Abstract

PROCESS SYSTEMS ENGINEERING

Effects of mass transfer on MHD flow of casson fluid with chemical reaction and suction

S. A. ShehzadI, ^* * E-mail: ali_qau70@yahoo.com ; T. Hayat^I; M. Qasim^II; S. Asghar^II

^IDepartment of Mathematics, Quaid-i-Azam University 45320, Islamabad 44000, Pakistan

^IIDepartment of Mathematics, COMSATS Institute of Information Technology, Park Road, Chak Shahzad, Islamabad 44000, Pakistan

ABSTRACT

Effect of mass transfer in the magnetohydrodynamic flow of a Casson fluid over a porous stretching sheet is addressed in the presence of a chemical reaction. A series solution for the resulting nonlinear flow is computed. The skin friction coefficient and local Sherwood number are analyzed through numerical values for various parameters of interest. The velocity and concentration fields are illustrated for several pertinent flow parameters. We observed that the Casson parameter and Hartman number have similar effects on the velocity in a qualitative sense. We further analyzed that the concentration profile decreases rapidly in comparison to the fluid velocity when we increased the values of the suction parameter.

Keywords: Casson fluid; Mass transfer; Chemical reaction.

INTRODUCTION

The analysis of boundary layer flow of viscous and non-Newtonian fluids has been the focus of extensive research by various scientists due to its importance in continuous casting, glass blowing, paper production, polymer extrusion, aerodynamic extrusion of plastic sheet and several others. Numerous studies have been presented on various aspects of stretching flows since the seminal work by Crane (1970). One may refer to recent investigations by Hayat and Qasim (2010), Fang et al. (2010), Khan and Pop (2010), Ahmad and Asghar (2011), Kandasamy et al. (2011), Rashidi et al. (2011), Hayat et al. (2011), Yao et al. (2011) and Makinde and Aziz (2011) in this direction. On the other hand, mass transfer is important due to its appearance in many scientific disciplines that involve convective transfer of atoms and molecules. Examples of this phenomenon are evaporation of water, separation of chemicals in distillation processes, natural or artificial sources etc. In addition, mass transfer with chemical reaction has special significance in chemical and hydrometallurgical industries. The formation of smog represents a first order homogeneous chemical reaction. For instance, one can take into account the emission of NO₂ from automobiles and other smoke-stacks. Thus, NO₂ reacts chemically in the atmosphere with unburned hydrocarbons (aided by sunlight) and produces peroxyacetylnitrate, which forms a layer of photo-chemical smog. Chemical reactions can be treated as either homogeneous or heterogeneous processes. It depends on whether they occur at an interface or as a single-phase volume reaction (Kandasamy et al., 2008). A few representative studies dealing with mass transfer in the presence of chemical reaction may be mentioned (Kandasamy et al., 2005; Hayat et al., 2010; Ziabaksh et al., 2010; Makinde, 2010; Ibrahim and Makinde, 2010; Bhattacharyya and Layek, 2011; Hayat et al., 2011; Makinde, 2011).

Previous studies on the topic show that little work is presented regarding the effect of mass transfer on the MHD flows of non-Newtonian fluids in the presence of chemical reaction. Constitutive equations of the Casson fluid model (Nakamura and Sawada, 1988; Eldabe and Silwa, 1995; Dash et al., 1996; Boyd et al., 2007) are employed in the mathematical modeling. The rest of the paper is organized as follows. The next section completes the problem formulation. Then, the next section develops the homotopic solutions. Convergence of the derived series solutions and the discussion of velocity and concentration fields are presented in the sequence. The last section summarizes main points.

GOVERNING PROBLEMS

Consider a magnetohydrodynamic (MHD) and incompressible flow of a Casson fluid over a porous stretching surface at _{y = 0}, as shown in Figure 1. We select the Cartesian coordinate system such that the axis be taken parallel to the surface and is perpendicular to the surface. The fluid occupies a half space _{y > 0}. The mass transfer phenomenon with chemical reaction is also retained. The flow is subjected to a constant applied magnetic field in the direction. The flow is taken to be steady and the magnetic Reynolds number is considered to be very small so that the induced magnetic field is negligible in comparison to the applied magnetic field. The fluid properties are constant.

The rheological equation of state for an isotropic flow of a Casson fluid can be expressed as (Eldabe and Silwa, 1995):

In the above equation and denotes the component of the deformation rate, the product of the component of deformation rate with itself, a critical value of this product based on the non-Newtonian model, the plastic dynamic viscosity of the non-Newtonian fluid and the yield stress of the fluid. The equations governing the steady boundary layer flow of the Casson fluid are (Mustafa et al., 2012)

along with the following boundary conditions:

in which and represent the velocity components in the x- and _y- directions, the non-Newtonian Casson parameter, ν = (µ_B / ρ) the kinematic viscosity, the mass diffusion, the concentration field and k₁ the reaction rate.

Equations (2)-(6) can be made dimensionless by introducing the following change of variables

The dimensionless problem satisfies:

where Eq. (2) is satisfied identically, the Hartman number, Sc = ν / D the Schmidt number, γ = k₁ - c the chemical reaction parameter and the suction parameter.

The skin friction coefficient and the local Sherwood number can be written as:

in which is the skin friction (or shear stress along the stretching surface) and the mass flux from the surface, defined by the following relations:

Now Eqs. (12) and (13) give:

HOMOTOPY ANALYSIS SOLUTIONS

The initial guesses and auxiliary linear operators for this problem are selected as follows:

such that:

where C_i (i = 1- 5) represent the arbitrary constants. Denoting the nonzero auxiliary parameters and the resulting zeroth order problems are developed as follows:

where is an embedding parameter; N_f and N_φ are nonlinear operators which can be defined as:

By setting and we have:

We observed that, when p changes from 0 to 1 , then f (η,p) and φ(η,p) vary from f₀ (η),φ₀ (η) to f (ç) and φ(η). In view of the Taylor series we can write:

The convergence of the series is strongly dependent upon and .We select and in such a way that the series converge at p = 1 and hence:

The m^th -order deformation equations are obtained by differentiating the Equations (18)-(20) times with respect to and then putting to obtain:

Our general solutions can be expressed in the form:

in which and represent the special solutions.

CONVERGENCE ANALYSIS

The developed series solutions Eqs. (24) and (25) contain and .The convergence and rate of approximation for the constructed series solutions depend upon these auxiliary parameters. Therefore the curves have been plotted for the _{20^th -}order of approximation in order to find the range of admissible values of and . Fig. 2 shows that the range of admissible values of and are .0.7 < < -0.1 and -0.8 < <. -0.3. The series solutions converge in the whole region of η when = = -0.5. Table 1 shows the convergence of our series solutions for different orders of approximation. It is very clear from this table that _10^th order deformations are enough for the velocity whereas _15^th order deformations are required for the concentration.

RESULTS AND DISCUSSION

The velocity (_f') and concentration (φ) fields are shown graphically in Figs. 3-10. Figs. 3-5 show the effects of the Casson parameter β Hartman number and the suction parameter _S, respectively, on the velocity profile _f'(η). From Fig. 3, we observed that the velocity field decreases when β increases. An increase in β leads to an increase in plastic dynamic viscosity that creates resistance in the flow of fluid and a decrease in fluid velocity is observed. The effects of Hartman number M and S the suction parameter on are seen in Figs. 4 and 5. These figs. show that both and decrease the velocity _f'(η). This is due to the fact that the applied magnetic field normal to the flow direction induces the drag in terms of a Lorentz force which provides resistance to flow; suction is an agent which causes resistance to the fluid flow and the fluid velocity also decreased. Figs. 6-10 show the plots of the effects of the Casson parameter β, Hartman number M suction parameter S, Schmidt number Sc and chemical reaction parameter γ on the concentration field φ(η). The concentration field and associated boundary layer thickness increase when β increases (Fig. 6). It is also noticed from Figs. 3 and 6 that the Casson parameter β has quite opposite effects on the velocity and concentration profiles. Fig. 7 depicts that, by increasing the Hartman number, both the concentration profiles and boundary layer thickness increase. Thus, Hartman number here decreases the resistive force when M increases. The influence of the suction parameter on the concentration profile is seen in Fig. 8. The concentration profile is a decreasing function of _S. This is in accordance with the fact that the fluid experiences a resistance upon increasing the friction between its layers. As a consequence, there is a decrease in concentration. Effects of the Schmidt number on are displayed in Fig. 9. Here both the concentration profile and the boundary layer thickness decrease when the Schmidt number increases. From a physical point of view, the Schmidt number φ(η) is dependent on mass diffusion D and an increase in Schmidt number corresponds to a decrease in mass diffusion and the concentration profile reduced. When γ=0 there is no chemical reaction. An increase in the chemical reaction parameter corresponds to an increase in the reaction rate parameter and an increase in the reaction rate parameter caused a reduction in concentrarion. From Fig. 10, one can see that an increase in the value of the chemical reaction parameter γ decreased the concentration field φ(η). Figs. 11 and 12 are sketched to visualise the influence of key parameters that are used in the present problems for . The influence of M against β is described in Fig. 11. It is obvious that is an increasing function of M. Similar effects can be seen in Fig. 12, which shows the influence of β against M. Figs. 13 and 14 are shown to present the influence of sundry parameters on Fig. 13 describes the influence of β vs γ on . This figure confirms that the Sherwood number is a decreasing function of and the effects on the Sherwood number of are the opposite (see Fig. 14). Table 2 shows the skin friction coefficient for the different values of and By increasing the values of the value of the skin friction coefficient decreases, but it increases upon increasing M and S Table 3 shows the numerical values of the local Sherwood numbers for the parameters β, M, S, Sc, and γ. This table concludes that the values of the local Nusselt number decrease upon increasing β and M_, but increase upon increasing S, Sc and γ. Table 4 shows the comparison with the previous limited studies in the literature. From this table one can see that our series solutions are in excellent agreement with the previous studies, validating the present series solutions.

CONCLUSIONS

Effects of mass transfer on the MHD boundary layer flow of a Casson fluid model with chemical reaction are addressed. The present analysis leads to the following observations.

§ The Casson parameter β and Hartman number have similar effects on the velocity profile f'(η).

§ β has opposite effects on the velocity and concentration profiles.

§ The concentration field φ(η) as well as the boundary layer thickness increase upon increasing the Hartman number _M.

§ An increase in the Schmidt number Sc causes a decrease in the concentration profile and the boundary layer thickness.

§ When γ=0 there is no chemical reaction. An increase in decreases φ(η).

(Submitted: January 25, 2012 ; Revised: May 9, 2012 ; Accepted: May 14, 2012)

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