Abstract
The effects of a homogeneousheterogeneous reaction on steady micropolar fluid flow
from a permeable stretching or shrinking sheet in a porous medium are numerically
investigated in this paper. The model developed by Chaudhary and Merkin (Fluid Dyn.
Res. 16:311333, 1995) for a homogeneousheterogeneous reaction in boundary layer
flow with equal diffusivities for reactant and autocatalysis is used and extended
in this study. The uniqueness of this problem lies in the fact that the solutions
are possible for all values of the stretching parameter
Keywords:
micropolar fluid; permeable stretching; shrinking sheet; homogeneousheterogeneous reactions; porous medium1 Introduction
Micropolar fluids are fluids with internal structures in which coupling between the spin of each particle and the microscope velocity field is taken into account. They represent fluids consisting of rigid, randomly oriented or spherical particles suspended in a viscous medium, where the deformation of fluid particles is ignored. Micropolar fluid theory was introduced by Eringen [1] in order to describe physical systems, which do not satisfy the NavierStokes equations. The equations governing the micropolar fluid involve a spin vector and a microinertia tensor in addition to the velocity vector. The potential importance of micropolar fluids in industrial applications has motivated many researchers to extend the study in numerous ways to include various physical effects. The essence of the theory of micropolar fluid lies in particle suspension (Hudimoto and Tokuoka [2]), liquid crystals (Lockwood et al.[3]); animal blood (Ariman et al.[4]), exotic lubricants (Erigen [5]), etc. An excellent review of the various applications of micropolar fluid mechanics was presented by Ariman et al.[6].
Boundary layer flow over a stretching surface is important as it occurs in several engineering processes, for example, materials manufactured by extrusion. During the manufacturing process, a stretching sheet interacts with the ambient fluid both thermally and mechanically. The study of boundary layer flow caused by a stretching surface was initiated by Crane [7]. Recently, several works on the dynamic of the boundary layer flow over a stretching surface have appeared in literature (Dutta et al.[8], Hayat et al.[9], Ishak [10]). The effect of surface conditions on the micropolar flow driven by a porous stretching sheet was studied by Kelson and Desseaux [11]. Mohammadein and Gorla [12] examined the flow of micropolar fluids bounded by a stretching sheet with prescribed wall heat flux, viscous dissipation and internal heat generation. The effect of suction or injection at a stretching surface was studied by Erickson et al.[13] and Fox et al.[14]. The process of suction is used in many engineering activities such as thermal oil recovery, removal of reactants etc. Elbashbeshy and Bazid [15] studied the flow and heat transfer in a porous medium over a stretching surface. Bhargava et al.[16] investigated the flow of a mixed convection micropolar fluid driven by a porous stretching sheet with uniform section. Later, Bhargava et al.[17] studied the same flow of a micropolar flow over a nonlinear stretching sheet. Abel et al.[18] carried out a numerical study of hydromagnetic micropolar fluid flow due to horizontal/vertical stretching sheet using a shooting method. They highlighted a scientific approach for the choice of the missing initial values on which the convergence of the shooting method highly depends. Recently, Narayana and Sibanda [19] studied the effects of laminar flow of a nanoliquid film over an unsteady stretching sheet. Kameswaran et al.[20] studied hydromagnetic nanofluid flow due to a stretching or shrinking sheet with viscous dissipation and chemical reaction effects. Recently, Kameswaran et al.[21] studied homogeneousheterogeneous reactions in a nanofluid flow over a permeable stretching sheet.
Many chemically reacting systems involve both homogeneous and heterogeneous reactions, with examples occurring in combustion, catalysis and biochemical systems. The interaction between the homogeneous reactions in the bulk of fluid and heterogeneous reactions occurring on some catalytic surfaces is generally very complex, involving the production and consumption of reactant species at different rates both within the fluid and on the catalytic surfaces. A simple mathematical model for homogeneousheterogeneous reactions in stagnationpoint boundarylayer flow was initiated by Chaudhary and Merkin [22]. They modeled the homogeneous (bulk) reaction by isothermal cubic kinetics and the heterogeneous (surface) reaction was assumed to have firstorder kinetics. Later Chaudhary and Merkin [23] extended their previous work to include the effect of loss of the autocatalyst. They studied the numerical solution near the leading edge of a flat plate. A model for isothermal homogeneousheterogeneous reactions in boundary layer flow of a viscous fluid flow past a flat plate was studied by Merkin [24]. Ziabakhsh et al.[25] studied the problem of flow and diffusion of chemically reactive species over a nonlinearly stretching sheet immersed in a porous medium. Chambre and Acrivos [26] studied an isothermal chemical reaction on a catalytic in a laminar boundary layer flow. They found the actual surface concentration without introducing unnecessary assumptions related to the reaction mechanism. The effects of flow near the twodimensional stagnation point flow on an infinite permeable wall with a homogeneousheterogeneous reaction was studied by Khan and Pop [27]. They solved the governing nonlinear equations using the implicit finite difference method. It was observed that the mass transfer parameter considerably affects the flow characteristics. Khan and Pop [28] studied the effects of homogeneousheterogeneous reactions on the viscoelastic fluid toward a stretching sheet. They observed that the concentration at the surface decreased with an increase in the viscoelastic parameter.
The purpose of the present study is to analyze the influence of the permeability, the homogeneous and heterogeneous reaction on the micropolar fluid towards a stretching/shrinking sheet. We transformed the governing momentum and concentration equations into a system of ordinary differential equations using a similarity variable and then numerically solved the equations for some values of the governing parameters. To the best of authors knowledge, such study has not been reported earlier in the literature.
2 Mathematical formulation
Consider steady, incompressible twodimensional boundary layer flow of a micropolar
fluid through a porous medium. The Cartesian coordinates x and y are taken along the surface and are normal to it, respectively, and u and v are the respective velocity components. The flow is generated due to stretching or
shrinking of the sheet caused by the simultaneous application of two equal forces
along the xaxis. Keeping the origin fixed, it is assumed that the surface is stretched/shrunk
with a linear velocity
Here a and b are concentrations of chemical species A and B, and
Under these assumptions, the governing equations lead to
where
where
where
the governing equations are written as the following system of ordinary differential equations:
where
where
It is expected that the diffusion coefficients of chemical species A and B are of comparable size, which leads us to further assumption that the diffusion coefficients
Thus, equations (2.13) and (2.14) reduce to
and the boundary conditions equation (2.16) take the form
The physical quantity of interest is the skin friction coefficient
The skin friction coefficient is defined as
Using the similarity variables in the above equation, we obtain
where
3 Results and discussion
The system of ordinary differential equations (2.10), (2.11) and (2.17) along with
the boundary conditions (2.15) and (2.19) are solved numerically for some values of
λ,
Table 1
. Comparison of
Table 2
. Comparison of
From Table 1, it is clear that the skin friction is a decreasing function of λ. All values of the skin friction coefficient are positive for
The effect of the stretching/shrinking parameter λ for
The variation of the velocity and concentration profiles is plotted as a function
of η for some values of λ in Figure 1. (i) For
Figure 1
. Effect ofλon (a) and (b) for
The effect of the micropolar parameter and suction parameter on the velocity and concentration
profile is shown in Figure 2. A comparison is made for the Newtonian fluid (
Figure 2
. Variation of
The effect of the homogeneous and heterogeneous reaction on the concentration profile
is shown in Figure 3. The effect of heterogeneous and homogeneous reactions separately is shown through
Figures 3(a) and 3(b), respectively. We considered
Figure 3
. Effect ofλon (a) and (b) for
The concentration of the reactants depends on the Schmidt number (Sc) and heterogeneous reaction parameter. The variation of the concentration with K for different values of the Schmidt number and
Figure 4
. Concentration at the surface effect on (a) and (b) for
4 Conclusions
The present analysis investigates the effect of the homogeneous and heterogeneous reaction on the micropolar fluid flow through a porous medium past a porous stretching/shrinking sheet with suction. The momentum and concentration equations were transformed into a set of coupled nonlinear differential equations using similarity transformations and solved numerically by Matlab bvp4c package. We compared our results with those in the literature for some limiting case. A dual solution appeared for the shrinking sheet case. The effect of the dual solution is shown by tables and graphs. The momentum boundary layer thickness increased for the case of the first solution, while an opposite phenomenon appeared for the second solution and a similar phenomenon was observed for concentration profile. It was observed that the concentration at the surface decreased with the strengths of the homogeneous and heterogeneous reactions. The solute concentration, however, increased with the permeability and stretching/shrinking parameters. The velocity of the fluid and the concentration of the reactants at the surface increase with the stretching/shrinking parameter. Also, velocity increases due to the increase in micropolar parameter. The concentration of the reactants decreases with the strength of the homogeneous and heterogeneous reaction.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All the authors participated in the design of the study and helped to draft and proofread the manuscript. SS and PKK carried out the numerical computations.
Acknowledgements
The authors wish to thank the University of KwaZuluNatal for financial support.
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