Axisymmetric Stagnation Flow of a Micropolar Nanofluid in a Moving Cylinder
An analysis is carried out for axisymmetric stagnation flow of a micropolar nanofluid in a moving cylinder with finite radius. The coupled nonlinear partial differential equations of the problem are simplified with the help of similarity transformations and the resulting coupled nonlinear differential equations are solved analytically by homotopy analysis method (HAM). The features of the flow phenomena, inertia, heat transfer, and nanoparticles are analyzed and discussed.
During the past years, the study of stagnation flows has become more and more important because of their applications in engineering and technology. Hiemenz  first discussed the steady flow of a Newtonian fluid impinging orthogonally on an infinite flat plate. Later on, a large number of papers have been done on orthogonal, nonorthogonal, and axisymmetric stagnation flows. Some important studies on the topic include references [2–10].
A large amount of literature is available on the viscous theory. However, only a limited attention has been given to the study of non-Newtonian fluids. There are two major reasons responsible for this. The main reason is that the additional nonlinear terms arising in the equation of motion rendering the problem more difficult to solve . The second reason is that a universal non-Newtonian constitutive relation that can be used for all fluids and flows are not available. The study of non-Newtonian fluids has many applications in various industries, such as nuclear paints, physiology, biomechanics, chemical engineering, and technology. There are many non-Newtonian fluid models. However, Eringen  proposed the theory of micropolar fluid that is capable to describe practical fluids by taking into account the effects arising from local structure and micromotion of the fluid elements. The study has attracted many researchers to investigate the non-Newtonian fluids with various aspects.
Recently, the study of convective transport of nanofluids has gained considerable importance due to its applications. According to Khan and Pop  most of the conventional transfer fluids like oil, water, and ethylene glycol are poor heat transfluids because the thermal conductivity of these fluids plays an important role on the heat transfer coefficient between the heat transfer medium and heat transfer surface. The importance of these nanofluids has been discussed by Choi . Kuznetsov and Nield  have studied the thermal instability in a porous medium layer saturated by nanofluids.
Keeping the above importance in mind, the aim of the present work is to discuss the axisymmetric stagnation flow of a micropolar nanofluid flow in a moving cylinder. To the best of the authors’ knowledge, no attempt has been made in this direction. The governing equations of the micropolar fluid along with heat transfer and nanoparticle equation are simplified by applying suitable similarity transformations and then the reduced highly nonlinear-coupled equations are solved analytically with the help of homotopy analysis method. The convergence of the HAM solution and the physical behavior of pertinent parameters of the problem are discussed through graphs.
Let us consider an incompressible flow of a micropolar nanofluid between two cylinders such that the flow is axisymmetric about -axis. The inner cylinder is of radius rotating with angular velocity and moving with velocity in the axial -direction. The inner cylinder is enclosed by an outer cylinder of radius . Further, the fluid is injected radially with velocity from the outer cylinder towards the inner cylinder. The geometry of the problem is shown in Figure 1. The governing equations of motion and microinertia in the presence of nanoparticles and the heat transfer are where (2.1) is the continuity equation, (2.2)–(2.4) are the , and components of momentum equation, (2.4)–(2.6) are the angular momentum, temperature, and nanoparticle concentration equations. Note that neglecting microrotation and nanoparticle concentration, the remaining system would be that solved by Hong and Wang . In the above equations are the velocity components along the axes, is the angular microrotation momentum, is the dynamic viscosity, is the vertex viscosity, is the microrotation density, is the micropolar constant, is density, is the specific heat at constant pressure, is the kinematic viscosity, is temperature, is the thermal conductivity, is the nanoparticle volume fraction, is the nanoparticle mass density, is the effective heat of the nanoparticle material, is the Brownian diffusion coefficient, is the thermophretic diffusion coefficient, and is pressure.
Let us define the following similarity transformations and nondimension variables as With the help of these above transformations, (2.1) is identically satisfied and (2.2) to (2.6) take the following forms: where is the cross-flow Reynolds number, is the Prandtl number, is the micropolar coefficient, and are the micropolar parameters, is the Lewis number, is the Brownian motion parameter, and is the thermophoresis parameter.
The boundary conditions in nondimensional form are
3. Solution of the Problem
The solution of the above boundary value problem is obtained with the help of HAM. For HAM solution we choose the initial guesses as [17–23] The corresponding auxiliary linear operators are These operators satisfy where are arbitrary constants. The zeroth-order deformation equations are defined as where The boundary conditions for the zeroth-order system are The th-order deformation equations can be obtained by differentiating the zeroth-order deformation equations (3.4) and the boundary conditions (3.6), -times with respect to , then dividing by , and finally setting , we get where With the help of MATHEMATICA, the solutions of (2.9) subject to the boundary conditions (2.10) can be written as where In which , , , , , , and are the special solutions, and the solutions can be written as
4. Results and Discussion
The governing nonlinear partial differential equations of the axisymmetric stagnation flow of micropolar nanofluid in a moving cylinder are simplified by using similarity transformation and then the reduced highly nonlinear-coupled differential equations are solved analytically by the help of homotopy analysis method. The velocity field for different values of , and are plotted in Figures 2 to 5. It is observed that increases with the increase in the parameters , and as (see Figures 2 and 3). The values of for different values of are shown in Figure 4. It is observed that the nondimensional velocity decreases with an increase in . The nondimensional velocity for different values of is plotted in Figure 5. It is depicted that the velocity field decreases with the increase in . The variation of microrotation functions and for different values of and are plotted inFigures 6 to 9. It is observed that for increase in both of these parameters increases (see Figures 6 and 7). The change in is similar to (see Figures 8 and 9). The variation of temperature for different values of , , , and are shown in Figures 10, 11, 12, and 13. It is observed from these figures that with an increase in these parameters, the temperature field increases. The variation of the nanoparticle concentration for different values of , , , and are shown in Figures 14, 15, 16 and 17. It is observed from these figures that with an increase in all the above-mentioned parameters the concentration increases. Physically, it means that the effect of the Prandtl number, micropolar parameter, thermophoresis parameter, and the Lewis number is to increase the nanoparticle concentration. It may be noted that the temperature and concentration functions are plotted for the case of strong concentration, that is, when .
The effects of various physical parameters on the velocity, temperature, and nondimensional nanoparticle parameter are summarized as the following.(1)With the increase in Reynold’s number the velocity , microrotation velocities and , temperature , and nanoparticle concentration increase, while the velocity profiles for and decrease.(2)With the increase in micropolar parameter , the profiles , , , , , , and all have shown increasing behavior.(3)With the increase in Prandtl number , the temperature profile and nanoparticle concentration increase.(4)With the increase in Brownian motion parameter , the temperature profile and nanoparticle concentration increase.(5)With the increase in thermophoresis parameter , the temperature profile and nanoparticle concentration increase.(6)With the increase in Lewis number , the nanoparticle concentration increases.
This paper was supported by WCU (World Class University) Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology R31-2008-000-10049-0.
K. Hiemenz, “Die Grenzschicht an einem in den gleichformingen Flussigkeitsstrom eingetauchten graden Kreiszylinder,” Dingler's Polytechnic Journal, vol. 326, pp. 321–324, 1911.View at: Google Scholar
A. C. Eringen, “Theory of micropolar fluids,” The Journal of Mathematical Fluid Mechanics, vol. 16, pp. 1–18, 1966.View at: Google Scholar
S. U. S. Choi, “Enhancing thermal conductivity of fluids with nanoparticles,” in Proceedings of the ASME International Mechanical Engineering Congress and Exposition, vol. 231, pp. 99–105, American Society of Mechanical Engineers, Fluids Engineering Division, San Francisco, Calif, USA, November 1995.View at: Google Scholar
S. J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC Press, Boca Raton, Fla, USA, 2003.