American Academic & Scholarly Research Journal Special Issue - January 2012

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1 Proceeding of 2 nd International Conference on Mathematics and Information Sciences, 9-13 Nov. 2011, Sohag, Egypt American Academic & Scholarly Research Journal Special Issue - January 2012 Heat and mass transfer analysis on the flow of non- Newtonian micropolar fluid with uniform suction/blowing, heat generation, chemical reaction and Thermophoresis effects. R. A. Mohamed 1, S. Z. Rida 2, A. A. M. Arfa 3 and M. Said 4 Mathematics Department, Faculty of Science, South Valley University, Qena, Egypt Corresponding author: M.elsaid10@yahoo.com Abstract: In this paper, the problem of heat and mass transfer on the flow of non-newtonian micropolar fluid with uniform suction/blowing, heat generation, radiation, thermophoresis and chemical reaction is presented and discussed. The Homotopy Analysis Method (HAM) is employed to compute an approximation to the solution of the system of nonlinear differential equations governing the problem. The effects of various physical parameters such as material parameter, suction parameter, heat generation/absorption parameter, Prandtl number, radiation parameter, thermophoretic parameter, chemical reaction parameter and Schmidt number on the velocity profile temperature profile and concentration profile are studied and shown in several plots. Keywords: Non-Newtonian micropolar fluid, Heat and mass transfer, Uniform suction/blowing, Heat generation, Chemical reaction,thermophoresis and HAM. 1.Introduction Micropolar fluids are fluids with microstructure belonging to a class of fluid with non-symmetrical stress tensor referred to as polar fluids. Physically they represent fluids consisting of randomly oriented particles suspended in a viscous medium. The classical theories of continuum mechanics are inadequate to explicate the microscopic manifestations of microscopic events, a new stage in the evolution of fluid dynamic theory is in progress. Eringen presented the earliest formulation of a general theory of fluid microcontinua taking into account the inertial characteristics of the substructure particles which are allowed to undergo rotation. Eringen's actual theory of a fluid microcontinuum was presented in 1964 in his paper on simple micro fluids [1]. This theory has been extended by Eringen [2] to take into account thermal effects. The theory of micropolar fluids and its extension thermo micropolar fluids [3] may form suitable non-newtonian fluid models which can be used to explain the flow of colloidal fluids, liquid crystals, polymeric suspensions, animal blood, etc. The theory of micropolar fluids developed by Eringen [1-3] describes some physical systems which do not satisfy the Navier-Stokes equations. This general theory of micropolar fluids deviates from that of Newtonian fluids by adding two new variables to the velocity. These variables are microrotations that are spin and microinertia tensors describing the distributions of atoms and molecules inside the microscopic fluid particles. This theory may be applied to the explanation for the phenomenon of the flow of colloidal fluids, liquid crystals, polymeric suspensions, animal blood, etc. An excellent review of micropolar fluids and their applications was given by Ariman et al.[4]. Gorla [5] discussed the steady state heat transfer in a micropolar fluid flow over a semi-infinite plate, and the analysis is based on similarity variables. Rees and Pop [6] studied the free convection boundary layer flow of micropolar fluid from a vertical flat plate. Singh [7] has studied the free convection flow of a micropolar fluid past an infinite vertical plate using the finite difference method.

2 86 The combined heat and mass transfer problems with chemical reactions are of importance in many processes, and therefore have received a considerable amount of attention in recent years. In processes, such as drying, evaporation at the surface of a water body, energy transfer in a wet cooling tower and the flow in a desert cooler, the heat and mass transfer occurs simultaneously. Chemical reactions can be codified as either homogeneous or heterogeneous processes. A homogeneous reaction is one that occurs uniformly through a given phase. In contrast, a heterogeneous reaction takes place in a restricted region or within the boundary of a phase. A reaction is said to be the first order if the rate of reaction is directly proportional to the concentration itself. In many chemical engineering processes, a chemical reaction between a foreign mass and the fluid does occur. These processes take place in numerous industrial applications, such as the polymer production, the manufacturing of ceramics or glassware, the food processing [8] and so on. Das et al.[9] considered the effects of a first order chemical reaction on the flow past an impulsively started infinite vertical plate with constant heat flux and mass transfer. Muthucumarswamy and Ganesan [11] and Muthucumarswamy [10] studied the first order homogeneous chemical reaction on the flow past an infinite vertical plate. Recently, Kandasamy et al.[12] discussed the heat and mass transfer effect along a wedge with a heat source and concentration in the presence of suction/injection taking into account the chemical reaction of the first order. The study of heat generation or absorption in moving fluids is important in problems dealing with chemical reactions and those concerned with dissociating fluids. Possible heat generation effects may alter the temperature distribution; consequently, the particle deposition rate in nuclear reactors, electronic chips and semiconductor wafers. In fact, the literature is replete with examples dealing with the heat transfer in laminar flow of micropolar fluids. The study of radiation effects on the various types of flows is quite complicated. In the recent years, many authors have studied radiation effects on the boundary layer of radiating fluids past a plate. Influence of chemical reaction and thermal radiation on the heat and mass transfer in MHD micropolar flow over a vertical moving porous plate in a porous medium with heat generation was studied by R.A. Mohamed and S.M. Abo-Dahab [13]. Raptis [14] studied the flow of a micropolar fluid past continuously moving plate by the presence of radiation. The radiation effect on heat transfer of a micropolar fluid past unmoving horizontal plate through a porous medium was studied by Abo- Eldahab and Ghonaim [15]. Kim and Fedorov [16] investigated the transient mixed radiative convection flow of a micropolar fluid past a moving semi-infinite vertical porous plate. Thermophoresis is a mechanism of migration of small particles in direction of decreasing thermal gradient [17]. It is an effective method for particle collection [18]. The velocity acquired by the particle is called thermophoretic velocity and the force experienced by the suspended particle is called thermophoretic force [19]. Thermophoresis causes small particles to deposit on the cold surfaces. It has many applications in aerosol technology, deposition of silicon thin films, and radioactive particle deposition in nuclear reactor safety simulations. For more detail on the topic, the readers may consult the studies [20-25]. Also convective free mixed and forced convection flows play an important role in petroleum extraction, in soils, storage of agricultural products, porous material heat exchanger etc [26-32]. Since there are some limitations with the common perturbation methods, and also because the basis of the common perturbation method is upon the existence of a small parameter, developing the method for different applications is very difficult. Therefore, many different methods have recently introduced to eliminate the small parameter. The Homotopy Analysis method (HAM) is one of the well-known methods to solve the nonlinear equations. This method has been first introduced in 1992 by Liao [33-38]. The method has been used by many authors in a wide variety of scientific and engineering applications to solve different types of governing differential equations: linear and nonlinear, homogeneous and non-homogeneous, and coupled and decoupled as well. The purpose of present paper is to study the problem of non-newtonian micropolar fluid flow with uniform suction/blowing, heat generation, radiation, thermophoresis and chemical reaction and to investigated the effect of the various dimensionless parameters of these non-newtonian micropolar fluid on the velocity, temperature and concentration. by means of an analytic technique, namely the Homotopy Analysis Method (HAM). 2. Mathematical description

3 87 Consider the two-dimensional stagnation point flow of an incompressible non-newtonian micropolar fluid impinging perpendicular on a permeable wall and flowing away along the -axis. And using the boundary layer approximation and neglecting the dissipation, the equation of energy for temperature with heat generation or absorption and thermal radiation, the equation of mass for concentration with thermophoresis and chemical reaction. The simplified two-dimensional equations governing the flow in the boundary layer of a steady, laminar, and incompressible micropolar fluid are governed by: (1) (2) (3) (4) (5) where is the microrotation or angular velocity whose direction of rotation is in the plane, is the viscosity of the fluid, is the density, is the specific heat capacity at constant pressure of the fluid, is the thermal conductivity of the fluid, is the heat generation/absorption coefficient and, and are the microinertia per unit mass, spin gradient viscosity, and vortex viscosity, respectively, which are assumed to be constant. The appropriate physical boundary conditions of Eqs. (1) (5) are (6) where is a constant and.the case indicates the vanishing of the ant symmetric part of the stress tensor and denotes weak concentration of microelements, which will be considered here. Using the transform function we have (7) After using the transformation (7), for micropolar fluid, there are two equations in which one is for angular velocity or microrotation and physically it is important in micropolar fluid. In this study, we have two equations and which equals to (see Ziabakhsh. Z, et al.[39]) So Eqs. (2) and (3) reduce to the single equation as Eq. (8a) (8a) subject to the boundary conditions (8b) (8c) (8d) where is the material parameter, is the suction parameter, and primes denote differentiation with respect to. is the Prandtl number and is the heat Generation/absorption parameter is the Schmidt number, is the chemical reaction parameter is the thermophoretic parameter and is the heat radiation parameter. For micropolar boundary layer flow, the wall skin friction. (9) Using as a characteristic velocity, the skin friction Coefficient, can be defined as (10) By using this definition we have (11) where is the local Reynolds number, The heat transfer from the surface to the fluid is computed by application of Fourier s law (12) Introducing the transformed variables, the expression for becomes and the heat transfer coefficient, in terms of the Nuselt number,, can be expressed as (13) then we have The definition of the local mass flux and the local Sherwood number are respectively given by 3. Application of ( HAM) to a problem (14) (15)

4 88 According to the boundary condition (8d), it is nature that, and can be expressed by the function (16) In the following form: Let denotes the embedding parameter and indicates non-zero auxiliary parameters. We then construct the following equations: (26) where are coefficients. The rule of solution expression provides us with a starting point. It is under the rule of solution expression that initial approximations, auxiliary linear operators, and the auxiliary functions are determined. So, according to the rule of solution expression, we choose the initial guess and auxiliary linear operator in the following form: subject to the boundary conditions: Obviously, when, the above (HAM ) deformation equations (26) have the solutions: (27) (28) As the initial approximations of, and we choose (20) if increases from then and vary from and By Taylor s theorem and using equations.(28), can be expanded in a power series of as follows: As the auxiliary linear operator, we have the following property: (22a) (22b) (22c) where are constants.based on, we are led to define the non linear operators: (24) (23) In which is chosen in such a way that these three series (29), (30) and (31) are convergent at, we have, using equations (28), the solutions series: (25) 3.1 Zeroth-order deformation equations 3.2 High order deformation equation

5 89 For the sake of simplicity, we define the vectors: Differentiation the zeroth-order deformation equations times with respect to then setting and finally dividing them by we obtain the th- order deformation equations subject to the boundary conditions: Where And (35) (36c) (37) (34) Let denote the particular solutions of equations (34). Using we have the general solution: (38) where to are constants that can be obtained by applying the boundary conditions in equations. (35) as discussed by Liao the rule of coefficient ergodicity and the rule of solution existence play important roles in determining the auxiliary function and ensuring that the high-order deformation equations are closed and have solutions. In many cases, by means of the rule of solution expression and the rule of coefficient ergodicity, auxiliary functions can be uniquely determined. So we define the auxiliary function which for both velocity field and temperature is true and same. It is in the following form: 4. Convergence of the (HAM) solution Liao [35] proved in general that, as long as a solution series given by homotopy analysis method is not divergent, it must converge to the exact solution of non linear problems under investigation. The convergence of the solution series depends upon the choice of initial approximations, the auxiliary linear operators and the nonzero auxiliary parameters. Once if the initial guess approximations and the auxiliary linear operators have been selected then the convergence of the solution series will strictly depend upon the auxiliary parameter only. Therefore, the convergence of the solution series is determined by the values of such kind of parameters. The admissible values of parameter is determined by the so-called curves. In order to find the allowed value of to make the series (32) convergent we have plotted the curves corresponding to Our analysis shows that the admissible value of for and are and, respectively. 5. Results and discussion To study the behavior of the velocity, temperature and concentration curves are drawn values of the parameters that describe the flow. The results of analytical computation are displayed in figures from Fig.1 to Fig 15. Results are obtained for. Fig. 1 and Fig. 2. display results for the velocity It is seen that increases with increasing the suction parameter and decreases with increasing the material parameter respectively. Fig. 3 and Fig. 4 display results for the temperature distribution, it is seen that decreases with increasing the suction parameter and increases with increasing the heat generation/absorption parameter respectively. Fig. 5 and Fig. 6 describes the behavior of the temperature distribution with changes in the values of the material parameter and radiation parameter it is seen that the temperature distribution decreases with increasing the material parameter, but it increase with increasing the radiation parameter. The effect of prandtl number on the dimensionless temperature distributions is displayed in Fig. 7. The effect of suction is to decrease temperature distribution. Fig. 8 and Fig. 9 show that the concentration distribution decreases with increasing the material parameter and with increasing prandtl number. Fig. 10 and Fig. 11 represents the effect of radiation parameter on the

6 f' f' 90 concentration profiles while no effect appears in the velocity. As the radiation parameter increases the concentration distribution decreases and it decreases with increasing the heat generation/absorption parameter too. Fig 12 and Fig. 13 describes the behavior of the concentration distribution with changes in the values of the thermophortic parameter and chemical reaction parameter, respectively. It is seen that concentration distribution decreases with increasing both thermophortic parameter and chemical reaction parameter. Fig. 14 and Fig. 15 show that the concentration distribution decreases with increasing the suction parameter and decreases with increasing the Schmidt number far from the wall but increases near from the wall. The governing fundamental are approximated by a system on non-linear ordinary differential equations by similarity transformation and it solved analytically by means of an analytic technique, namely the homotopy analysis method, results are presented graphically to illustrate the variation of velocity, temperature and concentration with various values of parameters for the problem, e.g. suction parameter, prandtl number, radiation, Schmidt number, chemical reaction parameter, heat generation/absorption parameter, thermophortic parameter and material parameter. The analytical results indicate that the velocity increases with increasing and decreases with increasing but and not affected on it The temperature distribution increases with increasing but decrease with increasing and The concentration distribution increases with increasing, but decrease with increasing, and Fig.2. The Fig.3. The Fig.4. The K= K=0.5 K= K=1.5 K=2.0 A = -2.0 A = - A = A = A = 2.0 B = -0.5 B = -0.1 B = B = 0.1 B = 0.5 A = -2.0 A = - A = A = A = 2.0 Fig.1. The

7 K = K = K = 2.0 K = 3.0 K = 5.0 pr = 0.1 pr = 0.5 pr = 0.71 pr = pr = 7.0 pr = 10 Fig. 5. The Fig. 8. The R = R = 0.5 R = R = 1.5 R = 2.0 K = K = 3.0 K = 7.0 K = 1 Fig. 6. The Fig.9. The Pr=0.1 Pr=0.5 Pr=0.71 Pr= Pr=7.0 B = -2.0 B = - B = B = B = 2.0 Fig. 7. The Fig.10. The

8 92 R=0 R=1 R=2 R=4 R=6 Sc = 6 Sc = 0.78 Sc = Sc = 2.0 Sc = 3.0 Fig.11. The Fig.14. The = =0.5 =1.5 =3.0 =5.0 A = -2.0 A = - A = A = A = 2.0 Fig.12. The Fig.15. The 6. Conclusions Fig.13. The =- =- = = = In this paper, the effect of chemical reaction and thermophoresis of a micropolar fluid in the presence of heat generation or absorption and thermal radiation are studied by means of an analytical technique, namely the homotopy analysis method. The governing equations for the problem are changed to dimension less ordinary differential equations by similarity transformation. The effect of the various dimensionless parameters are investtigated. The proposed analytic approach has general meaning and thus may be applied in a similar way to other unsteady nonlinear problems to get accurate analytic solutions valid for all dimensionless time. References [1] Eringen A C. Simple microfluids. Int J Engng Sci, 2 (1964)

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