Similarity Solutions to Unsteady Pseudoplastic. Flow Near a Moving Wall

Transcription

1 Iteratioal Mathematical Forum, Vol. 9, 04, o. 3, HIKARI Ltd, Similarity Solutios to Usteady Pseudoplastic Flow Near a Movig Wall W. Robi Egieerig Mathematics Group Ediburgh Napier Uiversity 0 Colito Road, EH0 5DT, UK Copyright 04 W. Robi. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Abstract It is show that similarity solutios to a partial differetial boudary value problem for power law fluids may be geerated by the applicatio of a multiparameter stretchig group. Furthermore, the resultig ordiary differetial boudary value problem arisig from the iitial group aalysis is itself trasformed to a iitial value problem by the applicatio of a further stretchig (oeparameter) group trasformatio. The resultat iitial value problem is solved by a stadard forward marchig method. Mathematics Subject Classificatio: 34B40, 35C06, 35G30, 76M55, 76M60 Keywords: Partial differetial equatios, boudary value problems, stretchig groups, similarity aalysis. Itroductio I this paper, we preset a similarity group aalysis of the equatio of motio of a semi-ifiite body of pseudoplastic (power law) fluid occupyig the halfspace y 0 [, 8], that is

2 466 W. Robi u u m t y y 0 (.) where u ( t) is the x-compoet of velocity i a cartesia system. The fluid is set i motio at time t 0 by impartig a costat velocity U o the boudary y 0; sice the effects of this velocity is expected to decay ad evetually vaish as we move ito the body of the fluid, equatio (.) is to be solved with the boudary coditios u ( 0,t) U, t 0; u(,t) 0, t 0 (.) I equatio (.), / m is the ratio of the fluid desity to the empirical costat m, characteristic of the fluid, defied via equatio () of []; the parameter also is characteristic of the fluid []. The solutio process proposed here for the partial differetial equatio (.) with boudary coditios (.), ad where the ovelty lies, ivolves the complete reductio of the origial problem (.)/(.) to a iitial value problem etirely by stretchig group trasformatios. The solutio process ad paper are orgaized as follows. The first part of the process, preseted i sectio, ivolves the idetificatio of a 3-parameter stretchig group trasformatio that the origial problem (.)/(.) is ivariat uder [5, 6]. This is achieved by demadig [5, 6] that the origial problem (.)/(.) be ivariat uder the more basic stretchig [9] 5-parameter group y a t a t, u a 3 u, a4 m m, U 5 a U (.3) a k 0, k,,,5. Next, oce the 3-parameter stretchig group is determied, it is the used, i sectio 3, to trasform the partial differetial boudary value problem (.)/(.) ito a ordiary differetial boudary value problem. To do this, we assume the existece of a implicit solutio [4] to the origial problem (.)/(.) of the form g( u, / m), U) 0 (.4) with g a arbitrary fuctio, which is to be ivariat uder the 3-parameter group of sectio. This ivariace requiremet o (.4) leads to two systems of partial differetial equatios havig idetifiable [5, 6] similarity solutios, which, i tur, allows a similarity trasformatio of the partial differetial boudary value problem (.)/(.) ito a ordiary differetial boudary value problem. The fial part of the reductio process, preseted i sectio 4, uses aother [, 3, 7]

3 Similarity solutios to usteady pseudoplastic flow 467 stretchig group trasformatio to trasform the ordiary differetial boudary value problem, obtaied from the ivariat solutio aalysis of sectio 3, ito a ordiary differetial iitial value problem. Fiall i sectio 5, the resultat iitial value problem is solved by a stadard fourth-order Ruge-Kutta forward marchig method ad the results of our aalysis compared with previous work o the problem, that is, (.)/(.).. Determiig the r-parameter group Followig Mora [5] ad Mora ad Marshek [6], we will use the 5-parameter stretch trasformatio (with the group parameters a,, a5 positive real variables) y a t a t, u a 3 u, a4 m m, U 5 a U (.) to costruct a r-parameter group trasformatio, which we write as a group, G r, ad a subgroup, S r [5,6], that is G r b b b y a a a r r y b b b t a a a r r t : Sr : b b b 3 3 ( / m) a a a 3r r ( m / ) (.) b b b 4 4 U a a a 4r r U c c c u a a a r r u The demad that equatio (.) ad the boudary coditios (.) be ivariat uder the 5-parameter group trasformatio (.), leads to relatios that determie both the order r of the trasformatio group (.) ad the values of the group idices b,, c. It will be apparet that the r-parameter group (.) is simply r aother way of writig the 5-parameter group (.), that is, the 5 parameters are ot idepedet ad so r 5. This meas that equatio (.) ad the boudary coditios (.) will be ivariat uder the r-parameter group trasformatio a fortiori. Applyig the trasformatios (.) ad the chai rule to equatio (.), we fid (subscripts o u or u implyig partial differetiatio)

4 468 W. Robi a a3 [( / m) ut (( u y) ) y ] a a a 3 a4( / m) ut (( u ) ) (.3) If we require equatio (.) to be ivariat uder the trasformatio (.), the from (.3) we must have y y a a a 3 a4 (.4) Also, if we demad that the boudary coditios (.) be ivariat uder the trasformatios (.), the we require a5 a 3 (.5) From (.4) ad (.5), we see that we may take a, a ad a 3 as the basic idepedet group parameters, with the trasformatio (.) ow takig the form of a 3-parameter group trasformatio, which we write as a group, G 3, ad a subgroup, S 3 [5, 6], that is G 0 0 y a aa3 y 0 0 t a a a3t : S3 : ( ) ( / m) a a a3 ( m / (.6) 0 0 U a aa3 U 0 0 u a aa3 u 3 ) I the ext sectio we show that by determiig the group ivariats associated with S 3 ad G 3 [5, 6], that the partial differetial boudary problem (.)/(.) ca be trasformed to a correspodig ordiary differetial boudary problem, (3.3) below. 3. The 3-parameter group aalysis The ext stage of the aalysis, is the reductio of the partial differetial boudary value problem (.)/(.) ito a ordiary differetial boudary value problem. Critical to this aalysis is the determiatio of the group ivariats of S 3 ad G 3 [5, 6] ad ivariat solutios of (.)/(.) uder G 3 [5, 6]. If we express the solutio to our partial differetial boudary value problem implicitl the we may itroduce the fuctio g such that the implicit solutio is writte as [4, 8]

5 Similarity solutios to usteady pseudoplastic flow 469 g( u, / m), U) 0 (3.) For (3.) to be ivariat uder the group G 3, we seek ivariats, g, of the form g( u, / m), U ) g( u, / m), U) (3.) So, by partial differetiatio of (3.) with respect to a,a ad a 3, ay (absolute [5, 6]) group ivariat for the group G 3 must satisfy the partial differetial system (where we may drop the bars) g g y ( )( / m) 0 (3.3a) y ( / m) g g t ( / m) 0 (3.3b) t ( / m) g u u g g ( )( / m) U 0 (3.3c) ( / m) U That is, we have three idepedet liear homogeeous partial differetial equatios, i five variables, which has two solutios [6]. However, if we cosider the subgroup S 3, we seek a ivariat,, of the form ( / m), U ) ( / m), U) (3.4) leadig to three idepedet liear homogeeous partial differetial equatios, i four variables, that is, by partial differetiatio of (3.4) with respect to a,a ad a 3, must satisfy y ( )( / m) 0 y ( / m) t ( / m) 0 t ( / m) (3.5a) (3.5b) ( )( / m) U 0 (3.5c) ( / m) U (where we drop the bars agai) which has oe solutio [6]. Apparetly the sole solutio to (3.5) is also oe of the two solutios to the G 3 equatios (3.3).

6 470 W. Robi From the geeral theory of the solutio of equatios such as (3.3) ad (3.5) [6], we kow that we may look for particular group ivariats of the form b b b y[ t] ][ / m] [ U] 3, gˆ u[ t] ][ / m] [ U ] 3 (3.6) The parameters b,, c3 are evaluated by substitutig that the relatios (3.6) ito equatios (3.3) ad (3.5): this leads to two sets of simultaeous equatios for b, b, ) ad c, c, ), which are solved to give the particular ivariats ( b3 ( c3 c c c y mtu, u gˆ (3.7) U As the ivariats (3.7) are fuctioally idepedet, the geeral solutio, g, of (3.3) is a (differetiable) fuctio [6], f sa of ad ĝ, so that g ( u, / m), U) f ( gˆ, ) 0 (3.8) Solvig the implicit relatio (3.8) for ĝ i terms of,, we see that we may rewrite (3.7) i the form with F ( ) (curretly) a arbitrary fuctio. y, u F( ) (3.9) mtu U Fiall sice u UF( ), we ca ow reduce the origial partial differetial equatio (.), with boudary coditios (.), to a boudary value problem i oe variable oly. That is, a ordiary differetial equatio with appropriately trasformed boudary coditios. Ideed, uder the similarity trasformatio (3.9), equatios (.) ad (.) are trasformed ito the followig ordiary differetial boudary value problem ( F) F F 0; F ( 0), F( ) 0 (3.0) ( ) where the dash deotes differetiatio with respect to. The relatios (3.9) ad (3.0) are essetially those obtaied by Bird [] from ad hoc dimesioal cosideratios. I fact, ( ) r, where r is defied by

7 Similarity solutios to usteady pseudoplastic flow 47 equatio (5) of referece []. I what follows, it proves coveiet to trasform the system (3.0) ito the form obtaied by Bird [] by makig the trasformatio /( ), to get (the dash deotes differetiatio with respect to ew ) ( F) ( ) F F 0; F ( 0), F( ) 0 (3.) 4. Reductio to a iitial value problem: further group aalysis We wish to solve the system (3.) to fid the dimesioless velocit F, from which we ca get the x-compoet of the velocit u, from UF. That is, we wish to solve the ordiary differetial boudary value problem ( F) ( ) F F 0; F ( 0), F( ) 0 (4.) Bird solved (4.) usig a semi-aalytic method.[]. I what follows, we solve the system (4.) for F, followig Pha-Thie [8], by turig (4.) ito a iitial value problem. To do this we itroduce, first, the chage of variable G F, whe the system (4.) becomes (the variable G is ot to be cofused with the trasformatio group) ( G) ( ) G G 0; G ( 0) 0, G( ) (4.) The boudary value problem (4.) may be trasformed ito a iitial value problem ow by a oe-parameter stretchig group trasformatio (see, for example, the books by Na [7] or Bluma ad Coles [3] or Ames []). If we defie G A G, A (4.3) the the equatio for G A ( ) [( G ) G G] F i (4.) will trasform as A ad, for ivariace, we require ( ) ( ) ( G) G G (4.4) ( ) α

8 47 W. Robi ( ) α ( ) 0 (4.5) To determie α ad, we require aother equatio, ad we obtai this by forcig a coditio o the ukow iitial coditio G (0). Let G ( 0) A (4.6) the, trasformig (4.6), we will have Now, by settig A G(0) A (4.7) α (4.8) equatio (4.7) becomes G (0) ad the simultaeous equatios (4.5) ad (4.8) may be solved, for α ad, to give α, α (4.9) It remais oly to fid the group parameter A. To fid A, we cosider the boudary coditio at ifiit G ( ). Trasformig the boudary coditio at ifiit we fid that A ( ) (4.0) G With everythig ow i place, the solutio procedure is as follows. First, for ay, we solve the iitial value problem ( G) ( ) G G 0; G ( 0) 0, G(0) (4.) to fid (a approximatio) to G (). Next, we use the iformatio from the solutio to the iitial value problem (4.), that is, G (), to solve the required system for F, from (4.) with F G, that is

9 Similarity solutios to usteady pseudoplastic flow 473 ( F) ( ) F F 0; F (0), F(0) ( ) (4.) G 5. Results ad coclusios The iitial value problems (4.) ad (4.) were solved usig a stadard fourthorder Ruge-Kutta routie ad the results of this solutio process are preseted i Table ad Figure, for the rage of values of cosidered by Bird []. Table. Numerical values for calculatig velocity profiles. G () F (0) 0. 0 G ( ) (Bird []) (6.57) (4.30) (3.05) (.9) (.83) The results of Table provide a compariso with the work of Bird [], the colum headed 0. 0 beig the values of the reduced variable for which the fluid

10 474 W. Robi velocity has falle-off to % of the velocity of the movig wall (the xz-plae). The umbers i brackets i Table refer directly to Bird s results. The compariso with Bird s results i Table help to validate the solutio method. Figure presets two represetative solutio curves, for the extreme values ( F - solid lie) ad /3( F - dashed lie ). Figure. Typical reduced velocity curves for ad /3( F - dashed lie ). ( F - solid lie) The approach here is restricted to a basic similarity aalysis through the medium of stretchig groups. Pha-Thie [8] has developed a Lie-group solutio process for problems slightly more geeral tha (.)/(.), from which similarity solutios may be extracted. The method developed here is, hopefull simpler. I coclusio, a example of a similarity approach to the group aalysis of partial differetial equatios has bee preseted, the partial differetial equatio, describig usteady flow i a pseudo-plastic fluid, beig reduced to a ordiary differetial equatio. The complete trasformatio, icludig boudary coditios, has led to a ordiary differetial boudary value problem which i tur was trasformed to a iitial value problem ad solved i a stadard maer. Refereces [] Ames W,F.: Noliear Partial Differetial Equatios i Egieerig. Academic Press, Lodo, Vol. I (965), Vol. II (970). [] Bird R. B.: Usteady pseudoplastic flow ear a movig wall. AIChE J. 5 (959) 565.

11 Similarity solutios to usteady pseudoplastic flow 475 [3] Bluma G. W. ad Cole J. D.: Similarity Methods for Differetial Equatios. Spriger-Verlag, New York (974). [4] Dreser L.: Applicatio of Lie s Theory of Ordiary ad partial Differetial Equatios. IOP Publishig, Lodo (999). [5] Mora M. J.: A geeralizatio of dimesioal aalysis. J. Frakli Ist. 6 (97) 43. [6] Mora M. J. ad Marshek K. M.: Some matrix aspects of geeralized dimesioal aalysis. J. Eg. Math. 6 (97) 9. [7] Na T. Y.: Computatioal Methods i Egieerig Boudary Value Problems. Academic Press, Lodo (979). [8] Pha-Thie N.: A method to obtai some similarity solutios to the geeralized Newto fluid. J. Appl. Math. Phys. (ZAMP) 3 (98) 609. [9] Seshadri R. ad Na T. Y.: Group Ivariace i Egieerig Boudary Value Problems. Spriger-Verlag, New York (985). Received: Jauary 5, 04