Fluid Mechanics
Volume 2, Issue 1, September 2016, Pages: 1-7

Axisymmetric Stagnation Flow of a Micropolar Fluid in a Moving Cylinder: An Analytical Solution

Abdul Rehman1, *, Saleem Iqbal1, Syed Mohsin Raza2

1Department of Mathematics, University of Balochistan, Quetta, Pakistan

2Department of Physics, University of Balochistan, Quetta, Pakistan

(A. Rehman)

*Corresponding author

Abdul Rehman, Saleem Iqbal, Syed Mosin Razan. Axisymmetric Stagnation Flow of a Micropolar Fluid in a Moving Cylinder: An Analytical Solution. Fluid Mechanics. Vol. 2, No. 1, 2016, pp. 1-7. doi: 10.11648/j.fm.20160201.11

Received: July 12, 2016; Accepted: July 22, 2016; Published: August 26, 2016

Abstract: In this paper, we have presented the axisymmetric stagnation flow of a micropolar fluid in a moving cylinder. The governing equations of motion, microrotation and energy are simplified with the help of suitable similarity transformations. System of six nonlinear coupled differential equations has been solved analytically with the help of strong analytical tool known as homotopy analysis method. The physical features of various parameters have been discussed through graphs.

Keywords: Series Solution, Axisymmetric Stagnation Flow, Micropolar Fluid, Moving Cylinder

Contents

1. Introduction

Numerous applications of stagnation flows in engineering and scientific interest have attracted the attention of number of researchers [1-5]. In some situations flow is stagnated by a solid wall, while in others a free stagnation point or line exist interior to the fluid domain [6]. The stagnation point flows can be viscous or inviscid, steady or unsteady, two dimensional or three dimensional, normal or oblique and forward or reverse. The stagnation flows were initiated by Hiemenz [7] and Homann [8]. Recently, Hong and Wang [9] have discussed the annular axisymmetric stagnation flow on a moving cylinder. According to Hong and Wang [9], in the previous literature the researchers have considered a stagnation flow originated from infinity. But there are certain situations in which finite geometry is more realistic and attractive for high speed and miniature rotating systems [10,11].

In the situations like polymeric fluids or certain naturally occurring fluids such as blood, the classical Navier Stokes theory does not hold [22]. Therefore, Erigen [23] has given the idea of micropolar fluid which describes both the effect of couple stresses and the microscopic effects arising from local structure and microrotation of the fluid elements. Also, the micropolar fluids consist of a suspension of small, rigid, cylindrical elements such as large dumbbell shaped molecules. Erigen [24] has also developed the theory of thermomicropolar fluids by extending the theory of micropolar fluids. Because of importance of this theory a large amount of literature on micropolar fluids with different geometries are now available. Few of them are cited in the Ref [25-27].

Motivated from the above highlights, the purpose of the present work is to extend the idea of Hong and Wang [9] for micropolar fluid. To the best of author's knowledge, not a single article is available in literature which discusses the axisymmetric stagnation flow of non-Newtonian fluid with a finite geometry. The problem has been first simplified with the help of suitable similarity transformations and then solved with the analytical technique known as homotopy analysis method (HAM), some relevant work on HAM are given in the Ref [28-34]. The convergence of the HAM solution has been discussed through -curves. A comparison of our HAM solution and previous numerical solutions for viscous fluid is also presented. At the end, the physical behavior of pertinent parameters is discussed through graphs. Few important works concerning fluid flow through cylindrical geometry are cited in [36-40].

2. Formulation

Let us consider an incompressible flow of a micropolar fluid between two cylinders. We are considering cylindrical geometry assuming 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 . The fluid is injected radially with velocity  from the outer cylinder towards the inner cylinder. The equations for micropolar fluid in the presence of heat transfer analysis are stated as

(1)

(2)

(3)

(4)

(5)

(6)

where  are the velocity components along the  directions,  is the angular microrotation momentum,  is the dynamic viscosity,  is the vertex viscosity,  is the density,  is the microrotation density,  is the micropolar constant,  is the specific heat at constant pressure,  is the temperature,  is the kinematic viscosity,  is the thermal conductivity and  is the pressure.

Defining the following similarity transformations and non-dimensional variables

(7)

(8)

(9)

With the help of these above transformations,  is identically satisfied and  to  take the following form

(10)

(11)

(12)

(13)

(14)

(15)

in which  is the cross-flow Reynolds number,  is the micropolar parameter,  and  are the micropolar coefficients and   is the Prandtl number.

The boundary conditions in nondimentional form are defined as

(16)

(17)

(18)

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

(19)

(20)

(21)

with the corresponding auxiliary linear operators

(22)

(23)

satisfying

(24)

(25)

(26)

where  are arbitrary constants. The zeroth-order deformation equations are defined as

(27)

(28)

(29)

(30)

(31)

(32)

where

(33)

(34)

(35)

(36)

(37)

(38)

The boundary conditions for the zeroth order system are

(39)

(40)

(41)

The  order deformation equations can be obtained by differentiating the zeroth-order deformation equations  and the boundary conditions , - times with respect to  then dividing by  and finally setting  we get

(42)

(43)

(44)

(45)

(46)

(47)

(48)

(49)

(50)

Further details of the HAM solution are presented in the next section.

4. Results and Discussion

The HAM solutions for the differential system are heavily dependent upon the selection of involved auxiliary parameters for the respective profiles. Figures 1 and  contain  curves for the convergence regions of different velocity profiles at the surface of the inner cylinder. Figure  shows the  curves for the linear velocity profiles  and  for specified values of the involved parameters. It is noticed from Figure  that the convergence region for  is the least. Figure  shows the convergence region for linear velocity profile  and angular velocity profiles  and  for presented values of the other parameters. From Figure  it is noted that the convergence region for angular velocity profiles is much larger than that for linear velocity profiles. From Figure  it is also observed that the suitable choice of auxiliary convergence parameter  for the nondimensional linear velocity profile  is  Figure  tweets the influence of Reynolds numbers  over the linear velocity and acceleration profiles  and  for specified values of the involved parameters. Figure  dictates that with increase in Reynolds numbers  the nondimensional linear velocity profile  increases, while  decreases, whereas the nondimensional acceleration profile  has shown dual behavior that is near the surface of inner cylinder the acceleration profile  increases,  has a turning point somewhere  and in the neighborhood of outer cylinder the acceleration profile decreases. Figure  predicts the influence of the micropolar parameter  over the velocity and acceleration profiles  and  It is observed from Figure  that with increase in  the nondimensional velocity profile  increases while  decreases, whereas the nondimensional acceleration profile has dual behavior that is  decreases near the surface of the inner cylinder while near the surface of the outer cylinder the nondimensional acceleration profile  increases. Figures  and  gives the behavior of linear velocity profiles  and  for different values of the micropolar parameter  and the Reynolds numbers  respectively. From these sketches it is evident that both velocity profiles  and  exhibits decreasing behavior with respect to the specified parameters. The influence of micropolar parameter  and micropolar coefficient  over the angular velocity profile  are presented in Figures  and  for the case of weak concentration with  From these plates it is observed that with respect to both micropolar parameter  and micropolar coefficient  the micropolar velocity profile  decreases. The effects of micropolar parameter  and micropolar coefficient  over the microrotation profile  are portrayed in Figures  and  respectively. It is seen from Figures  and  that with increase in micropolar parameter  the microrotaion profile  increases, while with increase in micropolar coefficient  the microrotaion profile  decreases. The influence of micropolar parameter  over micropolar velocites  and  for the case of strong concentration with  is presented in Figures  and  respectively. The observed behavior indicates that the micropolar velocity  has a sinusoidal behavior while  exhibits increasing influence. The influence of Prandtl numbers  and Reynolds numbers  over the temperature profile  is presented in Figures  and  From these figures it is observed that with increase in both Prandtl numbers  and Reynolds numbers  the temperature profile increases.

A comparison of our HAM solutions with the available numerical solutions in [9] without microrotation effects are shown in Table 1. It is seen that both solutions are almost identical. The value of skinfriction coefficient is presented in Table. 2. It is seen that with the increase in Re, the skinfriction coefficient decreases, however the magnitude of skinfriction increases with the increase in α.

Figure 1.  curves for velocity profiles  and .

Figure 2.  curves for velocity profile  and microrotation profiles  and .

Figure 3. Influence of  over the velocity profiles   and .

Figure 4. Influence of  over the velocity profiles   and .

Figure 5a. Influence of  over .

Figure 5b. Influence of  over .

Figure 6a. Influence of  over .

Figure 6b. Influence of  over .

Figure 7a. Influence of  over .

Figure 7b. Influence of  over .

Figure 8a. Influence of  over  for .

Figure 8b. Influence of  over  for .

Figure 9a. Influence of  over the temperature profile .

Figure 9b. Influence of  over the temperature profile .

Table 1. Comparison of boundary derivatives of present results with the available work of [9].

 Re\b 1.1 2 10 [9] Present [9] Present [9] Present f´´(1) 0.1 650.3526 650.3526 11.0010 11.0010 0.667 0.667 1 654.7679 654.7679 11.6772 11.6772 0.863 0.863 10 698.6176 698.6176 17.5348 17.5348 1.867 1.867 -f´´´(1) 0.1 13883 13883 36.1443 36.1443 0.9172 0.9172 1 14117 14117 41.0797 41.0797 1.3924 1.3924 10 16507 16507 93.5670 93.5670 5.2400 5.2400 -g´(1) 0.1 10.5382 10.5382 1.4963 1.4963 0.5082 0.5082 1 10.9489 10.9489 1.9309 1.9309 0.9040 0.9040 10 14.6586 14.6586 4.3856 4.3856 2.2168 2.2168 -h´(1) 0.1 10.5151 10.5151 1.5151 1.5151 0.6235 0.6235 1 10.6511 10.6511 1.6554 1.6554 0.7570 0.7570 10 12.0407 12.0407 3.0517 3.0517 1.5941 1.5941

Table 2. Variation of skin friction coefficient for different values of Re/α.

 K = 0, n = 0, ξ = 1 K = 1, n = 1/2, ξ = 1 K = 3, n = 1/2, ξ = 1 Re\α 0 1 2 0 1 2 0 1 2 0.1 353.767 412.605 389.131 432.517 -806.22 -2027.39 438.586 -2241.65 -2257.97 1 35.7919 43.1997 39.2168 44.6699 -1754.12 -3550.89 44.7221 -3167.30 -3154.96 5 7.52430 10.0520 9.37830 10.1557 -5764.42 -11538.4 9.70790 -6364.88 -6328.78 10 3.98710 8.35250 5.52620 5.77890 -13304.9 -26615.1 5.20750 -11903.3 -11820.3

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