Homogeneous-heterogeneous reactions in micropolar fluid flow from a permeable stretching or shrinking sheet in a porous medium
© Shaw et al.; licensee Springer. 2013
Received: 18 December 2012
Accepted: 18 March 2013
Published: 8 April 2013
The effects of a homogeneous-heterogeneous 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. 16:311-333, 1995) for a homogeneous-heterogeneous 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 , while for (shrinking surface), solutions are possible only for a limited range of values. The effects of physical and fluid parameters such as the stretching parameter, micropolar parameter, permeability parameter, Schmidt number, strength of homogeneous and heterogeneous reaction parameter on the skin friction, velocity and concentration are analyzed, and these results are presented through graphs. The solute concentration at the surface is found to decrease with the strength of the homogeneous reaction, and to increase with heterogeneous reactions, the permeability parameter and stretching or shrinking parameters. The velocity at the surface was found to increase with the micropolar parameter.
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  in order to describe physical systems, which do not satisfy the Navier-Stokes 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 ), liquid crystals (Lockwood et al. ); animal blood (Ariman et al. ), exotic lubricants (Erigen ), etc. An excellent review of the various applications of micropolar fluid mechanics was presented by Ariman et al. .
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 . Recently, several works on the dynamic of the boundary layer flow over a stretching surface have appeared in literature (Dutta et al. , Hayat et al. , Ishak ). The effect of surface conditions on the micropolar flow driven by a porous stretching sheet was studied by Kelson and Desseaux . Mohammadein and Gorla  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.  and Fox et al. . The process of suction is used in many engineering activities such as thermal oil recovery, removal of reactants etc. Elbashbeshy and Bazid  studied the flow and heat transfer in a porous medium over a stretching surface. Bhargava et al.  investigated the flow of a mixed convection micropolar fluid driven by a porous stretching sheet with uniform section. Later, Bhargava et al.  studied the same flow of a micropolar flow over a nonlinear stretching sheet. Abel et al.  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  studied the effects of laminar flow of a nanoliquid film over an unsteady stretching sheet. Kameswaran et al.  studied hydromagnetic nanofluid flow due to a stretching or shrinking sheet with viscous dissipation and chemical reaction effects. Recently, Kameswaran et al.  studied homogeneous-heterogeneous 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 homogeneous-heterogeneous reactions in stagnation-point boundary-layer flow was initiated by Chaudhary and Merkin . They modeled the homogeneous (bulk) reaction by isothermal cubic kinetics and the heterogeneous (surface) reaction was assumed to have first-order kinetics. Later Chaudhary and Merkin  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 homogeneous-heterogeneous reactions in boundary layer flow of a viscous fluid flow past a flat plate was studied by Merkin . Ziabakhsh et al.  studied the problem of flow and diffusion of chemically reactive species over a nonlinearly stretching sheet immersed in a porous medium. Chambre and Acrivos  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 two-dimensional stagnation point flow on an infinite permeable wall with a homogeneous-heterogeneous reaction was studied by Khan and Pop . 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  studied the effects of homogeneous-heterogeneous 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
Here a and b are concentrations of chemical species A and B, and () are the rate constants. We also assume that both reaction processes are isothermal. It is also assumed that the ambient fluid moves with a velocity , where is a constant, in which there is a uniform concentration of reactant A and in which there is no autocatalyst B over a flat surface.
where is the mass transfer parameter with for suction, for injection and for an impermeable surface. is the stretching parameter where corresponds to a stretching surface and corresponds to a shrinking surface. measures the strength of the heterogeneous (surface) reaction and is the Reynolds number.
where represents the local Reynolds number defined by . In the present paper, we consider only the case (suction) and (weak concentration).
3 Results and discussion
Comparison of for a stretching sheet obtained for different values of λ , for fixed values of , and
Ishak et al. 
Rosali et al. 
Comparison of for a shrinking sheet obtained for different values of λ , for fixed values of , and
Rosali et al. 
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 , while they are negative when . Physically, the negative values of the skin friction coefficient correspond to the surface exerting a drag force on the fluid and the opposite sign implies the inverse phenomenon. The skin friction coefficient is zero when regardless of the values of other parameters. This is because for , there is no shear stress at the surface as the surface and fluid move with the same velocity.
The effect of the stretching/shrinking parameter λ for is shown in Table 2. It is evident that initially the skin friction is an increasing function of λ, but it decreases after a certain value of λ. As mentioned earlier, the solution of the equations is possible only in the range of . It is evident that the value of depends on other parameters , s and n. The value of is approximate to −1.2465 when and , while introducing and , we get . It is also observed that the first solution is a decreasing function of λ, whereas the second solution is an increasing function of λ. It is interested to note that these both solutions coincide at .
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.
The authors wish to thank the University of KwaZulu-Natal for financial support.
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