Research Article Unsteady Helical Flows of a Size-Dependent Couple-Stress Fluid

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1 Hindawi Publishing Copoaion Advances in Mahemaical Physics Volume 7, Aicle ID 97438, pages hp://dx.doi.og/.55/7/97438 Publicaion Yea 7 Reseach Aicle Unseady Helical Flows of a Size-Dependen Couple-Sess Fluid Qamma Rubbab, Ia Abbas Miza, Iman Siddique, 3 and Saadia Ishad 4 Depamen of Compue Sciences, Ai Univesiy Mulan Campus, Mulan, Pakisan Abdus Salam School of Mahemaical Sciences, GC Univesiy, Lahoe, Pakisan 3 Depamen of Mahemaics, Univesiy of Managemen and Technology, Sialko Campus, Pakisan 4 Depamen of Managemen Sciences, Ai Univesiy Mulan Campus, Mulan, Pakisan Coespondence should be addessed o Qamma Rubbab; ubabqamma@gmail.com Received Decembe 6; Acceped 5 Januay 7; Published 8 Febuay 7 Academic Edio: John D. Clayon Copyigh 7 Qamma Rubbab e al. This is an open access aicle disibued unde he Ceaive Commons Aibuion License, which pemis unesiced use, disibuion, and epoducion in any medium, povided he oiginal wok is popely cied. The helical flows of couple-sess fluids in a saigh cicula cylinde ae sudied in he famewok of he newly developed, fully deeminae linea couple-sess heoy. The fluid flow is geneaed by he helical moion of he cylinde wih ime-dependen velociy. Also, he couple-sess veco is given on he cylindical suface and he nonslip condiion is consideed. Using he inegal ansfom mehod, analyical soluions o he axial velociy, azimuhal velociy, nonsymmeic foce-sess enso, and couple-sess veco ae obained. The obained soluions incopoae he chaaceisic maeial lengh scale, which is essenial o undesand he fluid behavio a micoscales. If chaaceisic lengh of he couple-sess fluid is zeo, he esuls o he classical fluid ae ecoveed. The influence of he scale paamee on he fluid velociy, axial flow ae, foce-sess enso, and couple-sess veco is analyzed by numeical calculus and gaphical illusaions. I is found ha he small values of he scale paamee have a significan influence on he flow paamees.. Inoducion The exisence of couple sesses in coninuum mechanics is a consequence of he discee chaace of he mae a he fines scale. Also, he noncenal popey of foces beween elemenay paicles of mae leads o he appeaing of couple sesses []. E. Cossea and F. Cossea [] wee hefisohaveincopoaedhecouplesessesbyconsideing he oiened maeial poin iads and independen micooaion in heoy of coninuum solids. Toupin [3], Mindlin and Tiesen [4], and Koie [5] have developed heoies wih couple sesses, in which he igid body moion of he infiniesimal elemen of mae a each poin of he coninuum is descibed by six degees of feedom. Based on hese heoies wih couple sesses, Sokes [6] has developed he heoy of couple-sess fluids in ode o sudy he size-dependen behavio of flows. Sokes heoy epesens he simples genealizaion of he classical heoy of fluids, which consides he pesence of couple sesses. All he above models have vaious inconsisencies. In he conex of Sokes heoy, he indeeminacy of he spheical pa of couple-sess enso is a majo shocoming. Anohe poblem appeas in he heoy of Mindlin and Tiesen; namely, in hei heoy, he consiuive elaion fo he focesess enso conains he body couples. The indeeminae spheical pa of couple-sess enso is ignoed in Sokes heoy, wihou jusificaions. Also, he pesence of body couples in he consiuive elaions was negleced by Sokes in his heoy. Hadjesfandiai e al. [, 7, 8] have solved hese issues by using agumens based upon he enegy equaion along wih kinemaical consideaions. They developed a fully consisen couple-sess heoy fo solids and fluids, which esablishes he couple-sess enso and mean cuvaue ae enso as skew-symmeic enegy conjugae measues. Thei esuls could be useful fo examining fluid flow poblems influenced by he mechanics a small scales. Significan applicaion aeas fo size-dependen fluid mechanics involve modeling of blood flows, lubicaion poblems, liquid cysals, and polymeic suspensions. This banch of fluid mechanics has aaced a gowing inees fom eseaches in he field.

2 Advances in Mahemaical Physics Bakhi and Aza [9] sudied he seady flow of a couplesess fluid hough consiced apeed aey unde influence of a ansvese magneic field, moving cahee, and slip velociy. Soluions o velociy and shea sess ae expessed wih Bessel s funcions. Palhad and Schulz [] used he couple-sess fluid model fo he sudy of he seady flow of blood hough senosed aey. Blood velociy, he esisance o flow, and shea sess disibuion have been obained. Vemaeal.[]havesudiedhebloodflowinasenosed ube consideing he blood as a couple-sess fluid. Effecs of slip velociy in senosed ube wee highlighed. Shenoy and Pai [] have made he saic analysis of a misaligned exenally adjusable fluid-film beaing including ubulence and couple-sess effecs in lubicans blended wih polyme addiives. Naduvinamani and Pail [3] obained a numeical soluion o finie modified Reynolds equaion fo couplesess squeeze film lubicaion of poous jounal beaings. The unseady hee-dimensional flow of couple-sess fluid ove a seched suface wih mass ansfe and chemical eacion was invesigaed by Haya e al. [4]. Devaka and Iyenga [5] invesigaed he genealized Sokes poblems fo incompessible couple-sess fluids. The effecs of kineic heliciy (velociy-voiciy coelaion) on ubulen momenum anspo wee invesigaed by Yokoi e al. [6, 7]. Ohe ineesing opics can be found in efeences [8 ]. We mus menion ha he models consideed in he aboveaiclesaebasedonheheoydevelopedbymindlin, Tiesen, and Sokes. In he pesen pape we conside he consisen heoy of couple-sess fluids elaboaed by Hadjesfandiai and his cowokes [, 8] and, we sudy he helical flows of couplesess fluids wihin a saigh cicula cylinde unde geneal bounday condiions. In he sudied poblem, he fluid moion is geneaed by he helical moion of he cylindical suface wih he ime-dependen axial and azimuhal velociies and by he ime-dependen couple-sess veco on he cylinde suface. The nonslip condiions ae, also, consideed. Using suiable nondimensional vaiables, we deemine analyical soluions o he axial and azimuhal velociies by means of he inegal ansfom mehod. The axial angula velociy and he flow ae ae also obained. Componens of he nonsymmeic foce-sess enso and he couple-sess veco ae deemined fom he consiuive elaions and velociy field. Obviously, if he chaaceisic lengh of couple-sess fluids is zeo, we ecove esuls fo he classical fluid. The obained analyical soluions ae used in ode o pefom numeical calculaions using he Mahcad sofwae fo paicula exenal loadings. The esuls ae gaphically pesened. I is found ha all flow paamees ae influenced by he fluid scale paamee. The significan influence is obained fo small values of he scale paamee.. Saemen of he Poblem We conside a homogeneous, incompessible, viscous couplesess fluid flowing inside a saigh cicula cylinde of adius R. The cylindical coodinae sysem (, θ, z) has he z-axis idenical wih he cylinde axis. The govening equaions of he couple-sess fluid ae as follows [, 8]: (i) Coninuiy equaion: V = V + V θ θ + V z =, () z whee V = V e + V θ e θ + V z e z is he fluid velociy. (ii) Equaion of linea momenum (he body foces ae negleced): ρ( V + V V) = p+μ V η V, () whee ρ is he fluid densiy, μ is he dynamic viscosiy, η is he viscosiy coefficien of couple-sess fluid, and p is he hemodynamic pessue. (iii) The consiuive equaions: The nonsymmeic foce-sess enso: whee T = pi + τ, τ =μd +ηω, (3) D = [ V + ( V)T ], (4) Ω = [ V ( V)T ] (5) epesen he sain ae enso and he angula velociy enso, especively. The pola couple-sess veco: M =ηδv. (6) In his pape, we conside ha he velociy field and pessue ae funcions of he fom V (, ) = V θ (, ) e θ + V z (, ) e z, p=p(, ). Equaion () is idenically saisfied and () (6) become p = ρ V θ, ρ V θ =μ(δ ) V θ η(δ )(Δ ) V θ, ρ V z =μδv z ηδ V z, τ =, τ θ =μ (V θ ) ηδ[ (V θ)], τ z =μ V z ηδ( V z ), τ θ =μ (V θ )+ηδ[ (V θ)], (7)

3 Advances in Mahemaical Physics 3 τ θθ =, τ θz =, τ z =μ V z +ηδ( V z ), τ zθ =, τ zz =, M =, M θ =η(δ ) V θ, M z =ηδv z. (8) V = RV z ], p = R p ρ], a= l R, τ = R ρ] τ, M = R ρ] M, f ( )= RV ] f ( R ] ), We conside he following iniial-bounday condiions: V θ (, ) =, V z (, ) =, V θ (, ) =, M θ (, ) =, V z =, = V θ (R, ) =V f (), M θ (R, ) =ηm f (), V z (R, ) =V g (), M z (R, ) =ηm g (). (9) () () () Funcions f (), f (), g (), and g () ae piecewise coninuous funcions on [, ]; fo evey >,hey have exponenial ode a infiniy and f () = f () = g () = g (). We define he chaaceisic maeial lengh l= η μ [m], (3) which is absen in classical fluid mechanics bu is fundamenal fo couple-sess fluids. Inoducing he nondimensional vaiables f ( )= m R 3 f ] ( R ] ), g ( )= RV ] g ( R ] ), g ( )= m R 3 g ] ( R ] ) (4) ino (8) () and dopping he sa noaion, we obain he following nondimensional poblem: p = u, (5) u =(Δ )u a(δ )(Δ )u, (6) V =ΔV aδ V, (7) τ =, τ θ = (u ) aδ[ τ z = V aδ( V ), τ θ = (u )+aδ[ τ θθ =, (u)], (u)], (8) (9) = R, = ] R, u = RV θ ], τ θz =, τ z = V +aδ( V ), τ zθ =, τ zz =, ()

4 4 Advances in Mahemaical Physics M =, M θ =a(δ )u, M z =aδv, u (, ) =, V (, ) =, u (, ) =, M θ (, ) =, V = =, u (, ) =f (), M θ (, ) =af (), V (, ) =g (), M z (, ) =ag (). () () (3) (4) (5) In he end of his secion, we give wo lemmas egading some popeies o he opeaos fom (6) and (7). These popeies will be used in ode o find soluions of he above poblem. Lemma. If α n, n =,,... ae he posiive oos of Bessel funcion J (x), V n () = V(, )J (α n )d, and ( V(, )/ ) = =,hen, (i) (ii) (ΔV (, )) J (α n ) d =α n J (α n ) V (, ) α n V n () ; (Δ V (, ))J (α n ) d = α 3 n J (α n ) V (, ) +α n J (α n )ΔV(, ) = +α 4 n V n (). (6) (7) Lemma. If β n, n =,,... ae he posiive oos of Bessel funcion J (x), u n () = u(, )J (β n )d, and(δ / )u(, ) = =,hen, (i) ((Δ )u(, ))J (α n ) d = β n J (α n )u(, ) β n u n () ; (8) (ii) ((Δ ) u (, ))J (β n ) d =β 3 n J (β n )u(, ) +β n J (β n )(Δ )u(, ) = +β 4 n u n (). (9) The demonsaion of elaions (6) (9) is made easy using inegaion by pas and popeies of Bessel funcions [, 3]. I is noed ha V n () is zeo-ode finie Hankel ansfom of funcion V(, ) and u n () is he finie Hankel ansfom of he fis ode of funcion u(, ),especively. 3. Soluion of he Poblem In ode o find he soluion of poblem (5) (5), we use he Laplace ansfom wih espec o he vaiable ime and finie Hankelansfomwihespecheadialcoodinae[4,5]. 3.. Velociy Field. Applying he Laplace and finie Hankel ansfoms o (6) and (7), using he iniial and bounday condiions () (5) and Lemmas and, we obain he following ansfomed equaions: u n = β nj (β n )(+aβ n ) s+βn ( + aβ n ) f + aβ n J (β n ) s+β n ( + aβ n )f, V n = α nj (α n )(+aα n ) s+αn ( + aα n ) g aα n J (α n ) s+α n ( + aα n )g, (3) whee u n = u n ()e s d, V n = V n ()e s d, f, f, g,andg ae he Laplace ansfoms of funcions u n (), V n (), f (), f (), g (), andg (), especively.equaions(3)canbewieninhesuiablefoms u n = J (β n ) β n f + J (β n ) β n sf s+βn ( + aβ n ) + J (β n ) J βn 3 f (β n ) βn 3 ( + aβ n )f aj (β n ) sf β n ( + aβn ) s+βn ( + aβ n ), V n = J (α n ) α n g J (α n ) α n (4 + α n )J (α n ) 4αn 3 g sg s+αn ( + aα n )

5 Advances in Mahemaical Physics α n ( + aα n ) 4α 3 n ( + aα n ) J (α n ) g aj + (α n ) sg α n ( + aαn ) s+αn ( + aα n ). (3) Now, using he inegals J (β n ) d = J (β n ), β n J (β n ) d = J (β n ) βn 3, J (α n ) d = J (α n ), α n J (α n ) d = 4+α n 4αn 3 J (α n ), (3) and applying he invese Hankel and Laplace ansfoms, we obain closed foms o azimuhal velociy and axial velociies as u (, ) =f () + 3 f 8 () f () J (β n ) n=βn 3 ( + aβ n )J (β n ) + J (β n ) n=β n J (β n ) f (τ) exp [ β n ( + aβ n ) ( τ)]dτ J a (β n ) n=β n (+aβn ) J (β n ) f (τ) exp [ β n ( + aβ n ) ( τ)]dτ, V (, ) =g () + g 4 () +g () 4+α n ( + aα n )J (α n ) J n= 4αn 3 ( + aα n )J (α n ) (α n ) n=α n J (α n ) g (τ) exp [ α n ( + aα n ) ( τ)]dτ J +a (α n ) n=α n ( + aαn )J (α n ) g (τ) exp [ α n ( + aα n ) ( τ)]dτ. Inheaboveelaionswehaveusedhenoaion df()/d. By inoducing noaions A n () = f () β 3 n ( + aβ n )J (β n ) + β n J (β n ) f (τ) exp [ β n ( + aβ n ) ( τ)]dτ (33) f() = a β n ( + aβn )J (β n ) f (τ) exp [ β n ( + aβ n ) ( τ)]dτ, B n () = g () (4 + α n ( + aα n )) 4αn 3 ( + aα n )J (α n ) (33) ae wien as α n J (α n ) g (τ) exp [ α n ( + aα n ) ( τ)]dτ a + α n ( + aαn )J (α n ) g (τ) exp [ α n ( + aα n ) ( τ)]dτ, u (, ) =f () V (, ) =g () + 4 f () + A n () J (β n ), n= g () + B n () J (α n ). n= (34) (35) I is obseved ha funcions A n () and B n () have he popey A n () = B n () =. The axial angula velociy (he spin veco) is given by ω= V = V e θ +( u + u ) = ( α n B n () J (α n ) g ()) n= + (f () + f 4 () + β n A n () J (β n )) e z, n= and he flow ae is Q () = πv (, ) d =πg () π 8 g B () +π n () J α (α n ). n n= e z e θ (36) (37) 3.. Foce-Sess Tenso and Couple-Sess Veco. Replacing (35) in (8) () and pefoming calculaions, we ge he following expessions fo he foce-sesses and couple-sess veco: τ θ =( 4 a)f () + β n A n () ( + aβ n )J (β n ) n= A n () J (β n ), n=

6 6 Advances in Mahemaical Physics τ θ =( 4 +a)f () + β n A n () ( aβ n )J (β n ) n= A n () J (β n ), n= τ z = a g () α n B n () [ + a (α n )] J (α n ), n= τ z = +a g () α n B n () [ a (α n )] J (α n ), n= τ =τ θθ =τ θz =τ zθ =τ zz =, M =, M θ =a[f () β n A n () J (β n )], n= M z =a[g () α n B n () J (α n )]. n= (38) I is easy o see ha, fo a=(he odinay Newonian fluid), he foce-sess enso becomes a symmeic enso and he couple-sess veco is zeo Paicula Case (Consan Velociy and Couple Sess on he Bounday). Le us conside he following bounday condiions: f () =p H (), f () =p H (), g () =q H (), g () =q H (), (39) whee p, p, q,andq ae consans and H() = (/)sign()( + sign()) is he Heaviside uni sep funcion. In his case, he deivaives of funcions given by (39) ae f () =p δ (), f () =p δ (), g () =q δ (), g () =q δ (), (4) δ() being he Diac disibuion. Funcions A n () and B n () ae wien in simple foms, as A n () = p H () β 3 n ( + aβ n )J (β n ) H () + β n J (β n ) [p ap +aβn ] exp [ β n ( + aβ n )], B n () = q H () (4 + α n ( + aα n )) 4α 3 n ( + aα n )J (α n ) H () α n J (α n ) [q aq +aαn ] exp [ α n ( + aα n )]. 4. Numeical Resuls and Discussion (4) Unseady helical flows in he consisen heoy of couplesess fluids wee consideed, unde geneal bounday condiions. By using suiable nondimensional vaiables, he govening flow equaions ae obained in he dimensionless fom. I is impoan o noe ha hese equaions conain as paamee he dimensionless scale flow paamee a, defined as he squae of he ae beween he chaaceisic maeial lengh and he adius of cicula cylinde. As a esul fom hesudiedpaiculapoblems,hescalepaameehasa significan influence on he fluid behavio. Obviously, if he scale paamee equals zeo, esuls coesponding o he Newonian fluid ae obained. Soluions fo fluid velociy, nonsymmeic foce-sess enso, and couple-sess veco wee obained using inegal ansfoms mehod (Laplace ansfom wih espec o he ime vaiable and finie Hankel ansfom wih espec o he adial coodinae). The axial angula velociy and he flow ae ae also deemined. The obained soluions conain in hei expessions he posiive oos of he Bessel funcions J (x) and J (x), denoed by α n and β n, oos which wee geneaed by means of Mahcad subouine oo(f(x), x, a, b). In he numeical simulaions, we have used n [5, ], valuesfowhich he numeical appoximaion accuacy is vey good. In ou sudy, he azimuhal velociy, axial velociy, and hecomponensofhecouple-sessvecoaegivenon he cylinde suface as abiay funcions of he ime ; heefoe, he obained soluions can geneae soluions o vaious poblems wih pacical applicaions. The numeical esuls fo Figues 3 and fo Table wee geneaed unde condiions f () = f () = g () = g () = H() = (/)sign()( + sign()). Figues 4 and 5 wee ploed unde condiions f () =g () =.H () =. sign () ( + sign ()), f () =, g () =. (4)

7 Advances in Mahemaical Physics 7 =. =..5 Velociy u(, ).5 Velociy (, ) a= a =. a =. a =.4 a =.6 =.5 =.5 (a).5 a= a =. a =. a =.4 a =.6 Velociy u(, ).6.4 Velociy (, ) a= a =. a =. a =.4 a =.6 (b) a= a =. a =. =.9 =.9..4 a =.4 a =.6 Velociy u(, ).6.4 Velociy (, ) a= a =. a =. a =.4 a =.6 a= a =. a =. a =.4 a =.6 (c) Figue : Pofiles of azimuhal velociy u(, ) and axial velociy V(, ) fo small ime and diffeen values of scale paamee a.

8 8 Advances in Mahemaical Physics Velociy u(, ) = = Velociy (, ) a= a =. a =. a =.4 a =.6 a= a =. a =. a =.4 a =.6 Figue : Pofiles of azimuhal velociy u(, ) and axial velociy V(, ) fo =and diffeen values of he scale paamee a Flow ae Q 3.5 Flow ae Q() a =. =. =. (a) =.5 =. a= a =. a =. (b) a =.4 a =.6 Figue 3: Vaiaion of flow ae wih he scale paamee a, especively, wih he ime. Shea sesses τ θ and τ θ 4 3 =. =. Shea sesses τ z and τ z 4 3 a 3 3 a τ θ =.5 =.5 =.85 τ θ =.5 =.5 =.85 τ z =.5 =.5 =.85 τ z =.5 =.5 =.85 Figue 4: The vaiaion of shea sesses wih he scale paamee a.

9 Advances in Mahemaical Physics 9 Couple-sess M θ =. 4 =. 3 Couple-sess M z. 3 a 3 a =.5 =.5 =.85 =.5 =.5 Figue 5: Vaiaion of he couple sess wih he scale paamee a. =.85 Table : The influence of ime values on fluid velociy componens. (a) a =. u(, ), = u(, ), = 5 u(, ), = (b) a =. V(, ), = V(, ), = 5 V(, ), = Figue shows pofiles of boh azimuhal and axial velociies, vesus adial coodinae fo diffeen values of he scale paamee a andfoheesmallvaluesofheime. The influence of he scale paamee a on oaional velociy is significan only fo vey small values of he ime.i is obseved fom Figue (a) ha azimuhal velociy inceases wih he scale paamee a. The oaional velociy of couplesess fluid is bigge han he velociy of classical fluid, excep he case of vey small values of ime.inhispaiculacase, hee ae values of he scale paamee, fo which he couplesess fluid flows moe slowly han he Newonian fluid in he cenal aea of he flow domain. The influence of he scale paamee on he axial velociy is moe significan han on he azimuhal velociy. I is seen fom Figue (b) ha he axial velociy inceases wih he scale paamee. As is appaen fom Figue and values of Table, he fluid azimual velociy is almos seady and he axial velociy has a small ime-vaiaion, fo >. These popeies ae due o he exponenial ems in (34) which end fas o zeo, because, α n ( + aα n ) ( a), β n ( + aβ n ) 4.68 ( a), n, α.45, n, β 3.83 (43) (Fo > and a he exponenial ems in (34) ae negligible). The influence of he scale paamee a on he flow ae Q() in axial diecion is pesened in Figue 3. Obviously, he significan vaiaion of he flow ae is fo small values of he ime. I is impoan o noe ha, fo lage values of he scale paamee a o fo lage values of he ime, he flow ae in he axial diecion becomes consan. The influence of he scale paamee a on he componens τ θ, τ θ, τ z, and τ z of he foce-sess enso and on he componens M θ and M z of he couple-sess veco is analyzed in Figues 4 and 5. The significan influence occus fo small values of he scale paamee. Fo hese values he shea sesses τ θ and τ θ and he couple-sess componen M θ have an exeme value and end o appoach zeo fo lage values of he scale paamee.

10 Advances in Mahemaical Physics Shea sesses τ z and τ z and he couple-sess componen M z ae monoone inceasing/deceasing fo small values of he paamee a andendoaconsanvalueifhe paamee values ae lage. Compeing Ineess I is heeby declaed ha none of he auhos have any compeing ineess in he manuscip. Refeences [] A.R.Hadjesfandiai,G.F.Dagush,andA.Hajesfandiai, Consisen skew-symmeic couple sess heoy fo size-dependen ceeping flow, Jounal of Non-Newonian Fluid Mechanics,vol. 96, pp , 3. [] E. Cossea and F. Cossea, Theoie des Cops Defomables, A. Hemann e fils, Pais, Fance, 99. [3] R. A. Toupin, Elasic maeials wih couple-sesses, Achive fo Raional Mechanics and Analysis,vol.,pp ,96. [4] R. D. Mindlin and H. F. Tiesen, Effecs of couple-sesses in linea elasiciy, Achive fo Raional Mechanics and Analysis, vol., pp , 96. [5] W. T. Koie, Couple-sesses in he heoy of elasiciy I and II, Poceedings of he Koninklijke Nedelandse Akademie van Weenschappen, Seies B, Physical Sciences, vol.67,pp.7 44, 967. [6] V. K. Sokes, Couple sesses in fluids, Physics of Fluids,vol.9, no. 9, pp , 966. [7] A. R. Hadjesfandiai and G. F. Dagush, Couple sess heoy fo solids, Inenaional Jounal of Solids and Sucues, vol. 48, no. 8, pp ,. [8] A. R. Hadjesfandiai, A. Hajesfandiai, and G. F. Dagush, Skew-symmeic couple-sess fluid mechanics, Aca Mechanica,vol.6,no.3,pp ,5. [9] H. Bakhi and L. Aza, Seady flow of couple-sess fluid in consiced apeed aey: effecs of ansvese magneic field, moving cahee, and slip velociy, Jounal of Applied Mahemaics,vol.6,AicleID989684,pages,6. [] R. N. Palhad and D. H. Schulz, Modeling of aeial senosis and is applicaions o blood diseases, Mahemaical Biosciences,vol.9,no.,pp.3,4. [] V. K. Vema, M. P. Singh, and V. K. Kaiya, Mahemaical modeling of blood flow hough senosed ube, Jounal of Mechanics in Medicine and Biology,vol.8,no.,pp.7 3,8. [] B. S. Shenoy and R. Pai, Effec of ubulence on he saic pefomance of a misaligned exenally adjusable fluid film beaing lubicaed wih couple sess fluids, Tibology Inenaional,vol.44,no.,pp ,. [3] N. B. Naduvinamani and S. B. Pail, Numeical soluion of finie modified Reynolds equaion fo couple sess squeeze film lubicaion of poous jounal beaings, Compues and Sucues,vol.87,no.-,pp.87 95,9. [4] T. Haya, M. Awais, A. Safda, and A. A. Hendi, Unseady hee dimensional flow of couple sess fluid ove a seching suface wih chemical eacion, Nonlinea Analysis: Modelling and Conol,vol.7,no.,pp.47 59,. [5] M.DevakaandT.K.V.Iyenga, GenealizedSokes poblems fo an incompessible couple sess fluid, Inenaional Jounal of Mahemaical, Compuaional, Physical, Elecical and Compue Engineeing,vol.8,no.,pp.3 6,4. [6] N. Yokoi and A. Yoshizawa, Saisical analysis of he effecs of heliciy in inhomogeneous ubulence, Physics of Fluids A,vol. 5, no., pp , 993. [7] N. Yokoi and A. Bandenbug, Lage-scale flow geneaion by inhomogeneous heliciy, Physical Review E Saisical, Nonlinea, and Sof Mae Physics, vol.93,no.3,aicleid 335, 6. [8] F.Awad,N.A.H.Haoun,P.Sibanda,andM.Khumalo, On couple sess effecs on unseady nanofluid flow ove seching sufaces wih vanishing nanopaicle flux a he wall, Jounal of Applied Fluid Mechanics,vol.9,no.4,pp ,6. [9] S. Ahmed, O. Anwa Bég, and S. K. Ghosh, A couple sess fluid modeling on fee convecion oscillaoy hydomagneic flow in an inclined oaing channel, Ain Shams Engineeing Jounal, vol. 5, no. 4, pp , 4. [] Y. Kwon, Incompessible limi fo he compessible flows of nemaic liquid cysals in he whole space, Advances in Mahemaical Physics, vol. 5, Aicle ID 47865, 7 pages, 5. [] B. Panicaud and E. Rouhaud, Deivaion of Cossea s medium equaions using diffeen muli-dimensional famewoks, Aca Mechanica,vol.7,no.,pp ,6. [] Y. A. Bychkov, Handbook of Special Funcions. Deivaives, Inegals, Seies and Ohe Fomulas, Chapman and Hall/CRC, Boca Raon, Fla, USA, 8. [3] W. Magnus, F. Obeheinge, and R. P. Soni, Fomulas and Theoems fo he Special Funcions of Mahemaical Physics, Spinge,NewYok,NY,USA,966. [4] L. Debnah and D. Bhaa, Inegal Tansfoms and Thei Applicaions, Chapman & Hall/CRC, Boca Raon, Fla, USA, nd ediion, 7. [5] J. Kang, Y. Liu, and T. Xia, Unseady flows of a genealized facional Buges fluid beween wo side walls pependicula o a plae, Advances in Mahemaical Physics, vol.5,aicle ID 569, 9 pages, 5.

Lecture 17: Kinetics of Phase Growth in a Two-component System:

Lecture 17: Kinetics of Phase Growth in a Two-component System: Lecue 17: Kineics of Phase Gowh in a Two-componen Sysem: descipion of diffusion flux acoss he α/ ineface Today s opics Majo asks of oday s Lecue: how o deive he diffusion flux of aoms. Once an incipien