Similarity Solutions for the Boundary Layer Flow and Heat Transfer of Viscous Fluids, Nanofluids, Porous Media, and Micropolar Fluids

294 Pages·2021·4.642 MB·English

by  John H. Merkin, Ioan Pop, Yian Yian Lok, Teodor Grosan

SIMILARITY
SOLUTIONS FOR THE
BOUNDARY LAYER
FLOW AND HEAT
TRANSFER OF
VISCOUS FLUIDS,
NANOFLUIDS,
POROUS MEDIA,
AND MICROPOLAR
FLUIDS
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SIMILARITY
SOLUTIONS FOR THE
BOUNDARY LAYER
FLOW AND HEAT
TRANSFER OF
VISCOUS FLUIDS,
NANOFLUIDS,
POROUS MEDIA,
AND MICROPOLAR
FLUIDS
JOHN H. MERKIN
Department of Applied Mathematics, University of Leeds, Leeds,
United Kingdom
IOAN POP
Department of Mathematics, Faculty of Mathematics and
Computer Science, Babeş-Bolyai University, Cluj-Napoca, Romania
YIAN YIAN LOK
MathematicsSection,SchoolofDistanceEducation,UniversitiSains
Malaysia, Pulau Pinang, Malaysia
TEODOR GROSAN
Department of Mathematics, Faculty of Mathematics and
Computer Science, Babeş-Bolyai University, Cluj-Napoca, Romania
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Contents
Preface ix
1. Basic equations and mathematical methods 1
1.1 Basicequations 1
1.2 Similarity solutions 9
1.3 Somenumerical methods 11
1.4 Analyticalsolutionmethods 15
Nomenclature 17
Greekletters 18
Subscript 18
References 18
2. Viscous fluids 23
2.1 Unsteadymixedconvectionflowatathree-dimensional
stagnation point 23
2.2 Mixedconvectionboundarylayerflownearthestagnation pointon
avertical surfacewithslip 34
2.3 MixedconvectionnonaxisymmetricHomannstagnation-point flow 41
Nomenclature 46
Greekletters 47
Subscript 47
Superscript 47
References 47
3. Stretching/shrinking sheets near a stagnation-point flow in
viscous fluids 49
3.1 Introduction 49
3.2 Unsteadyseparatedstagnation-point flowtowardstretching/shrinking
sheet 49
3.3 Axisymmetric rotationalstagnation-point flowoverapermeable
stretching/shrinking rotatingdisk 55
3.4 Magnetohydrodynamic obliquestagnation-point flowtowarda
stretching/shrinking surface 73
Nomenclature 83
Romanletters 83
v
vi Contents
Greeksymbols 83
Subscripts 84
References 84
4. Nanofluids 87
4.1 Forcedconvectionboundarylayerflowpastnonisothermal thin
needlesinnanofluids 87
4.2 Axisymmetric mixedconvectionboundarylayerflowpastavertical
cylinderinananofluid 92
4.3 Blasius andSakiadis problemsinnanofluids 105
Nomenclature 110
Greekletters 111
Subscript 111
Superscript 111
References 111
5. Stretching/shrinking sheets in nanofluids and hybrid nanofluids 113
5.1 Flowandheattransferoveranunsteadyshrinking sheetwith
suctioninananofluidusingBuongiorno’smodel 113
5.2 Axisymmetric rotationalstagnation-point flowimpingingradially a
permeablestretching/shrinking surfaceinananofluid 119
5.3 Flowandheattransferoverapermeablebiaxialstretching/shrinking
sheetinananofluid 127
5.4 Numericalsolutions ofnonalignment stagnation-point flowandheat
transfer ofananofluidoverastretching/shrinkingsurface inananofluid 134
5.5 Flowandheattransferalongapermeablestretching/shrinking curved
surfaceinahybridnanofluid 140
5.6 MHDflowandheattransfer overapermeable stretching/shrinking
sheetinahybridnanofluidwithaconvective boundarycondition 151
Nomenclature 157
Greekletters 158
Subscript 158
Superscript 159
References 159
6. Mixed convection flow in porous medium 163
6.1 Introduction 163
6.2 Mixedconvectionboundarylayerflowonaverticalsurface ina
saturatedporousmedium 164
6.3 Steadymixedconvectionflowoverapermeablevertical thin
cylinderinaporousmedium 167
Contents vii
6.4 Mixedconvectionboundarylayerflowfromaverticalflatplate
embedded inaporousmediumfilledwithnanofluids 172
6.5 Mixedconvectionboundarylayerflowalongavertical cylinder
embedded inaporousmediumfilledbyananofluid 178
6.6 Mixedconvectionboundarylayerflowoveraverticalplate
embedded inaporousmediumfilledwithasuspensionof
nano-encapsulated phasechangematerials 186
Nomenclature 195
Greekletters 196
Subscript 196
Superscript 197
References 197
7. Convective flows with internal heat generation in porous media 205
7.1 Introduction 205
7.2 Flowswithtemperaturedependent heatgeneration 205
7.3 Flowswithspatiallydependentheatgeneration 217
7.4 Concludingremarks 220
Nomenclature 221
Greeksymbols 221
References 221
8. Micropolar fluids over the moving surface 225
8.1 Introduction 225
8.2 Mixedconvectionflowofamicropolar fluidnearastagnation-point
flowoverastretchingsurface 227
8.3 Obliquestagnationeslip flowofamicropolarfluidtowarda
stretching/shrinking surface 231
8.4 Moving wedgeandflatplateinamicropolarfluid 241
Nomenclature 248
Greeksymbols 248
Subscripts 249
References 249
9. Jets 255
9.1 Introduction 255
9.2 Wall jet 258
9.3 JetprofilesolutionsoftheFalknereSkanequation 263
9.4 Numerical modelingofGlauerttypeexponentially decaying
wall jetflowsofnanofluidsusingTiwariandDas’nanofluidmodel 266
viii Contents
Nomenclature 273
Greekletters 274
Subscripts 274
Superscripts 274
References 274
Index 277
Preface
Fluid mechanics is a major part of applied mathematics, of physics, and in
many branches of engineering,particularly civil, mechanical, chemical, and
aeronautical engineering. It also plays an important role in naval architec-
ture,geophysics,aswellasinastrophysics,biologicalandphysiologicalfluid
dynamics. Significant contributions to the theory of airfoils came early in
the 20th century, and during the whole of the first-half of this century
applied aerodynamics was the major incentive, but also dealing with
questions that were important in both mechanical and civil engineering.
Partial differential equations are the basis for nearly all technical pro-
cesses in heat transfer and fluid mechanics. In our research activity, we
became aware of the fact that a lot of researchers focus increasingly on
numerical methods for solving the partial differential equations involved in
their problems. Analytical methods, taught in fluid mechanics and heat
transfer courses, are often quickly discarded, because of the common belief
that almost all problems appearing in real applications can be solved easily
bynumericalmethods.Thebasicideaofthisbookistoshowsomeselected
analytical and numerical methods and to explain their application to more
complicated problems, which are technically relevant. Of course, this
means that some of the standard analytical methods might not be discussed
andpresentinthisbook(forexample,integraltransforms,Liegroups,etc.).
Ontheotherhand,theanalyticalmethodsdiscussedhereareapplicabletoa
range of interesting problems and to applied mathematicians. Also, PhD
students or engineers could learn how to solve useful technical problems
analytically. There is no doubt that the knowledge of numerical solution
methods is very important, and there is an opportunity in using both nu-
merical and analytic tools to gain useful insights into flow physics and heat
transfer characteristics. However, numerical methods are always dependent
on grid quality and grid size and on a lot of implementation features.
Perhaps the most important development in fluid mechanics during the
20thcenturywastheconceptofboundarylayerflowintroducedbyPrandtl
in 1904. A boundary layer is a layer of fluid which forms on a surface
bounding the moving fluid. Every time a fluid moves along a surface a
boundarylayerformsonthesurface.Therefore,boundarylayersexistinthe
interiorofwaterpipes,insewerpipes,inirrigationchannels,neartheearth’s
surface, and around buildings due to winds, near aeroplane wings, around a
ix