3.1. Temperature distribution along the fibre

Previous studies (Liu et al. Reference Liu, Ding and Chen2018) of the thermal effects on liquid films have typically assumed a constant temperature gradient. In this work, we instead represent the streamwise temperature variations under our experimental conditions using a solution to a lumped capacitance model

(3.1)\begin{equation} \varTheta = \exp \left[ - \left(\frac{{h}_{b} A_s}{m_b v_b c}\right) x^*\right]. \end{equation}

The works of Hattori et al. (Reference Hattori, Ishikawa and Mori1994) and Zeng et al. (Reference Zeng, Sadeghpour, Warrier and Ju2017) showed that the radial temperature variation within each liquid bead is negligible owing to efficient mixing associated with internal circulation within the bead. Axial heat conduction is neglected. In the model (3.1), $\varTheta = {(T^*-T_0^*)}/{\varDelta T}$ is the dimensionless temperature, where $\varDelta T = T^*_{{IN}} - T^*_0$ is the temperature scale set by the difference between the inlet temperature $T^*_{{IN}}$ and the room temperature $T^*_0$. Here $A_s$ represents the surface area of the liquid beads, $c$ is the specific heat of the liquid, $v_{b}$ is the liquid bead velocity, $m_b$ is the mass of each bead, $x^*$ is the distance from the inlet and ${h}_{b}$ is the liquid bead-to-air heat transfer coefficient. We use the empirical relationship for the heat transfer coefficient of flow around a sphere (Whitaker Reference Whitaker1972; Mills Reference Mills1995) to obtain the value of ${h}_{b}$,

(3.2)\begin{equation} \frac{{h}_{b}D_{b}}{k_{air}} = 2 + (0.4 Re_{b}^{1/2} + 0.06 Re_{b} ^ {2/3}) Pr_{air}^{0.4}. \end{equation}

Here, $Re_{b} = \rho _{air} v_{b} D_{b}/\mu _{air}$ is the bead Reynolds number, $D_{b}$ is the bead diameter and $Pr_{air}$ is the Prandtl number of air. In addition, $\mu _{air}$, $\rho _{air}$ and $k_{air}$ are the viscosity, density and thermal conductivity of air, respectively. The temperature profiles obtained from the lumped capacitance model (3.1) are used later in the lubrication model to represent the axial variations in the surface tension and viscosity along the fibre.

As shown in figure 2, the model captures the experimentally measured liquid temperature profiles well. For later convenience, we rewrite the dimensionless temperature distribution in a rescaled form

(3.3)\begin{equation} \varTheta_I(x) = 1 - \frac{1-\exp(-\chi x)}{1-\exp(-\chi L_0)}, \end{equation}

where $x$ is the dimensionless streamwise spatial variable and $L_0$ is the dimensionless downstream position where the room temperature $T_0^*$ is nominally reached. The temperature profile (3.3) satisfies $\varTheta _I(0) = 1$ and $\varTheta _I(L_0) = 0$, and the dimensionless parameter $\chi$ specifies the temperature gradient. In all simulations, we set $L_0 = L_0^*/\mathcal {L}$, where $L_0^* = 0.5\ \textrm {m}$ and $\mathcal {L}$ is the length scale in the streamwise direction, which will be defined in § 3.3.

Figure 2.

Temperature profiles measured in experiments along the fibre for silicone oil v50 with different inlet temperatures compared with the prediction based on the model (3.1). The parameters are $h_b = 44\ \textrm {W}\,(\textrm {m}^{2}\,\textrm {K})^{-1}$, $A_s =13.2\ \textrm {mm}^2$, $c = 1510\ \textrm {J}\,(\textrm {kg}\,\textrm {K})^{-1}$, $m_b=2.04\times 10^{-6}\ \textrm {kg}$ and $v_b=19.2\ \textrm {mm}\,\textrm {s}^{-1}$.

3.2. Temperature-dependent liquid properties

Liquid properties can change significantly as the temperature is varied. In this work, we focus on the influence of temperature on the surface tension and viscosity of the liquid.

To characterize the temperature dependence of the surface tension of our liquid, we measured the rising height, $h_{{rise}}$, in a capillary tube with an inner diameter, $r_{{tube}}$, of 0.5 mm at various temperatures ($20\text {--}110\,^\circ \textrm {C}$). We then used $\sigma = ({ r_{{tube}} h_{{rise}} \rho g})/({2 \cos \theta })$ to estimate the surface tension at each temperature (Washburn Reference Washburn1921). Here, $\theta$ is the contact angle on the capillary tube wall that we measured independently. All the experiments were performed in an isothermal container.

Figure 3(a) shows the experimental results for the surface tension, $\sigma$ ($\textrm {mN}\,\textrm {m}^{-1}$), of silicone oil v50. The experimental uncertainty in the measured surface tension is estimated to be $0.3\ \textrm {mN}\,\textrm {m}^{-1}$ and the uncertainty in the measured temperatures is estimated to be ${\pm }0.5\,^\circ \textrm {C}$.

Figure 3.

(a) Surface tension and (b) viscosity of silicone oil v50 as functions of temperature $T^*$.

The kinematic viscosity $\nu (T^*)\ (\text {mm}^2\,\text {s}^{-1})$ of silicone oil v50 as a function of the temperature $T^*\ (^\circ \textrm {C})$ was provided by the manufacturer (see Shin-etsu 2005). A plot of the relation between viscosity and temperature is included in figure 3(b).

For simplicity, in the model derived here, we make an approximation that the kinematic viscosity $\nu$ and the surface tension $\sigma$ of the liquid are linearly dependent on temperature,

(3.4a,b)\begin{equation} \nu(T^*) = \nu_0 - \nu_T (T^*-T_0^*) ,\quad \sigma(T^*) = \sigma_0 - \sigma_T (T^*-T_0^*) , \end{equation}

where $\nu _0,\nu _T,\sigma _0, \sigma _T > 0$ are constants. Using the dimensionless temperature variable, one can also write $\nu = \nu _0 - \nu _T \varDelta T\varTheta$ and $\sigma = \sigma _0 - \sigma _T \varDelta T \varTheta .$

In this paper, we have the reference room temperature $T_0^*=20\,^\circ \textrm {C}$, $\sigma _T = 0.0504\ \textrm {mN}\,(\textrm {m}\,^{\circ }\textrm {C})^{-1}$ and $\nu _T = (\nu _0-\nu (T^*_{{IN}}))/\varDelta T$. The density of the liquid is assumed to be constant.

3.3. Lubrication model

Next we derive the governing equations following the works of Craster & Matar (Reference Craster and Matar2006), Ruyer-Quil et al. (Reference Ruyer-Quil, Treveleyan, Giorgiutti-Dauphiné, Duprat and Kalliadasis2008) and Ji et al. (Reference Ji, Falcon, Sadeghpour, Zeng, Ju and Bertozzi2019). We choose the length scale in the radial direction $y$ as $\mathcal {H}$ and the length scale in the streamwise direction $x$ as $\mathcal {L} = \mathcal {H}/\epsilon$. The scale ratio $\epsilon$ is set by the balance between the surface tension and the gravity $g$, and is given by $\epsilon = (\rho g \mathcal {H}^2/\sigma _0)^{1/3}$. This scale ratio is small (approximately $0.35$) in typical experiments and can also be rewritten as $\epsilon = We^{-1/3}$, where the Weber number $We = (l_c/\mathcal {H})^2$ compares the capillary length $l_c = \sqrt {\sigma _0/(\rho g)}$ with the radial length scale $\mathcal {H}$. The characteristic streamwise velocity is $\mathcal {U}=(g\mathcal {H}^2)/{\nu _0}$, and the pressure and time scales are given by $\rho g \mathcal {L}$ and $(\nu _0\mathcal {L})/(g\mathcal {H}^2)$, respectively.

Following the approach in Ji et al. (Reference Ji, Falcon, Sadeghpour, Zeng, Ju and Bertozzi2019, Reference Ji, Sadeghpour, Ju and Bertozzi2020a), we include a film stabilization term $\varPi (h)$,

(3.5)\begin{equation} \varPi(h) ={-}\frac{A}{h^3}, \end{equation}

where $A>0$ is a stabilization parameter. This term takes the functional form of the long-range disjoining pressure of the van der Waals model that characterizes the microscopic quantities for wetting liquids, and $A$ is typically referred as a Hamaker constant (de Gennes Reference de Gennes1985).

The dynamics of the axisymmetric flows is governed by the Navier–Stokes equations, the continuity equation and the energy equation. This system is coupled with the temperature-dependent viscosity and surface tension given by (3.4a,b). With these scales, we write the non-dimensional Navier–Stokes equations using dimensionless variables as

(3.6a)\begin{gather} \epsilon^2Re\left(u_{t} + u u_{x} + vu_{y}\right) ={-}p_{x}-\varPi_x+1+(1-\kappa\varTheta)\left(\frac{u_{y}}{y}+u_{yy}+\epsilon^2 u_{xx}\right), \end{gather}

(3.6b)\begin{gather} \epsilon^4Re\left( v_{t} + v v_{y} + uv_{x}\right) ={-}p_{y}+\epsilon^2(1-\kappa\varTheta)\left(\frac{v_{y}}{y} +v_{yy}-\frac{v}{y^{2}}+\epsilon^2 v_{xx}\right), \end{gather}

(3.6c)\begin{gather} v_{y}+\frac{v}{y} + u_{x} = 0, \end{gather}

where the Reynolds number $Re = \mathcal {UL}/\nu _0$, and the constants $\kappa = {\nu _T\varDelta T}/{\nu _0}$ and $\omega = {\sigma _T\varDelta T}/{\sigma _0}$ scale the relative change in viscosity and surface tension as the temperature is varied. The balances of normal and tangential stresses at $y = h+R$ are expressed as

(3.6d)\begin{gather} (1-\epsilon^2 h_x^2)(\epsilon^2 v_x+u_y) + 2\epsilon^2 h_x(v_y-u_x) ={-}Ma \frac{(1+\epsilon^2 h_x^2)^{1/2}}{1-\kappa\varTheta}(h_x\varTheta_y + \varTheta_x), \end{gather}

(3.6e)\begin{gather} p+\varPi = \left(\frac{\alpha}{\epsilon^2(1+\alpha h)(1+\epsilon^2 h_x^2)^{1/2}} - \frac{h_{xx}}{(1+\epsilon^2 h_x^2)^{3/2}}-\frac{A}{h^3}\right)(1-\omega \varTheta)\nonumber\\ +\frac{2\epsilon^2(1-\kappa\varTheta)}{1+\epsilon^2 h_x^2}\left[\epsilon^2(h_x^2 u_x - h_x v_x)-h_x u_y + v_y\right], \end{gather}

where $Ma = (\sigma _T \varDelta T)/(\rho g \mathcal {H} \mathcal {L})$ is the Marangoni parameter, $\alpha = \mathcal {H}/R^*$ is the aspect ratio of the characteristic radial length scale and the fibre radius, and the dimensionless fibre radius is $R = R^*/\mathcal {H}$. At the interface between the solid substrate and the liquid, $y = R$, we impose the no slip and no penetration boundary conditions,

(3.6f)\begin{equation} v = u = 0, \quad \text{at} \ y = R. \end{equation}

The kinematic boundary condition at $y = R+h$ is given by

(3.6g)\begin{equation} h_t + u h_{x} = v, \quad \text{at} \ y = R+h. \end{equation}

Next, we simplify the above set of governing equations following Ruyer-Quil et al. (Reference Ruyer-Quil, Treveleyan, Giorgiutti-Dauphiné, Duprat and Kalliadasis2008). Under the lubrication approximation, we neglect the inertial contributions by assuming that $Re = O(1)$ and $\epsilon \ll 1$. Omitting the terms of order $O(\epsilon ^2)$, we rewrite the leading order non-dimensional reduced momentum and continuity equations for the velocity field $(u,v)$ and the dynamic pressure $p$ as

(3.7a)\begin{gather} 1 -\frac{\partial p}{\partial x}-\frac{\partial \varPi}{\partial x}+(1-\kappa\varTheta)\left(\frac{\partial^2 u}{\partial y^2} +\frac{u_y}{y}\right)= 0, \end{gather}

(3.7b)\begin{gather} {-}\frac{\partial p}{\partial y} = 0, \end{gather}

(3.7c)\begin{gather}\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y} +\frac{v}{y} = 0. \end{gather}

Based on our discussion in § 3.1, the radial temperature variations within liquid droplets and across the inter-bead precursor layer are negligible compared with the streamwise temperature variations. Therefore, we assume that the temperature is constant in the radial direction and only varies in the axial direction in the leading order, $\varTheta = \varTheta _I(x)$. The balance of tangential stresses at the free surface $y = h + R$ reduces to

(3.7d)\begin{equation} u_y ={-}\frac{Ma}{1-\kappa\varTheta}(h_x\varTheta_y + \varTheta_x) ={-}\frac{Ma\varTheta_x}{1-\kappa\varTheta}. \end{equation}

The balance of normal stresses at the free surface $y = h + R$ becomes

(3.7e)\begin{equation} p+\varPi = \left(\frac{\alpha}{\epsilon^2(1+\alpha h)}-\frac{\partial^2 h}{\partial x^2}-\frac{A}{h^3}\right) (1-\omega\varTheta). \end{equation}

The two terms on the right-hand side of (3.7e), $\alpha /(\epsilon ^2(1+\alpha h))$ and $\partial ^2 h/\partial x^2$, describe both the destabilizing and stabilizing roles of the surface tension that originate from the azimuthal and axial curvature of the free surface, respectively. The balance between the azimuthal and axial scales is characterized by $\alpha$ and $\epsilon$. Here we use linear forms for both curvature terms; a discussion of other appropriate forms for the curvature terms can be found in Ji et al. (Reference Ji, Falcon, Sadeghpour, Zeng, Ju and Bertozzi2019).

Combining the kinematic boundary condition (3.6g), the no slip and no penetration boundary conditions (3.6f) and the continuity equation (3.7c) lead to the mass conservation equation

(3.8)\begin{equation} (1+\alpha h)\frac{\partial h}{\partial t} + \frac{\partial q}{\partial x} = 0, \quad \text{where } q = \frac{1}{R}\int_{R}^{h+R} uy\,{\textrm{d}y}. \end{equation}

Equation (3.7b) indicates that the pressure $p$ is a function of $x$ only. Integrating (3.7a) twice and using (3.7d) yields

(3.9)\begin{gather} u =\frac{1}{1-\kappa\varTheta(x)}\left[\frac{1-p_x-\varPi_x}{2}\left( -\frac{1}{2}(y^2-R^{2})+(h+R)^2\ln\left(\frac{y}{R}\right)\right)\right.\nonumber\\ \left. - Ma\varTheta_x (R+h)\ln\left(\frac{y}{R}\right)\vphantom{\frac{1-p_x-\varPi_x}{2}}\right]. \end{gather}

Because $\alpha = \mathcal {H}/R^* = 1/R$, by using (3.8), we obtain the form of flux $q$

(3.10)\begin{equation} q = \frac{1}{1-\kappa\varTheta}\left(\frac{h^3}{3}\phi(\alpha h)(1-p_x-\varPi_x)-\frac{1}{2}Ma\varTheta_x h^2\psi(\alpha h)\right), \end{equation}

where the shape factors $\phi$ and $\psi$ are functions defined by

(3.11)\begin{gather} \phi(X) = \frac{3}{16X^3}[(1+X)^4(4\ln(1+X)-3)+4(1+X)^2-1], \end{gather}

(3.12)\begin{gather} \psi(X) = \frac{1+X}{X^2}\left[(1+X)^2\ln(1+X)-X\left({\frac{1}{2}} X + 1\right)\right]. \end{gather}

In the limit of $X \to 0$, we have

(3.13a,b)\begin{equation} \phi(X) = 1+X+\tfrac{3}{20}X^2+O(X^3),\quad \psi(X)=1+\tfrac{4}{3}X+\tfrac{1}{4}X^2+O(X^3). \end{equation}

Given a dimensional volumetric flow rate $Q_m$ and fibre radius $R^*$, we define the volumetric flow rate per circumference unit $q_0^*$ as $q_0^* = Q_m/(2{\rm \pi} \rho R^*)$. We set the characteristic axial length scale $\mathcal {H}$ by solving (3.10) for $h$ with $q = q_0^*$, $\varTheta \equiv 0$, $\varPi_x\equiv 0$ and $p_x\equiv 0$. That is, we set the length scale $\mathcal {H}$ by the film thickness $h_N^*$ of a uniform Nusselt flow at room temperature $T_0^*$ with constant dynamic viscosity $\nu _0$ (Ruyer-Quil et al. Reference Ruyer-Quil, Treveleyan, Giorgiutti-Dauphiné, Duprat and Kalliadasis2008; Duprat et al. Reference Duprat, Ruyer-Quil and Giorgiutti-Dauphiné2009).

Finally, by rescaling the time scale $t \to {t}/{\phi (\alpha )}$, and combining (3.8), (3.10) and (3.7e), we obtain the dimensionless governing equation for $0 \le x \le L$,

(3.14a)\begin{equation} \frac{\partial}{\partial t} \left(h+\frac{\alpha}{2}h^2\right) + \frac{\partial q}{\partial x} = 0, \end{equation}

where the flux takes the form

(3.14b)\begin{equation} q = \mathcal{M}(h)\left(1-\frac{\partial }{\partial x}\left[(1-\omega\varTheta)\left(\mathcal{Z}(h)-h_{xx}\right)\right]\right) -\frac{h^2}{2}\frac{\varTheta_xMa}{1-\kappa\varTheta} \frac{\psi(\alpha h)}{\phi(\alpha)}, \end{equation}

where the mobility function is

(3.14c)\begin{equation} \mathcal{M}(h) = \frac{h^3}{3(1-\kappa\varTheta)}\frac{\phi(\alpha h)}{\phi(\alpha)}, \end{equation}

and the function $\mathcal {Z}(h)$ consists of the destabilizing azimuthal curvature term $\alpha /(\eta (1+\alpha h))$ and the film stabilization term $\varPi (h)$,

(3.14d)\begin{equation} \mathcal{Z}(h) = \frac{\alpha}{\eta (1+\alpha h)} + \varPi(h), \quad \varPi(h) ={-}\frac{A}{h^3}, \end{equation}

where the scaling parameter $\eta = \epsilon ^2$.

This choice of time scale leads to a normalized mobility function that satisfies $\mathcal {M} = 1/3$ for $h = 1$ and $\varTheta \equiv 0$. The model (3.14) can be written as a fourth-order nonlinear partial differential equation (PDE) for the thickness $h(x,t)$. It accounts for temperature-dependent viscosity and surface tension gradients, gravity and azimuthal instabilities, but neglects the inertia and streamwise viscous dissipation that are included in the work of Ruyer-Quil et al. (Reference Ruyer-Quil, Trevelyan, Giorgiutti-Dauphiné, Duprat and Kalliadasis2009). We note that because $0 \le \kappa < 1$ and $0\le \varTheta \le 1$, we have $0<1-\kappa \varTheta \le 1$ in (3.14b) and (3.14c).

Compared with the prior work in Liu et al. (Reference Liu, Ding and Chen2018) that focuses on the thermocapillarity effects in similar dynamics, this new model includes additional physics through the temperature-dependent viscosity and does not require the aspect ratio $\alpha$ to satisfy $\alpha \ll 1$. Our model with $\kappa = Ma = \omega=A = 0$ is also consistent with the model derived in the work of Craster & Matar (Reference Craster and Matar2006) except for a scaling difference.

In the limit $\alpha \to 0$, the model (3.14) describes the dynamics of a draining film flowing down an incline under thermal effects. If we assume that the viscosity is constant and set $\kappa=Ma=\omega=A = 0$, (3.14) reduces to (3.15), which describes thin film dynamics driven by the gravity and a surface tension gradient owing to a temperature gradient,

(3.15)\begin{equation} h_t + ({\tfrac{1}{3}}h^3-{\tfrac{1}{2}}h^2\varTheta_xMa)_x={-}{\tfrac{1}{3}}(h^3 h_{xxx})_x. \end{equation}

The cubic term and quadratic term on the left-hand side of (3.15) originate from gravity and the Marangoni stress, respectively. In this paper, we consider a decreasing temperature field, $\varTheta _x < 0$, and the surface tension gradient is expected to promote the downward movement of the film.

4.1. Isothermal films

We begin by considering the flow instability of a uniformly heated thin fluid film flowing down a vertical fibre. Specifically, we study an isothermal film with spatially-constant surface tension and viscosity at a temperature $\varTheta (x) \equiv \varTheta _0$ and $\textrm {d}\varTheta /{\textrm {d}x} = 0$, where $0 \le \varTheta _0 \le 1$. Then the governing equation (3.14) reduces to

(4.1)\begin{align} \frac{\partial}{\partial t} \left(h+\frac{\alpha}{2}h^2\right) + \frac{\partial}{\partial x}\left[\mathcal{M}(h)\left(1-(1-\omega\varTheta_0)\frac{\partial }{\partial x}\left(\frac{\alpha}{\eta (1+\alpha h)}-h_{xx} + \varPi(h)\right)\right)\right] = 0, \end{align}

where the mobility function takes the form $\mathcal {M}(h) = {[h^3\phi (\alpha h)]}/{[3(1-\kappa \varTheta _0)\phi (\alpha )]}.$ We perturb the uniform base state $\bar {h} = 1$ by an infinitesimal Fourier mode,

(4.2)\begin{equation} h = \bar{h} + \bar{\psi} \,\textrm{e}^{\mathrm{i}(k x - \varLambda t)}, \end{equation}

where $k$ is the wavenumber, $\varLambda$ is the wave frequency and $\bar {\psi } (\ll 1)$ is the initial amplitude. Expanding the PDE (4.1) then gives the dispersion relation

(4.3)\begin{equation} \varLambda = C(\alpha,\kappa, \varTheta_0)k + \mathrm{i} k^2 D(\alpha,\kappa, w, \varTheta_0)\left[{-}k^2+\frac{\alpha^2}{\eta(1+\alpha)^2}-3A\right], \end{equation}

where

(4.4a,b)\begin{align} C(\alpha,\kappa,\varTheta_0) = \frac{\alpha\phi'(\alpha)+3\phi(\alpha)}{3(1-\kappa\varTheta_0)(1+\alpha)\phi(\alpha)},\quad D(\alpha,\kappa, w, \varTheta_0) = \frac{1-\omega\varTheta_0}{3(1-\kappa\varTheta_0)(1+\alpha)}. \end{align}

The relation (4.3) indicates that the azimuthal curvature term $\alpha /(\eta (1+\alpha h))$ is destabilizing, and both the streamwise curvature term $h_{xx}$ and the film stabilization term $\varPi (h)$ are stabilizing. For $\kappa >0$, the temporal instability is enhanced by the thermal-viscosity influence.

Following the work of Ji et al. (Reference Ji, Falcon, Sadeghpour, Zeng, Ju and Bertozzi2019), we select the stabilization parameter $A$ based on the dimensional thickness $\epsilon _p = \mathcal {H}h_c$ of a stable uniform layer based on experiments, where $h_c$ is the dimensionless thickness of the stable coating film. Specifically, we pick $A = A_c$, where

(4.5)\begin{equation} A_c = \frac{\alpha^2 h_c^4}{3\eta(1+\alpha h_c)^2}, \end{equation}

which ensures that any thin flat film $h\le h_c$ is linearly stable ($\text {Im}(\varLambda ) < 0$) for all real wave numbers $k > 0$.

To understand the spatiotemporal stability of the uniform state, we consider the peak of a localized wave packet that travels with a speed given by the group velocity $v_g = \textrm {d} \varLambda /\textrm {d} k$, where both $\varLambda$ and $k$ are complex. The merging of two disconnected spatial branches at a point in the complex $k$-plane leads to a vanishing group velocity, $v_g|_{k=k_0} = 0$, which defines the absolute wavenumber $k_0$ and the corresponding absolute frequency $\varLambda _0 = \varLambda (k_0)$ (Duprat et al. Reference Duprat, Ruyer-Quil, Kalliadasis and Giorgiutti-Dauphiné2007; Scheid, Kofman & Rohlfs Reference Scheid, Kofman and Rohlfs2016). For $\text {Im}(\varLambda _0) >0$, the system presents absolute instability; for $\text {Im}(\varLambda _0) <0$, the fluid film shows convective instability. The absolute/convective (A/C) instability transition corresponds to a real absolute frequency, $\text {Im}(\varLambda _0) = 0$.

Following the approach in the work of Duprat et al. (Reference Duprat, Ruyer-Quil, Kalliadasis and Giorgiutti-Dauphiné2007), we introduce the transformation

(4.6a–c)\begin{equation} k = \tilde{k}\left(\frac{C}{3D}\right)^{1/3},\quad \varLambda = \tilde{\varLambda} C\left(\frac{C}{3D}\right)^{1/3}, \quad \frac{\alpha^2}{\eta(1+\alpha)^2}-3A = \beta\left(\frac{C}{3D}\right)^{2/3} \end{equation}

and reduce the dispersion relation (4.3) to the equation

(4.7)\begin{equation} \tilde{\varLambda} = \tilde{k} + \mathrm{i} \frac{\tilde{k}^2}{3}(\beta-\widetilde{k^2}). \end{equation}

This corresponds to the dispersion relation for a weakly nonlinear lubrication model studied in Frenkel (Reference Frenkel1992). Based on the calculation in Duprat et al. (Reference Duprat, Ruyer-Quil, Kalliadasis and Giorgiutti-Dauphiné2007), the instability of the system becomes absolute when $\beta > \beta _{ca}\equiv [9/4(-17+7\sqrt {7})]^{1/3}\approx 1.507$. This leads to the A/C threshold for the instability of isothermal films,

(4.8)\begin{equation} \frac{3(1-\omega\varTheta_0)\phi(\alpha)}{\alpha\phi'(\alpha)+3\phi(\alpha)}\left(\frac{\alpha^2}{\eta(1+\alpha)^2}-3A\right)^{3/2}=\beta_{ca}^{3/2}, \end{equation}

which shows that the A/C marginal curve is influenced by the thermal effects only through the prefactor $1-\omega \varTheta _0$, and the temperature-dependent viscosity does not play a role. We note that this conclusion may change if moderate inertia effects or streamwise heat diffusion is included in the model (Ruyer-Quil et al. Reference Ruyer-Quil, Treveleyan, Giorgiutti-Dauphiné, Duprat and Kalliadasis2008; Scheid et al. Reference Scheid, Kofman and Rohlfs2016; Ding et al. Reference Ding, Liu, Wong and Yang2018).

Following the studies of Duprat et al. (Reference Duprat, Ruyer-Quil and Giorgiutti-Dauphiné2009) and Ruyer-Quil & Kalliadasis (Reference Ruyer-Quil and Kalliadasis2012), we rewrite the threshold (4.8) in terms of $R^*/l_c$ and $\alpha$ and obtain

(4.9)\begin{equation} \frac{1}{C_k(\alpha)(1+\alpha)^4}\left[\alpha^{2/3}-3(\alpha+1)^2\left(\frac{R^*}{l_c}\right)^{4/3}A\right]^{3/2} = \beta_{ca}^{3/2}\mathcal{S}^2, \end{equation}

where $R^*/l_c$ is the ratio of the fibre radius $R^*$ and the capillary length $l_c$, $C_k(\alpha )$ is the linear wave speed $C(\alpha ,\kappa ,\varTheta _0)$ in (4.4a,b) for $\kappa \varTheta _0=0$, and $\mathcal {S}=({R^*}/{l_c})[1-\omega \varTheta _0]^{-1/2}$. For $\omega \varTheta _0 = 0$ and $A=0$, this threshold (4.9) is consistent with the A/C threshold proposed in the works of Ruyer-Quil & Kalliadasis (Reference Ruyer-Quil and Kalliadasis2012) and Duprat et al. (Reference Duprat, Ruyer-Quil and Giorgiutti-Dauphiné2009).

Figure 4 presents the A/C instability regimes predicted by (4.9) for isothermal films with $A = 0$. Downstream flow dynamics for the experiments in table 1 are shown in figure 4 with circles representing the RP flow regime and crosses representing droplet coalescence dynamics. This analysis suggests that for all the experimental cases in the present study, the flow dynamics of the uniformly heated isothermal liquid film at the temperature $T^*_{{IN}}$ fall in the absolute instability regime. Therefore, the bead coalescence observed in our experiments cannot take place without the presence of a temperature gradient. In particular, for $\omega = 0$, we note that for a spatially-constant temperature field $\varTheta (x) \equiv \varTheta _0$, a direct transformation $t \to t(1-\kappa \varTheta )$ in (3.14) leads to a PDE without any temperature-dependent terms. Therefore, for any trains of travelling beads in the absolute RP regime, we expect the flow to stay in the RP regime when the temperature is uniformly elevated across the fibre. This indicates that the non-uniform temperature field $\varTheta (x)$ is crucial for the bead coalescence observed in our experiments.

Figure 4.

The absolute and convective (A/C) instability regimes in the parameter plane of $\alpha$ and $\mathcal {S}$ predicted by (4.9) with $\beta _{ca} = 1.507$ and $A = 0$. The circle symbols represent experiments that are in RP regime and the cross symbols represent experiments with downstream bead coalescence.

4.2. Weakly non-isothermal films

Next, we show that for the weakly non-isothermal case, where the fibre is non-uniformly heated, both temperature-dependent viscosity and surface tension play an important role in determining the instability of the film. For simplicity, we consider the temperature $\varTheta =O(\delta )\ll 1$ that linearly decreases in space in the leading order,

(4.10)\begin{equation} \varTheta(x) = \delta\bar{\varTheta}(x) = \delta(1-\theta_1 x) + O(\delta^2) ,\quad \text{where }\theta_1 = 1/L \ll 1. \end{equation}

This temperature $\varTheta (x)$ satisfies $\varTheta (0) \approx \delta$ and $\varTheta (L) \approx 0$. Substituting (4.10) into the governing equation (3.14), for $\delta \ll 1$, we obtain the expansion for flux

(4.11)\begin{equation} q =(1+\kappa\delta\bar{\varTheta})\left\{ M_0(h)\left(1-\frac{\partial }{\partial x}[(1-\omega\delta\bar{\varTheta})\left(\mathcal{Z}(h)-h_{xx}\right)]\right) +\delta\theta_1Ma M_1(h)\right\} + O(\delta^2), \end{equation}

where

(4.12a,b)\begin{equation} M_0(h) = \frac{h^3 \phi(\alpha h)}{3\phi(\alpha)}, \quad M_1(h) = \frac{h^2}{2}\frac{\psi(\alpha h)}{\phi(\alpha)}. \end{equation}

For $\delta \ll 1$, the uniform film $\bar {h} = 1$ is a quasi-steady state of the system. Similar to the isothermal case, we apply the perturbation (4.2) to the governing equation and obtain the relation

(4.13)\begin{equation} \varLambda = \varLambda_0(k) + \delta \varLambda_1(k,x) + O(\delta^2), \end{equation}

where the leading-order term is given by

(4.14)\begin{equation} \varLambda_0(k)=k\frac{\alpha\phi'(\alpha)+3\phi(\alpha)}{3(1+\alpha)\phi(\alpha)} + \mathrm{i} k^2 \frac{1}{3(1+\alpha)}\left[{-}k^2+\frac{\alpha^2}{\eta(1+\alpha)^2}-3A\right], \end{equation}

which is consistent with the dispersion relation (4.3) for the isothermal case for $\kappa =\omega =0$. The $O(\delta )$ term $\varLambda _1$ depends on both the wavenumber $k$ and the spatial variable $x$,

(4.15)\begin{align} \varLambda_1(k,x) &= {\frac{k}{3(\alpha+1)}}{ ( {k}^{2}+\mathcal{Z}'(1) ) [ \mathrm{i} ( -\kappa+\omega ) \bar{\varTheta}(x) k+\theta_1\, ( \kappa-2\,\omega ) ] } \nonumber\\ &\quad + {\frac{1}{\alpha+1}}[( -\theta_1\,\omega\, \mathcal{Z} ( 1 ) k+ ( \bar{\varTheta}(x) k+\mathrm{i}\theta_1 ) \kappa ) M_0'(1) +\text{Ma}\,\theta_1\,kM_1'(1) ]. \end{align}

This analysis reveals the combined effects of the temperature-dependent liquid properties on the film stability in the non-isothermal scenario through parameters $\kappa$, $\omega$ and $Ma$.