ABSTRACT. and heat transfer

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1 41 v ABSTRAT The study o viscoelastic luids gives many opportunities to mathematicians, numerical analysts and simulationists to introduce suitable algorithms or computing the lo due to ulil many application o it such as in paints, coating, inks, and jet uels. In this thesis, the mixed convection boundary layer lo o a viscoelastic luid past a sphere subjected to constant temperature has been studied. The constitutive equations o viscoelastic luids usually generate a higher-order derivative term in the momentum equation than equations o Netonian luid. Thus, e are acing the problem here the boundary conditions insuicient to solve the problems o viscoelastic luid completely. Thereore, e need an extra boundary condition by augmenting an extra boundary condition at ininity. The governing boundary layer equations are irst transormed into a non-dimensional orm, and then, into a set o non similar boundary layer equations, hich are solved numerically using an eicient implicit inite-dierence method knon as Keller-box method. Numerical results are presented or dierent values o the viscoelastic and mixed convection parameters K and, respectively, and ith the Prandtl number Pr = 0.7, 1 and 7. It is ound that both skin riction and heat transer Q s decrease as K is increased. Further, or cases o cooling sphere ( 0) and heating sphere ( 0), the boundary layer separates rom the sphere. It is orth mentioning that the result obtained in viscoelastic luid hen e set the value o K = 0 (Netonian luid), are in excellent agreement ith those obtained in viscous luid.

2 42 vi ABSTRAK Kajian tentang bendalir likat-kenyal memberi banyak peluang kepada ahliahli matematik, ahli-ahli analisis berangka dan ahli pakar penyelaku memperkenalkan algoritma-algoritma yang sesuai dalam aliran ini untuk memenuhi banyak aplikasi seperti pembuatan cat, penyaduran, dakat dan minyak untuk jet. Dalam tesis ini, olakkan campuran aliran lapisan sempadan dalam bendalir likatkenyal melepasi sera bergantung kepada suhu yang tetap di kaji. Persamaanpersamaan juzuk bendalir likat-kenyal ini terjana dengan sebutan terbitan peringkat tinggi di dalam persamaan momentumnya berbanding persamaan bendalir Netonan. Oleh itu, permasalahan yang di hadapi ialah ketidakcukupan syarat-syarat sempadan untuk menyelesaikan masalah bendalir likat-kenyal ini. Oleh yang demikian, kita memerlukan tambahan syarat sempadan dengan menambahkan syarat sempadan di ininiti. Persamaan-persamaan lapisan sempadan menakluk, pada mulanya diubah kepada bentuk tak bermatra, selepas itu, diubah kepada set persamaan lapisan sempadan tak serupa dimana penyelesaian secara berangka dengan menggunakan skim beza terhingga tersirat yang eekti dikenali sebagai kaedah kotak-keller. Keputusan-keputusan berangka yang meliputi nilai-nilai parameter likat-kenyal dan parameter olakkan campuran masing-masing dipaparkan dengan nombor Prandtl yang berbeza iaitu Pr = 0.7,1 and 7. Dapati kedua-dua pekali geseran kulit pekali pemindahan haba dan Q menurun apabila nilai K meningkat. Untuk kes penyejukan dan pemanasan, lapisan sempadan terpisah dari sera. Dengan kata lain, di dalam bendalir likat-kenyal, keputusan yang diperolehi apabila menetapkan nilai K = 0 (bendalir Netonan) menunjukkan keputusan yang memuaskan setanding dengan keputusan yang diperolehi menerusi bendalir likat.

3 43 vii TABLE OF ONTENTS HAPTER TITLE PAGE REPORT STATUS DELARATION SUPERVISOR S DELARATION TITLE PAGE DELARATION DEDIATION AKNOWLEDGEMENTS ABSTRAT ABSTRAK TABLE OF ONTENTS LIST OF TABLES LIST OF FIGURES LIST OF SYMBOLS LIST OF APPENDIES i ii iii iv v vi vii ix x xv xvii 1 INTRODUTION Research Background Problem Statement Objectives o the Study Scope o the Study Signiicant o the Study Outline o the Dissertation 4

4 44 viii 2 LITERATURE REVIEW Introduction Mixed onvection Boundary Layer Flo Viscoelastic Fluid 8 3 THE DERIVATION OF THE GOVERNING EQUATION Introduction The ontinuity Equation The Momentum Equation The Energy Equation 17 4 MATHEMATIAL FORMULATION Introduction Basic Equations Solution 29 5 RESULTS AND DISUSSION Introduction The Validity o the results Results and Discussion 43 6 ONLUSION onclusion Suggestion or Future Research 66 REFERENES 68 APPENDIES A 71

5 45 ix LIST OF TABLES TABLE NO. TITLE PAGE 5.1 omparison the values o ''(0) and '(0) or various values o ith Pr =0.7 and K = omparison the values o ''(0) and '(0) or various values o ith Pr =7 and K = Values o local skin riction or K = 0.2, Pr = 1 and various values o Values o local heat transer Q or K = 0.2, Pr = 1 and various values o Values o local skin riction or K = 1, Pr = 1 and various values o Values o local heat transer Q or K = 1, Pr = 1 and various values o. 64

6 46 x LIST OF FIGURES FIGURE NO. TITLE PAGE 4.1 Physical model and coordinate system omparison o the local skin riction or K = 0 (Netonian luid) Pr = 0.7 and various values o omparison o the local heat transer Q or K = 0 (Netonian luid) Pr = 0.7 and various values o omparison o the local skin riction or K = 0 (Netonian luid) Pr = 7 and various values o omparison o the local heat transer Q or K = 0 (Netonian luid) Pr = 7 and various values o Velocity proile '( ) at x = 0 or various values o ith Pr =0.7 and K = Temperature proile ( ) at x = 0 or various values o ith Pr =0.7 and K = 0. 41

7 47 xi 5.7 Velocity proile '( ) at x = 0 or various values o ith Pr = 6.8 and K = Temperature proile ( ) at x = 0 or various values o ith Pr = 6.8 and K = Variation o the local skin riction or K=0.2, Pr = 0.7 and various values o Variation o the local heat transer Q or K=0.2, Pr = 0.7 and various values o Variation o the local skin riction or K=0.2, Pr = 1 and various values o Variation o the local heat transer Q or K=0.2, Pr = 1 and various values o Variation o the local skin riction or K=0.2, Pr = 7 and various values o Variation o the local heat transer Q or K=0.2, Pr = 7 and various values o Variation o the local skin riction or K=1, Pr =0.7 and various values o. 48

8 48 xii 5.16 Variation o the local heat transer Q or K=1, Pr =0.7 and various values o Variation o the local skin riction or K=1, Pr =1 and various values o Variation o the local heat transer Q or K=1, Pr =1 and various values o Variation o the local skin riction or K=1, Pr =7 and various values o Variation o the local heat transer Q or K=1, Pr =7 and various values o Variation o the boundary layer separation point X s ith or Pr = 1 and K = Variation o the boundary layer separation point X s ith or Pr = 1 and K = Variation o the boundary layer separation point X s ith or Pr = 1 and K = Velocity proiles '( ) or various values o ith Pr = 0.7 and K =

9 49 xiii 5.25 Temperature proiles ( ) or various values o ith Pr = 0.7 and K = Velocity proiles '( ) or various values o ith Pr = 1 and K = Temperature proiles ( ) or various values o ith Pr = 1 and K = Velocity proiles '( ) or various values o ith Pr = 7 and K = Temperature proiles ( ) or various values o ith Pr = 7 and K = Velocity proiles '( ) or various values o ith Pr = 0.7 and K = Temperature proiles ( ) or various values o ith Pr = 0.7 and K = Velocity proiles '( ) or various values o ith Pr = 1 and K = Temperature proiles ( ) or various values o ith Pr = 1 and K = Velocity proiles '( ) or various values o ith Pr = 7 and K = Temperature proiles ( ) or various values o ith Pr = 7 and K = 1. 58

10 xiv Velocity proiles '( ) or various values o K ith = 1 and Pr = Temperature proiles ( ) or various values o K at = 1 and Pr = Velocity proiles '( ) or various values o K ith = -1 and Pr = Temperature proiles ( ) or various values o K at = -1 and Pr =1. 60

11 xv 51 LIST OF SYMBOL/NOTATIONS a - radius o sphere - local skin riction - dimensionless stream unction K - viscoelastic m - velocity exponent parameter n - temperature exponent parameter g - gravitational acceleration Gr - Grassho number Pr - Prandtl number Re - Reynolds number Q - heat transer T - luid temperature u - velocity component in x-direction u e (x) - non-dimensional velocity outside boundary layer U 0 - dimensional constant U - ree stream velocity v - velocity component in y-direction x - coordinate in direction o surace motion y - coordinate in direction normal to surace motion Greek symbols - thermal diusivity - thermal expansion - dynamic viscosity

12 52 xvi - mixed convection parameter - dimensionless similarity variable - dimensionless temperature - kinematic viscosity - stream unction - luid density Subscripts - condition at the surace - condition at ambient medium Superscripts - dierentiation ith respect to

13 53 xvii LIST OF APPENDIES APPENDIES TITLE PAGE Appendix A Fortran 77 Programming or Finding the Solutions o Mixed onvection Boundary Layer Flo o a Viscoelastic Fluid Past a Sphere ith onstant Temperature. 70

14 54 HAPTER 1 INTRODUTION 1.1 Research Background It is ell knon that the Netonian equations do not adequately describe the lo properties o some naturally occurring luids such as animal blood. Thereore, non-netonian luids have become more important industrially such as polymer solution, polymer melts, blood, paints and certain oils. Amongst the several models o non-netonian luids, the viscoelastic luids have attracted much attention rom researchers. The lo o viscoelastic luids gives many opportunities to mathematicians, numerical analysts and simulationists to introduce suitable algorithms or computing the lo. There are to special categories o viscoelastic luids, second-order luid and Walters luid, hich have particularly attracted the attention o researchers during the last to decades. The boundary value problem (BVP) characteristic eature s o these luids is that the presence o viscoelasticity o the luid hikes the order o dierential equation. Hoever, there is no consequent increase in the number o boundary conditions. This is proved by Rivlin and Ericksen (1955), here they introduced classiication o the viscoelastic luids in simple ays. Hoever, the main diiculty

15 55 2 hich arises in the solution o the lo problems o these luids is that the constitutive equations o viscoelastic luids usually generate a higher-order derivative term in the momentum equations than equations o Netonian luids. Thus, e are acing the problem here the boundary conditions insuicient to solve the problems o this viscoelastic luid completely. Thus, e need an extra boundary condition to the usual obedience boundary conditions. Ariel (2002) introduced on extra boundary condition in the stagnation point lo o a second grade luid due to ork out the solution on this lo. Motivated by the ork above, this study aims to obtain mixed convection boundary layer lo o a viscoelastic luid over a sphere subjected to a constant surace temperature by augmenting the extra boundary condition introduced by Ariel (2002). The coupled non-linear partial dierential equations governing the lo have been solved numerically. The eects o the mixed convection and viscoelastic parameters on the skin riction and heat transer around the sphere are studied. The results have also been compared ith Nazar et al. (2003). We also ish to mention to this end that to our best knoledge this classical very important problem has not been studied beore or a viscoelastic luid so that the results are ne or these luids. 1.2 Problem Statement The study ill investigate the olloing questions. Ho the viscoelastic mathematical models o mixed convection boundary layer lo past a sphere? What are the eects o viscoelastic luids parameter on the skin riction and heat transer s?

16 Objectives o the Study The main objectives o this study are to carry out the mathematical ormulation o the governing equations o the boundary layer lo o a viscoelastic luid and to study the eects o skin riction and heat transer s o mixed convection boundary layer lo o a viscoelastic luid past a sphere. No experiments ill be conducted, but e are going to compare our result ith Nazar et al. (2003) to validate the results. 1.4 Scope o the Study This study ill take into consideration o to-dimensional incompressible viscoelastic luid model. Problem ill be narro don to boundary layer lo over a sphere subjected to constant temperature. We just considered the mixed convection problem. 1.5 Signiicance o the Study The boundary layer problem o viscoelastic luids theory has generated a lot o interest, and become important in recent years because o their applications in several industrial-manuacturing processes concerning petroleum drilling, manuacturing o oods and paper. In engineering problem, viscoelastic luids possible to reduce rictional drag on the hulls o ships and submarines. Some typical applications or viscoelastic boundary layer lo over a stretching sheet are polymer sheet extrusion rom a dye, glass iber and paper production, and draing o plastic ilms. There are

17 57 4 also many applications involving atomization o viscoelastic luids such as paints, coating, inks, and jet uels. The relationship beteen viscoelasticity and drop ormation aimed at the production o mono-disperse colloidal sized droplets used the same approach as ink jet printing and particle production. More to the point, the mixed convection (combined orced and ree convection) lo ith and ithout mass transer occurs in many technological and industrial applications such as solar central receivers exposed to ind currents, nuclear reactors cooled during emergency shutdon, heat exchangers placed in lovelocity environments, boundary-layer control on airoil, lubrication o ceramic machine parts and ood processing. Mixed convection los arise hen the ree stream, inertial and near all buoyant orces have strong eects on the resulting convective heat transport. Thus, the study o mixed convection o viscoelastic boundary layer lo problems is important due to the strong applications in real lie. The result or output o this research ill enhanced the understanding o the luids lo phenomena and improved the development o related industries, or example the manuacturing industries. Besides that, the generation o eicient algorithm o the viscoelastic problem ill help in solving the problem o omputational Fluid Dynamics in uture 1.6 Outline o Dissertation This dissertation consists o six chapters; hapter 1 discusses the background o research, the problem statement, objectives, scope and signiicance o the study. The literature revie or the research problem is given in hapter 2. Next, in hapter 3 e ill discuss the constitutive equation o the mixed convection boundary layer lo o a viscoelastic luid in sphere. The important o

18 585 viscoelastic term also discuss in this chapter. Then, e proceed to our mathematical ormulation in hapter 4. Further, hapter 5 includes the result and discussion o a problem. The results are presented both in the orm o tables and graphs. Lastly, hapter 6 contains a summary o the dissertation and recommendation or uture research.