3.1 Derivation of the dispersion relation

First of all, it is important to emphasise that in the present work we are only concerned with axisymmetric disturbances, since they are the only unstable modes in the case of clean interfaces. Therefore, the axisymmetric continuity, radial momentum and axial momentum equations,

are satisfied by the velocity and pressure fields in the liquid thread. In addition to the stress balances (2.9) and (2.10), the kinematic condition has to be satisfied at the free surface, $r=a(z,t)$ ,

Since we also assume isothermal flow, $\unicode[STIX]{x1D70E}$ , $\unicode[STIX]{x1D707}_{s}$ and $\unicode[STIX]{x1D705}_{s}$ are functions of only the surface concentration of surfactant $\unicode[STIX]{x1D6E4}(z,t)$ , which satisfies the following transport equation at  $r=a(z,t)$ :

Note that the time derivative in (3.5) is referred to a frame of reference fixed in space, i.e. the laboratory frame – for details on the subtleties of the time derivative of a surface quantity, see Stone (Reference Stone1990), Wong, Rumschitzki & Maldarelli (Reference Wong, Rumschitzki and Maldarelli1996) and Pereira & Kalliadasis (Reference Pereira and Kalliadasis2008). Furthermore, the surface diffusion has been neglected due to the large typical values of the associated Péclet number (Timmermans & Lister Reference Timmermans and Lister2002). Indeed, the typical values of surfactant surface diffusivities $D_{s}$ are in the range of $10^{-10}{-}10^{-9}~\text{m}^{2}~\text{s}^{-1}$ (Tricot Reference Tricot, Schweizer and Kistler1997; Valkovska & Danov Reference Valkovska and Danov2000; Liao, Franses & Basaran Reference Liao, Franses and Basaran2006), which leads to surface Péclet numbers, defined as $Pe_{s}=R^{2}/(D_{s}t_{c})=[\unicode[STIX]{x1D70E}_{0}R/(\unicode[STIX]{x1D70C}D_{s}^{2})]^{1/2}$ , of order $Pe_{s}\sim 10^{6}{-}10^{5}$ for a water thread of 1 mm radius and taking the capillary time as characteristic time scale, $t_{c}=(\unicode[STIX]{x1D70C}R^{3}/\unicode[STIX]{x1D70E}_{0})^{1/2}$ .

All the flow variables are assumed to be slightly perturbed around a stationary and uniform state in which the liquid thread is considered infinitely long with an initial cylindrical shape of radius $R$ (see figure 1),

where $\unicode[STIX]{x1D716}\ll 1$ and $\unicode[STIX]{x1D70E}_{0}$ , $\unicode[STIX]{x1D705}_{s0}$ and $\unicode[STIX]{x1D707}_{s0}$ are the uniform values of surface tension, dilatational and shear surface viscosity, respectively, associated with an initially uniform superficial surfactant concentration, $\unicode[STIX]{x1D6E4}_{0}$ . Note that the reference frame has been selected such that the unperturbed thread is at rest. The linearised continuity, radial momentum and axial momentum equations read, respectively,

with a harmonic pressure disturbance field,

Finally, the linearised boundary conditions at the free surface, $r=R$ , are

representing, respectively, the normal and tangential stress conditions, the kinematic condition, the surfactant transport equation and the constitutive equation for the dependence of surface tension on surfactant concentration, $\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D6E4})$ , where $E=-\unicode[STIX]{x1D6E4}_{0}(\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4})|_{\unicode[STIX]{x1D6E4}_{0}}$ is the Gibbs elasticity. The linearised versions of the stress boundary conditions taking into account the effect of surface viscosities, i.e. (3.11) and (3.12), are identical to those of Whitaker (Reference Whitaker1976). To derive a dispersion relation $D(k,\unicode[STIX]{x1D714})=0$ between the axial wavenumber $k$ and the temporal growth rate $\unicode[STIX]{x1D714}$ , the disturbances are decomposed as normal modes,

where the pressure amplitude $\hat{p}$ satisfies the modified Bessel equation,

with regular solution

where $A$ is an integration constant and $\text{I}_{n}(kr)$ is the nth-order modified Bessel function of the first kind. The linearised system of equations (3.7)–(3.9) and (3.13)–(3.15) can be straightforwardly solved for $\hat{u} (r)$ , ${\hat{w}}(r)$ , $\hat{a}$ , $\hat{\unicode[STIX]{x1D6E4}}$ and $\hat{\unicode[STIX]{x1D70E}}$ , to give

where $B$ is another integration constant and $\tilde{k}^{2}=k^{2}+\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}/\unicode[STIX]{x1D707}$ . Finally, the normal and tangential stress conditions given by equations (3.11) and (3.12) provide, respectively,

Once the normal-mode variables evaluated at $r=R$ are substituted in (3.24) and (3.25), these provide a homogeneous linear system for $A$ and $B$ , namely $\unicode[STIX]{x1D648}\boldsymbol{\cdot }\unicode[STIX]{x1D753}=0$ , where $\unicode[STIX]{x1D753}=(A,B)^{\text{T}}$ . The four entries of $\unicode[STIX]{x1D648}$ can be found in appendix B. Non-trivial solutions require that $\det (\unicode[STIX]{x1D648})=0$ , finally leading to the desired dispersion relation,

where $\tilde{k}^{2}=k^{2}+\mathscr{R}\unicode[STIX]{x1D714}$ and $F(x)=x\text{I}_{0}(x)/\text{I}_{1}(x)$ . Equation (3.26) has been non-dimensionalised taking $R$ as characteristic length and the viscocapillary time, $\unicode[STIX]{x1D707}R/\unicode[STIX]{x1D70E}_{0}$ , as characteristic time, where, for the sake of clarity, we have kept the same notation $\unicode[STIX]{x1D714}$ and $k$ for the dimensionless growth rate and wavenumber, respectively. According to (3.26), the dispersion relation depends on four dimensionless parameters: the capillary Reynolds number, $\mathscr{R}=\mathit{Oh}^{-2}=\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70E}_{0}R/\unicode[STIX]{x1D707}^{2}$ , where $\mathit{Oh}$ is the Ohnesorge number; the elasticity parameter $\unicode[STIX]{x1D6FD}=E/\unicode[STIX]{x1D70E}_{0}$ ; and the shear and dilatational Boussinesq numbers, $\mathscr{B}_{\unicode[STIX]{x1D707}}=\unicode[STIX]{x1D707}_{s0}/(\unicode[STIX]{x1D707}R)$ and $\mathscr{B}_{\unicode[STIX]{x1D705}}=\unicode[STIX]{x1D705}_{s0}/(\unicode[STIX]{x1D707}R)$ , respectively. In presenting the results below, the parameter $\mathscr{B}_{\unicode[STIX]{x1D705}}$ will be substituted by the ratio of surface shear to dilatational viscosity, $\unicode[STIX]{x1D707}_{s0}/\unicode[STIX]{x1D705}_{s0}=\mathscr{B}_{\unicode[STIX]{x1D707}}/\mathscr{B}_{\unicode[STIX]{x1D705}}$ . Note that the dispersion relation (3.26) reduces to that of Timmermans & Lister (Reference Timmermans and Lister2002) in the limit $\mathscr{B}_{\unicode[STIX]{x1D707}}\rightarrow 0$ and $\mathscr{B}_{\unicode[STIX]{x1D705}}\rightarrow 0$ , and to the Rayleigh–Chandrasekhar dispersion relation when, in addition, $\unicode[STIX]{x1D6FD}\rightarrow 0$ (Rayleigh 1892; Chandrasekhar Reference Chandrasekhar1961). However, there are several differences between (3.26) and the dispersion relation deduced by Whitaker (Reference Whitaker1976) in the insoluble limit due to several mistakes made in the latter work (Hansen et al. Reference Hansen, Peters and Meijer1999; Timmermans & Lister Reference Timmermans and Lister2002).

3.2 Temporal stability analysis

Figure 2.

Growth rate $\unicode[STIX]{x1D714}$ as a function of wavenumber $k$ for $\mathscr{R}=0.01$ and $\unicode[STIX]{x1D6FD}=0$ (a,b), $\unicode[STIX]{x1D6FD}=1$ (c,d), $\unicode[STIX]{x1D707}_{s0}/\unicode[STIX]{x1D705}_{s0}=0.1$ (a,c) and $\unicode[STIX]{x1D707}_{s0}/\unicode[STIX]{x1D705}_{s0}=10$ (b,d). The values of $\mathscr{B}_{\unicode[STIX]{x1D707}}$ are indicated near each curve. The symbols are extracted from figure 4 of Timmermans & Lister (Reference Timmermans and Lister2002).

Figure 3.

Same as figure 2 for $\mathscr{R}=100$ .

To illustrate the effect of surface viscosities on the growth rate of infinitesimal disturbances on the liquid cylinder, figures 2 and 3 show the amplification function $\unicode[STIX]{x1D714}(k)$ for $\mathscr{R}=0.01$ and $\mathscr{R}=100$ , respectively, with the value of $\mathscr{B}_{\unicode[STIX]{x1D707}}$ indicated near each curve. In figures 2 and 3, panels (a,b) and (c,d) correspond to values of the elasticity parameter $\unicode[STIX]{x1D6FD}=0$ and $\unicode[STIX]{x1D6FD}=1$ , while panels (a,c) and (b,d) show the results obtained with values of the surface viscosity ratio $\unicode[STIX]{x1D707}_{s0}/\unicode[STIX]{x1D705}_{s0}=0.1$ and $10$ , respectively. Our results were validated by comparing them with four amplification curves extracted from figure 4 of Timmermans & Lister (Reference Timmermans and Lister2002), represented by symbols, for the particular case $\mathscr{B}_{\unicode[STIX]{x1D707}}=0$ , finding perfect agreement.

As revealed by figures 2 and 3, surface viscosity stabilises the liquid cylinder, since the values of $\unicode[STIX]{x1D714}$ decrease monotonically for all values of $k$ as $\mathscr{B}_{\unicode[STIX]{x1D707}}$ increases. The stabilising effect is more pronounced for small values of $\mathscr{R}$ and $\unicode[STIX]{x1D6FD}$ . Indeed, in the particular case with $\mathscr{R}=0.01$ and $\unicode[STIX]{x1D6FD}=0$ , figure 2(a,b) shows that, for a value of $\mathscr{B}_{\unicode[STIX]{x1D707}}=1$ , the maximum growth rate decreases by factors of approximately 3.8 and 2.4, respectively, compared with the case with $\mathscr{B}_{\unicode[STIX]{x1D707}}=0$ . These factors are reduced to approximately 1.5 in figure 2(c,d), showing that an increase in the surface elasticity decreases the relative importance of surface viscosity.

It is important to point out that increasing the value of $\unicode[STIX]{x1D6FD}$ also has a stabilising effect, as studied in detail by Timmermans & Lister (Reference Timmermans and Lister2002), and also clearly seen in figures 2 and 3. However, in contrast with the fact that a finite value of $\unicode[STIX]{x1D714}$ is achieved when $\unicode[STIX]{x1D6FD}\rightarrow \infty$ , our results reveal that $\unicode[STIX]{x1D714}\rightarrow 0$ when $\mathscr{B}_{\unicode[STIX]{x1D707}}\rightarrow \infty$ , a behaviour similar to that observed for $\mathscr{R}\rightarrow 0$ , and clearly due to the dissipative nature of the surface shear viscosity (see § 3.2.1). In particular, the effect of $\mathscr{B}_{\unicode[STIX]{x1D707}}$ is important for both low- and high-viscosity threads, although, as happens with the effect of $\unicode[STIX]{x1D6FD}$ (Timmermans & Lister Reference Timmermans and Lister2002), the effect is more pronounced at low capillary Reynolds numbers, as can be deduced by comparing the results of figure 2 for $\mathscr{R}=0.01$ with those of figure 3 for $\mathscr{R}=100$ . Indeed, the reduction factors in the maximum growth rate between the cases $\mathscr{B}_{\unicode[STIX]{x1D707}}=0$ and $\mathscr{B}_{\unicode[STIX]{x1D707}}=1$ are substantially smaller for $\mathscr{R}=100$ compared with those stated before for $\mathscr{R}=0.01$ . In particular, these factors are approximately 1.3 and 1.2 in figures 3(a) and 3(b), respectively. The latter limit of dominant inertia, $\mathscr{R}\gg 1$ , was studied by Whitaker (Reference Whitaker1976), whose conclusions about the effect of the surface viscosities on $\unicode[STIX]{x1D714}$ were correct and coincide with the ones presented herein.

By comparing the left and right columns of figures 2 and 3, it is deduced that $\unicode[STIX]{x1D714}$ is only slightly affected by $\unicode[STIX]{x1D707}_{s0}/\unicode[STIX]{x1D705}_{s0}$ , with the general trend that an increase in $\mathscr{B}_{\unicode[STIX]{x1D705}}$ decreases the growth rate, as expected. In particular, for $\unicode[STIX]{x1D6FD}\lesssim 1$ the maximum growth rate, $\unicode[STIX]{x1D714}_{m}$ , is smaller when $\mathscr{B}_{\unicode[STIX]{x1D705}}$ is greater than $\mathscr{B}_{\unicode[STIX]{x1D707}}$ , as depicted by figures 2(a,b) and 3(a,b), this phenomenon being more noticeable for $\mathscr{R}\ll 1$ . This result is expected, since for a fixed $\mathscr{B}_{\unicode[STIX]{x1D707}}$ the overall surface viscosity increases when $\unicode[STIX]{x1D707}_{s0}/\unicode[STIX]{x1D705}_{s0}$ decreases, as already noticed by Whitaker (Reference Whitaker1976) in the limit $\mathscr{R}\gg 1$ . Figures 2 and 3 also reveal that the wavenumber of maximum amplification, $k_{m}$ , depends non-monotonically on $\mathscr{B}_{\unicode[STIX]{x1D707}}$ for a fixed value of the surface viscosity ratio, most notably at low capillary Reynolds numbers and large values of $\unicode[STIX]{x1D707}_{s0}/\unicode[STIX]{x1D705}_{s0}$ .

4.1 Leading-order model

Introducing (4.2)–(4.4) into the dimensionless versions of (3.4) and (3.3), the leading-order kinematic condition and the axial momentum equation read, respectively,

where $p_{0}$ and $w_{2}$ can be obtained from (2.9) and (2.10), respectively, which at leading order yield the following normal and tangential stress balances at $r=\unicode[STIX]{x1D700}a$ :

Additionally, the transport equation (3.5) for the surfactant concentration, $\unicode[STIX]{x1D6E4}(z,t)$ , reads, at leading order,

Although it does not affect the linear regime, (4.9) retains the full curvature, ${\mathcal{C}}$ , unlike the leading-order expressions for the surfactant concentration derived by Timmermans & Lister (Reference Timmermans and Lister2002), Liao et al. (Reference Liao, Franses and Basaran2006) and Craster et al. (Reference Craster, Matar and Papageorgiou2009), which include the approximation ${\mathcal{C}}\simeq a^{-1}$ . Finally, eliminating the small parameter $\unicode[STIX]{x1D700}$ from the formulation via the substitutions $z\rightarrow \unicode[STIX]{x1D700}z$ , $t\rightarrow \unicode[STIX]{x1D700}t$ and $w_{2j}\rightarrow w_{2j}/\unicode[STIX]{x1D700}^{2j}$ , the leading-order 1D model consists of the following three coupled equations:

These must be complemented with three equations of state, $\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D6E4})$ , $\unicode[STIX]{x1D707}_{s}(\unicode[STIX]{x1D6E4})$ and $\unicode[STIX]{x1D705}_{s}(\unicode[STIX]{x1D6E4})$ , relating the surface tension coefficient and the two surface viscosities with the surfactant concentration. However, note that explicit expressions for the three equations of state are not needed in the linearised analysis presented herein. Equations (4.10)–(4.12) recover the limit when only gradients of $\unicode[STIX]{x1D70E}$ are considered by setting $\mathscr{B}_{\unicode[STIX]{x1D707}}\rightarrow 0$ and $\mathscr{B}_{\unicode[STIX]{x1D705}}\rightarrow 0$ (Timmermans & Lister Reference Timmermans and Lister2002; Liao et al. Reference Liao, Franses and Basaran2006; Craster et al. Reference Craster, Matar and Papageorgiou2009), and also the limit of a clean liquid thread when $\unicode[STIX]{x1D6E4}\rightarrow 0$ , $\unicode[STIX]{x1D70E}\rightarrow 1$ , $\mathscr{B}_{\unicode[STIX]{x1D707}}\rightarrow 0$ and $\mathscr{B}_{\unicode[STIX]{x1D705}}\rightarrow 0$ (Eggers & Dupont Reference Eggers and Dupont1994; García & Castellanos Reference García and Castellanos1994).

4.2 Parabolic model

The accuracy of the 1D approximation deduced in § 4.1 can be improved by retaining terms of order $O(\unicode[STIX]{x1D700}^{2})$ , leading to the so-called parabolic model, whose relative error is of order $O(\unicode[STIX]{x1D700}^{4})$ (García & Castellanos Reference García and Castellanos1994). At order $O(\unicode[STIX]{x1D700}^{2})$ , the kinematic condition, the surfactant transport equation and the axial momentum equation read, respectively,

Here $w_{4}(z,t)$ can be obtained from the second-order truncation of the tangential stress balance,

and $p_{2}(z,t)$ can be deduced from the first-degree approximation in $\unicode[STIX]{x1D700}$ of the radial momentum (3.2),

Equation (4.15) is coupled to (4.6), i.e. the first-order momentum equation for $w_{0}(z,t)$ , whose unique unknown is $p_{0}(z,t)$ . A higher-order approximation of $p_{0}$ can be obtained from the second-order normal stress balance,

Hence, introducing the expressions for $p_{2}$ and $w_{4}$ into (4.15), and those for $p_{0}$ and $p_{2}$ into (4.6), and eliminating the small parameter $\unicode[STIX]{x1D700}$ from the formulation via the substitutions $z\rightarrow \unicode[STIX]{x1D700}z$ , $t\rightarrow \unicode[STIX]{x1D700}t$ and $w_{2j}\rightarrow w_{2j}/\unicode[STIX]{x1D700}^{2j}$ , the four equations of the parabolic model read

Note that equations (4.19)–(4.22), together with the three equations of state $\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D6E4})$ , $\unicode[STIX]{x1D707}_{s}(\unicode[STIX]{x1D6E4})$ and $\unicode[STIX]{x1D705}_{s}(\unicode[STIX]{x1D6E4})$ , form a closed system which determines the temporal evolution of $a(z,t)$ , $w_{0}(z,t)$ , $w_{2}(z,t)$ and $\unicode[STIX]{x1D6E4}(z,t)$ .

4.3 Temporal stability analysis of the 1D models

Performing numerical simulations of the temporal evolution of the liquid thread using the 1D models derived above is beyond the scope of the present work. Hence, as a first step, it is interesting to check the validity of both 1D approximations in the linear regime by comparing their temporal stability properties with those provided by the full axisymmetric dispersion relation derived in § 3. Linearising equations (4.10)–(4.12) and (4.19)–(4.22), and expanding in normal modes, the dispersion relations associated with the leading-order model and the parabolic model are straightforwardly obtained, as follows.

1. (i) Leading-order model:

   (4.23) $$\begin{eqnarray}\frac{2\unicode[STIX]{x1D714}^{2}\mathscr{R}}{k^{2}}-1+k^{2}+\unicode[STIX]{x1D6FD}+\unicode[STIX]{x1D714}(6+9\mathscr{B}_{\unicode[STIX]{x1D707}}+\mathscr{B}_{\unicode[STIX]{x1D705}})=0.\end{eqnarray}$$

2. (ii) Parabolic model:

   (4.24) $$\begin{eqnarray}\displaystyle & & \displaystyle \mathscr{R}\unicode[STIX]{x1D714}^{3}(16+4k^{2})+\unicode[STIX]{x1D714}^{2}[128+k^{2}(96+40\mathscr{B}_{\unicode[STIX]{x1D705}}+104\mathscr{B}_{\unicode[STIX]{x1D707}})+k^{4}(18+9\mathscr{B}_{\unicode[STIX]{x1D705}}+25\mathscr{B}_{\unicode[STIX]{x1D707}})]\nonumber\\ \displaystyle & & \displaystyle \quad +\,\unicode[STIX]{x1D714}k^{2} [-8+40\unicode[STIX]{x1D6FD}+384\,\mathscr{R}^{-1}\,(1+3\mathscr{B}_{\unicode[STIX]{x1D707}}/2+\mathscr{B}_{\unicode[STIX]{x1D705}}/6)\nonumber\\ \displaystyle & & \displaystyle \quad +\,k^{2} (\!7+9\unicode[STIX]{x1D6FD}+\mathscr{R}^{-1}\, (\!96+80\mathscr{B}_{\unicode[STIX]{x1D705}}\nonumber\\ \displaystyle & & \displaystyle \quad +\,144\mathscr{B}_{\unicode[STIX]{x1D707}}+64\mathscr{B}_{\unicode[STIX]{x1D705}}\mathscr{B}_{\unicode[STIX]{x1D707}}\!)\!)+k^{4}(1+\mathscr{R}^{-1}\,(14+9\mathscr{B}_{\unicode[STIX]{x1D705}}+25\mathscr{B}_{\unicode[STIX]{x1D707}}))\!]\nonumber\\ \displaystyle & & \displaystyle \quad +\,\mathscr{R}^{-1}\,k^{2} [\!64(\unicode[STIX]{x1D6FD}-1)\nonumber\\ \displaystyle & & \displaystyle \quad +\,k^{2}(48+80\unicode[STIX]{x1D6FD}-16\mathscr{B}_{\unicode[STIX]{x1D705}}-16\mathscr{B}_{\unicode[STIX]{x1D707}}+64\unicode[STIX]{x1D6FD}\mathscr{B}_{\unicode[STIX]{x1D707}})\nonumber\\ \displaystyle & & \displaystyle \quad +\,k^{4}(15+9\unicode[STIX]{x1D6FD}+16\mathscr{B}_{\unicode[STIX]{x1D705}}+16\mathscr{B}_{\unicode[STIX]{x1D707}})+k^{6}\! ]\nonumber\\ \displaystyle & & \displaystyle \quad -\,\frac{16\unicode[STIX]{x1D6FD}}{\mathscr{R}\,\unicode[STIX]{x1D714}}k^{4}(1-k^{2})=0.\end{eqnarray}$$

Figure 7.

Comparison between the full axisymmetric dispersion relation (3.26) (solid lines) and the dispersion relation associated with the leading-order 1D model (4.23) (dashed lines) for $\mathscr{R}=0.01$ and varying the value of $\unicode[STIX]{x1D6FD}$  (a),  $\mathscr{B}_{\unicode[STIX]{x1D707}}$  (b) and $\mathscr{B}_{\unicode[STIX]{x1D705}}$  (c). The values of the parameters are indicated near each curve.

To illustrate the performance of the two different 1D models deduced in the present work, figures 7 and 8 compare, respectively, the amplification curves obtained with the leading-order model (4.23) and with the parabolic model (4.24), with those given by the full axisymmetric dispersion relation (3.26), for several values of $\unicode[STIX]{x1D6FD}$ , $\mathscr{B}_{\unicode[STIX]{x1D707}}$ and $\mathscr{B}_{\unicode[STIX]{x1D705}}$ . In the case of the elasticity parameter, $\unicode[STIX]{x1D6FD}$ , figure 7(a) shows that the leading-order 1D model fails at predicting the linear behaviour of the liquid column as $\unicode[STIX]{x1D6FD}$ increases, as was already pointed out by Timmermans & Lister (Reference Timmermans and Lister2002). In fact, according to this 1D approximation, the liquid column is stable for every value of $k$ if $\unicode[STIX]{x1D6FD}\geqslant 1$ . With regard to $\mathscr{B}_{\unicode[STIX]{x1D707}}$ and $\mathscr{B}_{\unicode[STIX]{x1D705}}$ , figure 7(b,c) shows that the leading-order 1D approximation also fails at predicting the growth rate $\unicode[STIX]{x1D714}(k)$ , especially when $\mathscr{B}_{\unicode[STIX]{x1D707}}\gg 1$ and $\mathscr{B}_{\unicode[STIX]{x1D705}}\gg 1$ .

In contrast, the parabolic model captures the linear regime with high accuracy for every value of $\unicode[STIX]{x1D6FD}$ , $\mathscr{B}_{\unicode[STIX]{x1D707}}$ and $\mathscr{B}_{\unicode[STIX]{x1D705}}$ , as evidenced by the results of figure 8. The agreement between both $\unicode[STIX]{x1D714}(k)$ curves is slightly worse when $\unicode[STIX]{x1D6FD}\rightarrow \infty$ , whereas it improves when $(\mathscr{B}_{\unicode[STIX]{x1D705}},\mathscr{B}_{\unicode[STIX]{x1D707}})\gg 1$ independently of the value of $\unicode[STIX]{x1D6FD}$ . In particular, we have checked that in the limit $\mathscr{B}_{\unicode[STIX]{x1D707}}\rightarrow \infty$ , the dispersion relation (4.24) is identical, at leading order, to equation (3.27) deduced from the exact dispersion relation (3.26).

Figure 8.

Comparison between the full axisymmetric dispersion relation (3.26) (solid lines) and the dispersion relation associated to the parabolic 1D model (4.24) (dashed lines) for $\mathscr{R}=0.01$  (a,b) and $\mathscr{R}=100$  (c,d). Panels (a,c) show the amplification curves for $\mathscr{B}_{\unicode[STIX]{x1D707}}=\mathscr{B}_{\unicode[STIX]{x1D705}}=0$ and several values of $\unicode[STIX]{x1D6FD}$ , and (b,d) those for $\unicode[STIX]{x1D6FD}=5$ , $\unicode[STIX]{x1D707}_{s0}/\unicode[STIX]{x1D705}_{s0}=1$ and several values of $\mathscr{B}_{\unicode[STIX]{x1D707}}$ indicated near each curve.

It is worth mentioning that Timmermans & Lister (Reference Timmermans and Lister2002) also derived a higher-order approximation usually known as the averaged-parabolic model, but only accounting for gradients of $\unicode[STIX]{x1D70E}$ and not surface viscosities. These authors demonstrated that, for a very viscous thread, i.e. $\mathscr{R}\rightarrow 0$ , the averaged-parabolic model shows very good agreement with the full expression of the dispersion relation (3.26) for $\mathscr{B}_{\unicode[STIX]{x1D707}}=\mathscr{B}_{\unicode[STIX]{x1D705}}=0$ and different values of $\unicode[STIX]{x1D6FD}$ . Indeed, the accuracy of their averaged-parabolic model is slightly better than that of the parabolic model developed herein in the limit $(\mathscr{B}_{\unicode[STIX]{x1D707}},\,\mathscr{B}_{\unicode[STIX]{x1D705}})\rightarrow 0$ , when only surface elasticity is considered. Note that, in the case of a clean interface, a similar conclusion was obtained by García & Castellanos (Reference García and Castellanos1994), who found that the 1D averaged-parabolic model performs slightly better than the 1D parabolic model in predicting the linear amplification curve.

In summary, the leading-order 1D equations are suitable for small values of $\unicode[STIX]{x1D6FD}$ , $\mathscr{B}_{\unicode[STIX]{x1D707}}$ and $\mathscr{B}_{\unicode[STIX]{x1D705}}$ , including the limit of a clean interface (Eggers & Dupont Reference Eggers and Dupont1994; García & Castellanos Reference García and Castellanos1994). However, when $(\unicode[STIX]{x1D6FD},\mathscr{B}_{\unicode[STIX]{x1D707}},\mathscr{B}_{\unicode[STIX]{x1D705}})\gtrsim 1$ , a higher-order approximation like the parabolic model derived herein, or the averaged-parabolic model due to Timmermans & Lister (Reference Timmermans and Lister2002), is needed to obtain good quantitative results in the linear regime.