Motion of a spherical particle in a fluid forced vortex by DQM and DTM

ARTICLE IN PRESSG ModelPARTIC-639; No. of Pages 9

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Particuology xxx (2014) xxx–xxx

Contents lists available at ScienceDirect

Particuology

jo ur nal home page: www.elsev ier .com/ locate /par t ic

otion of a spherical particle in a fluid forced vortex by DQMnd DTM

. Hatamia,b,∗, D.D. Ganjib

Esfarayen University, Department of Mechanical Engineering, Esfarayen, North Khorasan, IranMechanical Engineering Faculty, Babol University of Technology, Babol, P.O. Box 484, Iran

r t i c l e i n f o

rticle history:eceived 17 July 2013eceived in revised form9 November 2013ccepted 8 January 2014

eywords:pherical particleorced vortexifferential transformation method (DTM)ngular velocityifferential quadrature method (DQM)

a b s t r a c t

In this study, coupled equations of the motion of a particle in a fluid forced vortex were investigatedusing the differential transformation method (DTM) with the Padé approximation and the differentialquadrature method (DQM). The significant contribution of the work is the introduction of two new,fast and efficient solutions for a spherical particle in a forced vortex that are improvements over theprevious numerical results in the literature. These methods represent approximations with a high degreeof accuracy and minimal computational effort for studying the particle motion in a fluid forced vortex. Inaddition, the velocity profiles (angular and radial) and the position trajectory of a particle in a fluid forcedvortex are described in the current study.

© 2014 Published by Elsevier B.V. on behalf of Chinese Society of Particuology and Institute of ProcessEngineering, Chinese Academy of Sciences.

. Introduction

A vortex arises when a fluid flows along a rotating device. The formation of a vortex in such fluid flows is due to the effects of pressureariations and shearing flows. If the inertia of the fluid is small and the device rotates at a high speed, the device will transfer partf its rotational energy to the fluid. This type of vortex is called a forced vortex. In scientific laboratories, centrifugation and Couetteiscometer techniques are encountered; both techniques depend on vortex motion. In industry, the separation of different dispersedpecies is performed by centrifugation, and centrifugal filters are also used. Centrifugation is characterized by an increasing tangentialelocity for increasing values of the radius. Many environmental phenomena involve vortices, such as oceans, airplanes, centrifugation,entrifugal filters, and industrial hoppers.

Computational fluid dynamics (CFD) methods have been widely used in modeling the particle transport and distribution in vortices.enerally, the particle models can be classified as either Eulerian or Lagrangian methods, with each method exhibiting its own pros andons. The Eulerian method treats the particle phase as a continuum and develops its conservation equation on a control volume basisnd in a similar form as that for the fluid phase. The Lagrangian method considers particles as a discrete phase and tracks the pathway ofach individual particle, so it is also able to calculate the particle concentration and the other phase data. The following presents selectedesearch studies in this area.

Prevel, Vinkovic, Doppler, Pera, and Buffat (2013) used direct numerical simulation (DNS) of the fluid flow and simultaneous Lagrangianracking of particles to describe the transport of particles in vortices. A viscous vortex particle method (VVPM) was presented for computinghe fluid dynamics of two-dimensional rigid bodies by Eldredge (2007). In another research article, Verkhoglyadova and le Roux (2005)umerically studied the particle transport in a two-dimensional coherent vortex field.

Although CFD results are completely reliable, the convergence of the calculations is related to the mesh numbers used. Further-

Please cite this article in press as: Hatami, M., & Ganji, D.D. Motion of a spherical particle in a fluid forced vortex by DQM and DTM.Particuology (2014), http://dx.doi.org/10.1016/j.partic.2014.01.001

ore, a test of the independence of the calculations to mesh number should be performed, which requires additional computationalime and high-accuracy computer devices. As a result, an analytical solution is usually the more preferred and convenient methodn engineering applications (Hatami & Ganji, 2013a, 2013b, 2014; Hatami, Hasanpour, & Ganji, 2013; Hatami, Nouri, & Ganji,

∗ Corresponding author. Tel.: +98 111 3234205; fax: +98 111 3234205.E-mail addresses: [email protected] (M. Hatami), ddg [email protected] (D.D. Ganji).

ttp://dx.doi.org/10.1016/j.partic.2014.01.001674-2001/© 2014 Published by Elsevier B.V. on behalf of Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences.

dx.doi.org/10.1016/j.partic.2014.01.001


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2 M. Hatami, D.D. Ganji / Particuology xxx (2014) xxx–xxx

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Fig. 1. Schematic view of spherical particles in a sample fluid forced vortex (industrial hopper).

013; Hatami, Hatami, & Ganji, 2014; Sheikholeslami, Hatami, & Ganji, 2013; Sheikholeslami, Hatami, & Ganji, 2014) because ofhe reduced computational work as well as high accuracy; such analytical solutions are widely used for predicting the motion of

particle in Couette flow or in the sedimentation process. Selected examples of the uses of analytical solutions are introducedelow.

Salari, Karmozdi, Maddahian, and Firoozabadi (2013) considered a single solid particle motion through free and forced vortices using thenalytical method. Jalaal et al. (2011) solved a spherical particle’s motion in Couette flow using the Homotopy perturbation method (HPM)nd achieved comparable results to the numerical ones. The unsteady rolling motion of a spherical particle restricted to a tube was studiednalytically by Jalaal and Ganji (2011); they obtained an exact solution of the particle velocity and acceleration motion under some practicalonditions by applying the HPM. Jalaal and Ganji (2010) proposed an analytical solution for the acceleration motion of a spherical particleolling down an inclined plate using the HPM; they studied various inclination angles, and observed that the settling velocity, accelerationuration and displacement are proportional to the inclination angle, while for a constant inclination angle, the settling velocity andcceleration duration are decreased by increasing the fluid viscosity. Torabi and Yaghoobi (2011) investigated the improved performancef a combination of He’s polynomials and the diagonal Padé approximants compared to that of the HPM for calculating the approximateolution of the acceleration motion of a single spherical particle moving in a continuous fluid phase. In another study (Yaghoobi & Torabi,012a), they obtained the acceleration trajectory of a non-spherical particle moving in a continuous fluid phase using variational iterationethod (VIM)-Padé approximants with acceptable accuracy compared to that of numerical results. Recently, Ghasemi, Palandi, Hatami,

nd Ganji (2012) discussed the convergence and accuracy of VIM and Adomian decomposition method (ADM) for solving the motion of apherical particle in Couette fluid flow. Additionally, Hamidi, Rostamiyan, Ganji, and Fereidoon (2013) and Torabi, Yaghoobi, and Boubaker2013) confirmed that the HPM-Padé is more accurate than the HPM for solving these types of problems. Differential quadrature methodDQM) is a fast technique, which was first developed by Richard Bellman and his associates in the early 1970s (Bellman, Kashef, & Casti,972). The DQM applications were rapidly developed due to the innovative works in computation of the weighting coefficient by othercientists, and recently this method has been applied in many engineering problems (Akbari Alashti & Khorsand, 2011; Akbari Alashti,horsand, & Tarahhomi, 2013; Hosseini-Hashemi, Akhavan, Rokni Damavandi Taher, Daemi, & Alibeigloo, 2010; Sun & Zhu, 2000; Yucel &oubaker, 2012). The advantages and disadvantages of these methods are presented in Section 3 of the present study.

As seen in mentioned works, many analytical methods, due to their simplicity and accuracy, are used for describing the free falling ofpherical particles or particles in Couette fluid flow, but for a particle in a free or forced vortex, fewer analytical methods exist (EL-Naggar

Kholeif, 2012). The main objective of the present paper is to introduce both differential transformation method (DTM)-Padé and DQM asigh accuracy and efficient methods for use in describing the 2D motion of a spherical particle in forced vortex. The results indicate that,ompared to previous numerical works, these methods are notably fast, accurate, and, for all constant values in the coupled equations ofhe problem, produce results that exhibit excellent agreement with those of numerical methods.

. Statement of the problem

Consider the polar coordinates r − � with the pole at the origin r = 0. When the rotating speed is approximately constant, the use of 2D model for a particle in a free or a forced vortex is an acceptable assumption and simplification, which has been considered in theiterature (Eldredge, 2007; EL-Naggar & Kholeif, 2012; Salari et al., 2013; Verkhoglyadova & le Roux, 2005). So, in this study, the motion ofhe particle in a plane is studied according to Fig. 1. The mass of the fluid is rotating in the counterclockwise direction around the originsee Fig. 1). The stream lines are concentric circles with a common center at the origin. For the forced vortex, the tangential velocity isiven by u� = ˝r, whereas for free vortex u� = c/r, and c are constants. In both cases, the radial components of the velocity of the fluidre zero. At the instant of t = 0, a small spherical particle is released in the flow field with an initial radius of r0, zero angular velocity, andhe radial component of velocity u . The fluid will drive the particle to rotate, by exerting a drag force on the particle (EL-Naggar & Kholeif,


0012)

FD = cDA12

�u2, (1)



wamd

w(ω

wynt

3

3

dx

t

a

ww

fTTcfie

M. Hatami, D.D. Ganji / Particuology xxx (2014) xxx–xxx 3

here u is the relative velocity between the particle and the flow; � is the mass density of the fluid; A is the projected area of the spherend cD is the drag coefficient, which is a function of the Reynolds number. The equations of motion of the particle can be written as ma = F;

is the mass of the particle; a is the acceleration and F is the exerted external force on the particle. Equations in the radial and tangentialirections are described by Eqs. (2) and (3), respectively (EL-Naggar & Kholeif, 2012),

m(r − r�2) = −cDA12

�r2, (2)

m(r� + 2r�) = cDA12

�(r� − u�fluid)2, (3)

here u�fluid = ω0r for the forced vortex and u�fluid = c/r for the free vortex. The parenthesized quantities on the left hand sides of Eqs.2) and (3) are respectively the radial and tangential components of particle acceleration in polar coordinates. Now, by defining =/ω0, U = u/u0, R = r/r0 and � = tω0, the non-dimensionalized equations are:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

R(�) − u0

r0ω0U(�) = 0,

u0

r0ω0U(�) − R(�)˝(�)2 + ˛2

(u0

r0ω0

)2U(�)2 = 0,

R(�) ˙ (�) + u0

r0ω02U(�)˝(�) = ˛2R(�)2

⎧⎪⎪⎨⎪⎪⎩

(˝(�) − 1)2 for forced vortex,

(˝(�) − 1

R(�)2

)2

for free vortex,

(4)

here the non-dimensional parameter ˛2 = cDA 12 �r0/m represents the drag to inertia ratio. Due to importance of particles motion anal-

sis, as described in the introduction, this system of equations should be solved simultaneously and, according to its rational functiononlinearity form, analytical and numerical methods should be applied. By obtaining the particle position and velocities, its motion andreatment in vortices is completely determined.

. Principle of the methods

.1. Differential transformation method with the Padé approximant (DTM-Padé)

For understanding the method’s concept, suppose that x(t) is an analytic function in domain D, and t = ti represents any point in theomain. The function x(t) is then represented by one power series, whose center is located at ti. The Taylor series expansion function of(t) is in form of (Zhou, 1986)

x(t) =∞∑

k=0

(t − ti)k

k!

[dkx(t)

dtk

]t=ti

∀t ∈ D. (5)

he Maclaurin series of x(t) can be obtained by taking ti = 0 in Eq. (5) expressed as:

x(t) =∞∑

k=0

tk

k!

[dkx(t)

dtk

]t=0

∀t ∈ D, (6)

s explained in (Zhou, 1986), the differential transformation of the function x(t) is defined as follows:

X(k) =∞∑

k=0

Hk

k!

[dkx(t)

dtk

]t=0

, (7)

here X(k) represents the transformed function (T-function) and x(t) is the original function. The differential spectrum of X(k) is confinedithin the interval t ∈ [0, H], where H is a constant value. The differential inverse transform of X(k) is defined as follows:

x(t) =∞∑

k=0

(t

H

)k

X(k). (8)

From Eq. (8), it can be seen that the theory of differential transformation is based upon the Taylor series expansion. The values ofunction X(k) at the values of argument k are referred to as discrete, i.e., X(0) is known as the zero discrete, X(1) as the first discrete, etc.he more the discretes that are available, the more likely it is possible to restore the unknown function. The function x(t) consists of the-function X(k), and its value is given by the sum of the T-function with (t/H)k as its coefficient. In real applications, for a right choice ofonstant H, as the value of argument k increases, the discreteness of the spectrum is rapidly reduced. The function x(t) is expressed by anite series, and Eq. (8) can be written as (Pourmahmoud, Mansoor, Eosboee, & Khameneh, 2011; Yaghoobi & Torabi, 2012b; Yahyazadeht al., 2012)


x(t) =n∑

k=0

(t

H

)k

X(k). (9)




Table 1Some fundamental operations of the differential transform method.

Original function Transformed function

x(t) = ˛f (x) ± ˇg(t) X(k) = ˛F(k) ± ˇG(k)x(t) = dmf (t)

dtm X(k) = (k+m)!F(k+m)k!

x(t) = f (t)g(t) X(k) =k∑

l=0

F(l)G(k − l)

x(t) = tm X(k) = ı(k − m) ={

1 if k = m0 if k /= m

x(t) = exp(t) X(k) = 1k!

x(t) = sin(ωt + ˛) X(k) = ωksin(

k� + ˛)

ti

f2

pat

(

(

(

(

3

nncaoaqbf

w

k! 2

x(t) = cos(ωt + ˛) X(k) = ωk

k! cos(

k�2 + ˛

)

Some of the important mathematical operations performed by differential transformation method are listed in Table 1. There are someechniques to increase the convergence of the DTM series. Among them, the Padé technique is widely applied. Suppose that a function f(�)s represented by a power series as:

f (�) =∞∑

i=0

ci�i. (10)

This expansion is the fundamental starting point of any analysis using the Padé approximant. The notation ci, i = 0, 1, 2,. . . is reservedor the given set of coefficients and f(�) is the associated function. The [L,M] Padé approximant is a rational function (Yaghoobi & Torabi,012b)

P[L, M] = a0 + a1� + . . . + aL�L

b0 + b1� + . . . + bM�M. (11)

This approximant has a Maclaurin expansion, which approximates Eq. (10) as far as possible. Although analytical methods cannotredict the particle trajectory contour versus CFD, they are fast and mesh-independent. Many advantages of DTM compared to othernalytical and numerical methods make it more valuable and motivate researchers to use it for solving the nonlinear problems. Some ofhese advantages are listed below:

1) DTM is a mesh-less method, so it does not require a mesh independency study. In addition, DTM solves the problem quickly andaccurately.

2) DTM does not require any perturbation, linearization, or small parameter, unlike the homotopy perturbation method (HPM) and theparameter perturbation method (PPM).

3) DTM is simple and powerful compared to numerical methods such as forth-order Runge–Kutta (RK4) and achieves the final resultsfaster than numerical procedures, while its results are acceptable and exhibit excellent agreement with numerical methods.

4) DTM does not need to determine the auxiliary function and parameter, unlike the homotopy analysis method (HAM).

.2. Differential quadrature method (DQM)

DQM is one of the efficient numerical methods that are widely used in engineering application. DQM rapidly converges to a fairly accurateumerical solution of differential equations in engineering problems. There are numerous (numerical) approximations of derivatives inumerical analysis; however, most of them lack the simplicity and efficiency in algorithmic construction and computational aspects. Unlikeonventional methods, such as finite difference (FD) and finite element (FE) methods, DQM requires fewer grid points to obtain acceptableccuracy. Another advantage of DQM is that it transforms the differential equations into a set of analogous algebraic equations in termsf the unknown function values at the resembled points in the solution domain. Perhaps the most important advantages of the DQMre its inherent conceptual simplicity and the fact that it is easily programmable. In this study, polynomial-expansion based differentialuadrature, as introduced by Quan and Chang (1989a, 1989b), is applied for solving the vortex problem. Several attempts have been madey researchers to develop polynomial based differential quadrature methods. One of the most useful approaches involves the use of theollowing Lagrange interpolation polynomials as test functions:

gk (x) = M(x)(x − xk)M(1)(xk)

k = 1, 2, . . ., N, (12)

here

M(x) = (x − x )(x − x ). . .(x − x ), M(1)(x ) =N∏

(x − x ). (13)


1 2 N i

k = 1

k /= i

i k



c

wrea

a

wAr

4

w

f

v


By applying the above equation at N grid points, the following algebraic formulations are developed to compute the weighting coeffi-ients:

A(1)ij

= 1xj − xi

N∏k = 1

k /= i, j

xi − xk

xj − xk, i /= j; A(1)

ij=

N∑k = 1

k /= i

1xi − xk

, i = j;

A(2)ij

=N∑

k=1

A(1)ik

A(1)kj

,

(14)

here A(1) and A(2) denote the weighting coefficients of the first and second order derivatives, respectively, of the function f(r) withespect to the r-direction. N is the number of grid points chosen in the r-direction. The differential quadrature approximation can beasily extended from the above formulation to other coordinates. The first order derivatives in the two-dimensional formulation arepproximated by (Akbari Alashti & Khorsand, 2011; Akbari Alashti et al., 2013):

∂f

∂r

)ij

≈N∑

l=1

A(1)il

flj,∂f

∂z

)ij

≈M∑

m=1

B(1)jm

fim (15)

nd the second order derivatives can be approximated by:

∂f

∂r

)ij

≈N∑

l=1

A(1)il

flj,∂2f

∂r2

)ij

≈N∑

l=1

A(1)il

flj;

∂f

∂z

)ij

≈P∑

m=1

B(1)jm

fim,∂2f

∂z2

)ij

≈P∑

m=1

B(2)jm

fim;

∂2f

∂r∂z

)ij

≈N∑

l=1

A(1)il

P∑m=1

B(1)jm

fim,

(16)

here A(1) and B(1) denote the weighting coefficients of the first order derivatives with respect to the r- and z-directions, respectively;(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,espectively; N and P are the number of grid points chosen in the r- and z-directions, respectively (Akbari Alashti et al., 2013).

. Results and discussion

To solve the particle motion in a forced vortex the system of Eq. (4) is solved using the DTM- Padé approach as follows.⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(k + 1)R(k + 1) − u0

r0ω0U(k) = 0,

u0

r0ω0(k + 1)U(k + 1) −

k∑k1

k1∑l=0

R(l) ¯ (k1 − l) ¯ (k − k1) + ˛2

(u0

r0ω0

)2 k∑l=0

U(l)U(k − l) = 0,

k∑l=0

R(l)(k + 1 − l) ¯ (k − l + 1) + 2u0

r0ω0

k∑l=0

U(l) ¯ (k − l)

−˛2

(k∑

k2=0

k2∑k1=0

k1∑l=0

R(l)R(k1 − l) ¯ (k2 − k1) ¯ (k − k2) − 2

k∑k1=0

k1∑l=0

R(l)R(k1 − l) ¯ (k − k1) +k∑

l=0

R(l)R(k − l)

)= 0,

(17)

here U(k), R(k), ¯ (k) are the DTM transformed forms of U(k), R(k), ˝(k) in Eq. (4). The initial conditions are

U(0) = 1, R(0) = 1, ¯ (0) = 0. (18)

Because the procedure of solving Eq. (17) involves the constants u0, r0, ω0, and ˛2, for generalization and simplification of the problemor future cases with different physical conditions, the constants that represent physical properties are assumed to be as follows,

u0 = r0 = ω0 = ˛2 = 1. (19)

After solving Eq. (17) by using initial conditions Eq. (18), the other terms of particle position, R(�), radial velocity, U(�), and angularelocity, ˝(�) will appear as follows:

R(1) = u0

r0ω0, U(1) = −˛2u0

r0ω, ¯ (1) = ˛2;

R(2) = −1 u20˛2

, U(2) = u20˛4

, ¯ (2) = −1 ˛2(u0 + 2˛2r0ω0);


2 r20 ω2

0 r20 ω2

02 r0ω0

R(3) = −13

u30˛4

r30 ω3

0

, U(3) = 13

˛4(r40 ω4

0 − 3˛2u40)

r30 ω3

0u0, ¯ (3) = 1

6˛2(6˛4r2

0 ω20 + 6u2

0 + 2˛2u0r0ω0 + 3u20˛2)

r20 ω2

0

. . .

(20)




(atf

E

Fig. 2. Calculated position trajectories (R) for a particle in a forced vortex obtained by different methods.

For example, when using constant values in Eq. (19), the final functions will be calculated as:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

R(t) = 1 + t − 12

t2 + 13

t3 − 16

t4 + 115

t5 + 17360

t6 − 4992520

t7 + 814720160

t8 − 2107930240

t9 + 509813453600

t10 − 175051619979200

t11 + 8053627129937600

t12 − 63553234631556755200

t13

+ 103136216031676505600

t14;

U(t) = 1 − t + t2 − 23

t3 + 13

t4 + 1760

t5 − 499360

t6 + 81472520

t7 − 210793360

t8 + 50981345360

t9 − 17505161907200

t10 + 805362712494800

t11 − 6355323463119750400

t12 + 10313621603119750400

t13

− 552091929139916800

t14;

˝(t) = t − 32

t2 + 176

t3 − 112

t4 + 414

t5 − 6641360

t6 + 3398105

t7 − 112201920160

t8 + 243964125920

t9 − 28445111181440

t10 + 45874001178200

t11 − 624104432914968800

t12

+ 10370857074591556755200

t13 − 2288364060278921794572800

t14.

(21)

Eq. (21) is a DTM solution of Eq. (4), results of which are depicted in Fig. 2 and compared with the numerical forth-order Runge–KuttaRK4) solution (EL-Naggar & Kholeif, 2012). As seen in this figure, the DTM solution, after a short agreement with numerical result, suddenlypproaches infinity (for all functions, R(t), U(t), and ˝(t)). To overcome this shortcoming and improve the convergence of this method,he Padé approximation was applied. Using the Padé approximant with accuracy of [6,6] results in Eq. (21) being transformed into theollowing form:

R(t) = (1 + 5.5828532t + 10.728458t2 + 7.9406046t3 + 1.7311221t4 + 0.48787532t5 + 0.27840808t6)(1 + 4.5828532t + 6.6456053t2 + 3.2530926t3 + 0.43988108t4 + 0.15648097t5 + 0.01235856t6)

;

U(t) = (1 + 3.6270954t + 5.2449498t2 + 3.6857199t3 + 1.2403949t4 + 1.4403629t5 + 0.58851084t6)(1 + 4.6270954t + 8.8720452t2 + 8.5973364t3 + 3.7170831t4 + 0.64910789t5 + 0.3698456t6)

;

˝(t) = (0.091822563t6 + 0.44660102t5 + 1.2839632t4 + 2.4523533t3 + 2.4126985t2 + t)(1 + 3.9126985t + 5.4880679t2 + 3.93008583t3 + 2.06204617t4 + 0.576084t5 + 0.2009531t6)

.

(22)

The polynomial-expansion based numerical differential quadrature is used for the non-dimensional time (�) to obtain the solution ofq. (4) with boundary conditions Eq. (18). After applying this method, the following set of equations was obtained:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪

(N∑

l=1

Ai,lRl,j

)− u0

r0ω0Ui,j = 0,

u0

r0ω0

(N∑

l=1

Ai,lUl,j

)− Ri,j˝

2i,j + ˛2

(u0

r0ω0

)2U2

i,j = 0,

( )(23)


⎪⎪⎪⎪⎪⎩ Ri,j

N∑l=1

Ai,l˝l,j + 2u0

r0ω0Ui,j˝i,j − ˛2R2

i,j(˝i,j − 1)2 = 0,




Table 2Comparison between the applied methods in non-uniform and non-dimensional time for the angular velocity results.

� ˝(�)

RK4a DTM-Padé [6,6] DQM

0.0000000000 0.000 0.000 0.0000.0301536896 0.0288632031758779 0.02886319921 0.03155900378703640.1169777784 0.100142863870931 0.1001428582 0.09956370644741570.2500000000 0.185921658986676 0.1859216559 0.1912457420765930.4131759114 0.267081663618700 0.2670816367 0.2700994189995330.5868240886 0.334822176009758 0.3348218891 0.3424470695101790.7500000000 0.386168789277334 0.3861657749 0.3902611734143340.8830222216 0.421185866197261 0.4211728745 0.4265683515874050.9698463104 0.441257247764227 0.4412281585 0.4463105657080621.0000000000 0.447764342038054 0.4477267112 0.453066160230719

a Data from EL-Naggar and Kholeif (2012).

Table 3Comparison between the applied methods in non-uniform and non-dimensional time for the radial velocity results.

� U(�)

RK4a DTM-Padé [6,6] DQM

0.0000000000 1.000 1.000 1.0000.0301536896 0.970737559394717 0.9707375589 0.9698607936576300.1169777784 0.895704744070214 0.8957047452 0.8953675797818970.2500000000 0.803455212921360 0.8034552207 0.8032069257292270.4131759114 0.720357770547646 0.7203578045 0.7231998898020200.5868240886 0.660545915322201 0.6605462648 0.6674546307593410.7500000000 0.625835951196769 0.6258387861 0.6347128496643660.8830222216 0.610250219603200 0.6102609542 0.6194288765556330.9698463104 0.605365864743497 0.6053880716 0.6139844755054961.0000000000 0.604553915451281 0.6045818949 0.612909563556282

a Data from EL-Naggar and Kholeif (2012).

wc1

i

Fig. 3. Calculated radial velocities (U) for a particle in a forced vortex obtained by different methods.

here R, U, and represent the particle position, the axial velocity, and the radial velocity, respectively. To proceed with the numericalomputations, non-uniform points in the � coordinate are defined using the Chebyshev–Gauss–Lobatto polynomials (Shu & Richards,992):

�k1= L

2

{1 − cos

(k1 − 1N − 1

�)}

, k1 = 1, . . ., N. (24)


These non-uniform points for � are presented in Table 2.The results obtained using the DQM (Eq. (23)) and the DTM-Pade (Eq. (22)) approaches are compared with those obtained using the RK4

n Tables 2 and 3 for ˝(�) and U(�), respectively. In addition, Figs. 2–4 compare the accuracy of the applied methods. In all cases, excellent




atat

5

qTstav

R

A

A

B

EEG

H

H

H

H

H

HH

H

JJJ

P

Fig. 4. Calculated angular velocities (˝) for a particle in a forced vortex obtained by different methods.

greement with RK4 is observed for both applied methods. Fig. 2 demonstrates the position trajectory (R) of the particle in the vortex. Forhe sample particle, the radial velocity of the particle is depicted in Fig. 3, and angular velocity is shown in Fig. 4. As observed in Figs. 2–4nd Tables 2 and 3, the DTM-Padé and DQM outcomes are in excellent agreement with that of RK4 (EL-Naggar & Kholeif, 2012), so thesewo methods are completely reliable and can be used in predicting the motion of particles in vortices or sedimentation processes.

. Conclusions

In this paper the analytical approaches, differential transformation method with Padé approximation (DTM-Padé) and differentialuadrature method (DQM), were successfully applied to produce accurate analytical solution for the motion of a particle in a forced vortex.he radial position, angular velocity and radial velocity of the particle were calculated and depicted. As a primary outcome of the presenttudy, the results of the DQM were in excellent agreement with the numerical ones. In addition this method is a simple, fast, and efficientechnique for finding problem solution in science and engineering with coupled non-linear differential equations. The proposed methodlso reduces the burden of the calculations. The results show that over time, the particle recedes from the vortex center and its radialelocity decreases, while its angular velocity increases.

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Motion of a spherical particle in a fluid forced vortex by DQM and DTM