Axisymmetric Stagnation Flow of a Micropolar Fluid in a Moving Cylinder: An Analytical Solution
Abdul Rehman^{1, *}, Saleem Iqbal^{1}, Syed Mohsin Raza^{2}
^{1}Department of Mathematics, University of Balochistan, Quetta, Pakistan
^{2}Department of Physics, University of Balochistan, Quetta, Pakistan
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To cite this article:
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
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 .
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 |
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 |
References