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Couette flow of a third grade fluid with rotating frame and slip condition
Nonlinear Analysis: Real World Applications 10 (2009) 3329–3334
Contents lists available at ScienceDirect
Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa
Couette flow of a third grade fluid with rotating frame and slip condition S. Abelman a,∗ , E. Momoniat a , T. Hayat a,b a
Centre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050, South Africa b
Department of Mathematics, Quaid-i-Azam University, Islamabad, 45320, Pakistan
article
info
Article history: Received 30 April 2008 Accepted 6 October 2008 Keywords: Third grade fluid Partial slip Porous space Rotating flow
abstract An incompressible third grade fluid occupies the porous space between two rigid infinite plates. The steady rotating flow of this fluid due to a suddenly moved lower plate with partial slip of the fluid on the plate is analysed. The fluid filling the porous space between the two plates is electrically conducting. The flow modeling is developed by employing a modified Darcy’s law. A numerical solution of the governing problem consisting of a non-linear ordinary differential equation and non-linear boundary conditions is obtained and discussed. Several limiting cases of the arising problem can be obtained by choosing suitable parameters. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction It is a well-known fact that non-Newtonian fluids are more appropriate than Newtonian fluids in industrial and technological applications. A large variety of consumer goods containing high concentration of glass or carbon fibres, paints and lubricants containing polymer additives, many foodstuffs and biological fluids are non-Newtonian. The constitutive equations of such fluids vary greatly in complexity. The class of Newtonian flows for which there exists analytic solution of the Navier–Stokes equations is very restricted. This class is further narrowed down when non-Newtonian fluids are taken into account. In fact, the resulting equations of non-Newtonian fluids are of higher order [1–3], more non-linear and complicated than the Navier–Stokes equations and thus make the task of obtaining accurate solutions difficult. Being scientifically appealing and challenging, non-Newtonian fluids in a non-rotating frame have been studied extensively. Some recent studies may be found in references [4–12]. Attempts have been also made on flows of non-Newtonian fluids in a rotating frame [13–22]. Rotating flows of non-Newtonian fluids in a porous space and partial slip have not received much attention in the literature. A viscous non-Newtonian fluid normally sticks to the boundary. In such cases the no-slip condition is no longer valid. There are numerous situations where the no-slip condition between the fluid and the boundary does not hold. For example, the fluid may be particulate or such that it could be a rarefied gas with a suitable value of the Kundsen number. Also, rotating flows of non-Newtonian fluids in a porous space have geophysical applications. In our paper [23] we considered a numerical solution for steady state rotating and magnetohydrodynamic (MHD) flow of a third grade fluid past a rigid plate with slip. In this paper we present a numerical solution for the steady Couette flow of a thermodynamic compatible third grade fluid filling the porous space in a rotating frame. Partial slip effects are taken into account. For a third grade fluid, both the governing equation and boundary conditions are non-linear, whereas the corresponding problem for a viscous fluid is linear. The paper is organized as follows. The problem is formulated in Section 2. In Section 3 the numerical solution is discussed and in Section 4 concluding remarks are presented.
∗
Corresponding author. E-mail addresses: [email protected] (S. Abelman), [email protected] (E. Momoniat), [email protected] (T. Hayat).
1468-1218/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2008.10.068
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S. Abelman et al. / Nonlinear Analysis: Real World Applications 10 (2009) 3329–3334
Fig. 1. Graphical representation of the problem.
2. Formulation of the problem Consider an incompressible third grade fluid occupying the porous space between two rigid infinite plates. The lower plate at z = 0 is suddenly jerked and the upper plate at z = h is kept stationary. A graphical representation of the problem is shown in Fig. 1. The fluid is electrically conducting in the presence of a constant magnetic field of strength B0 parallel to the z-axis. The magnetic Reynolds number is taken small and hence the induced magnetic field is neglected. The applied and induced electric fields are absent. The space 0 < z < h between the two plates is porous. Furthermore, the plates and the ˆ where kˆ is a unit vector in the z-direction. The hydromagnetic steady fluid rotate with constant angular velocity Ω = Ω k rotating flow in a porous space is governed by the following equations: div V = 0,
(1)
ρ [(V · ∇) V + 2Ω × V] = −∇ pˆ + div S − σ
B20 V
+ R.
(2)
In the above equations, V is the velocity, ρ is the fluid density, σ is the electrical conductivity of the fluid, R is Darcy’s ρ resistance in a porous space, S is the extra stress tensor and pˆ = p − 2 Ω 2 x2 + y2 . Since the plates are rigid, the incompressibility condition (1) requires that V = (u (z ) , v (z ) , 0) ,
(3)
in which u and v are the velocity components in the x- and y-directions respectively. Note that for rigid infinite plates and steady flow, u and v are dependent on z. The constitutive equation of the extra stress tensor is [24] S = µ + β3 trA21
A1 + α1 A2 + α2 A21 + β1 A3 + β2 (A1 A2 + A2 A1 ) ,
(4)
where µ is the dynamic viscosity, αi (i = 1, 2) and βi (i = 1 − 3) are material constants and the first three Rivlin–Ericksen tensors Ai (i = 1 − 3) are defined through the following relationships [25]: A1 = L + LT , An+1 =
dAn dt
L = ∇ V,
+ An L + LT An
(n = 1, 2) .
(5)
Fosdick and Rajagopal [26] have shown that the Clausius–Duhem inequality and the assumption that the Helmholtz free energy is minimum in equilibrium requires that
µ ≥ 0,
α1 ≥ 0, β1 = β2 = 0, p |α1 + α2 | ≤ 24µβ3 .
β3 ≥ 0, (6)
Employing the same procedure as in references [8,9,21], the following expression of R has been proposed in the present case R=−
φ k
" µ + 2β3
(
du dz
2
+
dv dz
2 )#
V,
in which φ (0 < φ < 1) and k are the porosity and permeability of the porous space.
(7)
S. Abelman et al. / Nonlinear Analysis: Real World Applications 10 (2009) 3329–3334
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In view of Eqs. (3)–(7), Eq. (2) in the absence of a modified pressure gradient yields
" ( 2 )# 2 σ B20 φ dv du + − u− + u, −2Ω v = ν 2 + µ + 2β3 dz ρ dz dz dz dz ρ kρ dz dz " ( " ( 2 )# 2 )# 2 2 d2 v 2β3 d dv dv σ B20 φ dv du du 2Ω u = ν 2 + + − v− + v, µ + 2β3 dz ρ dz dz dz dz ρ kρ dz dz 2β3 d
d2 u
"
du
(
du
2
dv
2 )#
(8)
(9)
where ν is the kinematic viscosity. On account of the slip velocities, the boundary conditions on the plates take the form
" u − U0 − γ
"
dv
v−γ
dz
" u+γ
du dz
" v+γ
dv dz
du
2β3 du
(
du
2
dv
2 )#
+ = 0, at z = 0, µ dz dz dz ( 2 )# 2 2β3 dv dv du + + = 0, at z = 0, µ dz dz dz ( 2 )# 2 du 2β3 du dv + + = 0, at z = h, µ dz dz dz ( 2 )# 2 2β3 dv du dv + + = 0, at z = h, µ dz dz dz dz
+
(10)
(11)
(12)
(13)
in which γ is a slip parameter having dimension of length and U0 is a mainstream reference velocity. Defining u
u∗ =
N =
U0
σ
,
v
v∗ =
B20 h2
,
ρν
,
U0
z∗ =
β ρν
2 3 U0 h2
M =
,
z h
,
H =
Ω∗ =
φ h2 k
,
Ω h2 , ν
γ∗ =
γ h
,
(14)
the dimensionless flow problem after omitting the asterisks can be written as d2 F dz 2
+ 2M "
F −γ
"
dz
dF dz
" F +γ
d
dF dz
2
dF dz
+ 2M
# − (2iΩ + N + H ) F − 2HMF
dz dF
2
dz
+ 2M
dF
dF dz
dF
dF dz
dz dz
= 0,
(15)
#
dz
2
dF dF
= 1,
at z = 0,
(16)
= 0,
at z = 1,
(17)
#
where F = u + iv,
F = u − iv.
(18)
The formulation of this problem as defined in Eq. (15) is novel in that it includes terms in the porosity parameter H. 3. Results and discussion r
Eq. (15) is solved numerically using MATLAB subject to conditions (16) and (17). For the chosen values of the emerging flow parameters M , N , H , Ω and γ , the variation of velocity profiles u and v are illustrated graphically. Fig. 2 shows the effect of the slip parameter γ (= 0.5, 2.0, 5.0) on the velocity profiles u and v for the case M = N = Ω = H = 1.0. As γ increases, the effect of the magnitude of the non-linearity for u and v decreases. In Fig. 3 the effect of the angular velocity Ω (= 0.2, 1.0, 5.0) on the velocity profiles u and v is shown when M = N = H = 1.0 and γ = 0.5. As Ω increases, the effect of the magnitude of the non-linearity for u is reduced, while that for v initially increases and then decreases further away from the lower plate. Fig. 4 elucidates the influence of the porosity parameter H (= 0.5, 2.0, 5.0) for the cases M = N = Ω = 1.0 and γ = 0.5. Near the lower plate, u and v are almost linear, whereas further away from the lower plate, the effect of the
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Fig. 2. Variation of the velocity profiles u and v for varying values of the slip parameter γ and fixed values of Ω , M , N and H.
Fig. 3. Variation of the velocity profiles u and v for varying values of the angular velocity Ω and fixed values of γ , M , N and H.
magnitude of the non-linearity for u and v is evident. Movement towards the upper plate results in the values for both u and v tending to 0. In Fig. 5 the velocity profiles u and v are shown for the Newtonian case (M = 0.0) and the non-Newtonian cases (M = 1.0, 2.0). The choice of the fixed parameters is N = Ω = H = 1.0 and γ = 0.5. As M increases, the effect of the magnitude of the non-linearity for u and v decreases. Near the lower plate (z = 0), u and v are almost linear, whereas nearer the upper plate (z = h), the effect of the magnitude of the non-linearity for u and v is evident. Movement towards the upper plate results in small values of both u and v . Fig. 6 denotes how the velocity profiles u and v change with the Hartman number N (= 0.0, 0.5, 1.0) and the choice of the other flow parameters M = Ω = H = 1.0 and γ = 0.5. The magnetic effect reduces the effect of the non-linearity for u and v . Nearer the upper plate the values of u and v are small.
S. Abelman et al. / Nonlinear Analysis: Real World Applications 10 (2009) 3329–3334
3333
Fig. 4. Variation of the velocity profiles u and v for varying values of the porosity parameter H and fixed values of γ , Ω , M and N.
Fig. 5. Variation on the velocity profiles u and v for varying values of M and fixed values of γ , Ω , N and H.
4. Concluding remarks In this paper we considered an incompressible third grade fluid that occupies the porous space between two rigid infinite plates. When there is partial slip of the fluid on the plates, we analysed the steady rotating flow of this fluid when the lower plate is suddenly moved. The effect of the emerging flow parameters on the velocity profiles u and v was illustrated graphically. It is interesting to note that for no-slip, that is γ = 0.0, (i) the velocity profile u is a linear function of z with a gradient of −1 for the porosity parameter H ∈ [0.5, 1], whereas for H ∈ [2, 5] the velocity profile u decreases non-linearly to zero; (ii) the velocity profile v remains constant equal to zero for all values of z ∈ [0, 1], with H ∈ [0.5, 5]. It is worth mentioning that for γ = 0.0, the flow analysis corresponding to no-slip is recovered. For M = 0.0, one obtains the flow problem for a viscous fluid. When H = 0.0, then flow in a non-porous medium as obtained in [23] is deduced. For N = 0.0 and Ω = 0.0, respective problems corresponding to hydrodynamic and non-rotating frame situations are obtained.
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S. Abelman et al. / Nonlinear Analysis: Real World Applications 10 (2009) 3329–3334
Fig. 6. Variation of the velocity profiles u and v for varying values of the Hartman number N and fixed values of γ , Ω , M and H.
Another interesting observation is that bifurcations are not present. Bifurcations observed in the velocity profiles for u and v in [23] are not observed here. Acknowledgments TH thanks DECMA and CAM (University of the Witwatersrand, Johannesburg) for their financial support and warm hospitality during his visit. EM acknowledges support from the National Research Foundation, South Africa, under Grant No. 2053745. The authors thank the referees for their encouraging comments. References [1] K.R. Rajagopal, On boundary conditions for fluids of the differential type, in: A. Sequira (Ed.), Navier–Stokes equations and related nonlinear problems, (Funchal, 1994), Plenum, New York, 1995, pp. 273–278. [2] K.R. Rajagopal, A.S. Gupta, An exact solution for the flow of a non-Newtonian fluid past an infinite porous plate, Meccanica 19 (1984) 158–160. [3] K.R. Rajagopal, A.Z. Szeri, W. Troy, An existence theorem for the flow of a non-Newtonian fluid past an infinite porous plate, Internat. J. Non-Linear Mech. 21 (1986) 279–289. [4] C. Fetecau, C. Fetecau, Unsteady flow of Oldroyd-B fluid in a channel of rectangular cross-section, Internat. J. Non-Linear Mech. 40 (2005) 1214–1219. [5] C. Fetecau, C. Fetecau, Starting solutions for some unsteady unidirectional flows of a second grade fluid, Int. J. Eng. Sci. 43 (2005) 781–789. [6] C. Fetecau, C. Fetecau, D. Vieru, On some helical flows of Oldroyd-B fluids, Acta Mech. 189 (2007) 53–63. [7] I. Chen, C.K. Chen, Y.T. Yang, Unsteady unidirectional flow of an Oldroyd-B fluid in a circular duct with different given volume flow rate conditions, Heat Mass Trans. 40 (2004) 203–209. [8] W.C. Tan, T. Masuoka, Stokes’ first problem for a second grade fluid in a porous half-space with heated boundary, Internat. J. Non-Linear Mech. 40 (2005) 515–522. [9] W.C. Tan, T. Masuoka, Stokes’ first problem for an Oldroyd-B fluid in a porous half-space, Phys. Fluids 17 (2005) 023101–023107. [10] M. Sajid, T. Hayat, S. Asghar, On the analytic solution of the steady flow of a fourth grade fluid, Phys. Lett. A 355 (2006) 18–26. [11] M. Khan, T. Hayat, S. Asghar, Exact solution for MHD flow of a generalized Oldroyd-B fluid with modified Darcy’s law, Int. J. Eng. Sci. 44 (2006) 333–339. [12] T. Hayat, N. Ahmed, M. Sajid, Analytic solution for MHD flow of a third grade fluid in a porous channel, J. Comput. Appl. Math. 214 (2008) 572–582. [13] P. Puri, Rotating flow of an elastico-viscous fluid on an oscillating plate, Z. Angew. Math. Mech. 54 (1974) 743–745. [14] P. Puri, P.K. Kulshrestha, Rotating flow of non-Newtonian fluids, Appl. Anal. 4 (1974) 131–140. [15] T. Hayat, K. Hutter, Rotating flow of a second-order fluid on a porous plate, Internat. J. Non-Linear Mech. 39 (2004) 767–777. [16] T. Hayat, S. Nadeem, S. Asghar, A.M. Siddiqui, Fluctuating flow of a third order fluid on a porous plate in a rotating medium, Internat. J. Non-Linear Mech. 36 (2001) 901–916. [17] T. Hayat, K. Hutter, S. Asghar, A.M. Siddiqui, MHD flows of an Oldroyd-B fluid, Math. Comput. Modeling 36 (2002) 987–995. [18] T. Hayat, M.I. Hameed, S. Asghar, A.M. Siddiqui, Some steady MHD flows of the second order fluid, Meccanica 39 (2004) 345–355. [19] T. Hayat, Y. Wang, K. Hutter, Hall effects on the unsteady hydromagnetic oscillatory flow of a second grade fluid, Int. J. Non-Linear Mech. 39 (2004) 1027–1037. [20] T. Hayat, K. Hutter, S. Nadeem, S. Asghar, Unsteady hydromagnetic rotating flow of a conducting second grade fluid, Z. Angew. Math. Phys. 55 (2004) 626–641. [21] T. Hayat, S.B. Khan, M. Khan, The influence of Hall current on the rotating oscillating flows of an Oldroyd-B fluid in a porous medium, Nonlin. Dyn. 47 (2007) 353–362. [22] T. Hayat, S. Abelman, A numerical study of the influence of slip boundary condition on rotating flow, Int. J. Comput. Fluid Dyn. 21 (2007) 21–27. [23] S. Abelman, E. Momoniat, T. Hayat, Steady MHD flow of a third grade fluid in a rotating frame and porous space, Nonlinear Anal. RWA 10 (6) (2009) 3322–3328. [24] C. Truesdell, W. Noll, The non-linear field theories of mechanics, in: Handbuch der Physik, III/3, Springer, Berlin, Heidelberg, New York, 1965. [25] R.S. Rivlin, J.L. Ericksen, Stress-deformation relations for isotropic materials, J. Rat. Mech. Anal. 4 (1955) 323–325. [26] R.L. Fosdick, K.R. Rajagopal, Thermodynamics and stability of fluids of third grade, Proc. R. Soc. Lond. A 339 (1980) 351–377.
Contents lists available at ScienceDirect
Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa
Couette flow of a third grade fluid with rotating frame and slip condition S. Abelman a,∗ , E. Momoniat a , T. Hayat a,b a
Centre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050, South Africa b
Department of Mathematics, Quaid-i-Azam University, Islamabad, 45320, Pakistan
article
info
Article history: Received 30 April 2008 Accepted 6 October 2008 Keywords: Third grade fluid Partial slip Porous space Rotating flow
abstract An incompressible third grade fluid occupies the porous space between two rigid infinite plates. The steady rotating flow of this fluid due to a suddenly moved lower plate with partial slip of the fluid on the plate is analysed. The fluid filling the porous space between the two plates is electrically conducting. The flow modeling is developed by employing a modified Darcy’s law. A numerical solution of the governing problem consisting of a non-linear ordinary differential equation and non-linear boundary conditions is obtained and discussed. Several limiting cases of the arising problem can be obtained by choosing suitable parameters. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction It is a well-known fact that non-Newtonian fluids are more appropriate than Newtonian fluids in industrial and technological applications. A large variety of consumer goods containing high concentration of glass or carbon fibres, paints and lubricants containing polymer additives, many foodstuffs and biological fluids are non-Newtonian. The constitutive equations of such fluids vary greatly in complexity. The class of Newtonian flows for which there exists analytic solution of the Navier–Stokes equations is very restricted. This class is further narrowed down when non-Newtonian fluids are taken into account. In fact, the resulting equations of non-Newtonian fluids are of higher order [1–3], more non-linear and complicated than the Navier–Stokes equations and thus make the task of obtaining accurate solutions difficult. Being scientifically appealing and challenging, non-Newtonian fluids in a non-rotating frame have been studied extensively. Some recent studies may be found in references [4–12]. Attempts have been also made on flows of non-Newtonian fluids in a rotating frame [13–22]. Rotating flows of non-Newtonian fluids in a porous space and partial slip have not received much attention in the literature. A viscous non-Newtonian fluid normally sticks to the boundary. In such cases the no-slip condition is no longer valid. There are numerous situations where the no-slip condition between the fluid and the boundary does not hold. For example, the fluid may be particulate or such that it could be a rarefied gas with a suitable value of the Kundsen number. Also, rotating flows of non-Newtonian fluids in a porous space have geophysical applications. In our paper [23] we considered a numerical solution for steady state rotating and magnetohydrodynamic (MHD) flow of a third grade fluid past a rigid plate with slip. In this paper we present a numerical solution for the steady Couette flow of a thermodynamic compatible third grade fluid filling the porous space in a rotating frame. Partial slip effects are taken into account. For a third grade fluid, both the governing equation and boundary conditions are non-linear, whereas the corresponding problem for a viscous fluid is linear. The paper is organized as follows. The problem is formulated in Section 2. In Section 3 the numerical solution is discussed and in Section 4 concluding remarks are presented.
∗
Corresponding author. E-mail addresses: [email protected] (S. Abelman), [email protected] (E. Momoniat), [email protected] (T. Hayat).
1468-1218/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2008.10.068
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S. Abelman et al. / Nonlinear Analysis: Real World Applications 10 (2009) 3329–3334
Fig. 1. Graphical representation of the problem.
2. Formulation of the problem Consider an incompressible third grade fluid occupying the porous space between two rigid infinite plates. The lower plate at z = 0 is suddenly jerked and the upper plate at z = h is kept stationary. A graphical representation of the problem is shown in Fig. 1. The fluid is electrically conducting in the presence of a constant magnetic field of strength B0 parallel to the z-axis. The magnetic Reynolds number is taken small and hence the induced magnetic field is neglected. The applied and induced electric fields are absent. The space 0 < z < h between the two plates is porous. Furthermore, the plates and the ˆ where kˆ is a unit vector in the z-direction. The hydromagnetic steady fluid rotate with constant angular velocity Ω = Ω k rotating flow in a porous space is governed by the following equations: div V = 0,
(1)
ρ [(V · ∇) V + 2Ω × V] = −∇ pˆ + div S − σ
B20 V
+ R.
(2)
In the above equations, V is the velocity, ρ is the fluid density, σ is the electrical conductivity of the fluid, R is Darcy’s ρ resistance in a porous space, S is the extra stress tensor and pˆ = p − 2 Ω 2 x2 + y2 . Since the plates are rigid, the incompressibility condition (1) requires that V = (u (z ) , v (z ) , 0) ,
(3)
in which u and v are the velocity components in the x- and y-directions respectively. Note that for rigid infinite plates and steady flow, u and v are dependent on z. The constitutive equation of the extra stress tensor is [24] S = µ + β3 trA21
A1 + α1 A2 + α2 A21 + β1 A3 + β2 (A1 A2 + A2 A1 ) ,
(4)
where µ is the dynamic viscosity, αi (i = 1, 2) and βi (i = 1 − 3) are material constants and the first three Rivlin–Ericksen tensors Ai (i = 1 − 3) are defined through the following relationships [25]: A1 = L + LT , An+1 =
dAn dt
L = ∇ V,
+ An L + LT An
(n = 1, 2) .
(5)
Fosdick and Rajagopal [26] have shown that the Clausius–Duhem inequality and the assumption that the Helmholtz free energy is minimum in equilibrium requires that
µ ≥ 0,
α1 ≥ 0, β1 = β2 = 0, p |α1 + α2 | ≤ 24µβ3 .
β3 ≥ 0, (6)
Employing the same procedure as in references [8,9,21], the following expression of R has been proposed in the present case R=−
φ k
" µ + 2β3
(
du dz
2
+
dv dz
2 )#
V,
in which φ (0 < φ < 1) and k are the porosity and permeability of the porous space.
(7)
S. Abelman et al. / Nonlinear Analysis: Real World Applications 10 (2009) 3329–3334
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In view of Eqs. (3)–(7), Eq. (2) in the absence of a modified pressure gradient yields
" ( 2 )# 2 σ B20 φ dv du + − u− + u, −2Ω v = ν 2 + µ + 2β3 dz ρ dz dz dz dz ρ kρ dz dz " ( " ( 2 )# 2 )# 2 2 d2 v 2β3 d dv dv σ B20 φ dv du du 2Ω u = ν 2 + + − v− + v, µ + 2β3 dz ρ dz dz dz dz ρ kρ dz dz 2β3 d
d2 u
"
du
(
du
2
dv
2 )#
(8)
(9)
where ν is the kinematic viscosity. On account of the slip velocities, the boundary conditions on the plates take the form
" u − U0 − γ
"
dv
v−γ
dz
" u+γ
du dz
" v+γ
dv dz
du
2β3 du
(
du
2
dv
2 )#
+ = 0, at z = 0, µ dz dz dz ( 2 )# 2 2β3 dv dv du + + = 0, at z = 0, µ dz dz dz ( 2 )# 2 du 2β3 du dv + + = 0, at z = h, µ dz dz dz ( 2 )# 2 2β3 dv du dv + + = 0, at z = h, µ dz dz dz dz
+
(10)
(11)
(12)
(13)
in which γ is a slip parameter having dimension of length and U0 is a mainstream reference velocity. Defining u
u∗ =
N =
U0
σ
,
v
v∗ =
B20 h2
,
ρν
,
U0
z∗ =
β ρν
2 3 U0 h2
M =
,
z h
,
H =
Ω∗ =
φ h2 k
,
Ω h2 , ν
γ∗ =
γ h
,
(14)
the dimensionless flow problem after omitting the asterisks can be written as d2 F dz 2
+ 2M "
F −γ
"
dz
dF dz
" F +γ
d
dF dz
2
dF dz
+ 2M
# − (2iΩ + N + H ) F − 2HMF
dz dF
2
dz
+ 2M
dF
dF dz
dF
dF dz
dz dz
= 0,
(15)
#
dz
2
dF dF
= 1,
at z = 0,
(16)
= 0,
at z = 1,
(17)
#
where F = u + iv,
F = u − iv.
(18)
The formulation of this problem as defined in Eq. (15) is novel in that it includes terms in the porosity parameter H. 3. Results and discussion r
Eq. (15) is solved numerically using MATLAB subject to conditions (16) and (17). For the chosen values of the emerging flow parameters M , N , H , Ω and γ , the variation of velocity profiles u and v are illustrated graphically. Fig. 2 shows the effect of the slip parameter γ (= 0.5, 2.0, 5.0) on the velocity profiles u and v for the case M = N = Ω = H = 1.0. As γ increases, the effect of the magnitude of the non-linearity for u and v decreases. In Fig. 3 the effect of the angular velocity Ω (= 0.2, 1.0, 5.0) on the velocity profiles u and v is shown when M = N = H = 1.0 and γ = 0.5. As Ω increases, the effect of the magnitude of the non-linearity for u is reduced, while that for v initially increases and then decreases further away from the lower plate. Fig. 4 elucidates the influence of the porosity parameter H (= 0.5, 2.0, 5.0) for the cases M = N = Ω = 1.0 and γ = 0.5. Near the lower plate, u and v are almost linear, whereas further away from the lower plate, the effect of the
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Fig. 2. Variation of the velocity profiles u and v for varying values of the slip parameter γ and fixed values of Ω , M , N and H.
Fig. 3. Variation of the velocity profiles u and v for varying values of the angular velocity Ω and fixed values of γ , M , N and H.
magnitude of the non-linearity for u and v is evident. Movement towards the upper plate results in the values for both u and v tending to 0. In Fig. 5 the velocity profiles u and v are shown for the Newtonian case (M = 0.0) and the non-Newtonian cases (M = 1.0, 2.0). The choice of the fixed parameters is N = Ω = H = 1.0 and γ = 0.5. As M increases, the effect of the magnitude of the non-linearity for u and v decreases. Near the lower plate (z = 0), u and v are almost linear, whereas nearer the upper plate (z = h), the effect of the magnitude of the non-linearity for u and v is evident. Movement towards the upper plate results in small values of both u and v . Fig. 6 denotes how the velocity profiles u and v change with the Hartman number N (= 0.0, 0.5, 1.0) and the choice of the other flow parameters M = Ω = H = 1.0 and γ = 0.5. The magnetic effect reduces the effect of the non-linearity for u and v . Nearer the upper plate the values of u and v are small.
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Fig. 4. Variation of the velocity profiles u and v for varying values of the porosity parameter H and fixed values of γ , Ω , M and N.
Fig. 5. Variation on the velocity profiles u and v for varying values of M and fixed values of γ , Ω , N and H.
4. Concluding remarks In this paper we considered an incompressible third grade fluid that occupies the porous space between two rigid infinite plates. When there is partial slip of the fluid on the plates, we analysed the steady rotating flow of this fluid when the lower plate is suddenly moved. The effect of the emerging flow parameters on the velocity profiles u and v was illustrated graphically. It is interesting to note that for no-slip, that is γ = 0.0, (i) the velocity profile u is a linear function of z with a gradient of −1 for the porosity parameter H ∈ [0.5, 1], whereas for H ∈ [2, 5] the velocity profile u decreases non-linearly to zero; (ii) the velocity profile v remains constant equal to zero for all values of z ∈ [0, 1], with H ∈ [0.5, 5]. It is worth mentioning that for γ = 0.0, the flow analysis corresponding to no-slip is recovered. For M = 0.0, one obtains the flow problem for a viscous fluid. When H = 0.0, then flow in a non-porous medium as obtained in [23] is deduced. For N = 0.0 and Ω = 0.0, respective problems corresponding to hydrodynamic and non-rotating frame situations are obtained.
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Fig. 6. Variation of the velocity profiles u and v for varying values of the Hartman number N and fixed values of γ , Ω , M and H.
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