Figure captions:
Fig.S1 S1a: nanofluid image for Fe3O4 in PEG at x1=0.3995×10-2. S1b: nanofluid image for Fe3O4 nanoparticles
coated with oleic acid - PEG with PEG: oleic acid ratio of 1:1 at different volume fractions of Fe3O4.
Fig. S2 S2a: Shear stress versus shear rate for PEG-oleic acid solution with PEG mole fraction of 0.44446. S2b:
Shear viscosity versus shear rate for PEG-oleic acid solution with PEG mole fraction of 0.44446.
Fig. S3 S3a: Shear stress versus shear rate for nanofluid of Fe3O4 nanoparticles coated with oleic acid dispersed
in PEG (PEG: oleic acid ratio of 1:1) at φ1=0.301%, different magnetic field and T=298.15 K. S3b: Shear
viscosity versus shear rate for nanofluid of Fe3O4 nanoparticles coated with oleic acid dispersed in PEG (PEG:
oleic acid ratio of 1:1) at φ1=0.301%, different magnetic field and T=298.15 K.
Fig. S4 S4a: Shear stress versus shear rate for nanofluid of Fe 3O4 nanoparticles coated with oleic acid dispersed
in PEG (PEG: oleic acid ratio of 1:1) at φ1=1.502%, different magnetic field and T=298.15 K. S4b: Shear
viscosity versus shear rate for nanofluid of Fe3O4 nanoparticles coated with oleic acid dispersed in PEG (PEG:
oleic acid ratio of 1:1) at φ1=1.502%, different magnetic field and T=298.15 K.
Fig. S5 S5a: Shear stress versus shear rate for nanofluid of Fe 3O4 nanoparticles coated with oleic acid dispersed
in PEG (PEG: oleic acid ratio of 1:1) at φ1=4.440%, different magnetic field and T=298.15 K. S5b: Shear
viscosity versus shear rate for nanofluid of Fe3O4 nanoparticles coated with oleic acid dispersed in PEG (PEG:
oleic acid ratio of 1:1) at φ1=4.440%, different magnetic field and T=298.15 K.
Fig. S6 S6a: Shear stress versus shear rate for nanofluid of Fe3O4 nanoparticles coated with oleic acid dispersed
in PEG (PEG: oleic acid ratio of 1:1) at φ1=6.483%, different magnetic field and T=310.15 K. S6b: Shear
viscosity versus shear rate for nanofluid of Fe3O4 nanoparticles coated with oleic acid dispersed in PEG (PEG:
oleic acid ratio of 1:1) at φ1=6.483%, different magnetic field and T=310.15 K.
Fig. S7 S7a: Shear stress versus shear rate for nanofluid of Fe 3O4 nanoparticles coated with oleic acid dispersed
in PEG (PEG: oleic acid ratio of 1:1) at φ1=6.483%, different magnetic field and T=323.15 K. S7b: Shear
viscosity versus shear rate for nanofluid of Fe3O4 nanoparticles coated with oleic acid dispersed in PEG (PEG:
oleic acid ratio of 1:1) at φ1=6.483%, different magnetic field and T=323.15 K.
1
Fig. S8 Experimental and calculated isentropic compressibility, κs, plotted against mole fraction of Fe 3O4
nanoparticle, x1, or PEG, x2, for nanofluids of Fe3O4 – PEG and Fe3O4 coated with oleic acid – PEG (PEG: oleic
acid ratio of 1:1) and PEG-oleic acid solution at different temperatures.
2
S1a
S1b
Fig. S1
3
S2a
shear stress (Pa)
90
80
experimental
70
Herschel-Bulkley model
60
50
40
30
20
10
0
0
200
400
600
800
1000
shear rate (s-1 )
S2b
shear viscosity (Pa.s)
2.5
experimental
2
Carreau-Yasuda model
1.5
1
0.5
0
0
200
400
600
shear rate (s-1 )
Fig. S2
4
800
1000
S3a
120
0
18.090 kA/m
36.347 kA/m
54.586 kA/m
91.013 kA/m
72.808 kA/m
shear stresss (Pa)
100
80
60
40
20
0
0
200
400
600
800
1000
shear rate (s-1 )
S3b
0.3
120
shear viscosity (Pa.s)
0.25
80
0.2
40
0
0.15
0
0.2
0.4
0.6
0.8
1
0.1
0
36.347 kA/m
72.808 kA/m
0.05
18.090 kA/m
54.586 kA/m
91.013 kA/m
0
0
200
400
shear rate
Fig. S3
5
600
(s-1 )
800
1000
S4a
300
0
18.090 kA/m
36.347 kA/m
54.586 kA/m
91.013 kA/m
72.808 kA/m
shear stress (Pa)
250
200
150
100
50
0
0
200
400
600
800
1000
shear rate (s-1 )
S4b
0
36.347 kA/m
91.013 kA/m
shear viscosity (Pa.s)
1
18.090 kA/m
72.808 kA/m
54.586 kA/m
700
0.9
600
0.8
500
400
0.7
300
0.6
200
0.5
100
0
0.4
0
0.2
0.4
0.6
0.8
1
0.3
0.2
0.1
0
0
200
400
600
shear rate (s-1 )
Fig. S4
6
800
1000
S5a
0
36.347 kA/m
91.013 kA/m
800
18.090 kA/m
72.808 kA/m
181.774 kA/m
shear stresss (Pa)
700
600
500
400
300
200
100
0
0
200
400
600
800
1000
shear rate (s-1 )
S5b
0
36.347 kA/m
72.808 kA/m
3
18.090 kA/m
54.586 kA/m
91.013 kA/m
shear viscosity (Pa.s)
8000
2.5
6000
4000
2
2000
1.5
0
0
1
0.1
0.2
0.3
0.4
0.5
0
0
200
400
600
shear rate (s-1 )
Fig. S5
7
800
1000
S6a
1200
shear stress (Pa)
1000
800
600
0
181.774 kA/m
272.099 kA/m
361.988 kA/m
91.013 kA/m
400
200
0
0
200
400
600
800
1000
shear rate (s-1 )
S6b
0
272.099 kA/m
91.013 kA/m
5
14000
12000
10000
8000
6000
4000
2000
0
4.5
shear viscosity (Pa.s)
181.774 kA/m
361.988 kA/m
4
3.5
3
2.5
0
2
0.1
0.2
0.3
0.4
1.5
1
0.5
0
0
200
400
600
shear rate
Fig. S6
8
(s-1 )
800
1000
S7a
1200
shear stress (Pa)
1000
800
600
91.013 kA/m
181.774 kA/m
272.099 kA/m
361.988 kA/m
0
400
200
0
0
200
400
600
800
1000
(s-1 )
shear rate
S7b
0
272.099 kA/m
181.774 kA/m
10
91.013 kA/m
361.988 kA/m
16000
14000
shear viscosity (Pa.s)
8
12000
10000
8000
6
6000
4000
2000
4
0
0
0.1
0.2
0.3
0.4
2
0
0
200
400
600
shear rate
Fig. S7
9
(s-1 )
800
1000
700
390
298.15 K
318.15 K
293.15 K
650
600
308.15 K
polynomial
385
380
Fe3O4-PEG
κs (TPa-1)
500
450
400
370
365
PEG-oleic acid
360
25
35
355
polynomial
45
350
350
345
300
0
0
0.2
0.4
x2
0.6
0.8
0.5
1
1
510
500
Fe3O4-oleic acid-PEG
490
480
293.15 K
308.15 K
470
298.15 K
318.15 K
460
450
440
430
0
0.3
0.6
1.5
100.x 1
520
κ s (TPa-1 )
κ s (TPa-1 )
375
550
0.9
1.2
1.5
100.x 1
Fig. S8
10
1.8
2.1
2.4
2
2.5
3
Eyring-NRTL-Carreau-Yasuda model used in this work is as follow
V  [1  ( ) ]
a
A
JI
(
n1
)
a
[exp(  ln( V )   ln( V )    (
1
 a JI  bJI T , G
JI
1 1
 exp( 
2
1 2 
1
2 2
A G
21 21
 G

2 21
A G
))   V ]   V (S1a)
12 12

2
G
1 12
A
JI
RT
).
(S1b)
In above relations V1and V2 are the molar volume of Fe3O4 nanoparticles and PEG, respectively; η1and η2 are the
viscosity of Fe3O4 nanoparticles and PEG, respectively;
xV
of component I, equal to
 is high shear rate viscosity, φI is the volume fraction
in which xI is the mole fraction of component I. T is temperature and R is
I I
2
 ( x J VJ )
J 1
the universal constant of gases. a12, b12, a21, b21, a, n and λ are empirical parameters of Eyring-NRTL-CarreauYasuda model. Subscripts 1 and 2 stand for Fe3O4 nanoparticles and PEG molecules, respectively. α is the non
randomness factor which was set to 0.2 in this work. Viscosity of Fe 3O4 nanoparticles is treated as an adjustable
parameter and set to 1000 for obtaining good result.
Eyring-mNRF-Carreau-Yasuda model is as following equation
V  [1  ( ) ]
a
X 2
X
s s
(

n1
)
a
1
[exp( X ln( V )  X ln( V )  X
1
s
X X
2
 X 2
s s
1 1
X X
2
s s
X X
2
2

s
X X
2
2 2
2 w
1
1
X X
X 
2 s w X  X 
2 w
X X
2
2
s w
X X
2
2
s w
s
X
s s
s s
)   V ]   V
s
(S2a)
 w

 

  w  ,  s  exp   s   s 
 Z

 Z

(S2b)
rJ x J
(S2c)
 w  exp  
Xs 
x2
, XJ 
3
 r xI
I 1
I
3
 r xI ,
I 1
I
11
where r2 approximates the ratio of the molar volume of the PEG and Fe3O4 nanoparticles; and
s ,  s
r1  1 ; w , w ,
a, n and λ are the parameters of Eyring-mNRF-Carreau-Yasuda model. Z is the nonrandom factor which
was set to 8 in this work.
The Krieger-Dougherty-Carreau-Yasuda model is as follow
  [1  ( ) ]
a
(
n1
)
a
[ 2 (1 
1
 max
)
[ ]max
  ]  
(S3)
where  max is the maximum particle volume fraction and [ ] is the intrinsic viscosity. These quantities along
with a, n and λ were set as empirical parameters in this work.
Table S1
Parameters of Eyring-NRTL-Carreau-Yasuda, Eyring-mNRF-Carreau-Yasuda and Krieger-Dougherty-CarreauYasuda models along with absolute average relative deviation, AARD,a and standard deviation, σ, b obtained
from fitting the viscosity values of Fe3O4-PEG nanofluids
Eyring-NRTL-Carreau-Yasuda model
b21
λ
a
a12
b12
a21
0.06305
0.3766
0.06305
m
s
m
26.54
0.2555
6.284
0.6404
0.5569
0.1193
Eyring-mNRF-Carreau-Yasuda model
λ
a
s
0.3691
1.505
0.1467
Krieger-Dougherty- Carreau-Yasuda model
[ ]
λ
a
 max
0.03733
a
1
AARD 
N Y
N
 iexp   ical
i 1
 iexp

N
parameters.  
b
 (
i 1
exp
i
in which N
1.945
880.8
-0.7828
n
-0.964
n
8229
100.AARD
(σ (Pa.s))
9.32
(0.012)
100.AARD
(σ (Pa.s))
9.64
(0.013)
100.AARD
(σ (Pa.s))
3.89
(0.009)
is the total number of data points and Y is the number of
 ical ) 2
N Y
-0.808
n
.
12