On cross-diffusion effects on flow over a vertical surface using a linearization method
- Precious Sibanda^{1}Email author,
- Ahmed Abdalmgid Khidir^{1} and
- Faiz Gadelmola Awad^{1}
https://doi.org/10.1186/1687-2770-2012-25
© Sibanda et al; licensee Springer. 2012
Received: 18 October 2011
Accepted: 24 February 2012
Published: 24 February 2012
Abstract
In this article, we explore the use of a non-perturbation linearization method to solve the coupled highly nonlinear system of equations due to flow over a vertical surface subject to a magnetic field. The linearization method is used in combination with an asymptotic expansion technique. The effects of Dufour, Soret and magnetic filed parameters are investigated. The velocity, temperature and concentration distributions as well as the skin-friction, heat and mass transfer coefficients have been obtained and discussed for various physical parametric values. The accuracy of the solutions has been tested using a local non-similarity method. The results show that the non-perturbation technique is an accurate numerical algorithm that converges rapidly and may serve as a viable alternative to finite difference and finite element methods for solving nonlinear boundary value problems.
Mathematics Subject Classification (2000): 76D05; 74S30; 76E06; 76M25.
Keywords
cross-diffusion free convection linearization method incompressible flow magneto-hydrodynamics (MHD)1 Introduction
Convection driven by density variations caused by two different components which have different rates of diffusion plays an important role in fluid dynamics since such flows occur naturally in many physical and engineering processes. Heat and salt in sea water provide perhaps the best known example of double-diffusive convection, Stern [1]. Other examples of double diffusive convection are encountered in diverse applications such as in chemical and petroleum industries, filtration processes, food processing, geophysics and in the modeling of solar ponds and magma chambers. A review of the literature in this subject can be found in Nield and Bejan [2].
One of the earliest studies of double diffusive convection was by Nield [3]. Baines and Gill [4] investigated linear stability boundaries, while Rudraiah et al. [5] used the nonlinear perturbation theory to investigate the onset of double diffusive convection in a horizontal porous layer. Poulikakos [6] presented the linear stability analysis of thermosolutal convection using the Darcy-Brinkman model. Bejan and Khair [7] presented a multiple scale analysis of heat and mass transfer about a vertical plate embedded in a porous medium. They considered concentration gradients which aid or oppose thermal gradients. Related studies on double diffusive convection have been undertaken by, among others, Lai [8], Afify [9], and Makinde and Sibanda [10].
Investigations by, among others, Eckert and Drake [11] and Mortimer and Eyring [12] have provided examples of flows such as in the geosciences, where diffusion-thermo and thermal-diffusion effects are quite significant. Anjalidevi and Devi [13] showed that diffusion-thermo and thermal-diffusion effects are significant when density differences exist in the flow regime. In general, Diffusion-thermo and thermal-diffusion effects have been found to be particularly important for intermediate molecular weight gases in binary systems that are often encountered in chemical engineering processes. Theoretical studies of the Soret and Dufour effects on double diffusive convection have been made by many researchers, among them, Kafoussias and Williams [14], Postelnicu [15], Mansour et al. [16], Narayana and Sibanda [17], and Awad et al. [18].
In this article, we investigate convective heat and mass transfer along a vertical flat plate in the presence of diffusion-thermo, thermal diffusion effects, and an external magnetic field. The governing momentum, heat and mass transfer equations are, in general, strongly coupled and highly nonlinear. Apart from numerical methods, a number of semi-analytical techniques have in recent years been proposed to find approximate solutions of nonlinear boundary value problems. Some of these techniques, such as the Adomian decomposition method, the variational iteration method, the homotopy analysis method, the homotopy perturbation method, the differential transform method, etc, are now well known and their strengths and weaknesses well understood. Some of these weaknesses include, for example, small regions of convergence and the use of artificially inserted parameters (Liao [19], Geng [20]). The recent spectral homotopy analysis method (see Motsa et el. [21, 22]) sought to improve the accuracy and efficiency of the homotopy analysis method while retaining its essence. The use of spectral methods also provided greater flexibility in the choice of basis functions. Nonetheless, the challenge to find more accurate, robust and computationally efficient solution techniques for nonlinear problems in engineering and science still remains.
A recent method that remains to be generalized and whose robustness remains to be tested in the case of highly nonlinear equations with a strong coupling is the successive linearization method (Makukula et al. [23, 24]). This method has been used in a limited number of studies by, for example, Awad et al. [25]) and Motsa et al. [26] to solve fluid flow problems. In this study the coupled set of differential equations that describe convective heat and mass transfer flow along a vertical flat plate in the presence of diffusion-thermo, thermal diffusion effects and an external magnetic field are solved using the successive linearization method. A non-similarity technique is used to validate the linearization method.
2 Problem formulation
where u and v are the velocity components along the x- and y- axes, respectively T and C are the fluid temperature and solute concentration across the boundary layer, ν is the kinematic viscosity, ρ is the fluid density, σ is the electrical conductivity, B_{0} is the uniform magnetic field, β_{ T }and β_{ C }are the coefficients of thermal and solutal expansions, D_{ m }is the thermal diffusivity, k_{ T }is the thermal diffusion ratio, c_{ s }is the concentration susceptibility, c_{ p }is the fluid specific heat capacity, T_{ m }is the mean fluid temperature, U_{∞} is the free stream velocity and g is the gravitational acceleration.
where Re_{ x }= U_{∞}x/ν.
3 Method of solution
The coupled system of equations (18)-(20), (22)-(24), and (26)-(28) together with the associated boundary conditions (21), (25), and (29) may be solved independently pairwise one after another. These equations may now be solved using the successive linearization method in the manner described in Makukula et al. [23, 24]). We begin by solving equations (18)-(20) with boundary conditions (21).
where F_{ i }, Θ_{ i }, and Φ_{ i }(i ≥ 1) are unknown functions and F_{ m }, Θ_{ m }, and Φ_{ m }are successive approximations which are obtained by recursively solving the linear part of the system that is obtained from substituting equations (30) in (18)-(20). In choosing the form of the expansions (30), prior knowledge of the general nature of the solutions, as is often the case with perturbation methods, is not necessary.
Equation (53) gives a solution of (18)-(20) for f_{0}, θ_{0} and ϕ_{0}. The procedure is repeated to obtain the O(ξ^{1}) and O(ξ^{2}) solutions using equations (22)-(24) and (26)-(28), respectively.
4 Results and discussion
In generating the results in this article, we determined through numerical experimentation that L = 15, N = 40, and M_{2} = 5 gave sufficient accuracy for the linearization method. The value of the Prandtl number used is Pr = 0.71 which physically corresponds to air. The Schmidt number used Sc = 0.22 is for hydrogen at approximately 25° and one atmospheric pressure (see Afify [27]). In concert with previous related studies, the Dufour and Soret numbers are chosen in such a way that their product is constant, provided the mean temperature T_{ m }is also kept constant.
To determine the accuracy and validate the linearization method, equations (8)-(10) were further solved using a local non-similarity method (LNSM) developed by Sparrow and Yu [28] and Sparrow et al. [29]. Previous studies have consistently used the Matlab bvp4c solver to evaluate the accuracy of the successive linearization method. However, as with other BVP solvers, the accuracy and convergence of the bvp4c algorithm depends on a good initial guess and works better for systems involving few equations (Shampine et al. [30]).
The effect of the magnetic field parameter Ha_{ x }on f"(0), θ' (0) and ϕ'(0) when Gr_{ x }= 0.5, Gc_{ x }= 2, D_{ f }= 0.2, and Sr = 0.3
ξ= 0.005 | ξ= 0.01 | ||||||
---|---|---|---|---|---|---|---|
SLM results | LNS Method | SLM results | LNS Method | ||||
Profile | Ha _{ x } | M _{ 2 } = 2 | M _{ 2 } = 3 | M _{ 2 } = 2 | M _{ 2 } = 3 | ||
0.0 | 3.045324 | 3.045324 | 3.045324 | 3.045324 | 3.045324 | 3.045324 | |
0.5 | 2.299370 | 2.299370 | 2.299370 | 2.299370 | 2.299370 | 2.299370 | |
f"(0) | 1.0 | 1.940291 | 1.940291 | 1.940291 | 1.940291 | 1.940291 | 1.940291 |
2.0 | 1.532041 | 1.532041 | 1.532041 | 1.532041 | 1.532041 | 1.532041 | |
2.5 | 1.404033 | 1.404033 | 1.404033 | 1.404033 | 1.404033 | 1.404033 | |
0.0 | 0.513493 | 0.513493 | 0.513493 | 0.513493 | 0.513493 | 0.513493 | |
0.5 | 0.435640 | 0.435640 | 0.435640 | 0.435640 | 0.435640 | 0.435640 | |
-θ'(0) | 1.0 | 0.392869 | 0.392869 | 0.392869 | 0.392869 | 0.392869 | 0.392869 |
2.0 | 0.335307 | 0.335307 | 0.335307 | 0.335307 | 0.335307 | 0.335307 | |
2.5 | 0.314405 | 0.314405 | 0.314405 | 0.314405 | 0.314405 | 0.314405 | |
0.0 | 0.289548 | 0.289548 | 0.289548 | 0.289548 | 0.289548 | 0.289548 | |
0.5 | 0.231673 | 0.231673 | 0.231673 | 0.231673 | 0.231673 | 0.231673 | |
-ϕ'(0) | 1.0 | 0.202778 | 0.202778 | 0.202778 | 0.202778 | 0.202778 | 0.202778 |
2.0 | 0.169217 | 0.169217 | 0.169217 | 0.169217 | 0.169217 | 0.169217 | |
2.5 | 0.157771 | 0.157771 | 0.157771 | 0.157771 | 0.157771 | 0.157771 |
The effect of the Soret parameter Sr on f"(0), θ'(0) and ϕ'(0) when Gr_{ x }= 0.5, Gc_{ x }= 2, and Ha_{ x }= 1
ξ= 0.005 | ξ= 0.01 | ||||||
---|---|---|---|---|---|---|---|
SLM results | LNS Method | SLM results | LNS Method | ||||
Profile | Sr | M _{ 2 } = 2 | M _{ 2 } = 3 | M _{ 2 } = 2 | M _{ 2 } = 3 | ||
0.1 | 1.933303 | 1.933303 | 1.933302 | 1.933303 | 1.933303 | 1.933302 | |
0.4 | 1.946426 | 1.946426 | 1.946426 | 1.946426 | 1.946426 | 1.946426 | |
f"(0) | 0.6 | 1.959632 | 1.959632 | 1.959632 | 1.959632 | 1.959632 | 1.959632 |
1.5 | 2.023004 | 2.023004 | 2.023004 | 2.023004 | 2.023004 | 2.023004 | |
2.0 | 2.059313 | 2.059313 | 2.059313 | 2.059313 | 2.059313 | 2.059313 | |
0.1 | 0.368159 | 0.368159 | 0.368159 | 0.368159 | 0.368159 | 0.368159 | |
0.4 | 0.396905 | 0.396905 | 0.396905 | 0.396905 | 0.396905 | 0.396905 | |
-θ'(0) | 0.6 | 0.402187 | 0.402187 | 0.402187 | 0.402187 | 0.402187 | 0.402187 |
1.5 | 0.416430 | 0.416430 | 0.416430 | 0.416430 | 0.416430 | 0.416430 | |
2.0 | 0.422788 | 0.422788 | 0.422788 | 0.422788 | 0.422788 | 0.422788 | |
0.1 | 0.213565 | 0.213565 | 0.213565 | 0.213565 | 0.213565 | 0.213565 | |
0.4 | 0.197708 | 0.197708 | 0.197708 | 0.197708 | 0.197708 | 0.197708 | |
-ϕ'(0) | 0.6 | 0.187601 | 0.187601 | 0.187601 | 0.187601 | 0.187601 | 0.187601 |
1.5 | 0.141063 | 0.141063 | 0.141063 | 0.141063 | 0.141063 | 0.141063 | |
2.0 | 0.114262 | 0.114262 | 0.114262 | 0.114262 | 0.114262 | 0.114262 |
Soret and Dufour effects of the skin friction coefficient C_{ f }, Nusselt number Nu and Sherwood number Sh when Gr_{ x }= 0.5, Gc_{ x }= 2, Sc = 0.22, and Ha_{ x }= 0.5
S _{ r } | D _{ f } | C _{ f } | Nu | Sh |
---|---|---|---|---|
0.1 | 0.60 | 1.933303 | 0.368159 | 0.213565 |
0.2 | 0.30 | 1.935047 | 0.386060 | 0.207941 |
0.4 | 0.15 | 1.946426 | 0.396905 | 0.197708 |
0.6 | 0.10 | 1.959632 | 0.402187 | 0.187601 |
1.5 | 0.04 | 2.023004 | 0.416430 | 0.141063 |
2.0 | 0.03 | 2.059313 | 0.422788 | 0.114262 |
Table 1 shows the effect of increasing the magmatic field parameter Ha_{ x }on the local skin friction, heat and mass transfer coefficients. We observe that increasing the magnetic field parameter reduces the local skin friction as well as the heat and mass transfer coefficients. In Table 2, we present the effect of the Soret parameter on f"(0), -θ'(0), and -ϕ'(0) which are, respectively, proportional to the local skin friction coefficient, the local Nusselt number and Sherwood number. We observe that f"(0) and -θ'(0) increase with increases in Sr, while -ϕ'(0) decreases as Sr increases. These results are confirmed in Table 3. Here the Nusselt number increases as the Soret number increases, while the opposite trend occurs as the Dufour number increases. The recent study by El-Kabeir [31] shows that these results may be modified by injection, suction or the presence of a chemical reaction.
Figure 5 shows the effect of increasing the Soret parameter (reducing the Dufour parameter) on the fluid velocity f'(η). The fluid velocity is found to increase with the Soret parameter.
The effect of Soret parameter on the temperature within the thermal boundary layer and the solute concentration is shown in Figures 6 and 7, respectively. An increase in the Soret effect reduces the temperature within the thermal boundary layer leading to an increase in the temperature gradient at the wall and an increase in heat transfer rate at the wall. On the other hand, increasing the Soret effect increases the concentration distribution which reduces the concentration gradient at the wall. These results are similar to the earlier findings by El-Kaberir [31] and Alam and Rahman [32], although the latter studies were subject to injection/suction.
5 Conclusions
In this article, we have investigated MHD and cross-diffusion effects on double-diffusive convection from a vertical flat plate in a viscous incompressible fluid. Numerical approximations for the governing equations were found using a combination of a regular perturbation expansion and the successive linearization method. The solutions were validated by using a local similarity, non-similarity method. We determined the effects of various parameters on the fluid properties as well as on the skin-friction coefficient, the heat and the mass transfer rates. We have shown that the magnetic field parameter enhances the temperature and concentration distributions within the boundary layer. The effect of thermo-diffusion is to reduce the temperature and enhance the velocity and the concentration profiles. The diffusion-thermo effect enhances the velocity and temperature profiles while reducing the concentration distribution. The skin-friction, heat and mass transfer coefficients decrease with an increase in the magnetic field strength. The skin-friction and heat transfer coefficients increase whereas the mass transfer coefficient decreases with increasing Soret numbers.
Declarations
Acknowledgements
The authors would like to thank the referees for helpful comments and suggestions. This study was supported by the National Research Foundation (NRF). AAK was supported by the government of Sudan.
Authors’ Affiliations
References
- Stern ME: The 'salt fountain' and thermohaline convection. Tellus 1960, 12: 172-175. 10.1111/j.2153-3490.1960.tb01295.xView ArticleGoogle Scholar
- Nield DA, Bejan A: Convection in Porous Media. Springer Verlag, New York; 1999.View ArticleGoogle Scholar
- Nield DA: Onset of thermohaline convection in a porous medium. Water Resour Res 1968, 4: 553-560. 10.1029/WR004i003p00553View ArticleGoogle Scholar
- Baines PG, Gill AE: On thermohaline convection with linear gradients. J Fluid Mech 1969, 37: 289-306. 10.1017/S0022112069000553View ArticleGoogle Scholar
- Rudraiah N, Srimani PK, Friedrich R: Finite amplitude convection in a two component fluid saturated porous layer. Int J Heat Mass Transfer 1982, 25: 715-722. 10.1016/0017-9310(82)90177-6MATHView ArticleGoogle Scholar
- Poulikakos D: Double diffusive convection in a horizontally sparsely packed porous layer. Int Commun Heat Mass Transfer 1986, 13: 587-598. 10.1016/0735-1933(86)90035-7View ArticleGoogle Scholar
- Bejan A, Khair KR: Heat and mass transfer by natural convection in porous medium. Int J Heat Mass Transfer 1985, 28: 909-918. 10.1016/0017-9310(85)90272-8MATHView ArticleGoogle Scholar
- Lai FC: Coupled heat and mass transfer by natural convection from a horizontal line source in saturated porous medium. Int Commun Heat Mass Transfer 1990, 17: 489-499. 10.1016/0735-1933(90)90067-TView ArticleGoogle Scholar
- Afify AA: MHD free convective flow and mass transfer over a stretching sheet with chemical reaction. Heat Mass Transfer 2004, 40: 495-500.View ArticleGoogle Scholar
- Makinde OD, Sibanda P: MHD mixed-convective flow and heat and mass transfer past a vertical plate in a porous medium with constant wall suction. J Heat Transfer 2008, 130: 1-8.View ArticleGoogle Scholar
- Eckert ERG, Drake RM: Analysis of heat and mass transfer. McGraw-Hill, New York; 1972.Google Scholar
- Mortimer RG, Eyring H: Elementary transition state theory of the Soret and Dufour effects. Proc Natl Acad Sci 1980, 77: 1728-1731. 10.1073/pnas.77.4.1728View ArticleGoogle Scholar
- Anjalidevi SP, Devi RU: Soret and Dufour effects on MHD slip flow with thermal radiation over a porous rotating infnite disk. Commun Nonlinear Sci 2011, 16: 1917-930. 10.1016/j.cnsns.2010.08.020View ArticleGoogle Scholar
- Kafoussias NG, Williams EW: Thermal-diffusion and diffusion thermo effects on mixed free-forced convective and mass transfer boundary layer flow with temperature dependent viscosity. Int J Eng Sci 1995, 33: 1369-1384. 10.1016/0020-7225(94)00132-4MATHView ArticleGoogle Scholar
- Postelnicu A: Influence of a magnetic field on heat and mass transfer by natural convection from vertical surfaces in porous media considering Soret and Dufour effects. Int J Heat Mass Transfer 2004, 47: 1467-1472. 10.1016/j.ijheatmasstransfer.2003.09.017MATHView ArticleGoogle Scholar
- Mansour MA, El-Anssary NF, Aly AM: Effects of chemical reaction and thermal stratification on MHD free convective heat and mass transfer over a vertical stretching surface embedded in a porous media considering Soret and Dufour numbers. Chem Eng J 2008, 145: 340-345. 10.1016/j.cej.2008.08.016View ArticleGoogle Scholar
- Lakshmi-Narayana PA, Sibanda P: Soret and Dufour effects on free convection along a vertical wavy surface in a fluid saturated Darcy porous medium. Int J Heat Mass Transfer 2010, 53: 3030-3034. 10.1016/j.ijheatmasstransfer.2010.03.025View ArticleGoogle Scholar
- Awad FG, Sibanda P, Motsa SS: On the linear stability analysis of a Maxwell fluid with double-diffusive convection. Appl Math Model 2010, 34: 3509-3517. 10.1016/j.apm.2010.02.038MATHMathSciNetView ArticleGoogle Scholar
- Liao SJ: Beyond Perturbation: Introduction to the Homotopy Analysis Method. CRC Press, Boca Raton; 2003.View ArticleGoogle Scholar
- Geng F: Modified variational iteration method for second order initial value problems. Appl Appl Math 2010, 1: 73-81.Google Scholar
- Motsa SS, Sibanda P, Shateyi S: A new spectral-homotopy analysis method for solving a nonlinear second order BVP. Commun Nonlinear Sci Numer Simulat 2010, 15: 2293-2302. 10.1016/j.cnsns.2009.09.019MATHMathSciNetView ArticleGoogle Scholar
- Motsa SS, Sibanda P, Awad FG, Shateyi S: A new spectral-homotopy analysis method for the MJD Jeffery-Hamel problem. Comput. Fluids 2010, 39: 1219-1225. 10.1016/j.compfluid.2010.03.004MATHMathSciNetGoogle Scholar
- Makukula ZG, Sibanda P, Motsa SS: A note on the solution of the Von Karman equations using series and chebyshev spectral methods. Volume 2010. Boundary Value Problems; 2010:17.Google Scholar
- Makukula ZG, Sibanda P, Motsa SS: A novel numerical technique for two-dimensional laminar flow between two moving porous walls. Math Problems Eng 2010, 2010: 15. Article ID 528956MathSciNetView ArticleGoogle Scholar
- Awad FG, Sibanda P, Motsa SS, Makinde OD: Convection from an inverted cone in a porous medium with cross-diffusion effects. Comput Math Appl 2011, 61: 1431-1441. 10.1016/j.camwa.2011.01.015MATHMathSciNetView ArticleGoogle Scholar
- Motsa SS, Sibanda P, Shateyi S: On a new quasi-linearization method for systems of nonlinear boundary value problems. Math Methods Appl Sci 2011, 34: 1406-1413. 10.1002/mma.1449MATHMathSciNetView ArticleGoogle Scholar
- Afify AA: Similarity solution MHD: Effects of thermal and diffusion thermo on free convective heat and mass transfer over a stretching surface considering suction or injection. Commun. Nonlinear Sci Numer Simulat 2009, 14: 2202-2214. 10.1016/j.cnsns.2008.07.001MATHView ArticleGoogle Scholar
- Sparrow EM, Yu HS: Local non-similarity thermal boundary layer solutions. J Heat Transfer 1971, 93: 328-334. 10.1115/1.3449827View ArticleGoogle Scholar
- Sparrow EM, Quack H, Boerner CJ: Local non-similarity boundary layer solutions. AIAA 1970, 8: 1936-1942. 10.2514/3.6029MATHView ArticleGoogle Scholar
- Shampine LF, Ketzscher R, Forth SA: Using AD to solve BVPs in Matlab. ACM Trans Math Softw 2005, 31(1):79-94. 10.1145/1055531.1055535MATHMathSciNetView ArticleGoogle Scholar
- El-Kabeir SMM: Soret and Dufour effects on heat and mass transfer by mixed convection over a vertical surface saturated porous medium with temperature dependent viscosity. Int J Numer Meth Fluids 2011.Google Scholar
- Alam MS, Rahman MM: Dufour and Soret effects on mixed convection flow past a vertical porous flat plate with variable suction. Nonlinear Anal: Model Control 2006, 11: 3-12.MATHMathSciNetGoogle Scholar
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