Choudhary, R  C

Rajvanshi, S C

English

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Laminar source flow of second order fluid between two parallel coaxial stationary infinite porous disksiS. C Rajvanshi ajsd R. C Ohaudhaby Department of Mathematic,s, M. E Engineering College, Jaipur Rajasthan{Received 15 Mag 1972, revised 29 Alag 1972)lijinuiiaj' Rouicjo flow of .second oidei fluid between two atatioiiary inliiiite jjorona disks Jias been studied The sohiticjii lias been determined in the form of a double sia-ies exjDansion. I'lie ellect ol' nou-N(^wtoniaii parameters on velocity prolile, pressure and .shear stress has been dis­cussed.Indian J. Phys. 47, 24-30 (1973)1. TntroduottonLivesey (1902) investigated the incompressible radial flow between two parallel stationary disks using tlu' integral approach and the asumjition of a parabolic velocity profile Savage (1904) olitainc'd the solution by expanding velocity components and pre.ssure in terms of the downstream coordinate. But he has omitted the no-slip condition on the disk r.. -- 0. Similar jiroblems have also been studied by Peube (1903), Chen & Peube (1904), Geiger ei al (1904).t^ low between porous boundaiies is of practical as well as theoretical interest. 4’he problems of gaseous diffusion, boundary cooling and lubrication of porous boaimgs are a few examples. Elkouh (1909) obtained a solution for source flow between parallel stationary porous disks Ho has given the effect of cither equal suction or equal injection The solution is in the form of a perturbation from the creeping flow solution, valid for small values of the v^ all Reynolds number {Hw), and of the reduced source Reynolds number {Re*). {Rw) is based on equal suction or injection velocities at the disks. In a subsequeni. paper Elkouh (1971) has investigated laminar source flow between parallel porous disks with equal suction and injection. The solution is in the form of an infinite series expansion -about the solution at infinite radius and is valid for all suction and injection rates. Khan (1968) has investigated the laminar source flow between two parallel coaxial porous disks rotating at tlui same speed taking into account suction at one disk and an equal injection at the other.Ill the present pajicr, the source flow of the second order fluid between two parallel coaxial inlinitc porous disks with eijual permeability has been investigated.24Laminar source flow etc. 25The constitutive equations of an incompressible second order fluid as suggested by Riviliii & Erickson (1955) are ;andAij =- vt,j-hvj,i,(1)(2)(3)whore Tij is the stress sensor, gtj is the metric tensor, Vi is the velocity vector, ai Ls the acceleration v ector, is the pressure, o^, 0a are the fluid parameters and comma in tlie suflix denotes eovariant differentiation.The series expansion method used by Elkouh (1909) has been adopted to obtain the solution. The effect of non-Newtonian parameters on radial velocity jiertiirbatioiis and pressure drop has been exhibited graphically. It is noted that the visco-elasticity and cross viscosity increase the magnitude ol the radial velo­city perturbatioiio. The lesults obtained in this paper are valid at a distance /• >  -\/{Re) from the source, whore (Be) denotes source Reynolds number.2. E quations of MotionThe momentum equation for the incompressible flow ari'pVjVi,} Ttj,j ... (4)and the equation of continuity is'y«,i =  0. ... (5)We shall use cylindrical polar coordinates (f, %) and consider blie axially syni-metne stoad^  ^ flow bolN\een two infimte stationary poious disks z — —a and 3 — -\-a. The line aouicc is situat ed on the a;-axis. Let the volumetric floAv rate of the source and magnitude of the constant injection velocity at the disks be Q and W, respectively The boundary conditions are .h((r, i a )  0,di«) =  Tand J 2TTf^^—27Tr‘^ W ~  Q,which is the overall conservation of mass equationWe introduce the following non-dimensional quantities :(6)r =  fja, z =  zfa^  u =  puaj^^, w — pwaf(f>^\j(7)Equations (6) aiid (7) ^ve the houndary eondiiionM in the following formu{r, -i: 1) =  0,w[r, ± 1 )/  uL\z-{Rw)r ;26 S. C. Rajvanshi and E. C. Ohaudhary2(726)(72if;)(wall Reynolds number) =  -(/2e)(soiuv-o flow Reynolds number) ~  *where {Itw) is taken positive for nijeobiou and negative for suction. The radial and axial velocity components are related to a stream function »// in the following form(«)1U  W  —r dz1 ^r dr (9)For small values ol {Re)jr^, let us expand ijj and p as follows ://• =  J rHRto)f_,{z) f  («e)[/„(s)-|- );■ (-)+  ( &  ) / , ( * )+ . . .  ] .  (10)andl> =  i  l■\Rw)h_^^z)+h{z)+{Jie)[K^z)\nr^ ) V ) + - ]  (H)I'Vom, (10) and (11) Ave infer that the solution so obtained will be valid for r >  y/iRe) We transform eipiations (1) to (5) in cylindrical polar coordinate flyst-om, and use (7), (9), (10), (11) and equate the coefficients of like poAA^ ers of r. This gives us an infinite set of simultaneous ordinary differentia] equa­tions. For' the sake of brevity we rccoi’d only two systems below. However solution in lolloAviug section has been given upto system IV.System, I  ,=  4- (12a)7t'_, =  2 (fi,«)(2 /!:+8 )r_ir '^ „ (12b)iuidft '- - (B « , ) ( f  _ ,4 -(K «>)f-i/.i)4 -i:(K <a)[(il«> )(-/-.r '-x+8/'-,r -i + 8 r '- . / ' - i + 8 r “- i+ 2 / ‘ '' i / - , )4 -2 (B a )(r - ,r o + r _ ,r 'o )]+  .S '(K .«)[7 (B .«)f„,r.i+ (B e)(f'.,f„4 -/"-.r '„)J - (13)Laminar source fiow eic. 27(15)jSiysfem J1 :and ^  0. ... (14b)Tn the above equations the primes denote the differential coefficients "witb respect to The equation of continuity is identically satisfied bjr equation (9) Equations (8), (9) and (10) give the modified conditions as. r „ (d r l ) - 0 ,  - 1 ,0 ,1 ,2 , . . .  ,/ n ( ± l ) - 0 .  7 / . -1 ,2 , . . . ,/- i (d il)  ^  ± 1 ,and / o ( l ) - . / o ( - l )  =  2-We choose /o(-“ l) — —1, so that /„ (!)  — 1We note that the differential equations in above systems and the governing boundary eouditions are independent of (-Be), hence the solutions of the above sybtems will be independent of (Br). However, (Re) appears as a parameter in the solution of syatom TV.3. Solutions fob Small (Rw)The sets of differential equations in the above systems have been solved by expanding the different functions in powers of Eiv (for small values of the suction or injection Reynolds number) Thus(Rw)^u,.a-oin which are indepoudent of (Rw).The corresponding boundary conditions for/^,# are.r» )fl(il)  =  d, for 71 — —1, 0, 1,2, and all at fn,aii  1) == 0, 1,2, and all a,/ W ± l ) - ± 1 ,  7 ? - - 1 , 0 ,/n ,«(±l) =  0, 71 = —1,0 and a >  1.Solution of System I  :Integration of equation (12b) gives-  G_^+(2ir+.8f)(Bu;)(f _j®),(10)ill)(IS)wlK'vr () _^ IS a oonslanl liquations (12a) anrl (18) giver  -  < ? - , 4 r - , i+ 8 (« ,« ) r i .r . i i  -  (19)28 S. a  RajVanshi and B . C. ChaudbaryLot us further asBiinu'a_, =  2 {Jiwra ,.,. (20)Wo suhstituto ocimiti.ms (10) and (20) in (19) and equate tlui coefficients of like |jowors of (Rw). Wo get a set of ordinary linear differential equations. The solution of these differential equations subjected to the modified boundary condi­tions (17) is(21)3(2A' |-,S')(z« - 2 s=41). ... (22)/  -.j_;-gi^s(fi3^'‘’ -ir>40,^«-[-110H2;=«-15708s^-f2215r;-= f 3288)4  I ~  /Cs(4.'5.r»-(i8fc«+2022s''-2172s''‘ -|-789)4 22-3j) f 1 3 3 2 s '  |-228(l:‘'-n u i:2 _ 1 2 3 ) - ~  if'‘5(10s«-4!)2^-| ()8s=—29)'T4(H) '''’ ‘ ■('02“- 2 l 4  +  12«“- l ) - j ^ ’^jjA'&(l!3(te“-367« ‘ -| 684=='-267), ...andh . -  -3 + (A « .)  [_^ *+ ^ (2 A + ,S ')(1 0 .s* -l)]  I (fiw)“ [ - 1512695(23)■’'35d 195.':;3-54-)-( -^^X315;:;«-3150s^H 1756^2-136)18 9-  j ^-Z2(35(bj^~210?j2H-51)- ^  i82(175s«^105s2+3)-  KS (7002^-4202^2+57) ] + 0 (Ewf . . .  (24)It is observed that this solution is affected by non-Newtonian parameters lilkouh (1969) pointed out that second order perturbation solution of system I in ease of Newtonian fluid is reasonably accurate for |(Ww>)|i^  3 0 approxi­mately.Laminar source flow etc. 29i^ ohiHon of Sf/fttem II :Proceeding in a manner similar to tliat. adopted in obtaining the solution of system T, we get/o»o “1280 10'-(26)(26)/o.2 =" — ,^-^ -^jjj^ ,•-(812^ “ -lS475s«-|-7(K)92:«-108570^‘ +56704s2-5fi63)+ ^-T^^^^=(23S^“--: ‘^l84^»+(i83+5'-'7156z“+2571)++  &(245z*-2610z«4-6408z’ -59e6s2+1923)-j^j^7ir«zX(205z»-987z*+1359«2-677)- ^  -S%(56z«-189z^-|-2ia5®-79)-^|j^«'.9z(fl852“-2919z*+37832“-15 4 9 ), (27)andK  -  - 3 - | g ( i ; » ) | i - 7 ( / i : 4  .s’ ) J-(« i« )^ [  ,-^ ^ g  + ^ ( 3 0 9 i : . |  ii,s-d-1404A2-280>SH-1224A/S) j +0(iM®- (28)Solutio7i of Sysfevi TIT :A, „ =  -  |-12z“-5 )+ ^ z(^+ -S )(z> -2z“+ ) ), ... (20)■A.1 =  -f5:il„sZ(85zi»-1155z'+4960z“-8778z«+690U“-2003) lo4:nUUA>.{335s8-279br.«-| 76022^-81 SGs^ -f 3015)+ Sz X(1702«-1377s« + 33392'> - 3 2 2 7 2 2 + 1 0 4 5 ) -A%( 1152®-540z<^+74722-310)~  -^?j^>S'^2(1652«~5072i+0392‘i -237)-^-^^ AaS\2(3052’ -49772H 6399z“-2607),(30)h.-18900z*+9400z*+507)+-j*j^<S''(44l2"-189z‘ +9452»+59)7f:»(75600z‘ -4536n2!!-1877)-^ljS"(15120s*-9072z'+27) -  j^£:S(226800z*-130480?«-2619) ] +0(lfi<>)*. (31)SofvhoH of St/sinh f\ ' 'I'm  - -,„,L,r;(l(iSs'»-l!)2rK»-7524;«-1524f>2'' |-UTMfc^ -fiSOI) 4o I 2 (h)30 S. C. Rajvanshi and H . C. Chaudhary]28001Kz{ I ■ - 096.^ « I- 2706z4 -  3004^ 2 4-114)I 144()s4-1364;c2 | 400)-.^-^ 7i:22:(2r>54'-1197;:-*4-1021-<>S7)^ *823(1052"-507;=“4 03022-237)-^  A'&(302«-1202“H 16222-00),und(32)78 3■2695+175 ft'(31S~''-10'')(te'>+445s‘>- 13)I- .9(315s«-1050s'' I 495s2 | 42)-• -^^A:''(175s«-105 s2 ( 6) 3oO 1 /o.S»(175s''-105z"-l-2)-|| 0 ( 7 5 2 ' '- 15s"4 2)+0(R)(.) (33)OIkmi Clic-Pcn (I960) in Ins experimental investigation tor flow between non- ])orous disks, assumed (fie) >  1000 On the assmnptioii that {Re) is of the sameordei- as used by Che-Pen, the terms eontamin^ in (32) and (33) have been * ne^leetedOn integrating equation (13) we haveh -= -(ft.i')(f_ ,+ J /_ ,''(fi.«))+ Jr(JJw )[(fl« ')(2 /„,f'_ ,-/_ ,f_ ,+6 /'_ ,.r  .h3r‘ - i ) -hi(fie)/"-i/"oJ-h S(Rw)\ \( fiw )/'“_ ,+ , |-|-coiiBt ... (34)>vher(‘ the constant is determined fi om a knowm pressure at a point in the flenvAbove expressions are in agreement wdth Elkouh (1909) on taking K “  S — 0. Equations (21), (25), (29) and (32) represent the solution of laminar source flow between non-porous disks, while equations (22). (23), (26), (27) and (30) are due Lo interaction betv\'(*en the flow" of system T and the sonreo flow4. V elocity IDistiubutionThe component, of radial velocity, in terms of the average radial velocity 18 given byLaminar source flow etc. 31whoro < u >  =  ] w h  -^ l  r{Iiw)+^^^As we have taken {Re) to be very large, that is, is indepeiidont of (i^e), the radial velocity distribution is only a function of and {Re*) (~  {Re)lr^). Equation (35) will be valid at largo distance from the source. The series repre­sentation for u* can converge only for r j> \^{Re). For r <  ^/{Re), that is, in the neighbourhood of the source, the solution given by (35) will not be valid In the absence of porosity the flow in the neighbourhood ol source vill be entirely radial. The offoefc of non-Newtonian parameters has been shown graphically on variousFiq I AGAINbT Z -----------NON-NEWTONlAN.-Figure 1.F ig u r e  2 .32 S. C. Rajvanshi and tl. C. OhaudharyJ?’iguro :t.Fig. 5 AGAINST Z ----------- -NON-NEWTONIAN,-----------NEWTONIANFigure 6.Laminar source flow etc. 33perturbation terms. For numerical work {Rw) has been taken to be I Figure ] ■sbo^  ^H the eiffoct of visco-elasticity and cross-viscosity on and compared with IS'cwLtmiaii profile The magnitude of/'^,i in second order fluid is always greater than that in Newtonian viscous fluids This difference goes on decreasing as we move from the central region towards the upj>er disk, till it becomes zero, and then it again shows a variation and the iirofllcs coincide again at thi‘ upper disk d’h(i effects of non-Newtonian parameters on other' ladial I'clocity perturbations f'lH)' 2M) depicted in figures 2,3.4 and f> respectively.5- tSTai; AMLIN JiS'riic stream function i// is givrsi l>y equation (10) This equation is rrnvnttcn ill the Ibllowing loriii .. . (3h)where i/r -  iJfliRv) aiul R -  rjy/'{Re).The Htreainliiios have been drawn in figure (> for {Rw) ----- 0.5, K  41,L and S 0 3 from R -  1 to it — 0. it is noted that the injected fluid part ides vill lirsi move towards the mid-plaii(‘ and then will be moving parallel t*o the mid- plane.u S. 0. Rajvanshi and R. 0. Ohaudhary0. Pkessuue DxstkibutionKquatioiib (11), (24), (28), (31), (33) and (34) give the presaure diatribution p{r,z). The constant of integration has been determined by the assumption tbafc the pressure at a point {H, -f-1) in the- How is known The pressure drop in the j adial direction is dehnod an: 'P{t’,Z)^p{M, Z). (37)Fig 7 PRESSURE DROP IN RROIAL DIRECTION FOR Rw=-OSO,R,=IO*-----------NON-NEWTONIAN (K=-0 I, S=Z5),----------- NEWTONIANFigure 7.Here we note a significant difference between Newtonian and noii-Nowtonian case In Newtonian case the pressure drop in radial direction is independent of axial distance, that is, remain constant on all planes perpendicular to ,':-axiH, as shown by Elkoiih (19(19). In the present case it is dependent on axial distance and therefore will Ihj different on different planes iierpendicular to s-axis The difference, ol pressure drop in Newtonian and non-Newtonian case has been shown ligure 7 For numerical w^ ork we have taken (Me) = 10^ , It 200 and [Ew)— -0.50, K — —0 1, S 0.25 The magnitude of pressure drop in the prc'sent case i.s im)]“o than that in ih(‘ NowTonian fluids.7. Skin FrictionThe iimi-dimenHional radial slioav Bt^ vess is bym (dll , dw\ . [ d ( Ou , dii\ , D / dw , dw\’ -  ^  + . i r ) - '-M  0. ( “ flr- I + n i  (“ a7-< 5 . )dn Ou- dw ()u'i I dll / dw dun dwjdv. , dw\-i+ ^ d 7  dz'^^Tr +  L -> l f e + a f )  J-wborrFrom equal iojiB (0) and (38) Tfg at upper disk is given byLaminar source flow etc^ 3.’5(3S)(38)44777~30A''-h^5A'-342/r^-l-27.^2-117A^^^|'j-{- ™ 3 - ( / e 7 e ) | - ^ ^ ( / i + A ') j~l4ir I 'Isi ' 8'*'S’--12*»«A'.Sf I +<-^  I Ip4 - ( « « ’) ( - _^,  -306A'-70.S(-2412/P-4;)2iS'2-2844A'.S'j |+  y i  (^^^2--)”|~+294A:-|-r)4,S+I728A:i>+288,S'2+2016K-A' j j  -  -/<:(««.)[ J4«»>){-34 ! '7 (^2A' ( ,S)J|+ -3+ (A .r)•I-’ I  (A' f  .S')| +  - ^ 7 7 * -5fi4A'-791 A’-|-3000K:“~13!)r>A’}^}1Tlie skin friction t at the upper disk is defined as :T — —{TrT)z=i.. (39)... (40)The expression for skin friction agrees with Elkouh (1969) if K — S ^  0. The following table shows the effect of non-Newtonian parameters on t * {  =  r/r) for different values of r*{— r/y'(JSc)). For numerical work we have taken {Bw) ^  0.25 and JT -  =  0, ^  -0.01, S -  0.025,aci iS. (’. Rajvanshi and H. CJ. CliaudliaryT ol)l(i I{ K  - S  (1) -0.01. S1 r, 1.5822 1 57892 0 1.0090 1 00712 n (1 8210 0 8200:i 0 0 OS 10 0 0841:i .5 <t 0013 0 00091 (1 0 .5109 0 51570 025)IfoK' WO find Hull lih(' skin friction iu thc casc of noii-NeMdonian is less thcji iJml of Newtonian case Ac KN DGEMEN'VOne of Iho aiiiliors (ECO) wishes to thank the Ministry oC Education and Social AVclfariy (ilovemmont of India, for awarding* a Scholarship uliich cnahlcd him to carry out this u'ork. Authors are grateful to D r. P . D  Vernia for sug­gestions. iriianks are also due to the referee for suggestions to improve the original draft of tiic paperR e f e r e n c e sdim Che-Pi'll 190(i ,/ <lv M P c . 5, 215.Chi'ii. rlio-PiMi (S. Pmhi\ ,I n. 1964 C om jil. J in itl A roH  Svi (Piiiiw) 258, r);{.5;< l-]lki.iili A l'\ 1909 A2J2>!. S r i  21, 285.Klkouh A. K 1971 A p 2>l. S m . l i e s  23. 431.G'oigor D , Fam 11. D & Stveot N. 1964 J .  M r c h , 31E. :15 |.Khan Molvl A A 19GS J .  th  M i ‘ C 7, 570.Livoscy, .1 L. 1902 .1 M c ch  S c i . 1. 8^1.Poubo .! h 1903 ,1 dr M h o . 2 , 377.Kiviliii K, S «Sr EijcUhch J S. 1955 J .  H a t. M e c h . A m i! 4, 323.Savage S. H. 1904 ./. A 2)p L  M e c h . 31E, 594

Laminar Source Flow of Second Order Fluid Between Two Parallel Coaxial Stationary Infinite Porous Disks

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Laminar Source Flow of Second Order Fluid Between Two Parallel Coaxial Stationary Infinite Porous Disks

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