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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