,20 (999 ) Applied Mathematics and Mechanics 000-0887(999) -49-2 Ξ (, 6004) ( ) Macsma Darmstadt Macsma, Oldrod B Kantorovich, 3 2 2 3,, ; Oldrod B ; ; ; O373 A,,Macsma,Maple,Mathematica,,, [7 ],,,, [83 Kantorovich, ] Darmstadt (THD), Macsma, [4, Maxwell Jeffre ], Ξ 998-03-06 ; 999-06-25 (9672063) (AvH) (935),,,, 80,,. 49 995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.
50 Oldrod B, Weisnerberg,, 3 THD Macsma Kantorovich, 3, 2, 2 3 ( ),,,, [57,4,56 ] Oldrod B ( r,, z) v r = v z = 0, v = v ( r, t), () Oldrod B g ik S S ik + = 0 ( A ik g + ik 2 A ), (2) g, ik S g ik S = 5 S ik 5 t + v m 5 S ik 5 x m - S im 5 v k 5 x m - S mk 5 v i, S ik 5 x m (3) () (2) S rr = S zz = S rz = 0, S z = 0 (4) 5 S r 5 v S r+ = 5 t 0 5 r - v + r 5 t 5 r - v (5) r 2 5 5 v 5 v = 5 5 t r 2 5 r ( r2 S r ) (6) (6) (5), 5 v 5 v 5 2 v 5 2 v + 5 t 5 t 2 = 0 5 2 r 2 + r 5 r - v + r 2 5 t 5 r 2 + r (7) L ( F) = F r + Ha F rr - F + F - F 2 2 5 + Ka 5 5 r 5 2 v 5 v 5 r - v r 2 (7) F + F - F 2 = 0, (8) F = v, = 0 t U 0 R 2, = r, R (9) Ha = We Re, Re = R U 0, We = U 0, Ka = 2 0 R Ha (0) (0), Ha Ka, Ha, Ka 995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.
5 2, (6), r = R, v = R, r = R 2, v 2 = 2 R 2 () v = C r + C 2 r, m C = 2 - n - m 2, C 2 = - R 2 n - - m 2 (3) m = r2, n = 2 R (4), 2, R R 2 (9), (2) F 0 = a 0 + b 0, (5) a 0 = m2 - n m 2 -, b 0 = - (2) - n m 2 (6) - 3 Oldrod B, F 0, a 0, b 0 n 0, F, a, b n F 0, F 0, F - F 0 = a -,, a = n 0 - n - m 2, (7),, (7),,,,,,,,n n 0 n,,,, n 0 n,, Kantorovich,,, -, - 2, - 4, (8) (8) 995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.
52 F N (, ) - F 0 = N k = f k (, ) (9) f k = - 2 ( k - ) g k ( ) (20), H[ v ] = 0, Kantorovich Kantorovich N, N - f, f 2,, f N -, N,, F N H N f j - (2) 2 ( j - ) g j ( ) dd = 0, ( j =,2,, N) (22), N, f N, H[ f N ] - J k - = H[ f ] - H[ f ] - H[ f 2 ] - H( f k - ) - 2 ( N - ) g N ( ) dd + N J k - = 0, (23) 2 ( k - ) g k ( ) dd ( k =,2,, N), (24) g ( ) dd = 0 (25) 4 2 g 2 ( ) dd = 0, (26) 2 g 2 ( ) dd = 0 (27), Macsma, (8) N F(, ) = F 0 + - [ g ( ) + 2 g 2 ( ) + + 2 ( N - ) g N ( ) ] (28) (28) (23), N, d 2 g N ( ) a N d 2 + b d g ( ) N N d 2 + C N g N ( ) + N k =3 J k - = 0, (29) a N = HaP N ( m), (29) b N = P N ( m) + KaQ N ( m), (29) c N = Q N, (29) g P N ( m) = m ( p n3 m 4 n - + p n2 m 4 + p n m 2 + p n0 ) 995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.
53 p n3 = 8 (4 n - 7) ( n = N), p n2 = - 64 n 3 + 76 n 2-24 n + 2, p n = 4 (32 n 3-04 n 2 + 94 n - 5), p n0 = - 64 n 3 + 240 n 2-284 n + 05, Q N ( m) = m 3 [ q n3 m 4 n - 3 + q n2 m 4 + q n m 2 + q n0 ], q n0 = 4 (64 n 5-272 n 4 + 396 n 3-223 n 2 + 35), q n = 8 ( - 64 n 5 + 304 n 4-540 n 3 + 445 n 2-66 n + 2), q n2 = 4 (64 n 5-336 n 4 + 652 n 2-579 n 2 + 229 n - 30) q n3 = 6 (8 n 3-22 n 2 + 7 n - 3) (29), J k -, d 2 g k - ( ) d 2 g k - ( ) J k - = j k - d 2 + l k - d + m k - g k - ( ), (30) j k - = HaS k ( m), l k - = T k ( m) + KaS k ( m), m k - = T k ( m), S k ( m) = ( s k3 m 4 k - 3 + s k2 m 4 + s k m 2 + s k0 ), s k0 = - 64 k 3 + 336 k 3 + 572 k + 35, s k = 2 (64 k 3-304 k 2 + 444 k - 89), s k2 = - 64 k 3 + 272 k 2-348 k + 35, s k3 = 8 (4 k - 9), T k ( m) = Ham 5 ( t k3 m 4 k - 5 + t k2 m 4 + t k m 2 + t k0 ), t k0 = 4 (64 k 5-496 k 4 + 484 k 3-229 k 2 + 455 k - 378) t k = 8 ( - 64 k 5 + 528 k 4-692 k 3 + 265 k 2-932 k + 540), t k2 = 4 (64 k 5-560 k 4 + 868 k 3-2965 k 2 + 2229 k - 630), t k3 = 8 (6 k 3-68 k 2 + 90 k - 36) -, F + F - F 2 = 0 (3) F = g ( ) -, L ( F ) - 2 ( k - ) d = 0 (32), k J = 0, F(, ) = F 0 + - g ( ) (34) (25), (33) 995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.
54 Ha d2 g ( ) d 2 + d g ( ) = 0 (35) d = 0, g (0) = 0 ;, g (36) (35), g ( ) = A, g 2 ( ) = A 2 e - Ha (37), (35), (35) g ( ) = a ( - e - / Ha ) (38) F(, ) = F 0 + a - [ g ( ) + 2 g 2 ( ) ] (39) (39) (27), g ( ), d 2 g 2 ( ) d g 2 ( ) a 2 d 2 + b 2 d + c 2 g 2 ( ) = 0, (40) a 2 = HaP 2 ( m), m = R, R 2 (40) b 2 = P 2 ( m) + KaQ 2 ( m), (40) c 2 = P 2 ( m) = 8 m 7-35 m 4 + 42 m 2-5, (40) g Q 2 = 56 m 2 (2 m 5-5 m 2 + 3) (40) (40) g 2 ( ) = Ae k (4) (40), k a 2 k 2 + b 2 k + c 2 = 0 (42), k,2 = - [ P 2 ( m) + KaQ 2 ( m) ] [ P 2 2 ( m) + 2 ( Ka - 2 Ha) P 2 ( m) Q 2 ( m) + Ka 2 Q 2 2 ( m) ] 2 [2 HaP 2 ( m) ] - (43) (43), P 2 ( m) 7 ((40) g), 7, P 2 ( m) = 0, 3,4, m = (3 ) m 2,3 = i 2-2 05 4-3 4, 2 + 2 05 3 m 4,5 = i - 4 4, M Ha Ka, k k 2, M Ha Ka, k k 2 g 2 (4) (42) 995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.
55 g 2 ( ) = A 3 e k + A 4 e k 2 (44) = 0, g 2 (0), = 0 ;, g 2 ( ) 0 (45) A 3 = - A 4 F(, ) = F 0 + a - [ ( - e - Ha + A 3 2 (e k - e k 2 ) ] (47) k k 2 (43) A 3 5 =0, = / m = 0, (48) A 3 = - m 2 Ha ( k - k 2 ) 2 k = k 2, (43) P 2 2 ( m) + 2 ( Ka - 2 Ha) P 2 ( m) Q 2 ( m) + Ka 2 Q 2 2 ( m) = 0, g 2 = ( B + B 2 ) sh ( k ) + ( B 3 + B 4 ) ch ( k ) (49) (45),B 3 = 0, g 2 ( ) 0, B 4 = - B 2, B = 0, g 2 ( ) = B 2 [sh ( k ) - ch ( k ) ] (50) F(, ) = F 0 + a - B 2 (48), B 2 = m2 Ha 3 F(, ) = F 0 + a - (23), (46) [ ( - e - Ha) + B 2 2 (sh ( k ) - ch ( k ) ) ], (5) (5) [ g ( ) + 2 g 2 ( ) + 4 g 3 ( ) ] (52) d 2 g 3 ( ) a 3 d ) 3 d + c 3 g 3 ( ) + J 2 = 0, (53) a 3 = HaP 3 ( m), (54a) b 3 = P 3 ( m) + KaQ 3 ( m), (54b) c 3 = Q 3, (54c) P 3 ( m) = 5 (8 m - 99 m 4 + 54 m 2-63) (54d) Q 3 ( m) = 24 m 2 (44 m 7 + 23 m 4-660 m 2 + 385), (54e) 995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.
56 d 2 g 2 ( ) d g 2 ( ) J 2 = j 2 d + l 2 d + m 2 g 2 ( ), (54f) j 2 = HaS 3 ( m), (54g) l 2 = S 3 ( m) + KaT 3 ( m), (54h) m 2 = T 3 ( m), (54i) S 3 ( m) = (8 m 9-63 m 4 + 90 m 2-35), (54j) T 3 ( m) = 72 m 2 (2 m 7-7 m 2 + 5), (54k) 2, g 2 ( ) = e k - e k 2, J 2 J 2 = a 22 e k - b 22 e k 2, (55) a 22 = j 2 k 2 + l 2 k + m 2, b 22 = j 2 k 2 2 + l 2 k 2 + m 2 (53) g 3 ( ) = g s ( ) + g h ( ), (56), g s (52), g s ( ) = D e k + D 2 e k 2 (57) (57) (53), D D 2 D = - a 22 c 3 + b 3 k + a 3 k 2, (58a) b 22 D 2 = - c 3 + b 3 k 2 + a 3 k 2, (58b) 2 g h ( ) d 2 g h ( ) a 3 d 2 + b d g ( ) h 3 d 2 + c 3 g h ( ) = 0, (59) (59) g h ( ) = Eexp ( q ) (60) (59), q a 3 q 2 + b 3 q + c 3 = 0 (6) q,2 = [ - [ P 3 ( m) + KaQ 3 ( m) ] [ P 2 3 ( m) + 2 ( Ka - 2 Ha) P 3 ( m) Q 3 ( m) + Ka 2 Q 2 3 ( m) ] 2 ] [2 HaP 3 ( m) ] -, (62), g h ( ) g h ( ) = D 3 e q + D 4 e q 2 (63) = 0, g 3 (0) = 0, (63), g 3 ( ) = 0 (63) q > 0, q 2 < 0 (6) 995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.
= 6 2, Ha = 6 Ka, q > 0, D 3 = 0, (64) g h g h = D 4 e q 2, (65) g 3 g 3 ( ) = D e k + D 2 e k 2 + D 4 e q 2, (66) (63a), D 4 = - ( D + D 2 ) (67) g 3 ( ) = D e k + D 2 e k 2 - ( D + D 2 ) e - q 2 (68) k, k 2, q 2, (63b), 3, 2 q = q = q 2 < 0 P 2 3 ( m) + 2 ( Ka - 2 Ha) P 2 ( m) Q 3 ( m) + Ka 2 Q 2 3 ( m) = 0 (69), m 3 q = - [ P 3 ( m 3 ) + KaQ 3 ( m 3 ) ] 2 HaP 3 ( m 3 (70) ) g h = ( D2 3 + D3 3 ) e - q + ( D 5 + D 6 ) e q (7) (63b), D2 3 D 3 3, g h = ( D 5 + D 6 ) e q (72) g 3 ( ) = D e k + D 2 e k 2 + ( D 5 + D 6 ) e q (73) (63a) D 5 = - ( D + D 2 ) (74) (73) D 6 g 3 ( ) ] 5 =0, = / m = 0 (75) D 6 = D ( q - k ) + D 2 ( q - k 2 ) (76),(47) A 3, (5) B 2, (73) D 6, (48) (5a) (76),,(47) =, = / m, F( m, ) = F 0 ( m) - a m2 - - [ ( - e / Ha m ) + m - 2 (e k - e k 2 ) ], (77) 2,(5) =, = / m, F( m, ) = F 0 ( m) - 57 m 2 - m B 2 A 3 - [ ( - e / Ha ) + m - 2(sh ( k ) - ch ( k ) ) ], (78) 995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.
58 (77), (52) (76) F( m, ) = F 0 ( m) - a m2 - ( - e - / Ha ) + m - 2 (e k - e k 2 ) + m m - 4 ( D e k + D 2 e k 2 - [ D + D 2 - ]e q ) (79) (77) (79), A 3 = B 2 = D 6 = 5, Kantorvich Oldroa B, 3 MACSY2 MA, 2 2 6 (47) (5) (73) ( A 3 = B 2 = D 6 = ) 2 (28) 2 3 g ( ) g 2 ( ) g 3 ( ) 3 4 Y = 32 Y = 4 F M = / m, F F 2 F 2 F 3, 5 (47), A 3 F A 3 5, = 0,,,, 6 F, 2 3 5 6, (47) (5) (73), Oldrod B,,,, Bodart Crochet [ 5 ],,,,,, 2 Bodart Crochet, Ol2 drod B, Oldrod B, 995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.
59 3 4 5 6 [ ] [ ] Han Shifang. A variational method for research on an unstead flow of upper- convected Maxwell flu2 id[j ]. Acta Mathematica Scietia,992,2 (2) 59525. [2 ] Han Shifang, Roesner K G. Time dependent flow of upper- convected J effre fluid between two ro2 tating clinders [A ]. In Proc of XI Inter n Congress on Rheolog[ C ]. Brussels,992,2628. [ 3 ] Han Shifang, Roesner K G. The time dependent flow of upper- convected J effre fluid between coax2 ial clinders [J ]. Comp utational Fluid Dna mics J,J apan,993,2 () 904. [4 ] Han Shifang, Roesner K G. Unnstead flow of polmer fluid between coaxial clinders [J ]. J of H2 drodnamics,993,ser B2 () 5262. [5 ] Han Shifang. Computational simulation of non-newtonian fluid flow [ A ]. In Proc of Fi rst CFD Conference[ C ]. Hong Kong,995,900. [6 ]. ( ),, [A ]. [ C ].,990,6. [7 ].,, [A ]. 993,66, [ C ].,993. [8 ] Han Shifang. Unstead flows of Maxwell-Oldrod fluid in tube [ A ]. In Proc 4- th Asia n Congress on Fluid Mech[ C ]. Hong Kong,989. 7376. [9 ]. [M ].,988. 995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.
60 [ 0 ] Han Shifang, et al. Non- stead flow of upper- convected Maxwell fluid in circular tube [J ]. Acta Me2 cha nica Si nica,990,6 (3) 22226. [ ],. Oldrod B [J ].,995, 6 (2) 6978. [2 ]. [J ].,993,3 (4) 92. [ 3 ] Ramkissoon H, Han Shifang. Unstead motion of a spehere in elasticviscous fluid[j ]. Inter n J En2 gng Sci,993,3 () 926. [ 4 ] Roesner K G. The impact of computer algebra on fluid dnamics [A ]. In Proc 2 nd Europea n Fluid Mecha nics Conference General Review Lect ure[ C ]. Warsaw 994,04. [5 ] Kantorovich L W, Krlov W I. App roxi mate Methods of Higher Analsis [ M ]. Interscience, New York Elsevier,958,256258. [6 ] Gorla R S R, Modden P E. A variational approach to non- stead non-newtonian flow in circular tube [J ]. J Non-Newtonia n Fluid Mech,984,6 () 25265. [7 ] Ch Bodart, Crochet M J. Numerical stud of the stabilit of viscoelastic flows [A ]. In Theoretical a nd Applied Rheolog, Proc[ C ]. XI th Intern Congress on Rheolog,992,256257. Comp ut a ti onal I nt ellect ual Analtical The or of Comp ut a ti onal Analtical App r oac h t o Rot a ti ng Fl ow of Non- Newt onia n Fl ui d Han Shifang ( Cha ngdu Bra nch, Academia Si nica, Chengdu 6004, P R Chi na) Abs t ract A combination of the computational smbolic calculation, mathematical approach and phsico- mechanical model leads to a computational intellectual analtical approach developed b the author. There is a principal difference between the computer proof and the computer derivation com2 pleted b the computer, also difference between the numerical and smbolic calculations. In this in2 vestigation the comutatiinal analtical approach is extended, and an unstead flow of non- Newtonian fluid is the gap between two rotating coaxial clinders is studied. The Oldrod fluid B model is used b which the Weissenberg effects are explained in a good comparison with the experiments. The gov2 erning equations are reduced to a partial differential equation of 3rd order for the dimensionless veloci2 t. Using the computer software Macsma and an improved variational approach the problem with the initial and boundar conditions are then reduced to a problem of an ordinar differential equation for different approximations. The analtical solutions are given for the st, 2nd 3rd approximations. The present investigation shows the abilit of the computational smbolic manipulation in solving the prob2 lems of non- Newtonian fluid flows. There is a possibilit of that to solve the problems in the mathe2 matics and the mechanics. An important conclusion can be drawn from the results that the transition from a stead state to another stead state is non- unique. Ke words time- dependent rotating flow ; non- Newtonian fluid ; Oldrod fluid B ; computational smbolic manipulation ; computational analtical approach 995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.