Unsteady laminar flow of visco-elastic fluid of second order type between two parallel plates

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1 Indian Jornal of Engineering & Maerials Sciences Vol., Febrary 5, pp Unseady laminar flow of visco-elasic flid of second order ype beween wo parallel plaes Ch V Ramana Mrhy & S B Klkarni Deparmen of Applied Sciences, Finolex Academy of Managemen & Technology, Ranagiri 5 639, India Received 6 April ; acceped 9 Sepember In his paper a general solion of he nseady sae laminar flow of a visco-elasic flid of second order ype beween wo parallel plaes has been obained, nder sal assmpion of no-slip condiion a he bondary. The solion is obained no by sing any of he reglar mehods, b by employing he conceps of Dhamel's heorem. The velociy profiles are fond o be more seep frher, as ime passes hrogh, he profiles are significanly disribed. Frher, as ime elapses, he skin fricion on he plaes also increases. The applicaions are many are of ineres, as i approximaes o he flows ha are commonly enconered in echnological engineering pracices. IPC Code: F5D A simple maerial can be defined as a sbsance for which sress can be deermined wih he enire knowledge of he hisory of he srain. Frher, i has he propery ha all local saes, wih he same mass densiy, are inrinsically eqal in response, wih all observable differences in response being de o definie differences in he hisory. For any given hisory g(s, a rearded hisory g ( s can be defined as: g ( s g( s; < s<, < ( being ermed as a reardaion facor. Assming ha he sress is more sensiive o recen deformaion ha o he deformaions a disan pas, i has been proved ha he heory of simple flids yields he heory of perfec flids as ha of Newonian flid as a correcion (p o he order of o heory of he perfec flids. Neglecing all he erms of he order of higher han wo in, we have incompressible second order flids, governed by he consiive relaion: ( ( ( 3 S PI φe φ E φ E ( E U U... (3 i, i,, i E A A U U ( i, i,, i m, i m, In he above eqaions, S is he sress-ensor, Ui, Ai are he componens of velociy acceleraion in he direcion of he i h coordinae Xi, P is indeerminae hydrosaic pressre he coefficiens φ, φ, φ 3, are maerial consans. The consiive relaion for general Rivlin- Ericksen flid also redces o Eq. ( when he sqares higher orders of E are negleced, he coefficiens being consans while φ, naming φ 3 as he coefficien of cross-viscosiy. Wih reference o he Rivlin-Ericksen flids, φ may be called as he coefficien of elasico-viscosiy. I has been repored ha a solion of poly-iso-bylene in ceane behaves as a second order flid ha Markoviz deermined he consans φ, φ, φ 3. The problem of viscos laminar flow beween wo parallel plaes is of ineres as i approximaes o he flows ha are commonly enconered in echnological engineering pracices. Moreover, he eqaions of moion become remarkably simplified yielding an exac solion. The cases of nseady moion of a Newonian flid conained beween concenric roaing cylinders had been examined by several research workers, while Paabhi Ramacharyl sdied he case of laminar flow of a viscos liqid beween wo parallel plaes.

2 5 INDIAN J. ENG. MATER. SCI., FEBRUARY 5 Sbseqenly, Ramana Mrhy sdied he flow of visco-elasic flid of second order ype beween wo poros parallel plaes. I has been fond ha he presence of elasico-viscos parameer increases he order of he governing eqaion of moion from wo o hree while only wo naral permeable bondary condiions are available for solion. A hird bondary condiion was proposed by aking ino effec he nare of he bonding srfaces for a meaningfl solion. Three-dimensional free convecive MHD flow hea ransfer of viscos incompressible flid hrogh a poros medim which is bonded by a verical infinie poros plae had been sdied by Ja & Jhankal 3 in which he effec of magneic field permeabiliy parameers were fond o have significan effecs on he flow hea ransfer. Saring from he known vales of flow fncions for small vales of he Reynolds nmber, he solion is exended for larger Reynolds nmber while sdying he effecs of radial oflow inflow of a second order flid nder an enclosed roaing disc by Sharma Biradar. In all he above invesigaions hogh he bonding srfaces are of differen geomeries, he effec of elasico-viscos nare of he flid nder consideraion on he flow variables has no been aken ino accon. Frher, he nare of he field variables are examined obained by employing he classical mehods someimes by finie difference approximaions. The aim of he presen invesigaion is o examine he general problem of nseady flow beween wo infiniely long parallel plaes nder he sal assmpion of no-slip on he solid bondary. This problem has been sdied by several invesigaors for over a decade. However, in he presen case, insead of obaining he solion in radiional way, a novel concep of Dhamel's heorem was applied o analyse he flow variables. The general nseady sae problem is divided ino hree classes of physical problems, each corresponding o a se of condiions a he bondaries. These relae o he specificaion of (i flid velociy on boh bondaries, (ii flid velociy on one bondary shear sress on he second bondary, (iii shear sress on boh bondaries as fncions of ime, in general, i is noiced ha he firs wo classes of problems are well posed, while hird class amons o he specificaion of velociy gradien a boh he bondaries. Sch simplificaion is incompaible wih he momenm conservaion consies a problem which is no well se-p no solion is fond for his case. These hree classes of bondary condiions, when expressed mahemaically, are fond o be special cases of one more general se of bondary condiions. Each of he original ses corresponds o a pariclar choice for he consans appearing in he more general se. Formlaion of he Problem A se of recanglar caresian co-ordinaes (X, Y, Z wih he axis of he Z along one of he wo plaes separaed by a disance L he axis of he Y perpendiclar o he plane of he plaes, he plaes being Y Y L in he dimensional form. The wo plaes are assmed o be infinie so ha all he physical qaniies are independen of Z. Now he velociy componens characerizing he recilinear flow beween he plaes are given by {, U(Y, T} in he Y Z direcions. The eqaion of moion is given by: U P φ U U ρ φ ( T X Y T Y Inrodcing he non-dimensional parameers as: φ i Ui X T ρ φ i Lxi Yi Lyi φ ai Ai S 3 ρ L ( φ ( ei, Ei, ρ L L i, i, L P φ ρ φ υ φ s 3 c E φ e ( ( i, i, φ p... (5 The eqaion of moion in he non-dimensional form is: p x wih he iniial condiion:... (6 y (, f( y... (7 ogeher wih he bondary condiions:

3 MURTHY & KULKARNI: UNSTEADY LAMINAR FLOW OF VISCO-ELASTIC FLUID OF SECOND ORDER TYPE 53 (, (, ψ(, (, 3(, ψ(, (8 (9,, 3, are consans ψ, ψ are prescribed fncions of he ime parameer while f ( y is a prescribed fncion of y. So as o mainain he generaliy of he problem, i is assmed ha he p pressre gradien ( x φ. Solion of he Problem The bondary vale problem formlaed above involves a non-homogenos differenial eqaion (6 wih wo varying bondary condiions a non-zero iniial condiion. The lineariy of he governing Eq. (6, he bondary he iniial condiions allow s a linear ransformaion of ype: ( y, ( y, ( y, ( y, ( y,... ( 3 F(, 3F(, (3 Leing F (, ( (, y y F y ( y, F( y, are seady nseady sae solions. we ge, F, sbec o: a... ( ( ( ( 3 ( For which seady sae solion is (5 (6 ( y Ay B... (7 he fncions ( y, ( associaed wih respecive line varying bondary condiions on y y, he fncion 3 ( y, wih he viscos decay of he iniial velociy disribion wih he exclsion of he pressre erm in he Eq. (5 ( y, corresponding o he sdden applicaion sbseqen mainenance of he pressre gradien φ (, i.e., he fncions (, 3 (, y are o be deermined. Deerminaion of ( y, ( y, The axiliary problem which deermines F (, y is : [ ( 3 ] A ( 3 3 ( 3 B ( 3 3 The nseady sae moion is governed by: F F (8 (9 ( F F F ( The iniial condiion: F( y, ogeher wih he bondary condiions: F (, F(, ( ogeher wih he bondary condiions: F (, F (, ( F (, 3F (, ( F ( y F ( y, (3 s

4 5 INDIAN J. ENG. MATER. SCI., FEBRUARY 5 Assming solion λ F e [ Acosy Bsin y] as he rial λ F ( y, C p( ye... ( he eigen fncions are given by p ( y λ cosλ y sinλ y... (5 he eigen vales λ are he roos of ( anλ λ ( λ (6 I is noiced ha he seqence [ p ( y ] is orhogonal in [,] wih in he norm p( y [( λ ( ( ] 3 3 ( 3 λ... (7 The consans C are deermined from he iniial condiion of he eqaion F ( y, C p( y... (8 which yields C... (9 λ p ( y Hence F (, ( (, y y F y... (3 F ( y, A y B C Employing Dhamel's heorem p ( y e λ (3 F( y, τ (, ψ( τ dτ... (3 y λ τλ (, ( ψ( τ y C p y e e dτ (33 Proceeding for ( y, in he same lines as ha for ( y, λ τλ (, ( ( y D p y e ψ τ e dτ D (3... (35 p 3 ( y Deerminaion of ( y, 3 I is seen ha (, 3 y can be wrien as 3 λ ( y, E p ( y e... (36 he consans iniial condiion. E f ( y p ( y dy p ( y Deerminaion of ( y, F p F F x E can be obained by sing... (37 (38 leing he solion as: F ( y, ( y F( y, ( y, F( y, are seady nseady sae solions. The bondary iniial condiions will now be ( ( (39

5 MURTHY & KULKARNI: UNSTEADY LAMINAR FLOW OF VISCO-ELASTIC FLUID OF SECOND ORDER TYPE 55 ( 3 ( ( The nseady sae solion is given by: λ F ( y, Fp( ye...(5 F ( y F ( y, ( s for which he seady sae solion is he consans iniial condiion. F can be obained by sing y ( y Ay B 3 3 A ( (3 F Hence, f ( yf ( y, p( ydy p ( y... (6 λ y (, ( (7 y Ay B F p y e (3 3 B ( A, B are given he relaions (3 ( respecively Fig. Velociy verss plaes separaion for.3

6 56 INDIAN J. ENG. MATER. SCI., FEBRUARY 5 Employing Dhamel's heorem, hen ( y, ( C D p ( y e τλ φτ ( e dτ λ... (8 Therefore, he complee solion of he velociy filed saisfying he condiions (7-9 are given by: λ y (, p ( ye [ E { λ( Cψ( τ Dψ( τ τλ ( C D φ( τ}] e dτ (9 Skin-fricion on lower pper plaes are given by respecively. y y y y Resls Discssion The velociy profiles for varios vales of are presened graphically in Figs. I is observed ha as he elasico-viscosiy parameer increases, he velociy profiles are fond o be more seep frher, as increases, he profiles are significanly disribed. The skin fricion on lower pper plaes for differen vales of elasico-viscosiy parameers are presened graphically in Fig. 3. I is observed ha as increases, he skin fricion on he plaes also increases. Frher, as he elasico-viscosiy parameer, he resls coincide wih ha of a Newonian flid. Conclsions From his sdy, i is fond ha as he elasico viscosiy of he flid increases while reaining he maerial consans he velociy profiles are more parabolic in nare. De o he sdden applicaion of pressre gradien, he inra moleclar forces ends o pll he neighboring clser of flid elemen as a resl of which he flid velociy a he boom plae is high gradally redces. A a shorer draion Fig. Velociy verss plaes separaion for.

7 MURTHY & KULKARNI: UNSTEADY LAMINAR FLOW OF VISCO-ELASTIC FLUID OF SECOND ORDER TYPE 57 Fig. 3 Skin fricion verss elasico-viscosiy parameer (i.e as redces, he velociy profiles are fond o be more parabolic. This is de o he fac ha immediaely afer he applicaion of he pressre gradien, he velociy of he flid near he bonding srfaces is high as ime passes, he inra moleclar forces redces which is in accordance wih he physical observable phenomena. I is also fond ha for a consan-elasico viscosiy of he flid, he profiles are more seep wih increase of ime. Acknowledgemens The ahors express heir sincere hanks o Dr R A Mashelkar, Direcor General of CSIR for his criical commens sggesions o Dr K L Asanare, Direcor, Finolex Academy of Managemen Technology, Ranagiri, for providing necessary compaional faciliies consan encoragemen. References Paabhi Ramacharyl & Narasimhacharyl V, J Mah Phys Sci, 3 ( Ramana Mrhy Ch V, Some Second Order Flids, Ph.D Thesis, Naional Insie of Technology, Warangal, Ja R N & Jhankal An K, Indian J Eng Maer Sci, (3 38- Sharma H G & Biradar K S, Indian J Eng Maer Sci, ( 9-5