Motion of a spherical solid particle in Couette flow: exact ... - arXiv

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Abstract. The motion of a spherical solid particle in plane Couette flow is governed by a linear problem that has a simple exact solution. As such, there is no need ...
Motion of a spherical solid particle in Couette flow: exact solution vs. homotopy perturbation approximation with and without Padé approximants Tarek M. A. El-Mistikawy Department of Engineering Mathematics and Physics, Cairo University, Giza 12211, Egypt

Abstract The motion of a spherical solid particle in plane Couette flow is governed by a linear problem that has a simple exact solution. As such, there is no need for an approximate analytical representation of the solution; specially when it is tedious, complicated, and requires hairy terms to give accurate results only at small or moderate values of the time. Keywords: Couette flow; solid particle; exact solution; homotopy perturbation; Padé approximants Introduction The motion of a spherical solid particle in plane Couette flow is governed by the following linear problem x  Ay  B( x  y)  0

(1)

y  By  (C  A)( x  y)  0

(2)

x(0)  0, x (0)  u

(3)

y(0)  0, y (0)  v

(4)

where dots denote differentiation with respect to time t. This problem was first formulated by Vander Werff [1], who also obtained its closed form solution when u=0. The solution when u≠0 can be easily obtained using Laplace transform. It is x  t    e  Bt cosh t  (u    B )e  Bt y    e  Bt cosh t  (v  B )e  Bt

sinh t



sinh t

(5)

(6)



where

1

C  C  A,   C (  A)  B,  

vB  uC (uB  vA)  2 B ,  2 2 B  B 2  2

(7)

For =B, the solution becomes

x

(vB  uC ) 2 2 B(uB  vA)  (vB  uC ) t  t 4B 4B 2 2 B(uB  vA)  (vB  uC )  (1  e  2 Bt ) 3 8B

(5')

(vB  uC ) vB  uC t (1  e 2 Bt ) 2B 4B 2

(6')

y

Jalaal et al. [2] handled the problem, for the special case A=B=C= =u=v=1, using the homotopy perturbation method (HPM). They stated 5 terms of the expansions for x and y, which combine to give

x  (e t  1)  (e t  t  1)  (2te t  5e t  12 t 2  3t  5)  (t 2 e t  8te t  19e t  16 t 3  52 t 2  11t  19)  ( 13 t 3 e t  6t 2 e t  36te t  81e t  241 t 4  76 t 3 

t  45t  81)

21 2 2

(8a)

 y  (e t  1)  (2te t  3e t  t  3)  (t 2 e t  6te t  11e t  12 t 2  5t  11)  ( 13 t 3 e t  5t 2 e t  24te t  45e t  16 t 3  72 t 2  21t  45)  ( 121 t 4 e t  73 t 3 e t  22t 2 e t  100te t  191e t 

t  32 t 3  352 t 2  91t  191)

1 4 24

(8b)

 They also presented results for up to 12 terms with reasonable accuracy achieved for t up to 6, at most; as their Fig. 1 indicates. Hamidi et al. [3] extended the accuracy of this homotopy perturbation solution to higher values of t by representing more terms by Padé approximants; as their Figs. 5 and 6 indicate. They gave the [8/8] Padé approximant for the x-velocity component and the [10/10] Padé approximant for the y- velocity component as follows.

2

80630168 94677997 2 130919939 3 t t  t 150869313 258633108 1508693130 1097649283 4 352444733 5 573969527  t  t  t6 94142451312 353034192420 10355669644320 29057689 213287989  t7  t8 1553350446 6480 7117169319 187200 Vx  80630168 34638557 2 10391023 3 1 t t  t 150869313 258633108 502897710 203299561 4 55668097 27350129  t  t5  t6 94142451312 353034192420 3451889881440 786311 24009973  t7  t8 3106700893296 6022220193158400 1

(9a)

617044058999277705 28813203934779614633 2 t t 383360519991315667 29135399519339990692 706520298740306571137 3 158825071969373483159 4  t  t 1485905375486339525292 1485905375486339525292 8660567629020972911 5 16420817408783354039 6  t  t 782055460782283960680 71323458023344297214016 87188708731614445057 1062628804263470249167  t7  t8 1248160515408525201245280 129808693602486620929509120 56968766216441448797 260048516288380772873  t9  t 10 1460347803 0279744854 5697760 3671731619 0417644205 775436800 Vy  149676980983353629 7861005765239380203 2 1 t t 383360519991315667 29135399519339990692 215683905534451894091 3 22009213655154354015 4  t  t 1485905375486339525292 495301791828779841764 42598825941687607239 5 404907655928398513021 6  t  t 4953017918287798417640 356617290116721486070080 43862642271172589551 883706266259083441033  t7  t8 416053505136175067081760 129808693602486620929509120 16590820145764536325 307482708619550706343  t9  t 10 58413912121118979418279104 51404242666584701888085611520 1

(9b)

The simple exact solution given by Eqs. (5)-(7) reduces for this special case {using sinh t lim  t } to  0  x  2  2e t  te t , y  te t

(10a,b)

so that 3

Vx  (1  t )e t , Vy  (1  t )e t

(11a,b)

Comparing Expressions (8) to (10) and (9) to (11) one realizes how futile it is to handle this particular problem with an approximate analytical approach. References [1] T. J. Vander Werff, Critical Reynolds number for a spherical particle in plane Couette flow, Zeitschrift fur Angewandte Mathematik und Physik 21 (1970) 825-830. [2] M. Jalaal, et al., Homotopy perturbation method for motion of a spherical solid particle in plane Couette fluid flow, Computers and Mathematics with Applications, 61 (2011), pp. 2267-2270. [3] S.M. Hamidi, Y. Rostamiyan, D.D. Ganji, , A. Fereidoon, A novel and developed approximation for motion of a spherical solid particle in plane coquette fluid flow, Advanced Powder Technology (2012), http://dx.doi.org/10.1016/j.apt.2012.07.00.

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