Abstract
The paper discusses the effects of homogeneousheterogeneous reactions on stagnationpoint flow of a nanofluid over a stretching or shrinking sheet. The model presented describes mass transfer in copperwater and silverwater nanofluids. The governing system of equations is solved numerically, and the study shows that dual solutions exist for certain suction/injection, stretching/shrinking and magnetic parameter values. Comparison of the numerical results is made with previously published results for special cases.
Keywords:
homogeneousheterogeneous reactions; nanofluids; volume fraction model; stretching or shrinking sheet; magnetic field strength1 Introduction
Problems involving fluid flow over stretching or shrinking surfaces can be found in many manufacturing processes such as in polymer extrusion, wire and fiber coating, foodstuff processing, etc. Crane [1] was the first to consider the steady twodimensional flow of a Newtonian fluid driven by a stretching elastic flat sheet which moved in its own plane with velocity varying linearly with the distance from a fixed point. This study was subsequently extended by many authors to explore various aspects of heat transfer in a fluid surrounding a stretching sheet (Tsou et al.[2], Erickson et al.[3], Mucoglu and Chen [4], Grubka and Bobba [5], Karwe and Jaluria [6], Chen [7], AboEldahab and ElAziz [8], Salem and ElAziz [9], Ali [10], Ishak et al.[11]).
The magnetohydrodynamic effect has important engineering applications in electrical
motors. Heat transfer over a stretching or shrinking sheet subject to an external
magnetic field, viscous dissipation and joule effects was studied by Jafar et al.[12]. They observed that the flow and heat transfer characteristics for a shrinking sheet
were quite different from those of a stretching sheet. Lok et al.[13] analyzed MHD stagnationpoint flow from a shrinking sheet. They found that dual solutions
existed for small values of the magnetic parameter. The stagnationpoint flow over
a stretching or shrinking sheet in a nanofluid was investigated by Bachok et al.[14]. They showed that adding nanoparticles to a base fluid increased the skin friction
and heat transfer coefficients. Recently, Narayana and Sibanda [15] investigated the laminar flow of a nanoliquid film over an unsteady stretching sheet.
They noticed that the effect of an increase in the nanoparticle volume fraction was
to reduce the axial velocity and free stream velocity in the case of a Cuwater nanoliquid.
However, the opposite appeared to be true in the case of an
Most chemically reacting systems involve both homogeneous and heterogeneous reactions (combustion, catalysis and biochemical systems). The simple combustion model helps us to understand the combustion phenomenon in many complex engineering applications such as in aircraft and rocket engines. A model for isothermal homogeneousheterogeneous reactions in the boundary layer flow of a viscous fluid past a flat plate was presented by Merkin [17]. He modeled the homogeneous reaction by a cubic autocatalysis process and the heterogeneous reaction by a firstorder process. Chaudhary and Merkin [18] analyzed homogeneousheterogeneous reactions in boundary layer flow. They presented a numerical solution of the boundary layer equations near the leading edge of a flat plate. Ziabakhsh et al.[19] studied the diffusion of a chemically reactive species into a nonlinearly stretching sheet immersed in a porous medium. Chambre and Acrivos [20] studied isothermal chemical reactions on laminar boundary layer flow. The twodimensional stagnationpoint flow near an infinite permeable wall with a homogeneousheterogeneous reaction was studied by Khan and Pop [21], while Khan and Pop [22] and Bachok et al.[23] studied the effects of homogeneousheterogeneous reactions on fluid flow due to a stretching sheet. Recently, the effects of homogeneousheterogeneous reactions in nanofluid flow due to a porous stretching sheet were studied by Kameswaran et al.[24]. They found that the concentration at the surface decreased with the strength of the heterogeneous reaction. In the case of a shrinking sheet, they showed that the velocity profiles decreased with increasing nanoparticle volume fraction in the case of a Cuwater nanofluid.
This article presents a study of homogeneousheterogeneous reactions on MHD nanofluid stagnation point flow due to a stretching or shrinking sheet. The transformed nonlinear conservation equations are solved numerically.
2 Mathematical formulation
Consider a twodimensional steady boundary layer flow of an incompressible nanofluid
over a stretching or shrinking sheet. A Cartesian coordinate system is used with
the xaxis along the sheet and the yaxis normal to the sheet. The flow configuration and the coordinate system are shown
in Figure 1. The velocity of the outer flow is of the form
while on the catalyst surface we have the single, isothermal, firstorder reaction
where a and b are the concentrations of the chemical species A and B, and
The boundary conditions for equations (3)(6) are given in the form
where u, v are the velocity components in the x and y directions, respectively,
where ϕ is the solid volume fraction of nanoparticles. The effective density of the nanofluids is given as
Figure 1. Physical model and coordinate system.
Here, the subscripts nf, f and s represent the thermophysical properties of the nanofluid, the base fluid and nanoparticles, respectively.
The continuity equation (3) is satisfied by introducing a stream function ψ such that
where
The velocity components are given by
The concentrations of the chemical species A and B are represented as
where
The nondimensional parameters in equations (13)(18) are the magnetic parameter
M, the Schmidt number Sc, the measure of the strength of the homogeneous reaction K, the ratio of diffusion coefficients δ, the mass transfer parameter
where
In most applications, we expect the diffusion coefficients of chemical species A and B to be of a comparable size. This leads us to making a further assumption that the
diffusion coefficients
Thus equations (14) and (15) reduce to
subject to the boundary conditions
It is quite straightforward to show that the skin friction coefficient
where
3 Results and discussion
The system of ordinary differential equations (13) and (23) with boundary conditions (16) and (24) was solved numerically using Matlab bvp4c routine. We considered Cuwater and Agwater nanofluids. The thermophysical properties of the nanofluids used in this paper are given in Table 1.
In order to determine the accuracy of our numerical results, the present results for
the skinfriction coefficient
Table 2. Comparison of
Tables 2 and 3 give the coefficient
Table 4 gives the values of
The effects of the Schmidt number, stretching, magnetic and chemical reaction parameters are shown in Figures 212.
Figure 2. Effects ofλon velocity, when
Figure 2 shows the effects of both stretching and shrinking on the velocity profiles in the
case of a Cuwater nanofluid. We observe that in both cases, the velocity profiles
increase with the parameter λ. Further, we note that for a shrinking sheet, the velocity in the case of a Cuwater
nanofluid is larger than that of a clear fluid. The opposite is, however, true for
the case of a stretching sheet. The momentum boundary layer thickness decreases as
λ increases and the flow has an inverted boundary layer structure when
Figures 3(a) and 3(b) illustrate the effect of the magnetic parameter, nanoparticle volume fraction
and stretching or shrinking parameters on the velocity profiles. We note that for
both stretching and shrinking sheets, the fluid velocity increases with λ and M. Furthermore, increasing the value of M also causes thinning of the boundary layer. This implies an increase in the velocity
gradient
Figure 3. Effects ofMon velocity when (a)
Figures 4(a) and 4(b) illustrate the effects of the stretching or shrinking parameter and volume fraction
on the solute concentration when
Figure 4. Variation of concentration for different values ofλ, when (a)
The variation of
Figure 5. Effects of
Figure 6. Effects ofKand
Figures 7 and 8 show the effects of K and λ on the concentration when the other parameters are fixed. We note, as expected, that the wall concentration decreases as the strength of the homogeneous reaction increases. The level of decrease is, however, more significant in the case of Agwater than for Cuwater. Figure 8 shows the influence of stretching on the concentration profiles. It can be seen that the concentration decreases when the sheet is shrunk and increases with stretching.
Figure 7. Effects ofKon concentration, when
Figure 8. Effects ofλon concentration, when
The effects of suction/injection parameter
Figure 9. Effects of
Figure 10 shows the effect of the magnetic parameter on
Figure 10. Effects of magnetic parameterMon (a) velocity and (b) concentration profiles, when
The variation of the velocity and concentration profiles with the stretching/shrinking
parameter is shown in Figure 11. The dual velocity
Figure 11. Effects ofλon (a) velocity and (b) concentration profiles, when
The variation of the reduced skin friction coefficient
Figure 12. Effects ofλon (a)
4 Conclusions
The effects of homogeneousheterogeneous reactions in MHD nanofluid flow due to a stretching or shrinking sheet have been studied. The transformed governing nonlinear differential equations have been solved numerically. Dual solutions for the velocity and concentration distributions have been obtained for some values of the stretching/shrinking, suction/injection and magnetic parameters. The effects of physical and fluid parameters on the velocity, concentration and skin friction have been analyzed. It was observed that for both the cases of shrinking and stretching sheets, the fluid velocity increased with the magnetic parameter. The concentration at the surface decreased as the strength of heterogeneous reactions increased for both Cuwater and Agwater nanofluids. The boundary layer thickness for the first solution is always thinner than that for the second solution.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The work including proof reading was done by all the authors.
Acknowledgements
The authors are grateful to the University of KwaZuluNatal for financial support.
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