p ocesses
A icle
Resea ch o Flow S abili y o Non-New onian Magne o heological
Fluid Flow in he Gap be ween Two Cylinde s †
Milada Kozubko á1,*, Jana Jablonská1, Ma ian Bojko 1, F an išek Pochylý2and Simona Fialo á2


Ci a ion: Kozubko á, M.; Jablonská,
J.; Bojko, M.; Pochylý, F.; Fialo á, S.
Resea ch o Flow S abili y o
Non-New onian Magne o heological
Fluid Flow in he Gap be ween Two
Cylinde s. P ocesses 2021,9, 1832.
h ps://doi.o g/10.3390/p 9101832
Academic Edi o : Richa d Lenha d
Recei ed: 30 Augus 2021
Accep ed: 11 Oc obe 2021
Published: 15 Oc obe 2021
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Copy igh : © 2021 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
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A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
1VSB—Depa men o Hyd omechanics and Hyd aulic Equipmen , Facul y o Mechanical Enginee ing,
Technical Uni e si y o Os a a, 17. Lis opadu 2172/15, 708 00 Os a a-Po uba, Czech Republic;
[email p o ec ed] (J.J.); [email p o ec ed] (M.B.)
2BUT—Vic o Kaplan Depa men o Fluid Enginee ing, Facul y o Mechanical Enginee ing,
B no Uni e si y o Technology, Technická2896/2, 616 69 B no, Czech Republic; [email p o ec ed].cz (F.P.);
[email p o ec ed].cz (S.F.)
*Co espondence: [email p o ec ed]
† This pape is an ex ended e sion o pape Mul iphase low in he gap be ween wo o a ing cylinde s
published in he in e na ional con e ence: “XXII. In e na ional Scien i ic Con e ence—The Applica ion o
Expe imen al and Nume ical Me hods in Fluid Mechanics and Ene gy 2020 (AEaNMiFMaE-2020), Pieš ’any,
Slo akia, 7–9 Oc obe 2020.
Abs ac :
This pape deals wi h a ma hema ical modeling o low s abili y o New onian and non-
New onian luids in he gap be ween wo concen ic cylinde s, one o which o a es. A ypical ea u e
o he low is he o ma ion o a o ex low, so-called Taylo o ices. Vo ex s uc u es a e a ec ed
by he speed o he o a ing cylinde and he physical p ope ies o he luids, i.e., iscosi y and
densi y. Analogy in e ms o iscosi y is assumed o non-New onian and magne o heological luids.
Ma hema ical models o lamina , ansien and u bulen low wi h cons an iscosi y and iscosi y
as a unc ion o he de o ma ion g adien we e o mula ed and nume ically sol ed o analyze he
s abili y o single-phase low. To e i y hem, a physical expe imen was pe o med o New onian
luids using isualiza ions o o ex s uc u es—Taylo o ices. Based on he ag eemen o selec ed
nume ical and physical esul s, he expe ience was used o nume ical simula ions o non-New onian
magne o heological luid low.
Keywo ds:
Taylo o ices; non-New onian iscosi y; magne o heological luids; expe imen ;
nume ical simula ion; CFD
1. In oduc ion
Immiscible liquids a e speci ied as a sys em o wo (o mo e) componen s, e.g., liquid–
liquid o liquid–solid phase. The solid phase is ep esen ed by pa icles dispe sed in he
ca ie luid. Thei in e ac ion in he low depends on hei chemical composi ion and
physical p ope ies. In case o magne o heological luids, we mus accoun o he in luence
o he magne ic ield, which can change he New onian iscosi y o a non-New onian
one [1–6].
Viscosi y is conside ed cons an o New onian luids. Fo non-New onian and mag-
ne o heological luids, i depends on he de o ma ion g adien [
2
,
6
–
9
]. Non-New onian
luids a e widely used in he indus y, especially in he hyd aulic gaps o o a y machines.
The use o magne o heological luid and e o luid has ecen ly been in es iga ed in he
applica ion o hyd aulic lub ica ion.
The aim o his wo k is o de ine and e i y he ma hema ical model o lamina ,
ansien and u bulen low o non-New onian and magne o heological luids in he
gap be ween wo concen ic cylinde s. The low in he annulus is closely connec ed wi h
p ac ical applica ions. In addi ion, he low is mos ly lamina , so a en ion is ocused on
he s udy o lamina , ansien and incipien u bulen low.
P ocesses 2021,9, 1832. h ps://doi.o g/10.3390/p 9101832 h ps://www.mdpi.com/jou nal/p ocesses
P ocesses 2021,9, 1832 2 o 16
The low o non-New onian luids has many o he indus ial applica ions. I has been
in es iga ed mainly in connec ion wi h lamina low in a ious geome ies [
10
], in po ous
media [
11
], in pipelines and hyd aulic lub ica ion gaps [
12
], in chemical indus y [
13
] and
o he s. A p esen , magne o heological luids can be also included in his ca ego y. These
luids a e mos ly a suspension o me al pa icles wi h a diame e o he o de o se e al
µ
m
dispe sed in he ca ie luid (wa e o oil).
These luids change hei physical p ope ies unde he ac ion o a magne ic ield
o a ious in ensi ies. The luid, in e ms o iscosi y o iginally o he New onian ype,
changes o a non-New onian luid. The iscosi y depends nonlinea ly on he shea s ain
a e. The e a e many heological models, which a e used o app oxima e he heog am
o non-New onian luids o some deg ee, such as he Bingham, powe law, Ca eau and
He schel–Bulkley, bu hey do no cap u e he na u e o iscosi y in he equi ed ange o
applica ions. I is ecommended o de e mine he iscosi y by expe imen al measu emen s.
Ma hema ical models o he ansien low be ween he lamina and u bulen egime
a e p oblema ic, especially in a eas wi h he o ma ion o o ex s uc u es. These s uc u es
can be well obse ed in he gaps be ween he o a ing cylinde s (see Figu e 1).
P ocesses 2021, 9, x FOR PEER REVIEW 2 o 17
be ween wo concen ic cylinde s. The low in he annulus is closely connec ed wi h p ac-
ical applica ions. In addi ion, he low is mos ly lamina , so a en ion is ocused on he
s udy o lamina , ansien and incipien u bulen low.
The low o non-New onian luids has many o he indus ial applica ions. I has been
in es iga ed mainly in connec ion wi h lamina low in a ious geome ies [10], in po ous
media [11], in pipelines and hyd aulic lub ica ion gaps [12], in chemical indus y [13] and
o he s. A p esen , magne o heological luids can be also included in his ca ego y. These
luids a e mos ly a suspension o me al pa icles wi h a diame e o he o de o se e al
μm dispe sed in he ca ie luid (wa e o oil).
These luids change hei physical p ope ies unde he ac ion o a magne ic ield o
a ious in ensi ies. The luid, in e ms o iscosi y o iginally o he New onian ype,
changes o a non-New onian luid. The iscosi y depends nonlinea ly on he shea s ain
a e. The e a e many heological models, which a e used o app oxima e he heog am o
non-New onian luids o some deg ee, such as he Bingham, powe law, Ca eau and He -
schel–Bulkley, bu hey do no cap u e he na u e o iscosi y in he equi ed ange o
applica ions. I is ecommended o de e mine he iscosi y by expe imen al measu e-
men s.
Ma hema ical models o he ansien low be ween he lamina and u bulen egime
a e p oblema ic, especially in a eas wi h he o ma ion o o ex s uc u es. These s uc-
u es can be well obse ed in he gaps be ween he o a ing cylinde s (see Figu e 1).
Coue e low Ro a ion o inne
cylinde Taylo o ices
Figu e 1. Vo ex s uc u es be ween wo concen ic cylinde s.
In lamina and ansien low, Coue e low wi hou o ex s uc u es can be ob-
se ed. Wi h inc easing he o a ional speed o he cylinde , s able Taylo o ices o m,
hen he o ices change o wa e mode, spi al mode, e c. In u bulen mode, o ices a e
o med analogous o s able o ices. Va ian s o o ex s uc u es can be in es iga ed by
s abili y me hods applied o ma hema ical models and e alua ed by s abili y diag ams
[14–16].
Wi h he de elopmen o ma hema ical models o u bulence, a ew low models
ha e been de eloped. Thei co ec ness in he case o Taylo o ices is e i ied expe i-
men ally. The mos accu a e model o lamina i y and u bulence is he DNS model, which
sol es he low in he h ee-dimensional egion desc ibed by Na ie –S okes equa ions and
a con inui y equa ion unde he assump ion o a su icien ly ine compu a ional g id
[10,17].
Resea ch has also ex ended o he s udy o o ex s uc u es in connec ion wi h he
empe a u e g adien [18], a ious bounda y condi ions [19] and o he s. RANS ( ime a -
e aging) me hods a e commonly used in se e al publica ions, [20]. Basic RANS models
Figu e 1. Vo ex s uc u es be ween wo concen ic cylinde s.
In lamina and ansien low, Coue e low wi hou o ex s uc u es can be obse ed.
Wi h inc easing he o a ional speed o he cylinde , s able Taylo o ices o m, hen he
o ices change o wa e mode, spi al mode, e c. In u bulen mode, o ices a e o med
analogous o s able o ices. Va ian s o o ex s uc u es can be in es iga ed by s abili y
me hods applied o ma hema ical models and e alua ed by s abili y diag ams [14–16].
Wi h he de elopmen o ma hema ical models o u bulence, a ew low models ha e
been de eloped. Thei co ec ness in he case o Taylo o ices is e i ied expe imen ally.
The mos accu a e model o lamina i y and u bulence is he DNS model, which sol es
he low in he h ee-dimensional egion desc ibed by Na ie –S okes equa ions and a
con inui y equa ion unde he assump ion o a su icien ly ine compu a ional g id [
10
,
17
].
Resea ch has also ex ended o he s udy o o ex s uc u es in connec ion wi h he
empe a u e g adien [
18
], a ious bounda y condi ions [
19
] and o he s. RANS ( ime
a e aging) me hods a e commonly used in se e al publica ions, [
20
]. Basic RANS models
a e no he mos app op ia e o low in he ansi ion om lamina o u bulen low.
Newly de eloped me hods a e used, which a e especially sui able o low wi h a low
Reynolds numbe (SST k-om, SA model). In his case, i is necessa y o add ess he issue o
g id quali y nea he wall [4].
A less demanding a ian wi h espec o he g id equi emen s in compa ison wi h
DNS is he LES app oach [
21
]. F om he poin o iew o applica ions, i is e ec i e o
moni o he i s s able o ex s uc u es in he low be ween wo concen ic cylinde s. Then,
i is possible o use a wo-dimensional model. In his case, RANS models a e su icien .
P ocesses 2021,9, 1832 3 o 16
Ma hema ical low models a e embedded in a ew so wa e, such as ANSYS Fluen CFX,
which has he ad an age o e sa ili y. Howe e , i s use equi es e i ica ion o he esul s
and he c ea ion o p ocedu es ha a e necessa y o he applica ion.
Se e al me hods a e used o in es iga e low ins abili ies expe imen ally. The mos
sui able o compa ison wi h nume ical expe imen s is he PIV me hod. This me hod
enables o moni o he de elopmen o o ex s uc u es in wo-dimensional c oss-sec ional
planes. Howe e , i is e y demanding, bo h in he p epa a ion o he expe imen and
e alua ion o he measu emen s. To e i y he ma hema ical model [
9
,
22
,
23
], an expe imen
was cons uc ed o isualize Taylo o ices in he gap be ween concen ic cylinde s wi h a
o a ing inne one.
Fo di e en ypes o luid and o a ional speeds o he inne cylinde , he basic
low cha ac e is ics a e e y well-obse able, including di e en ypes o o ex s uc u es,
especially in he a ea o eme ging s able Taylo o ices. The p oblem o oil s abili y analysis
has been in es iga ed in he pas and is desc ibed in su icien de ail in he li e a u e o
single-phase New onian luids [15].
Vo ex s uc u es in he low o wo di e en immiscible liquids (oil and e hanol)
o di e en densi ies and iscosi ies we e in es iga ed in Re e ence [
24
]. The di usion
be ween he luids was e y small in he case o lamina low, and he luids o med o ex
s uc u es sepa a ely. Thei shapes co esponded o a single-phase low a a gi en eloci y
o he inne cylinde . A highe o a ional speeds, he low was al eady u bulen , he liquids
s i ed and a mix u e o med. The nume ical model showed a homogeneous mix u e nea
he walls, and a pa ially mixed mix u e copying he o ex s uc u es could be obse ed
inside. The esul s o ma hema ical modeling co esponded o he expe imen al esul s.
The aim o his wo k is o ex end he applica ion o he single-phase ma hema ical
model o New onian luid (oil, wa e and e hanol) o he low o non-New onian magne-
o heological luid. Expe ience ob ained du ing he modeling o he oil low [
15
] and wa e
and e hanol low [
24
] is he backg ound o modeling he ins abili y o a non-New onian
magne o heological luid wi h a iscosi y de e mined expe imen ally.
2. Expe imen , Reynolds and Taylo Numbe
Flow isualiza ion is a e y sui able ool o expe imen al luid mechanics and is
necessa y in he s udy o complex low cases. A e y in e es ing case is he low s uc u es
in connec ion wi h he Taylo –Coue e low, de ined by he low o a iscous liquid in an
annulus be ween wo cylind ical su aces ha mo e ela i e o each o he .
2.1. Expe imen al Equipmen
The expe imen al equipmen o he isualiza ion o low ins abili ies is shown in
Figu e 2. The base o he measu ing de ice is a s able suppo ing cons uc ion wi h an
elec ic mo o enabling o each he maximum speed o he inne cylinde a 2840 pm. The
elec ic mo o is con olled by a equency con e e .
P ocesses 2021, 9, x FOR PEER REVIEW 4 o 17
Figu e 2. Flow be ween concen ic cylinde s du ing o a ion o he inne cylinde , expe imen and
pho os o he Coue e and Taylo low [15].
The uppe pa o he de ice consis s o wo concen ic cylinde s. The ou e cylinde
is made o glass, and he inne o a ing cylinde is made o s eel.
The pa ame e s o he measu ing a ea a e as ollows:
R
1
= 65 mm adius o he inne cylinde
R
2
= 80 mm adius o he ou e cylinde
s = 15 mm hickness o he annulus (R
2
−R
1
)
L = 170 mm leng h o he inne cylinde
The isualiza ion [15] was ealized by sui able illumina ion o he lowing medium
wi h dispe sed aluminum powde ; see he scheme in Figu e 3. The wo-dimensional low
was gi en by an illumina ion uni , i.e., a ligh kni e, which illumina es only pa icles in
he selec ed sec ion. When isualizing a h ee-dimensional low, he en i e space can be
illumina ed, bu e alua ion is di icul . Using isualiza ion, he indi idual limi modes o
Taylo o ices can be ecognized, and he co esponding speed anges can be eco ded.
I is also possible o obse e a di e en numbe o s ips ha cha ac e ize he numbe o
o ices o med.
Figu e 3. Flow isualiza ion scheme (LA—lase , C—cylind ical lens c ea ing a ligh kni e, K—cam-
e a and P
1
and P
2
—windows).
2.2. Physical P ope ies o Liquids
The basic physical p ope ies o common luids a e he densi y, dynamic o kinema ic
iscosi y; su ace ension and o he s [2,5]. The iscosi y o mos gases, apo s, liquids and,
especially, wa e is go e ned by he so-called New on′s law o iscosi y, whe e he s ess
enso 𝜏
 (Pa) is p opo ional o he eloci y g adien , and dynamic iscosi y is a coe i-
cien o p opo ionali y [25]. The luids a e called New onian luids. Gene aliza ion o he
New on law o iscosi y

(Pas) is a se o nine ela ions:
𝜏
=

∇𝑢
→+󰇡∇𝑢
→󰇢−
di 󰇡𝑢
→󰇢𝛿
, (𝑃𝑎) (1)
Figu e 2.
Flow be ween concen ic cylinde s du ing o a ion o he inne cylinde , expe imen and
pho os o he Coue e and Taylo low [15].
P ocesses 2021,9, 1832 4 o 16
The uppe pa o he de ice consis s o wo concen ic cylinde s. The ou e cylinde
is made o glass, and he inne o a ing cylinde is made o s eel.
The pa ame e s o he measu ing a ea a e as ollows:
R1= 65 mm adius o he inne cylinde
R2= 80 mm adius o he ou e cylinde
s= 15 mm hickness o he annulus (R2−R1)
L= 170 mm leng h o he inne cylinde
The isualiza ion [
15
] was ealized by sui able illumina ion o he lowing medium
wi h dispe sed aluminum powde ; see he scheme in Figu e 3. The wo-dimensional low
was gi en by an illumina ion uni , i.e., a ligh kni e, which illumina es only pa icles in
he selec ed sec ion. When isualizing a h ee-dimensional low, he en i e space can be
illumina ed, bu e alua ion is di icul . Using isualiza ion, he indi idual limi modes o
Taylo o ices can be ecognized, and he co esponding speed anges can be eco ded.
I is also possible o obse e a di e en numbe o s ips ha cha ac e ize he numbe o
o ices o med.
P ocesses 2021, 9, x FOR PEER REVIEW 4 o 17
Figu e 2. Flow be ween concen ic cylinde s du ing o a ion o he inne cylinde , expe imen and
pho os o he Coue e and Taylo low [15].
The uppe pa o he de ice consis s o wo concen ic cylinde s. The ou e cylinde
is made o glass, and he inne o a ing cylinde is made o s eel.
The pa ame e s o he measu ing a ea a e as ollows:
R
1
= 65 mm adius o he inne cylinde
R
2
= 80 mm adius o he ou e cylinde
s = 15 mm hickness o he annulus (R
2
−R
1
)
L = 170 mm leng h o he inne cylinde
The isualiza ion [15] was ealized by sui able illumina ion o he lowing medium
wi h dispe sed aluminum powde ; see he scheme in Figu e 3. The wo-dimensional low
was gi en by an illumina ion uni , i.e., a ligh kni e, which illumina es only pa icles in
he selec ed sec ion. When isualizing a h ee-dimensional low, he en i e space can be
illumina ed, bu e alua ion is di icul . Using isualiza ion, he indi idual limi modes o
Taylo o ices can be ecognized, and he co esponding speed anges can be eco ded.
I is also possible o obse e a di e en numbe o s ips ha cha ac e ize he numbe o
o ices o med.
Figu e 3. Flow isualiza ion scheme (LA—lase , C—cylind ical lens c ea ing a ligh kni e, K—cam-
e a and P
1
and P
2
—windows).
2.2. Physical P ope ies o Liquids
The basic physical p ope ies o common luids a e he densi y, dynamic o kinema ic
iscosi y; su ace ension and o he s [2,5]. The iscosi y o mos gases, apo s, liquids and,
especially, wa e is go e ned by he so-called New on′s law o iscosi y, whe e he s ess
enso 𝜏
 (Pa) is p opo ional o he eloci y g adien , and dynamic iscosi y is a coe i-
cien o p opo ionali y [25]. The luids a e called New onian luids. Gene aliza ion o he
New on law o iscosi y

(Pas) is a se o nine ela ions:
𝜏
=

∇𝑢
→+󰇡∇𝑢
→󰇢−
di 󰇡𝑢
→󰇢𝛿
, (𝑃𝑎) (1)
Figu e 3.
Flow isualiza ion scheme (LA—lase , C—cylind ical lens c ea ing a ligh kni e, K—came a
and P1and P2—windows).
2.2. Physical P ope ies o Liquids
The basic physical p ope ies o common luids a e he densi y, dynamic o kinema ic
iscosi y; su ace ension and o he s [
2
,
5
]. The iscosi y o mos gases, apo s, liquids
and, especially, wa e is go e ned by he so-called New on
0
s law o iscosi y, whe e he
s ess enso
=
τ(Pa)
is p opo ional o he eloci y g adien , and dynamic iscosi y is a
coe icien o p opo ionali y [
25
]. The luids a e called New onian luids. Gene aliza ion
o he New on law o iscosi y µ(Pa.s)is a se o nine ela ions:
=
τ=µ∇→
u+∇→
uT−2
3di →
u=
δ,(Pa)(1)
whe e
∇→
u
(s
−1
) is a enso o he eloci y g adien wi h componen s
∂
∂xiujs−1
,
∇→
uT
s−1
is a ansponse enso o he eloci y g adien wi h componen s
∂
∂xjui
,
di →
u
s−1
is a di e gence o he eloci y ec o and
=
δ
is he uni enso . Fo an incomp essible
luid, di →
u=0s−1, and he equa ion has a simple o m:
=
τ=µ∇→
u+∇→
uT=−µ.
γ,(Pa)(2)
Symbol
.
γ=∇→
u+∇→
uTs−1
is he s ain a e enso o he de o ma ion
a e enso .
P ocesses 2021,9, 1832 5 o 16
I he beha io o he luids does no comply wi h his law, hey a e non-New onian
luids, such as suspensions, highe polyme s, e c. In magne o heological luids, he shea
s ess may change depending on he di ec ion o he magne ic ield unde he load, and
he luids become non-New onian luids. Fo a non-New onian luid, he analogous
gene alized shea s ess equa ion applies as o New onian luids, i.e., he shea s ess
depends nonlinea ly on he eloci y g adien (25). The mos used empi ically de e mined
dependence is he powe dependence o he iscosi y on he s ain a e. The mos commonly
used o mula ion is he Os wald and de Waele (25) o mula ion o iscosi y
µe (Pa s)
in
he o m:
µe =m.
γn−1, (3)
whe e mand na e cons an s cha ac e izing he liquid. Powe unc ions a e only in e pola-
ion unc ions and a e no de i ed om a physical model o he in e nal s uc u e o he
luids. Fo his eason, hei use has been c i icized, bu he powe unc ions, despi e his
sho coming, cap u e mos o he ac ual low cu es e y well and ail only o liquids
whose heog ams ha e in lec ion poin s. In hese cases, howe e , i is possible o use a
powe unc ion wi h su icien accu acy o pa s o he low cu e.
Commonly a ailable liquids (oil, wa e and e hanol) we e es ed in he expe imen al
equipmen desc ibed abo e; see Figu e 2. The ansi ion s a e and o ex s uc u es a e
bes obse ed wi h hyd aulic oil using aluminum powde . The iscosi y o he oil wi h
he addi ion o aluminum powde was measu ed and compa ed wi h he iscosi y o he
pu e oil. I di e ed by abou 2% (15). Visualiza ion o he low by means o aluminum
powde is impossible wi h wa e , because aluminum powde is poo ly dispe sed in wa e
and o ms clumps, so oil and e hanol we e used in he expe imen . The physical p ope ies
o common liquids a e gi en in Table 1.
Table 1. Physical p ope ies o common liquids.
Uni Wa e E hanol Oil
Densi y kg/m3998 790 876
Kinema ic iscosi y m2/s 1.002 ×10−61.5209 ×10−68.2182 ×10−5
Dynamic iscosi y Pa.s 0.001 0.0012 0.072
The physical p ope ies o magne o heological and e omagne ic luids di e om
con en ional New onian luids, because he nanopa icles change hei o ien a ion in space
when exposed o an ex e nal magne ic ield. Thus, he physical p ope ies change, and he
luids become non-New onian. The physical p ope ies o magne o heological luids aken
om he li e a u e [2,6] a e gi en in Table 2.
Table 2. Physical p ope ies o e omagne ic liquids
Uni EMG 900 EMG 905
Concen a ion o nanopa icles % ol. 17.7 7.8
Sa u a ion magne iza ion mT 99 44
Densi y kg/m31.74·× 1031.2·× 103
Dynamic iscosi y mPa.s 60 3
Mel ing poin (a pn)◦C−94 −94
Flash poin (a pn)◦C 89 89
Ini ial magne ic suscep ibili y 18.6 3.52
Magne o heological luids EMG900 and EMG905 con ain e omagne ic pa icles
(combina ion o magne i e–maghemi e) wi h a mean pa icle diame e o 10 nm. The ca ie
liquid consis s o ligh hyd oca bon (ke osene wi h addi i es [
5
]), which ensu es a low

P ocesses 2021,9, 1832 6 o 16
iscosi y. Liquids do no di e in he ype o pa icles, only in he concen a ion o pa icles,
addi i es and su ac an s. The physical p ope ies o he magne o heological luids a e
gi en in Table 2, whe e he iscosi y is speci ied o a ce ain alue o he magne ic ield.
Viscosi y measu emen s we e pe o med o di e en alues o he magne ic ield,
using a An onPaa MCR502 (An on Paa GmbH, G az, Aus ia) o a y heome e in a
pla e–pla e con igu a ion (PP20/MRD/TI). The shea loading o he liquid ook place in a
hin laye be ween wo plana , ci cula pla es [
5
]. A (homogeneous) magne ic ield wi h a
selec able in ensi y o 0–432 kA/m ac ed pe pendicula o he di ec ion o loading. Du ing
he expe imen , he de ice e alua ed he dependences o he shea s ess in he liquid (o
iscosi y) on he s ain a e. Samples o EMG 900 and EMG 905 luids om Fe o ec Co p.
we e measu ed. (Fe o ec (USA) Co po a ion, San a Cla a, CA, USA).
Due o he s ong in luence o he magne ic ield on he iscosi y in he a ea o e y low
s ain a es, he iscosi y was in e spe sed wi h wo di e en powe unc ions in
he o m
:
µe =m1.
γN1+c1µe =m2.
γN2+c2, (4)
The in e sec ion o he iscosi y cu es o he a ea highly and less a ec ed by he
magne ic ield was almos he same o di e en magne ic ield in ensi ies. The in e sec ion
alue o he s ain a e is app oxima ely 15 s
−1
. The esul is he dependences o he shea
s ess and iscosi y o he in es iga ed luids on he s ain a e; see Figu e 4.
P ocesses 2021, 9, x FOR PEER REVIEW 6 o 17
Sa u a ion magne iza ion mT 99 44
Densi y kg/m
3
1.74·× 10
3
1.2·× 10
3
Dynamic iscosi y mPa.s 60 3
Mel ing poin (a pn) °C −94 −94
Flash poin (a pn) °C 89 89
Ini ial magne ic suscep ibili y 18.6 3.52
Magne o heological luids EMG900 and EMG905 con ain e omagne ic pa icles
(combina ion o magne i e–maghemi e) wi h a mean pa icle diame e o 10 nm. The ca -
ie liquid consis s o ligh hyd oca bon (ke osene wi h addi i es [5]), which ensu es a
low iscosi y. Liquids do no di e in he ype o pa icles, only in he concen a ion o
pa icles, addi i es and su ac an s. The physical p ope ies o he magne o heological
luids a e gi en in Table 2, whe e he iscosi y is speci ied o a ce ain alue o he mag-
ne ic ield.
Viscosi y measu emen s we e pe o med o di e en alues o he magne ic ield,
using a An onPaa MCR502 (An on Paa GmbH, G az, Aus ia) o a y heome e in a
pla e–pla e con igu a ion (PP20/MRD/TI). The shea loading o he liquid ook place in a
hin laye be ween wo plana , ci cula pla es [5]. A (homogeneous) magne ic ield wi h a
selec able in ensi y o 0–432 kA/m ac ed pe pendicula o he di ec ion o loading. Du ing
he expe imen , he de ice e alua ed he dependences o he shea s ess in he liquid (o
iscosi y) on he s ain a e. Samples o EMG 900 and EMG 905 luids om Fe o ec Co p.
we e measu ed. (Fe o ec (USA) Co po a ion, San a Cla a, CA, USA).
Due o he s ong in luence o he magne ic ield on he iscosi y in he a ea o e y
low s ain a es, he iscosi y was in e spe sed wi h wo di e en powe unc ions in he
o m:
𝜇 = 𝑚
γ
󰇗 +𝑐 𝜇 = 𝑚
γ
󰇗 +𝑐, (4)
The in e sec ion o he iscosi y cu es o he a ea highly and less a ec ed by he
magne ic ield was almos he same o di e en magne ic ield in ensi ies. The in e sec-
ion alue o he s ain a e is app oxima ely 15 s
−1
. The esul is he dependences o he
shea s ess and iscosi y o he in es iga ed luids on he s ain a e; see Figu e 4.
Figu e 4. Viscosi y dependence on he s ain a e o oil, e hanol, wa e and EMG 900 [5].
Concen a ed liquid EMG900 has a s ong esponse in he a ea o low s ain a es
(mos p onounced up o abou 100 s
−1
), bu o high s ain a es, he e ec o he magne ic
ield is negligible, and he e is e en an appa en dec ease in he iscosi y below i s basic
Figu e 4. Viscosi y dependence on he s ain a e o oil, e hanol, wa e and EMG 900 [5].
Concen a ed liquid EMG900 has a s ong esponse in he a ea o low s ain a es
(mos p onounced up o abou 100 s
−1
), bu o high s ain a es, he e ec o he magne ic
ield is negligible, and he e is e en an appa en dec ease in he iscosi y below i s basic
alue. Bo h EMG 900 and EMG 905 beha e like non-New onian luids unde he ac ion o
a magne ic ield.
2.3. Ta, a Re Numbe
Reynolds numbe Re [13, 04] is used o de e mine whe he he luid low is lamina
o u bulen .
Re =ΩR1(R2−R1)
ν,(1)(5)
whe e
Ω
( ad s
−1
) is he angula eloci y, R
1
(m) is he inne adius, R
2
(m) is he ou e
adius and
ν
(m
2
s
−1
) is he kinema ic iscosi y. The c i ical alue o he ansi ion om
lamina o u bulen low is in he in e al om 1100 o 1400.
P ocesses 2021,9, 1832 7 o 16
Taylo numbe Ta [
25
,
26
] is a dimensionless quan i y ha cha ac e izes he impo ance
o cen i ugal o ces o so-called ine ial o ces due o he o a ion o a luid abou an axis
ela i e o he iscous o ces.
Ta =ResR2−R1
R1=ΩR1s
νsR2−R1
R1,(1)(6)
whe e s = R
2−
R
1
(m). The c i ical alue cha ac e izing he o ma ion o s a iona y o ices
is Ta
c
= 41.3 (1), and he pe iodic wa e modes o o ices occu a up o 100 imes he
c i ical Taylo numbe . This alue di e s sligh ly in he case o a eal expe imen wi h he
ini e leng h o he cylinde and he de o ma ion o he o ices nea he closings.
The e a e o he , mo e complica ed a ian s o hese o ices (wa e mode, spi al
mode, e c.), whe e de e mining he c i ical alues o he Taylo numbe is di icul and no
unambiguous [
15
].
¨
The dependences o he Reynolds numbe (5) and Taylo numbe (6) on
he o a ional speed o he inne cylinde and he ype o liquid ( iscosi y)
we e e alua ed
.
Fo magne o heological luids, he dependence o he Taylo numbe and Reynolds
numbe on speed was sol ed as ollows. Fo a gi en magne ic ield in ensi y, he depen-
dence o shea s ess and iscosi y on he s ain a e, which was ob ained expe imen ally [
5
],
is known. To de e mine he dependence, he minimum and maximum iscosi y we e cho-
sen as a cons an . Subsequen ly, he minimum and maximum Taylo and Reynolds numbe s
we e calcula ed. Fo iscosi ies wi hin his ange, i is possible o oughly es ima e Ta and
Re o o calcula e hem in he same way.
The ollowing g aphs (Figu e 5) show he dependence o he Taylo numbe on he
speed o he inne cylinde o oil, e hanol, wa e and magne o heological luid EMG 900
unde he in luence o he magne ic ield o he maximum and minimum iscosi y alues;
see Figu e 5. F om he g aph, Ta = (n), and i is e iden ha , o wa e and e hanol, i is
possible o expec he o ma ion o o ices p ac ically a he small speed ha he physical
de ice is able o ealize. Fo oil, Taylo o ices occu a app oxima ely 100 pm. F om he
g aph, Re = (n), and i is e iden ha he low is in he lamina , ansien and u bulen
egime due o he possibili ies o he expe imen ; see Figu e 6. Wa e and e hanol low a
smalle speeds al eady in he u bulen mode, espec i ely, and ansi ion om lamina i y
o u bulence. The oil low is lamina and changes he u bulence up o speed alues o
he o de o 1000 pm. The magne o heological luid shows he ini ial exis ence o Taylo
o ices e en a a e y low speed and high iscosi y. The low iscosi y liquid has p ope ies
be ween he es oil and he e hanol, and hen, i changes in o he uns able mode.
P ocesses 2021, 9, x FOR PEER REVIEW 8 o 17
Figu e 5. Taylo numbe s. speed o oil, e hanol, wa e and EMG 900 (Ta
c
= 41).
Figu e 6. Reynolds numbe s. speed o oil, e hanol, wa e and EMG 900 (Re
c
= 1100 o 1300).
3. Ma hema ical Model
The equa ions ha desc ibe he low o eal luids a e an exp ession o he basic phys-
ical conse a ion laws o mass and momen um. Physically, hese laws exp ess he balance
o a quan i y J (physical uni depends on he ype o a iable J) in a gi en olume [25].
Acco ding o his balance, he ime change o he p ese ed quan i y in he gi en olume
V is equal o he low o his quan i y h ough he a ea S, which ci cumsc ibes his olume,
and i s p oduc ion wi hin he olume V.

𝐽𝑑𝑉
+𝐹(𝐽)𝑛𝑑𝑆
=𝑃(𝐽)𝑑𝑉
, (7)
whe e J is he balanced quan i y, 𝐹(𝐽) is he j h componen o he low densi y ec o o
he quan i y J by he a ea dS, 𝑛 is he j h componen o he no mal ec o and P (J) is he
p oduc ion densi y o he quan i y J (p oduc ion pe uni ime in olume V). Fo index j,
Eins ein′s summa ion ule is used.
The low o a quan i y J is de ined as he ans e o his quan i y h ough a uni o
a ea pe uni ime, e.g., mass low and momen um low. The h ee-dimensional model can
be simpli ied in he case o piping sys ems o he wo-dimensional axially symme ical
model. The de ined ma hema ical model cha ac e izes he low o luids in gene al spa ial
geome y, which is p esen ed he e by he in e nal space be ween wo cylinde s, whe e he
inne one o a es. All bounda y condi ions on he ec angula a ea we e o he wall ype.
The a ious o a ional speeds om 10 pm o 1000 pm, in acco dance wi h he physical
Figu e 5. Taylo numbe s. speed o oil, e hanol, wa e and EMG 900 (Tac= 41).
P ocesses 2021,9, 1832 8 o 16
P ocesses 2021, 9, x FOR PEER REVIEW 8 o 17
Figu e 5. Taylo numbe s. speed o oil, e hanol, wa e and EMG 900 (Ta
c
= 41).
Figu e 6. Reynolds numbe s. speed o oil, e hanol, wa e and EMG 900 (Re
c
= 1100 o 1300).
3. Ma hema ical Model
The equa ions ha desc ibe he low o eal luids a e an exp ession o he basic phys-
ical conse a ion laws o mass and momen um. Physically, hese laws exp ess he balance
o a quan i y J (physical uni depends on he ype o a iable J) in a gi en olume [25].
Acco ding o his balance, he ime change o he p ese ed quan i y in he gi en olume
V is equal o he low o his quan i y h ough he a ea S, which ci cumsc ibes his olume,
and i s p oduc ion wi hin he olume V.

𝐽𝑑𝑉
+𝐹(𝐽)𝑛𝑑𝑆
=𝑃(𝐽)𝑑𝑉
, (7)
whe e J is he balanced quan i y, 𝐹(𝐽) is he j h componen o he low densi y ec o o
he quan i y J by he a ea dS, 𝑛 is he j h componen o he no mal ec o and P (J) is he
p oduc ion densi y o he quan i y J (p oduc ion pe uni ime in olume V). Fo index j,
Eins ein′s summa ion ule is used.
The low o a quan i y J is de ined as he ans e o his quan i y h ough a uni o
a ea pe uni ime, e.g., mass low and momen um low. The h ee-dimensional model can
be simpli ied in he case o piping sys ems o he wo-dimensional axially symme ical
model. The de ined ma hema ical model cha ac e izes he low o luids in gene al spa ial
geome y, which is p esen ed he e by he in e nal space be ween wo cylinde s, whe e he
inne one o a es. All bounda y condi ions on he ec angula a ea we e o he wall ype.
The a ious o a ional speeds om 10 pm o 1000 pm, in acco dance wi h he physical
Figu e 6. Reynolds numbe s. speed o oil, e hanol, wa e and EMG 900 (Rec= 1100 o 1300).
This in o ma ion will be used o es ima e a ma hema ical low model. The g aphs in
Figu es 5and 6a e a good o ien a ion o nume ical calcula ions o he low o magne o he-
ological luids be ween cylinde s, and i should be no ed ha dimensionless quan i ies a e
only an indica i e indica o o he ansi ion.
3. Ma hema ical Model
The equa ions ha desc ibe he low o eal luids a e an exp ession o he basic physi-
cal conse a ion laws o mass and momen um. Physically, hese laws exp ess he balance
o a quan i y J(physical uni depends on he ype o a iable J) in a gi en olume [
25
].
Acco ding o his balance, he ime change o he p ese ed quan i y in he gi en olume V
is equal o he low o his quan i y h ough he a ea S, which ci cumsc ibes his olume,
and i s p oduc ion wi hin he olume V.
∂
∂ ZVJdV +ZSFj(J)njdS =ZVP(J)dV, (7)
whe e Jis he balanced quan i y,
Fj(J)
is he j h componen o he low densi y ec o o
he quan i y Jby he a ea dS,
nj
is he j h componen o he no mal ec o and P (J) is he
p oduc ion densi y o he quan i y J(p oduc ion pe uni ime in olume V). Fo index j,
Eins ein0s summa ion ule is used.
The low o a quan i y Jis de ined as he ans e o his quan i y h ough a uni o
a ea pe uni ime, e.g., mass low and momen um low. The h ee-dimensional model can
be simpli ied in he case o piping sys ems o he wo-dimensional axially symme ical
model. The de ined ma hema ical model cha ac e izes he low o luids in gene al spa ial
geome y, which is p esen ed he e by he in e nal space be ween wo cylinde s, whe e he
inne one o a es. All bounda y condi ions on he ec angula a ea we e o he wall ype.
The a ious o a ional speeds om 10 pm o 1000 pm, in acco dance wi h he physical
expe imen , we e gi en on o o , closing1 and closing2. The s a o was s a iona y; see
Figu e 7. The low was assumed as he iso he mal low, and he physical p ope ies o he
luids a e gi en in Tables 1and 2and Figu e 4.
P ocesses 2021, 9, x FOR PEER REVIEW 9 o 17
expe imen , we e gi en on o o , closing1 and closing2. The s a o was s a iona y; see Fig-
u e 7. The low was assumed as he iso he mal low, and he physical p ope ies o he
luids a e gi en in Tables 1 and 2 and Figu e 4.
Figu e 7. Axisymme ic egion and bounda y condi ions.
The low can be lamina , ansien o u bulen , depending on he ype o luid and
he speed o he inne cylinde . The ma hema ical model was applied in ANSYS Fluen
so wa e, whe e he lamina (DNS) model was used o he lamina low. Fo ansien
and u bulen low, he one-equa ion u bulen model Spala –Allma as and he wo-
equa ion SST k-ω u bulen model we e used. These u bulen models a e sui able o he
low o low Reynolds numbe s and conside he low in he bounda y laye . In addi ion
o molecula iscosi y 𝜇 (Pa⋅s), he u bulen iscosi y 𝜇 (Pas) is in oduced om
om he gene al heo y o u bulence. I s alue a ull u bulence can g ea ly exceed he
molecula iscosi y. The concep o e ec i e iscosi y 𝜇 (Pas) can be in oduced and
is gi en as ollows [4]:
𝜇 =𝜇 +𝜇,(Pas) (8)
The ini e olume me hod was used o sol e he ma hema ical models.
The ma hema ical models in MATLAB and ANSYS Fluen we e sol ed o simpli ied
geome ies (one-dimensional and wo-dimensional) and a simpli ied iscosi y de ini ion
[27,28]. A comp ehensi e solu ion using ANSYS Fluen will be p esen ed in Sec ion 5.
4. Expe imen al Resul s
Hyd aulic oil and e hanol we e used o isualize he single-phase low. In he expe -
imen , he lowes speed is app oxima ely 10 pm, and he maximal speed is pm 1000. The
numbe o o ices (s ips) is a ec ed by he ype o luid and he bounda y condi ions. I
is also used o e i y he nume ical expe imen (0). In Table 3, he low esul s o e hanol
and he oil low o a speed o 10 pm a e shown o illus a ion. Fu he esul s a e used
in Table 4, whe e hey a e also compa ed wi h he nume ical esul s.
Table 3. Flow isualiza ion o a 10- pm speed.
Speed
/
Liquid E hanol Oil
10
A Coue e low is e y e iden in he oil, and s able Taylo o ices a e o med o
speeds g ea e han 130 pm. E hanol shows simila p ope ies; only he c i ical speed o
he o ma ion o Taylo ins abili ies is e y small.
The ollowing di e ences could be obse ed om he expe imen due o signi ican ly
di e en physical p ope ies, especially iscosi y:
Figu e 7. Axisymme ic egion and bounda y condi ions.
The low can be lamina , ansien o u bulen , depending on he ype o luid and
he speed o he inne cylinde . The ma hema ical model was applied in ANSYS Fluen
P ocesses 2021,9, 1832 9 o 16
so wa e, whe e he lamina (DNS) model was used o he lamina low. Fo ansien and
u bulen low, he one-equa ion u bulen model Spala –Allma as and he wo-equa ion
SST k-
ω
u bulen model we e used. These u bulen models a e sui able o he low
o low Reynolds numbe s and conside he low in he bounda y laye . In addi ion o
molecula iscosi y µmol (Pa.s), he u bulen iscosi y µ u b (Pa.s)is in oduced om he
gene al heo y o u bulence. I s alue a ull u bulence can g ea ly exceed he molecula
iscosi y. The concep o e ec i e iscosi y
µe (Pa.s)
can be in oduced and is gi en as
ollows [4]:
µe =µmol +µ u b,(Pa.s)(8)
The ini e olume me hod was used o sol e he ma hema ical models.
The ma hema ical models in MATLAB and ANSYS Fluen we e sol ed o simpli-
ied geome ies (one-dimensional and wo-dimensional) and a simpli ied iscosi y de ini-
ion [
27
,
28
]. A comp ehensi e solu ion using ANSYS Fluen will be p esen ed
in Sec ion 5
.
4. Expe imen al Resul s
Hyd aulic oil and e hanol we e used o isualize he single-phase low. In he expe i-
men , he lowes speed is app oxima ely 10 pm, and he maximal speed is pm 1000. The
numbe o o ices (s ips) is a ec ed by he ype o luid and he bounda y condi ions. I
is also used o e i y he nume ical expe imen (0). In Table 3, he low esul s o e hanol
and he oil low o a speed o 10 pm a e shown o illus a ion. Fu he esul s a e used in
Table 4, whe e hey a e also compa ed wi h he nume ical esul s.
Table 3. Flow isualiza ion o a 10- pm speed.
Speed/Liquid E hanol Oil
10
P ocesses 2021, 9, x FOR PEER REVIEW 9 o 17
expe imen , we e gi en on o o , closing1 and closing2. The s a o was s a iona y; see Fig-
u e 7. The low was assumed as he iso he mal low, and he physical p ope ies o he
luids a e gi en in Tables 1 and 2 and Figu e 4.
Figu e 7. Axisymme ic egion and bounda y condi ions.
The low can be lamina , ansien o u bulen , depending on he ype o luid and
he speed o he inne cylinde . The ma hema ical model was applied in ANSYS Fluen
so wa e, whe e he lamina (DNS) model was used o he lamina low. Fo ansien
and u bulen low, he one-equa ion u bulen model Spala –Allma as and he wo-
equa ion SST k-ω u bulen model we e used. These u bulen models a e sui able o he
low o low Reynolds numbe s and conside he low in he bounda y laye . In addi ion
o molecula iscosi y 𝜇 (Pa⋅s), he u bulen iscosi y 𝜇 (Pas) is in oduced om
om he gene al heo y o u bulence. I s alue a ull u bulence can g ea ly exceed he
molecula iscosi y. The concep o e ec i e iscosi y 𝜇 (Pas) can be in oduced and
is gi en as ollows [4]:
𝜇 =𝜇 +𝜇,(Pas) (8)
The ini e olume me hod was used o sol e he ma hema ical models.
The ma hema ical models in MATLAB and ANSYS Fluen we e sol ed o simpli ied
geome ies (one-dimensional and wo-dimensional) and a simpli ied iscosi y de ini ion
[27,28]. A comp ehensi e solu ion using ANSYS Fluen will be p esen ed in Sec ion 5.
4. Expe imen al Resul s
Hyd aulic oil and e hanol we e used o isualize he single-phase low. In he expe -
imen , he lowes speed is app oxima ely 10 pm, and he maximal speed is pm 1000. The
numbe o o ices (s ips) is a ec ed by he ype o luid and he bounda y condi ions. I
is also used o e i y he nume ical expe imen (0). In Table 3, he low esul s o e hanol
and he oil low o a speed o 10 pm a e shown o illus a ion. Fu he esul s a e used
in Table 4, whe e hey a e also compa ed wi h he nume ical esul s.
Table 3. Flow isualiza ion o a 10- pm speed.
1
A Coue e low is e y e iden in he oil, and s able Taylo o ices a e o med o
speeds g ea e han 130 pm. E hanol shows simila p ope ies; only he c i ical speed o
he o ma ion o Taylo ins abili ies is e y small.
The ollowing di e ences could be obse ed om he expe imen due o signi ican ly
di e en physical p ope ies, especially iscosi y:
P ocesses 2021, 9, x FOR PEER REVIEW 9 o 17
expe imen , we e gi en on o o , closing1 and closing2. The s a o was s a iona y; see Fig-
u e 7. The low was assumed as he iso he mal low, and he physical p ope ies o he
luids a e gi en in Tables 1 and 2 and Figu e 4.
Figu e 7. Axisymme ic egion and bounda y condi ions.
The low can be lamina , ansien o u bulen , depending on he ype o luid and
he speed o he inne cylinde . The ma hema ical model was applied in ANSYS Fluen
so wa e, whe e he lamina (DNS) model was used o he lamina low. Fo ansien
and u bulen low, he one-equa ion u bulen model Spala –Allma as and he wo-
equa ion SST k-ω u bulen model we e used. These u bulen models a e sui able o he
low o low Reynolds numbe s and conside he low in he bounda y laye . In addi ion
o molecula iscosi y 𝜇 (Pa⋅s), he u bulen iscosi y 𝜇 (Pas) is in oduced om
om he gene al heo y o u bulence. I s alue a ull u bulence can g ea ly exceed he
molecula iscosi y. The concep o e ec i e iscosi y 𝜇 (Pas) can be in oduced and
is gi en as ollows [4]:
𝜇 =𝜇 +𝜇,(Pas) (8)
The ini e olume me hod was used o sol e he ma hema ical models.
The ma hema ical models in MATLAB and ANSYS Fluen we e sol ed o simpli ied
geome ies (one-dimensional and wo-dimensional) and a simpli ied iscosi y de ini ion
[27,28]. A comp ehensi e solu ion using ANSYS Fluen will be p esen ed in Sec ion 5.
4. Expe imen al Resul s
Hyd aulic oil and e hanol we e used o isualize he single-phase low. In he expe -
imen , he lowes speed is app oxima ely 10 pm, and he maximal speed is pm 1000. The
numbe o o ices (s ips) is a ec ed by he ype o luid and he bounda y condi ions. I
is also used o e i y he nume ical expe imen (0). In Table 3, he low esul s o e hanol
and he oil low o a speed o 10 pm a e shown o illus a ion. Fu he esul s a e used
in Table 4, whe e hey a e also compa ed wi h he nume ical esul s.
Table 3. Flow isualiza ion o a 10- pm speed.
1
A Coue e low is e y e iden in he oil, and s able Taylo o ices a e o med o
speeds g ea e han 130 pm. E hanol shows simila p ope ies; only he c i ical speed o
he o ma ion o Taylo ins abili ies is e y small.
The ollowing di e ences could be obse ed om he expe imen due o signi ican ly
di e en physical p ope ies, especially iscosi y:
A Coue e low is e y e iden in he oil, and s able Taylo o ices a e o med o
speeds g ea e han 130 pm. E hanol shows simila p ope ies; only he c i ical speed o
he o ma ion o Taylo ins abili ies is e y small.
The ollowing di e ences could be obse ed om he expe imen due o signi ican ly
di e en physical p ope ies, especially iscosi y:
•
Taylo o ices can be obse ed mainly in he a ea o lamina low, so, in e hanol,
hey appea a a lowe speed, while o ex s uc u es in oil a e o med a he speed o
130 pm;
• he wa e mode has no ye mani es ed;
• u bulence in e hanol causes o ex s uc u es o be illegible;
•
he expe imen wi h he EMG 900 luid was no pe o med. I was no possible o
ensu e he low in he annulus and, a he same ime, o in luence i by means o a
magne ic ield ac ing pe pendicula o he di ec ion o load.
P ocesses 2021,9, 1832 16 o 16
10.
Lenci, A.; Chiapponi, L. An Expe imen al Se up o In es iga e Non-New onian Fluid Flow in Va iable Ape u e Channels. Wa e
2020,12, 1284. [C ossRe ]
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Tong, T.A.; Yu, M.; Ozbayoglu, E.; Takach, N. Nume ical simula ion o non-New onian luid low in pa ially blocked eccen ic
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12.
Lacassagne, T.; Cagney, N.; Balabani, S. Shea - hinning media ion o elas o-ine ial Taylo –Coue e low. J. Fluid Mech.
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Kaushik, V.; Wu, S.; Jang, H.; Kang, J.; Kim, K.; Suk, J.W. Scalable Ex olia ion o Bulk MoS2 o Single- and Few-Laye s Using
To oidal Taylo Vo ices. Nanoma e ials 2018,8, 587. [C ossRe ] [PubMed]
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Teng, H.; Liu, N.; Lu, X.; Khomami, B. Di ec nume ical simula ion o Taylo -Coue e low subjec ed o a adial empe a u e
g adien . Phys. Fluids 2015,27, 125101. [C ossRe ]
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Compu a ional Mechanics o Sma Applica ions. Sma Inno a ion, Sys ems and Technologies; Jain, L.C., Fa o skaya, M.N., Niki in,
I.S., Re izniko , D.L., Eds.; Sp inge : Singapo e, 2021; Volume 217. [C ossRe ]
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Wang, H. Expe imen al and Nume ical S udy o Taylo -Coue e Flow. PhD Theses, Iowa S a e Uni e si y, Ames, IA, USA, 2015; p. 14462.
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Baek, S.I.; Ahn, J. La ge Eddy Simula ion o Film Cooling In ol ing Compound Angle Holes: Compa a i e S udy o LES and
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Fialo á, S.; Pochylý, F.; Volko , A.V.; Ryzhenko , A.V.; D uzhinin, A.A. The Ma hema ical Model Simpli ica ion Me hods o
Cacula ing Flows in he Hyd aulic Tu bines Flow Pa h. Teploene ge ika 2021,12, 1–11. (In Russian)
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