Singh, O P

Tandon, P N

English

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I n d ia n  J .  P h y s . 47. 334-340. (1973)P. N. Tandon and 0  P SinghDepartmeM of Mathematics, H B. Technological Institute, Kanpur {India) {Received 13 June 1972, revised 1 September 1972)H eat and m ass tran sfer and their th erm od yn am ic  cou p lin g  inb oundary layer flow  o f  a v isco ela stic  fluid past a flat p lateStudies on the steady state heat and mass transfer of a viscoelastic fluid past a flat plate are presented in this paper. The plate is either porous such that blowing or suction of a diffusing fluid through the surface can be effected or it is made of such material, as sublimes into a hot boundary layer fluid, thus introducing a mass transfer process. According to Thermodynamics of Irreversible Processes, the fluxes of heat and mass arc couifled, The effects of mass transfer on the temperature profiles have been studied, neglecting the effects of toimporature gradient on concentration fields. It is observed that the skin friction, heat transfer and mass transfer coefficients increase with suction and decrease with injection The viscoelasticity of“ the fluid increases the skin friction and decreases the heat and mass transfer cooffioionts. Tin? effect of thermodiffusion coefficient is to increase the temperature at a point.1. IntuodtjctionThe importance of the effects of thermodynamic coupling between heat and mass transfer on heat transfer and adiabatic wall temperature in binary systems have been discussed in some recent papers (Baron 1962, Tewfik & Shirliffe 1962). Measurements, with helium injection into an incompressible turbulent boundary layer (Tewfik et al 1962) clearly demonstrated experimentally significant effects of the coupling in an important engineering appfication of injection process Subsequent moasuromonts (Tewfik 1964) showed that the heat flux due to coupl­ing was of the same order of magnitude and even exceeded the famihar Fourier’s condition of heat flux in opposite directionsFollowing Ibbs & Grow (1952), we have neglected the Soret effect, i.e., the effect of temperature on mass, which is very small. The general conservation equations, taking such coupling into consideration, were developed by Baron (1959) for two dimensional flow Those equations have been extended in this paper for the viscoelastic fluid represented by Walters B" model (Beard & Walters 1964). It is thought that the boundary layer flows are mostly developed in visco­elastic fluids, that are mobile and highly elastic. We shall regard the flow as a perturbation of a Newtonian viscous flow so that usual concept of boundary layer can be assumed to apply.334Heat and mass transfer etc. 3362. G o v e r n in g  E quationsFollowing Baron (1959), Beard & Walters (1964) and Tewfik & Shirtiffo (1962) and neglecting the Soret effect, the governing equations for two dimen­sional boundary layer flow, hoat and mass transfer may bo obtained as :du . dv(1)- p dx dy^  L dxdy  ^ dy^  'dx  dy^dx dydu d^ u dy dxdyJ’ (2)du>xdxdw^dy. p  ^^ ^^ 1' V  ’dT , dT — 4-V - Ox Dy dif ' dy\~ dy with the following boundary conditionsu — 0, V ^  Vq, T — Tw =■ constant, a-f V ■u — Uj, T — T „, Wy =  at y -> oo... . (3)(4)„ }(5)The value v — Vq shows injection if Fj, >  0 and suction if Fq <  0. In the above u, v are the velocities of the fluid along x and y axis, p isotropic pressure, k \iscoelastic parameter, velocity outside the boundary, layer, D, and a are coefficients of diffusion, thormodiffusion and the thermal diffusivity {k*jpc).3. Sim il a r it y  T ra n sform ation sA search for the similarity solutions of the non-linear partial differential equations reveal that the only possible form for the pressure gradient in these circumstances is of tho form C.x whore G is a constant, defined by (7 =  17®. Introducing the stream function (f>{x, y) such that,dy ’ dx (6 )whore4, =  (vtr,M(>/)and using the following nondimensional tiansformationb :. T{x, y ) -T ^T aa—Tn)mr ( « ) =  , ,  =  ( ^ W  .. .  (76)336 P. N . Tandon and O. P . Singhtho governing equations (1-4) and the boundary conditions (5) become;, r " + / r + i - r + ^ f c [ F ^ - 2 /T " + r ‘*] =  o ... (s)fr'MPr)f0’ -]-/i{Sc){0'W'~{-eW") =  O ... (9)=  0 ... (10)and/ '  =  0, / =  -  = / » ,  F  =  0, 0 =  0 at ?  =  0. . .  (11)f  — I, 0 — I, W =  I at 7j =  CO, where {Pr) — {ficjK) Prandtl number, (>Sc) — (v/Z)) Schmict number, fi thermo- diffusion coefficient, fc viscoelastic parameter.It may be noted that / “ <  0 indicates injection or blowing of a foreign or C(Jolant fluid across the porous wall into tho mainstream fluid, whereas / “ >  0 depicts suction from tho boundary layer fluid or mass transfer towards the wall.4. Solution o r  the P roblemA. Momentum Transfer : It has boon observed that the presence of elasti­city in tho fluid yields a fourth order differential tMpiation (8), whereas in viscous case {k — 0), the order of the equation is tliree. It would thus appear that an additional boundary condition must bo imposed to obtain a solution. However, in tlie derivation of equation (2). we have implicitly neglected the terms of tho order of Jc^ . It is therefore reasonable to seek a solution for f  in the form/  =  / o + ¥ i + W .  ... (12)Substituting in equation (8) and equating the coefficients of k and independent terms on both sides we got :andwith the boundary conditions,/o(0) = / “, /o '(0 ) -  0, / ; (o o )  =  1A ( 0 ) - / i ' ( 0 ) - 0 ,  //(oo) =  0.and... (13) ... (14)... (15) ... (16)The nonlinear part of the equation (13) is separated from the linear port and sequenoes of apjiroximations are thus constnicted using the Laplace transforma­tion as follows :U i = r + 21 ‘ O ■ 'O 1and= ^a_ ( l _  ^ j^  ^  rfi— m«4- — 1J -^ 2 ! ' 31 g j V ^+71^(17)(18)Heat arid mass transfer etc. 337using two boundary conditions at  ^=  0 ^  ^  0) and an additional condi­tion fo’iO) =  at — 0. The value of is determined by using the condition /„' =  1, t^ %->cx) by Meksyn’s (1961) method of steepest descent.Proceeding exactly in a similar manner, we construct the sequences for fi from the equation (15) as follows ;=  ...(19)and3 1 5 !2a2 6 ! ^ (20) ^ The unknown parameter /i"(0) =  has been evaluated by using the method \ of steepest descent (Meksyn 1961). ^ The values of and a,^  are given in table 1. The corresponding values givenhy Hartree (1937) for viscous fluid and Beard & Walters (1964) for visco-elastic fluids are also given for comparison.Table 1For the present analysisBeard &;Walters’ Hartroe’ssolution for solution for visoo-elastio viscousfluid fluidp 0 .1 .2 .5 - .1 - . 2 - . 51.2348 1.4155 1.6163 2.2617 1.0557 0.8946 0.04676 1.2326 1.2321.2976 1.1769 1.4490 0.9710 1.1649 1.0647 0.2724 1.1390 _Skin friction : The dependence of skin friction on blowing or suction values arc given bym  = /o " (0 )+ W 9 )  =  -  (21)The values of skin friction [i.e., /"(O)] for various values of suction and injection are given in table 2.Table 2P 0 .1 .2 .6 - . 1 - . 2 - . 6H =  0 1.2348 1.4166 1.6166 2.2617 1.0657 0.8046 0.46762 1.4943 1.7694 1.9161 2.4569 1.2867 1.1047 0.6221338 P . N . T andon and O . P . SinghB. M o s h  Transfer ; Separating nonlinear terms from linear ones of the equation (10), one gets the sequences of approximations using the Laplace trans­formation as follows :W  ^=  d7j, ...(22)andW ,: ( ( l - a^fO)+Z(a,=^+/og,)} ^4 ! 5 !I I ( g i + 2g « i « a )  ,  2( t t i + g a 2) 8 1 2 , ]- r g i / T  7 1 / f j p l-ajwliore d is an unknown constant and the value of 5 has been evaluated by Meksyn’s steepest descent method (1961). The values of B arc given in table 3.Table 30 .1 .2 6 - . 1 -  2 - . 6=  0 0.6666 0.6622 0.6031 0 8339 0.4859 0 3878 0.1022J c =  . 2 0.6016 0.6025 0 6902 0 7905 0.5191 0 4761 0 1234Mass transfer coefficient : Ratio of mass transfer coefficient [^q] with that of zero blowing or suction [fco]/°=o given by_ K  ^  W \ 0 )_[Kifo^o l1^'(0).Ko.. (24)Values of hJ[ho\fO. □ foi' various values of suction and injection are given in table 4Table 4P 0 1 ,2 . 5 - . 1 - . 2 - . 5k  =  0 1.0000 1 115 1.223 1 471 0.8572 0.6810 0.1803=  .2 1.0000 1.101 1.148 1.314 0.8630 0.7914 0.2061C. Heat Trarisfer : In the present case, we disregard the terms -duo to dissipation and uicludo the thorinodynamic coupling effect on temperature due to concentration fields. Using the transformation0 =  6o-^/3m0, ... (25)in the equation (9) and equating the terms indepondent of {Sc) and its coefficients on both sides wo get ■V + (P 0 /< 9 ( /= 0  ... (26)and=: 0, ... (27)with boundary conditions0o(O) =  0, d i^oo) =  1 and dj{0) =  0, 0j(oo) =  0 ... (28)Assuming =  7 i a'lid ^i'(O) =  and using the same technique, weconstruct sequences, {^o.n} a-nd as follows :^o,i^7iV> ...(29 )iPr)P , (Pr)((l-aijF«)4fc(ai“+/«a,)) 5«'2 7i L/ 2 ! ' 4 !  ^ ^  &] ^Heat and mass transfer etc. 339_  {Pr)P e _  iPr){cc, -^^2Ka,oc,) ,^ {a ,^K a ,){P r) . 2 { ^  3^6!  / - 7 -t- g , n g , w ^and1^.1 =  7zV^(i - r r - f o ^ z  , 2/?(/5c)7j« (Pr)y^(a^-l-Kcc.J f^ i,3 “  72V — 2“i----  ^ ^  “  2\ ■  ^ -----  41--------(31)+ (P r)ya (l-« if)+ ^ (a i^  +  /X)+^<^(>S'c)yi(oti+^o^2) «55 ! L_" 7^0(Pr)y,f+5g(^c)yi{(l - g ,  P )K joc^^+poc )^] „6 !{Pr)y,{oi^-{-2K<^,cc,)--6pd{Sc)y, ,- , /, 2(Pr)y2(ai+Za2)4-7iy(/Sfc)y3(ai2+2/Caia2) o-h g j ' - /_2(Pr)y2-[ 16*!i(.S^yi(ai+A:a2) g _95(^c)yi“ 9 ! ' ..........   ^ 10"! ‘  ’ ... (32)The values of y  ^and y  ^are evaluated by the same method Maksyn (1961) et al 1962), using the conditions Oq{co) — 1 and ^i(co) — 0. The values of y  ^and y  ^are given in table 5 for {Pr) — {Sc) =  1.Table 6y 0 .1 .2 .5 - . 1 - . 2 - . 6Vi k  =  0 0.5666 0.6322 0.6931 0.8339 0.4869 0.3878 0.10227 i Jfc =  .2 0.6016 0.6625 0.6902 0 7906 0.6191 0.4761 0.123472 /c =  0 0.0000 0 0287 0.0966 0.2433 1.1728 0.0439 -0.00597 i k  =  ,2 0.0000 0.0178 0.0764 0.2494 0.2408 0.0282 -0.0086340 P. N. Tandon and O. P. SinghBejdi transfer coefficient: Ratio of heat transfer coefficient [h] to that at zero blowing or suction i.e..h ^  0'{O)[AJ/o„o [0'(O)]/o=oare given for various values of suction and injection in table 6.Table 6... (33)r 0 .1 .2 .5 - . 1 - . 2 - . 6Without k =  0 1.0000 1.115 1.223 1.471 0.8672 0.6810 0,1803coupling effect k .2 1.0000 1.101 1.148 1.314 0.8630 0.7914 0.2061With k =  0 1.0000 1.180 1.240 1.516 0.8360 0.6766 0.1776ooupliug effect k =  .2 1.0000 1.165 1 160 1 355 0.8460 0.7368 0.20365. DiSOTTSBION and RESUIiTSThe results of above analysis are shown in tables (1-6). It may be observed that the skin friction, heat transfer and mass transfer coefficients increase with suction and decrease with injection. The effect of viscoelastic parameter {k) is to increase akin friction and to decrease heat and mass transfer coefficients. It has also been observed that the thermodynamic coupling increases the tem­perature at a point.AcknfwledoembntbfThe authors gratefully acknowledge the award of a C.S.I.R. fellowship to one of us (O.P.S.).ReeerencesBaron J. R. 1902 A R S J .  32. 1063.Boron J. R. 1959 M . I  T .,  N a v a l  S u p e r s o n ic  L a b o ra to ry  T .R . , 160.Beard D. W. & Walters K. 1904 P r o c .  G am b. P h i l .  S o c . 60, 667.Be Groot S. R. & Mazur P. 1962 N o n -e q u il ib r iu m  T h erm o d y n a m ics , North Holland PublishingCompany, Amstordam.Hortroe B. R. 1937 P r o c ,  G am b . P h i l .  S o c . 33, 223.Ibbs T. L. & Grew K. E. 1952 T h erm a l D i f f u s io n  in  C h ses , Cabridge University Press. Meiksyn B. 1961 N e w  M e th o d s  in  L a m in a r  B o u n d a r y  L a y e r  T h e o r y , Pergamon Press, NewYork. Oxford.Tewflk O. E. & Shirtliffe G. J. 1962 J .  A e r o -s p a c e  S c i .  2 9 , 1009.Tewfik O. E.i Eoker E. R. G. & Shirtliffo C. J. 1962 P r o c e e d in g s  H e a t  a n d  F lu id  M e ch a m i^In s t i tu te ,Tevrfik O. E. 1064, I n te r n . J ,  H e a l  a n d  M a s s  tr a n s fe r , 7.

Heat and Mass Transfer and their Thermodynamic Coupling in Boundary Layer Flow of a Viscoelastic Fluid Past a Flat Plate

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Heat and Mass Transfer and their Thermodynamic Coupling in Boundary Layer Flow of a Viscoelastic Fluid Past a Flat Plate

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