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Flow of micropolar fluid over an off centered rotating disk with modified Darcy's law
Propulsion and Power Research 2017;6(4):285–295
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ORIGINAL ARTICLE
Flow of micropolar fluid over an off centered rotating disk with modified Darcy's law N.A. Khana,n, S. Khana, A. Arab a
Department of Mathematics, University of Karachi, Karachi, Sindh 75270, Pakistan Department of Computer Science, Mohammad Ali Jinnah University, Karachi, Sindh 75400, Pakistan
b
Received 15 January 2016; accepted 10 January 2017 Available online 6 December 2017
KEYWORDS Micropolar fluid; Stagnation point; Modified Darcy's law; Porous medium
Abstract The problem of the steady, incompressible, three dimensional stagnation point flow of a micropolar fluid over an off centered infinite rotating disk in a porous medium is studied in this article. Injection/suction is applied uniformly throughout the surface of porous disk. The Darcy's resistance for the micropolar fluid is also formulated. The partial differential equations are converted into the set of ordinary differential equation by utilizing the suitable transformation. The system of equations is analytically solved by the means of a nonperturbative technique, homotopy analysis method (HAM). The influence of rotational parameter, material parameter, spin gradient viscosity parameter, micro-inertia density parameter, porosity parameter and suction/injection parameter on velocity functions is presented in graphical form and discussed in detail. Verification of the solutions is made by a numerical comparison with the previous study. & 2017 National Laboratory for Aeronautics and Astronautics. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction For a long time, the non-Newtonian fluids captivated the attention of the researchers. The great interest in non-Newtonian n
Corresponding author. Tel.: þ923333012008.
E-mail address: [email protected] (N.A. Khan). Peer review under responsibility of National Laboratory for Aeronautics and Astronautics, China.
fluids is because of their great commercial importance. In reality, many fluids like biological fluids, slurries, shampoo, yoghurt, tomato sauce, grease, cosmetic products, paints, lubricants, polymers, custard, blood, and several other fluids do not obey the linear stress-velocity gradient relationship, which is the Newtonian fluid theory. Numerous non-Newtonian fluid models were therefore proposed to explain the complex behavior. Usually, the stress constitutive relations of such models inherit complexities, which lead to highly nonlinear
2212-540X & 2017 National Laboratory for Aeronautics and Astronautics. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). https://doi.org/10.1016/j.jppr.2017.11.006
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Nomenclature x; y; z u; v; w r; θ p r˜ N I j κ d a M c1 c2
Cartesian coordinates Velocity components Cylindrical coordinates Pressure Darcy resistance Micro rotation body couple per unit mass micro-inertia vortex viscosity Distance between the axis of flow and disk axis strength of the stagnation flow Torque experienced by disk material parameter spin gradient viscosity parameter
equations of motion with many terms. Most of the fluids used in the industrial applications are non-Newtonian in nature, especially, in polymer processing and chemical engineering processes etc. [1–3]. Among all the non-Newtonian fluids, the micropolar fluid has received special attention due to its additional angular momentum equation. Micropolar fluids are physically based on microstructure; these fluids may represent the fluids consisting of rigid, randomly oriented or spherical particles suspended in a viscous medium, where the deformation of fluid particles is ignored [4]. This fluid model was proposed by Eringen [5,6]. Many researchers focused on this fluid and studied it in different ways. Khan et al. [7] calculated the analytical solution of the creeping flow of two dimensional, unsteady micropolar fluid in Cartesian coordinates. Nazir and Shafique [8] investigated the numerical solution of the steady micropolar fluid flow caused by stretching cylinder, by utilizing the SOR method. Ahmed et al. [9] studied the three dimensional laminar boundary layer flow of a micropolar fluid due to a stretching surface, the analysis and discussion of stretching and material parameter have also been made. ElKabeir et al. [10] analyzed the heat and mass transfer flow of a micropolar fluid, which is passing through a permeable continuously moving surface, two different cases: plane surface moving parallel to the free stream and surface moving opposite to the free stream are considered. Rahman and AlLawatia [11] founded the influence of higher order chemical reaction and heat transfer flow of micropolar fluid past a permeable stretching sheet in a porous medium, the Darcy parameter shows the decreasing rate of the surface mass. Turkyilmazoglu [12] studied the flow of a micropolar fluid due to a stretching sheet with heat transfer. Nadeem et al. [13] examined the unsteady two dimensional magnetohydrodynamics (MHD) boundary layer flow of incompressible micropolar fluid through a porous medium near a forward stagnation point of a plane wall. Ashraf and Batool [14] numerically studied the MHD and heat transfer flow of the axisymmetric micropolar fluid passed through the stretching disk.
c3 e S k1
micro-inertia density parameter porosity parameter Uniform injection parameter permeability
Greek letters λ1 ; α 1 ; β 1 μ γ1 ω ρ τ α ϕ
material constant of micropolar fluid Viscosity of micropolar fluid spin gradient viscosity angular velocity Density extra stress tensor rotational parameter porosity of the porous medium
In the history of fluid mechanics, the most significant article in the literature of fluid dynamics is Von Karman [15]. He was the first one who calculated the solution of the laminar flow of a viscous fluid over a rotating disk. This paper captivated the attention of many researchers due to its several technical and industrial applications. Many researchers reinvestigated this article in various dynamics with MHD effect, porous medium, chemical reactions, thermal effect, uniform suction/injection effects, etc. [16]. The fluid passing through porous media has many applications, such as aerogels, injection of mud's, porous rocks, alloys, slurries or cement grouts to reinforce soils, foams and foamed solids, micro emulsions, polymer blends, and most important application is drilling fluids through injection in rocks for the reinforcement of the wells and also for enhancing oil recovery, etc. [17,18]. In 1856, Darcy [19] proposed a Darcy law in which he established the relation between the pressure drop and flow rate relation in a porous medium. Tan and Masuoka [20,21] examined the Stokes’ first problem for the second grade and Oldroyd-B fluids by using the modified Darcy's law. Hayat et al. [22] re-examined the Stoke's first problem for non-Newtonian forth order fluid by using the modified Darcy's law in a porous medium. To the best of authors' knowledge, the micropolar fluid model has never been investigated for an off centered rotating disk in a porous medium with Darcy resistance effect. In this article, the problem of Wang [23] is extended to study the influence of micropolar fluid parameters (material parameter, spin gradient viscosity parameter, and micro-inertia density parameter) based on the vortex viscosity, spin gradient viscosity, the material constant, and microinertia of the micropolar fluid over an off centered rotating porous disk with modified Darcy law, which has not been investigated previously. To attain the analytical solution of the current problem, the similarity transformations from [23] are used to reduce the partial differential equations of motion into non-linear ordinary differential equations (ODEs). The ODEs are then solved by a well-known non perturbative
Flow of micropolar fluid over an off centered rotating disk with modified Darcy's law
287
technique, homotopy analysis method (HAM), proposed by Liao [24]. The influence of the rotational parameter, suction/ injection parameter, porosity parameter, and micropolar fluid parameters has been examined and discussed in detail by plotting the velocity curves in radial, azimuthal, induced, and angular directions. The structure of this paper is based on 5 sections. Section 1 is dedicated to the introduction of the study, Section 2 contains the basic equations of the model, in Section 3 the mathematical model of the problem is formulated, Section 4 shows the HAM procedure, and the effect of the pertinent parameters on velocity profile is offered in Section 5. In the last Section 6, the summary of the study is exhibited.
2. Basic equations Figure 1
In a porous medium, the governing equations for micropolar fluid flow are defined as: Continuity ∂ρ þ ð∇⋅ρV Þ ¼ 0 ∂t Momentum
ð1Þ
ðλ þ 2μ þ κÞ∇ð∇⋅V Þ − ðμ þ κÞ ∇ð∇⋅V Þ − ∇ V −κ∇ N−∇p−˜r ¼ ρV̇ 2
Angular momentum α1 þ β1 þ γ 1 ∇ð∇⋅N Þ − γ 1 ∇ð∇⋅N Þ − ∇2 N þ κ∇ V−2κN þ ρI ¼ ρjṄ
ð2Þ
ð3Þ
where, V is the velocity vector, ρ is the density, p is the pressure, r˜ is the Darcy resistance for micropolar fluid, N is the micro-rotation, I is the body couple per unit mass, j is the micro-inertia, ∇ is the gradient operator, μ is the viscosity of the micropolar fluid, κ is the vortex viscosity, γ 1 is the spin gradient viscosity, operator λ1 ; α1 and β1 is the material constant of micropolar fluid.
3. Mathematical model Consider the steady, incompressible, stagnation point flow of a micropolar fluid over an off-centered rotating disk in a porous medium with Darcy resistance. The flow axes of the disk are parallel to z axis at a distance d. The disk is rotating with constant angular velocity ω. Let u; v; and w be the velocity components along the direction of Cartesian coordinates x; y; and z, respectively. In Figure 1, the physical geometry of the model is presented. Eqs. (1)–(3) are reduced into the following partial differential equations governing the three dimensional micropolar flow over an off centered rotating disk in a porous medium. ∂u ∂v ∂w þ þ ¼0 ∂x ∂y ∂z
ð4Þ
∂u ∂u 1 ∂p u ∂u ∂x þ v ∂y þ w ∂z ¼ − ρ ∂x þ ϕ μþκ 2 þ ρκ ∂N − ∂z k1 ρ u ∂v ∂v 1 ∂p u ∂v ∂x þ v ∂y þ w ∂z ¼ − ρ ∂y þ ϕ μþκ 1 − ρκ ∂N ∂z − k 1 ρ v
Physical model.
∂w ∂w 1 ∂p u ∂w ∂x þ v ∂y þ w ∂z ¼ − ρ ∂z þ ∂N 1 2 þ ρκ ∂N ∂x þ ∂y
μþκ ρ
μþκ ρ
μþκ ρ
∂2 u ∂x2
þ ∂∂yu2 þ ∂∂zu2
∂2 v ∂x2
þ ∂∂yv2 þ ∂∂z2v
2
∂2 w ∂x2
2
2
2
ð5Þ
ð6Þ
þ ∂∂yw2 þ ∂∂zw2 2
2
2 ∂N 1 ∂N 1 ∂ N1 ∂2 N 1 ∂2 N 1 1 þ v þ w þ þ ρj u ∂N ¼ γ 1 ∂x2 ∂x ∂y ∂z ∂y2 ∂z2 ∂v −κ 2N 1 þ ∂w ∂y − ∂z 2 2 2 ∂N 2 ∂N 2 2 ρj u ∂N ¼ γ 1 ∂∂xN22 þ ∂∂yN22 þ ∂∂zN22 ∂x þ v ∂y þ w ∂z ∂w −κ 2N 2 þ ∂u − ∂z ∂x
ð7Þ
ð8Þ
ð9Þ
where u; v; and w are the velocity components along the x; y; and z axes, ϕ is the porosity of the porous medium, k1 is the permeability. N 1 and N 2 are the components of microrotation vector N normal to the planes xz and yz respectively. We shall solve Eqs. (4)–(9) subjected to the following boundary conditions. u ¼ −ωy;
v ¼ ωðx−d Þ;
N 1 ¼ n ∂v ∂z ;
N 2 ¼ −n ∂u ∂z
u ¼ ax; N 1 →0;
v ¼ ay; N 2 →0 at
w ¼ w0 ; at z→0
ð10Þ
w ¼ −2az; z→∞
ð11Þ
Where a represents the strength of the stagnation flow, n is a boundary constant such that 0 r nr 1. Guram and Smith [25] studied the case n ¼ 0; which they named as strong
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concentration; the microelements close to a wall are unable to rotate, near the wall N ¼ 0 [26]. The weak concentration is observed in case n ¼ 12 due to the disappearance of the antisymmetric part of the stress tensor [27]. The case n ¼ 1 was proposed by Peddieson [28] in which he studied the turbulent boundary layer flows. By using the Wang [23] similarity transformations, u ¼ axf ′ðξÞ − ωygðξÞ þ dωkðξÞ; v ¼ ayf ′ðξÞ þ ωxgðξÞ þ dωhðξÞ; pffiffi pffiffiffiffiffi w ¼ −2 aνf ðξÞ; ξ ¼ z aν; pffiffi pffiffi N 2 ¼ ax aνF ðξÞ N 1 ¼ ay aνGðξÞ;
ð12Þ
Using the above similarity transformations in Eqs. (5)–(8), the equation of continuity is satisfied whereas the governing partial differential equations are reduced to the following form: ð1 þ c1 Þðf ′′′−ef ′Þ þ c1 F′ þ 2f f ′′ −f ′2 þ α2 g2 þ 1 ¼ 0;
ð13Þ
ð1 þ c1 Þ ðg′′−egÞ þ 2g′f −2gf ′ ¼ 0;
ð14Þ
ð1 þ c1 Þðk′′−ek Þ þ αgh −f ′ k þ 2f k′ ¼ 0;
ð15Þ
ð1 þ c1 Þðh′′−ehÞ−αgk −f ′ h þ 2f h′ ¼ 0;
ð16Þ
F′′−2c2 F−c3 Ff ′ þ 2c3 F′f −c2 f ′′ ¼ 0;
ð17Þ
G′′−2c2 G−c3 Gf ′ þ 2c3 G′f þ c2 f ′′ ¼ 0;
ð18Þ
and the boundary conditions of Eqs. (10)–(11) become, f ð0Þ ¼ S;
f ′ð0Þ ¼ 0;
gð0Þ ¼ 1;
F ð0Þ ¼ −f ′′ð0Þ; Gð0Þ ¼ f ′′ð0Þ f ′ð∞Þ ¼ 1; gð∞Þ ¼ 0; kð∞Þ ¼ 0; F ð∞Þ ¼ 0;
kð0Þ ¼ 0;
hð0Þ ¼ 1;
τyz ¼ ρ
μþκ ρ
2
∂v κ ∂z − ρ N 1
z¼0
yðf ′′ð0Þ þ c1 f ′′ð0Þ−c1 Gð0ÞÞ
pffiffiffiffiffi6 τyz ¼ ρa νa4 þαxðg′ð0Þ þ c1 g′ð0ÞÞ þdαðh′ð0Þ þ c1 h′ð0ÞÞ
3
ð22Þ
7 5
The shear stress at the center is zero, the shear stress on the surface of the disk can be obtained by setting Eqs. (21) and (22) to zero and solving for ðx; yÞ. The torque experienced by the disk of radius R is given by: 2 M ¼ ∫R0 ∫2π ð23Þ 0 τyz cos θ − τxz sin θ r dθdr; where, ðr; θÞ are the cylindrical coordinates, τ is the extra stress tensor, τxz and τyz are the xz and yz components of the extra stress tensor. Since, x ¼ r cos θ−b, the torque can be found at: π pffiffiffiffiffi ð24Þ M ¼ a aναR4 ρðg′ð0Þ þ c1 g′ð0ÞÞ; 2 Which is unaffected by the non-aligned disk and flow axis.
4. Analytical approximations by means of HAM The auxiliary linear operators, L1 ½f and L2 ½g; k; h; F; G are selected for the Eqs. (14)–(19) as: L1 ½ f ¼
∂3 ∂ − ; ∂ξ3 ∂ξ
L2 ½g; h; k; F; G ¼
∂2 −1; ∂ξ2
ð25Þ
Also, the initial guesses satisfying the initial conditions in Eq. (19) are obtained as:
hð∞Þ ¼ 0;
Gð∞Þ ¼ 0: ð19Þ ω a
where, α ¼ is the non-dimensional rotational parameter, c1 ¼ μκ is the material parameter, c2 ¼ aγκν is the spin gradient 1
viscosity parameter, c3 ¼ is the micro-inertia density 0ffi pffiffiffi parameter, e ¼ akνφ1 is the porosity parameter, and S ¼ 2−w aν is the uniform injection parameter obtained from Eqs. (10) and (12). Here, S represents a uniform injection S40. The pressure p can be recovered from third momentum Eq. (7) as:
2 1 2 2 w μþκ 2 p ¼ p0 − ρa x þ y −ρ − wz ð20Þ 2 ρ 2 jνρ γ1
where, p0 is the pressure at the origin. The shear stress on the off centered rotating disk of a micropolar fluid in porous medium is given by: ∂u κ τxz ¼ ρ μþκ þ N 2 ρ ∂z ρ z¼0 2 3 xðf ′′ð0Þ þ c1 f ′′ð0Þ þ c1 F ð0ÞÞ ð21Þ pffiffiffiffiffi6 7 τxz ¼ ρa νa4 −αyðg′ð0Þ þ c1 g′ð0ÞÞ 5 þdαðk′ð0Þ þ c1 k′ð0ÞÞ
f 0 ðξÞ ¼ ξ þ S − 1 þ e−ξ ; g0 ðξÞ ¼ e−ξ ; F 0 ¼ −f ″0 ð0Þe−ξ ; G0 ¼ f ″0 ð0Þe−ξ
k 0 ðξÞ ¼ 0;
h0 ðξÞ ¼ e−ξ ;
ð26Þ satisfying the following properties
L1 C 1 þ C 2 eξ þ C 3 e−ξ ¼ 0;
L2 C 4 eξ þ C 5 e−ξ ¼ 0;
ð27Þ
Where, C 1 ; C 2 ; :::; C 5 are arbitrary constants. The zeroth order deformation equations can be constructed as: " # h i f^ ðξ; qÞ; g^ ðξ; qÞ; ^ ð1−qÞL1 f ðξ; qÞ − f 0 ðξÞ ¼ qℏf N 1 ð28Þ F^ ðξ; qÞ h i ð1−qÞL2 g^ ðξ; qÞ − g0 ðξÞ ¼ qℏg N 2 f^ ðξ; qÞ; g^ ðξ; qÞ " i f^ ðξ; qÞ; ^ ð1−qÞL2 kðξ; qÞ − k 0 ðξÞ ¼ qℏk N 3 ^ kðξ; qÞ; h
"
h i f^ ðξ; qÞ; ð1−qÞL2 h^ ðξ; qÞ − h0 ðξÞ ¼ qℏh N 4 ^ k ðξ; qÞ;
g^ ðξ; qÞ; h^ ðξ; qÞ g^ ðξ; qÞ; h^ ðξ; qÞ
ð29Þ #
#
ð30Þ
ð31Þ
Flow of micropolar fluid over an off centered rotating disk with modified Darcy's law
h i ð1−qÞL2 F^ ðξ; qÞ − F 0 ðξÞ ¼ qℏF N 5 f^ ðξ; qÞ; F^ ðξ; qÞ h i ^ ðξ; qÞ − G0 ðξÞ ¼ qℏG N 6 f^ ðξ; qÞ; G ^ ðξ; qÞ ð1−qÞL2 G
ð32Þ
L2 gm ðξÞ − χ m gm−1 ðξÞ ¼ ℏg R2;m ðξÞ
ð33Þ
L2 km ðξÞ − χ m km−1 ðξÞ ¼ ℏk R3;m ðξÞ L2 hm ðξÞ − χ m hm−1 ðξÞ ¼ ℏh R4;m ðξÞ L2 F m ðξÞ − χ m F m−1 ðξÞ ¼ ℏF R5;m ðξÞ
ð44Þ
L2 Gm ðξÞ − χ m Gm−1 ðξÞ ¼ ℏG R6;m ðξÞ
ð46Þ
where
h i 3^ ^ Þ N 1 f^ ðξ; qÞ; g^ ðξ; qÞ; F^ ðξ; qÞ ¼ ð1 þ c1 Þ ∂ f∂ξðξ;q − e ∂f ð∂ξξ;qÞ 3
2^ ^ ðξ;qÞ Þ þ 2f^ ðξ; qÞ ∂ f∂ξðξ;q þc1 ∂F∂ξ 2 ^ 2 − ∂f ð∂ξξ;qÞ þ α2 ðg^ ðξ; qÞÞ2 þ 1
ð34Þ
^ ∂^gðξ;qÞ gðξ; qÞ ∂f ð∂ξξ;qÞ ∂ξ −2^
−
^ k^ ðξ; qÞ ∂f ð∂ξξ;qÞ
ð35Þ
þ αh^ ðξ; qÞ^gðξ; qÞ
ð36Þ h i 2^ Þ N 4 f^ ðξ; qÞ; g^ ðξ; qÞ; k^ ðξ; qÞ; h^ ðξ; qÞ ¼ ð1 þ c1 Þ ∂ h∂ξðξ;q − eh^ ðξ; qÞ 2 ^ ðξ;qÞ ^ þ2f^ ðξ; qÞ ∂h∂ξ − h^ ðξ; qÞ ∂f ð∂ξξ;qÞ −αk^ ðξ; qÞ^gðξ; qÞ
∂2 F^ ðξ;qÞ ∂ξ2
^ − 2c2 F^ ðξ; qÞ − c3 F^ ðξ; qÞ ∂f ð∂ξξ;qÞ
2^ ^ ðξ;qÞ Þ þ2c3 f^ ðξ; qÞ ∂F∂ξ − c2 ∂ f∂ξðξ;q 2
ð38Þ h i ^ ðξ; qÞ ¼ N 6 f^ ðξ; qÞ; G
km ð∞Þ ¼ 0;
hm ð∞Þ ¼ 0;
ð45Þ
and χm ¼
(
0; 1;
mr1 mZ2
ð48Þ
R1;m ðξÞ ¼ ð1 þ c1 Þ f ′″m−1 ðξÞ−ef ′m−1 ðξÞ þ c1 F ′m−1 ðξÞ m−1
m−1
þ α2 ∑ gi ðξÞgm−1−i ðξÞ þ 2 ∑ f i ðξÞf ″m−1−i ðξÞ i¼0
i¼0
− ∑ f ′i ðξÞf ′m−1−i ðξÞ þ 1−χ m ; i¼0
R2;m ðξÞ ¼ ð1 þ c1 Þ g″m−1 ðξÞ − egm−1 ðξÞ
ð49Þ
m−1
m−1
i¼0
i¼0
þ 2 ∑ f i ðξÞg′m−1−i ðξÞ − 2 ∑ gi ðξÞf ′m−1−i ðξÞ;
ð50Þ
i¼0
2^ ^ ðξ;qÞ Þ þ2c3 f^ ðξ; qÞ ∂G∂ξ þ c2 ∂ f∂ξðξ;q 2
ð39Þ in which, q∈½0; 1 is the embedding parameter and ℏ is the auxiliary non-zero parameter. By Taylor's theorem, ∞
f^ ðξ; qÞ ¼ f 0 ðξÞ þ ∑ f m ðξÞqm ;
m−1
m−1
i¼0
i¼0
− ∑ ki ðξÞf ′m−1−i ðξÞ þ α ∑ hi ðξÞgm−1−i ðξÞ; ð51Þ m−1 R4;m ðξÞ ¼ ð1 þ c1 Þ h″m−1 ðξÞ − ehm−1 ðξÞ þ 2 ∑ f i ðξÞh′m−1−i ðξÞ i¼0
m¼1 ∞
m−1
m1
i¼0
i¼0
− ∑ hi ðξÞf ′m−1−i ðξÞ − α ∑ ki ðξÞgm−1−i ðξÞ;
g^ ðξ; qÞ ¼ g0 ðξÞ þ ∑ gm ðξÞqm ; m¼1 ∞
ð52Þ
^ qÞ ¼ k 0 ðξÞ þ ∑ km ðξÞqm ; kðξ; m¼1 ∞
Gm ð∞Þ ¼ 0
m−1 R3;m ðξÞ ¼ ð1 þ c1 Þ k″m−1 ðξÞ − ek m−1 ðξÞ þ 2 ∑ f i ðξÞk′m−1−i ðξÞ
^ ðξ;qÞ ∂2 G ^ ðξ; qÞ−c3 G ^ ðξ; qÞ ∂f^ ðξ;qÞ −2c2 G ∂ξ ∂ξ2
^ qÞ ¼ h0 ðξÞ þ ∑ hm ðξÞqm ; hðξ;
F m ð∞Þ ¼ 0;
ð47Þ
m−1
ð37Þ h i N 5 f^ ðξ; qÞ; F^ ðξ; qÞ ¼
ð43Þ
f m ð0Þ ¼ 0; f ′m ð0Þ ¼ 0; gm ð0Þ ¼ 0; km ð0Þ ¼ 0; hm ð0Þ ¼ 0; ″ F m ð0Þ ¼ −f m ð0Þ; Gm ð0Þ ¼ f ″m ð0Þf ′m ð∞Þ ¼ 0; gm ð∞Þ ¼ 0;
2^ h i Þ N 3 f^ ðξ; qÞ; g^ ðξ; qÞ; k^ ðξ; qÞ; h^ ðξ; qÞ ¼ ð1 þ c1 Þ ∂ k∂ξðη;q − ek^ ðξ; qÞ 2 ^ ðξ;qÞ þ2f^ ðξ; qÞ ∂k∂ξ
ð42Þ
with the following boundary conditions
2 h i ^ ðξ;qÞ N 2 f^ ðξ; qÞ; g^ ðξ; qÞ ¼ ð1 þ c1 Þ ∂ g∂ξ − e^gðξ; qÞ 2 þ2f^ ðξ; qÞ
289
ð40Þ
m¼1 ∞
^ qÞ ¼ F 0 ðξÞ þ ∑ F m ðξÞqm ; Fðξ; m¼1 ∞
^ qÞ ¼ G0 ðξÞ þ ∑ Gm ðξÞqm ; Gðξ; m¼1
The general HAM equations for mth order can be given by: L1 f m ðξÞ − χ m f m−1 ðξÞ ¼ ℏf R1;m ðξÞ ð41Þ
R5;m ðξÞ ¼ F ″m−1 ðξÞ − 2c2 F m−1 ðξÞ − c2 f ″m−1 ðξÞ m−1
m−1
i¼0
i¼0
−c3 ∑ F i ðξÞf ′m−1−i ðξÞ þ 2c3 ∑ f i ðξÞF ′m−1−i ðξÞ; ð53Þ
290 Table 1
N.A. Khan et al. Comparision of skin-friction coefficient ðf ′′ð0Þ; g′ð0Þ; k′ð0Þ and h′ð0ÞÞ with Wang [23].
α
0
0:5
1
2
3
f ′′ð0Þ Wang [23] HAM
1.31194 1.30439
1.3787 1.37001
1.5739 1.56213
2.2951 2.27341
3.3657 3.36438
g′ð0Þ Wang [23] HAM
−1.07467 −1.07071
−1.0839 −1.07949
−1.1100 −1.10451
−1.1968 −1.18571
−1.3055 −1.26544
k′ð0Þ Wang [23] HAM
0 0
0.1380 0.136905
0.2700 0.222618
0.5040 0.54914
0.6972 0.847817
h′ð0Þ Wang [23] HAM
−0.9387 −0.9343
−0.9495 −0.944477
−0.9787 −0.971984
–1.0787 -1.04035
−1.2003 −1.2015
R6;m ðξÞ ¼ G″m−1 ðξÞ − 2c2 Gm−1 ðξÞ þ c2 f ″m−1 ðξÞ m−1
m−1
i¼0
i¼0
−c3 ∑ Gi ðξÞf ′m−1−i ðξÞ þ 2c3 ∑ f i ðξÞG′m−1−i ðξÞ; ð54Þ According to the above defined method, the linear Eqs. (34)–(39) with the boundary conditions in Eq. (40) in the order m ¼ 1; 2; 3; :::, can be solved easily by means of computational software Mathematica 10. Figure 2 Convergence region of ℏf .
5. Results and discussion In this section the findings of the study are discussed and presented in graphical form. Also, the comparison has been made in Table 1 as a special case for Newtonian fluid studied by Wang [23]. Table 1 shows the promising agreement between the present and previous solution, which shows the validity of the current analytical solution. The results of series solution of Eq. (40) are computed at a valid convergence range of auxiliary parameters −1:0oℏf o0:2; −1:0oℏg o0:5; −1:6oℏk o0:6; −1:0oℏh o0:4; −1:0oℏF o0:6; and −1:0oℏG o0:6 as presented in Figures 2–7. The influence of rotational parameter α on velocity profiles is displayed in Figures 8–11. Figure 8 shows the exponential rise in the radial function f ′ðξÞ near the disk due to the presence of the centrifugal force. Radial velocity is reached at its maximum point in the neighborhood of ξ ¼ 0:7 and after reaching its maximum point, it starts converging to unity as ξ→∞ far from the disk. Figure 9 demonstrates the effect of α on azimuthal velocity profile gðξÞ; as the rotation of the disk increases, it shows the negative impact on gðξÞ and starts showing the decrease in boundary layer thickness near the disk, later on away from the disk, it starts converging to zero. The influence of rotational parameter on the induced velocity profile kðξÞ expresses the rapid increase near the disk at ξ ¼ 0:6 (see
Figure 3 Convergence region of ℏg .
Figure 4 Convergence region of ℏk .
Flow of micropolar fluid over an off centered rotating disk with modified Darcy's law
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Figure 10). The effect of rotational parameter of another induced velocity function hðξÞ is observed in Figure 11, which illustrates that the rotation of the disk distracts the motion of fluid particles and shows the decreasing effect in
Figure 9 Effect of rotational parameter α on velocity function gðξÞ at c1 ¼ 0:1; c2 ¼ 0:1; c3 ¼ 0:1; e ¼ 0:1; S ¼ 0:1.
Figure 5 Convergence region of ℏh .
Figure 10 Effect of rotational parameter α on velocity function k ðξÞ at c1 ¼ 0:1; c2 ¼ 0:1; c3 ¼ 0:1; e ¼ 0:1; S ¼ 0:1. Figure 6 Convergence region of ℏF .
Figure 7 Convergence region of ℏG .
Figure 8 Effect of rotational parameter α on velocity function f ′ðξÞ at c1 ¼ 0:1; c2 ¼ 0:1; c3 ¼ 0:1; e ¼ 0:1; S ¼ 0:1.
Figure 11 Effect of rotational parameter α on velocity function hðξÞ at c1 ¼ 0:1; c2 ¼ 0:1; c3 ¼ 0:1; e ¼ 0:1; S ¼ 0:1.
Figure 12 Effect of material parameter c1 on velocity function f ′ðξÞ at α ¼ 0:1; c2 ¼ 0:1; c3 ¼ 0:1; e ¼ 0:1; S ¼ 0:1.
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Figure 13 Effect of material parameter c1 on velocity function gðξÞ at α ¼ 0:1; c2 ¼ 0:1; c3 ¼ 0:1; e ¼ 0:1; S ¼ 0:1.
Figure 14 Effect of material parameter c1 on velocity function k ðξÞ at α ¼ 0:1; c2 ¼ 0:1; c3 ¼ 0:1; e ¼ 0:1; S ¼ 0:1.
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Figure 17 Effect of spin gradient viscosity parameter c2 on velocity function GðξÞ at α ¼ 0:1; c1 ¼ 1; c3 ¼ 0:1; e ¼ 0:1; S ¼ 0:1.
Figure 18 Effect of microinertia density parameter c3 on velocity function F ðξÞ at α ¼ 0:1; c1 ¼ 1; c2 ¼ 1; e ¼ 0:1; S ¼ 0:1.
Figure 19 Effect of microinertia density parameter c3 on velocity function GðξÞ at α ¼ 0:1; c1 ¼ 1; c2 ¼ 1; e ¼ 0:1; S ¼ 0:1. Figure 15 Effect of material parameter c1 on velocity function hðξÞ at α ¼ 0:1; c2 ¼ 0:1; c3 ¼ 0:1; e ¼ 0:1; S ¼ 0:1.
Figure 16 Effect of spin gradient viscosity parameter c2 on velocity function F ðξÞ at α ¼ 0:1; c1 ¼ 1; c3 ¼ 0:1; e ¼ 0:1; S ¼ 0:1.
the induced direction, which is the same pattern previously observed in Figure 9. The effects of material parameter on velocity functions are presented in Figures 12–15. Figure 12 demonstrates the impact of vortex viscosity parameter c1 on radial function f ′ðξÞ. The increasing values of viscosity parameter cause distortion in radial function and this distortion don’t allow a radial function to converge to unity far from the disk. The increasing values of material parameter c1 cause acceleration in the boundary layer thickness of azimuthal function gðξÞ in the azimuthal direction (See Figure 13). The effect of fluid parameter c1 on induced velocity function kðξÞ in Figure 14 shows the decreasing-increasing behavior near the disk, but as it starts moving away from the disk in induced direction, it converges to zero in the neighborhood of ξ ¼ 5. Figure 15 demonstrates the same increasing pattern of micropolar fluid
Flow of micropolar fluid over an off centered rotating disk with modified Darcy's law
parameter c1 in induced velocity function hðξÞ as we have observed in azimuthal component gðξÞ. The impact of spin gradient viscosity parameter c2 on microrotation profiles F ðξÞ and GðξÞ is illustrated in Figures 16 and 17. In Figure 16, the influence of spin gradient viscosity parameter c2 on the microrotation function F ðξÞ shows acceleration, whereas, another microrotation function GðξÞ expresses the opposite deceleration behavior. The increasing values of c2 decreased the spin gradient viscosity of the fluid, which cause distortion in the movement of fluid particles in the microrotation profile GðξÞ. The impact of the micro-inertia density parameter of micropolar fluid c3 is demonstrated in Figures 18 and 19. The microrotation profiles F ðξÞ and GðξÞ manifested the same pattern as we have seen in Figures 16 and 17. The profile of microrotation F ðξÞ is antisymmetric, and shows that it is an increasing
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Figure 23 Effect of porosity parameter e on velocity function hðξÞ at α ¼ 1; c1 ¼ 0:01; c2 ¼ 0:01; c3 ¼ 0:01; S ¼ 0:01.
Figure 24 Effect of suction/injection parameter S on velocity function f ′ðξÞ at α ¼ 0:1; c1 ¼ 0:1; c2 ¼ 0:1; c3 ¼ 0:1; e ¼ 0:1. Figure 20 Effect of porosity parameter e on velocity function f ′ðξÞ at α ¼ 1; c1 ¼ 0:01; c2 ¼ 0:01; c3 ¼ 0:01; S ¼ 0:01.
Figure 21 Effect of porosity parameter e on velocity function gðξÞ at α ¼ 1; c1 ¼ 0:01; c2 ¼ 0:01; c3 ¼ 0:01; S ¼ 0:01.
Figure 22 Effect of porosity parameter e on velocity function k ðξÞ at α ¼ 1; c1 ¼ 0:01; c2 ¼ 0:01; c3 ¼ 0:01; S ¼ 0:01.
Figure 25 Effect of suction/injection parameter S on velocity function gðξÞ at α ¼ 0:1; c1 ¼ 0:1; c2 ¼ 0:1; c3 ¼ 0:1; e ¼ 0:1.
Figure 26 Effect of suction/injection parameter S on velocity function k ðξÞ at α ¼ 0:1; c1 ¼ 0:1; c2 ¼ 0:1; c3 ¼ 0:1; e ¼ 0:1.
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Figure 27 Effect of suction/injection parameter S on velocity function hðξÞ at α ¼ 0:1; c1 ¼ 0:1; c2 ¼ 0:1; c3 ¼ 0:1; e ¼ 0:1.
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support the micropolar fluid particles to move in induced direction due to which, it shows decreasing effect on velocity profile kðξÞ, whereas, another induced function hðξÞ present the decreasing-increasing effect of porosity parameter close to the disk as shown in Figure 23. Figures 24–29 present the influence of the injection parameter S on the velocity functions. Figure 24 shows the increasing function of radial velocity f ′ðξÞ, injection parameter S helps the fluid particles to move in the radial direction. The effect of parameter S on azimuthal velocity gðξÞ is displayed in Figure 25. For the injection S40, the boundary layer is increasingly blown away from the disk to form an interlayer between the injection and the outer flow regions. From Figures 26 and 27, it is noticed that the injection in porous disk causes the disturbance in the flow motion due to which, it is observed that k ðξÞ and hðξÞ are the decreasing functions of the parameter. The impact of the parameter S on the microrotation functions F ðξÞ and GðξÞ is presented in Figures 28 and 29. The microrotation function F ðξÞ is antisymmetric and shows a decreasing behavior as the injection on the disk increases. The microrotation function GðξÞ is symmetric and shows an increasing behavior as the injection on the disk increases near the disk (see Figure 29).
Figure 28 Effect of suction/injection parameter S on velocity function F ðξÞ at α ¼ 0:1; c1 ¼ 0:1; c2 ¼ 0:1; c3 ¼ 0:1; e ¼ 0:1.
6. Conclusions
Figure 29 Effect of suction/injection parameter S on velocity function GðξÞ at α ¼ 0:1; c1 ¼ 0:1; c2 ¼ 0:1; c3 ¼ 0:1; e ¼ 0:1.
The present study focuses on the analytical solution of steady, incompressible, stagnation point flow of a micropolar fluid over an off centered rotating disk in a porous medium with Darcy's resistance. The convergence region of the homotopy analysis method (HAM) is plotted in Figures 2–7 on which the optimal serious solution is obtained. The effects of rotational parameter α, material parameter c1 , spin gradient viscosity parameter c2 , micro-inertia density parameter c3 , porosity parameter e, and suction/injection parameter S on velocity functions (radial, azimuthal, and induced) are presented in Figures 8–29. The comparison with the previous study has also been made for the verification of current study. The following conclusions can be drawn as a result of the computations:
function of parameter c3 , which means that the rotation becomes vigorous as the micro-inertia density of the fluid increases (see Figure 18), whereas in Figure 19, the microrotation profile of GðξÞ being symmetric appeared to be a decreasing function of the parameter c3 . Figures 20–23 display the influence of the porosity parameter on velocity profiles (radial, azimuthal and induced). The porosity parameter e shows the decreasing radial function f ′ðξÞ away from the disk in Figure 20. The impact of parameter e on azimuthal function gðξÞ is exhibited in Figure 21, which expresses the reduction of boundary layer thickness near the disk that converges to zero after ξ ¼ 5 away from the disk. Figure 22 illustrates the effect of porosity parameter e on induced velocity function kðξÞ, which shows that the porosity parameter does not
1) The rotational parameter α shows the maximum radial velocity f ′ðξÞ near the disk due to the presence of the centrifugal force. Velocity function kðξÞ expresses the decreasing function, whereas, the azimuthal velocity gðξÞ and the induced velocity hðξÞ shows the same decreasing patterns near the disk surface. 2) The radial velocity shows the decreasing function of material parameter c1 of a micropolar fluid away from the disk, but azimuthal gðξÞ and induced velocity function hðξÞ shows the increasing boundary layer thickness with the increasing values of c1 , whereas, another induced velocity function kðξÞ is the decreasing-increasing function of c1 . 3) The microrotation profiles F ðξÞ and GðξÞ of a micropolar fluid exhibits the same increasing and decreasing
Flow of micropolar fluid over an off centered rotating disk with modified Darcy's law
function of spin gradient viscosity parameter c2 and micro-inertia density parameter c3 . The microrotation profile F ðξÞ is antisymmetric, whereas, the microrotation profile GðξÞ is symmetric. 4) The porosity parameter e of Darcy's resistance is the decreasing function of radial, azimuthal and induced velocity profiles. 5) The increasing values of suction/injection parameter S give rapid acceleration in radial function near the disk, but causes distortion in the azimuthal and induced velocity functions, which shows the decreasing behavior. The microrotation functions F ðξÞ and GðξÞ show the opposite, decreasing and increasing function of the parameter S.
References [1] M. Patel, M.G. Timol, Numerical treatment of MHD PowellEyring fluid flow using the method of satisfaction of asymptotic boundary conditions, Int. J. Math. Sci. Comput. 1 (2) (2011) 71–78. [2] T. Hayat, M. Pakdemirli, Y. Aksoy, Similarity solutions for boundary layer equations of a Powell-Eyring fluid, Math. Comput. Appl. 18 (1) (2013) 62–70. [3] K.V. Prasada, P.S. Datti, B.T. Raju, Momentum and heat transfer of a non-Newtonian Eyring-Powell fluid over a non-isothermal stretching sheet, Int. J. Math. Arch. 4 (2013) 230–241. [4] G. Lukaszewicz, Micropolar Fluids Theory and Applications, Birkhauser, Basel, 1999. [5] A.C. Eringen, Simple micro fluids, Int. J. Eng. Sci. 2 (2) (1964) 205–217. [6] A.C. Eringen, Theory of thermomicrofluids, J. Math. Anal. Appl. 38 (2) (1972) 480–496. [7] N.A. Khan, M. Jamil, Analytical solution for creeping flow of an unsteady micropolar fluid, Int. J. Appl. Math. Mech. 5 (2009) 39–47. [8] A. Nazir, M. Shafique, Numerical study of micropolar fluids flow due to a stretching cylinder by SOR iterative procedure, Univers. J. Comput. Math. 3 (1) (2015) 8–13. [9] K. Ahmad, R. Nazar, A. Ishak, I. Pop, Unsteady threedimensional boundary layer flow due to a stretching surface in a micropolar fluid, Int. J. Numer. Methods Fluids 68 (2012) 1561–1573. [10] S.M.M. El-Kabeir, A.M. Rashad, R.S.R. Gorla, Heat transfer in a micropolar fluid flow past a permeable continuous moving surface, J. Appl. Math. Mech. 91 (2011) 360–370. [11] M.M. Rahman, M. Al-Lawatia, Effects of higher order chemical reaction on micropolar fluid flow on a power law permeable stretching sheet with variable concentration in a porous medium, Can. Soc. Chem. Eng. 88 (2010) 23–32.
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[12] M. Turkyilmazoglu, Flow of a micropolar fluid due to a porous stretching sheet and heat transfer, Int. J. Non-Linear Mech. 83 (2016) 59–64. [13] S. Nadeem, M. Hussain, M. Naz, MHD stagnation flow of a micropolar fluid through a porous medium, Meccanica 45 (6) (2010) 869–880. [14] M. Ashraf, K. Batool, MHD and heat transfer of a micropolar fluid over a stretchable disk, J. Theor. Appl. Mech. 51 (1) (2013) 25–38. [15] T. Von Kármán, Uber laminare und turbulente Reibung, Z. für Angew. Math. und Mech. 1 (4) (1921) 233–252. [16] M. Turkyilmazoglu, Analytic approximate solutions of rotating disk boundary layer flow subject to a uniform suction or injection, Int. J. Mech. Sci. 52 (12) (2010) 1735–1744. [17] E.M. Abo-Eldahab, A.F. Ghonaim, Radiation effect on heat transfer of a micropolar fluid through a porous medium, Appl. Math. Comput. 169 (1) (2005) 500–510. [18] P. Nuske, O. Ronneberger, N.K. Karadimitriou, R. Helmig, S.M. Hassanizadeh, Modeling two-phase flow in a micromodel with local thermal non equilibrium on the Darcy scale, Int. J. Heat. Mass Transf. 88 (2015) 822–835. [19] M. Turkyilmazoglu, Three dimensional MHD stagnation flow due to a stretchable rotating disk, Int. J. Heat. Mass Transf. 55 (23–24) (2012) 6959–6965. [20] W. Tan, T. Masuoka, Stokes' first problem for a second grade fluid in a porous half space with heated boundary, Int. J. Non-Linear Mech. 40 (4) (2005) 515–522. [21] W. Tan, T. Masuoka, Stokes' first problem for an Oldroyd-B fluid in a porous half space, Phys. Fluids 17 (2) (2005) 023101–023107. [22] N.A. Khan, S. Aziz, N.A. Khan, MHD flow of PowellEyring fluid over a rotating disk, J. Taiwan Inst. Chem. Eng. 45 (6) (2014) 2859–2867. [23] C.Y. Wang, Off-centered stagnation flow towards a rotating disc, Int. J. Eng. Sci. 46 (4) (2008) 391–396. [24] S.J. Liao, The Proposed Homotopy Analysis Techniques for the Solution of Nonlinear Problems (Ph.D. Dissertation), Shanghai Jiao Tong University, Shanghai, 1992. [25] G.S. Guram, A.C. Smith, Stagnation flows of micropolar fluids with strong and weak interactions, Comput. Math. Appl. 6 (2) (1980) 213–233. [26] S.K. Jena, M.N. Mathur, Similarity solutions for laminar free convection flow of a thermomicropolar fluid past a nonisothermal vertical flat plate, Int. J. Eng. Sci. 19 (11) (1981) 1431–1439. [27] G. Ahmadi, Self-similar solution of incompressible micropolar boundary layer flow over a semi-infinite plate, Int. J. Eng. Sci. 14 (7) (1976) 639–646. [28] J. Peddieson, An application of the micropolar fluid model to the calculation of turbulent shear flow, Int. J. Eng. Sci. 10 (1) (1972) 23–32.
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ORIGINAL ARTICLE
Flow of micropolar fluid over an off centered rotating disk with modified Darcy's law N.A. Khana,n, S. Khana, A. Arab a
Department of Mathematics, University of Karachi, Karachi, Sindh 75270, Pakistan Department of Computer Science, Mohammad Ali Jinnah University, Karachi, Sindh 75400, Pakistan
b
Received 15 January 2016; accepted 10 January 2017 Available online 6 December 2017
KEYWORDS Micropolar fluid; Stagnation point; Modified Darcy's law; Porous medium
Abstract The problem of the steady, incompressible, three dimensional stagnation point flow of a micropolar fluid over an off centered infinite rotating disk in a porous medium is studied in this article. Injection/suction is applied uniformly throughout the surface of porous disk. The Darcy's resistance for the micropolar fluid is also formulated. The partial differential equations are converted into the set of ordinary differential equation by utilizing the suitable transformation. The system of equations is analytically solved by the means of a nonperturbative technique, homotopy analysis method (HAM). The influence of rotational parameter, material parameter, spin gradient viscosity parameter, micro-inertia density parameter, porosity parameter and suction/injection parameter on velocity functions is presented in graphical form and discussed in detail. Verification of the solutions is made by a numerical comparison with the previous study. & 2017 National Laboratory for Aeronautics and Astronautics. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction For a long time, the non-Newtonian fluids captivated the attention of the researchers. The great interest in non-Newtonian n
Corresponding author. Tel.: þ923333012008.
E-mail address: [email protected] (N.A. Khan). Peer review under responsibility of National Laboratory for Aeronautics and Astronautics, China.
fluids is because of their great commercial importance. In reality, many fluids like biological fluids, slurries, shampoo, yoghurt, tomato sauce, grease, cosmetic products, paints, lubricants, polymers, custard, blood, and several other fluids do not obey the linear stress-velocity gradient relationship, which is the Newtonian fluid theory. Numerous non-Newtonian fluid models were therefore proposed to explain the complex behavior. Usually, the stress constitutive relations of such models inherit complexities, which lead to highly nonlinear
2212-540X & 2017 National Laboratory for Aeronautics and Astronautics. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). https://doi.org/10.1016/j.jppr.2017.11.006
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Nomenclature x; y; z u; v; w r; θ p r˜ N I j κ d a M c1 c2
Cartesian coordinates Velocity components Cylindrical coordinates Pressure Darcy resistance Micro rotation body couple per unit mass micro-inertia vortex viscosity Distance between the axis of flow and disk axis strength of the stagnation flow Torque experienced by disk material parameter spin gradient viscosity parameter
equations of motion with many terms. Most of the fluids used in the industrial applications are non-Newtonian in nature, especially, in polymer processing and chemical engineering processes etc. [1–3]. Among all the non-Newtonian fluids, the micropolar fluid has received special attention due to its additional angular momentum equation. Micropolar fluids are physically based on microstructure; these fluids may represent the fluids consisting of rigid, randomly oriented or spherical particles suspended in a viscous medium, where the deformation of fluid particles is ignored [4]. This fluid model was proposed by Eringen [5,6]. Many researchers focused on this fluid and studied it in different ways. Khan et al. [7] calculated the analytical solution of the creeping flow of two dimensional, unsteady micropolar fluid in Cartesian coordinates. Nazir and Shafique [8] investigated the numerical solution of the steady micropolar fluid flow caused by stretching cylinder, by utilizing the SOR method. Ahmed et al. [9] studied the three dimensional laminar boundary layer flow of a micropolar fluid due to a stretching surface, the analysis and discussion of stretching and material parameter have also been made. ElKabeir et al. [10] analyzed the heat and mass transfer flow of a micropolar fluid, which is passing through a permeable continuously moving surface, two different cases: plane surface moving parallel to the free stream and surface moving opposite to the free stream are considered. Rahman and AlLawatia [11] founded the influence of higher order chemical reaction and heat transfer flow of micropolar fluid past a permeable stretching sheet in a porous medium, the Darcy parameter shows the decreasing rate of the surface mass. Turkyilmazoglu [12] studied the flow of a micropolar fluid due to a stretching sheet with heat transfer. Nadeem et al. [13] examined the unsteady two dimensional magnetohydrodynamics (MHD) boundary layer flow of incompressible micropolar fluid through a porous medium near a forward stagnation point of a plane wall. Ashraf and Batool [14] numerically studied the MHD and heat transfer flow of the axisymmetric micropolar fluid passed through the stretching disk.
c3 e S k1
micro-inertia density parameter porosity parameter Uniform injection parameter permeability
Greek letters λ1 ; α 1 ; β 1 μ γ1 ω ρ τ α ϕ
material constant of micropolar fluid Viscosity of micropolar fluid spin gradient viscosity angular velocity Density extra stress tensor rotational parameter porosity of the porous medium
In the history of fluid mechanics, the most significant article in the literature of fluid dynamics is Von Karman [15]. He was the first one who calculated the solution of the laminar flow of a viscous fluid over a rotating disk. This paper captivated the attention of many researchers due to its several technical and industrial applications. Many researchers reinvestigated this article in various dynamics with MHD effect, porous medium, chemical reactions, thermal effect, uniform suction/injection effects, etc. [16]. The fluid passing through porous media has many applications, such as aerogels, injection of mud's, porous rocks, alloys, slurries or cement grouts to reinforce soils, foams and foamed solids, micro emulsions, polymer blends, and most important application is drilling fluids through injection in rocks for the reinforcement of the wells and also for enhancing oil recovery, etc. [17,18]. In 1856, Darcy [19] proposed a Darcy law in which he established the relation between the pressure drop and flow rate relation in a porous medium. Tan and Masuoka [20,21] examined the Stokes’ first problem for the second grade and Oldroyd-B fluids by using the modified Darcy's law. Hayat et al. [22] re-examined the Stoke's first problem for non-Newtonian forth order fluid by using the modified Darcy's law in a porous medium. To the best of authors' knowledge, the micropolar fluid model has never been investigated for an off centered rotating disk in a porous medium with Darcy resistance effect. In this article, the problem of Wang [23] is extended to study the influence of micropolar fluid parameters (material parameter, spin gradient viscosity parameter, and micro-inertia density parameter) based on the vortex viscosity, spin gradient viscosity, the material constant, and microinertia of the micropolar fluid over an off centered rotating porous disk with modified Darcy law, which has not been investigated previously. To attain the analytical solution of the current problem, the similarity transformations from [23] are used to reduce the partial differential equations of motion into non-linear ordinary differential equations (ODEs). The ODEs are then solved by a well-known non perturbative
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technique, homotopy analysis method (HAM), proposed by Liao [24]. The influence of the rotational parameter, suction/ injection parameter, porosity parameter, and micropolar fluid parameters has been examined and discussed in detail by plotting the velocity curves in radial, azimuthal, induced, and angular directions. The structure of this paper is based on 5 sections. Section 1 is dedicated to the introduction of the study, Section 2 contains the basic equations of the model, in Section 3 the mathematical model of the problem is formulated, Section 4 shows the HAM procedure, and the effect of the pertinent parameters on velocity profile is offered in Section 5. In the last Section 6, the summary of the study is exhibited.
2. Basic equations Figure 1
In a porous medium, the governing equations for micropolar fluid flow are defined as: Continuity ∂ρ þ ð∇⋅ρV Þ ¼ 0 ∂t Momentum
ð1Þ
ðλ þ 2μ þ κÞ∇ð∇⋅V Þ − ðμ þ κÞ ∇ð∇⋅V Þ − ∇ V −κ∇ N−∇p−˜r ¼ ρV̇ 2
Angular momentum α1 þ β1 þ γ 1 ∇ð∇⋅N Þ − γ 1 ∇ð∇⋅N Þ − ∇2 N þ κ∇ V−2κN þ ρI ¼ ρjṄ
ð2Þ
ð3Þ
where, V is the velocity vector, ρ is the density, p is the pressure, r˜ is the Darcy resistance for micropolar fluid, N is the micro-rotation, I is the body couple per unit mass, j is the micro-inertia, ∇ is the gradient operator, μ is the viscosity of the micropolar fluid, κ is the vortex viscosity, γ 1 is the spin gradient viscosity, operator λ1 ; α1 and β1 is the material constant of micropolar fluid.
3. Mathematical model Consider the steady, incompressible, stagnation point flow of a micropolar fluid over an off-centered rotating disk in a porous medium with Darcy resistance. The flow axes of the disk are parallel to z axis at a distance d. The disk is rotating with constant angular velocity ω. Let u; v; and w be the velocity components along the direction of Cartesian coordinates x; y; and z, respectively. In Figure 1, the physical geometry of the model is presented. Eqs. (1)–(3) are reduced into the following partial differential equations governing the three dimensional micropolar flow over an off centered rotating disk in a porous medium. ∂u ∂v ∂w þ þ ¼0 ∂x ∂y ∂z
ð4Þ
∂u ∂u 1 ∂p u ∂u ∂x þ v ∂y þ w ∂z ¼ − ρ ∂x þ ϕ μþκ 2 þ ρκ ∂N − ∂z k1 ρ u ∂v ∂v 1 ∂p u ∂v ∂x þ v ∂y þ w ∂z ¼ − ρ ∂y þ ϕ μþκ 1 − ρκ ∂N ∂z − k 1 ρ v
Physical model.
∂w ∂w 1 ∂p u ∂w ∂x þ v ∂y þ w ∂z ¼ − ρ ∂z þ ∂N 1 2 þ ρκ ∂N ∂x þ ∂y
μþκ ρ
μþκ ρ
μþκ ρ
∂2 u ∂x2
þ ∂∂yu2 þ ∂∂zu2
∂2 v ∂x2
þ ∂∂yv2 þ ∂∂z2v
2
∂2 w ∂x2
2
2
2
ð5Þ
ð6Þ
þ ∂∂yw2 þ ∂∂zw2 2
2
2 ∂N 1 ∂N 1 ∂ N1 ∂2 N 1 ∂2 N 1 1 þ v þ w þ þ ρj u ∂N ¼ γ 1 ∂x2 ∂x ∂y ∂z ∂y2 ∂z2 ∂v −κ 2N 1 þ ∂w ∂y − ∂z 2 2 2 ∂N 2 ∂N 2 2 ρj u ∂N ¼ γ 1 ∂∂xN22 þ ∂∂yN22 þ ∂∂zN22 ∂x þ v ∂y þ w ∂z ∂w −κ 2N 2 þ ∂u − ∂z ∂x
ð7Þ
ð8Þ
ð9Þ
where u; v; and w are the velocity components along the x; y; and z axes, ϕ is the porosity of the porous medium, k1 is the permeability. N 1 and N 2 are the components of microrotation vector N normal to the planes xz and yz respectively. We shall solve Eqs. (4)–(9) subjected to the following boundary conditions. u ¼ −ωy;
v ¼ ωðx−d Þ;
N 1 ¼ n ∂v ∂z ;
N 2 ¼ −n ∂u ∂z
u ¼ ax; N 1 →0;
v ¼ ay; N 2 →0 at
w ¼ w0 ; at z→0
ð10Þ
w ¼ −2az; z→∞
ð11Þ
Where a represents the strength of the stagnation flow, n is a boundary constant such that 0 r nr 1. Guram and Smith [25] studied the case n ¼ 0; which they named as strong
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concentration; the microelements close to a wall are unable to rotate, near the wall N ¼ 0 [26]. The weak concentration is observed in case n ¼ 12 due to the disappearance of the antisymmetric part of the stress tensor [27]. The case n ¼ 1 was proposed by Peddieson [28] in which he studied the turbulent boundary layer flows. By using the Wang [23] similarity transformations, u ¼ axf ′ðξÞ − ωygðξÞ þ dωkðξÞ; v ¼ ayf ′ðξÞ þ ωxgðξÞ þ dωhðξÞ; pffiffi pffiffiffiffiffi w ¼ −2 aνf ðξÞ; ξ ¼ z aν; pffiffi pffiffi N 2 ¼ ax aνF ðξÞ N 1 ¼ ay aνGðξÞ;
ð12Þ
Using the above similarity transformations in Eqs. (5)–(8), the equation of continuity is satisfied whereas the governing partial differential equations are reduced to the following form: ð1 þ c1 Þðf ′′′−ef ′Þ þ c1 F′ þ 2f f ′′ −f ′2 þ α2 g2 þ 1 ¼ 0;
ð13Þ
ð1 þ c1 Þ ðg′′−egÞ þ 2g′f −2gf ′ ¼ 0;
ð14Þ
ð1 þ c1 Þðk′′−ek Þ þ αgh −f ′ k þ 2f k′ ¼ 0;
ð15Þ
ð1 þ c1 Þðh′′−ehÞ−αgk −f ′ h þ 2f h′ ¼ 0;
ð16Þ
F′′−2c2 F−c3 Ff ′ þ 2c3 F′f −c2 f ′′ ¼ 0;
ð17Þ
G′′−2c2 G−c3 Gf ′ þ 2c3 G′f þ c2 f ′′ ¼ 0;
ð18Þ
and the boundary conditions of Eqs. (10)–(11) become, f ð0Þ ¼ S;
f ′ð0Þ ¼ 0;
gð0Þ ¼ 1;
F ð0Þ ¼ −f ′′ð0Þ; Gð0Þ ¼ f ′′ð0Þ f ′ð∞Þ ¼ 1; gð∞Þ ¼ 0; kð∞Þ ¼ 0; F ð∞Þ ¼ 0;
kð0Þ ¼ 0;
hð0Þ ¼ 1;
τyz ¼ ρ
μþκ ρ
2
∂v κ ∂z − ρ N 1
z¼0
yðf ′′ð0Þ þ c1 f ′′ð0Þ−c1 Gð0ÞÞ
pffiffiffiffiffi6 τyz ¼ ρa νa4 þαxðg′ð0Þ þ c1 g′ð0ÞÞ þdαðh′ð0Þ þ c1 h′ð0ÞÞ
3
ð22Þ
7 5
The shear stress at the center is zero, the shear stress on the surface of the disk can be obtained by setting Eqs. (21) and (22) to zero and solving for ðx; yÞ. The torque experienced by the disk of radius R is given by: 2 M ¼ ∫R0 ∫2π ð23Þ 0 τyz cos θ − τxz sin θ r dθdr; where, ðr; θÞ are the cylindrical coordinates, τ is the extra stress tensor, τxz and τyz are the xz and yz components of the extra stress tensor. Since, x ¼ r cos θ−b, the torque can be found at: π pffiffiffiffiffi ð24Þ M ¼ a aναR4 ρðg′ð0Þ þ c1 g′ð0ÞÞ; 2 Which is unaffected by the non-aligned disk and flow axis.
4. Analytical approximations by means of HAM The auxiliary linear operators, L1 ½f and L2 ½g; k; h; F; G are selected for the Eqs. (14)–(19) as: L1 ½ f ¼
∂3 ∂ − ; ∂ξ3 ∂ξ
L2 ½g; h; k; F; G ¼
∂2 −1; ∂ξ2
ð25Þ
Also, the initial guesses satisfying the initial conditions in Eq. (19) are obtained as:
hð∞Þ ¼ 0;
Gð∞Þ ¼ 0: ð19Þ ω a
where, α ¼ is the non-dimensional rotational parameter, c1 ¼ μκ is the material parameter, c2 ¼ aγκν is the spin gradient 1
viscosity parameter, c3 ¼ is the micro-inertia density 0ffi pffiffiffi parameter, e ¼ akνφ1 is the porosity parameter, and S ¼ 2−w aν is the uniform injection parameter obtained from Eqs. (10) and (12). Here, S represents a uniform injection S40. The pressure p can be recovered from third momentum Eq. (7) as:
2 1 2 2 w μþκ 2 p ¼ p0 − ρa x þ y −ρ − wz ð20Þ 2 ρ 2 jνρ γ1
where, p0 is the pressure at the origin. The shear stress on the off centered rotating disk of a micropolar fluid in porous medium is given by: ∂u κ τxz ¼ ρ μþκ þ N 2 ρ ∂z ρ z¼0 2 3 xðf ′′ð0Þ þ c1 f ′′ð0Þ þ c1 F ð0ÞÞ ð21Þ pffiffiffiffiffi6 7 τxz ¼ ρa νa4 −αyðg′ð0Þ þ c1 g′ð0ÞÞ 5 þdαðk′ð0Þ þ c1 k′ð0ÞÞ
f 0 ðξÞ ¼ ξ þ S − 1 þ e−ξ ; g0 ðξÞ ¼ e−ξ ; F 0 ¼ −f ″0 ð0Þe−ξ ; G0 ¼ f ″0 ð0Þe−ξ
k 0 ðξÞ ¼ 0;
h0 ðξÞ ¼ e−ξ ;
ð26Þ satisfying the following properties
L1 C 1 þ C 2 eξ þ C 3 e−ξ ¼ 0;
L2 C 4 eξ þ C 5 e−ξ ¼ 0;
ð27Þ
Where, C 1 ; C 2 ; :::; C 5 are arbitrary constants. The zeroth order deformation equations can be constructed as: " # h i f^ ðξ; qÞ; g^ ðξ; qÞ; ^ ð1−qÞL1 f ðξ; qÞ − f 0 ðξÞ ¼ qℏf N 1 ð28Þ F^ ðξ; qÞ h i ð1−qÞL2 g^ ðξ; qÞ − g0 ðξÞ ¼ qℏg N 2 f^ ðξ; qÞ; g^ ðξ; qÞ " i f^ ðξ; qÞ; ^ ð1−qÞL2 kðξ; qÞ − k 0 ðξÞ ¼ qℏk N 3 ^ kðξ; qÞ; h
"
h i f^ ðξ; qÞ; ð1−qÞL2 h^ ðξ; qÞ − h0 ðξÞ ¼ qℏh N 4 ^ k ðξ; qÞ;
g^ ðξ; qÞ; h^ ðξ; qÞ g^ ðξ; qÞ; h^ ðξ; qÞ
ð29Þ #
#
ð30Þ
ð31Þ
Flow of micropolar fluid over an off centered rotating disk with modified Darcy's law
h i ð1−qÞL2 F^ ðξ; qÞ − F 0 ðξÞ ¼ qℏF N 5 f^ ðξ; qÞ; F^ ðξ; qÞ h i ^ ðξ; qÞ − G0 ðξÞ ¼ qℏG N 6 f^ ðξ; qÞ; G ^ ðξ; qÞ ð1−qÞL2 G
ð32Þ
L2 gm ðξÞ − χ m gm−1 ðξÞ ¼ ℏg R2;m ðξÞ
ð33Þ
L2 km ðξÞ − χ m km−1 ðξÞ ¼ ℏk R3;m ðξÞ L2 hm ðξÞ − χ m hm−1 ðξÞ ¼ ℏh R4;m ðξÞ L2 F m ðξÞ − χ m F m−1 ðξÞ ¼ ℏF R5;m ðξÞ
ð44Þ
L2 Gm ðξÞ − χ m Gm−1 ðξÞ ¼ ℏG R6;m ðξÞ
ð46Þ
where
h i 3^ ^ Þ N 1 f^ ðξ; qÞ; g^ ðξ; qÞ; F^ ðξ; qÞ ¼ ð1 þ c1 Þ ∂ f∂ξðξ;q − e ∂f ð∂ξξ;qÞ 3
2^ ^ ðξ;qÞ Þ þ 2f^ ðξ; qÞ ∂ f∂ξðξ;q þc1 ∂F∂ξ 2 ^ 2 − ∂f ð∂ξξ;qÞ þ α2 ðg^ ðξ; qÞÞ2 þ 1
ð34Þ
^ ∂^gðξ;qÞ gðξ; qÞ ∂f ð∂ξξ;qÞ ∂ξ −2^
−
^ k^ ðξ; qÞ ∂f ð∂ξξ;qÞ
ð35Þ
þ αh^ ðξ; qÞ^gðξ; qÞ
ð36Þ h i 2^ Þ N 4 f^ ðξ; qÞ; g^ ðξ; qÞ; k^ ðξ; qÞ; h^ ðξ; qÞ ¼ ð1 þ c1 Þ ∂ h∂ξðξ;q − eh^ ðξ; qÞ 2 ^ ðξ;qÞ ^ þ2f^ ðξ; qÞ ∂h∂ξ − h^ ðξ; qÞ ∂f ð∂ξξ;qÞ −αk^ ðξ; qÞ^gðξ; qÞ
∂2 F^ ðξ;qÞ ∂ξ2
^ − 2c2 F^ ðξ; qÞ − c3 F^ ðξ; qÞ ∂f ð∂ξξ;qÞ
2^ ^ ðξ;qÞ Þ þ2c3 f^ ðξ; qÞ ∂F∂ξ − c2 ∂ f∂ξðξ;q 2
ð38Þ h i ^ ðξ; qÞ ¼ N 6 f^ ðξ; qÞ; G
km ð∞Þ ¼ 0;
hm ð∞Þ ¼ 0;
ð45Þ
and χm ¼
(
0; 1;
mr1 mZ2
ð48Þ
R1;m ðξÞ ¼ ð1 þ c1 Þ f ′″m−1 ðξÞ−ef ′m−1 ðξÞ þ c1 F ′m−1 ðξÞ m−1
m−1
þ α2 ∑ gi ðξÞgm−1−i ðξÞ þ 2 ∑ f i ðξÞf ″m−1−i ðξÞ i¼0
i¼0
− ∑ f ′i ðξÞf ′m−1−i ðξÞ þ 1−χ m ; i¼0
R2;m ðξÞ ¼ ð1 þ c1 Þ g″m−1 ðξÞ − egm−1 ðξÞ
ð49Þ
m−1
m−1
i¼0
i¼0
þ 2 ∑ f i ðξÞg′m−1−i ðξÞ − 2 ∑ gi ðξÞf ′m−1−i ðξÞ;
ð50Þ
i¼0
2^ ^ ðξ;qÞ Þ þ2c3 f^ ðξ; qÞ ∂G∂ξ þ c2 ∂ f∂ξðξ;q 2
ð39Þ in which, q∈½0; 1 is the embedding parameter and ℏ is the auxiliary non-zero parameter. By Taylor's theorem, ∞
f^ ðξ; qÞ ¼ f 0 ðξÞ þ ∑ f m ðξÞqm ;
m−1
m−1
i¼0
i¼0
− ∑ ki ðξÞf ′m−1−i ðξÞ þ α ∑ hi ðξÞgm−1−i ðξÞ; ð51Þ m−1 R4;m ðξÞ ¼ ð1 þ c1 Þ h″m−1 ðξÞ − ehm−1 ðξÞ þ 2 ∑ f i ðξÞh′m−1−i ðξÞ i¼0
m¼1 ∞
m−1
m1
i¼0
i¼0
− ∑ hi ðξÞf ′m−1−i ðξÞ − α ∑ ki ðξÞgm−1−i ðξÞ;
g^ ðξ; qÞ ¼ g0 ðξÞ þ ∑ gm ðξÞqm ; m¼1 ∞
ð52Þ
^ qÞ ¼ k 0 ðξÞ þ ∑ km ðξÞqm ; kðξ; m¼1 ∞
Gm ð∞Þ ¼ 0
m−1 R3;m ðξÞ ¼ ð1 þ c1 Þ k″m−1 ðξÞ − ek m−1 ðξÞ þ 2 ∑ f i ðξÞk′m−1−i ðξÞ
^ ðξ;qÞ ∂2 G ^ ðξ; qÞ−c3 G ^ ðξ; qÞ ∂f^ ðξ;qÞ −2c2 G ∂ξ ∂ξ2
^ qÞ ¼ h0 ðξÞ þ ∑ hm ðξÞqm ; hðξ;
F m ð∞Þ ¼ 0;
ð47Þ
m−1
ð37Þ h i N 5 f^ ðξ; qÞ; F^ ðξ; qÞ ¼
ð43Þ
f m ð0Þ ¼ 0; f ′m ð0Þ ¼ 0; gm ð0Þ ¼ 0; km ð0Þ ¼ 0; hm ð0Þ ¼ 0; ″ F m ð0Þ ¼ −f m ð0Þ; Gm ð0Þ ¼ f ″m ð0Þf ′m ð∞Þ ¼ 0; gm ð∞Þ ¼ 0;
2^ h i Þ N 3 f^ ðξ; qÞ; g^ ðξ; qÞ; k^ ðξ; qÞ; h^ ðξ; qÞ ¼ ð1 þ c1 Þ ∂ k∂ξðη;q − ek^ ðξ; qÞ 2 ^ ðξ;qÞ þ2f^ ðξ; qÞ ∂k∂ξ
ð42Þ
with the following boundary conditions
2 h i ^ ðξ;qÞ N 2 f^ ðξ; qÞ; g^ ðξ; qÞ ¼ ð1 þ c1 Þ ∂ g∂ξ − e^gðξ; qÞ 2 þ2f^ ðξ; qÞ
289
ð40Þ
m¼1 ∞
^ qÞ ¼ F 0 ðξÞ þ ∑ F m ðξÞqm ; Fðξ; m¼1 ∞
^ qÞ ¼ G0 ðξÞ þ ∑ Gm ðξÞqm ; Gðξ; m¼1
The general HAM equations for mth order can be given by: L1 f m ðξÞ − χ m f m−1 ðξÞ ¼ ℏf R1;m ðξÞ ð41Þ
R5;m ðξÞ ¼ F ″m−1 ðξÞ − 2c2 F m−1 ðξÞ − c2 f ″m−1 ðξÞ m−1
m−1
i¼0
i¼0
−c3 ∑ F i ðξÞf ′m−1−i ðξÞ þ 2c3 ∑ f i ðξÞF ′m−1−i ðξÞ; ð53Þ
290 Table 1
N.A. Khan et al. Comparision of skin-friction coefficient ðf ′′ð0Þ; g′ð0Þ; k′ð0Þ and h′ð0ÞÞ with Wang [23].
α
0
0:5
1
2
3
f ′′ð0Þ Wang [23] HAM
1.31194 1.30439
1.3787 1.37001
1.5739 1.56213
2.2951 2.27341
3.3657 3.36438
g′ð0Þ Wang [23] HAM
−1.07467 −1.07071
−1.0839 −1.07949
−1.1100 −1.10451
−1.1968 −1.18571
−1.3055 −1.26544
k′ð0Þ Wang [23] HAM
0 0
0.1380 0.136905
0.2700 0.222618
0.5040 0.54914
0.6972 0.847817
h′ð0Þ Wang [23] HAM
−0.9387 −0.9343
−0.9495 −0.944477
−0.9787 −0.971984
–1.0787 -1.04035
−1.2003 −1.2015
R6;m ðξÞ ¼ G″m−1 ðξÞ − 2c2 Gm−1 ðξÞ þ c2 f ″m−1 ðξÞ m−1
m−1
i¼0
i¼0
−c3 ∑ Gi ðξÞf ′m−1−i ðξÞ þ 2c3 ∑ f i ðξÞG′m−1−i ðξÞ; ð54Þ According to the above defined method, the linear Eqs. (34)–(39) with the boundary conditions in Eq. (40) in the order m ¼ 1; 2; 3; :::, can be solved easily by means of computational software Mathematica 10. Figure 2 Convergence region of ℏf .
5. Results and discussion In this section the findings of the study are discussed and presented in graphical form. Also, the comparison has been made in Table 1 as a special case for Newtonian fluid studied by Wang [23]. Table 1 shows the promising agreement between the present and previous solution, which shows the validity of the current analytical solution. The results of series solution of Eq. (40) are computed at a valid convergence range of auxiliary parameters −1:0oℏf o0:2; −1:0oℏg o0:5; −1:6oℏk o0:6; −1:0oℏh o0:4; −1:0oℏF o0:6; and −1:0oℏG o0:6 as presented in Figures 2–7. The influence of rotational parameter α on velocity profiles is displayed in Figures 8–11. Figure 8 shows the exponential rise in the radial function f ′ðξÞ near the disk due to the presence of the centrifugal force. Radial velocity is reached at its maximum point in the neighborhood of ξ ¼ 0:7 and after reaching its maximum point, it starts converging to unity as ξ→∞ far from the disk. Figure 9 demonstrates the effect of α on azimuthal velocity profile gðξÞ; as the rotation of the disk increases, it shows the negative impact on gðξÞ and starts showing the decrease in boundary layer thickness near the disk, later on away from the disk, it starts converging to zero. The influence of rotational parameter on the induced velocity profile kðξÞ expresses the rapid increase near the disk at ξ ¼ 0:6 (see
Figure 3 Convergence region of ℏg .
Figure 4 Convergence region of ℏk .
Flow of micropolar fluid over an off centered rotating disk with modified Darcy's law
291
Figure 10). The effect of rotational parameter of another induced velocity function hðξÞ is observed in Figure 11, which illustrates that the rotation of the disk distracts the motion of fluid particles and shows the decreasing effect in
Figure 9 Effect of rotational parameter α on velocity function gðξÞ at c1 ¼ 0:1; c2 ¼ 0:1; c3 ¼ 0:1; e ¼ 0:1; S ¼ 0:1.
Figure 5 Convergence region of ℏh .
Figure 10 Effect of rotational parameter α on velocity function k ðξÞ at c1 ¼ 0:1; c2 ¼ 0:1; c3 ¼ 0:1; e ¼ 0:1; S ¼ 0:1. Figure 6 Convergence region of ℏF .
Figure 7 Convergence region of ℏG .
Figure 8 Effect of rotational parameter α on velocity function f ′ðξÞ at c1 ¼ 0:1; c2 ¼ 0:1; c3 ¼ 0:1; e ¼ 0:1; S ¼ 0:1.
Figure 11 Effect of rotational parameter α on velocity function hðξÞ at c1 ¼ 0:1; c2 ¼ 0:1; c3 ¼ 0:1; e ¼ 0:1; S ¼ 0:1.
Figure 12 Effect of material parameter c1 on velocity function f ′ðξÞ at α ¼ 0:1; c2 ¼ 0:1; c3 ¼ 0:1; e ¼ 0:1; S ¼ 0:1.
292
Figure 13 Effect of material parameter c1 on velocity function gðξÞ at α ¼ 0:1; c2 ¼ 0:1; c3 ¼ 0:1; e ¼ 0:1; S ¼ 0:1.
Figure 14 Effect of material parameter c1 on velocity function k ðξÞ at α ¼ 0:1; c2 ¼ 0:1; c3 ¼ 0:1; e ¼ 0:1; S ¼ 0:1.
N.A. Khan et al.
Figure 17 Effect of spin gradient viscosity parameter c2 on velocity function GðξÞ at α ¼ 0:1; c1 ¼ 1; c3 ¼ 0:1; e ¼ 0:1; S ¼ 0:1.
Figure 18 Effect of microinertia density parameter c3 on velocity function F ðξÞ at α ¼ 0:1; c1 ¼ 1; c2 ¼ 1; e ¼ 0:1; S ¼ 0:1.
Figure 19 Effect of microinertia density parameter c3 on velocity function GðξÞ at α ¼ 0:1; c1 ¼ 1; c2 ¼ 1; e ¼ 0:1; S ¼ 0:1. Figure 15 Effect of material parameter c1 on velocity function hðξÞ at α ¼ 0:1; c2 ¼ 0:1; c3 ¼ 0:1; e ¼ 0:1; S ¼ 0:1.
Figure 16 Effect of spin gradient viscosity parameter c2 on velocity function F ðξÞ at α ¼ 0:1; c1 ¼ 1; c3 ¼ 0:1; e ¼ 0:1; S ¼ 0:1.
the induced direction, which is the same pattern previously observed in Figure 9. The effects of material parameter on velocity functions are presented in Figures 12–15. Figure 12 demonstrates the impact of vortex viscosity parameter c1 on radial function f ′ðξÞ. The increasing values of viscosity parameter cause distortion in radial function and this distortion don’t allow a radial function to converge to unity far from the disk. The increasing values of material parameter c1 cause acceleration in the boundary layer thickness of azimuthal function gðξÞ in the azimuthal direction (See Figure 13). The effect of fluid parameter c1 on induced velocity function kðξÞ in Figure 14 shows the decreasing-increasing behavior near the disk, but as it starts moving away from the disk in induced direction, it converges to zero in the neighborhood of ξ ¼ 5. Figure 15 demonstrates the same increasing pattern of micropolar fluid
Flow of micropolar fluid over an off centered rotating disk with modified Darcy's law
parameter c1 in induced velocity function hðξÞ as we have observed in azimuthal component gðξÞ. The impact of spin gradient viscosity parameter c2 on microrotation profiles F ðξÞ and GðξÞ is illustrated in Figures 16 and 17. In Figure 16, the influence of spin gradient viscosity parameter c2 on the microrotation function F ðξÞ shows acceleration, whereas, another microrotation function GðξÞ expresses the opposite deceleration behavior. The increasing values of c2 decreased the spin gradient viscosity of the fluid, which cause distortion in the movement of fluid particles in the microrotation profile GðξÞ. The impact of the micro-inertia density parameter of micropolar fluid c3 is demonstrated in Figures 18 and 19. The microrotation profiles F ðξÞ and GðξÞ manifested the same pattern as we have seen in Figures 16 and 17. The profile of microrotation F ðξÞ is antisymmetric, and shows that it is an increasing
293
Figure 23 Effect of porosity parameter e on velocity function hðξÞ at α ¼ 1; c1 ¼ 0:01; c2 ¼ 0:01; c3 ¼ 0:01; S ¼ 0:01.
Figure 24 Effect of suction/injection parameter S on velocity function f ′ðξÞ at α ¼ 0:1; c1 ¼ 0:1; c2 ¼ 0:1; c3 ¼ 0:1; e ¼ 0:1. Figure 20 Effect of porosity parameter e on velocity function f ′ðξÞ at α ¼ 1; c1 ¼ 0:01; c2 ¼ 0:01; c3 ¼ 0:01; S ¼ 0:01.
Figure 21 Effect of porosity parameter e on velocity function gðξÞ at α ¼ 1; c1 ¼ 0:01; c2 ¼ 0:01; c3 ¼ 0:01; S ¼ 0:01.
Figure 22 Effect of porosity parameter e on velocity function k ðξÞ at α ¼ 1; c1 ¼ 0:01; c2 ¼ 0:01; c3 ¼ 0:01; S ¼ 0:01.
Figure 25 Effect of suction/injection parameter S on velocity function gðξÞ at α ¼ 0:1; c1 ¼ 0:1; c2 ¼ 0:1; c3 ¼ 0:1; e ¼ 0:1.
Figure 26 Effect of suction/injection parameter S on velocity function k ðξÞ at α ¼ 0:1; c1 ¼ 0:1; c2 ¼ 0:1; c3 ¼ 0:1; e ¼ 0:1.
294
Figure 27 Effect of suction/injection parameter S on velocity function hðξÞ at α ¼ 0:1; c1 ¼ 0:1; c2 ¼ 0:1; c3 ¼ 0:1; e ¼ 0:1.
N.A. Khan et al.
support the micropolar fluid particles to move in induced direction due to which, it shows decreasing effect on velocity profile kðξÞ, whereas, another induced function hðξÞ present the decreasing-increasing effect of porosity parameter close to the disk as shown in Figure 23. Figures 24–29 present the influence of the injection parameter S on the velocity functions. Figure 24 shows the increasing function of radial velocity f ′ðξÞ, injection parameter S helps the fluid particles to move in the radial direction. The effect of parameter S on azimuthal velocity gðξÞ is displayed in Figure 25. For the injection S40, the boundary layer is increasingly blown away from the disk to form an interlayer between the injection and the outer flow regions. From Figures 26 and 27, it is noticed that the injection in porous disk causes the disturbance in the flow motion due to which, it is observed that k ðξÞ and hðξÞ are the decreasing functions of the parameter. The impact of the parameter S on the microrotation functions F ðξÞ and GðξÞ is presented in Figures 28 and 29. The microrotation function F ðξÞ is antisymmetric and shows a decreasing behavior as the injection on the disk increases. The microrotation function GðξÞ is symmetric and shows an increasing behavior as the injection on the disk increases near the disk (see Figure 29).
Figure 28 Effect of suction/injection parameter S on velocity function F ðξÞ at α ¼ 0:1; c1 ¼ 0:1; c2 ¼ 0:1; c3 ¼ 0:1; e ¼ 0:1.
6. Conclusions
Figure 29 Effect of suction/injection parameter S on velocity function GðξÞ at α ¼ 0:1; c1 ¼ 0:1; c2 ¼ 0:1; c3 ¼ 0:1; e ¼ 0:1.
The present study focuses on the analytical solution of steady, incompressible, stagnation point flow of a micropolar fluid over an off centered rotating disk in a porous medium with Darcy's resistance. The convergence region of the homotopy analysis method (HAM) is plotted in Figures 2–7 on which the optimal serious solution is obtained. The effects of rotational parameter α, material parameter c1 , spin gradient viscosity parameter c2 , micro-inertia density parameter c3 , porosity parameter e, and suction/injection parameter S on velocity functions (radial, azimuthal, and induced) are presented in Figures 8–29. The comparison with the previous study has also been made for the verification of current study. The following conclusions can be drawn as a result of the computations:
function of parameter c3 , which means that the rotation becomes vigorous as the micro-inertia density of the fluid increases (see Figure 18), whereas in Figure 19, the microrotation profile of GðξÞ being symmetric appeared to be a decreasing function of the parameter c3 . Figures 20–23 display the influence of the porosity parameter on velocity profiles (radial, azimuthal and induced). The porosity parameter e shows the decreasing radial function f ′ðξÞ away from the disk in Figure 20. The impact of parameter e on azimuthal function gðξÞ is exhibited in Figure 21, which expresses the reduction of boundary layer thickness near the disk that converges to zero after ξ ¼ 5 away from the disk. Figure 22 illustrates the effect of porosity parameter e on induced velocity function kðξÞ, which shows that the porosity parameter does not
1) The rotational parameter α shows the maximum radial velocity f ′ðξÞ near the disk due to the presence of the centrifugal force. Velocity function kðξÞ expresses the decreasing function, whereas, the azimuthal velocity gðξÞ and the induced velocity hðξÞ shows the same decreasing patterns near the disk surface. 2) The radial velocity shows the decreasing function of material parameter c1 of a micropolar fluid away from the disk, but azimuthal gðξÞ and induced velocity function hðξÞ shows the increasing boundary layer thickness with the increasing values of c1 , whereas, another induced velocity function kðξÞ is the decreasing-increasing function of c1 . 3) The microrotation profiles F ðξÞ and GðξÞ of a micropolar fluid exhibits the same increasing and decreasing
Flow of micropolar fluid over an off centered rotating disk with modified Darcy's law
function of spin gradient viscosity parameter c2 and micro-inertia density parameter c3 . The microrotation profile F ðξÞ is antisymmetric, whereas, the microrotation profile GðξÞ is symmetric. 4) The porosity parameter e of Darcy's resistance is the decreasing function of radial, azimuthal and induced velocity profiles. 5) The increasing values of suction/injection parameter S give rapid acceleration in radial function near the disk, but causes distortion in the azimuthal and induced velocity functions, which shows the decreasing behavior. The microrotation functions F ðξÞ and GðξÞ show the opposite, decreasing and increasing function of the parameter S.
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