THE NUMERICAL SOLUTION OF THE NEWTONIAN FLUIDS FLOW DUE TO A STRETCHING CYLINDER BY SOR ITERATIVE PROCEDURE ABSTRACT

Transcription

1 Europea Joural of Egieerig ad Techology Vol. 3 No., 5 ISSN THE NUMERICAL SOLUTION OF THE NEWTONIAN FLUIDS FLOW DUE TO A STRETCHING CYLINDER BY SOR ITERATIVE PROCEDURE Atif Nazir, Tahir Mahmood ad Mohammad Shafique 3 Mathematics Group Coordiator, Departmet of Geeral Studies, Yabu Idustrial College, SAUDI ARABIA Departmet of Mathematics, Islamia Uiversity, Bahawalpur, PAKISTAN 3 Ex-Assistat Professor of Mathematics, Gomal Uiversity, D I Kha, PAKISTAN ABSTRACT The umerical solutio has bee obtaied of the goverig equatios for the steady, icompressible fluid flow due to a stretchig cylider. The umerical results are calculated, by usig SOR method ad Simpso's (/3) rule, for the rage. to of the parameter R. The accuracy of the results is checked very carefully by performig calculatios o three differet grid sizes ad comparig them with the kow results. AMS Subject Classificatio: 76D99, 76M, 65N. Keywords: Newtoia Fluids, Stretchig Cylider ad SOR Iterative Procedure. INTRODUCTION The fluid dyamics due to a stretchig surface is importat i extrusio processes. Crae [], discussed a closed form exact solutio of the Navier-Stokes equatios subject to two dimesioal stretchig of a flat surface. Brady ad Acrivos [], examied the exact similarity i solutios of a flow iside a stretchig chael ad iside a stretchig cylider. Crae [3], agai foud the boudary layer solutio outside a stretchig cylider. Kuike [4] ad Baks [5], studied the two-dimesioal boudary layer due to o-uiform stretchig. Wag [6] cosidered the two-dimesioal stretchig of a surface i a rotatig fluid. The threedimesioal flow subject to a stretchig flat surface was studied by Wag [7]. Also, Wag [8] solved the problem of the exterior fluid flow due to the extrusio of hollow tubes. I the preset paper, Wag s [8] work is exteded to large values of the Reyold umber R usig differet umerical techiques. The umerical techiques used i the preset work are straightforward ad easy to program. Wag [8] has used the Ruge-Kutta method, which is quite laborious ad ot straightforward for solvig the boudary value problems. The basic aalysis of this problem is preseted. The fiite differece equatios are obtaied ad solved by usig SOR iterative procedure ad the Simpso s (/3) rule, subject to the appropriate boudary coditios. The results are give i tabular ad graphical form, ad compared with previous results. BASIC ANALYSIS The Navier-Stokes equatio ad cotiuity equatio for steady ad icompressible flow i the absece of the body force are give by p V ( V ) V, () V, () where, ad V are desity, kiematics viscosity ad velocity of the fluid respectively. Progressive Academic Publishig, UK Page 45

2 Europea Joural of Egieerig ad Techology Vol. 3 No., 5 ISSN Wag [8] obtaied the equatios of motio for the flow of fluid due to a stretchig cylider. He used the cylidrical coordiate system (r,,z) such that the cylider is described with the radius r = a. Let u ad w are the velocity compoets i the r ad z directios, respectively the the equatio () ad () take the followig form: u u u u u u p ( ) ( u w ) r r r r z r r z (3) w w w w w p ( ) ( u w ) z r r r z r z (4) with cotiuity equatio u w r r z (5) where the subscripts deote the partial differetiatio with respect to space coordiates, is the desity, p the pressure ad the coefficiet of kiematics viscosity. The equatios (3) to (5) are to be solved subject to the followig boudary coditios: whe r= a, u=, w=kz whe r=, u= (6) where k is a positive costat of dimesio [/Time]. We ow use the similarity trasformatios to make the equatios of motio i dimesioless form as follows. u= -ka(f ()/ ) ad w= kf ()z, (7) where =(r/a) is the dimesioless variable. The equatio (5) is satisfied ad the equatio (3) ad (4) by usig (7) become: f f R( f ff ), (8) p p k a f kf. (9) Here R ka is the Reyolds umber, where a is the radius of the cylider, the coefficiet of kiematics viscosity ad k the give costat. I view of (7), the boudary coditios (6) take the form: =: f =, f = : f =. () I order to solve umerically, it is coveiet to reformulate the problem by usig the followig trasformatio: x e () Thus the equatio (8) ad boudary coditios (), due to (), become ad f f f R( f ff ff ) () x=: f =, f = Progressive Academic Publishig, UK Page 46

3 Europea Joural of Egieerig ad Techology Vol. 3 No., 5 ISSN x: f =, (3) where here the prime deotes differetiatio with respect to x. I order to treat the equatio () umerically we write it ito two equatios as follows: f = p, (4) p p p R( p fp fp) (5) The boudary coditios (3) ow become: x=: f =, p= x: p=. (6) Now, if we approximate the equatio (5) by cetral differece approximatio at a typical poit x = x of the iterval [,), we obtai Rhf ( h ) p Rhf ( h ) p ( h Rh f Rh p ) p (7) where h deotes a grid size. For computatioal purposes, we shall replace the iterval [,) by [, ), where is a sufficietly large. We ow solve umerically the first order ordiary differetial equatios (4) ad the fiite differece equatio (7) at each iterior grid poit of the iterval. The equatio (4) is itegrated by the Simpso s (/3) rule, with the formula give i Mile, whereas the equatio (7) is solved by usig SOR iterative procedure, subject to the appropriate coditios. The pressure p ca be calculated by itegratig (9). DISCUSSION ON RESULTS Calculatios have bee carried out to obtai umerical solutios of the equatios (4) ad (7) by usig Simpso s (/3) Rule ad SOR iterative procedure. I order to check the accuracy of the umerical results, they have bee calculated o three differet grid sizes. Also, the results have bee compared with the previous results by Wag [8] ad are foud to be i good agreemet. The effects of the flow parameters amely R ad Pr have bee examied for the velocity ad temperature profiles. The results have bee preseted i tabular as well graphical forms. The compariso of the preset results with the previous results by Wag [8] is give i Table to Table 3. The Table shows that all the values of f () are egative that meas, the fluid is uder the actio of a draggig force due to stretchig surface. Figure demostrate f ( ) for various values of R. Figure show f ( ) for differet values of R. The velocity gradiet also icreases for icreasig values of R as ca be see i Figure. It is worth metioig that the velocity field is ot affected by Pradtle umber Pr. The Figure 3 ad Figure 4 demostrate the temperature distributios for Pr.7 (such as air) ad Pr 7 (such as water) for a fixed value of R. I both the figures ( ) decreases ad the becomes zero at a large i both the cases. This situatio causes the icrease i wall temperature gradiet ad thus the surface heat trasfer rate is icreased. Progressive Academic Publishig, UK Page 47

4 Europea Joural of Egieerig ad Techology Vol. 3 No., 5 ISSN Table : Compariso of f () for possible values of Reyolds umber R. R=. R=5. R=. Preset Wag Preset Wag Preset Wag Table : Compariso of f ( ) for possible values of Reyolds umber R. R=. R=5. R=. Preset Wag Preset Wag Preset Wag Table 3: Compressio of () for possible value of R=. Pr Preset Wag Progressive Academic Publishig, UK Page 48

5 Europea Joural of Egieerig ad Techology Vol. 3 No., 5 ISSN f'.4.. f Figure : The velocity profile f ( ) for R=.,, 5 ad from top to bottom Figure : The similarity profile f ( ) for R=.,, 5, ad from top to bottom Figure 3: Temperature profile ( ) for R=, Pr= Figure 4: Temperature profile ( ) for R=, Pr=.7. REFERENCE [] I.J. Crae, Zeit. Agew. Math. Phys.,, pp645, (97). [] J.F. Bardy ad A. Acrivos, J.Fluid Mech.,, pp7, (98). [3] I.J. Crae, Zeit. Agew. Math. Phys., 6, pp69, (975). [4] H.K. Kuike, IMA J. Appl. Math., 7, pp387, (98). [5] W.H.H. Baks, J. Mech. Theor. Appl.,, pp375, (983). [6] C.Y.Wag, Zeit. Agew. Math. Phys., 39, pp7, (988). [7] C.Y.Wag, J. Phys. Fluids, 7(8), pp95, (984). [8] C.Y.Wag, J. Phys. Fluids, 3(3), pp466, (988). [9] R.L. Burde, Numerical Aalysis, Pridle, Weber & Schmidt, Bosto (985). Progressive Academic Publishig, UK Page 49