EA:
Activation energy
h:
Heat generation parameter
k:
Thermal conductivity
Cp:
Specific heat at constant pressure
U0:
Uniform velocity of the plate
RG:
Universal gas constant
n:
Order of the chemical reaction
Sc:
Schmidt number
ΔH:
Enthalpy change
M:
Magnetic field parameter.
Greek Symbolsθ:
Nondimensional fluid temperature
ϕ:
Nondimensional fluid concentration
θw:
Dimensionless wall temperature
ϕw:
Dimensionless wall concentration
η:
Similarity variable
γ:
Activation energy parameter
βT:
Volumetric thermal-expansion coefficient
βC:
Volumetric solutal-expansion coefficient
σ:
Stefan-Boltzmann constant
α:
Absorption coefficient
ρ:
Density of the fluid
υ:
Kinematic viscosity
μ:
Fluid viscosity
′:
Prime represents the derivative with respect to.
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