UNSTEADY MAGNETOHYDRODYNAMICS FLOW OF A MICROPOLAR FLUID WITH HEAT AND MASS TRANSFER AURANGZAIB MANGI UNIVERSITI TEKNOLOGI MALAYSIA

Transcription

1 UNSTEADY MAGNETOHYDRODYNAMICS FLOW OF A MICROPOLAR FLUID WITH HEAT AND MASS TRANSFER AURANGZAIB MANGI UNIVERSITI TEKNOLOGI MALAYSIA

2 UNSTEADY MAGNETOHYDRODYNAMICS FLOW OF A MICROPOLAR FLUID WITH HEAT AND MASS TRANSFER AURANGZAIB MANGI A thesis submitted in ulilment o the requirements or the award o the degree o Doctor o Philosophy (Mathematics) Faculty o Science Universiti Teknologi Malaysia MAY 213

3 To My Beloved Family iii

4 iv ACKNOWLEDEMENT First, and oremost, All Almighty Allah SWT is with me at each step and I am very thankul o His Great Gratitude. I oer my humblest, sincerest and billons o Darood to Holy Prophet Hazrat Muhammad (PBUH) who exhorts his ollowers to seek knowledge rom cradle to grave. I would like to sincere thanks to my honorable and devoted supervisor Associate Proessor Dr. Sharidan Shaie or his supervision and valuable suggestion and extraordinary eorts rom beginning o my study. Without his eorts it is not possible or me to complete this diicult task. Lastly and most importantly, I thank my amily or love and encouragement throughout the whole period o my study.

5 v ABSTRACT The unsteady boundary layer low has become o great interest in the ield o luid mechanics including the area o convective double diusion. This is due to the complexity o the problem by including extra independent time variable, especially in the study o magnetohydrodynamic low immersed in a micropolar luid. In this thesis, the unsteady two-dimensional laminar boundary layer and mixed convection stagnation point low towards a stretching or shrinking sheet immersed in magnetohydrodynamic micropolar luid are considered. Speciic problems are considered with dierent eects such as Soret and Duour eects, thermophoresis eect and slip eect. Along with these eects, the micropolar parameter, the magnetic parameter and the suction or injection parameter are also considered. The governing non-linear equations are transormed into a system o dierential equations by using appropriate non-dimensional variables which are then solved numerically using an implicit inite dierence scheme. Numerical results or the skin riction, the Nusselt number and the Sherwood number as well as the velocity, microrotation, temperature and concentration proiles or dierent physical parameters are presented graphically and in tabular orm. The results obtained show that there is a smooth transition rom small time solution to large time solution. It is also ound that with an increase o Soret and Duour numbers, the momentum boundary layer thickness increases whereas the microrotation boundary layer thickness decreases or assisting low while a reverse trend is observed or opposing low. The thermal and concentration boundary layer thicknesses increase in both cases. By increasing the values o the slip parameter, all the boundary layer thicknesses decrease. In addition, by increasing the values o thermophoresis, the concentration boundary layer thickness decreases.

6 vi ABSTRAK Aliran lapisan sempadan tak mantap telah menjadi suatu kajian yang amat menarik di dalam bidang mekanik bendalir termasuklah juga bidang resapan kembar berolak. Ini disebabkan oleh penambahan pembolehubah tak bersandar masa yang menjadikan masalah ini semakin rumit, terutamanya di dalam kajian aliran hidrodinamik magnet di dalam bendalir mikropolar. Di dalam tesis ini, aliran lapisan sempadan lamina dan olakan campuran titik genangan dua matra tak mantap ke arah kepingan meregang atau mengecut di dalam bendalir mikropolar hidrodinamik magnet dipertimbangkan. Masalah yang dipertimbangkan melibatkan pelbagai kesan, khusususnya kesan Soret dan Duour, kesan termooresis dan kesan gelincir. Bersama dengan kesan ini, parameter mikropolar, parameter magnetik dan parameter sedutan atau suntikan juga dipertimbangkan. Persamaan menakluk tak linear diubah ke sistem persamaan pembezaan dengan menggunakan pembolehubah tak bermatra yang bersesuaian yang kemudiannya diselesaikan secara berangka menggunakan skim beza terhingga tersirat. Keputusan berangka bagi geseran kulit, nombor Nusselt dan nombor Sherwood beserta proil halaju, mikroputaran, suhu dan kepekatan bagi pelbagai parameter izikal yang berbeza dipersembahkan secara graik dan berjadual. Keputusan yang diperoleh menunjukkan adanya peralihan yang lancar daripada penyelesaian pada masa kecil kepada penyelesaian pada masa besar. Keputusan juga menunjukkan bahawa dengan meningkatnya nombor Soret dan Duour, ketebalan lapisan sempadan momentum meningkat sedangkan ketebalan lapisan sempadan mikroputaran berkurangan bagi aliran berbantu manakala keadaan sebaliknya berlaku bagi aliran bertentang. Ketebalan lapisan sempadan terma dan kepekatan diperhatikan meningkat untuk kedua-dua kes. Peningkatan nilai parameter gelincir menyebabkan ketebalan kesemua lapisan sempadan berkurangan. Di samping itu, peningkatan nilai termooresis juga mengakibatkan ketebalan lapisan sempadan kepekatan berkurangan.

7 vii TABLE OF CONTENTS CHAPTER TITLE PAGE DECLARATION DEDICATION ACKNOWLEDEMENT ABSTRACT ABSTRAK TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES LIST OF SYMBOLS LIST OF APPENDICES ii iii iv v vi vii x xii xx xxv 1 INTRODUCTION Research Background Problem Statements Objectives and Scope Signiicance o Study Thesis Outline 8 2 LITERATURE REVIEW Introduction Flow over a Stretching Sheet in a Micropolar Fluid Flow over a Shrinking Sheet in a Micropolar Fluid Micropolar Fluid in a Porous Medium 2

8 viii 3 MATHEMATICAL FORMULATION Introduction Governing Equations Boussinesq Approximation Non-dimensional Equations Boundary Layer Approximation Non-similar Transormation Physical Quantities 38 4 THERMOPHORESIS AND SUCTION OR INJECTION 4 EFFECTS ON THE STAGNATION POINT FLOW TOWARDS A HORIZONTAL SHEET 4.1 Introduction Governing Equations Non-dimensional Equations Non-similar Transormation Solution Procedure Results and Discussions 46 5 THERMOPHORESIS AND SLIP EFFECTS ON THE 64 STAGNATION POINT FLOW TOWARDS A SHRINKING SHEET 5.1 Introduction Governing Equations Non-similar Transormation Results and Discussions 67 6 MIXED CONVECTION STAGNATION POINT FLOW 85 TOWARDS A PERMEABLE SHRINKING SHEET WITH SLIP EFFECT 6.1 Introduction Governing Equations Non-dimensional Equations and Boundary Layer Approximation 87

9 ix 6.4 Non-similar Transormation Results and Discussions 9 7 MIXED CONVECTION STAGNATION POINT FLOW 17 TOWARDS A STRETCHING SHEET IN A POROUS MEDIUM WITH SORET AND DUFOUR EFFECTS 7.1 Introduction Governing Equations Non-dimensional Equations and Boundary Layer Approximation Non-similar Transormation Results and Discussions CONCLUSION Summary o Research Suggestion or Future Research 136 REFERENCES 138 Appendices A-C

10 x LIST OF TABLES TABLE NO. TITLE PAGE 4.1 Comparison o values o the skin riction o inal steady-state low ( 1) or various values o when K M n Results o the skin riction or various values o K 1, M 1, 1, n.5,.6 with dierent step sizes The skin riction, the Nusselt number and the Sherwood number or various values o and when K 1, M 1,Pr.71, Sc.94,.2, n Comparison o the values o ''() or stretching sheet in the absence o concentration, thermophoresis and slip parameter Comparison o the values o ''() or shrinking sheet when K in the absence o concentration, thermophoresis and slip parameter The reduced skin riction, the reduced Nusselt number and the reduced Sherwood number or various values o and when K 1, M 1,Pr.71, Sc.94,.2, n, The reduced skin riction, the reduced Nusselt number and the reduced Sherwood number or various values o and when K 1, M 1,Pr.71, Sc.94,.2, n, Comparison o the values o ''() or stretching sheet in the absence o concentration, magnetic parameter M and slip parameter Comparison o the values o ''() or shrinking sheet when K in the absence o concentration, magnetic parameter

11 xi M and slip parameter The reduced skin riction, the reduced Nusselt number and the reduced Sherwood number or various values o, and when K 1, M 1,Pr.71, Sc.94, n,.2, 1, The reduced skin riction, the reduced Nusselt number and the the reduced Sherwood number or various values o, and when K 1, M 1,Pr.71, Sc.94, n,.75, 1, / Comparison o the values o C Re and x x Nu / Re or dierent values o Pr in absence o concentration, Soret S and Duour D eects when 1 and r 1/ Comparison o the values o C Re and x x x 1/ 2 x Nu / Re or dierent values o Pr in absence o concentration, Soret S and Duour eects D when 1 and r 1/ Values o C Re, x x Nu / Re and x 1/ 2 x x 1/ 2 x 1/ 2 x Sh / Re or dierent values o and MD when K 1,Pr.71, Sc.94, n, 1, Sr.2, D.2, 1,.6 or assisting and opposing low 118 1/ Values o C Re, x values o and x r Nu / Re and x 1/ 2 x x Sh / Re or dierent x 1/ 2 x S when K 1, MD 1.5, Pr.71, Sc.94, n.5, 1, D.2, 1,.6 and opposing low 119 or assisting 1/ Values o C Re, x values o and x Nu / Re and x 1/ 2 x Sh / Re or dierent x 1/ 2 x D when K 1, MD 1.5, Pr.71, Sc.94, n.5, 1, Sr.2, 1,.6 and opposing low 12 or assisting

12 xii LIST OF FIGURES FIGURE NO. TITLE PAGE 4.1 Physical model and coordinate system The velocity proiles or various values o when K 1, M 1, n.5 (a) suction and (b) injection The microrotation proiles or various values o when K 1, M 1,.5 1 and (b) injection 1 51 n (a) suction 4.4 The temperature proiles or various values o when K 1, M 1, Pr.71, Sc.94, n.5 1 and (b) injection 1 52 (a) suction 4.5 The concentration proiles or various values o when K 1, M 1, Pr.71, Sc.94,.2, n.5 (a) suction 1 and (b) injection The velocity proiles or various values o when K 1, M 1, and (b) injection 54 n (a) suction The microrotation proiles or various values o when K 1, M 1, 1 and (b) injection 1 55 n (a) suction 4.8 The temperature proiles or various values o when K 1, M 1, Pr.71, Sc.94, n 1 and (b) injection 1 56 (a) suction

13 xiii 4.9 The concentration proiles or various values o when K 1, M 1, Pr.71, Sc.94,.2, n (a) suction 1 and (b) injection The velocity proiles or various values o when K 1, M 1, n, The microrotation proiles or various values o when K 1, M 1, n, The temperature proiles or various values o when K 1, M 1, Pr.71, Sc.94, n,.4 59 The concentration proiles or various values o when K 1, M 1, Pr.71, Sc.94,.2, n, The velocity proiles or various values o K or suction or injection when M 1, n, The microrotation proiles or various values o K or suction or injection when M 1, n, The concentration proiles or various values o or suction or injection when K 1, M 1, Pr.71, Sc.94, n, The velocity proiles or various values o M or suction or injection when K 1, n, The microrotation proiles or various values o M or suction suction or injection when K 1, n, Variation o the skin riction with or dierent values o suction or injection when K 1, M 1, n Variation o the Nusselt number with or dierent values o suction or injection when K 1, M 1, Pr.71, Sc.94, n Variation o the Sherwood number with or dierent values o suction or injection when K 1, M 1, Pr.71, Sc.94,.2, n Physical model and coordinate system 65

14 xiv 5.2 The velocity proiles or various values o when K 1, M 1, n,.2, The microrotation proiles or various values o when K 1, M 1, n,.2, The temperature proiles or various values o when K 1, M 1, Pr.71, Sc.94, n, The concentration proiles or various values o when K 1, M 1, Pr.71, Sc.94,.2, n,.2, The velocity proiles or various values o when K 1, M 1, n,.75, The microrotation proiles or various values o when K 1, M 1, n,.75, The temperature proiles or various values o when K 1, M 1, Pr.71, Sc.94, n,.75, The concentration proiles or various values o when K 1, M 1, Pr.71, Sc.94,.2, n,.75, The velocity proiles or various values o when K 1, M 1,.75,.2 or (a) n.5 and (b) n The microrotation proiles or various values o when K 1, M 1,.75,.2 or (a) n.5 and (b) n The temperature proiles or various values o when K 1, M 1, Pr.71, Sc.94,.75,.2 or (a) n.5 and (b) n The concentration proiles or various values o when K 1, M 1, Pr.71, Sc.94,.2,.75,.2 or (a) n.5 and (b) n The concentration proiles or various values o when K 1, M 1, Pr.71, Sc.94, n,.75,.2, Variation o the reduced Sherwood number with or dierent values o when K 1, M 1,Pr.71, Sc.94, n,.75,.2 81

15 xv 5.16 Variation o the reduced skin riction with or dierent values o when K 1, M 1, n, Variation o the reduced Nusselt number with or dierent values o when K 1, M 1,Pr.71, Sc.94,.2, n, Variation o the reduced Sherwood number with or dierent values o when K 1, M 1,Pr.71, Sc.94, n, Variation o the reduced skin riction with or dierent values o when K 1, M 1, n, Variation o the reduced Nusselt number with or dierent values o when K 1, M 1,Pr.71, Sc.94, n, Variation o the reduced Sherwood number with or dierent values o when K 1, M 1,Pr.71, Sc.94,.2, n, Physical model and coordinate system The velocity proiles or various values o when K 1, M 1, Pr.71, Sc.94, n, 1,.75,.2, 1, The microrotation proiles or various values o when K 1, M 1, Pr.71, Sc.94, n, 1,.75,.2, 1, The temperature proiles or various values o when K 1, M 1, Pr.71, Sc.94, n, 1,.75,.2, 1, The concentration proiles or various values o when K 1, M 1, Pr.71, Sc.94, n, 1,.75,.2, 1, The velocity proiles or various values o when K 1, M 1, Pr.71, Sc.94, n, 1,.2, 1, The microrotation proiles or various values o when K 1,

16 xvi M 1, Pr.71, Sc.94, n, 1,.2, 1, The temperature proiles or various values o when K 1, M 1, Pr.71, Sc.94, n, 1,.2, 1, The concentration proiles or various values o when K 1, M 1, Pr.71, Sc.94, n, 1,.2, 1, The velocity proiles or various values o M when K 1, Pr.71, Sc.94, n, 1,.75,.2, 1, The microrotation proiles or various values o M when K 1, Pr.71, Sc.94, n, 1,.75,.2, 1, The temperature proiles or various values o M when K 1, Pr.71, Sc.94, n, 1,.75,.2, 1, The concentration proiles or various values o M when K 1, Pr.71, Sc.94, n, 1,.75,.2, 1, The velocity proiles or various values o K when M 1, Pr.71, Sc.94, n, 1,.75,.2, 1, The microrotation proiles or various values o K when M 1, Pr.71, Sc.94, n, 1,.75,.2, 1, The temperature proiles or various values o K when M 1, Pr.71, Sc.94, n, 1,.75,.2, 1, The concentration proiles or various values o K when M 1, Pr.71, Sc.94, n, 1,.75,.2, 1, Variation o the reduced skin riction with or dierent values o and when K 1, M 1,Pr.71, Sc.94, n,.2, 1,.6 14

17 xvii 6.19 Variation o the reduced Nusselt number with or dierent values o and when K 1, M 1, Pr.71, Sc.94, n,.2, 1, Variation o the reduced Sherwood number with or dierent values o and when K 1, M 1,Pr.71, Sc.94, n,.2, 1, Variation o the reduced skin riction with or dierent values o and when K 1, M 1, Pr.71, Sc.94, n,.75, 1, Variation o the reduced Nusselt number with or dierent values o and when K 1, M 1, Pr.71, Sc.94, n,.75, 1, Variation o the reduced Sherwood number with or dierent values o and when K 1, M 1,Pr.71, Sc.94, n,.75, 1, Physical model and coordinate system The velocity proiles or various values o when K 1, MD 1.5, Pr.71, Sc.94, n.5, 1, S.2, D.2, 1, The microrotation proiles or various values o when K 1, MD 1.5, Pr.71, Sc.94, n.5, 1, S.2, D.2, 1, The temperature proiles or various values o when K 1, MD 1.5, Pr.71, Sc.94, n.5, 1, S.2, D.2, 1, The concentration proiles or various values o when K 1, MD 1.5, Pr.71, Sc.94, n.5, 1, S.2, D.2, 1, The velocity proiles or various values o S r when K 1, MD 1.5, Pr.71, Sc.94, n.5, 1, D.2, 1, The microrotation proiles or various values o S r when K 1, r r r r

18 xviii MD 1.5, Pr.71, Sc.94, n.5, 1, D.2, 1, The temperature proiles or various values o S r when K 1, MD 1.5, Pr.71, Sc.94, n.5, 1, D.2, 1, The concentration proiles or various values o S r when K 1, MD 1.5, Pr.71, Sc.94, n.5, 1, D.2, 1, The velocity proiles or various values o D when K 1, MD 1.5, Pr.71, Sc.94, n.5, 1, S.2, 1, r The microrotation proiles or various values o D when K 1, MD 1.5, Pr.71, Sc.94, n.5, 1, S.2, 1, r The temperature proiles or various values o D when K 1, MD 1.5, Pr.71, Sc.94, n.5, 1, S.2, 1, r The concentration proiles or various values o D when K 1, MD 1.5, Pr.71, Sc.94, n.5, 1, S.2, 1, r Variation o the skin riction with or dierent values o MD when K 1, M 1,Pr.71, Sc.94, n.5, 1, S r.2, D.2, 1, Variation o the Nusselt number with or dierent values o MD when K 1, M 1,Pr.71, Sc.94, n.5, 1, S.2, D.2, 1, r 7.16 Variation o the Sherwood number with or dierent values o MD when K 1, M 1,Pr.71, Sc.94, n.5, 1, S.2, D.2, 1, r 7.17 Variation o the skin riction with or dierent values o S r when K 1, MD 1.5,Pr.71, Sc.94, n.5, 1, D.2, 1,.6 128

19 xix 7.18 Variation o the Nusselt number with or dierent values o S when K 1, MD 1.5,Pr.71, Sc.94, n.5, 1, r D.2, 1, Variation o the Sherwood number with or dierent values o S r when K 1, MD 1.5,Pr.71, Sc.94, n.5, 1, D.2, 1, Variation o the skin riction with or dierent values o D when K 1, MD 1.5,Pr.71, Sc.94, n.5, 1, S.2, 1,.6 13 r 7.21 Variation o the Nusselt number with or dierent values o D when K 1, MD 1.5,Pr.71, Sc.94, n.5, 1, S.2, 1,.6 13 r 7.22 Variation o the Sherwood number with or dierent values o D when K 1, MD 1.5,Pr.71, Sc.94, n.5, 1, S.2, 1, r

20 xx LIST OF SYMBOLS Roman Letters ab, - positive constants * b - induced magnetic ield B - magnetic ield vector B - externally imposed magnetic strength in the y direction c s - concentration susceptibility C - luid concentration C - nondimensional luid concentration C - external concentration C w - surace concentration C p - speciic heat at constant pressure C x - skin riction coeicient C - concentration dierence 1 Da - inverse Darcy number D - Duour number D m - mass diusivity E - electric ield vector - non-dimensional velocity - suction or injection parameter F - body orce F x - scalar orce in x component g - magnitude o the acceleration due to gravity Gc - concentration Grasho number

21 xxi Gr - thermal Grasho number h - non-dimensional microrotation j - microinertia density J - electric current density vector k - thermal conductivity k 1 - permeability o porous medium * k 1 - vortex viscosity k T - thermal diusion ratio K - micropolar material parameter L - characteristic length L 1 - slip length m w - non-dimensional mass lux rom the surace o the wall M - magnetic parameter MD - eective Darcy number n - ratio o the microrotation vector component to the luid skin riction at the wall N - component o the microrotation vector normal to x y plane N - non-dimensional component o the microrotation vector normal to x y plane Nu x - Nusselt number O - order o magnitude p - pressure p - non-dimensional pressure p d - dynamic pressure p h - hydrostatic pressure Pr - Prandtl number q w - non-dimensional heat lux rom the surace o the wall Re - Reynolds number Sc - Schmidt number Sh x - Sherwood number

22 xxii S r - Soret number t - time t - non-dimensional time T - luid temperature T - non-dimensional luid temperature T m - mean luid temperature T r - reerence temperature T w - surace temperature T - external temperature T - temperature dierence u ( ) e x - dimensional external velocity u ( ) e x - non-dimensional external velocity u ( x ) w - dimensional velocity along the sheet u ( x ) w - non-dimensional velocity along the sheet uv, - velocity components along x, y axes uv, - non-dimensional velocity components along x and y U - reerence velocity V - dimensional velocity vector V T - dimensional thermophoretic velocity v - velocity suction or injection x, y - cartesian coordinates along the plate and normal to it, respectively xy, - non-dimensional cartesian coordinates along the wall and - gradient normal to it, respectively Greek Letters - thermal diusivity - coeicient o thermal expansion

23 xxiii * - coeicient o concentration expansion - partial derivative - spin gradient viscosity,, - transormed coordinate - slip parameter * - boundary layer thickness - non-dimensionl temperature - thermal conductivity * - thermophoretic coeicient - dynamic viscosity m - magnetic permeability - kinematic viscosity - density - luid density in the ambient medium - shrinking parameter * - convergence tolerance - non-dimensionl concentration - stream unction - thermophoretic parameter w - non-dimensional wall shear stress or skin riction * - concentration buoyancy parameter - mixed convection parameter - electrical conductivity - velocity ratio parameter - buoyancy ratio Subscripts,e - ar ield or ree stream condition w - wall condition d - dynamic pressure

24 xxiv p - constant pressure condition Superscripts k - number o iteration ' - dierentiation with respect to

25 xxv LIST OF APPENDICES APPENDIX TITLE PAGE A The Keller-Box Method 15 B FORTRAN Program or the Problem o the Thermophoresis and Suction or Injection Eects on the Stagnation Point Flow towards a Horizontal Sheet 164 C Status o Publications 18

26 CHAPTER 1 INTRODUCTION 1.1 Research Background The large volume o published studies on the boundary layer low indicates the importance o the subject in engineering applications, such as hot rolling, skin riction drag reduction, grain storage, glass iber and paper production. The importance o unsteady boundary layer low is increasing in the ield o luid mechanics especially in the area o convective double diusion. One o the main reason or such a importance is result o the complexity o the problem by including extra independent time variable. Generally, the ideal low environment around the device is steady, although there are numerous situations or instances sel-induced motions o the body, luctuations, non-uniormities in the surrounding luid, where undesirable unsteady eects arises. The study o unsteady boundary layer owes its importance to the act that all the boundary layers, are, in sense, unsteady. Unsteady lows are generally observed in technological and environmental situations. Some o these observed in geophysical and biological lows, the processing o the materials, and in the spread o pollutants and ires. In the same vein, notable serious discussion on the unsteady boundary layer low were previously carried out by researcher s like Riley (1975) and Telionis (1981). The study o luids uses two lenses to explain the concept at length. The two categories are Newtonian and non-newtonian. The Newtonian luid indicates that when the luids shear stress is directly and linearly proportional to the rate o angular deormation, whereas in the non-newtonian luid, shearing stress is not related to the

27 2 rate o shearing strain. The equations which govern the low o Newtonian luid are the Navier-Stokes equations. There are ew exceptions where Newtonian luid ails to describe the properties o the luid, these includes industrial colloids luid, polymeric suspensions, liquid crystals and dust in air and blood low in arteries and capillaries. Limitation in the explanation o these luids led development o theories on non-newtonian luids. A classic example o the non-newtonian luids, that became centre o researcher s attention, is o micropolar luid. Eringen (1964) earlier developed the luid mechanics o deormable microelements, which were termed as Simple Microluids. Eringen deined the simple microluid as: A viscous medium whose behavior and properties are aected by the local motion o particles in its microvolume. These luids are characterized by 22 viscosity and material constants and when applied to low problems the result is a system o 19 partial dierential equations with 19 unknown that may not be amenable to be solved. Eringen (1966) subsequently introduced a subclass o luids which he named micropolar luids that ignores the deormation o the microelements but still allows or the particle micromotion to take place. The theory o micropolar luids, which consist o rigid, randomly oriented particles suspended in a viscous medium. These special eatures o micropolar luids were discussed comprehensively by Ariman et al. (1973). In general, as part o the momentum is lost in rotating o particles, the low o a micropolar luid is less prone to instability than that o a classical luid. The stability o micropolar luids problems have been investigated by Lakshmana Rao (197) as well as Sastry and Das (1985). The study o micropolar luid with heat and mass transer has many engineering applications. These applications include rerigerator coils, power generators, metal and plastic extrusion, paper production, crystal growing, electric transormers and transmission lines. Eringen (21) earlier demonstrated the adequacy o applying micropolar luid theory in order to describe the liquid crystal behavior. Eringen in-addition indicated, other possible substances that may be modeled by micropolar luid, these are magnetic luids, clouds with dust, anisotropic luids and biological luids. It seems worth to mention importance o the study o

28 3 non-newtonian luid as a result o the behavior that is not described by the Newtonian relationships. Recent evidences highlight increasing interest o researchers in the stagnation point low in micropolar luid. A review o luid dynamics history reveals stagnation point lows as one o the unique issue o the ield. These problems can take any orm, such as steady or unsteady, viscous or inviscid, two dimensional or threedimensional, orward or reverse, and normal or oblique. The impetus or studying convective lows near the stagnation point region is due to the act that the heat transer is maximum at the stagnation point. Similarly, Sharidan (25) urther added that solutions at stagnation point may also serve as a starting solution or the solution over the entire body. The study o dynamics o electrically conducting luid is known as magnetohydrodynamics (MHD). The study o MHD low o an electrically conducting luid is o considerable interest in modern metallurgical and metalworking processes. There has been a great interest in the study o MHD low with heat and mass transer in any medium. This is largely, because o, the eect o magnetic ield on the boundary layer low control and on the perormance o many systems using electrically conducting luids. This type o low has attracted the interest o numerous researchers. One o the most signiiciant reason o this importance is its applications in engineering problems such as MHD generators, plasma studies, nuclear reactors, geothermal energy extractions. However as a result o the application o magnetic ield, hydromagnetic techniques are used or the extraction o pure molten metals rom non-metallic inclusions. This is why; the type o problems that are dealing with is very useul or polymer technology and metallurgy. Despite o the importance o MHD micropolar luid low near the stagnation point, recent eorts in this regard help identiy new eects on the low such as Soret and Duour eects, slip eect and thermophoresis eect. Critical review o thermophoresis causes small particles to deposit on cold suraces. Thermophoresis plays a central role in the iber optical synthesis. Kishan and Maripala (212) ound

29 4 that this importance is the result o its identiication as the principal mechanism o mass transer which used in the technique o modiied chemical vapor deposition (MCVD). Similarly, the mathematical modeling o the deposition o silicon thin ilms, using MCVD methods, has been accelerated by the quality control measures enorced by the micro-electronics industry. These topics involve variety o complex luid dynamical processes including thermophoretic transport o particles deposits, heterogeneous/homogenous chemical reactions, homogenous particulate nucleation and coupled heat and energy transer. Other notable example relating to thermophoresis is the blackening o glass globe o kerosene lanterns, chimneys and industrial urnace walls by carbon particles, corrosion o heat exchanger, which reduces heat transer coeicient, and ouling o gas turbine blades (Kandasamy et al. 21). In case o heat and mass transer, Soret and Duour eects are signiicant when the temperature and concentration gradients are highs. Thermal diusion (thermo diusion or Soret eect) corresponds to species dierentiation developing in an initial homogeneous mixture submitted to a thermal gradient (Soret, 198) while the energy lux caused by a composition gradient is called Duour (diusionthermo) eect. These eects are considered as second order phenomena, on the basis that they are o smaller order o magnitude than the eects described by Fourier s and Fick s laws, but they may become signiicant in areas such as geosciences or hydrology (Benano-Melly et al. 21). Whilst discussing the ield o luid mechanics it is worth to mention important aspects like partial slip condition. One o the important pillars on which the luid mechanics is based is the no-slip condition. Although, there are situations where the conditions role is not signiicant or it is no more valid. These are the conditions where partial slip between the luid and the moving surace may occur. Generally in case o no-slip condition it is noted that the molecule o the luid lowing near the boundary stiks with the surace. In numerous practical situations it is important to replace the no-slip condition by the partial slip condition. This is because th no-slip condition at the solid luid interace is no longer applicable when luid lows in micro electro mechanical system (MEMS). The non-equilibrium

30 5 region near the interace is more accurately described by the slip low model. In order to deal with the problem, Navier recommended general boundary condition which shows the luid slip at the surace. Navier similarly suggested that, the dierence o the luid velocity and the velocity o the boundary is proportional to the shear stress at that boundary. 1.2 Problem Statements Interest in the magnetohydrodynamic low o micropolar luid has increased substantially over the past ew decades due to the occurrence o these luids in many applications. The behavior o this low near the stagnation point towards a stretching or shrinking sheet has been studied theoretically by many researchers. The phenomenon o this luid aected by some important eects such as thermophoresis eect, suction or injection eect, slip eect, Soret and Duour eects are not yet explore but interesting to be investigated. Thereore, this research is conducted to explore the ollowing questions. What is the behavior o this luid in nature near the orward stagnation point with the eect o thermophoresis and suction or injection towards a horizontal sheet? In-addition eorts are also required to see how do the micropolar luids models compared with the Newtonian luids models on the stagnation point low towards a shrinking sheet with thermophoresis and slip eects? Apart rom these need also exists to view how are the reduced skin riction, the reduced Nusselt number and the reduced Sherwood number aected due to the presence o magnetic parameter, slip eect and mixed convection parameter on stagnation point low towards a permeable shrinking sheet? How are the skin riction, the Nusselt number and the Sherwood number aected due to the presence o Soret and Duour eects towards a stretching sheet in a porous medium?

31 6 1.3 Objectives and Scope The purpose o this study is to investigate theoretically the unsteady boundary layer low o a micropolar luid near the stagnation point towards a stretching or shrinking sheet. This involves with developing the mathematical ormulation and numerical simulation or computation, in order to calculate the low characteristics as well as analyzing the numerical results o the ollowing MHD heat and mass transer problems: 1. The thermophoresis and suction or injection eects on the stagnation point low towards a horizontal sheet; 2. The thermophoresis and slip eects on the stagnation point low towards a horizontal shrinking sheet; 3. The slip eect on the mixed convection stagnation point low towards a permeable shrinking sheet; 4. The Soret and Duour eects on the mixed convection stagnation point low towards a stretching sheet in a porous medium. This investigation examines the laminar two-dimensional incompressible low o a micropolar luid. These problems are solved numerically by using an implicit inite dierence scheme, namely Keller s box. The Keller box method was earlier introduced by Keller (197) and was used widely in solving the parabolic dierential equations. The Newton s method can be used i the dierential equations that need to be solved are nonlinear. The implicit nature o the Keller s Box method has generated a tridiagonal matrix, like other implicit method, however, the speciality o the Keller s Box method is, the entries are expressed in blocks rather than scalars. The work o Cebeci and Bradshaw (1984), Cebeci (22), Nazar (23), Sharidan (25) and Lok (28) provide details about the method. No real experiments have been conducted to validate the numerical results.

32 7 1.4 Signiicance o Study The Newtonian luids have ew limitations, one o such shortcomings is its incapability to describe some engineering and industrial processes which are made up o materials having an internal structure. The theory o micropolar luid model introduced by Eringen (1966) exhibits the local eects arising rom the microstructure and micro motion o the luid elements. The presence o smoke or dust particularly in gas may also modeled using micropolar luid dynamics. Vogel and Patterson (1964) and Hoyt and Fabula (1964) conducted experiments with luids that containing the amounts o minute polymeric additives. It was observed that the skin riction reduced near a rigid body. Gray and Hilliard (1966) in his invention, introduce relatively small amounts o a non-newtonian luid, a long-chain polymer such as polyethylene oxide, into the water adjacent the bow o the ship. This alters the shear characteristics o the luid in boundary layer o the ship which decreases the overall rictional drag o the vessel. This leads to the increasing ship speed and it decreases the required power to maintain a given vessel speed. There are some advantages in the ields o aeronautics and submarine navigation. It is becoming extremely diicult to ignore importance o micropolar luid with heat and mass transer in areas like aeronautics and submarine. Similarly it have signiiciant importance in engineering applications, such as exothermic reaction in packed-bed reactors, heat transer associated with storage o nuclear waste, cooling metallic plate in a bath and heat removal rom nuclear uel debris. Besides, the study o micropolar luids, there are other applications in several industrial and technical processes such as nuclear reactors cooled during emergency shutdown, solar central receivers exposed to wind current, electronic devices cooled by an and heat exchangers placed in a low-velocity environment. Moreover, due to the use o the micropolar luid in dierent manuacturing and processing industries, considerable attention has been given towards the understanding the important phenomena involving in heat exchange devices (Elbarbary and Elgazery, 25). This importance require scientists and engineers to

33 8 be amiliar with the low behavior and properties o such luids or the way to use such kind o properties to predict low behavior in the process equipment. Inaddition it is also beneicial to know the nature o the low, heat and mass transer o micropolar luid towards a stagnation point and the inluence o the material properties to the stagnation point heat and mass transer problems. 1.5 Thesis Outline This thesis is comproised o eight chapters. The irst chapter is introduction, which provides detail account o research background, problem statement, objectives, scope and signiicance o research. The study then moves to conduct critical review o the previous literature. Based on the eorts o chapter two in the Chapter 3 eorts was made or the mathematical ormulation. The ourth chapter is concerned with the thermophoresis and suction or injection eects on an unsteady MHD stagnation point low in a micropolar luid towards a horizontal sheet. The governing boundary layer equations are solved numerically by using Keller-box method. Graphical results presented includes velocity, microrotation, temperature and concentration proiles as well as the physical quantities, namely the skin riction, the Nusselt number and the Sherwood number, which signiicance in characterizing the heat and mass transer. Apart rom these dierent physical parameters such as MHD parameter, micropolar parameter, suction or injection parameter, thermophoresis parameter and time variable are also considered in this chapter. In-addition both weak concentration and strong concentration are considered in this chapter. The case n represents concentrated particle low in which micro-element to wall surace are unable to rotate and denote strong concentration; n.5 present the vanishing o the anti-symmetric part o the stress tensor and denote weak concentration while n 1 indicates the turbulent boundary layer lows. A FORTRAN program or the eect o thermophoresis and suction or injection on the stagnation point low towards a horizontal sheet is given in Appendix B.

34 9 In Chapter 5, the unsteady MHD boundary layer low near the stagnation point on a shrinking sheet with thermophoresis and slip eects is considered. Based on the explanation provided in Chapter 4, both weak n.5 concentration and strong n concentration are considered. Graphs are plotted and discussed or various emerging parameters such as slip parameter, shrinking parameter, thermophoresis parameter, micropolar parameter and time variable. The study, in Chapter 6, then looks at the numerical solution o the unsteady MHD mixed convection low near the stagnation point on a shrinking sheet with slip eect. The eect o mixed convection parameter which is involved in the momentum equation is studied or both assisting and opposing lows. The mixed convection parameter is a measure o the relative importance o ree convection in relation to orced convection. When 1, the ree and orced convection are o the same order o magnitude. I 1, low is primary by orced convection while 1 Khramtsov, 25). ree convection become dominant (Martynenko and In Chapter 7, the unsteady MHD mixed convection low near the stagnation point in micropolar luid on a stretching sheet in a porous medium with Soret and Duour eects is considered. As in the Chapter 6, both cases o assisting and opposing lows are also considered. The novel aspect o this study is the ocus on the porosity and Soret and Duour eects. The numerical results have been plotted or the indispensable dimensionless parameters to show the inluences on the velocity, microrotation, temperature as well as the three physical quantities. Finally, the summary o this thesis and the suggestions or uture research are given in Chapter 8. The list o the publication and current status are given in Appendix C.

35 138 REFERENCES Abo-Eldahab, E.M. and Ghonaim, A.F. (25). Radiation eect on heat transer o a micropolar luid through a porous medium. Appl. Mathematics Computation 169: Ali, F.M., Nazar, R., Ariin, N.M. and Pop, I. (211). Unsteady low and heat transer past an axisymmetric permeable shrinking sheet with radiation eect. Int. J. Numer. Meth. Fluids 67: Ariman, T., Turk, M.A. and Sylvester, N.D. (1973). Microcontinuum Fluid Mechanics A Review Int. J. Eng. Sci. 11: Ashra, M. and Bashir, S. (211). Numerical simulation o MHD stagnation point low and heat transer o a micropolar luid towards a heated shrinking sheet. Int. J. Num. Meth. Fluids DOI: 1.12/ld Ashra, M. and Shahzad, A. (211). Radiation eects on MHD boundary layer stagnation point low towards a heated shrinking sheet. World Appl. Sci. J. 13(7): Attia, H.A. (26). Heat transer in a stagnation point low o a micropolar luid over a stretching surace with heat generation/absorption. Tamkang J. Science Eng. 9(4): Attia, H.A. (28). Stagnation point low and heat transer o a micropolar luid with uniorm suction or blowing. J. Braz. Soc. Mech. Sci. Eng. XXX(1): Aziz, A. (29). A similarity solution or laminar thermal boundary layer over a lat plate with a convective surace boundary condition. Commun. Nonlinear Sci. Num. Simul. 14: Bayley, F.J., Owen, J.M. and Turner, A.B. (1972). Heat Transer. London: Thomas Nelson and Sons Ltd. Bejan, A. (24). Convection Heat Transer. 3 rd edition. New York: John Wiley.

36 139 Benano-Melly, L.B., Caltagirone, J.P., Faissat, B., Montel, F. and Costesque, P. (21). Modelling Soret coeicient measurement experiments in porous media considering thermal and solutal convection. Int. J. Heat Mass Transer 44: Bhargava, R., Kumar, L. and Takhar, H.S. (23). Finite element solution o mixed convection micropolar low driven by a porous stretching sheet. Int. J. Eng. Sci. 4: Bhargava, R., Sharma, S., Takhar, H.S., Bég, O.A. and Bhargava, P. (27). Numerical solutions or micropolar transport phenomena over a nonlinear stretching sheet. Nonlinear Analysis: Modelling Control 12(1): Bhargava, R. and Takhar, H.S. (2). Numerical study o heat transer characteristics o the micropolar boundary layer near a stagnation point on a moving wall. Int. J. Eng. Sci. 38: Bhattacharyya, K. and Layek, G.C. (211). Eects o suction/blowing on steady boundary layer stagnation-point low and heat transer towards a shrinking sheet with thermal radiation. Int. J. Heat Mass Transer 54: Bhattacharyya, K. Mukhopadhyay, S., Layek, G.C. Pop, I. (212). Eects o thermal radiation on micropolar luid low and heat transer over a porous shrinking sheet. Int. J. Heat Mass Transer 55: Bhattacharyya, K. and Vajravelu, K. (212). Stagnation-point low and heat transer over an exponentially shrinking sheet. Commun. Nonlinear Sci. Numer. Simulat. 17: Bird, R.B., Stewart, W.E. and Lightoot, E.N. (26). Transport Phenomena. Second Edition: Wiley India Edition. Cebeci, T. (22). Convective Heat Transer. Caliornia: Horizon Publishing Inc. Cebeci, T. and Bradshaw, P. (1988). Physical and Computational Aspects o Convective Heat Transer. New York: Springer. Chamkha, A.J. and Camille, I. (2). Eects o heat generation/absorption and the thermophoresis on hydromagnetic low with heat and mass transer over a lat plate. Int. J. Numerical Methods Heat Fluid Flow 1(4): Chen, C.K. and Hsu, T.H. (1991). Heat transer o a thermomicropolar luid past a porous stretching sheet. Computers Math. Applic. 21(8):

37 14 Das, K. (212). Slip eects on MHD mixed convection stagnation point low o a micropolar luid towards a shrinking vertical sheet. Comp. Math. Appli. 63: El-Arabawy, H.A.M. (23). Eect o suction/injection on the low o a micropolar luid past a continuously moving plate in the presence o radiation. Int. J. Heat Mass Transer 46: Elbashbeshy, E.M.A. and Aldawody, D.A. (21). Heat transer over an unsteady stretching surace with variable heat lux in the presence o a heat source or sink. Comp. Math. Appl. 6: Elbarbary, E.M.E. and Elgazery, N.S. (25). Flow and heat transer o a micropolar luid in an axisymmetric stagnation low on a cylinder with variable properties and suction (Numerical Study). Acta Mechanica 176: Eringen, A.C. (1964). Simple microluids. Int. J. Eng. Sci. 2: Eringen, A.C. (1966). Theory o Micropolar luid. Journal Mathematics Mechanics 16: Eringen, A.C. (21). Microcontinuum Field Theories. II: Fluent Media. New York, Springer. Fan, T., Hang X. and Pop, I. (21). Unsteady stagnation low and heat transer towards a shrinking sheet. Int. Commun. Heat Mass Transer 37: Fang, T. and Zhang, Ji. (29). Closed-orm exact solutions o MHD viscous low over a shrinking sheet. Commun Nonlinear Sci Numer Simulat. 14: Fang, T.G., Zhang, Ji. and Yao, S.S. (29). Viscous low over an unsteady shrinking sheet with mass transer. CHIN. PHYS. LETT. 26(1): Fang, T.G., Tao, H. and Zhong, Y.F. (212). Non-Newtonian power-law luid low over a shrinking sheet. CHIN. PHYS. LETT. 29(11): Gorla, R.S.R. (1983). Micropolar boundary layer low at a stagnation point on a moving wall. Int. J. Eng. Sci. 21: Gorla, R.S.R. and Gorla, P. (1996). Unsteady planar stagnation point heat transer in micropolar luids. Can. J. Phys. 74: Gray, W.O. and Hilliard, B.A. (1966). U.S. Patent No. 3, 289, 623. Washington DC: United States Patent Oice. Guram, G.S. and Smith, A.C. (198). Stagnation lows o micropolar luids with strong and weak interactions. Comp. Maths. Appls. 6:

38 141 Hassanien, I.A. and Gorla, R.S.R. (199). Combined orced and ree convection in stagnation lows o micropolar luids over vertical non-isothermal suraces. Int. J. Eng. Sci. 28: Hayat, T., Abbas, Z. and Javed, T. (28). Mixed convection low o a micropolar luid over a non-linearly stretching sheet. Physics Letters A 372: Hayat, T., Javed, T. and Abbas, Z. (29). MHD low o a micropolar luid near a stagnation-point towards a non- linear stretching surace. Nonlinear Analysis: Real World Appls. 1(3): Hayat, T., Hussain, M., Hendi, A.A. and Nadeem, S. (212a). MHD stagnation point low towards heated shrinking surace subjected to heat generation/ absorption. Appl. Math. Mech. -Engl. Ed. 33(5): Hayat, T., Awais, M. and Alsaedi, A. (212b). Newtonian heating and magnetohydrodynamic eects in low o a Jeery luid over a radially stretching surace. Int. J. Physical Sci. 7(21): Hayat, T. and Qasim, M. (21). Inluence o thermal radiation and Joule heating magnetohydrodynamics low o a Maxwell luid in the presence o thermophoresis. Int. J. Heat Mass Transer 53: Heruska, M.W., Watson, L.T. and Kishore, K.S. (1986). Micropolar low past a stretching sheet. Computers Fluids 14(2): Hoyt, J.W. and Fabula, A.G. (1964). The eect o additives on luid riction. US Naval Ordinance Test Station Report. Hsiao, K.L. (212). Multimedia physical eature or unsteady MHD mixed convection viscoelastic luid over a vertical stretching sheet with viscous dissipation. Int. J. Physical Sci. 7(17): Ibrahim, F.S., Hassanien, I.A. and Bakr, A.A. (24). Unsteady magnetohydrodynamic micropolar luid low and heat transer over a vertical porous plate through a porous medium in the presence o thermal and mass diusion with a constant heat source. Canadian J. Phys. 82(1): Itikhar, A. (28). Solution o some unsteady lows over a stretching sheet using homotopy analysis method. Quaid-i-Azam University: Ph.D Thesis. Ishak, A. (21). Similarity solution or low and heat transer over a permeable surace with convective boundary condition. Appl. Math. Comp. 217:

39 142 Ishak, A. and Nazar, R. (21). Eects o suction or injection on the stagnation point low over a stretching sheet in a micropolar luid. Proc. 2 nd Int. Con. Mathematical Sciences 1-7. Ishak, A., Nazar, R. and Pop, I. (28a). Mixed convection stagnation point low o a micropolar luid towards a stretching sheet. Meccanica 43: Ishak, A., Nazar, R. and Pop, I. (28b). Magnetohydrodynamic (MHD) low o a micropolar luid towards a stagnation point on a vertical surace. Comput. Math. Appls. 56: Ishak, A., Lok, Y. Y. and Pop, I. (21). Stagnation-point low over a shrinking sheet in a micropolar luid. Chem. Eng. Comm. 197: Ishak, A., Yacob, N.A. and Bachok, N. (211). Radiation eects on the thermal boundary layer low over a moving plate with convective boundary condition. Meccanica 46: Jadidi, M., Moallemi, N. Shaieenejad, I. and Alaee, J. (211). An analytical approximation o stagnation point low and heat transer o a micropolar luid in a porous medium. Adv. Theor. Appl. Mech. 4(2): Jena, S.K. and Mathur, M.N. (1981). Similarity solution or laminar ree convection low o a thermomicropolar luid past a nonisothermal lat plate. Int. J. Eng. Sci. 19: Joshi, N., Kumar, M. and Saxena, P. (21). Chemical reaction in steady mixed convection MHD viscous Flow over shrinking sheet. Int. J. Stability Fluid Mech. 1(1): Kandasamy, R., Muhaimin, I. and Hashim B.S. (21). Lie group analysis or the eect o temperature-dependent luid viscosity with thermophoresis and chemical reaction on MHD ree convective heat and mass transer over a porous stretching surace in the presence o heat source/sink. Commun. Nonlinear Sci. Numer. Simulat. 15: Katagiri, M. (1969). Magnetohydrodynamic low with suction or injection at the orward stagnation point. J. Phy. Soc. Japan 27(6): Kays, W.M., Craword, M.E. and Weigand, B. (25). Convective Heat and Mass Transer. 4 th ed. New Jersey: McGraw-Hill. Keller, H.B. (197). A New Dierence Scheme or Parabolic Problems. In: Bramble, J. Numerical Solution o Partial Dierential Equations. New York: Academic Press.

40 143 Keller, H.B. and Cebeci, T. (1972). Accurate numerical methods or boundary layer lows, II: Two dimensional turbulent lows. AIAA Journal 1: Kelson, N.A. and Farrell, T.W. (21). Micropolar low over a porous stretching sheet with strong suction or injection. Int. Comm. Heat Mass Transer 28(4): Khan, Md.S., Karim, I. and Biswas, Md.H.A. (212). Non-Newtonian MHD mixed convective power-law luid low over a vertical stretching sheet with thermal radiation, heat generation and chemical reaction eects. Natural Appl. Sci. 3(2): Kim, J.K. (21). Unsteady convection low o micropolar luids past a vertical porous plate embedded in a porous medium. Acta Mechanica 148: Kishan, N. and Deepa, G. (212). Viscous dissipation eects on stagnation point low and heat transer o a micropolar luid with uniorm suction or blowing. Advances Applied Science Research 3(1): Kishan, N. and Maripala, S. (212). Thermophoresis and viscous dissipation eects on Darcy-Forchheimer MHD mixed convection in a luid saturated porous media. Advances Applied Science Research 3 (1): Koichi, A. (26). Mass Transer. Wiley-VCH: Japan. Kumari, M. and Nath, G. (1984). Unsteady incompressible boundary layer low o a micropolar luid at a stagnation point. Int. J. Eng. Sci. 22(6): Kumari, M. and Nath, G. (1986). Unsteady sel-similar stagnation point boundary layers or micropolar luids. Indian J. Pure Appl. Math. 17(2): Kumar, L., Singh, B., Kumar, L. and Bhargava, R. (211). Finite element solution o MHD low o micropolar luid towards a stagnation point on a vertical stretching sheet. Int. J. Appl. Math. Mech. 7(3): Lakshmana Rao, S.K. (197). Stability o micropolar luid motions. Int. J. Eng. Sci. 8: Leal, L.G. (1992). Laminar Flow and Convective Transport Process: Scaling, Principles and Asymptotic Analysis. London: Heinemann. Lienhard IV, J.H. and Lienhard V, J.H. (211). A Heat Transer Textbook. 4 th Edition, Mineola NY: Dover Publications. Lok, Y.Y. (28). Mathematical Modelling o a Micropolar Fluid Boundary Layer near a Stagnation Point. Universiti Teknologi Malaysia: Ph.D Thesis.