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Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 257161, 5 pages
Micropolar Fluid Flow and Heat Transfer over a Nonlinearly Stretching Plate with Viscous Dissipation
1Department of Science in Engineering, Kulliyyah of Engineering, International Islamic University Malaysia, Gombak, 50728 Kuala Lumpur, Malaysia
2School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia
Received 1 March 2013; Accepted 12 June 2013
Academic Editor: Xing-Gang Yan
Copyright © 2013 Kartini Ahmad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The flow and heat transfer of a micropolar fluid past a nonlinearly stretching plate is studied numerically, by taking into account the viscous dissipation effect. It is assumed that the plate is stretched nonlinearly from the slot where it is issued. The governing system of partial differential equations is transformed into ordinary differential equations, which are then solved numerically using a finite-difference scheme known as the Keller-box method. The effects of the governing parameters, namely, the material parameter , the Eckert number Ec, the Prandtl number Pr, and the nonlinear stretching parameter , on the flow field and the heat transfer characteristics are obtained and discussed. The velocity and the temperature profiles are also illustrated to aid the validity of the numerical results obtained. It is found that both the local Nusselt number and the magnitude of the skin friction coefficient increase with the nonlinear stretching parameter , and the opposite trend occurs as increases for fixed .
The initial study by Crane  has attracted many researchers to investigate similar problems on the boundary layer flow caused by a stretching sheet, as it has a number of applications in industry such as the extrusion of polymer sheet from a dye, continuous casting, drawing of plastic films, and crystal growing. The desired characteristics of the final product strictly depend on the rate of cooling and the process of stretching. This stretching sheet may not necessarily be linear, as it can also take nonlinear fashion, even though the problem might not have obvious technological relevance . In view of this, Vajravelu  studied the flow over a nonlinearly stretching sheet, and Cortell [4, 5] examined the flow and heat transfer on a nonlinear stretching sheet for two different types of thermal boundary conditions on the sheet, namely, constant surface temperature (CST) and prescribed surface temperature (PST). Ganji et al.  reported the analytical solution of the magnetohydrodynamic flow over a nonlinearly stretching sheet. Similar problems have been studied recently by Ishak et al. , Prasad et al. , Van Gorder et al. , Raftari et al. , Abbas and Hayat , and Abel et al. , among others.
The Navier-Stokes equations alone do not adequately describe the flow properties for fluids with diversity in the physical structure which occur in most non-Newtonian fluids. Hence, the theory of micropolar fluid which was proposed by Eringen [12, 13] perhaps gives an accurate model for fluids where the polymeric and rotating particles need to be taken into account, by including the microrotational momentum equation in addition to the classical momentum equation. Studies of micropolar fluids have recently received considerable attention due to their applications in a number of processes that occur in industry. Such applications include suspension solutions, solidification of liquid crystals, animal bloods, and exotic lubricants. Some of the researchers who investigated such fluid with nonlinear stretching sheet are Hayat et al. [14, 15] and Rahman et al. , who investigated the micropolar fluid flow past a nonlinear stretching sheet taking into account the effects of a temperature-dependent viscosity and variable surface temperature. El-Aziz  examined the micropolar boundary-layer flow and heat transfer characteristics associated with a heated exponential stretching continuous sheet being cooled by a mixed convection flow, and Hsiao  has done an analysis for the heat and mass transfer with radiation effect of a micropolar fluid past a nonlinearly stretching sheet. Ishak et al.  studied the behavior of micropolar fluid flow over a fixed or continuous moving surface. The flow over a stretching/shrinking sheet immersed in a micropolar fluid has been considered by Ishak et al. , Yacob and Ishak [21, 22], and Rosali et al. , among others. They showed the existence of dual solutions for the shrinking case.
Motivated by the above investigations, the present paper studies the problem of micropolar fluid flow and heat transfer over a nonlinearly stretching plate with viscous dissipation. It is worth mentioning that viscous dissipation plays an important role in the polymer processing where it is considered like an energy source for producing heat which in turn delays the process of solidification and as a result more time is required to cool the final product (Hassan et al. ). It will affect the heat transfer problem as the source term does appear in the energy equation when the viscosity of the fluid or the velocity gradient is high.
2. Problem Formulation
Consider a steady two-dimensional laminar boundary layer flow over a nonlinearly stretching plate immersed in an incompressible micropolar fluid of ambient temperature as shown in Figure 1. It is assumed that the plate is stretched with velocity , and the plate temperature varies like , where is the distance from the slot where the plate is issued and , , and are constants. The boundary layer equations are [25, 26] where and are the velocity components in the and directions, respectively. Further more, is the dynamic viscosity, is the vortex viscosity (or the microrotation viscosity), is the fluid density, is the thermal conductivity, is the specific heat at constant pressure, is the fluid temperature, is the microinertia density, is the microrotation (or angular velocity), and is the spin gradient viscosity. The boundary conditions are where is a constant with . The case indicates the vanishing of antisymmetric part of the stress tensor and denotes weak concentrations (Ahmadi ), which is considered in the present paper. Furthermore, we follow the work of many recent authors by assuming that , where is the micropolar or material parameter. This assumption is invoked to allow the field of equations to predict the correct behavior in the limiting case when the microstructure effects become negligible and the total spin reduces to the angular velocity (see Ahmadi  or Yücel ).
In order to solve (1)–(4) subject to the boundary conditions (5), we introduce the following similarity transformation (see Ziabakhsh et al.  and Ishak et al. ): where is the similarity variable, primes denote differentiation with respect to , is the kinematic viscosity, is the dimensionless stream function, is the dimensionless microrotation, and is the stream function defined as and which identically satisfies (1). Using transformation (6) and utilising the boundary condition from (5), (2) and (3) reduce to the single equation and the energy equation (4) becomes where is the Prandtl number and is the Eckert number defined as The transformed boundary conditions are
The physical quantities of interest are the skin friction coefficient and the local Nusselt number which are defined as where the surface shear stress and the surface heat flux are given by Using the dimensionless variables in (6), we obtain where is the local Reynolds number.
3. Results and Discussion
The transformed system of (7) and (8) with the boundary conditions (10) was solved numerically using the Keller-box method as described in  for some values of parameters. In order to validate the numerical results obtained, we compare our results with those obtained by Grubka and Bobba  as shown in Table 1, which shows a favorable agreement.
Figures 2 and 3 show the effect of the nonlinear stretching parameter on the skin friction coefficient and the local Nusselt number , respectively, as the material parameter varies when and . As discussed by Alam et al. , corresponds to air that may be considered as a micropolar fluid when it is heavily contaminated with suspended particles of dirt. It is found that all the values of the skin friction coefficients are negative and decrease slowly as increases from (Newtonian fluid) to (micropolar fluid) for all values of the nonlinearity parameter . The same trend is observed as the plate is being stretched ; that is, the values of the skin friction coefficient decrease with respect to . It is worth mentioning that the plate is being as more stretched, it will decrease the values of the skin friction coefficients. But opposite trend is observed for the local Nusselt number as depicted in Figure 3. As the plate is being stretched, the values of the local Nusselt number are greater and for a particular value of , the local Nusselt number decreases slowly as increases from 0 (Newtonian fluid) to positive values of (micropolar fluid).
The velocity and temperature distributions obtained by the finite-difference method for various values of the governing parameters are displayed in Figures 4 to 7. Figures 4 and 5 depict the effect of the micropolar/material parameter when , , and and on the velocity and temperature profiles. On the other hand, Figures 6 and 7 display the velocity and temperature distributions for different values of the Eckert number when , , and and . It is seen that an increase in the Eckert number increases the thermal boundary layer thickness, while stretching the plate decreases the thermal boundary layer. There are no changes in the velocity boundary layer thickness as illustrated in Figure 6. This is expected as the viscous dissipation only affects the thermal field. We can see that the velocity and the temperature profiles for all the figures subside monotonously to zero as increases which validate the boundary conditions (10), asymptotically.
We have done a numerical investigation on the effects of the material parameter , the viscous dissipation , the nonlinear stretching parameter , and the Prandtl number on the fluid flow and heat transfer characteristics toward a nonlinear stretching sheet immersed in a micropolar fluid. It is found that both the magnitude of the skin friction coefficient and the local Nusselt number increase with the nonlinear stretching parameter and decrease when increases for fixed values of .
The authors wish to express their thanks to the anonymous referees for their valuable comments and suggestions. The financial support received from the Universiti Kebangsaan Malaysia (Project code: DIP-2012-31) is gratefully acknowledged.
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