Thesis (PhD)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: Different generalized Newtonian fluids (where the normal stresses were neglected) were
considered in this study. Analytical expressions were derived for time independent,
fully developed velocity profiles of Herschel-Bulkley fluids (including the simplifications
thereof: Newtonian, power law and Bingham plastic fluids) and Casson fluids through
open channel sections. Both flow through cylindrical pipes (Hagen-Poiseuille flow) and
parallel plates (plane Poiseuille flow) were brought under consideration. Equations were
derived for the wall shear stresses in terms of the average channel velocities. These
expressions for plane Poiseuille flow were then utilized in the modelling of flow through
homogeneous, isotropic porous media.
Flow through parallel plates was extended and a possibility of a moving lower wall (plane
Couette-Poiseuille flow) was included for Herschel-Bulkley fluids (and the simplifications
thereof). The velocity of the wall was assumed to be opposite to the pressure gradient
(thus in the streamwise direction) yielding three different possible flow scenarios. These
equations were again revisited in the study on flow over porous structures. Averaging of the microscopic momentum transport equation was carried out by means
of volume averaging over an REV (Representative Elementary Volume). Flow through
parallel plates enclosing a homogeneous porous medium (assumed homogeneous up to
the external boundary) was studied at the hand of Brinkman’s equation. It was as-
sumed (also for non-Newtonian fluids) that the term dominating outside the external
boundary layer area is directly proportional to the superficial velocity that is, since only
the viscous flow regime was considered, referred to as the ‘Darcy’ velocity if the diffusive
Brinkman term is completely neglected. For a shear thinning or shear thickening fluid,
the excess superficial velocity term was included in the proportionality coefficient that
is constant for a particular fluid traversing a particular porous medium subjected to a
specific pressure gradient. For such fluids only the inverse functions could be solved. If the ‘Darcy’ velocity is not reached within the considered domain, Gauss’s hypergeo-
metric function had to be utilized. For Newtonian and Bingham plastic fluids, direct
solutions were obtained. The effect of the constant yield stress was embedded in the
proportionality coefficient.
For linear flow, the proportionality coefficient consists of both a Darcy and a Forch-
heimer term applicable to the viscous and inertial flow regimes respectively. Secondary
averaging for different types of porous media was accomplished by using an RUC
(Representative Unit Cell) to estimate average interstitial properties. Only homoge-
neous, isotropic media were considered. Expressions for the apparent permeability as
well as the passability in the Forchheimer regime (also sometimes referred to as the
non-Darcian permeability) were derived for the various fluid types.
Finally fluid flow in a domain consisting of an open channel adjacent to an infinite porous
domain is considered. The analytically derived velocity profiles for both plane Couette-
Poiseuille flow and the Brinkman equation were matched by assuming continuity in the
shear stress at the porosity jump between the two domains.
An in-house code was developed to simulate such a composite domain numerically. The
difference between the analytically assumed constant apparent permeability in a macro-
scopic boundary layer region as opposed to a dependency of the varying superficial
velocity was discussed. This code included the possibility to alter the construction of
the domain and to simulate axisymmetrical flow in a cylinder. / AFRIKAANSE OPSOMMING: Verskeie veralgemeende Newtoniese vloeistowwe (waarvan die normaalspannings ignoreer-
baar is) word in hierdie studie beskou. Analitiese uitdrukkings vir tyd-onafhanklike, ten
volle ontwikkelde snelheidsprofiele vir Herschel-Bulkley vloeistowwe (wat die vereen-
voudigde weergawes daarvan insluit: Newtoniese, magswet- en Bingham-plastiek vloei-
stowwe), sowel as Casson vloeistowwe, is afgelei vir vloei deur ‘n oop kanaal. Beide vloei
deur silindriese pype (Hagen-Poiseuille vloei) en parallelle plate (vlak-Poiseuille vloei)
is oorweeg. Vergelykings vir die skuifspannings op ‘n wand in terme van die gemiddelde
snelhede is afgelei. Hierdie uitdrukking wat vir vlak-Poiseuille vloei verkry is, is in die
modellering van vloei deur homogene, isotropiese poreuse media ook gebruik.
Vloei deur parallelle plate is uitgebrei en die moontlikheid van ‘n bewegende onderste
wand (vlak-Couette-Poiseuille vloei) is ondersoek vir Herschel-Bulkley vloeistowwe (en
die vereenvoudigings daarvan). Dit word aangeneem dat die snelheid van die wand in
die teenoorgestelde rigting as die drukgradiënt georiënteer is (dus in die stroomgewyse
rigting) wat dan tot drie verskillende moontlike vloeigevalle lei. Hierdie vergelykings is
weer in die studie van vloei oor poreuse strukture gebruik.
Die gemiddelde van die mikroskopiese momentum transportvergelyking is bereken oor
die volume van ‘n REV (“Representative Elementary Volume”). Vloei deur parallelle
plate wat ‘n homogene poreuse medium omsluit (waar die medium homogeen aanvaar
word tot by die eksterne grens) is bestudeer aan die hand van Brinkman se vergelyking.
Daar is aanvaar (ook vir nie-Newtoniese vloeistowwe) dat die dominante term buite
die eksterne grenslaaggebied direk eweredig is aan die oppervlaksnelheid en, aangesien
slegs vloei in die viskeuse gebied oorweeg word, daarna verwys word as die “Darcy”-
snelheid, indien die diffusiewe Brinkman-term heeltemal weglaatbaar is. Vir ‘n span-ningsverdunnende of -verdikkende vloeistof, word die oortollige oppervlaksnelheidsterm
ingesluit by die proporsionaliteitskoëffisiënt wat konstant is vir ‘n spesifieke vloeistof wat
deur ‘n sekere poreuse medium, onderhewig aan ‘n spesifieke drukgradiënt, vloei. Vir
sulke vloeistowwe kon slegs die inverse funksies opgelos word. As die “Darcy”- snelheid
nie binne die betrokke gebied bereik word nie, is daar van Gauss se hipergeometriese
funksie gebruik gemaak. Vir Newtoniese en Bingham-plastiek vloeistowwe is egter direkte oplossings verkry. Die effek van die konstante toegeespanning is ingebed in die
proporsionaliteitskoëffisiënt.
Vir lineêre vloei bestaan die proporsionaliteitskoëffisiënt uit beide ‘n Darcy- en ‘n Forch-
heimer-term wat van toepassing is in die viskeuse- en traagheidsvloeigebiede onder-
skeidelik. Sekondˆere gemiddeldes vir verskillende tipes poreuse media is verkry; deur
gebruik te maak van ‘n RUC (“Representative Unit Cell”) kan interstisiële gemiddelde
eienskappe geskat word. Slegs homogene, isotrope media is in oorweging gebring. Uit-
drukkings vir die o¨enskynlike deurlaatbaarheid sowel as die deurdringbaarheid in die
Forchheimer-gebied (ook soms na verwys as die nie-Darcy deurlaatbaarheid) is afgelei
vir die verskillende vloeistoftipes.
Ten slotte is vloeistofvloei in ‘n gebied wat bestaan uit ‘n oop kanaal aangrensend
aan ‘n oneindige poreuse domein ondersoek. Die analities-afgeleide snelheidsprofiele
vir beide vlak-Couette-Poiseuille vloei en die Brinkman-vergelyking is gekoppel deur
‘n kontinu¨ıteit in die skuifspanning by die poreuse-sprong tussen die twee gebiede te
aanvaar. ‘n Interne numeriese kode is ontwikkel om so ‘n saamgestelde domein numeries te
simuleer. Die verskil tussen die analities konstant-aanvaarde deurlaatbaarheid in ‘n
makroskopiese grenslaagstreek, eerder as ‘n afhanklikheid met die veranderende opper-
vlaksnelheid, is bespreek. Hierdie kode sluit ook die moontlikheid in om die domein
te herkonstrueer, asook om die simulasie van aksiaal-simmetriese vloei in ‘n silinder te
ondersoek.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/80240 |
| Date | 03 1900 |
| Creators | Cloete, Maret |
| Contributors | Smit, G. J. Francois, Stellenbosch University. Faculty Science. Dept. of Mathematical Sciences. |
| Publisher | Stellenbosch : Stellenbosch University |
| Source Sets | South African National ETD Portal |
| Language | en_ZA |
| Detected Language | Unknown |
| Type | Thesis |
| Format | 433 p. : ill. |
| Rights | Stellenbosch University |
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