The functions bei ν (x) and ber ν (x) are the Kelvin functions of order ν (de Andrade
Lima and Rey 2004a-e). Note that these results have the same pattern of the results
presented for simple shear flow of NLCs. The main viscoelastic parameter ratios that
the dimensionless storage modulus G
0 , the loss modulus G
00 , and the loss tangent
tanδ are respectively M 1 , M 2 and M given by:
M 1 ¼ 8
α 3
γ 1
2
¼ 2 1 À λ
ð
Þ
2
(39)
M 2 ¼
η 1
γ 1
(40)
M ¼
1
8
η 1
α 3
γ 1
α 3
(41)
The storage modulus G
0 increases with M 1 since this ratio increases the viscous
torques creating more elastic storage and less rotational dissipation; when λ = 1,
viscous torques are absent and no elastic storage is possible G
0
= 0. The loss
modulus G
00 increases with M 2 since this ratio increases translation dissipation and
is larger than rotational dissipation; when γ 1 ! 1 rotational dissipation dominates
G
00
= 0. The ratio M is the product of the two dissipation torques ratios and can be
rewritten in terms of λ as:
M 1 ¼
1
2
η 1
γ 1
1
1 À λ
ð
Þ
2
(42)
By using the known asymptotic behavior (de Andrade Lima and Rey 2004a-e) of
the Kelvin functions, the frequency dependence of the viscoelastic moduli for α 3 6 ¼ 0
is as follows: the loss modulus is always greater than the storage modulus, the low
frequency (terminal) regime is classic of a viscous fluid, and characteristic slopes are:
a
Á ω ! 0,G
0
$ ω
2 ,G
00
$ ω; b
Á ω ! 1,G
0
$ ω
1=2 ,G
00
$ ω
(43)
It follows from Eq. 43, the behavior corresponds to a Newtonian material and
G
0
= 0. In addition, at frequencies larger than resonance, the dependency of material
properties simplifies; resonance is found with ω r = 18.6522, and the large frequency
regime results in this section hold for ω > 10 ω r . The factorization of material
properties from frequency dependent functions is obtained by direct order of magnitude analysis. At frequencies larger than the resonance ω r , the terms F 1 /M and F 2 /
M – 1/ωM in Eqs. 32 and 33 are very small compared with one; in addition, in the
numerator of Eq. 33 F 2 /M is less than the unity; therefore, the asymptotic expressions for the viscoelastic functions for frequencies larger than ω r are:
G
0
M 1
¼ ωF 2 À 1 ¼ Φ
0
ω
ð Þ
(44)
300
A. D. Rey et al.
Lima and Rey 2004a-e). Note that these results have the same pattern of the results
presented for simple shear flow of NLCs. The main viscoelastic parameter ratios that
the dimensionless storage modulus G
0 , the loss modulus G
00 , and the loss tangent
tanδ are respectively M 1 , M 2 and M given by:
M 1 ¼ 8
α 3
γ 1
2
¼ 2 1 À λ
ð
Þ
2
(39)
M 2 ¼
η 1
γ 1
(40)
M ¼
1
8
η 1
α 3
γ 1
α 3
(41)
The storage modulus G
0 increases with M 1 since this ratio increases the viscous
torques creating more elastic storage and less rotational dissipation; when λ = 1,
viscous torques are absent and no elastic storage is possible G
0
= 0. The loss
modulus G
00 increases with M 2 since this ratio increases translation dissipation and
is larger than rotational dissipation; when γ 1 ! 1 rotational dissipation dominates
G
00
= 0. The ratio M is the product of the two dissipation torques ratios and can be
rewritten in terms of λ as:
M 1 ¼
1
2
η 1
γ 1
1
1 À λ
ð
Þ
2
(42)
By using the known asymptotic behavior (de Andrade Lima and Rey 2004a-e) of
the Kelvin functions, the frequency dependence of the viscoelastic moduli for α 3 6 ¼ 0
is as follows: the loss modulus is always greater than the storage modulus, the low
frequency (terminal) regime is classic of a viscous fluid, and characteristic slopes are:
a
Á ω ! 0,G
0
$ ω
2 ,G
00
$ ω; b
Á ω ! 1,G
0
$ ω
1=2 ,G
00
$ ω
(43)
It follows from Eq. 43, the behavior corresponds to a Newtonian material and
G
0
= 0. In addition, at frequencies larger than resonance, the dependency of material
properties simplifies; resonance is found with ω r = 18.6522, and the large frequency
regime results in this section hold for ω > 10 ω r . The factorization of material
properties from frequency dependent functions is obtained by direct order of magnitude analysis. At frequencies larger than the resonance ω r , the terms F 1 /M and F 2 /
M – 1/ωM in Eqs. 32 and 33 are very small compared with one; in addition, in the
numerator of Eq. 33 F 2 /M is less than the unity; therefore, the asymptotic expressions for the viscoelastic functions for frequencies larger than ω r are:
G
0
M 1
¼ ωF 2 À 1 ¼ Φ
0
ω
ð Þ
(44)
300
A. D. Rey et al.