Flow of micropolar fluid between two orthogonally moving porous disks

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1 Appl. Math. Mech. -Engl. Ed., 33(8), (2012) DOI /s c Shanghai University and Springer-Verlag Berlin Heidelberg 2012 Applied Mathematics and Mechanics (English Edition) Flow of micropolar fluid between two orthogonally moving porous disks Xin-hui SI ( ) 1, Lian-cun ZHENG ( ) 1, Xin-xin ZHANG ( ) 2, Xin-yi SI ( ) 3 (1. Department of Applied Mathematics, University of Science and Technology Beijing, Beijing , P. R. China; 2. Department of Mechanical Engineering, University of Science and Technology Beijing, Beijing , P. R. China; 3. College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing , P. R. China) Abstract The unsteady, laminar, incompressible, and two-dimensional flow of a micropolar fluid between two orthogonally moving porous coaxial disks is considered. The extension of von Karman s similarity transformations is used to reduce the governing partial differential equations (PDEs) to a set of non-linear coupled ordinary differential equations (ODEs) in the dimensionless form. The analytical solutions are obtained by employing the homotopy analysis method (HAM). The effects of various physical parameters such as the expansion ratio and the permeability Reynolds number on the velocity fields are discussed in detail. Key words porous disk homotopy analysis method (HAM), expansion ratio, orthogonally moving Chinese Library Classification O175.8, O Mathematics Subject Classification 76M99, 76D10 1 Introduction The flows between two disks have many technical applications in the fields of rotating machineries, computer storage devices, viscometries, lubricants, crystal growth processes, heat and mass exchangers, biomechanics, and oceanography. In thrust bearings, the disks are separated by means of a lubricant injected through the disks. Furthermore, the fluids with polymer additives have been used as improved lubricating oils in the modern lubrication technology [1]. Elcrat [2] proved the existence and uniqueness for non-rotational fluid motion between fixed porous disks with arbitrary uniform injection or suction. Rasmussen [3] investigated the steady viscous flow between two porous disks by an extension of Berman similarity transformation [4] Received Nov. 21, 2011 / Revised Mar. 26, 2012 Project supported by the National Natural Science Foundation of China (Nos , , , and ), the Research Foundation of Engineering Research Institute of University of Science and Technology Beijing (No. Yj ), and the Fundamental Research Funds for the Central Universities (No.T-RF-TP A) Corresponding author Xin-hui SI, Ph.D., sixinhui ustb@126.com

2 964 Xin-hui SI, Lian-cun ZHENG, Xin-xin ZHANG, and Xin-yi SI to reduce the governing equations to a set of non-linear coupled ordinary differential equations in the dimensionless form. The theory of micropolar fluids was introduced by Eringen [5] and Aero et al. [6] to describe a class of fluids consisting of rigid randomly oriented particles suspended in a viscous medium undergoing both translational and rotational motion, which are the generalization of Navier- Stokes equations. The applications of these fluids are in blood flows, lubricants, porous media, turbulent shear flows, and flows in capillaries and micro channels [7 8]. Anwar et al. [7] made a comparison between the results of micropolar and Newtonian fluids, and considered the numerical simulation of the symmetric flow of a micropolar fluid between two porous coaxial disks. Ashraf et al. [8] investigated the influence of the Reynolds number and the micropolar structure on the flow between two disks using the successive over relaxation (SOR) method. A thorough review of the subject and application of micropolar fluids can be found in Refs. [9 11]. The problem of the steady incompressible flow of a micropolar fluid between a rotating disk and a stationary disk was solved numerically by Guram and Anwar [12 13]. Using a suitable similarity transformation, Atif and Tahir [14] first considered the flow and heat transfer of the viscous fluid between two orthogonally moving rotating disks. However, they considered the disks impermeable. The special boundary condition was first proposed by Uchida and Aoki [15] in order to learn the transport of biological fluids through contracting or expanding vessels, the synchronous pulsation of porous diaphragms, and the air circulation in the respiratory system. Ohki [16] investigated the unsteady flow in a semi-infinite tube with a porous, the length of the elastic wall varied with time, but its cross section did not vary. Barron et al. [17] firstly achieved experimentally the time-dependent motion in a long rectangular channel with porous walls. They used the sublimation process of carbon dioxide to simulate the injection process at the walls. As a result, the walls of their channels expanded during the sublimation process. Majdalani et al. [18], Majdalani and Zhou [19], and Dauenhauer and Majdalani [20] obtained both numerical and asymptotical solutions for different permeability Reynolds numbers in a porous channel with expanding or contracting walls. Recently, Asghar et al. [21] discussed the flow in a slowly deforming channel with weak permeability using the adomian decomposition method (ADM). Dinarvand and Rashidi [22] got analytical approximate solutions for two-dimensional viscous flow through expanding or contracting gaps with permeable walls. Si et al. [23 24] extended the model to micropolar fluids and viscoelastic fluids. However, all the above works considered the expansion ratio α to be a constant. In seeking further generalization, Xu et al. [25] also extended the Dauenhauer-Majdalani model to the case in which the wall expansion ratio α is no longer a constant, but a time-dependent variable that transitions from α 0 to α 1. As a result, they found the time-dependent solutions approached to the steady state behavior very rapidly. Motivated by the above mentioned works, the objective of this analysis is to investigate the unsteady, laminar, and incompressible flow of a micropolar fluid between two orthogonally moving coaxial porous disks. We neglect the effects of body forces and body couples. The flow is symmetrically driven by equal injection or suction through the two porous disks and the motion of the two disks. A recently powerful technique developed by Liao [26 27] named the homotopy analysis method (HAM), which has been used to solve a lot of nonlinear problems successfully [28 34], is adopted to solve this problem for velocity fields. Using the HAM, the influence of the disk moving and the permeability of the two disks on the velocity of the fluid are taken into consideration. 2 Statement of problem and governing equations We consider the motion of the incompressible flow of a micropolar fluid between two orthogonally moving porous disks, and neglect the effects of body forces and body couples. The distance between the disks is 2a(t). The disks have the same permeability and move down or

3 Flow of micropolar fluid between two orthogonally moving porous disks 965 up uniformly at a time-dependent rate ȧ(t). As shown in Fig. 1, a cylindrical coordinate system may be chosen with the origin at the middle of the two disks. The velocity components u and w are taken to be in the r- and z-directions, respectively, and φ is the microrotation component. Fig. 1 Model for orthogonally moving porous disks Under these assumptions, the governing equations are given by the following relations [7 8] : u r + u r + w = 0, (1) z ( 2 u u t + u u r + w u z = 1 p ρ r + µ + κ ρ w t + u w r + w w z = 1 ρ φ t + u φ r + w φ z = γ ρj r r ( 2 w r w r u r u r u p z + µ + κ ρ ( 2 φ r φ r r φ r φ ) z 2 + κ ρj ) z 2 κ ρ r + 2 w ) z 2 + κ ( φ ρ r + φ r ( u z w r φ z, (2) ), (3) ) 2 κ φ, (4) ρj where ρ and ν are the density and kinematic viscosity, respectively, and j, γ, and κ are the microinertial per unit mass, spin gradient viscosity, and vortex viscosity, respectively. Here, γ is assumed to be [35] ( γ = µ + κ ) j, (5) 2 where µ is the dynamic viscosity, and we take j = a 2 as the reference length. According to Ref. [36], we also assume that there is strong concentration of microelements and the microelements close to the walls are unable to rotate. The boundary conditions are u = 0, w = 2v w = Aȧ, φ = 0 for z = a(t), (6) u = 0, w = 2v w = Aȧ, φ = 0 for z = a(t), (7) where A = 2v w /ȧ is the measure of the wall permeability [15,19]. By considering von Karman s similarity transformation [37] and motivated by the definition of the stream function [7,8,20], we define u = 1 ψ r z = νr a 2 F η(η, t), φ = νr a 3 G(η, t), η = z a. w = 1 r ψ r = 2ν F(η, t), a (8)

4 966 Xin-hui SI, Lian-cun ZHENG, Xin-xin ZHANG, and Xin-yi SI Substituting u, w, and φ into (1) (4) and eliminating pressure, we obtain the following nonlinear partial differential equations (PDEs): (1 + K)F ηηηη + α(3f ηη + ηf ηηη ) 2FF ηηη KG ηη a 2 υ 1 F ηηt = 0, (9) (1 + K/2)G ηη + α(3g + ηg η ) + KF ηη 2KG + F η G 2FG η a 2 ν 1 G t = 0. (10) The boundary conditions (6) and (7) become F (1) = 0, F(1) = Re, G(1) = 0, (11) F ( 1) = 0, F( 1) = Re, G( 1) = 0, (12) where K = κ aȧ[20] µ, α is the wall expansion ratio defined by α = ν, the negative shows that the distance of the two disks are approaching each other, and Re = avw υ is the permeability Reynolds number and positive for injection. Let f = F Re, g = G Re. (13) Substituting (13) into (9) and (10), the PDEs (9) and (10) become (1 + K)f ηηηη + α(3f ηη + ηf ηηη ) 2Reff ηηη Kg ηη a 2 υ 1 f ηηt = 0, (14) (1 + K/2)g ηη + α(3g + ηg η ) + Kf ηη 2Kg + Re(f η g 2fg η ) a 2 ν 1 g t = 0. (15) A similar solution with respect to both space and time can be developed into the transformations described by Uchida and Aoki [15], Majdalani et al. [18], and Dauenhauer and Majdalani [20]. This can be accomplished by considering the following case: α is a constant and f = f(η), which leads to f ηηt = 0. Similarly [23], in the present paper, we also assume that g = g(η). From a physical standpoint [15,19 20], our idealization is based on a decelerating expansion rate that follows a plausible model, according to which α = ȧa ν = ȧ0a 0 = constant, (16) ν where a 0 and ȧ 0 denote the initial height and the initial moving velocity of the disk, respectively. Integrating (16) with respect to time, the similar solution can be achieved. The result is a = v w(0) a 0 v w (t) = 1 + 2ναta 2 0. (17) Since 2v w = Aȧ and A a is constant [15,19 20], the expression for the injection velocity can also be determined. Under these provisions, the differential equations (14) and (15) become The boundary conditions are (1 + K)f + α(3f + ηf ) Kg 2Reff = 0, (18) (1 + K/2)g + α(3g + ηg ) + Kf 2Kg + Re(f g 2fg ) = 0. (19) f (1) = 0, f(1) = 1, g(1) = 0, (20) f ( 1) = 0, f( 1) = 1, g( 1) = 0. (21)

5 Flow of micropolar fluid between two orthogonally moving porous disks 967 Equations (18) and (19) can also be expressed as f + α 1 + K (3f + ηf ) c 1 g 2Rff = 0, (22) g + α 1 + K (3g + ηg ) + c 2 (f 2g) + c 3 (f g 2fg ) = 0, 2 (23) where R = ρav w /(µ + κ), c 1 = µ/(µ + κ), c 2 = κa 2 /γ, c 3 = av w j/(νγ). The boundary conditions are the same as (20) and (21). When α = 0, it is the case described by Anwar et al. [7]. 3 HAM solutions for velocity Here, we choose the initial guesses The auxiliary linear operators are which satisfy f 0 = 1 2 η η, g 0 = 0. (24) 1 (f) = f, 2 (f) = f, (25) { 1 (C 1 + C 2 η + C 3 η 2 + C 4 η 3 ) = 0, 2 (C 5 + C 6 η) = 0, (26) where C i (i = 1, 2,, 6) are constants. Making use of the above definitions, we construct the zero-order deformation problems as follows: (1 p) 1 ( ˆf f 0 ) = phℵ 1 ( ˆf, ĝ), (27) ˆf ( 1, p) = 0, ˆf( 1, p) = 1, ˆf (1, p) = 0, ˆf(1, p) = 1, (28) (1 p) 2 (ĝ g 0 ) = phℵ 2 ( ˆf, ĝ), (29) ĝ( 1, p) = 0, ĝ(1, p) = 0, (30) ℵ 1 ( ˆf, ĝ) = (1 + K) 4 ˆf(η, p) η 4 2Re ˆf(η, p) 3 ˆf(η, p) η 3 ℵ 2 ( ˆf, ĝ) = + 3α 2 ˆf(η, p) η 2 + αη 3 ˆf(η, p) η 3 K 2 ĝ(η, p) η 2, (31) ( 1 + K ) 2ĝ(η, p) ( ˆf(η, p) 2 η 2 + Re η ĝ(η, p) 2 ˆf(η, p) ĝ(η, p) ) η ĝ(η, p) + 3αĝ(η, p) + αη 2Kĝ(η, p) + K 2 ˆf(η, p) η η 2, (32)

6 968 Xin-hui SI, Lian-cun ZHENG, Xin-xin ZHANG, and Xin-yi SI where p [0, 1] is an embedding parameter, h is the auxiliary non-zero parameter. As p increases from 0 to 1, ˆf(η, p) and ĝ(η, p) vary from f0 (η), g 0 (η) to f(η), g(η), respectively. Using the Taylor s theorem, we can write ˆf(η, p) = f 0 (η) + ĝ(η, p) = g 0 (η) + m=1 m=1 f m (η)p m, f m (η) = 1 m ˆf(η, p) p=0 m! p m, (33) g m (η)p m, g m (η) = 1 m ĝ(η, p) p=0 m! p m. (34) The convergence of the two series is strongly dependent upon h. Assume that h is chosen so that the series (33) and (34) are convergent at p = 1. From (33) and (34), we have f = f 0 (η) + g = g 0 (η) + f m (η), (35) m=1 g m (η). (36) Differentiating (27) (29) m times with respect to p, then setting p = 0, and finally dividing them by m!, we obtain the following mth-order deformation problems: m=1 1 (f m (η) χ m f m 1 (η)) = hr f m(η), (37) f m ( 1) = 0, f m( 1) = 0, f m (1) = 0, f m(1) = 0, (38) R f m(η) = (1 + K)f (4) m 1 + α(3f m 1 + ηf m 1) m 1 Kg m 1 2Re f m k 1 f k, (39) k=0 2 (g m (η) χ m g m 1 (η)) = hr g m(η), (40) g m ( 1) = 0, g m (1) = 0, (41) ( R g m(η) = 1 + K ) g m 1 + α(3g m 1 + ηg 2 m 1) + K(f m 1 2g m 1 ) where m 1 + Re (f m k 1 g k 2f m k 1 g k ), (42) k=0 χ m = { 0, m 1, 1, m > 1. (43) The general solutions to (37) (40) are f m (η) = f m(η) + C 1 + C 2 η + C 3 η 2 + C 4 η 3, (44) g m (η) = g m (η) + C 5 + C 6 η, (45) where f m(η) and g m(η) denote the special solutions to (37) and (40), and the integral constants C i (i = 1, 2,, 6) are determined by employing the boundary conditions (38) and (41). In this way, it is easy to solve the linear non-homogeneous equations (37) and (40) by using Maple one after another in the order m = 1, 2, 3,.

7 Flow of micropolar fluid between two orthogonally moving porous disks Results and discussion As pointed out by Liao [26 27], the convergence of the two series (33) and (34) depends upon h. The value of h determines the convergence region and the rate of approximation for HAM. For this purpose, the h-curves are plotted in Fig. 2 for different approximations. Once the proper value of h is chosen, we can get the convergent HAM series solutions. Fig. 2 h-curves on 20th-order approximation for f (0) and g (0) as K = 0.2 and Re= 3 In theory [38], one also can define the exact square residual error for the kth-order approximation to prove the effectiveness of h-curve or to choose the best value of h as follows: k = 1 ( k (ℵ 1 f m (η), 0 m=0 which is shown in Fig.3. k m=0 ) 2 ( k g m (η) + ℵ2 m=0 f m (η), k m=0 ) 2 ) g m (η) dη, (46) Fig. 3 k for different h as K = 0.2 and Re= 3 Now, we give the graphical presentation of the streamwise and normal velocity profiles and the microrotation across the disks, and investigate the influence of some physics parameters. The influence of the permeability Reynolds number on the streamwise velocity is presented in Figs These figures show that, for different values of Re, the profile of the streamwise velocity is symmetric with respect to the central η = 0. For the streamwise velocity, the maximum

8 970 Xin-hui SI, Lian-cun ZHENG, Xin-xin ZHANG, and Xin-yi SI values decrease with the increase in Re. Figure6 depicts the behavior of the microrotation for different values of the permeability Reynolds number Re. As a general trend, there are two extreme points and one zero point between two disks. When α = 4, the magnitude of the extreme points is shifted towards the disks as the Reynolds number Re is increasing. However, when α = 2, the magnitude of the extreme points is shifted towards the central of two disks as the Reynolds number Re is increasing. Fig. 4 Characteristics of f and f for different Re as α = 4 and K = 0.2 Fig. 5 Characteristics of f and f for different Re as α = 2 and K = 0.2 Fig. 6 Characteristics of g for different Re The graphical presentation of the influence of negative α on velocity fields and microrotation fields are given in Figs. 7 9 for fixed K and Re. The normal velocity f(η) changes their concavity

9 Flow of micropolar fluid between two orthogonally moving porous disks 971 at the position which is at the central of the disks. The streamwise velocity is symmetric. The maximum of the streamwise velocity also lies at the central of the disks, which is an increasing function of α. However, for the normal velocity, there are no significant changes as Re = 5. For the microrotation g(η), the microrotation has opposite signs near the walls with one zero value and two extreme values across the disks. Opposite signs are due to the fact that the shear stresses at the two walls tend to rotate the fluid in opposite directions, and the point of zero microrotation marks the position across the walls where the effects of the opposite rotation balance each other. Fig. 7 Characteristics of f and f for different α as Re = 5 and K = 0.2 Fig. 8 Characteristics of f and f for different α as Re = 5 and K = 0.2 Fig. 9 Characteristics of g for different α

10 972 Xin-hui SI, Lian-cun ZHENG, Xin-xin ZHANG, and Xin-yi SI Figure 10 presents profiles of the microrotation to reflect the influence of variations in K for different Re and α on its behavior. The magnitude of the maximum point is an increasing function of K. For the case of Re = 5 and α = 2, the influence is larger than other cases. Fig. 10 Characteristics of g for different K 5 Conclusions The developmental characteristics of micropolar flows between orthogonally moving disks are studied. The problem of the symmetric flow with both stationary permeable disks occurs as a special case of the present problem corresponding to the parametric value α = 0. The influence of the permeation Reynolds number Re, the expansion ratio α, and the parameter K on the flow are discussed in detail. Acknowledgements The authors want to thank the anonymous reviewers for their valuable suggestions and comments. References [1] Connor, J. J., Boyd, J., and Avallone, E. A. Standard Handbook of Lubrication Engineering, McGraw-Hill, New York (1968) [2] Elcrat, A. R. On the radial flow of a viscous fluid between porous disks. Archive for Rational Mechanics and Analysis, 61, (1976) [3] Rasmussen, H. Steady viscous flow between two porous disks. Zeitschrift für Angewandte Mathematik und Physik, 21, (1970)

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