- Email: [email protected]
Numerical study of MHD two-phase Couette flow analysis for fluid-particle suspension between moving parallel plates
G Model
JTICE-927; No. of Pages 8 Journal of the Taiwan Institute of Chemical Engineers xxx (2014) xxx–xxx
Contents lists available at ScienceDirect
Journal of the Taiwan Institute of Chemical Engineers journal homepage: www.elsevier.com/locate/jtice
Numerical study of MHD two-phase Couette flow analysis for fluid-particle suspension between moving parallel plates M. Hatami a, Kh. Hosseinzadeh b, G. Domairry c,*, M.T. Behnamfar d a
Esfarayen University, Mechanical Engineering Department, Esfarayen, North Khorasan, Iran Department of Mechanical Engineering, Sari Branch, Islamic Azad University, Sari, Iran c Babol University of Technology, Department of Mechanical Engineering, Babol, Iran d Isfahan University of Technology, Department of Chemistry, Isfahan, Iran b
A R T I C L E I N F O
A B S T R A C T
Article history: Received 6 November 2013 Received in revised form 15 April 2014 Accepted 18 May 2014 Available online xxx
In this study, steady and unsteady magneto-hydrodynamic (MHD) Couette flows between two parallel infinite plates have been studied through numerical Differential Quadrature Method (DQM) and analytical Differential Transformation Method (DTM), respectively. Coupled equations by taking the viscosity effect of the two phases for fixed and moving plates have been introduced. The precious contribution of the present study is introducing new, fast and efficient numerical and analytical methods in a two-phase MHD Couette fluid flow. Results are compared with those previously obtained by using Finite Difference Method (FDM). The velocity profiles of two phases are presented and a parametric study of physical parameters involved in the problem is conducted. As an outcome, when magnetic source is fixed relative to the moving plate, by increasing the Hartmann number, velocity profiles for both phases increased, but when it is fixed relative to the fluid an inverse treatment is observed. ß 2014 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
Keywords: Two-phase flow Couette flow Differential Quadrature Method (DQM) Differential Transformation Method (DTM) Particle suspension
1. Introduction The study of fluid-particle two-phase flow is important in many science applications such as petroleum transport, waste water treatment, combustion, power plant piping, corrosive particles in engine oil flow and mining, smoke emission from vehicles, etc. The cases that particles come into contact with a solid or fluid boundary have been immense interest of many researchers such as nanofluids which can be found in Sheikholeslami et al. works [1–5]. Various processes such as filtration, combustion, air and water pollution, coal transport and cleaning, micro contamination control and xerography involve the particle transportation and deposition. Numerical solutions of equation derived from the balance equation of external forces applied to a particle in interaction with drag, virtual mass force, the Basset force, lift force, etc. have been broadly considered in literature. It is known that, a fast and high accurate solution is usually the preferred and convenient method in engineering area because of less computational work as well as high accuracy.
* Corresponding author. Tel.: +98 111 3234205; fax: +98 111 3234205. E-mail addresses: [email protected] (M. Hatami), [email protected] (G. Domairry).
Chamkha [6] studied the unsteady laminar flow and heat transfer of a particulate suspension in an electrically conducting fluid through channels and circular pipes in the presence of uniform transverse magnetic field. Jha and Apere [7] provided the semianalytical solution to the unsteady MHD two-phase Couette flow between two infinite parallel plates using the Laplace transform technique and the Riemann-sum approximation method and compared their results with those using Finite Difference Method (FDM). The unsteady magneto-hydrodynamic flow of an electrically conducting viscous incompressible non-Newtonian Bingham fluid bounded by two parallel non-conducting porous plates is studied with heat transfer considering the Hall effect by Attia et al. [8]. Gedik et al. [9] investigated the unsteady flow of two-phase (solid/liquid) flow for laminar, incompressible and electrically conducting fluid through a circular pipe in the presence of a uniform transverse external magnetic and electrical field. Coupled equations in velocity and magnetic field for unsteady MHD flow through a pipe of rectangular section have been solved by Shakeri and Dehghan [10] with using combined finite volume method and spectral element technique. Jalaal et al. [11] solved a spherical particle’s motion in Couette flow using Homotopy perturbation method (HPM) and got comparable results to numerical ones. The unsteady rolling motion of a spherical particle restricted to a tube was studied
http://dx.doi.org/10.1016/j.jtice.2014.05.018 1876-1070/ß 2014 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
Please cite this article in press as: Hatami M, et al. Numerical study of MHD two-phase Couette flow analysis for fluid-particle suspension between moving parallel plates. J Taiwan Inst Chem Eng (2014), http://dx.doi.org/10.1016/j.jtice.2014.05.018
G Model
JTICE-927; No. of Pages 8 2
M. Hatami et al. / Journal of the Taiwan Institute of Chemical Engineers xxx (2014) xxx–xxx
analytically by Jalaal and Ganji [12]. They obtained an exact solution of particle velocity and acceleration motion under some practical conditions through applying HPM. Jalaal and Ganji [13] proposed an analytical solution for acceleration motion of a spherical particle rolling down an inclined boundary with drag coefficient which is correlated linearly to Re in a specific range using HPM. Various inclination angles were studied and observed that settling velocity, acceleration duration and displacement are proportional to amount of inclination angle while for a constant inclination angle; settling velocity and acceleration duration are decreased by increasing the fluid viscosity. Recently, Ghasemi et al. [14] discussed about the convergence and accuracy of Variational Iteration Method (VIM) and Adomian Decomposition Method (ADM) for solving the motion of a spherical particle in Couette fluid flow. Differential Quadrature Method (DQM) is a numerical technique which was first developed by Richard Bellman and his associates in the early 1970s [15]. The DQM applications were rapidly developed thanks to the innovative works in computation of weighting coefficient by other scientists and recently this method was applied in many engineering problems [16–20]. The concept of differential transformation method (DTM) was first introduced by Zhou [21] in 1986 and it was used to solve both linear and nonlinear initial value problems in electric circuit analysis. This method can be applied directly for linear and nonlinear differential equation without requiring linearization, discretization, or perturbation and this is the main benefit of this method. Ghafoori et al. [22] used the DTM for solving the nonlinear oscillation equation. Many other papers which used these types of analytical and numerical methods for solving the engineering problems can be found in the literature [23–32]. Motivated by the above-mentioned works, the main objective of the present research paper is to introduce DQM as a numerical and high efficient technique on unsteady two-phase MHD Couette fluid flow for obtaining the velocity profiles of both the fluid and particle phases. Flow in the channel is generated by the impulsive movement of the lower plate and DTM is applied to problem for investigating the steady state behavior. Results show that these two methods are accurate and for all constant values in coupled equations have excellent agreement with numerical FDM method.
(a) It is assumed that the fluid and the particles are interacting as a continuum. (b) The fluid phase is assumed to be electrically conducting, but the channel walls are assumed to be electrically nonconducting. (c) Particles are not subject to electro-magnetic force. So, equation of motion for the particles does not contain the force term due to the electro-magnetic field. (d) Velocity of both the phases is considered to be irrotational. According to [7] and [33] model, the equation of motion with electromagnetic force added is given as shown below [7]
rf
¼0
~f r:V (1)
@V~ p ~f V ~ p Þ; r:V ~p ¼ 0 ~ p þ K 0 ðV ¼ f p P þ f p m p r2 V (2) @t ~ f ¼ ðu0 ; y0 ; w0 Þ and V ~ p ¼ ðu0 ; y0 ; w0 Þ denotes, respecwhere V p p p f f f rp
tively the velocity components of the fluid and the particles and f f ; f p represent the fluid and particle quantity in unit volume ~ f BÞ ~ which where f f þ f p ¼ 1 [7]. In general form, ~ J ¼ s ð~ EþV electric field is not considered in this paper and should be zero. Also, in the present study magnetic field is applied in one direction, ~ ¼ ð0; B0 ; 0Þ. Under the assumptions made in the present i.e.: B problem, Eqs. (1) and (2) become:
@u0f @ 2 u0f s B20 0 0 ¼ nf f f þ k ðu0p u0f Þ u 2 @t r f @z0
(3)
@u0p @2 u0p 0 ¼ np f p þ k ðu0f u0p Þ @t @z0 2
(4)
Eq. (3) is valid when the magnetic field is fixed relative to the fluid. If the magnetic field is also accelerated with the same velocity as the plate, we must account for the relative motion. Thus, Eq. (3) turns to
@u0f @ 2 u0f s B20 0 0 ¼ nf f f þ k ðu0p u0f Þ ðu f Utn Þ 2 0 @t r @z
2. Problem description The motion of an unsteady, laminar, viscous fluid/particle suspension between two infinite horizontal plates located at the z = 0 and z = h planes is considered. A uniform transverse magnetic field is applied to the flow and it is assumed that no applied or polarization voltage exists. When t > 0 the lower plate begins to move in its own plane with a velocity Utn where U is a constant and the upper plate remains fixed. A schematic view of described problem is shown in Fig. 1. Following assumptions are considered by Rahmatulin [33] and Jha and Apere [7] to simplification the problem:
@V~ f ~p V ~f Þ þ~ ~ ~ f þ K 0 ðV ¼ f f P þ f f m f r2 V J B; @t
(5)
Eqs. (3) and (5) are combined together to obtain
@u0f @ 2 u0f s B20 0 0 ¼ nf f f þ k ðu0p u0f Þ ðu f UKtn Þ r @t @z0 2 where 0 K¼ 1
when B0 is fixed relative to the fluid when B0 is fixed relative to the moving plate
(6)
(7)
The initial and boundary conditions for the present problem are 0
u f ¼ u0p ¼ 0; u0f ¼ u0p ¼ Utn ; u0f ¼ u0p ¼ 0;
0 z0 h z0 ¼ 0; z0 ¼ h;
and t 0; t > 0; t > 0:
(8)
by using the following equations, non-dimensionalised governing system of equations will be found. ðu0f ; u0p Þ z0 v1 ; ðu0f ; u0p Þ ¼ ; h ¼ t f f 2 ; Ha h U h sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2 s kh : ; k¼ ¼ B0 h nf f f nfrf f f
z¼
Fig. 1. Schematic view of the problem.
(9)
where Ha represents the Hartmann number which is the ratio of electromagnetic force to the viscous force. Considering the impulsive motion of the lower plate which corresponds to n = 0
Please cite this article in press as: Hatami M, et al. Numerical study of MHD two-phase Couette flow analysis for fluid-particle suspension between moving parallel plates. J Taiwan Inst Chem Eng (2014), http://dx.doi.org/10.1016/j.jtice.2014.05.018
G Model
JTICE-927; No. of Pages 8 M. Hatami et al. / Journal of the Taiwan Institute of Chemical Engineers xxx (2014) xxx–xxx
and substituting the dimensionless quantities into the governing equations and the boundary condition, the following dimensionless equation results 8 > @u f @2 u f > > ¼ þkðu p u f Þ Ha2 ðu f KÞ < @h @z2 (10) > @u @2 u p > > : p ¼ R f Rv þkðu u Þ p f @h @z2
Table 1 Some fundamental operations of the differential transform method. Origin function
Transformed function
xðtÞ ¼ a f ðxÞ bgðtÞ
XðkÞ ¼ aFðkÞ bGðkÞ
m
xðtÞ ¼
d f ðtÞ dt m
ðk þ mÞ!Fðk þ mÞ k! k X XðkÞ ¼ FðlÞGðk lÞ l¼0 1; if k ¼ m; XðkÞ ¼ dðk mÞ ¼ 0; if k 6¼ m: 1 XðkÞ ¼ k! vk kp sin þa XðkÞ ¼ k! 2 vk kp cos þa XðkÞ ¼ k! 2 XðkÞ ¼
xðtÞ ¼ f ðtÞgðtÞ xðtÞ ¼ t m
where R f ¼ f p = f f ; Ry ¼ y p =y f ; and the initial and boundary conditions (8) also become u f ¼ u p ¼ 0; u f ¼ u p ¼ 1; u f ¼ u p ¼ 0;
(11)
3. Principles of methods 3.1. Differential Quadrature Method (DQM) In this study, polynomial expansion based differential quadrature, as introduced by Quan and Chang [34,35], is applied for solving the problem. Several attempts have been made by researchers to develop polynomial based differential quadrature methods. One of the most useful approaches is the one that uses the following Lagrange interpolation polynomials as test functions: MðxÞ
g k ðxÞ ¼
ðx xk ÞM ð1Þ ðxk Þ
;
k ¼ 1; 2; :::; N
(12)
where:
N Y
¼
M
ð1Þ
ðxi Þ
ðxi xk Þ
(13)
k¼1 k 6¼ i By applying the above equation at N grid points, the following algebraic formulations to compute the weighting coefficients are developed: ð1Þ
N Y 1 xi xk ; x j xi k¼1 x j xk
ð1Þ
i 6¼ j;
Ai j
N X k¼1k 6¼ i
1 ; xi xk
N X ð1Þ ð1Þ Aik Ak j
ð2Þ
Ai j ¼
(14)
k¼1
where A(1) and A(2) denote the weighting coefficients of the first and second order derivatives of the function f(r) with respect to the r direction. N is the number of grid points chosen in the r direction. The differential quadrature approximation can be easily extended from the above formulation to other coordinates. The first order derivatives in the two-dimensional formulation are approximated by:
@f @r
ij
N X ð1Þ Ail f l j ; l¼1
@f @z
ij
M X
ð1Þ
B jm f im
@2 f @z2 2
! !i j
@ f @r@z
3.2. Differential Transformation Method (DTM) Basic definitions and operations of differential transformation are introduced as follows. Differential transformation of the function f ðhÞ is defined as follows: " # k 1 d f ðhÞ (17) FðkÞ ¼ k! dhk
l¼1
!i j ij
n¼1 N X
P X ð1Þ ð1Þ Ail B jm f lm
l¼1
m¼1
f ðhÞ ¼
1 X FðkÞðh h0 Þk
(18)
k¼0
by combining Eqs. (17) and (18) f ðhÞ can be obtained: " # k 1 X d f ðhÞ ðh h0 Þk f ðhÞ ¼ k k! dh k¼0
(19)
Eq. (19) implies that the concept of the differential transformation is derived from Taylor’s series expansion, but the method does not evaluate the derivatives symbolically. However, relative derivatives are calculated by an iterative procedure that is described by the transformed equations of the original functions. From the definitions of Eqs. (17) and (18), it is easily proven that the transformed functions comply with the basic mathematical operations shown in below. In real applications, the function f ðhÞ in Eq. (19) is expressed by a finite series and can be written as: f ðhÞ ¼
N X ð2Þ Ail f l j ;
P X ð2Þ Bkn f i jn ;
In Eq. (17), f ðhÞ is the original function and FðkÞ is the transformed function which is called the T-function (it is also called the spectrum of the f ðhÞ at h ¼ h0 , in the k domain). The differential inverse transformation of FðkÞ is defined as:
(15)
m¼1
And the second order derivatives can be approximated by:
@2 f @r 2
where A(1) and B(1) denote the weighting coefficients of the first order derivatives; A(2) and B(2) denote the weighting coefficients of the second order derivatives of the function f(r,z) with respect to the r and z-directions, respectively; N, and P are the number of grid points chosen in the r and z-directions, respectively.
h¼ h0
k 6¼ i; j
¼
xðtÞ ¼ cos ðvt þ aÞ
h¼h0
MðxÞ ¼ ðx x1 Þðx x2 Þ ðx xN Þ;
Ai j ¼
xðtÞ ¼ exp ðtÞ xðtÞ ¼ sin ðvt þ aÞ
and h 0; h > 0; h > 0:
0z1 z ¼ 0; z ¼ 1;
3
(16)
N X FðkÞðh h0 Þk
(20)
k¼0
P k Eq. (20) implies that f ðhÞ ¼ 1 k¼Nþ1 ðFðkÞðh h0 Þ Þ is negligibly small, where N is series size. Theorems to be used in the transformation procedure, which can be evaluated from Eq. (18), are given in Table 1. 4. Results and discussion The polynomial expansion based numerical differential quadrature is used for the non-dimensional time (h) and the z directions
Please cite this article in press as: Hatami M, et al. Numerical study of MHD two-phase Couette flow analysis for fluid-particle suspension between moving parallel plates. J Taiwan Inst Chem Eng (2014), http://dx.doi.org/10.1016/j.jtice.2014.05.018
G Model
JTICE-927; No. of Pages 8 4
M. Hatami et al. / Journal of the Taiwan Institute of Chemical Engineers xxx (2014) xxx–xxx
Table 2 Non-uniform grid points in numerical method (DQM) for h 2 [0,1]. Time step number
Non-dimensional time (h) (Eq. (22))
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.0000000000 0.0125360439 0.0495155661 0.1090842588 0.1882550991 0.2830581303 0.3887395332 0.5000000000 0.6112604668 0.7169418697 0.8117449009 0.8909157412 0.9504844339 0.9874639561 1.0000000000
Table 4 Effect of interpolate point numbers in DQM on the velocity profiles presented in Table 3. K
z
Table 3 Comparison between DQM and FDM results for k = 50, h = 0.5, Ha = 4, Rf = 0.25 and Rv = 1.
P = N = 15
P = N = 30
Fluid
Particle
Fluid
Particle
0.2
0 1
0.4822 0.9509
0.5132 0.9488
0.4831 0.9515
0.5125 0.9490
0.4
0 1
0.2339 0.8835
0.2494 0.8761
0.2336 0.8849
0.2491 0.8796
0.6
0 1
0.1089 0.7582
0.1161 0.7417
0.1090 0.7583
0.1160 0.7416
0.8
0 1
0.0426 0.5064
0.0457 0.4774
0.0425 0.5071
0.0457 0.4779
where U and V represent the fluid and particle velocities, respectively. In order to proceed with numerical computations, non-uniform points in the h and z directions are defined using the Chebyshev–Gauss-Lobatto polynomials [36]: H k1 1 1 cos p ; k1 ¼ 1; :::; P 2 P 1 L k2 1 1 cos ¼ p ; k2 ¼ 1; :::; N 2 N1
z
K
Fluid
Particle
Fluid
Particle
0.2
0 1
0.4822 0.9509
0.5132 0.9488
0.4857 0.9517
0.5111 0.9495
zk2
0.4
0 1
0.2339 0.8835
0.2494 0.8761
0.2333 0.8851
0.2487 0.8805
0.6
0 1
0.1089 0.7582
0.1161 0.7417
0.1090 0.7583
0.1160 0.7415
0.8
0 1
0.0426 0.5064
0.0457 0.4774
0.0425 0.5079
0.0457 0.4780
These non-uniform points for h are presented in Table 2 for P = 15. Obtained results by DQM are compared with FDM [7] in Table 3 (for fluid and particle phase for both when K = 0 and 1) and an excellent agreement between these two numerical methods is observed. But, solution of DQM for each run takes less than 1 min (approximately 30 s) while FDM results converge more than 1 min, so DQM is faster than FDM. Furthermore, unlike conventional methods such as Finite Element (FE) and FDM, DQM requires less grid points to obtain acceptable accuracy. Another advantage of DQM is that it transforms the differential equations into a set of analogous algebraic equations in terms of the unknown function values at the resembled points in the solution domain. Maybe the main advantage of DQM is its inherent conceptual simplicity, accuracy and the fact that easily programmable [37,38]. As described by Shu [37], Table 4 shows that DQM needs to less interpolation points and results of P = N = 15 are approximately similar to when grid points are twice, i.e. N = P = 30. Figs. 2–7 are related to the unsteady analysis. Fig. 2 demonstrates that an increase in time leads to an increase in the fluid and particle
DQM
FDM [7]
for obtaining the solution of Eq. (10) with boundary condition Eq. (11). After applying this method, following set of equations (Eq. (21)) will be obtained, 8 > > > > < > > > > :
P X m¼1 P X m¼1
! B j;m U i;m
!
B j;m V i;m
N X ð2Þ Ai;l U l; j l¼1
R f Rv
! kðV i; j U i; j Þ þ Ha2 ðU i; j KÞ ¼ 0 !
N X ð2Þ Ai;l V l; j
kðU i; j V i; j Þ ¼ 0
l¼1
(21)
hk1 ¼
(22)
Fig. 2. Effect of time (h) on velocity profiles for (a) fluid phase, (b) particle phase when K = 0, k = 50, Rv = 1, Rf = 0.25, Ha = 2.
Please cite this article in press as: Hatami M, et al. Numerical study of MHD two-phase Couette flow analysis for fluid-particle suspension between moving parallel plates. J Taiwan Inst Chem Eng (2014), http://dx.doi.org/10.1016/j.jtice.2014.05.018
G Model
JTICE-927; No. of Pages 8 M. Hatami et al. / Journal of the Taiwan Institute of Chemical Engineers xxx (2014) xxx–xxx
Fig. 3. Effect of time (h) on velocity profiles for (a) fluid phase, (b) particle phase when K = 1, k = 50, Rv = 1, Rf = 0.25, Ha = 2.
Fig. 4. Effect of Ha number on velocity profiles for (a) fluid phase, (b) particle phase when K = 0, k = 50, Rv = 1, Rf = 0.25, h = 0.5.
Fig. 5. Effect of Ha number on velocity profiles for (a) fluid phase, (b) particle phase when K = 1, k = 50, Rv = 1, Rf = 0.25, h = 0.5.
Please cite this article in press as: Hatami M, et al. Numerical study of MHD two-phase Couette flow analysis for fluid-particle suspension between moving parallel plates. J Taiwan Inst Chem Eng (2014), http://dx.doi.org/10.1016/j.jtice.2014.05.018
5
G Model
JTICE-927; No. of Pages 8 6
M. Hatami et al. / Journal of the Taiwan Institute of Chemical Engineers xxx (2014) xxx–xxx
Fig. 6. Effect of Rv number on velocity profiles for (a) fluid phase, (b) particle phase when K = 0, k = 50, Rv = 1, Rf = 0.25, z = 0.5.
velocities for K = 0. Also, the same trend is notice in Fig. 3 for both phases when K = 1. In fact, this effect is more pronounced on the particle field. Fig. 4 demonstrates the effect of Ha number on fluid and particle velocities. As seen by increasing Ha number, velocity profiles are decreased for both fluid and particle phases when K = 0 and h = 0.5. But, when K = 1, increasing the Ha number leads to increase in velocity profiles which are depicted in Fig. 5. Figs. 6 and 7, which show the velocity profiles versus time (h), demonstrate the effect of Rv on velocity profiles when K = 0 and K = 1, respectively. For K = 0 (Fig. 6), it is observed that an increase in Rv, result into an increase in velocities. For K = 1 (Fig. 7), at small times (h < 0.4) the same trend is observed, but as the time is increased further, the trend is completely reversed. It is found out that the fluid velocity is always greater than that of the particle for all the dimensional quantities considered. For the two phases by setting @ð Þ=@h in Eq. (10) to zero, the following ordinary differential equations then results for steady
state analysis, 8 2 > d uf > < þkðu p u f Þ Ha2 ðu f KÞ ¼ 0 dz2 2 > > : R Rv d u p þkðu u p Þ ¼ 0 f f dz2
(23)
For solving this coupled equation, DTM is applied due to its simplicity and high accuracy. From Table 1, Eq. (23) will be transformed to, ðmþ1Þðmþ 2ÞUðmþ2Þþ kðVðmÞ UðmÞÞ Ha2 ðUðmÞ K dm Þ ¼ 0 R f Rv ðm þ 1Þðm þ 2ÞVðm þ 2Þ þ kðUðmÞ VðmÞÞ ¼ 0 (24) and the boundary conditions will be, Uð0Þ ¼ 1;
Vð0Þ ¼ 1;
Uð1Þ ¼ a;
Vð1Þ ¼ b
(25)
Fig. 7. Effect of Rv number on velocity profiles for (a) fluid phase, (b) particle phase when K = 1, k = 50, Rv = 1, Rf = 0.25, z = 0.5.
Please cite this article in press as: Hatami M, et al. Numerical study of MHD two-phase Couette flow analysis for fluid-particle suspension between moving parallel plates. J Taiwan Inst Chem Eng (2014), http://dx.doi.org/10.1016/j.jtice.2014.05.018
G Model
JTICE-927; No. of Pages 8 M. Hatami et al. / Journal of the Taiwan Institute of Chemical Engineers xxx (2014) xxx–xxx Table 5 Comparison between DTM and FDM for k = 5, Ha = 1, Rf = 0.25 and Rv = 1 in steady state. DTM
Using Eq. (25) and solving Eq. (24), other DTM terms for velocities will be obtained as follows, (
z
K
FDM [7]
Fluid
Particle
Fluid
Particle
0.2
0 1
0.7603 0.8246
0.7792 0.8180
0.7609 0.8246
0.7772 0.8180
0.4
0 1
0.5477 0.6434
0.5728 0.6309
0.5493 0.6435
0.5665 0.6307
0.6
0 1
0.3522 0.4505
0.3853 0.4339
0.3564 0.4506
0.3692 0.4334
0.8
0 1
0.1676 0.2388
0.2115 0.2236
0.1753 0.2390
0.1819 0.2227
7
1 2 1 2 Ha Ha K 2 2 8 Vð2Þ ¼ 0 1 1 1 > > < Uð3Þ ¼ kb þ ka þ Ha2 a 6 6 6 1 kðb þ aÞ > > : Vð3Þ ¼ 6 R f Rv 8 1 1 1 > > Ha4 K þ kHa2 K þ Ha4 < Uð4Þ ¼ 24 24 24 2 1 kHa ðK 1Þ > > : Vð4Þ ¼ 24 R f Rv ::: Uð2Þ ¼
(26)
Results of DTM and FDM are compared in Table 5 for both K = 0 and K = 1 which a good agreement between them is observed. Figs. 8 and 9 show the Ha number effect in steady state on fluid and particle velocity profiles, respectively. As concluded from these
Fig. 8. Effect of Ha number on fluid phase velocity profiles when k = 50, Rv = 1, Rf = 0.2 for (a) K = 0, (b) K = 1 in steady state.
Fig. 9. Effect of Ha number on particle phase velocity profiles when k = 50, Rv = 1, Rf = 0.2 for (a) K = 0, (b) K = 1 in steady state.
Please cite this article in press as: Hatami M, et al. Numerical study of MHD two-phase Couette flow analysis for fluid-particle suspension between moving parallel plates. J Taiwan Inst Chem Eng (2014), http://dx.doi.org/10.1016/j.jtice.2014.05.018
G Model
JTICE-927; No. of Pages 8 8
M. Hatami et al. / Journal of the Taiwan Institute of Chemical Engineers xxx (2014) xxx–xxx
figures, increasing the Ha number makes a decrease in both velocity profiles for K = 0, but it makes an increase in velocity profiles when K = 1. 5. Conclusion In this paper, an analytical approach called Differential Transformation Method (DTM) and a numerical method, Differential Quadrature Method (DQM), have been successfully applied to find the accurate solution for the steady and unsteady MHD Couette flow between two parallel infinite plates. As a main outcome from the present study, it is observed that the results of DTM and DQM are in excellent agreement with FDM. Also, the effects of time h, Hartmann number Ha, the viscosity ratio Rv on fluid and particle velocity profiles are studied. It is concluded that velocities can be controlled by increasing or decreasing the magnetic field. References [1] Sheikholeslami M, Ganji DD, Gorji-Bandpy M, Soleimani S. Magnetic field effect on nanofluid flow and heat transfer using KKL model. J Taiwan Inst Chem Eng 2014;45:795–807. [2] Sheikholeslami M, Gorji-Bandpy M, Ganji DD, Soleimani S. Natural convection heat transfer in a cavity with sinusoidal wall filled with CuO-water nanofluid in presence of magnetic field. J Taiwan Inst Chem Eng 2014;45:40–9. [3] Sheikholeslami M, Gorji-Bandpy M, Soleimani S. Two phase simulation of nanofluid flow and heat transfer using heatline analysis. Int Comm Heat Mass Transf 2013;47:73–81. [4] Sheikholeslami M, Gorji-Bandpy M, Ganji DD. Numerical investigation of MHD effects on Al2O3-water nanofluid flow and heat transfer in a semi-annulus enclosure using LBM. Energ 2013;60:501–10. [5] Sheikholeslami M, Gorji Bandpy M, Ellahi R, Hassan M, Soleimani S. Effects of MHD on Cu-water nanofluid flow and heat transfer by means of CVFEM. J Magnet Magnetic Mater 2014;349:188–200. [6] Chamkha AJ. Unsteady laminar hydromagnetic fluid–particle flow and heat transfer in channels and circular pipe. Int J Heat Fluid Flow 2000;21:740–6. [7] Jha BK, Apere CA. Unsteady MHD two-phase Couette flow of fluid–particle suspension. App Math Model 2013;37:1920–31. [8] Attia HA, Sayed-Ahmed ME. Hall effect on unsteady MHD Couette flow and heat transfer of a Bingham fluid with suction and injection. App Math Model 2014;28:1027–45. [9] Gedik E, Kurt H, Recebli Z, Kec¸ebas A. Unsteady flow of two-phase fluid in circular pipes under applied external magnetic and electrical fields. Int J Thermal Sci 2012;53:156–65. [10] Shakeri F, Dehghan M. A finite volume spectral element method for solving magnetohydrodynamic (MHD) equations. Appl Numer Math 2011;61:1–23. [11] Jalaal M, Nejad MG, Jalili P, Esmaeilpour M, Bararnia H, Ghasemi E, et al. Homotopy perturbation method for motion of a spherical solid particle in plane Couette fluid flow. Comput Math With Appl 2011;61:2267–70. [12] Jalaal M, Ganji DD. On unsteady rolling motion of spheres in inclined tubes filled with incompressible Newtonian fluids. Adv Powder Technol 2011;22:58–67. [13] Jalaal M, Ganji DD. An analytical study on motion of a sphere rolling down an inclined plane submerged in a Newtonian fluid. Powd Technol 2010;198:82– 92. [14] Ghasemi SE, Palandi SJ, Hatami M, Ganji DD. Efficient analytical approaches for motion of a spherical solid particle in plane couette fluid flow using nonlinear methods. The J Math Comput Sci 2012;5:97–104.
[15] Bellman RE, Kashef BG, Casti J. Differential quadrature: a technique for rapid solution of non linear partial differential equations. J Comput Phys 1972;10:40– 52. [16] Yucel U, Boubaker K. Differential quadrature method (DQM) and Boubaker Polynomials Expansion Scheme (BPES) for efficient computation of the eigenvalues of fourth-order Sturm–Liouville problems. Appl Math Model 2012;36:158–67. [17] Hosseini-Hashemi Sh, Akhavan H, Rokni Damavandi Taher H, Daemi N, Alibeigloo A. Differential quadrature analysis of functionally graded circular and annular sector plates on elastic foundation. Mater Des 2010;31:1871–80. [18] Sun J, Zhu Zh. Upwind local differential quadrature method for solving incompressible viscous flow. Comput Methods Appl Mech Eng 2000;188: 495–504. [19] Akbari Alashti R, Khorsand M. Three-dimensional thermo-elastic analysis of a functionally graded cylindrical shell with piezoelectric layers by differential quadrature method. Int J Press Vess Pip 2011;88:167–80. [20] Akbari Alashti R, Khorsand M, Tarahhomi MH. Thermo-elastic analysis of a functionally graded spherical shell with piezoelectric layers by differential quadrature method. Sci Iranic B 2013;20:109–19. [21] Zhou JK. Differential transformation method and its application for electrical circuits. Wuhan, China: Hauzhang Univers. Press; 1986. [22] Ghafoori S, Motevalli M, Nejad MG, Shakeri F, Ganji DD, Jalaal M. Efficiency of differential transformation method for nonlinear oscillation: comparison with HPM and VIM. Curr Appl Phys 2011;11:965–71. [23] Hatami M, Ganji DD. Heat transfer and flow analysis for SA-TiO2 non-Newtonian nanofluid passing through the porous media between two coaxial cylinders. J Molec Liq 2013;188:155–61. [24] Hatami M, Ganji DD. Thermal performance of circular convective-radiative porous fins with different section shapes and materials. Energ Conver Manage 2013;76:185–93. [25] Hatami M, Hasanpour A, Ganji DD. Heat transfer study through porous fins (Si3N4 and AL) with temperature-dependent heat generation. Energ Convers Manage 2013;74:9–16. [26] Hatami M, Ganji DD. Investigation of refrigeration efficiency for fully wet circular porous fins with variable sections by combined heat and mass transfer analysis. Intl J Refrig 2014;40:140–51. [27] Hatami M, Ganji DD. Thermal flow analysis of microchannel heat sink (MCHS) cooled by Cu–water nanofluid using porous media approach and least square method. Energ Convers Manage 2014;78:347–58. [28] Hatami M, Nouri R, Ganji DD. Forced convection analysis for MHD Al2O3– water nanofluid flow over a horizontal plate. J Molec Liq 2013;187:294–301. [29] Hatami M, Ganji DD. Heat transfer and nanofluid flow in suction and blowing process between parallel disks in presence of variable magnetic field. J Molec Liq 2014;190:159–68. [30] Hatami M, Domairry G. Transient vertically motion of a soluble particle in a Newtonian fluid media. Powd Tech 2014;253:481–5. [31] Hatami M, Hatami J, Ganji DD. Computer simulation of MHD blood conveying gold nanoparticles as a third grade non-Newtonian nanofluid in a hollow porous vessel. Comput Meth Prog Biomed 2014;113:632–41. [32] Domairry G, Hatami M. Squeezing Cu–water nanofluid flow analysis between parallel plates by DTM-Pade´ method. J Molec Liq 2014;193:37–44. [33] Rahmatulin HA. Osnovi gidrodinamiki vzaimopronikayus_ih dvijeniy. Prikl Mat Mak 1956;20:56–65. [34] Quan JR, Chang CT. New insights in solving distributed system equations by quadrature methods-I. analysis. Comput Chem Eng 1989;13:779–88. [35] Quan JR, Chang CT. New insights in solving distributed system equations by quadrature methods-II. Numerical experiments. Comput Chem Eng 1989;13:1017–24. [36] Shu Ch, Richards BE. Application of generalized differential quadrature to solve two-dimensional incompressible Navier–Stokes equations. Int J Numer Methods Fluids 1992;15:791–8. [37] Shu Ch. Differential quadrature and its application in engineering. Singapore: Springer; 1999. [38] Hatami M, Ganji DD. Motion of a spherical particle in a fluid forced vortex by DQM and DTM. Partic 2014. http://dx.doi.org/10.1016/j.partic.2014.01.001.
Please cite this article in press as: Hatami M, et al. Numerical study of MHD two-phase Couette flow analysis for fluid-particle suspension between moving parallel plates. J Taiwan Inst Chem Eng (2014), http://dx.doi.org/10.1016/j.jtice.2014.05.018
JTICE-927; No. of Pages 8 Journal of the Taiwan Institute of Chemical Engineers xxx (2014) xxx–xxx
Contents lists available at ScienceDirect
Journal of the Taiwan Institute of Chemical Engineers journal homepage: www.elsevier.com/locate/jtice
Numerical study of MHD two-phase Couette flow analysis for fluid-particle suspension between moving parallel plates M. Hatami a, Kh. Hosseinzadeh b, G. Domairry c,*, M.T. Behnamfar d a
Esfarayen University, Mechanical Engineering Department, Esfarayen, North Khorasan, Iran Department of Mechanical Engineering, Sari Branch, Islamic Azad University, Sari, Iran c Babol University of Technology, Department of Mechanical Engineering, Babol, Iran d Isfahan University of Technology, Department of Chemistry, Isfahan, Iran b
A R T I C L E I N F O
A B S T R A C T
Article history: Received 6 November 2013 Received in revised form 15 April 2014 Accepted 18 May 2014 Available online xxx
In this study, steady and unsteady magneto-hydrodynamic (MHD) Couette flows between two parallel infinite plates have been studied through numerical Differential Quadrature Method (DQM) and analytical Differential Transformation Method (DTM), respectively. Coupled equations by taking the viscosity effect of the two phases for fixed and moving plates have been introduced. The precious contribution of the present study is introducing new, fast and efficient numerical and analytical methods in a two-phase MHD Couette fluid flow. Results are compared with those previously obtained by using Finite Difference Method (FDM). The velocity profiles of two phases are presented and a parametric study of physical parameters involved in the problem is conducted. As an outcome, when magnetic source is fixed relative to the moving plate, by increasing the Hartmann number, velocity profiles for both phases increased, but when it is fixed relative to the fluid an inverse treatment is observed. ß 2014 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
Keywords: Two-phase flow Couette flow Differential Quadrature Method (DQM) Differential Transformation Method (DTM) Particle suspension
1. Introduction The study of fluid-particle two-phase flow is important in many science applications such as petroleum transport, waste water treatment, combustion, power plant piping, corrosive particles in engine oil flow and mining, smoke emission from vehicles, etc. The cases that particles come into contact with a solid or fluid boundary have been immense interest of many researchers such as nanofluids which can be found in Sheikholeslami et al. works [1–5]. Various processes such as filtration, combustion, air and water pollution, coal transport and cleaning, micro contamination control and xerography involve the particle transportation and deposition. Numerical solutions of equation derived from the balance equation of external forces applied to a particle in interaction with drag, virtual mass force, the Basset force, lift force, etc. have been broadly considered in literature. It is known that, a fast and high accurate solution is usually the preferred and convenient method in engineering area because of less computational work as well as high accuracy.
* Corresponding author. Tel.: +98 111 3234205; fax: +98 111 3234205. E-mail addresses: [email protected] (M. Hatami), [email protected] (G. Domairry).
Chamkha [6] studied the unsteady laminar flow and heat transfer of a particulate suspension in an electrically conducting fluid through channels and circular pipes in the presence of uniform transverse magnetic field. Jha and Apere [7] provided the semianalytical solution to the unsteady MHD two-phase Couette flow between two infinite parallel plates using the Laplace transform technique and the Riemann-sum approximation method and compared their results with those using Finite Difference Method (FDM). The unsteady magneto-hydrodynamic flow of an electrically conducting viscous incompressible non-Newtonian Bingham fluid bounded by two parallel non-conducting porous plates is studied with heat transfer considering the Hall effect by Attia et al. [8]. Gedik et al. [9] investigated the unsteady flow of two-phase (solid/liquid) flow for laminar, incompressible and electrically conducting fluid through a circular pipe in the presence of a uniform transverse external magnetic and electrical field. Coupled equations in velocity and magnetic field for unsteady MHD flow through a pipe of rectangular section have been solved by Shakeri and Dehghan [10] with using combined finite volume method and spectral element technique. Jalaal et al. [11] solved a spherical particle’s motion in Couette flow using Homotopy perturbation method (HPM) and got comparable results to numerical ones. The unsteady rolling motion of a spherical particle restricted to a tube was studied
http://dx.doi.org/10.1016/j.jtice.2014.05.018 1876-1070/ß 2014 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
Please cite this article in press as: Hatami M, et al. Numerical study of MHD two-phase Couette flow analysis for fluid-particle suspension between moving parallel plates. J Taiwan Inst Chem Eng (2014), http://dx.doi.org/10.1016/j.jtice.2014.05.018
G Model
JTICE-927; No. of Pages 8 2
M. Hatami et al. / Journal of the Taiwan Institute of Chemical Engineers xxx (2014) xxx–xxx
analytically by Jalaal and Ganji [12]. They obtained an exact solution of particle velocity and acceleration motion under some practical conditions through applying HPM. Jalaal and Ganji [13] proposed an analytical solution for acceleration motion of a spherical particle rolling down an inclined boundary with drag coefficient which is correlated linearly to Re in a specific range using HPM. Various inclination angles were studied and observed that settling velocity, acceleration duration and displacement are proportional to amount of inclination angle while for a constant inclination angle; settling velocity and acceleration duration are decreased by increasing the fluid viscosity. Recently, Ghasemi et al. [14] discussed about the convergence and accuracy of Variational Iteration Method (VIM) and Adomian Decomposition Method (ADM) for solving the motion of a spherical particle in Couette fluid flow. Differential Quadrature Method (DQM) is a numerical technique which was first developed by Richard Bellman and his associates in the early 1970s [15]. The DQM applications were rapidly developed thanks to the innovative works in computation of weighting coefficient by other scientists and recently this method was applied in many engineering problems [16–20]. The concept of differential transformation method (DTM) was first introduced by Zhou [21] in 1986 and it was used to solve both linear and nonlinear initial value problems in electric circuit analysis. This method can be applied directly for linear and nonlinear differential equation without requiring linearization, discretization, or perturbation and this is the main benefit of this method. Ghafoori et al. [22] used the DTM for solving the nonlinear oscillation equation. Many other papers which used these types of analytical and numerical methods for solving the engineering problems can be found in the literature [23–32]. Motivated by the above-mentioned works, the main objective of the present research paper is to introduce DQM as a numerical and high efficient technique on unsteady two-phase MHD Couette fluid flow for obtaining the velocity profiles of both the fluid and particle phases. Flow in the channel is generated by the impulsive movement of the lower plate and DTM is applied to problem for investigating the steady state behavior. Results show that these two methods are accurate and for all constant values in coupled equations have excellent agreement with numerical FDM method.
(a) It is assumed that the fluid and the particles are interacting as a continuum. (b) The fluid phase is assumed to be electrically conducting, but the channel walls are assumed to be electrically nonconducting. (c) Particles are not subject to electro-magnetic force. So, equation of motion for the particles does not contain the force term due to the electro-magnetic field. (d) Velocity of both the phases is considered to be irrotational. According to [7] and [33] model, the equation of motion with electromagnetic force added is given as shown below [7]
rf
¼0
~f r:V (1)
@V~ p ~f V ~ p Þ; r:V ~p ¼ 0 ~ p þ K 0 ðV ¼ f p P þ f p m p r2 V (2) @t ~ f ¼ ðu0 ; y0 ; w0 Þ and V ~ p ¼ ðu0 ; y0 ; w0 Þ denotes, respecwhere V p p p f f f rp
tively the velocity components of the fluid and the particles and f f ; f p represent the fluid and particle quantity in unit volume ~ f BÞ ~ which where f f þ f p ¼ 1 [7]. In general form, ~ J ¼ s ð~ EþV electric field is not considered in this paper and should be zero. Also, in the present study magnetic field is applied in one direction, ~ ¼ ð0; B0 ; 0Þ. Under the assumptions made in the present i.e.: B problem, Eqs. (1) and (2) become:
@u0f @ 2 u0f s B20 0 0 ¼ nf f f þ k ðu0p u0f Þ u 2 @t r f @z0
(3)
@u0p @2 u0p 0 ¼ np f p þ k ðu0f u0p Þ @t @z0 2
(4)
Eq. (3) is valid when the magnetic field is fixed relative to the fluid. If the magnetic field is also accelerated with the same velocity as the plate, we must account for the relative motion. Thus, Eq. (3) turns to
@u0f @ 2 u0f s B20 0 0 ¼ nf f f þ k ðu0p u0f Þ ðu f Utn Þ 2 0 @t r @z
2. Problem description The motion of an unsteady, laminar, viscous fluid/particle suspension between two infinite horizontal plates located at the z = 0 and z = h planes is considered. A uniform transverse magnetic field is applied to the flow and it is assumed that no applied or polarization voltage exists. When t > 0 the lower plate begins to move in its own plane with a velocity Utn where U is a constant and the upper plate remains fixed. A schematic view of described problem is shown in Fig. 1. Following assumptions are considered by Rahmatulin [33] and Jha and Apere [7] to simplification the problem:
@V~ f ~p V ~f Þ þ~ ~ ~ f þ K 0 ðV ¼ f f P þ f f m f r2 V J B; @t
(5)
Eqs. (3) and (5) are combined together to obtain
@u0f @ 2 u0f s B20 0 0 ¼ nf f f þ k ðu0p u0f Þ ðu f UKtn Þ r @t @z0 2 where 0 K¼ 1
when B0 is fixed relative to the fluid when B0 is fixed relative to the moving plate
(6)
(7)
The initial and boundary conditions for the present problem are 0
u f ¼ u0p ¼ 0; u0f ¼ u0p ¼ Utn ; u0f ¼ u0p ¼ 0;
0 z0 h z0 ¼ 0; z0 ¼ h;
and t 0; t > 0; t > 0:
(8)
by using the following equations, non-dimensionalised governing system of equations will be found. ðu0f ; u0p Þ z0 v1 ; ðu0f ; u0p Þ ¼ ; h ¼ t f f 2 ; Ha h U h sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2 s kh : ; k¼ ¼ B0 h nf f f nfrf f f
z¼
Fig. 1. Schematic view of the problem.
(9)
where Ha represents the Hartmann number which is the ratio of electromagnetic force to the viscous force. Considering the impulsive motion of the lower plate which corresponds to n = 0
Please cite this article in press as: Hatami M, et al. Numerical study of MHD two-phase Couette flow analysis for fluid-particle suspension between moving parallel plates. J Taiwan Inst Chem Eng (2014), http://dx.doi.org/10.1016/j.jtice.2014.05.018
G Model
JTICE-927; No. of Pages 8 M. Hatami et al. / Journal of the Taiwan Institute of Chemical Engineers xxx (2014) xxx–xxx
and substituting the dimensionless quantities into the governing equations and the boundary condition, the following dimensionless equation results 8 > @u f @2 u f > > ¼ þkðu p u f Þ Ha2 ðu f KÞ < @h @z2 (10) > @u @2 u p > > : p ¼ R f Rv þkðu u Þ p f @h @z2
Table 1 Some fundamental operations of the differential transform method. Origin function
Transformed function
xðtÞ ¼ a f ðxÞ bgðtÞ
XðkÞ ¼ aFðkÞ bGðkÞ
m
xðtÞ ¼
d f ðtÞ dt m
ðk þ mÞ!Fðk þ mÞ k! k X XðkÞ ¼ FðlÞGðk lÞ l¼0 1; if k ¼ m; XðkÞ ¼ dðk mÞ ¼ 0; if k 6¼ m: 1 XðkÞ ¼ k! vk kp sin þa XðkÞ ¼ k! 2 vk kp cos þa XðkÞ ¼ k! 2 XðkÞ ¼
xðtÞ ¼ f ðtÞgðtÞ xðtÞ ¼ t m
where R f ¼ f p = f f ; Ry ¼ y p =y f ; and the initial and boundary conditions (8) also become u f ¼ u p ¼ 0; u f ¼ u p ¼ 1; u f ¼ u p ¼ 0;
(11)
3. Principles of methods 3.1. Differential Quadrature Method (DQM) In this study, polynomial expansion based differential quadrature, as introduced by Quan and Chang [34,35], is applied for solving the problem. Several attempts have been made by researchers to develop polynomial based differential quadrature methods. One of the most useful approaches is the one that uses the following Lagrange interpolation polynomials as test functions: MðxÞ
g k ðxÞ ¼
ðx xk ÞM ð1Þ ðxk Þ
;
k ¼ 1; 2; :::; N
(12)
where:
N Y
¼
M
ð1Þ
ðxi Þ
ðxi xk Þ
(13)
k¼1 k 6¼ i By applying the above equation at N grid points, the following algebraic formulations to compute the weighting coefficients are developed: ð1Þ
N Y 1 xi xk ; x j xi k¼1 x j xk
ð1Þ
i 6¼ j;
Ai j
N X k¼1k 6¼ i
1 ; xi xk
N X ð1Þ ð1Þ Aik Ak j
ð2Þ
Ai j ¼
(14)
k¼1
where A(1) and A(2) denote the weighting coefficients of the first and second order derivatives of the function f(r) with respect to the r direction. N is the number of grid points chosen in the r direction. The differential quadrature approximation can be easily extended from the above formulation to other coordinates. The first order derivatives in the two-dimensional formulation are approximated by:
@f @r
ij
N X ð1Þ Ail f l j ; l¼1
@f @z
ij
M X
ð1Þ
B jm f im
@2 f @z2 2
! !i j
@ f @r@z
3.2. Differential Transformation Method (DTM) Basic definitions and operations of differential transformation are introduced as follows. Differential transformation of the function f ðhÞ is defined as follows: " # k 1 d f ðhÞ (17) FðkÞ ¼ k! dhk
l¼1
!i j ij
n¼1 N X
P X ð1Þ ð1Þ Ail B jm f lm
l¼1
m¼1
f ðhÞ ¼
1 X FðkÞðh h0 Þk
(18)
k¼0
by combining Eqs. (17) and (18) f ðhÞ can be obtained: " # k 1 X d f ðhÞ ðh h0 Þk f ðhÞ ¼ k k! dh k¼0
(19)
Eq. (19) implies that the concept of the differential transformation is derived from Taylor’s series expansion, but the method does not evaluate the derivatives symbolically. However, relative derivatives are calculated by an iterative procedure that is described by the transformed equations of the original functions. From the definitions of Eqs. (17) and (18), it is easily proven that the transformed functions comply with the basic mathematical operations shown in below. In real applications, the function f ðhÞ in Eq. (19) is expressed by a finite series and can be written as: f ðhÞ ¼
N X ð2Þ Ail f l j ;
P X ð2Þ Bkn f i jn ;
In Eq. (17), f ðhÞ is the original function and FðkÞ is the transformed function which is called the T-function (it is also called the spectrum of the f ðhÞ at h ¼ h0 , in the k domain). The differential inverse transformation of FðkÞ is defined as:
(15)
m¼1
And the second order derivatives can be approximated by:
@2 f @r 2
where A(1) and B(1) denote the weighting coefficients of the first order derivatives; A(2) and B(2) denote the weighting coefficients of the second order derivatives of the function f(r,z) with respect to the r and z-directions, respectively; N, and P are the number of grid points chosen in the r and z-directions, respectively.
h¼ h0
k 6¼ i; j
¼
xðtÞ ¼ cos ðvt þ aÞ
h¼h0
MðxÞ ¼ ðx x1 Þðx x2 Þ ðx xN Þ;
Ai j ¼
xðtÞ ¼ exp ðtÞ xðtÞ ¼ sin ðvt þ aÞ
and h 0; h > 0; h > 0:
0z1 z ¼ 0; z ¼ 1;
3
(16)
N X FðkÞðh h0 Þk
(20)
k¼0
P k Eq. (20) implies that f ðhÞ ¼ 1 k¼Nþ1 ðFðkÞðh h0 Þ Þ is negligibly small, where N is series size. Theorems to be used in the transformation procedure, which can be evaluated from Eq. (18), are given in Table 1. 4. Results and discussion The polynomial expansion based numerical differential quadrature is used for the non-dimensional time (h) and the z directions
Please cite this article in press as: Hatami M, et al. Numerical study of MHD two-phase Couette flow analysis for fluid-particle suspension between moving parallel plates. J Taiwan Inst Chem Eng (2014), http://dx.doi.org/10.1016/j.jtice.2014.05.018
G Model
JTICE-927; No. of Pages 8 4
M. Hatami et al. / Journal of the Taiwan Institute of Chemical Engineers xxx (2014) xxx–xxx
Table 2 Non-uniform grid points in numerical method (DQM) for h 2 [0,1]. Time step number
Non-dimensional time (h) (Eq. (22))
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.0000000000 0.0125360439 0.0495155661 0.1090842588 0.1882550991 0.2830581303 0.3887395332 0.5000000000 0.6112604668 0.7169418697 0.8117449009 0.8909157412 0.9504844339 0.9874639561 1.0000000000
Table 4 Effect of interpolate point numbers in DQM on the velocity profiles presented in Table 3. K
z
Table 3 Comparison between DQM and FDM results for k = 50, h = 0.5, Ha = 4, Rf = 0.25 and Rv = 1.
P = N = 15
P = N = 30
Fluid
Particle
Fluid
Particle
0.2
0 1
0.4822 0.9509
0.5132 0.9488
0.4831 0.9515
0.5125 0.9490
0.4
0 1
0.2339 0.8835
0.2494 0.8761
0.2336 0.8849
0.2491 0.8796
0.6
0 1
0.1089 0.7582
0.1161 0.7417
0.1090 0.7583
0.1160 0.7416
0.8
0 1
0.0426 0.5064
0.0457 0.4774
0.0425 0.5071
0.0457 0.4779
where U and V represent the fluid and particle velocities, respectively. In order to proceed with numerical computations, non-uniform points in the h and z directions are defined using the Chebyshev–Gauss-Lobatto polynomials [36]: H k1 1 1 cos p ; k1 ¼ 1; :::; P 2 P 1 L k2 1 1 cos ¼ p ; k2 ¼ 1; :::; N 2 N1
z
K
Fluid
Particle
Fluid
Particle
0.2
0 1
0.4822 0.9509
0.5132 0.9488
0.4857 0.9517
0.5111 0.9495
zk2
0.4
0 1
0.2339 0.8835
0.2494 0.8761
0.2333 0.8851
0.2487 0.8805
0.6
0 1
0.1089 0.7582
0.1161 0.7417
0.1090 0.7583
0.1160 0.7415
0.8
0 1
0.0426 0.5064
0.0457 0.4774
0.0425 0.5079
0.0457 0.4780
These non-uniform points for h are presented in Table 2 for P = 15. Obtained results by DQM are compared with FDM [7] in Table 3 (for fluid and particle phase for both when K = 0 and 1) and an excellent agreement between these two numerical methods is observed. But, solution of DQM for each run takes less than 1 min (approximately 30 s) while FDM results converge more than 1 min, so DQM is faster than FDM. Furthermore, unlike conventional methods such as Finite Element (FE) and FDM, DQM requires less grid points to obtain acceptable accuracy. Another advantage of DQM is that it transforms the differential equations into a set of analogous algebraic equations in terms of the unknown function values at the resembled points in the solution domain. Maybe the main advantage of DQM is its inherent conceptual simplicity, accuracy and the fact that easily programmable [37,38]. As described by Shu [37], Table 4 shows that DQM needs to less interpolation points and results of P = N = 15 are approximately similar to when grid points are twice, i.e. N = P = 30. Figs. 2–7 are related to the unsteady analysis. Fig. 2 demonstrates that an increase in time leads to an increase in the fluid and particle
DQM
FDM [7]
for obtaining the solution of Eq. (10) with boundary condition Eq. (11). After applying this method, following set of equations (Eq. (21)) will be obtained, 8 > > > > < > > > > :
P X m¼1 P X m¼1
! B j;m U i;m
!
B j;m V i;m
N X ð2Þ Ai;l U l; j l¼1
R f Rv
! kðV i; j U i; j Þ þ Ha2 ðU i; j KÞ ¼ 0 !
N X ð2Þ Ai;l V l; j
kðU i; j V i; j Þ ¼ 0
l¼1
(21)
hk1 ¼
(22)
Fig. 2. Effect of time (h) on velocity profiles for (a) fluid phase, (b) particle phase when K = 0, k = 50, Rv = 1, Rf = 0.25, Ha = 2.
Please cite this article in press as: Hatami M, et al. Numerical study of MHD two-phase Couette flow analysis for fluid-particle suspension between moving parallel plates. J Taiwan Inst Chem Eng (2014), http://dx.doi.org/10.1016/j.jtice.2014.05.018
G Model
JTICE-927; No. of Pages 8 M. Hatami et al. / Journal of the Taiwan Institute of Chemical Engineers xxx (2014) xxx–xxx
Fig. 3. Effect of time (h) on velocity profiles for (a) fluid phase, (b) particle phase when K = 1, k = 50, Rv = 1, Rf = 0.25, Ha = 2.
Fig. 4. Effect of Ha number on velocity profiles for (a) fluid phase, (b) particle phase when K = 0, k = 50, Rv = 1, Rf = 0.25, h = 0.5.
Fig. 5. Effect of Ha number on velocity profiles for (a) fluid phase, (b) particle phase when K = 1, k = 50, Rv = 1, Rf = 0.25, h = 0.5.
Please cite this article in press as: Hatami M, et al. Numerical study of MHD two-phase Couette flow analysis for fluid-particle suspension between moving parallel plates. J Taiwan Inst Chem Eng (2014), http://dx.doi.org/10.1016/j.jtice.2014.05.018
5
G Model
JTICE-927; No. of Pages 8 6
M. Hatami et al. / Journal of the Taiwan Institute of Chemical Engineers xxx (2014) xxx–xxx
Fig. 6. Effect of Rv number on velocity profiles for (a) fluid phase, (b) particle phase when K = 0, k = 50, Rv = 1, Rf = 0.25, z = 0.5.
velocities for K = 0. Also, the same trend is notice in Fig. 3 for both phases when K = 1. In fact, this effect is more pronounced on the particle field. Fig. 4 demonstrates the effect of Ha number on fluid and particle velocities. As seen by increasing Ha number, velocity profiles are decreased for both fluid and particle phases when K = 0 and h = 0.5. But, when K = 1, increasing the Ha number leads to increase in velocity profiles which are depicted in Fig. 5. Figs. 6 and 7, which show the velocity profiles versus time (h), demonstrate the effect of Rv on velocity profiles when K = 0 and K = 1, respectively. For K = 0 (Fig. 6), it is observed that an increase in Rv, result into an increase in velocities. For K = 1 (Fig. 7), at small times (h < 0.4) the same trend is observed, but as the time is increased further, the trend is completely reversed. It is found out that the fluid velocity is always greater than that of the particle for all the dimensional quantities considered. For the two phases by setting @ð Þ=@h in Eq. (10) to zero, the following ordinary differential equations then results for steady
state analysis, 8 2 > d uf > < þkðu p u f Þ Ha2 ðu f KÞ ¼ 0 dz2 2 > > : R Rv d u p þkðu u p Þ ¼ 0 f f dz2
(23)
For solving this coupled equation, DTM is applied due to its simplicity and high accuracy. From Table 1, Eq. (23) will be transformed to, ðmþ1Þðmþ 2ÞUðmþ2Þþ kðVðmÞ UðmÞÞ Ha2 ðUðmÞ K dm Þ ¼ 0 R f Rv ðm þ 1Þðm þ 2ÞVðm þ 2Þ þ kðUðmÞ VðmÞÞ ¼ 0 (24) and the boundary conditions will be, Uð0Þ ¼ 1;
Vð0Þ ¼ 1;
Uð1Þ ¼ a;
Vð1Þ ¼ b
(25)
Fig. 7. Effect of Rv number on velocity profiles for (a) fluid phase, (b) particle phase when K = 1, k = 50, Rv = 1, Rf = 0.25, z = 0.5.
Please cite this article in press as: Hatami M, et al. Numerical study of MHD two-phase Couette flow analysis for fluid-particle suspension between moving parallel plates. J Taiwan Inst Chem Eng (2014), http://dx.doi.org/10.1016/j.jtice.2014.05.018
G Model
JTICE-927; No. of Pages 8 M. Hatami et al. / Journal of the Taiwan Institute of Chemical Engineers xxx (2014) xxx–xxx Table 5 Comparison between DTM and FDM for k = 5, Ha = 1, Rf = 0.25 and Rv = 1 in steady state. DTM
Using Eq. (25) and solving Eq. (24), other DTM terms for velocities will be obtained as follows, (
z
K
FDM [7]
Fluid
Particle
Fluid
Particle
0.2
0 1
0.7603 0.8246
0.7792 0.8180
0.7609 0.8246
0.7772 0.8180
0.4
0 1
0.5477 0.6434
0.5728 0.6309
0.5493 0.6435
0.5665 0.6307
0.6
0 1
0.3522 0.4505
0.3853 0.4339
0.3564 0.4506
0.3692 0.4334
0.8
0 1
0.1676 0.2388
0.2115 0.2236
0.1753 0.2390
0.1819 0.2227
7
1 2 1 2 Ha Ha K 2 2 8 Vð2Þ ¼ 0 1 1 1 > > < Uð3Þ ¼ kb þ ka þ Ha2 a 6 6 6 1 kðb þ aÞ > > : Vð3Þ ¼ 6 R f Rv 8 1 1 1 > > Ha4 K þ kHa2 K þ Ha4 < Uð4Þ ¼ 24 24 24 2 1 kHa ðK 1Þ > > : Vð4Þ ¼ 24 R f Rv ::: Uð2Þ ¼
(26)
Results of DTM and FDM are compared in Table 5 for both K = 0 and K = 1 which a good agreement between them is observed. Figs. 8 and 9 show the Ha number effect in steady state on fluid and particle velocity profiles, respectively. As concluded from these
Fig. 8. Effect of Ha number on fluid phase velocity profiles when k = 50, Rv = 1, Rf = 0.2 for (a) K = 0, (b) K = 1 in steady state.
Fig. 9. Effect of Ha number on particle phase velocity profiles when k = 50, Rv = 1, Rf = 0.2 for (a) K = 0, (b) K = 1 in steady state.
Please cite this article in press as: Hatami M, et al. Numerical study of MHD two-phase Couette flow analysis for fluid-particle suspension between moving parallel plates. J Taiwan Inst Chem Eng (2014), http://dx.doi.org/10.1016/j.jtice.2014.05.018
G Model
JTICE-927; No. of Pages 8 8
M. Hatami et al. / Journal of the Taiwan Institute of Chemical Engineers xxx (2014) xxx–xxx
figures, increasing the Ha number makes a decrease in both velocity profiles for K = 0, but it makes an increase in velocity profiles when K = 1. 5. Conclusion In this paper, an analytical approach called Differential Transformation Method (DTM) and a numerical method, Differential Quadrature Method (DQM), have been successfully applied to find the accurate solution for the steady and unsteady MHD Couette flow between two parallel infinite plates. As a main outcome from the present study, it is observed that the results of DTM and DQM are in excellent agreement with FDM. Also, the effects of time h, Hartmann number Ha, the viscosity ratio Rv on fluid and particle velocity profiles are studied. It is concluded that velocities can be controlled by increasing or decreasing the magnetic field. References [1] Sheikholeslami M, Ganji DD, Gorji-Bandpy M, Soleimani S. Magnetic field effect on nanofluid flow and heat transfer using KKL model. J Taiwan Inst Chem Eng 2014;45:795–807. [2] Sheikholeslami M, Gorji-Bandpy M, Ganji DD, Soleimani S. Natural convection heat transfer in a cavity with sinusoidal wall filled with CuO-water nanofluid in presence of magnetic field. J Taiwan Inst Chem Eng 2014;45:40–9. [3] Sheikholeslami M, Gorji-Bandpy M, Soleimani S. Two phase simulation of nanofluid flow and heat transfer using heatline analysis. Int Comm Heat Mass Transf 2013;47:73–81. [4] Sheikholeslami M, Gorji-Bandpy M, Ganji DD. Numerical investigation of MHD effects on Al2O3-water nanofluid flow and heat transfer in a semi-annulus enclosure using LBM. Energ 2013;60:501–10. [5] Sheikholeslami M, Gorji Bandpy M, Ellahi R, Hassan M, Soleimani S. Effects of MHD on Cu-water nanofluid flow and heat transfer by means of CVFEM. J Magnet Magnetic Mater 2014;349:188–200. [6] Chamkha AJ. Unsteady laminar hydromagnetic fluid–particle flow and heat transfer in channels and circular pipe. Int J Heat Fluid Flow 2000;21:740–6. [7] Jha BK, Apere CA. Unsteady MHD two-phase Couette flow of fluid–particle suspension. App Math Model 2013;37:1920–31. [8] Attia HA, Sayed-Ahmed ME. Hall effect on unsteady MHD Couette flow and heat transfer of a Bingham fluid with suction and injection. App Math Model 2014;28:1027–45. [9] Gedik E, Kurt H, Recebli Z, Kec¸ebas A. Unsteady flow of two-phase fluid in circular pipes under applied external magnetic and electrical fields. Int J Thermal Sci 2012;53:156–65. [10] Shakeri F, Dehghan M. A finite volume spectral element method for solving magnetohydrodynamic (MHD) equations. Appl Numer Math 2011;61:1–23. [11] Jalaal M, Nejad MG, Jalili P, Esmaeilpour M, Bararnia H, Ghasemi E, et al. Homotopy perturbation method for motion of a spherical solid particle in plane Couette fluid flow. Comput Math With Appl 2011;61:2267–70. [12] Jalaal M, Ganji DD. On unsteady rolling motion of spheres in inclined tubes filled with incompressible Newtonian fluids. Adv Powder Technol 2011;22:58–67. [13] Jalaal M, Ganji DD. An analytical study on motion of a sphere rolling down an inclined plane submerged in a Newtonian fluid. Powd Technol 2010;198:82– 92. [14] Ghasemi SE, Palandi SJ, Hatami M, Ganji DD. Efficient analytical approaches for motion of a spherical solid particle in plane couette fluid flow using nonlinear methods. The J Math Comput Sci 2012;5:97–104.
[15] Bellman RE, Kashef BG, Casti J. Differential quadrature: a technique for rapid solution of non linear partial differential equations. J Comput Phys 1972;10:40– 52. [16] Yucel U, Boubaker K. Differential quadrature method (DQM) and Boubaker Polynomials Expansion Scheme (BPES) for efficient computation of the eigenvalues of fourth-order Sturm–Liouville problems. Appl Math Model 2012;36:158–67. [17] Hosseini-Hashemi Sh, Akhavan H, Rokni Damavandi Taher H, Daemi N, Alibeigloo A. Differential quadrature analysis of functionally graded circular and annular sector plates on elastic foundation. Mater Des 2010;31:1871–80. [18] Sun J, Zhu Zh. Upwind local differential quadrature method for solving incompressible viscous flow. Comput Methods Appl Mech Eng 2000;188: 495–504. [19] Akbari Alashti R, Khorsand M. Three-dimensional thermo-elastic analysis of a functionally graded cylindrical shell with piezoelectric layers by differential quadrature method. Int J Press Vess Pip 2011;88:167–80. [20] Akbari Alashti R, Khorsand M, Tarahhomi MH. Thermo-elastic analysis of a functionally graded spherical shell with piezoelectric layers by differential quadrature method. Sci Iranic B 2013;20:109–19. [21] Zhou JK. Differential transformation method and its application for electrical circuits. Wuhan, China: Hauzhang Univers. Press; 1986. [22] Ghafoori S, Motevalli M, Nejad MG, Shakeri F, Ganji DD, Jalaal M. Efficiency of differential transformation method for nonlinear oscillation: comparison with HPM and VIM. Curr Appl Phys 2011;11:965–71. [23] Hatami M, Ganji DD. Heat transfer and flow analysis for SA-TiO2 non-Newtonian nanofluid passing through the porous media between two coaxial cylinders. J Molec Liq 2013;188:155–61. [24] Hatami M, Ganji DD. Thermal performance of circular convective-radiative porous fins with different section shapes and materials. Energ Conver Manage 2013;76:185–93. [25] Hatami M, Hasanpour A, Ganji DD. Heat transfer study through porous fins (Si3N4 and AL) with temperature-dependent heat generation. Energ Convers Manage 2013;74:9–16. [26] Hatami M, Ganji DD. Investigation of refrigeration efficiency for fully wet circular porous fins with variable sections by combined heat and mass transfer analysis. Intl J Refrig 2014;40:140–51. [27] Hatami M, Ganji DD. Thermal flow analysis of microchannel heat sink (MCHS) cooled by Cu–water nanofluid using porous media approach and least square method. Energ Convers Manage 2014;78:347–58. [28] Hatami M, Nouri R, Ganji DD. Forced convection analysis for MHD Al2O3– water nanofluid flow over a horizontal plate. J Molec Liq 2013;187:294–301. [29] Hatami M, Ganji DD. Heat transfer and nanofluid flow in suction and blowing process between parallel disks in presence of variable magnetic field. J Molec Liq 2014;190:159–68. [30] Hatami M, Domairry G. Transient vertically motion of a soluble particle in a Newtonian fluid media. Powd Tech 2014;253:481–5. [31] Hatami M, Hatami J, Ganji DD. Computer simulation of MHD blood conveying gold nanoparticles as a third grade non-Newtonian nanofluid in a hollow porous vessel. Comput Meth Prog Biomed 2014;113:632–41. [32] Domairry G, Hatami M. Squeezing Cu–water nanofluid flow analysis between parallel plates by DTM-Pade´ method. J Molec Liq 2014;193:37–44. [33] Rahmatulin HA. Osnovi gidrodinamiki vzaimopronikayus_ih dvijeniy. Prikl Mat Mak 1956;20:56–65. [34] Quan JR, Chang CT. New insights in solving distributed system equations by quadrature methods-I. analysis. Comput Chem Eng 1989;13:779–88. [35] Quan JR, Chang CT. New insights in solving distributed system equations by quadrature methods-II. Numerical experiments. Comput Chem Eng 1989;13:1017–24. [36] Shu Ch, Richards BE. Application of generalized differential quadrature to solve two-dimensional incompressible Navier–Stokes equations. Int J Numer Methods Fluids 1992;15:791–8. [37] Shu Ch. Differential quadrature and its application in engineering. Singapore: Springer; 1999. [38] Hatami M, Ganji DD. Motion of a spherical particle in a fluid forced vortex by DQM and DTM. Partic 2014. http://dx.doi.org/10.1016/j.partic.2014.01.001.
Please cite this article in press as: Hatami M, et al. Numerical study of MHD two-phase Couette flow analysis for fluid-particle suspension between moving parallel plates. J Taiwan Inst Chem Eng (2014), http://dx.doi.org/10.1016/j.jtice.2014.05.018