On the flow of an Oldroyd B liquid through a straight circular tube performing longitudinal and torsional oscillations of different frequencies
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(2) \. . O % w `'Y$ 5L ¡ h ° NWQ S °ND7 !$ I °@ N) 'Y !$ e$ # -, @' 4 D 1 jNMQ¡ ! Õ Q ! °¡ ' b % k ) #^ ) ` hp 4 ,!$ !$ I !$# $ !$# $ A' $ 5J ¡ ° NQ T °Nu7 !$ I °@ $ !$ $ ì'' ' N)?7 4 B % &e D 1&6M ! ° $ C % #p $e$ ' " % ìQN # ! MNM' °kL ! àj$ % e$ !$+# Q T 4h(N$ Q 'I # CL @'N !$ I A , 'N 4Q¡ $ ! "6V$c D 1 ! °¡ ' ¡ " $ ½NM u 1^Õ -, !$ °>\ ! % N( °4J Q' % ¡ !$ °Ia $ , 4 °GJ W[M Q A °N ! °¡ '/° < ue$ !$+# (N$ G-H !$ ° ej e ! '/° BÕ 5)+ 4 oQ Q Qe 4 ° !$# $ e $ ½ ËD 1 ° %#l 4 J ! Q' $ ½N ;) !$#! ° N Q I °N9¡E °@ % E-, !$ °TfÒ e °¡ Ë ¡ $ ! = !$ °>g ! ° kb°¡ ! ° ! C ¡ 1Õ. #. $&%(' ü ')zøÍÿ.ø(* ' +ý,[ù ÿUû ø(. k #^ W0/ Y+6\jq4 #^ ) . Q ! z °k. "L ©Q k ). T = −pI + S S + λ1 S̆ = 2µD + 2λ2 D̆ ß . &6 k &6 $ T I S ' ! ' b 76 NJ ¡ ° 4 D = 1 (q + q ) !$ Q $I[M ° ¡ 76 2 ! 7i,jr#l! j,i 2 1 µ λ2 # ¡ % '' $ 5) ! °e !$ h $ e N( l%# ˘. 85X : 8J\ :. &6 p &6 λ1 ° ¡ 4 mW. ∂C ij ij j i C̆ ij = + q m C,m − C im q,m − C mj q,m ∂t $ Qe$ !$+# (N$ q -, q % ! MNM $ N L ! # NQ¡ ! ° dq ∇ · S − ∇p = ρ dt q °o ! ° +#p Q l. 8Ja :. 8e : ∇ · q = 0. ! ÒlN % NM\Q¡ I h ° $! !$ I % 76MlN $! # ! '/I ! ^ N ),[M M [M . !$# $ (r, θ, z) z Q b ¡ Qe$ !$+# (N$ 6 6 4 e % # q 8Jf : q = (0, v(r, t), w(r, t))..
(3) . a. - 2 Õ # 4V N 4 ! ° r# ° 8 e :D °¡ ! # 4 5(N % # 8Jf : ! SN 1e !$# $ % >' $ 5J ¡ 7 !$ I ° ! % % # e !$ 6 $ qb 8Jg : qb = q r=a = q0 cos(Ω1 t) cos(β)θ̂ + q0 cos(Ω2 t) sin(β)ẑ ! Õ !$ °4 ' ! ) $ Q e !$ Q ẑ θ− z− θ̂ e$ # .mW<'N 4Q¡ $ $ ¡ 9¡ 6 0 ! N q0 , Ω 1 , Ω 2 β q0 B ' !$ eBL !$ Q ËQ7 © ! B7 !$ I °&6 Ω ,Ω β % # e$ !$+# 1 6 2 e q 8J\[]g : 6°¡1e o !$ ° ! q θ̂ !$# $ ËNQb° #> °N07 !$ I °D $ ÕL 6 N β =0 β = π2 ° # ° Nj7 !$ I °Õ *BÕ6. % 8Jf :Q 8J\ : Q 8Ja : 6^jCe [ $ @ °¡ e % 46 "J Õ °4L Srθ , Srz Srθ + λ1. . 8/ :. ∂Srθ ∂v v ∂ ∂v v = µ( − ) + λ2 ( − ) ∂t ∂r r ∂t ∂r r ∂Srz ∂w ∂ ∂w Srz + λ1 =µ + λ2 ( ) ∂t ∂r ∂t ∂r. ρv 2 ∂p = ∂r r 1 ∂p ∂Srθ 2Srθ ∂v =− + + ρ ∂t r ∂θ ∂r r ∂p ∂Srz 1 ∂w =− + + Srz . ρ ∂t ∂z ∂r r d ¡ N Q L °¡ ß °¶ > ' S S °4J rθ Q rz DW m Ë °pL v(r, t) w(r, t) v(r, t). 8 : 8 : 85X : 85X7X : @L . ∂2v ∂v ∂ 2 ∂v v ∂ ∂v v − (µ + λ2 )( ( − )+ ( − )) +ρ 2 ∂t ∂t ∂t r ∂r r ∂r ∂r r ∂ ∂p 1 ∂p =− ( + λ1 ( )) r ∂θ ∂t ∂θ q °lJ Q l w(r, t) ρλ1. 85X&\ :. 85X&a : ∂2w ∂w ∂ 1 ∂w ∂ ∂w ∂p ∂ ∂p +ρ − (µ + λ2 )( + ( )) = −( + λ1 ( )). 2 ∂t ∂t ∂t r ∂r ∂r ∂r ∂z ∂t ∂z ! No ¡ N p = p(r, t) ! $ ¡E ∂p , ∂p % "' 4 ∂θ ∂z $ \ ! ¡¡ $ # % $ e N " jjA' λ = 0 j, ! Fe$ °BJ 2 @s"U _ `W]f[YÍQ o Í' j e 3 Ë °3J λ1 = λ 2 = 0 @UDI b ! c 6 W ! $ &W7YH ρλ1.
(4) e. !. rû.ýoù ÿ"ûlø-û ù %'zû . . '+ý,[ù ÿUû ø(-. Õ e© 85X&\ : 6H85X&a : % Z !$D mW % #l! ° ° . %. #P! ° °4. !$ °\[. vb (a, t) = q0 cos(β) cos(Ω1 t). !. wb (a, t) = q0 sin(β) cos(Ω2 t). N 1PL 3b °i "L ¡ h i v(r, t) = < f (r)exp(iΩ1 t). Q . h i w(r, t) = < g(r)exp(iΩ2 t) h Õk'N $ Q 4 5 # % # <[z] z f (r), g(r) ! ° ° f (a) = q0 cos(β), g(a) = q0 sin(β) 85X&\ : z % >J 6 6MjPe A J Õ v(r, t) p = p(r, t) °lJ f (r) f 00 +. $ . 1 f0 + (− 2 + K12 )f = 0 r r. K1 =. Q 1 λ2 K = µ+iΩ ρ mWb °oz 85X$e :u. r. 1 (λ1 Ω21 − iΩ1 ) K. 85X$e :. 85X&f :. f (r) = AJ1 (K1 r) + BY1 (K1 r). $ Q W 4$ !$ °D 2( Q ! ° 1 'Y $! e$ # 6 J Y Õ ! I01 $ Q°176rQ Q ! ° Q Õk*BÕ 6Q ! 6 A B v(r, t) f (r) ¡ 3% B( q ! DjD¡ ' 6Q b r=0. B=0. h i v(r, t) = < AJ1 (K1 r)exp(iΩ1 t) .. Q S j e 6j° ! '/ 4 K1 = iK̂1 J1 (iK̂1 r) = iI1 (K̂1 r) % #l! ° °c6 v(r, t) = <. h I (K̂ r) 1 1. I1 (K̂1 a). i exp(iΩ1 t) q0 cos(β). 85X&g :.
(5) . f. q¡E (N b$ !$ °> ( $ ρ(iΩ1 −Ω21 λ1 ) Q K̂1 = I1 1 λ2 1 $ i°µ+iΩ h i ¡ I #q V N 4 5(N q. w(r, t) = < g(r)exp(iΩ2 t). g(r). g0 + K22 g = 0 r 4 e %# 85X&f : \ $ 6 $'I ! o% # K2 Ω1 Ω2 ! % #q! ° ° mWb °lL w(r, t) g 00 +. w(r, t) = <. h I (K̂ r) 0 2. I0 (K̂2 a). i exp(iΩ2 t) q0 sin(β). 85X / :. 85X :. ½¡E (N W $ !$ °l i( 1 $ ρ(iΩ2 −Ω22 λ1 ) 6 I0 K̂2 = µ+iΩ2 λ2 $ $ Q DN e r) = J (iK̂2 r) = I0 (K̂2 r) ! ¶¡ ° w N 4 BJ ¡EJ40 (K 2Q 0 ° j¶N e 4 ¡ ' # N 6^Ë¡ ° T !$ °cX76Í ¡ 6 ¡E °Q S t=0 e$ !$+#l! °¡ 'H° (N$ ½N l BJ ¡E q. v(r, t) |t=0 = <. Q . h I (K̂ r) i 1 1. w(r, t) |t=0 = <. I1 (K̂1 a). q0 cos(β). h I (K̂ r) i 0 2. q0 sin(β). I0 (K̂2 a) ! N G °N7 !$ I °& !$ ° # θ) ! °¡ 'H° e$ $! +# 6 A ° N<7 !$ I °Ë !$ ° #o ) ! °¡ 'H° 0e$ $! +# 6 6Q h v z w. . %',\ú -VùÕú '--Íù,ø(* 'ø ù ÿ ,. WüÍú. , * ,øoü. àûNú ü^û ø('. o % S ,e$ !$+#S! °¡ 'H° jCNMÕ $ 5) v(r, t), w(r, t) ¡ pÕ 5)+ ! °¡ '/° mW a V S ,S r = a $ &6w $ 6 ½ ¡ $ ( C 46 6[ ° rθ!$# rz S·n n 5$ ! qQ Q F6[ ! o'Y$ 3 e % # 6 6[ $ S D. !. ¶. D = −2πa(Srθ θ̂ + Srz ẑ) |r=a .. 85X :. ©N eQ4V N 2 "Õ 5)+ ! °¡ 'H° i4 5 # °8 / : Q 8 : Õ 85X&g : 6485X : Q ' ' $ , sB NM' °QJ v(r, t), w(r, t) W $2 !$ ° jle 6D° Õ 8 / : 6 Jn (x), Jn0 (x), In (x), In0 (x) 8 : 6J 6 Srθ , Srz.
(6) g. Srθ |r=a = q0 cos(β)<. Q . . i 8J\ : h (µ + iΩ λ ) (K̂ I (K̂ a) − 2 I (K̂ a)) 1 0 1 1 1 2 a 1 exp(iΩ1 t) (1 + iΩ1 λ1 ) I1 (K̂1 a). 8J\X :. h I (K̂ a) (µ + iΩ λ ) i 1 2 2 2 Srz |r=a = q0 sin(β)< K̂2 exp(iΩ2 t) . I0 (K̂2 a) (1 + iΩ2 λ1 ). sB Õ 85X : 8J\X : DN e ". ! K̂1 I0 (K̂1 a) − a2 I1 (K̂1 a) µ + iΩ1 λ2 D = −2πa< q0 cos (β) exp(iΩ1 t)θ̂ 1 + iΩ1 λ1 I1 (K̂1 a) ! # K̂2 I1 (K̂2 a) µ + iΩ2 λ2 +q0 sin (β) exp(i Ω2 t)ẑ . 1 + iΩ2 λ1 I0 (K̂2 a) 8J\7\ : . mWìu 1 °76 6 % # 4 L ! 76 6"° W D °NÍQ ° j No¡E ° 4 e % #. Wj = −. Z. π Ωj. > . 'Y$ N ) !$#^! S. 8J\7a :. D · qb dt,. 0. $ 6 N6$ °NÍQ ° NÍ¡E °3 ' !$ e$ # j = 1, 2 % 8Jg : Q 8J\7\ :u 8J\7a : e. ". ! 2 I ( K̂ a) K̂ I ( K̂ a) − µ + iΩ λ 1 1 1 0 1 1 2 a ˆ j , Ω1 ) Wj = πaq02 < cos2 (β)I(Ω + 1 + iΩ1 λ1 I1 (K̂1 a) !# I ( K̂ a) µ + iΩ λ 1 2 2 2 ˆ j , Ω2 ) K̂2 + sin2 (β)I(Ω , 1 + iΩ2 λ1 I0 (K̂2 a) 8J\e : . $ ˆ j , Ω) = I(Ω =. Z. π Ωj. cos(Ωt)exp(iΩt)dt. 0. πΩ 2 πΩ Ωj cos( πΩ Ωj ) sin( Ωj ) + πΩ − iΩj cos ( Ωj ) + iΩj. ΩΩj. ..
(7) . /. aý 'wú°ÿUþ, 9ú '- ý Uù- ,ø^ü þeû øoþ .ý^üÍÿ.ø* ú ' ,\ú -. !. ! °¡ 'N 6 ¡ $ ! # 6 3e$ !$r#E! °¡ '/° &6 6N&j$ v(r, t), w(r, t) D 1 ° % #> 4PJ ! 76/J $ 7 !$ I °I !$ ! NMQ¡ $e$ 4 ! °¡ % N °A0L !$ &6rÕ$e$ &6j ! à Ω 1 , Ω 2 '' # < ËQ # ' !$ ' % ¡p6wj ! 7 'I # < D 'N !$ I B !$# &6 Q I Ω1 = 3.6, Ω2 = 1 II Ω1 = 0.5, Ω2 = 6 e $ . D ! 7 2.0 λ1 = 0.3, λ2 = 0.06, µ = 0.05, ν = 0.1 I ! = !$#o! 76 @° 6 ! ° $ 3.6 Ω = Ω2 = 3.6 %# QNQ¡QcW]fYHGmWB 2 i'N 4Q¡ $ 1 j$ D ! 74 λ , λ2 , µ, ν % k ¡ I 2° >W]fYrQ P @ % # ! 1° °4 P D 1 m=°¡E©Q h 4V % e7W YH 1. 1. 0.9. 0.9 Oldroyd. 0.8. 0.8. 0.7 0.6. 0.6 Newtonian. 0.5. r/a. r/a. UCM Oldroyd. 0.7. 0.4. 0.5 0.4 Newtonian. UCM. 0.3. 0.3. 0.2. 0.2. 0.1 0 −0.8. 0.1 −0.6. −0.4. −0.2 0 v(r,t)/q0cos(β). 0.2. 0.4. 0.6. θ !" $ #%& &. . mj 'I # 0, 1, 2, 3. = ^0' . 2mW" . t=. π. 0 0.2. 0.25. 0.3. 0.35. 0.4 0.45 w(r,T)/q0sin(β). 0.5. 0.55. 0.6. ' )( * +z !&" $#%& & v(r,t) w(r,t) q0 cos(β) , q0 sin(β). $ . ! 7 % ! . t=. π (Ω1 +Ω2 ) 2. 0.65. . nπ , n= , 2Ω i. Ω1 +Ω2 2. X 4 3 °¡ N 3! ° ! 'H° E Q t _ ! $ ! s"U Q © * ° Q BJ c &6 ( π 6 TW]f[Y+6 t= Ω \[uL j'N !$ I j j j N e % jV Ω1 , Ω 2 ! Q 6 i° # $ ! ½ % ¡ Ω >Ω Ω <Ω D1¡E (N2 W 1$H 2!$ ° ! h ! !$ v(r, t), w(r, t) mW"7 4'\4 v(r,T ) w(r,T ) L Ëk # 6sQU _ Q * ° Ql ) Í 'I # q 0 cos(β) . Xj,Qq 0 sin(β) . ]\3 'Y !$ e$ # J 9 % &e0° j Ω ∈ I i . ]aËQ . e $' !$0 4e$ !$+# ' ( <L v(r,t) w(r,t) π , T = Ω1 +Ω 2 q cos(β) q0 sin(β) 'Y !$ e$ 2# 60L ak #^ ° J q° E 0 Ωi ∈ II ¡ &6 3π omW D 1 ° % #` 4hL ! ° C t = 0, π i , Ωπi , 2Ω q¡ °2Ω 6.' $ àNi !$#^! P °NQ ° NW¡E °&6 6 'I # p . ]f Q . ]gA 'Y !$ e$ # Ωi ∈ I.
(8) . . . 1. 1 t=3*π/(2*Ω1). 0.9. t=π/(2*Ω1). 0.8 t=0. t=π/Ω1. t=π/(2*Ω ). 2. 2. t=0. t=π/Ω2. 0.8. 0.7. 0.7 0.6 r/a. 0.6 r/a. t=3*π/(2*Ω ). 0.9. 0.5. 0.5. 0.4. 0.4. 0.3. 0.3. 0.2. 0.2. 0.1. 0.1. 0 −1.5. −1. −0.5. 0 v(r,t)/q cos(β). 0.5. 1. 0 −1.5. 1.5. 0. & $θ#%& . . −1. −0.5. 0 w(r,t)/q sin(β). 0.5. 0. 1. ' z $# . 0.6. 1.5. . 2 UCM. 0.5. 1.8 UCM 1.6. 0.4 Newtonian. 1.4 0. W/q2. 0. W/q2. 0.3 1.2. Newtonian. Oldroyd. 0.2. 1 0.1. 0.8. 0. 0.6. −0.1. 0.4. 0. 0.5. 1. β. 1.5. $ #%& & + Ω ∈I . i. !. Oldroyd. 0. 0.5. β. 1. 1.5. & ' $# &) & Ωi ∈ I. > % $ e . X N $ ° ) ! °¡ 'H° e$ θ̂ 6 "7 Õ J Qs"U _ ËQ Õ L àk #^ Õ t=T °l ! !$ DL ) ! °¡ 'H° 4 . ]\[mW¡ ẑ ! Q % $ e q N =Õ $e$ 4c ! !$ ! °¡ 'H° θ̂ 'N !$ I ¡ . !$+# ¡ I $ 5) 4 . -, . ]aQ . e j4V % a¡E ° N k $ 2 $e$ 4zQ J ¡ , % $ e 6 P ' !$ eP' ( # ¡¡ ! % [M\ @ $ `e$ !$r# -, 'N !$ I &64 # ¡¡ # C % $ e C ¡ t = 0, Ωπ Q i π 3π t = 2Ω , i 2Ωi -, . ]f©j34V N < WD 1 °4'Y$ MN ;) !$#! 4 °N¡E °c698 j= 8J\e :: 6 27 Õ L 4sQU _ Q Õ J ½k # &6\Q 1 "4Q¡ 2 % $ e . ]g J u 1 °B'Y$ jN ) !$#^! Qj ° N a 8J\e :: ¡E °c6G8 j=2.
(9) . . . ' w' ú 'ø^þ'-. WXNY,UW0 $I6 _ MQ V 460RÍ à6&B 4S° Q 7 !$ I S °) # °N ! &6 X [ )bX7a6 N9Q h °Nz¡E °c6 N ^ X 7g [ W]\[Yà©[Z'N?6(q6 V ° N¶Q °Ni7 !$ I °P¶> °)+*© Qp 6 $! _ ! NQ ! 6 \ XN)r\ 7f6=X 7a[ W]a[Yà©Q¡ 1 b °c6 àNQ o_ [Z ¡ &6 ( à6 .G Õ Q ° NMQ °N<7 !$ I °OW $! # $ &6 N 0 k'' S_ ¡ ! qQ R& # F8 _ R : 6 ö f [)rg 7 a 6cX [ W eYà©Q¡ 1 b °c66d9 4Qc6U© eQ _ [Z ¡ &6 ( q6 V ° NYQ °NB7 !$ I ° àI S'/I C 4 6 -H $ N °NDZ Nk d& V$ ^!$ ! &6 # # \X&f[)r\7\X76cX X W]f[Yà©QNQ¡Qc66H-H $ N.Õ A ° NjQ > °N7 !$ ) I °CD !$# $ &6 B Q N i"-HJ ¡ ° mo ! \ # 6 ö 7g [) 7g7g6/\ e W]g[Yà©QNQ¡Qc6l6*k°)+*© ° Qo E 7 $! I C 6 B Q Nj-HJ ¡ °omo ! \ # 6 # \ e a[)r\e6/\ 7f[ W0/FYßk #^ 6ß6R< ! k # . ! \e f6\/[)r\ /^6cX7f [ . W[Y& ! $ &6ß(6 ½ -H !$ ° e$ +# R &6 s( 6cX7g/. .G . # NQ¡ ! &6UWQ¡ % els ). ! l' 'Y$ o W[Y@mj°¡E&6¶ àQ 4V % e76àN6QdI ! Q h °CQRG # ¡ # _ N ! # I wQ ! F6Dm= 4QÕ .l 4 #h ! X&\7\7f[)bX&\7a7\6jX7f7a[. ) 2 & + & 9. "!"!#%$'&)(* ,+.-0/,1'2435-' 6/87 *<. =. >:. BA. '?. *<. ,9. D=. @. ;:. =. @. C:. <.
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