Double stratification effects on heat and mass transfer in unsteady MHD nanofluid flow over a flat surface
 Winifred Nduku Mutuku^{1}Email author and
 Oluwole Daniel Makinde^{2}
DOI: 10.1186/s4054001700212
© The Author(s) 2017
Received: 27 March 2017
Accepted: 3 May 2017
Published: 12 May 2017
Abstract
The focus of this work is to theoretically investigate the problem of double stratification on heat and mass transfer in an unsteady hydromagnetic boundary layer flow of a nanofluid over a flat surface. The model employed for the nanofluid transport equations incorporate the effects of Brownian motion and thermophoresis in the presence of thermal and solutal stratification. The governing nonlinear partial differential equations and their associated boundary conditions are initially transformed into dimensionless form by using similarity variables, before being solved numerically by employing the Runge–Kutta–Fehlberg fourthorder method with shooting technique. The effects of different controlling parameters, viz. solutal and thermal stratification, Lewis number, thermophoresis, Brownian motion, magnetic field and unsteadiness on the fluid velocity, temperature, skin friction coefficient, the local Nusselt number, and the local Sherwood number are graphically depicted and quantitatively discussed in detail taking into account the practical applications of each profile. It is noted that thermal stratification reduces the fluid temperature, while the solutal stratification reduces the nanoparticle concentration.
Keywords
Heat transfer Mass transfer Unsteady MHD Nanofluid Double stratificationBackground
Magnetohydrodynamics (MHD) boundary layer flow of electrically conducting fluids has diverse industrial and engineering applications in fields such as nuclear reactors, geothermal engineering, liquid metals and plasma flows, petroleum industries, boundary layer control in aerodynamics and crystal growth. Several authors have studied the problem of MHD boundary layer flow, heat and mass transfer about different surface geometries in electrically conducting fluids. Makinde [1] numerically analysed the influence of magnetic field on the steady heat and mass flow of an electrically conducting fluid by mixed convection along a semiinfinite vertical porous plate with constant heat flux taking into account Soret and Dufour effects. The effect of the Hall current on the MHD natural convection flow from a vertical permeable flat plate with a uniform heat flux in the presence of a transverse magnetic field was analysed by Saha et al. [2]. Rout et al. [3] investigated MHD heat and mass transfer of chemically reacting fluid flow over a moving vertical plate in the presence of a heat source with convective boundary condition.
Stratification is the layering of a fluid system due to temperature differences and variations in concentration or the presence of different fluids with varying densities. In practical situations where the heat and mass transfer mechanisms occur simultaneously, it is important to analyse the effect of double stratification (stratification of medium with respect to thermal and concentration fields) on the convective transport in the fluid. The effect of double stratification is important in applications involving fluids’ convective transport where heat and mass transfer run concurrently. It is encountered in several classical problems, for example, reservoir mixing, internal waves, shear flow instability, internal hydraulics, turbulence and jets, plumes and wakes. Stratified fluids are omnipresent in nature and their occurrence is very common in heterogeneous fluid bodies, such as thermal stratification of reservoirs and oceans, salinity stratification in estuaries, rivers, groundwater reservoirs, and oceans, heterogeneous mixtures in industrial food manufacturing processing, density stratification of the atmosphere, and uncountable similar examples [4]. For instance, density variation causes thermal stratification in reservoirs which leads to a reduction in the vertical mixing of oxygen to the point that bottom water becomes anoxic through the action of biological processes. Preventing, predicting, and solving such a reservoir problem, though dependent on other limnological factors, require an understanding of the dynamics of stratified fluids. MHDconvective heat transfer in thermally stratified fluid occurs in many industrial applications and is an important aspect in the study of heat transfer. Higher energy efficiency and increased system performance can be achieved with better thermal stratification. This has led to intensified research in mass and heat transfer in thermally stratified medium [5–7]. However, the aforementioned investigations analysed natural convection flow of convectional heat transfer fluids.
Recently with the advent of nanofluids—a term coined by Choi [8] referring to a liquid containing a suspension of nanometresized solid particle (nanoparticles)—research has diversified to convective heat and mass transfer of nanofluids in doubly stratified medium. Although the effect of double stratification on heat and mass transfer in a fluid is significant, very little work has been reported in the literature [9–11]. On the other hand, studies on unsteady boundary layer flow, mass transfer, and convective heat transfer in a nanofluid are the topics of contemporary research areas in fluid science and engineering, owing to its novelty [12–17]. The mass and heat transfer in a doubly stratified medium in the presence of a magnetic field is not only fundamentally interesting but also finds wide range of natural, industrial, and engineering applications. These applications include heat rejection into the environment such as lakes, rivers, and seas; thermal energy storage systems such as solar ponds; heat transfer from thermal sources such as condensers of power plants.
Despite its numerous applications, it is evident from the literature herein that research has neglected mass and heat transfer in unsteady MHD nanofluid flow taking into account thermophoresis and Brownian motion in a doubly stratified medium. Hence, the motivation of this paper is to bridge this knowledge gap. This study aims to extend the recent work of Olanrewaju and Makinde [18] to incorporate unsteadiness, magnetic field, and the effects of double stratification on boundary layer nanofluid flow. The shooting iteration technique coupled with fourthorder Runge–Kutta integration scheme is employed to numerically solve the model nonlinear system of equations of this particular problem. Numerical results are displayed graphically and discussed for different values of the various dimensionless parameters controlling the flow regime. It is hoped that the results obtained will not only provide useful information for applications but also serve as a basis for studying other analogous systems that arise in engineering and industrial applications.
Model formulation
Numerical procedure
The system of nonlinear differential Eqs. (9)–(11) with the boundary conditions (12) has been solved numerically using shooting technique coupled with the fourthorder Runge–Kutta method and a modified version of the Newton–Raphson algorithm.
By applying the shooting method, the unspecified initial conditions S _{1}, S _{2}, and S _{3} in (19) are assumed and (18) integrated numerically as an initial valued problem to a given terminal point. The accuracy of the assumed missing initial conditions was checked by comparing the calculated value of the dependent variable at the terminal point with its given value there. If differences exist, improved values of the missing initial conditions are obtained and the process repeated. The method is programmed in MAPLE with the step size of ∆η = 0.001 selected to be satisfactory for a convergence criterion of 10^{−7} in nearly all cases. The maximum value of η _{∞} to each group of parameters was determined when the values of the unknown boundary conditions at η = 0 do not change to successful loop with error less than 10^{ −7}.
Results and discussion
Computations showing comparison with Aman et al. [20] for different values of Pr when A = β _{1} = β _{2} = Ha = ϕ = 0
Pr  \(f^{\prime\prime}\left( 0 \right) \,\) Amani et al. [20]  \( \theta^{\prime}(0) \,\) Amani et al. [20]  \(f^{\prime\prime}\left( 0 \right) \,\) present  \( \theta^{\prime}(0) \,\) present 

0.1  2.50997  0.26815  2.50997  0.26815 
0.3  2.50997  0.40338  2.50997  0.40338 
0.7  2.50997  0.52965  2.50997  0.52965 
6.2  2.50997  0.59014  2.50997  0.59014 
Velocity profiles
Figure 2 shows the effect of unsteadiness and magnetic field strength on the nanofluids’ velocity profiles. It can be seen that there is a gradual increase in the fluid velocity, from zero at the plate surface to a maximum value as it approaches the free stream far away from the plate surface thus satisfying the boundary conditions. It is observed that both the fluid velocity and the hydrodynamic boundary layer thickness decrease as the unsteadiness parameter (A) and Hartmann number (Ha) increases. It is well known that application of a transverse magnetic field orthogonal to the flow direction has a tendency to create a drag force known as Lorentz force which tends to resist the fluid flow and thus reducing its velocity.
Temperature profiles
Figures 3, 4, 5, 6 show the effects of various parameters on the temperature profile. Generally, the temperature is the maximum at the plate surface but decreases exponentially to zero far away from the plate surface satisfying the free stream conditions. As shown in Fig. 3, an increase in Brownian parameter N _{b}, thermophoresis parameter N _{t}, and Eckert number 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 to viscous dissipation, friction, and rapid collision of the nanoparticles as a result of Brownian motion and thermophoresis. Figure 4 shows that increasing Hartmann number Ha initially increases the fluid temperature at the plate surface, but as the fluid moves from the plate surface the temperature decreases. The increase in temperature is due to ohmic heating as a result of the magnetic effect. On the other hand, as illustrated in Figures 5 and 6, there is a decrease in both the temperature and the thermal boundary layer thickness with an increase in thermal stratification parameter β _{1} and unsteadiness parameter A. When the thermal stratification effect is taken into consideration, the effective temperature difference between the plate and the ambient fluid decreases; therefore, the temperature is reduced.
Nanoparticle concentration profiles
Figures 7, 8, 9 show the effects of various parameters on the nanoparticle concentration profile. The nanoparticle concentration is the highest at the plate surface but decreases exponentially to zero free stream value satisfying the boundary conditions. Unlike in the case of temperature, a decrease in the nanoparticle concentration is noted with an increase in unsteadiness parameter A, Lewis number Le, Brownian parameter N _{b}, thermophoresis parameter N _{t}, Hartmann number Ha, Eckert number Ec, and solutal stratification parameter β_{2}.
Effects of parameter variations on C _{f} , Nu, Sh
Figures 10, 11, 12, 13, 14 depict the various pertinent parameters at the plate surface for the skin friction coefficient, the local Nusselt number (rate of heat transfer), and the local Sherwood number with variation in the values of thermophysical parameters embedded in the problem. From Fig. 10, it can be seen that increasing both the unsteadiness parameter A and Hartmann number Ha leads to an increase in the skin friction coefficient. This is as expected since both parameters causes an overshoot of the fluid towards the plate surface. From Fig. 11, it is noted increasing the Lewis number Le, Brownian parameter N _{b}, thermophoresis parameter N _{t}, Hartmann number Ha, Eckert number Ec, and thermal stratification parameter β _{1} leads to a decrease in the local Nusselt number which implies that there is reduced heat transfer rate at the plate surface. As seen earlier, the same parameters led to an increase to the fluid temperature; thus, it is expected that with increased temperature within the fluid the rate of heat transfer from the plate to the fluid will be reduced. On the other hand, an increase in the unsteadiness parameter A increases the local Nusselt number, as observed in Fig. 12. It is noted in Figs. 13 and 14 that increasing the Hartmann number Ha, Eckert number Ec, Lewis number Le, thermophoresis parameter N _{t}, and thermal stratification parameter β _{1} increases the Sherwood number, while a decrease in the local Sherwood number is realized by increasing the Brownian parameter N _{b} and solutal stratification parameter β _{2}. It well known that at high Lewis number the nanoparticle concentration is low, since the concentration at the plate surface is higher than in the fluid, mass transfer from the plate to the fluid then occurs.
Conclusions

The fluid overshoots towards the plate surface with increasing A and Ha, thus reducing both the fluid velocity and the hydrodynamic boundary layer thickness.

An increase in Ec, Nt, Nb, and Ha yields an increase in both the thermal boundary layer thicknesses and temperature, whereas the reverse is noted with increases in A and β _{1}.

A decrease in the nanoparticle concentration is noted with increase in A, Le, Nb, Nt, Ha, Ec, and β _{2}.

Increasing A and Ha increase the skin friction.

A decrease in Nusselt number is realized with increased Ec, Ha, Nb, and β _{ 1 }, whereas it increases with increasing values of A.

The local Sherwood number increases with increase in Ha, β _{ 1 }, Le, Nt, and Ec but reduces with increased β _{ 2 } and Nb.
Abbreviations
(u, v): velocity components in x and ydirection; A: unsteadiness parameter (λ/a); B _{0}: magnetic field strength (T); C: nanoparticle volume fraction; C _{f}: skin friction coefficient (τ _{ w } / \(U_{\infty }^{2}\) ρ); C _{0,∞}: constant free stream volume fraction; D _{B}: Brownian diffusion coefficient; D _{T}: thermophoresis diffusion coefficient; Ec: Eckert number (\(U_{\infty }^{2}\) /C _{p}(T _{w} − T _{ 0,∞ })); Ha: magnetic parameter (σ \(B_{0}^{2}\) B _{ 0 } ^{ 2 } /αρ); k: thermal conductivity (W/mK); Le: Lewis number (υ/D _{m}); N _{b}: Brownian motion parameter (τD _{B} (C _{w} − C _{0,∞} )/υ); N _{t}: thermophoresis parameter (τD _{T} (T _{w} − T _{ 0,∞ } )/T _{0,∞} υ); Pr: Prandtl number (υ/α); Nu: Nusselt number (xq _{w} /k(T _{w} − T _{0,∞} )); Sh: Sherwood number (xq _{m} /D _{B} (C _{w} − C _{ 0,∞ } )); T: temperature of fluid (K)
Greek symbols
ψ: stream function; η: similarity variable; β _{1}: thermal stratification parameter; β _{2}: solutal stratification parameter; υ: kinematic viscosity of fluid (kg/m s); ρ: density of fluid (kg/m ^{ 3 }); φ: dimensionless concentration function; α: thermal diffusivity(W/m k); σ: electrical conductivity of fluid (S/m); θ: dimensionless temperature; (c _{ p })_{f}: heat capacity of fluid (J/kg K); (c _{p})_{p}: heat capacity of nanofluid; τ: parameter defined by (ρC) _{p} /(ρC) _{f}; µ: dynamic viscosity of fluid (kg/m s)
Subscripts
∞: condition at the free stream; w: condition at the surface
Declarations
Authors’ contributions
Both authors were actively involved in the formulation of the problem, literature review, and simulations. Both authors read and approved the final manuscript.
Acknowledgements
None.
Competing interests
The authors declare that they have no competing interests.
Funding
There is no funding towards the publication of this manuscript.
Ethics approval and consent to participate
This study does not involve human participants, human data, or human tissue.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Makinde OD (2011) On MHD mixed convection with Soret and Dufour effects past a vertical plate embedded in a porous medium. LAAR 41(1):63–68MathSciNetGoogle Scholar
 Saha LK, Siddiqa S, Hossain MA (2011) Effect of hall current on MHD natural convection flow from vertical permeable flat plate with uniform surface heat flux. Appl Math Mech Engl 3(9):1127–1146MathSciNetView ArticleMATHGoogle Scholar
 Rout BR, Panda SK, Panda S (2013) MHD heat and mass transfer of chemical reaction fluid flow over a moving vertical plate in presence of heat source with convective surface boundary condition. Int J Chem Eng. doi:10.1155/2013/296834 Google Scholar
 Socolofsky AS, Jirka GH (2004) Stratified flow and buoyant mixing. Eng Lect Notes. Texas A&M University, College StationGoogle Scholar
 Singh G, Sharma PR, Chamkha AJ (2010) Effect of thermally stratified ambient fluid on MHD convective flow along a moving nonisothermal vertical plate. Int J Phys Sci 5(3):208–215Google Scholar
 Saha SC, Hossain MA (2004) Natural convection flow with combined buoyancy effects due to thermal and mass diffusions in a thermally stratified media. NonLinear Anal 9(1):89–102MATHGoogle Scholar
 Mukhopadhyay S, Mondal C, Gorla SR (2012) Effects of thermal stratification on flow and heat transfer past a porous vertical stretching surface. Heat Mass Transf 48:915–921View ArticleGoogle Scholar
 Choi SUS (1995) Enhancing thermal conductivity of fluids with nanoparticle. In: Siginer DA, Wang HP (eds) Development and applications of nonNewtonian flow. The American Society of Mechanical Engineers, New York, pp 227–233Google Scholar
 Srinivasacharya D, Upendar M (2013) Effect of double stratification on MHD free convection in a micropolar fluid. J Egypt Math Soc. doi:10.1016/j.joems.2013.02.006 MathSciNetMATHGoogle Scholar
 Narayana L, Murthy PA (2006) Free convective heat and mass transfer in a doubly stratified nonDarcy porous medium. J Heat Transf 128:1204–1212View ArticleGoogle Scholar
 Ibrahim W, Shankar B (2013) MHD boundary layer flow and heat transfer of a nanofluid past a permeable stretching sheet with velocity, thermal and solutal slip boundary conditions. J Comput Fluids 75:1–10MathSciNetView ArticleMATHGoogle Scholar
 Makinde OD, Olanrewaju PO, Charles WM (2011) Unsteady convection with chemical reaction and radiative heat transfer past a flat porous plate moving through a binary mixture. Afr Mat 22(2011):65–78MathSciNetView ArticleMATHGoogle Scholar
 Chamkha AJ, Rashady AM, AlMeshaieiz E (2011) Melting effect on unsteady hydromagnetic flow of a nanofluid past a stretching sheet. Int J Chem Reactor Eng 9(113):1Google Scholar
 Makinde OD (2012) Computational modelling of MHD unsteady flow and heat transfer towards a flat plate with Navier slip and Newtonian heating. Braz J Chem Eng 29(1):159–166View ArticleGoogle Scholar
 Khan MS, Karim I, Ali LE, Islam A (2012) Unsteady MHD free convection boundary layer flow of a nanofluid along a stretching sheet with thermal radiation and viscous dissipation effects. Int Nano Lett 2:24View ArticleGoogle Scholar
 Vajravelu K, Prasad KV, Ng C (2013) Unsteady convective boundary layer flow of a viscous fluid at a vertical surface with variable fluid properties. Nonlinear Anal Real 14:455–464MathSciNetView ArticleGoogle Scholar
 Makinde OD, Chinyoka T (2013) Numerical investigation of buoyancy effects on hydromagnetic unsteady flow through a porous channel with suction/injection. J Mech Sci Tech 27(5):1557–1568View ArticleGoogle Scholar
 Olanrewaju AM, Makinde OD (2013) On boundary layer stagnation point flow of a nanofluid over a permeable flat surface with Newtonian heating. Chem Eng Commun 200:836–852View ArticleGoogle Scholar
 Ibrahim W, Shanker B (2012) Unsteady MHD boundary layer flow and heat transfer due to stretching sheet in the presence of heat source or sink by quasi linearization technique. Int J Appl Math Mech 8(7):18–30Google Scholar
 Aman F, Ishak A, Pop I (2011) Mixed convection boundary layer flow near stagnation point on vertical surface with slip. Appl Math Mech 32(12):1599–1606MathSciNetView ArticleMATHGoogle Scholar