DOI:
10.1039/D4SM00061G
(Paper)
Soft Matter, 2024,
20, 5456-5463
Plateau–Rayleigh instability of a soft layer coated on a rigid cylinder
Received
16th January 2024
, Accepted 7th June 2024
First published on 13th June 2024
Abstract
We study the Plateau–Rayleigh instability of a soft viscoelastic solid layer coated on a rigid cylinder i.e., a soft fibre with a rigid core. The onset of instability is examined using a linear stability analysis. We find that increasing the rigid cylinder radius or the stiffness of the layer reduces the growth rate of the fastest growing mode. For each rigid cylinder radius, a critical elastocapillary number is found below which all wavelengths of disturbances are stable. The critical value for a soft fibre with a thick rigid cylindrical core can be several orders of magnitudes larger than that for a totally soft fibre (no rigid core). This highlights the strong stabilizing effect of the rigid core on the system. Increasing the relaxation timescale of the viscoelastic material also slows down the growth of disturbances, but has no effect on the critical elastocapillary number. Interestingly, the wavelength of the fastest growing mode is independent of the rigid cylinder radius for the purely elastic case.
1 Introduction
Significant stress is usually required for a solid to deform, and as such the effects due to surface tension have often been ignored. Soft solids such as elastomers and gels,1 and biological materials such as protein condensates,2 cells and tissues,3 on the other hand, have elastic moduli ranging between kPa and MPa, which means that they can deform much easier. In recent years, there has been significant attention to how capillary effects can lead to soft solid deformations, and fascinating elastocapillary phenomena have been discovered.4–8 The Plateau–Rayleigh instability (PRI),9,10 namely the instability driven by surface tension and often illustrated by the breakup of liquid jets into droplets,11 has currently been examined for fibres made of soft solids.12–21
Studies of the PRI of soft fibres have focused on soft materials that demonstrate elastic or viscoelastic responses.12–21 An experimental study using soft agar gel fibres by Mora et al.14 has shown that the instability occurs when the elastocapillary length γ/μ ≥ 6R, where R is the radius of the soft fibre, γ is the solid surface tension and μ is the shear modulus. Other studies of the PRI on a soft fibre include, for example, the formation of bead-on-string structures.12,16,18,22 Another interesting factor that might significantly modify the instability is having an inner rigid core, i.e. a fibre consists of a soft-layer-coated on a rigid cylinder. Such a kind of setup, but with a coated liquid film instead of a soft solid layer, has been studied extensively23–32 since the early works in the 1960s by Goren.33,34 Recent studies on this problem have addressed the effects of liquid slip on the solid surface28,32 and the dynamics of the droplets formed on the fibre.25 However, investigations on situations in which the coated layer is a soft solid are still missing. How viscoelastic properties and the rigid core influence the PRI of a soft-layer-coated fibre remain unclear, which is addressed in this article.
Soft solids have recently been used in many applications, for example in 3D bioprinting,35 mimicking muscle tissues in biomedicine36 and water harvesting.37 In living organisms, soft fibrous-shaped structures are often found in cellular tubes and compartments of cells. Instability of these biological ingredients have been rationalised through the PRI.38–42 One example is a recent study on the undulation and droplet formation of a layer of condensed protein TPX2 on a microtubule.42 Hence, investigating the PRI of fibrous soft solids is becoming more important for both understanding the fundamental physical problems and the development of new technologies. In this study, we examine the onset of PRI of a soft layer coated on a rigid cylinder through a linear stability analysis.
2 Formulation
We consider a rigid cylinder of radius r0 coated with a layer of incompressible viscoelastic material of density ρ and uniform thickness H at an undeformed state. The whole fibre consisting of the rigid cylinder and the coated soft layer has a total radius R = r0 + H as shown in Fig. 1. Consider that the fiber is thin enough such that the Bond number Bo ≡ ρgR2/γ ≪ 1, where g is the gravitational acceleration, which allows us to neglect the effect of gravity. By only consider viscoelastic stresses, the equation of motion of a soft material element is given by |  | (1) |
where σij(x,t) and ui(x,t) are respectively the stress tensor and the displacement vector in index notation at a position vector x and time t. Assuming a linear viscoelastic response of the incompressible soft material, the relation between σij and the strain tensor εij is |  | (2) |
where μ(t) is the shear relaxation function, p is the pressure, δij is the Kronecker delta, and the strain tensor εij is related to the displacement as |  | (3) |
 |
| Fig. 1 Schematic representation of an infinitely long, rigid cylinder of radius r0 coated with a layer of incompressible viscoelastic material of uniform thickness H at an undeformed state. A random disturbance of the interface of the viscoelastic layer is decomposed into sinusoidal deformations of wavenumber k. In cylindrical coordinate systems, r and z are respectively the radial coordinate and the longitudinal coordinate. | |
There are different models that have been used to describe viscoelastic materials. The simplest models are the Kelvin–Voigt model or the Maxwell model, which consists of a spring and a viscous damper connected in parallel or in series respectively. In this study, we consider the soft material to behave as a gel described by the Chasset–Thirion model assuming a power law response given by
|  | (4) |
where
μ0 is the static shear modulus,
τ is the relaxation timescale of the viscoelastic response,
Γ is the gamma function and
n is a parameter typically smaller than or equal to unity.
To compute the growth rate of disturbance on the viscoelastic layer, we follow the approach delineated by previous studies.17,32 We decompose the time-dependent variables into normal modes est, where s is the growth rate. The amplitude of the normal mode of a function f(t) is obtained by the Laplace transform defined as
|  | (5) |
We apply the Laplace transform to the constitutive relation (2) and obtain
|  | (6) |
where
ij,
ũi and
![[p with combining tilde]](https://www.rsc.org/images/entities/i_char_0070_0303.gif)
are respectively the Laplace transform of
σij,
ui and
p, and
![[small mu, Greek, circumflex]](https://www.rsc.org/images/entities/i_char_e0b3.gif)
(
s) is the shear modulus in Laplace space defined as
|  | (7) |
Thus the governing eqn (1) in Laplace space can be written as
|  | (8) |
Next, we scale the lengths with the radius of the whole fibre R, the time with the inertial-capillary time
and the stresses by μo. We define the following dimensionless variables as
|  | (9) |
|  | (10) |
Note that variables in Laplace space have an additional dimension of time. The dimensionless form of the governing equation (eqn (8)) in Laplace space is
|  | (11) |
where
|  | (12) |
is the inverse of the dimensionless shear modulus in Laplace space which is a function of
![[s with combining macron]](https://www.rsc.org/images/entities/i_char_0073_0304.gif)
and consists of 2 parameters:
Σ ≡
γ/(
μ0R) and
![[small tau, Greek, macron]](https://www.rsc.org/images/entities/i_char_e0d4.gif)
≡
τ/
tc. In general, the expression of
β depends on the viscoelastic model. The parameter
Σ is called the elastocapillary number which compares the capillary pressure to the elastic stress. When
Σ → ∞, capillarity dominates over elasticity. The other parameter
![[small tau, Greek, macron]](https://www.rsc.org/images/entities/i_char_e0d4.gif)
is the ratio of the viscoelastic relaxation timescale to the inertial-capillary time and can be interpreted as the Deborah number since the material responds purely elastically when
![[small tau, Greek, macron]](https://www.rsc.org/images/entities/i_char_e0d4.gif)
= 0. When
![[small tau, Greek, macron]](https://www.rsc.org/images/entities/i_char_e0d4.gif)
→ ∞, viscoelastic effects are expected to dominate over inertial effects. To sum up, the dimensionless control parameters of the system are:
0,
Σ,
![[small tau, Greek, macron]](https://www.rsc.org/images/entities/i_char_e0d4.gif)
and
n.
We consider only the longitudinal disturbance and neglect the azimuthal disturbance as azimuthal normal modes always increase the surface energy.11 The deformation of the soft layer is axisymmetric. We hence use the cylindrical coordinate system (r,ϕ,z) and the corresponding unit vectors are denoted as (
,
,ẑ). The displacement vector is denoted as u(r,z,t) = ur(r,z,t)
+ uz(r,z,t)ẑ. Note that uϕ = 0 due to axisymmetry.
The governing eqn (11) can be solved by applying the Helmholtz decomposition of the displacement as described in ref. 17 and 32. The general solutions are given as
| ( ) = − 2[A1I0(![[k with combining macron]](https://www.rsc.org/images/entities/i_char_006b_0304.gif) ) +A3K0(![[k with combining macron]](https://www.rsc.org/images/entities/i_char_006b_0304.gif) )]ei![[k with combining macron]](https://www.rsc.org/images/entities/i_char_006b_0304.gif) , | (13) |
| ūr = [A1 I1(![[k with combining macron]](https://www.rsc.org/images/entities/i_char_006b_0304.gif) ) − A2(i )I1(α ) − A3 K1(![[k with combining macron]](https://www.rsc.org/images/entities/i_char_006b_0304.gif) ) − A4(i )K1(α )]ei![[k with combining macron]](https://www.rsc.org/images/entities/i_char_006b_0304.gif) , | (14) |
and
| ūz = [A1(i )I0(![[k with combining macron]](https://www.rsc.org/images/entities/i_char_006b_0304.gif) ) + A2αI0(α ) + A3(i )K0(![[k with combining macron]](https://www.rsc.org/images/entities/i_char_006b_0304.gif) ) − A4αK0(α )]ei![[k with combining macron]](https://www.rsc.org/images/entities/i_char_006b_0304.gif) ![[z with combining macron]](https://www.rsc.org/images/entities/i_char_007a_0304.gif) | (15) |
where
![[k with combining macron]](https://www.rsc.org/images/entities/i_char_006b_0304.gif)
≡
kR is the dimensionless wavenumber,
I and
K are respectively the modified Bessel functions of the first and second kind,

is a function of
![[s with combining macron]](https://www.rsc.org/images/entities/i_char_0073_0304.gif)
and
![[k with combining macron]](https://www.rsc.org/images/entities/i_char_006b_0304.gif)
, and is introduced here to simplify the expressions of equations, and
Am(
m = 1,2,3,4) are the undermined coefficients.
Next we impose the boundary conditions. At
=
0 ≡ r0/R, there is no penetration of material, thus
| ūr| = 0 = 0. | (16) |
For the z-direction, we impose a no-slip condition
| ūz| = 0 = 0. | (17) |
At
= 1, we assume the slope of the deformed interface to be small. Balancing the Laplace pressure due to solid surface tension and the viscoelastic stresses gives
|  | (18) |
in the
r-direction, and gives the vanishing shear stress
| rz| =1 = 0 | (19) |
in the
z-direction.
3 Dispersion relation
Using the expressions of the general solutions (eqn (13)–(15)) for the boundary conditions (eqn (16)–(19)) yields a set of linear equations for the unknowns Am which is given in Appendix 6.1. The solvability condition for non-trivial solutions of the linear equations with unknowns Am is that the determinant is vanishing, which gives the following dispersion relation |  | (20) |
where
when
0 → 0, the modified Bessel functions I0(
0) and I1(
0) are bounded, but K0(
0) and K1(
0) diverge. These unbounded terms are eliminated for the case of
0 = 0.17 To examine the feature of the dispersion relation for small
0, we expand the determinant of the dispersion relation (20) in series of
0. The result for the leading order terms is given in Appendix 6.2 which consists of logarithmic terms of
0.
4 Results
The validation of the dispersion relation (eqn (20)) is presented in Appendix 6.3 in which we compare our results in the Newtonian fluid limit with previous studies in the literature. In this section, we first consider the purely elastic solid limit and then we focus on the effects of viscoelasticity.
4.1 Purely elastic solid limit
When the viscoelastic relaxation timescale τ is small compared to the inertial-capillary time
, i.e. taking the limit
→ 0, our viscoelastic model reduces to the purely elastic model. In this limit,
and β = Σ in the dispersion relation (eqn (20)). We examine how the instability depends on the dimensionless parameters Σ and
0.
4.1.1 Dependence on the rigid core radius and the elastocapilliary number.
We show the dispersion relation (eqn (20)) by plotting the dimensionless growth rate
as a function of the dimensionless wavenumber
for different values of Σ and a fixed
0 = 0.1 in Fig. 2(a). We see that for each curve, the unstable modes (i.e.
> 0) lie within a certain range of dimensionless wavenumbers, i.e.
1 <
<
2, where
1 and
2 are defined as
(
1) = 0 and
(
2) = 0. Within each range of unstable modes, there is a dimensionless wavenumber
=
m that corresponds to the fastest growing mode with a maximum dimensionless growth rate
m ≡
(
=
m). To study how the characteristic quantities depend on the control parameters, we plot
m,
m, and
1 (and
2) as a function of Σ respectively in Fig. 2(b)–(d) for three different dimensionless rigid core radii, i.e.
0 = 10−9,
0 = 0.1 and 0.9. We also add the result of Tamim et al.17 for the situation of a soft fibre without a rigid cylindrical core (i.e.
0 = 0). We see in Fig. 2(b) that for a fixed value of Σ,
m is smaller for a larger
0. For all four different
0, the dimensionless growth rate of the fastest growing mode
m decreases with decreasing Σ. When Σ is reduced to a critical value Σc,
m drops to zero. There is no positive solution of
for Σ < Σc. This means that the coated elastic layer is stable under disturbance of any wavelength when Σ < Σc. We would also point out that, firstly, the critical value Σc for
0 = 0.9 is orders of magnitude larger than that for
0 = 0.1. This can be understood as follows: when the coated layer becomes thinner, the strain is larger and hence a softer layer can still generate strong enough elastic stress to resist the growth of disturbance. Secondly, even for the rigid cylindrical core radius as small as
0 = 10−9,
m starts to deviate significantly from that for the case of
0 = 0 when Σ is close to Σc. Moreover, the critical value Σc for
0 = 10−9 (Σc = 8.7) is not the same as that for
0 = 0 (Σc = 6)14,17 but larger. These features will be discussed in details in Section 4.1.2 where a stability phase diagram of Σcvs.
0 will be examined.
 |
| Fig. 2 (a) The dimensionless growth rate as a function of the dimensionless wavenumber for different values of Σ and a fixed 0 = 0.1 and = 0 (purely elastic). (b)–(d): The characteristic quantities m in (b), m in (c), and 1 and 2 in (d) as a function of Σ for four different dimensionless rigid core radius, i.e. 0 = 0 (from Tamim et al.17), 0 = 10−9, 0 = 0.1 and 0.9. Other parameters: = 0. | |
Interestingly, as shown in Fig. 2(c), we find that the dimensionless wavenumber of the fastest growing mode
m is independent of the dimensionless radius of the rigid core
0, and decreases with reducing Σ. It is thus expected that elastic beads formed on a softer coated layer are closer together. When Σ → ∞, we find
m = 0.7, which agrees with the dimensionless wavenumber of the fastest growing mode for the classical PRI of inviscid fluid threads.10,11 Another remarkable point is that
m starts to drop significantly with reducing Σ only when Σ ⪅ 102. For cases with Σc > 102, e.g. for
0 = 0.9, the dimensionless wavenumber of the fastest growing mode is always close to the asymptotic value, i.e.
m ≈ 0.7. Regarding the range of unstable modes, as we can see in Fig. 2(a) and (d), it shrinks when Σ is reduced. In contrast to the
0 = 0 case for which
1 remains zero, for non-zero
0 cases,
1 increases when Σ is reduced. This also means that the transition from unstable fibers to stable fibers (i.e. at Σ = Σc) occurs at
m ≠ 0, which needs to be verified experimentally. It is expected that the bead separation formed on a soft fiber with a rigid core is finite (
m ≠ 0) at the transition, instead of diverging for the
0 = 0 case.
4.1.2. Stability phase diagram.
To show how the onset of instability depends on Σ and
0, we plot Σc (in log scales) as a function of
0 in a phase diagram, see Fig. 3(a). We see that increasing the radius of the rigid cylindrical core or decreasing the elastocapillary number can make the coated film more stable. Hence both the stiffness and the rigid core have a stabilizing effect. In Fig. 3(b), we plot Σc as a function of the thickness of the coated layer, i.e.
= 1 −
0, in log–log scales. We find that when
0 ⪆ 0.2, the curve follows a power law Σc = 21.5
λ where λ ≈ −2.75. Hence the critical value Σc for a thin coated elastic film is orders of magnitudes larger than a thick film. In the opposite limit of
0 → 0, we find that our results can be approximately described by a logarithmic relation given as |  | (21) |
where Σco = 6 is the critical value for a soft elastic fibre without a rigid cylindrical core obtained by Mora et al.14 or Tamim et al.,17 and a = 37 is a fitting parameter. The comparison is shown in Fig. 3(c) for small
0. This result is consistent with the series expansion of the dispersion relation (eqn (20)) for small
0 given in Appendix 6.2.
 |
| Fig. 3 (a) Stability phase diagram. Line: the critical elastocapillary number Σc (log scale) as a function of 0. (b) The critical elastocapillary number Σc as a function of 1 − 0 in log–log scales. (c) A linear-log plot of the critical elastocapillary number Σc as a function of 0. | |
4.2. Viscoelastic effects
We study the instability when
≠ 0. We show results for two different rigid cylindrical core radii, namely
0 = 0.1 (a thick coated layer) and
0 = 0.9 (a thin coated layer). Since the critical elastocapillary numbers for
0 = 0.1 and
0 = 0.9 are different by orders of magnitude, we take a fixed value of Σ = 50 for the case of
0 = 0.1 and a fixed value of Σ = 15
000 for the case of
0 = 0.9. In Fig. 4(a) and (b), we present the dispersion relation from eqn (20) for different
, and in Fig. 4(c) and (d) for different n. We see that both
1 and
2 are independent of
and n. The dimensionless growth rate of the fastest growing mode
m decreases when
is enhanced. This means that the viscoelastic relaxation of the material slows down the growth of disturbance. We also see that when
is increased, the dimensionless wavenumber of the fastest growing mode becomes smaller for
0 = 0.1 but remains constant for
0 = 0.9. When varying the other viscoelastic parameter n, the dimensionless growth rate of the fastest growing mode
m decreases with decreasing
. Hence smaller values of n have a larger damping effect. The change of
m with varying
is more sensitive for
0 = 0.9 than for
0 = 0.1 as we can see in Fig. 4(c) and (d). The dimensionless wavenumber of the fastest growing mode
m decreases slightly when n is increased from n = 0.2 to n = 1 for
0 = 0.1 but remains constant for
0 = 0.9.
 |
| Fig. 4 The dimensionless growth rate as a function of the dimensionless wavenumber for different values of and a fixed value of exponent n = 1 with 0 = 0.1 and Σ = 50 in (a), and 0 = 0.9 and Σ = 15 000 in (b). The dimensionless growth rate vs. the dimensionless wavenumber for different values of exponent n and a fixed value of = 1 with 0 = 0.1 and Σ = 50 in (c), and 0 = 0.9 and Σ = 15 000 in (d). | |
Next, we take
= 100 and n = 0.5, and show the results for
m and
m as a function of Σ for three different
0 = 0, 0.1 and 0.9 in Fig. 5. We see that the behaviors of
m and
m for
= 100 are similar to those shown in Fig. 2(b) and (c) for the purely elastic case, i.e.
= 0, except that when Σ is close to the critical value Σc. When Σ is approaching Σc, the curves for both
m and
m bend concavely, and the value of
m becomes dependent on
0. However, varying the values of
or n does not change the critical values Σc, namely Σc remains the same as for the purely elastic case (
= 0). When Σ → ∞, viscoelastic effects disappear, and the results are the same as for the PRI of an inviscid fluid.11
 |
| Fig. 5 The characteristic quantities m in (a) and m in (b) as a function of Σ for three different dimensionless rigid core radii, i.e. 0 = 0 (from Tamim et al.17), 0 = 0.1 and 0.9. Other parameters: = 100 and n = 0.5. | |
It is also interesting to investigate the situation when keeping Σ finite, and taking the limit that the Deborah number
→ ∞. It is then expected that viscoelastic effects dominate over inertial effects. We then introduce a dimensionless growth rate using the viscoelastic timescale τ (instead of the inertial-capillary time tc) for the rescaling, i.e.
τ ≡ sτ = ![[s with combining macron]](https://www.rsc.org/images/entities/i_char_0073_0304.gif)
. The expression of β can be written as β = Σ/(1 +
nτ). In the limit of
→ ∞,
=
τ/
→ 0, and thus α =
. Hence the dispersion relation (eqn (20)) consists of only the quantities
0, β and
. Inertia plays no role in this limit. A plot of β vs.
from eqn (20) for
0 = 0.1 is shown in Fig. 6(a). The dimensionless maximum growth rate
τ,m(
m) can be computed from the value of βm, defined as the minimum of β. Namely,
τ,m = [Σ/βm − 1]1/n. We can then find the critical elastocapillary number by putting
τ,m = 0, and thus obtain Σc = βm. For
0 = 0.1, we find that Σc = 21.7 (see Fig. 6(a)), which is the same as for the purely elastic case.
 |
| Fig. 6 (a) The relation between β and for the limiting case of → ∞. (b) and (c) The characteristic quantities m in (b) and m in (c) as a function of Σ for different values of . In (c) for → ∞, the dashed-dotted curve represents τ,m/100 vs. Σ. Other parameters: 0 = 0.1 and n = 0.5. | |
Lastly, we compare the results when varying
. In Fig. 6(b), we plot
mvs. Σ for
= 0, 100 and when
→ ∞. We can see that the curve of
m for
= 100 becomes flat when Σ is approaching Σc, which collapses with the constant value
m = 0.586 for the case of
→ ∞. In Fig. 6(c), we plot
mvs. Σ for
= 0 and 100. To compare the results of
= 100 and the case of
→ ∞, we divide
τ,m for the case of
→ ∞ by the value
= 100 since
=
τ/
, and add a curve for
τ,m/100 vs. Σ in Fig. 6(c). We can see that for Σ close to Σc,
m for τ = 100 and
τ,m/100 collapse. Hence for
≠ 0, when Σ is approaching Σc,
m ≪ 1 and viscoelastic effects dominate over inertial effects.
5 Conclusions
We investigate the onset of the PRI of a soft layer coated on a rigid cylinder by analytically deriving the dispersion relation using the linear stability analysis. We implement the Chasset–Thirion model for the viscoelastic response of the soft layer. We find that the stiffness (characterized by 1/Σ) and the rigid cylindrical core (characterized by
0) have a stabilizing effect. The dimensionless growth rate of the fastest growing mode
m decreases with decreasing Σ or increasing
0. Importantly, there exists a critical elastocapillary number Σc for each
0 such that the coated layer is stable for any Σ < Σc. The critical value depends strongly on
0. For example, for
0 = 0.9, the soft layer becomes unstable only when the fibre is very soft, namely when Σ ≥ 12
100. While for a soft fibre without a rigid core (
0 = 0), the onset of instability occurs at Σ ≈ 6, which is three to four orders of magnitudes smaller. This remarkable result would be interesting to be verified by performing experiments for different soft coatings on a fibre. Regarding the dimensionless wavenumber of the fastest growing mode
m, we find that
m decreases with decreasing Σ. Interestingly,
m is independent of
0 for the purely elastic case.
Regarding the roles of the viscoelastic parameters
and n, we find that increasing the relaxation timescale of viscoelastic material
or reducing the power n can slow down the growth of disturbance. Changing either
or n has no effect on the critical elastocapillary number. However, for
≠ 0, the curves for both
mvs. Σ and
mvs. Σ bend concavely when Σ is approaching Σc, and viscoelastic effects become dominant over inertial effects. The value of
m becomes dependent on
0 when Σ is around Σc. On the other hand, viscoelastic effects disappear in the limit of Σ → ∞.
Our linear stability analysis predicts the threshold of instability14 and the fastest growing modes. When the undulation of the interface grows, elastic stresses become larger and could stop the growth.14,18 To study the evolution of the undulation and the equilibrium shape of the deformed soft layer, nonlinear elastic or viscoelastic models are required as large deformation is expected.18
6 Appendices
6.1 Expressions of the boundary conditions
Using the expressions of the general solutions (eqn (13)–(15)) for the boundary conditions (eqn (16)–(19)) yields the following set of linear equations for the unknowns Am. | A1 I1( 0 ) − A2i I1( 0α) − A3 K1( 0 ) − A4i K1( 0α) = 0, | (22) |
| A1(i )I0( 0 ) + A2αI0( 0α) + A3(i )K0( 0 ) − A4αK0( 0α) = 0, | (23) |
|  | (24) |
and | A12i 2I1( ) + A2( 2 + α2)I1(α) − A32i 2K1( ) + A4( 2 + α2)K1(α) = 0 | (25) |
6.2 Dispersion relation for small core radii
We expand the determinant D(
,
,
0,
,n) of the dispersion relation (eqn (20)) in series of
0 and the leading order terms are given as follows: | D( , ; 0, , ,n) = F1 + F2ln( 0) + …, | (26) |
where | F1 = 2(C2D1 − C1D2)γE(α − )(α + ) − 2iα2(B2D1 − B1D2) K0(α) + i(B2D1−B1D2) (α2+ 2)K0( ) + α[2α2(B2C1 − B1C2) − i(2 + β)(B2D1 − B1D2) + 2(B2C1 − B1C2) 2 + iβ(B2D1 − B1D2) 3]K1(α) + i 2[(2 + β)(B2D1 − B1D2) + 4i(B2C1 − B1C2) + β(−B2D1 + B1D2) 2]K1( ), | (27) |
| F2 = 2α2ln(α/2)(C2D1 − C1D2) + 2(−C2D1 + C1D2) 2ln( /2) | (28) |
and
B2 = αI0( 0α), |
D1 = 2i 2I1( ), |
D2 = ( 2 + α2)I1(α), |
γ
E is Euler's constant ≈ 0.57721.
6.3 Validation of our model: the Newtonian fluid limit
Our viscoelastic model reduces to the Newtonian fluid when n = 1 and in the limits of μo → 0 and μoτ → η, where η is the dynamic viscosity of the fluid. In terms of the dimensionless parameters, it means Σ → ∞ and
, where Oh is called the Ohnesorge number. In this limiting situation, the dimensionless governing eqn (11) is reduced to |  | (29) |
where we have introduced a new rescaled growth rate ŝ ≡ ηRs/γ = Oh
. The dispersion relation is obtained by substituting
and β = 1/ŝ in eqn (20). The dimensionless control parameters are Oh and
0.
6.3.1 Stokes flow case.
Taking further the limit that Oh → ∞, eqn (29) reduces to the Stokes equation in Laplace space. The remaining dimensionless control parameter is only
0. We validate the expression of our dispersion relation (eqn (20)) in this limiting case by comparing our results with that of Zhao et al.32 in which the dispersion relation is obtained using the normal mode method to solve the Stokes equations. Fig. 7 shows the comparison for the cases of very small fibre radius (
0 = 10−9), thick coated liquid film (
0 = 0.1) and thin coated liquid film (
0 = 0.9).
 |
| Fig. 7 The dispersion relation between the growth rate ŝ and the wavenumber in the Stokes flow limit for 0 = 10−9 in (a), 0 = 0.1 in (b) and 0 = 0.9 in (c). Solid lines: our results when substituting α = and β = 1/ŝ for eqn (20). Dashed lines: results of Zhao et al.32 | |
Data availability
Data for this article, including the codes used to compute the data shown in the figures, are available at https://doi.org/10.11582/2024.00099.
Conflicts of interest
There are no conflicts to declare.
Acknowledgements
The authors gratefully acknowledge financial support from the Research Council of Norway (Project No. 315110).
References
- Y. Guo, J. Bae, Z. Fang, P. Li, F. Zhao and G. Yu, Chem. Rev., 2020, 120, 7642–7707 CrossRef CAS PubMed.
- Z. G. Qian, S. C. Huang and X. X. Xia, Nat. Chem. Biol., 2022, 18, 1330–1340 CrossRef CAS PubMed.
- I. Levental, P. C. Georges and P. A. Janmey, Soft Matter, 2007, 3, 299–306 RSC.
- B. Roman and J. Bico, J. Phys.: Condens. Matter, 2010, 22, 493101 CrossRef CAS PubMed.
- A. Jagota, D. Paretkar and A. Ghatak, Phys. Rev. E, 2012, 85, 051602 CrossRef PubMed.
- M. L. Wei, Europhys. Lett. (EPL), 2014, 106, 14002 CrossRef.
- R. W. Style, A. Jagota, C. Y. Hui and E. R. Dufresne, Annual Rev. Cond. Matter Phys., 2017, 8, 99–118 CrossRef CAS.
- J. Bico, É. Reyssat and B. Roman, Annu. Rev. Fluid Mech., 2018, 50, 629–659 CrossRef.
- J. A. F. Plateau, Acad. Sci. Bruxelles Mem., 1843, 16, 3 Search PubMed.
- L. Rayleigh, Proc. Roy. Soc. London, 1879, 10, 4 Search PubMed.
- J. Eggers and E. Villermaux, Rep. Progr. Phys., 2008, 71, 036601 CrossRef.
- M. Taffetani and P. Ciarletta, Phys. Rev. E, 2015, E91, 032413 CrossRef PubMed.
- C. Xuan and J. Biggins, Phys. Rev. E, 2017, 95, 1–8 CrossRef PubMed.
- S. Mora, T. Phou, J. M. Fromental, L. M. Pismen and Y. Pomeau, Phys. Rev. Lett., 2010, 105, 1–4 CrossRef PubMed.
- C. Lestringant and B. Audoly, Proc. R. Soc. A, 2020, 476, 20200337 CrossRef.
- Y. Fu, L. Jin and A. Goriely, J. Mech. Phys. Solids, 2021, 147, 104250 CrossRef.
- S. I. Tamim and J. B. Bostwick, Soft Matter, 2021, 17, 4170–4179 RSC.
- A. Pandey, M. Kansal, M. A. Herrada, J. Eggers and J. H. Snoeijer, Soft Matter, 2021, 17, 5148–5161 RSC.
- B. Dortdivanlioglu and A. Javili, Extreme Mech. Lett., 2022, 55, 101797 CrossRef.
- G. Yang, C. F. Gao and C. Q. Ru, Int. J. Solids Struct., 2022, 241, 111491 CrossRef.
- C. Q. Ru, Mech. Res. Communicat., 2022, 124, 103959 CrossRef.
- P. P. Bhat, S. Appathurai, M. T. Harris, M. Pasquali, G. H. McKinley and O. A. Basaran, Nat. Phys., 2010, 6, 625–631 Search PubMed.
- D. Quéré, Annu. Rev. Fluid Mech., 1999, 31, 347–384 CrossRef.
- H.-C. Chang and E. A. Demekhin, J. Fluid Mech., 1999, 380, 233–255 CrossRef CAS.
- I. L. Kliakhandler, S. H. Davis and S. G. Bankoff, J. Fluid Mech., 2001, 429, 381–390 CrossRef CAS.
- C. Ruyer-quil, P. Treveleyan, F. Giorgiutti-dauphiné, C. Duprat and S. Kalliadasis, J. Fluid Mech., 2008, 603, 431–462 CrossRef.
- Y. Zheng, H. Bai, Z. Huang, X. Tian, F.-Q. Nie, Y. Zhao, J. Zhai and L. Jiang, Nature, 2010, 463, 640–643 CrossRef CAS PubMed.
- S. Haefner, M. Benzaquen, O. Bäumchen, T. Salez, R. Peters, J. D. McGraw, K. Jacobs, E. Raphaël and K. Dalnoki-Veress, Nat. Commun., 2015, 6, 7409 CrossRef CAS PubMed.
- A. Sadeghpour, Z. Zeng and Y. S. Ju, Langmuir, 2017, 33, 6292–6299 CrossRef CAS PubMed.
- H. Chen, T. Ran, Y. Gan, J. Zhou, Y. Zhang, L. Zhang, D. Zhang and L. Jiang, Nat. Mater., 2018, 17, 935–942 CrossRef CAS PubMed.
- H. Ji, C. Falcon, A. Sadeghpour, Z. Zeng, Y. S. Ju and A. L. Bertozzi, J. Fluid Mech., 2019, 865, 303–327 CrossRef CAS.
- C. Zhao, Y. Zhang and T. Si, J. Fluid Mech., 2023, 954, 1–22 Search PubMed.
- S. L. Goren, J. Fluid Mech., 1962, 12, 309–319 CrossRef.
- S. L. Goren, J. Colloid Sci., 1964, 19, 81–86 CrossRef CAS.
- Y. Zhang, S. T. Ellison, S. Duraivel, C. D. Morley, C. R. Taylor and T. E. Angelini, Bioprinting, 2021, 21, e00121 CrossRef.
- M. M. Fitzgerald, K. Bootsma, J. A. Berberich and J. L. Sparks, Biomacromolecules, 2015, 16, 1497–1505 CrossRef CAS PubMed.
- D. K. Nandakumar, Y. Zhang, S. K. Ravi, N. Guo, C. Zhang and S. C. Tan, Adv. Mater., 2019, 31, 1806730 CrossRef PubMed.
- R. Bar-Ziv and E. Moses, Phys. Rev. Lett., 1994, 73, 1392–1395 CrossRef PubMed.
- E. Hannezo, J. Prost and J.-F. M. C. Joanny, Phys. Rev. Lett., 2012, 109, 018101 CrossRef PubMed.
- D. Gonzalez-Rodriguez, S. Sart, A. Babataheri, D. Tareste, A. I. Barakat, C. Clanet and J. Husson, Phys. Rev. Lett., 2015, 115, 088102 CrossRef PubMed.
- M. R. King and S. Petry, Nat. Commun., 2020, 11, 270 CrossRef CAS PubMed.
- S. U. Setru, B. Gouveia, R. Alfaro-Aco, J. W. Shaevitz, H. A. Stone and S. Petry, Nat. Phys., 2021, 17, 493–498 Search PubMed.
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