2.1. Flow configuration

We consider an incompressible viscoelastic fluid film of density $\rho$ and dynamic viscosity $\mu$ flowing on an infinite flat plate. A Cartesian coordinate system $(x^{*},y^{*} )$ is introduced, as shown in figure 1, with the $x^{*}$ -axis aligned with the plate, where the dimensional variables are indicated by an asterisk. The plate oscillates in the $x^{*}$ -direction with its velocity magnitude given by

(2.1) \begin{equation} U_{p}^{*} \big(t^{*} \big) =U_{1}\cos \big(\omega _{1} t^{*}\big)+U_{2}\cos \big(\omega _{2} t^{*}\big), \end{equation}

where $U_{1}$ and $U_{2}$ are the superimposed constant velocity amplitudes, and $\omega _{1}$ and $\omega _{2}$ are considered two incommensurable frequencies. Two separate modes that contain no phase shifts are superimposed into the oscillation, and each mode has its own frequency and amplitude. The undisturbed configuration corresponds to a film thickness of $d$ . The upper free surface of the film is described by $y^{*}=\eta ^{*}(x^{*},t^{*})$ , and is covered by a monolayer of insoluble surfactant. We note ${\boldsymbol{V}}^{*}=(U^{*},V^{*})$ , where ${\boldsymbol{V}}^{*}$ is the velocity field, and $U^{*}$ and $V^{*}$ are the velocity components in the streamwise and vertical directions. For the two-dimensional stability problem, the surfactant concentration ${\varGamma }^{*}(x^{*},t^{*})$ obeys the transport equation (Halpern & Frenkel Reference Halpern and Frenkel2003)

(2.2) \begin{equation} \frac {\partial \left (H {\varGamma }^{*} \right ) }{\partial t^{*} } +\frac {\partial }{\partial x^{*} } \left (H{\varGamma }^{*} U^{*} \right )=D_{s}\frac {\partial }{\partial x^{*} }\Biggl (\frac {1}{H}\frac {\partial {\varGamma }^{*}}{\partial x^{*} } \Biggr ), \end{equation}

where $H=\sqrt {1+ ({\partial \eta ^{*} }/{\partial x^{*} } )^{2} }$ is the curvature parameter associated with the free upper surface of the fluid film. Here, $D_{s}$ is the diffusivity of the surfactant, which is usually negligible and is ignored below. In the present study, we restrict our attention to an insoluble surfactant with negligible surface diffusion and a linear relation between surface tension and concentration. These assumptions provide a standard small-disturbance framework in the linear stability problem. Under the assumption of small-amplitude surface disturbances, the precise nature of the dependence of surface tension $\sigma ^{*}$ and the surfactant concentration ${\varGamma }^{*}(x^{*},t^{*})$ is a linear relation

(2.3) \begin{equation} \sigma ^{*}=\sigma _{0}-E\left ({\varGamma } ^{*} -{\varGamma }_{0} \right ) \!, \end{equation}

where $E$ is the surface elasticity, and ${\varGamma }_{0}$ denotes the basic value of the surfactant concentration, associated with a uniform surface tension $\sigma _{0}$ .

The system of the governing equations is given by the following momentum and continuity equations in tensor form:

(2.4) \begin{equation} \partial _{i} U^*_{i} = 0, \end{equation}

(2.5) \begin{equation} \rho \big (\partial _{t^*} U^*_{i}+U^*_{\!j} \partial _{\!j} U^*_{i}\big ) = \rho g_{i}+\partial _{\!j} \tau ^*_{ij}. \end{equation}

Here, $\tau ^{*}_{ij}$ is the stress tensor, with the subscripts $i,j$ implying summation over the $x^{*}$ - and $y^{*}$ -directions. The fluid rheological model is described by Walter’s liquid $B''$ , a viscoelastic model representing a first-order elastic approximation of Newtonian behaviour and a rapidly fading memory. The constitutive equation of liquid $B''$ is (Beard & Walters Reference Beard and Walters1964; Andersson & Dahl Reference Andersson and Dahl1999)

(2.6) \begin{equation} \tau ^*_{i j} = -P^*\delta _{i j} + 2\mu e^*_{ij}-2M_0\dfrac {\delta }{\delta t^*}e^*_{ij}, \end{equation}

where $P^*$ is the isotropic pressure, $\delta _{i j}$ is the Kronecker symbol, $e^*_{ij}=(\partial _{i}U^*_j + \partial _j U^*_i)/2$ is the strain rate tensor, $2\mu e^*_{ij}$ is the Newtonian stress, $2M_0(\delta /\delta {t^*})e^*_{i j}$ is the elastic stress contribution and $M_0$ is the viscoelastic coefficient. The co-rotational derivative of the strain rate tensor $e^*_{ij}$ is defined as (Shrestha Reference Shrestha1970; Samanta Reference Samanta2017)

(2.7) \begin{equation} \dfrac {\delta }{\delta {t^*}} e^*_{ij}=\partial _{t^*} e^*_{ij}+U^*_k \partial _k e^*_{ij}- \partial _k U^*_j e^*_{ik}-\partial _k U^*_i e^*_{kj}. \end{equation}

Obviously, the constitutive equation of liquid $B''$ recovers the Newtonian behaviour when the viscoelastic coefficient $M_0$ is set to zero.

This study employs standard boundary conditions for a free-surface liquid film with insoluble surfactant driven by an oscillating flat plate. At the oscillating substrate, the no-slip and no-penetration conditions are imposed, so that the velocity in the $x^{*}$ -direction matches the given plate velocity (2.1). At the free surface, the kinematic condition requires that the normal velocity at the interface is equal to the normal velocity of the film at the interface. The dynamic conditions consist of the tangential stress balance condition, in which the interfacial viscoelastic stress is balanced by the Marangoni stress induced by surfactant concentration gradients, and the normal stress balance condition, in which the normal stress discontinuity is balanced by capillary pressure. Thus, the corresponding boundary conditions are

1. (i) At the bottom of the liquid film $y^*=-d$

   (2.8) \begin{equation} U^*=U^{*}_{p}(t^{*}), \quad V^*=0. \end{equation}

2. (ii) The kinematic boundary condition at the free surface

   (2.9) \begin{equation} \partial _{t^*}\eta ^* = V^* - U^* \partial _{x^*}\eta ^*. \end{equation}

3. (iii) The normal and tangential stresses at the free surface

   (2.10a) \begin{align} -\big (1+\big (\partial _{x^*}\eta ^*\big )^{2}\big )P^{*}+2\mu \big [\big (\partial _{x^*}\eta ^*\big )^{2}\partial _{x^*}U^* -\partial _{x^*}\eta ^*(\partial _{y^*}U^*+\partial &_{x^*}V^*)+\partial _{y^*}V^*\big]\notag \\ -2M_{0}\biggl [\frac {\delta e_{xx}^{*}}{\delta t^{*}}(\partial _{x^*}\eta ^*)^{2}-2\frac {\delta e_{xy}^{*}}{\delta t^{*}}\partial _{x^*}\eta ^*+&\frac {\delta e_{yy}^{*}}{\delta t^{*}} \biggr ]=\frac {\sigma ^{*}\partial _{x^*x^*}\eta ^*}{\sqrt {1+\left (\partial _{x^*}\eta ^*\right )^{2}} }, \end{align}
   (2.10b) \begin{align} \mu \big [\big (1-\big (\partial _{x^*}\eta ^*\big )^{2}\big )\big (\partial _{y^*}U^* +\partial _{x^*}V^* \big) + 2\partial _{x^*}\eta ^*\big (\partial _{y^*}V^*-\partial _{x^*}U^*\big )&\big ] +2M_{0}\biggl (\frac {\delta e_{xx}^{*}}{\delta t^{*}}\partial _{x^*}\eta ^* \notag \\ +\frac {\delta e_{xy}^{*}}{\delta t^{*}} \big (\big (\partial _{x^*}\eta ^*\big )^{2}-1\big )-\frac {\delta e_{yy}^{*}}{\delta t^{*}}\partial _{x^*}\eta ^* \biggr ) &=\partial _{x^*}\sigma ^*\sqrt {1+\left (\partial _{x^*}\eta ^*\right )^{2}} , \end{align}
   where $\sigma ^*$ is the surface tension.

To non-dimensionalise the governing equations and boundary conditions, we introduce the following dimensionless variables, and we choose the parameters of the first modes as the characteristic scales. The mean thickness of the liquid film $d$ is selected as the characteristic length scale, $\nu /d$ as the velocity scale, $\omega _{1}^{-1}$ as the time scale and $\rho \nu ^{2}/d^{2}$ as the pressure scale. The surfactant concentration and surface tension are normalised by ${\varGamma }_{0}$ and $\sigma _{0}$ .

Then, a set of dimensionless parameters is obtained as follows:

(2.11) \begin{align} \textrm{Re}=\frac {U_{1}d}{\nu }, \qquad \chi =\frac {gd^{3}}{2\nu ^{2}}, \qquad \beta =&\sqrt {\frac {\omega _{1}d^{2}}{2\nu }},\qquad \textit{Ca}=\frac {\rho \nu ^{2}}{\sigma _{0}d}, \qquad \textit{Ma}=\frac {E{\varGamma }_{0}}{\sigma _{0}},\notag \\ M=\frac {M_{0}}{\rho d^{2}}, \qquad M_{s}=&\frac {E{\varGamma }_{0}}{\rho U_{1}^{2}d}, \qquad \omega =\frac {\omega _{2}}{\omega _{1}},\qquad \gamma =\frac {U_{2}}{U_{1}}. \end{align}

Here, $\nu =\mu /\rho$ denotes the kinematic viscosity. The Reynolds number $\textrm{Re}$ quantifies the ratio of the inertial effect, due to the oscillation amplitude $U_{1}$ , to the viscous effect, while the Galileo number $\chi$ measures the ratio of the gravitational effect to the viscous effect; $\beta$ is the Womersley number, which is the dimensionless frequency parameter representing the ratio of the average film thickness to the thickness of the Stokes layer induced by the wall oscillation, and it is also interpreted as a measure of the relative oscillation frequency $\omega _{1}$ . The capillary number $\textit{Ca}$ characterises the ratio of the viscous force to the surface-tension force, whereas the Marangoni number $\textit{Ma}$ interprets the ratio of the surface-tension gradient force to the viscous force. The viscoelastic parameter $M$ accounts for the effect of the viscoelasticity, and $M_{s}$ here is introduced to quantify the influence of surface surfactant. Finally, $\gamma =U_{2}/U_{1}$ denotes the ratio of superimposed velocity amplitudes, and $\omega = \omega _{2}/\omega _{1}$ is the frequency ratio considered irrational. These two parameters play a central role in the description of quasi-periodic oscillations considered in this paper. The present parameters describe a class of thin-film configurations relevant to applications such as coating flows, thin polymer layers and related films with surfactants. For illustration, if one considers a representative film thickness $d \sim 5\times 10^{-4}\rm\,m$ and a kinematic viscosity $\nu \sim 10^{-6}\,\rm m^{2}\,s^{-1}$ , then the range $\beta =O(1-10)$ corresponds to oscillation frequencies $\omega _{1}=2\nu \beta ^{2}/d^{2}$ of approximately $8-800\,\rm s^{-1}$ , which lie within a laboratory-accessible range for oscillatory thin-film experiments. The relatively small values of $M$ considered are also consistent with the weakly viscoelastic regime appropriate to Walter’s liquid $B''$ .

The basic state corresponds to a plane interface flow with a unidirectional velocity parallel to the $x$ axis, and ${\boldsymbol{V}} _{b}=(U_{b}(y,t),\ 0)$ . The base state is associated with a zero mean pressure gradient, so that the characteristic pressure is fixed by the perturbation problem. By the method of separation of variables, we obtain the dimensionless form of the basic velocity profile

(2.12) \begin{equation} U_{b}={\rm Re} \left [ \frac {{\rm Re}\cosh [\varOmega _{1}(1+\mathrm{i}S_{1})y]}{\cosh [\varOmega _{1}(1+\mathrm{i}S_{1})]}\mathrm{e}^{{i}t} +\gamma \frac {{\rm Re}\cosh [\varOmega _{2}(1+\mathrm{i}S_{2})y]}{\cosh [\varOmega _{2}(1+\mathrm{i}S_{2})]}\mathrm{e}^{{i}\omega t} \right ]\!, \end{equation}

with

(2.13) \begin{equation} S_{1}=(1+4M^2\beta ^4)^{1/2}+2M\beta ^2, \qquad \varOmega _{1}=\beta \sqrt { \frac { (1+4M^2\beta ^4)^{1/2} -2M\beta ^2}{1+4M^2\beta ^4} }, \end{equation}

(2.14) \begin{align} S_{2}=(1+4M^2 \omega ^{2} \beta ^4)^{1/2}+2M\omega \beta ^2, \qquad \varOmega _{2}=\beta \sqrt {\omega } \sqrt { \frac { (1+4M^2\omega ^{2}\beta ^4)^{1/2} -2M\omega \beta ^2}{1+4M^2\omega ^{2}\beta ^4} }, \end{align}

where $-1 \leqslant y\leqslant 0$ . Here, $\mathrm{Re}[\boldsymbol{\cdot }]$ represents the real part of that complex function. The pressure is induced by gravity

(2.15) \begin{equation} \bar {P}\left (y\right ) =P_{0}-2\chi y. \end{equation}

Four parameters are involved in the expression of the base-flow profile: the ratio of the velocity amplitudes $\gamma$ , the frequency ratio $\omega$ , the viscoelastic parameter $M$ and the parameter $\beta$ . Notably, the absence of $\textit{Ma}$ in the expression for the basic state velocity $U_{b}$ indicates the fact that the base flow is unaffected by surfactants. This results from the assumption of a spatially uniform initial surfactant distribution along the interface. Under this assumption, no surfactant concentration gradients are generated in the base state, and consequently no Marangoni stresses arise to modify the flow field. Moreover, in the absence of the viscoelastic parameter ( $M=0$ ), the base-flow velocity (2.12) is identical to that of Jia et al. (Reference Jia, Xiao, Wang, Zhao and Xue2026)

2.2. Linear stability problem

In order to study the linear stability of the base flow, it is assumed that the disturbances are infinitesimal on the fundamental flow by introducing the following variables:

(2.16) \begin{equation} U=U_{b}+u',\qquad V=v', \qquad P=\bar {P}+p', \qquad \varGamma =1+\varGamma ', \qquad \sigma =1+\sigma '. \end{equation}

Here, $(u',v')$ are the components of the velocity disturbance in the Cartesian coordinates $ (x,y )$ , $p'$ and $\eta '$ designate pressure and interfacial disturbances and ${\varGamma }'$ and $\sigma '$ designate surfactant concentration and surface-tension disturbances, respectively.

Since only two-dimensional perturbations are considered, the continuity equation (2.4) is automatically satisfied by introducing a perturbed streamfunction $\psi ' (x,y )$ , such that

(2.17) \begin{equation} u'=\partial _{y}\psi ',\quad v'=-\partial _{x}\psi '. \end{equation}

Since the base flow is homogeneous in the $x$ -direction, perturbations of the streamfunction $\psi '$ , surfactant concentration ${\varGamma }'$ and interface $\eta '$ are expressed in normal modes, i.e.

(2.18) \begin{equation} \left (\psi ',\ {\varGamma }',\ \eta '\right )=\left (\phi \left (y,t\right ),\ \xi \left (t\right ),\ h\left (t\right )\right )\mathrm{e}^{{i}kx}, \end{equation}

where $\phi(y,t)$ , $\xi(t)$ and $h(t)$ are the normal-mode amplitudes of the streamfunction, surfactant concentration and interfacial disturbances, respectively, and $k\in \mathbb{R}$ is the wavenumber in the streamwise direction. It is worth noting that the perturbation pressure $p'$ can be eliminated by an appropriate combination of the Navier–Stokes equations and therefore does not enter the linearised system explicitly. Substituting (2.18) into the governing equations and retaining only linear terms, the resulting formulation reduces to a time-dependent Orr–Sommerfeld equation

(2.19) \begin{align} 2 {\beta} ^{2}(\mathscr{L}+M \mathscr{L}^{2}) \partial _{t}\phi =\mathscr{L}^{2} \phi -\mathrm{i} k[U_{b}(\mathscr{L}+M \mathscr{L}^{2})-(\mathscr{D}^{2} U_{b}+M \mathscr{D}^{4} U_{b})] \phi , \end{align}

with differential operators $\mathscr{D}=\partial _{y}$ and $\mathscr{L}=\mathscr{D}^{2}-k^{2}$ . Substituting the normal modes (2.18) and linearising, the boundary conditions at the wall $y=-1$ are

(2.20) \begin{equation} \phi =\mathscr{D}\phi =0, \end{equation}

which represent the linearised no-slip and no-penetration conditions. At the free surface $y=0$ , the linearised system contains the kinematic condition, the surfactant transport equation for the concentration disturbance and the linearised tangential and normal stresses conditions. These four conditions provide the closure required for the time-dependent Orr–Sommerfeld problem.

The linearised kinematic boundary condition and transport equation for surfactant at $y=0$ are expressed by

(2.21) \begin{align} &2\beta ^{2} \partial _{t}h+\mathrm{i} k U_{b} h+\mathrm{i}k\phi =0, \end{align}

(2.22) \begin{align} &2\beta ^{2} \partial _{t}\xi +\mathrm{i} k U_{b} \xi +\mathrm{i}k \mathscr{D} \phi =0. \end{align}

The linearised conditions for the normal and tangential stresses at $y=0$ are given by

(2.23) \begin{align} 2 \beta ^{2}[\mathscr{D}+M(\mathscr{L}-2 k^{2}) \mathscr{D}] \partial _{t}\phi &=-\mathrm{i} k[U_{b} \{\mathscr{D}+M(\mathscr{L}-2 k^{2}) \mathscr{D} \} \notag \\+M(\mathscr{D}^{2} U_{b} \mathscr{D}-\mathscr{D}^{3} U_{b})]& \phi +(\mathscr{L}-2 k^{2}) \mathscr{D} \phi -2 \mathrm{i} k\left (\chi +\frac {k^{2}}{2 \textit{Ca}}\right ) h, \end{align}

(2.24) \begin{align} 2 \beta ^{2} M(\mathscr{L}+2 k^{2}) \partial _{t}\phi =(\mathscr{L}+2 &k^{2}) \phi - \mathrm{i}k M [U_{b}(\mathscr{L}+2 k^{2})-\mathscr{D}^{2} U_{b}] \phi \notag \\ +[&\mathscr{D}^{2} U_{b}-2 \beta ^{2} M \mathscr{D}^{2} \partial _{t}U_{b}]h + M_s {\textrm{Re}}^2 \mathrm{i}k\xi . \end{align}

The time-dependent Orr–Sommerfeld equation (2.19) subject to conditions (2.20)–(2.24) forms a Floquet system. For finite-wavelength instabilities, the differential system should be solved numerically, whereas for the long-wave instabilities, analytical solutions can be obtained through an expansion in the wavenumber $k$ , as discussed in the following section.

3.1. Absence of gravity $\chi =0$

In certain microgravity environments, such as the space environment, gravitational effects may be negligible or entirely absent, making it necessary to examine the stability characteristics under zero-gravity conditions. The influences of the viscoelastic parameter $M$ , surface surfactant $M_{s}$ , the amplitude ratio $\gamma$ and the frequency ratio $\omega$ on the neutral stability conditions are therefore investigated through a direct analysis of (3.27). For the long-wavelength instability ( $k \ll 0$ ), the neutral condition in the absence of gravity becomes

(3.28) \begin{equation} \max \big (4I_{1}-2I_{2},I_{1}-M_{s}{\textrm{Re}}^{2}\big )=0. \end{equation}

While (3.28) shares the same functional form as that derived by Gao & Lu (Reference Gao and Lu2006), the influence of viscoelasticity and quasi-periodic oscillation is introduced into the expressions $I_{1}$ and $I_{2}$ .

Figure 2.

The real part of $\theta /{\textrm{Re}}^{2}$ as a function of $\beta$ when $\chi =0$ , $\textrm{Re}=200$ , $\omega =\sqrt {37}$ , $\gamma =1$ for ( $a$ ) $M_{s}=0.005$ and ( $b$ ) $M_{s}=0.001$ with $M=0$ (black dashed lines), $M=0.003$ (orange solid lines) and $M=0.007$ (blue solid lines). The black solid lines represent Newtonian fluids without surfactants.

In order to explore more clearly the influence of viscoelasticity on different Floquet modes in the presence of surfactants, the real part of $\theta /{\textrm{Re}}^{2}$ as a function of $\beta$ for typical values $M=0$ , $M=0.003$ and $M=0.007$ for different $M_{s}$ when $\chi =0$ , $\textrm{Re}=200$ , $\omega =\sqrt {37}$ and $\gamma =1$ is depicted in figure 2. In fact, for other frequency ratios $\omega$ and amplitude ratios $\gamma$ , the real part of $\theta /{\textrm{Re}}^{2}$ for different viscoelasticity parameters $M$ exhibits similar trends and is thus omitted, confirming the generality of the present conclusion. Rather than presenting the eigenvalue spectrum, we focus here on the two dominant Floquet roots, since the real part of $\theta /{\textrm{Re}}^{2}$ already captures the modal information most relevant to the long-wave instability discussed below. In addition, for comparison, the growth curves of the Newtonian fluid ( $M=0$ ) and the clean surface ( $M_{s}=0$ ) are also shown in black solid lines. In the presence of surfactants, the two roots of (3.24) are represented in the same colour in figure 2 and these two roots correspond to the two Floquet modes that we have obtained. Throughout most of the range of $\beta$ , the Floquet exponents are real and are shown as two branches of the growth rate curve, corresponding to standing-wave modes, whereas in the small frequency range, such as $2.06\lt \beta \lt 2.79$ when $M=0.003$ , the two branches merge into a curve. The Floquet exponents in this interval are a pair of conjugate complex numbers, corresponding to the travelling-wave disturbances. Additionally, in contrast to the periodic oscillations in Wang et al. (Reference Wang, Du, Xiao and Zhao2023), the curves for both clean and contaminated surfaces exhibit two distinct peaks of the unstable modes, corresponding to the two primary frequencies $\omega _{1}$ and $\omega _{2}$ of the quasi-periodic oscillations. Specifically, the peak at the low-frequency corresponds to $\omega _{2}$ , while $\omega _{1}$ corresponds to the high-frequency peak. This bimodal structure is a direct consequence of the quasi-periodic oscillation, since the two independent primary frequencies $\omega _{1}$ and $\omega _{2}$ each introduce a preferred amplification band. Furthermore, according to the result of Gao & Lu (Reference Gao and Lu2006), the existence of surfactants will inhibit the growth of disturbance and make the flow more stable. Therefore, in figure 2, the growth rate of a Newtonian fluid with surfactants (black dashed lines) is lower than that in the case of a clean surface (black solid lines). For a larger $M_{s}=0.005$ in figure 2( $a$ ), as the viscoelastic parameter $M$ increases from zero to $0.003$ , the growth rate of the two most unstable modes also increases. However, within the given parameter range, their maximum growth rate still does not exceed the growth rate under the clean surface condition. This indicates that, although viscoelasticity can promote the growth of disturbance, the destabilising effect of viscoelasticity is less obvious than the stabilising effect of surfactants when viscoelasticity is weak. Notably, the increase in viscoelasticity has no significant effect on the oscillation frequencies corresponding to the two peaks, which remain at $\beta =0.46$ and $\beta =1.06$ . As the viscoelastic parameter $M$ continues to increase to $0.007$ , the growth rate of the most unstable mode corresponding to $\omega _{2}$ exceeds that under the clean surface condition, while the peak growth rate corresponding to $\omega _{1}$ remains lower than that under the clean surface. This indicates that, when the viscoelasticity is sufficiently large, even in the presence of surfactants, the flow may still be more unstable than the Newtonian fluid with a clean surface, and the growth rate of the most unstable mode corresponding to $\omega _{2}$ increases more rapidly. Figure 2( $b$ ) further illustrates the case for a smaller surfactant concentration parameter $M_{s}=0.001$ . The results indicate that when the viscoelastic effect significantly exceeds the surfactant effect, the peaks of the unstable modes corresponding to the two frequencies $\omega _{1}$ and $\omega _{2}$ both exceed those under the clean surface condition. In summary, when viscoelasticity and surfactant exist at the same time, the stability of oscillating liquid film mainly depends on the competitive relationship between the two, with the peak of the unstable mode corresponding to $\omega _{2}$ being more significantly influenced by viscoelasticity.

Figure 3.

Effect of different viscoelastic parameter, $M$ , on the neutral curves on the $\beta -M_{s}$ plane when $\chi =0$ , $\textrm{Re}=200$ , $\omega =\sqrt {37}$ , $\gamma =1$ , ( $a$ ) $M=0$ , ( $b$ ) $M=0.001$ , ( $c$ ) $M=0.0015$ and ( $d$ ) $M=0.002$ . Stable (S) and unstable (U) regions are denoted by yellow and blue bands, respectively.

Figure 3 shows the effect of different viscoelastic parameters $M$ on the $\beta -M_{s}$ neutral curves under given parameters. These neutral curves divide the parameter plane into alternating stable and unstable intervals. As the value of $M_{s}$ increases from nearly zero, the neutral curve initially appears as vertical lines. When $M_{s}$ continues to increase, the majority of the neutral curve, with the exception of the high-frequency region in figure 3( $a$ ), becomes curved, accompanied by the narrowing of the unstable interval, indicating a stabilising effect of surfactant. Upon a further increase in $M_{s}$ , these neutral curves straighten back into a vertical line. For a Newtonian fluid ( $M=0$ ), the characteristics of the neutral curves within the interval $0\lt \beta \lt 10$ are consistent with those found by Gao & Lu (Reference Gao and Lu2006), i.e. with the increase of the surfactant concentration, the form of motion of the perturbed mode changes from standing waves (vertical straight lines controlled by $4I_{1}-2I_{2}$ ) to travelling waves (curve parts controlled by $I_{1}-{\textrm{Re}}^{2}M_{s}$ ) to standing waves again, and the width of unstable intervals changes from decreasing to eventually constant. However, the neutral curves within the interval $10\lt \beta \lt 16$ remain as vertical lines (standing waves), and the width of unstable intervals remains unchanged. This indicates that, in the high-frequency region, the effect of surfactant concentration on the neutral curve can be neglected. For clarity in subsequent discussion, the unstable regions in the figure are sequentially labelled as the first to the fifth unstable regions along the direction of increasing oscillatory frequency $\beta$ . This nomenclature will be used throughout the following discussion. When the viscoelastic parameter $M=0.001$ in figure 3( $b$ ), the first unstable region in the interval $0\lt \beta \lt 2.6$ remains almost unchanged, while the shrinkage positions of the second and third unstable regions are raised, leading to the expansion of these unstable regions. This indicates that viscoelasticity plays a role in promoting instability. The fourth unstable region undergoes significant changes. It shifts toward lower frequencies, and compared with the fourth unstable region under Newtonian fluid conditions in figure 3( $a$ ), a shrinkage phenomenon emerges, manifested as a travelling-wave perturbed mode, emerges. Similarly, the fifth unstable region also exhibits shrinkage, and the $\beta$ range it occupies increases significantly. These results indicate that the viscoelastic parameter has a more significant effect on instability in the high-frequency region. When the viscoelastic parameter $M$ increases further to 0.0015 in figure 3( $c$ ), the first three unstable regions remain relatively unchanged, indicating that the viscoelastic parameter has little effect on the neutral curve in the low-frequency region. However, the range of the fourth unstable region continues to expand, and the shrinkage position continues to be raised. Meanwhile, the fifth unstable region moves to higher frequencies, causing the stable regions between the fourth and fifth unstable regions to occupy most of the high-frequency range.

Figure 4.

The real part of $\theta /{\textrm{Re}}^{2}$ as a function of $\beta$ when $\chi =0$ , $\textrm{Re}=200$ , $\omega =\sqrt {37}$ and $\gamma =1$ with fixed $M_{s}=10^{-6}$ . The blue and orange curves represent the two roots of (3.24) when $M=0$ in figure 3( $a$ ), while the grey curves represent results when $M=0.0015$ in figure 3( $c$ ).

To further illustrate this phenomenon, figure 4 presents the variations of the real parts of the growth rates of two Floquet modes with the oscillation frequency $\beta$ for this parameter combination with a fixed $M_{s}=10^{-6}$ , which are represented in blue and orange. For comparison, the real part of the growth rate curves under the same parameter combination when viscoelasticity is absent, i.e. $M=0$ , is also shown in the figure. Overall, in the low-frequency range, the curve follows a trend similar to that shown in figure 2; however, some differences emerge in the high-frequency region. In the inset, when $\beta$ increases to $9.21$ , the growth rate corresponding to the blue solid line changes from negative to positive, marking the onset of the fourth unstable region in figure 3( $c$ ). As $\beta$ continues to increase to $10.19$ , the growth rate corresponding to the blue solid line changes from positive to negative and only becomes positive again when $\beta$ reaches $14.22$ . The interval $10.19\lt \beta \lt 14.22$ corresponds to the stable regions within the fourth and fifth unstable regions. In the interval where $\beta \gt 14.22$ , due to the very small range of the vertical axis, the growth rate curve in the inset approximates a vertical line. Within this interval, although the growth rate corresponding to the orange dashed line becomes negative, the growth rate corresponding to the blue solid line remains positive, resulting in an unstable flow. In addition, by comparing with the growth rate curves under non-viscoelastic condition, it can be seen that, under this parameter combination, the presence of viscoelasticity results in a broad range of negative growth rates in the high-frequency range, leading to a wider stable region on the $\beta -M_{s}$ plane in figure 3 $(c)$ . Therefore, the high-frequency region becomes more stable, and this stability transition is caused by the modal change induced by viscoelasticity, indicating that viscoelasticity exhibits a stabilising effect under specific parameters. Finally, as shown in figure 3 $(d)$ , when the viscoelastic parameter $M=0.002$ , the fourth and fifth unstable regions merge and occupy the entire high-frequency range, indicating that viscoelasticity once again exhibits a destabilising effect. In summary, in the absence of gravity, the increase in viscoelasticity has little effect on the neutral curve in the low-frequency region but has a significant impact on the high-frequency region, where it exhibits a dual effect. Here, the term dual effect refers to the non-monotonic influence of viscoelasticity in the high-frequency region.

Figure 5.

Effect of different frequency ratios, $\omega$ , on the neutral curves on the $\beta -M_{s}$ plane when $\chi =0$ , $\textrm{Re}=200$ , $\gamma =1$ and $M=0.001$ , ( $a$ ) $\omega =\sqrt {37}$ , ( $b$ ) $\omega =\sqrt {7}$ , ( $c$ ) $\omega =\sqrt {2}$ , ( $d$ ) $\omega =1/\sqrt {2}$ , ( $e$ ) $\omega =1/\sqrt {7}$ and ( $f$ ) $\omega =1/\sqrt {37}$ . Stable (S) and unstable (U) regions are denoted by yellow and blue bands, respectively.

Figure 6.

Effect of different amplitude ratios, $\gamma$ , on the neutral curves on the $\beta -M_{s}$ plane when $\chi =0$ , $\textrm{Re}=200$ , $\omega =\sqrt {37}$ and $M=0.001$ , ( $a$ ) $\gamma =0.1$ , ( $b$ ) $\gamma =1.5$ , ( $c$ ) $\gamma =2$ and ( $d$ ) $\gamma =3$ . Stable (S) and unstable (U) regions are denoted by yellow and blue bands, respectively.

Figure 6. Long description

The image contains four vertical bar graphs labeled (a), (b), (c), and (d). Each graph shows the relationship between the variable M_s on the vertical axis and beta on the horizontal axis. The vertical axis is logarithmic, ranging from 10^-10 to 10^0, and the horizontal axis ranges from 0 to 16. Stable (S) and unstable (U) regions are denoted by yellow and blue bands, respectively. Panel A: The graph shows the neutral curves for a specific amplitude ratio. The unstable regions are concentrated at lower values of beta and higher values of M_s. Panel B: This graph also depicts the neutral curves for a different amplitude ratio. The pattern of stable and unstable regions is similar to Panel A but with slight variations in the width and position of the bands. Panel C: The graph shows another set of neutral curves for a different amplitude ratio. The unstable regions are more pronounced at specific intervals of beta. Panel D: This graph depicts the neutral curves for yet another amplitude ratio. The stable and unstable regions follow a pattern similar to the previous panels but with distinct differences in the distribution and width of the bands.

To facilitate a more direct comparison with the results of previous studies, the frequency-ratio parameters $\omega$ employed in Jaouahiry & Aniss (Reference Jaouahiry and Aniss2025) and Jia et al. (Reference Jia, Xiao, Wang, Zhao and Xue2026) are adopted in the present study. Figure 5 displays the effect of different frequency ratios $\omega$ on the $\beta -M_{s}$ neutral curves under given parameters. Firstly, when the frequency ratio $\omega \gt 1$ , as the frequency ratio $\omega$ decreases from $\sqrt {37}$ to $\sqrt {7}$ , the stability characteristics of the system do not exhibit significant changes within the interval $0\lt \beta \lt 11.5$ . However, in the high-frequency interval of $11.5\lt \beta \lt 16$ , the range of the stable region expands markedly, indicating that a decrease in the frequency ratio $\omega$ can enhance the stability in the high-frequency range. As the frequency ratio $\omega$ further decreases to $\sqrt {2}$ in figure 5( $c$ ), the changes in stability characteristics are primarily confined to the interval $4.08\lt \beta \lt 8.49$ . Within this second unstable region, increasing $M_{s}$ causes the perturbed mode to transform from standing waves to travelling waves, without reverting back to the standing-wave mode again, and this leads to the range of the second unstable region shrinking to zero along the $M_{s}$ direction. Meanwhile, the shrinkage phenomenon of the third unstable region becomes more significant. Collectively, these changes promote stabilisation within this frequency range. The above results indicate that when both viscoelasticity and surfactants are present, a decrease in the frequency ratio $\omega$ can suppress flow instability when $\omega \gt 1$ . As the frequency ratio $\omega$ decreases, instability in the high-frequency region is suppressed first, followed by the suppression of instability in the mid-to-low-frequency regions. Furthermore, when the frequency ratio $\omega \lt 1$ , the neutral curves for different values of $\omega$ exhibit the same trend as the curve for $1/\omega$ . The only difference is that, at this point, the range of the stable and unstable regions on the $\beta -M_{s}$ plane expands significantly, occupying a larger range of frequency parameters $\beta$ .

Figure 6 illustrates the effect of different frequency ratios $\gamma$ on the $\beta -M_{s}$ neutral curves under given parameters. As the amplitude ratio $\gamma$ increases from $0.1$ to $1.5$ , the shrinkage position of the fifth unstable region within the high-frequency interval is raised. While in the low-frequency domain, an additional smaller stable region is observed in the interval $0.7\lt \beta \lt 1$ . The physical mechanism of this smaller stable region originates from the bimodal structure of the growth rate curves under quasi-periodic oscillations, and this phenomenon will be discussed in detail in the subsequent sections. With the further increase of $\gamma$ to $2$ in figure 6( $c$ ), the smaller stable region is gradually engulfed by the larger stable region on its right. As $\gamma$ increases further to $3$ in figure 6( $d$ ), the overall area of the stable region begins to decrease, demonstrating that for sufficiently large $\gamma$ , the stabilising effect of surfactants is weaker than the destabilising effect of high frequency ratio $\gamma$ . Moreover, when $\gamma$ exceeds 1.5, the evolution of the neutral curves in the high-frequency region for different amplitude ratios shows no significant characteristic trends.

Figure 7.

The real part of $\theta /{\textrm{Re}}^{2}$ as a function of $\beta$ when $\chi =0$ , $\textrm{Re}=200$ , $\omega =\sqrt {37}$ and $M=0.001$ for fixed $M_{s}=0.3$ in figure 6, with $\gamma =1.5$ (black solid line), $\gamma =2$ (orange solid line) and $\gamma =3$ (blue solid line).

To further clarify the smaller stable region observed in the low-frequency region in figure 6( $b$ ) and its subsequent evolution, figure 7 plots the variation of the real part of the growth rate with the oscillation frequency $\beta$ for the given parameters at different frequency ratios $\gamma$ , where $M_{s}=0.3$ is held constant. Since one of the two Floquet modes remains negative for all three prescribed frequency ratios, it is therefore neglected in figure 7. Even at a higher surfactant concentration of $M_{s}=0.3$ , the growth rate curves still exhibit the characteristic of two distinct peaks. However, owing to the larger amplitude ratio $\gamma$ , the growth rate of the most unstable mode corresponding to $\omega _{1}$ is correspondingly reduced (see the inset). When $\gamma =1.5$ , in contrast to the growth rate curves for $\gamma =1$ in figures 2 and 4, an interval with negative growth rates ( $0.7\lt \beta \lt 1$ ) emerges between the two peaks, corresponding to the smaller stable region observed in figure 6( $b$ ). As $\gamma$ increases to $2$ , the peak growth rate corresponding to $\omega _{1}$ further decreases to a negative value and merges with the adjacent interval with negative growth rates. This forms a broader band of negative growth rates, which corresponds to the merging of stable regions in figure 6( $c$ ). Finally, as the frequency ratio $\gamma$ continues to increase to $3$ , the range of oscillation frequencies covered by the interval with negative growth rates actually contracts, consistent with a reduction in the area of the stable region in figure 6( $d$ ). The above results indicate that increasing the amplitude ratio $\gamma$ enhances the growth rate of the most unstable mode associated with $\omega _{2}$ , while suppressing that corresponding to $\omega _{1}$ , even reducing it to negative. Moreover, a similarly smaller stable region was also observed in the work of Jia et al. (Reference Jia, Xiao, Wang, Zhao and Xue2026), suggesting that this phenomenon is not strongly related to viscoelasticity.

3.2. Presence of gravity $\chi =1$

Figure 8.

( $a$ ) The neutral curves in the $\textrm{Re}-\beta$ plane for different viscoelastic parameters with $\chi =1$ , $\omega =\sqrt {37}$ , $\gamma =1$ and $M_{s}=0.01$ when $M=0$ (black dashed lines), $M=0.001$ (orange solid lines), $M=0.0015$ (blue solid lines) and $M=0.002$ (red dotted lines). ( $b$ ) When $\textrm{Re}=10^{5}$ is fixed, the real part of $\theta /{\textrm{Re}}^{2}$ as a function of $\beta$ under identical parameters as in figure 8( $a$ ) in the interval $9\lt \beta \lt 15$ , and the black dashed lines, orange solid lines, blue solid lines and red dotted lines respectively represent $M=0$ , $M=0.001$ , $M=0.0015$ and $M=0.002$ .

Figure 8. Long description

Panel A: A line graph shows the relationship between the Reynolds number (Re) on the vertical axis and the parameter beta (β) on the horizontal axis. The graph includes multiple lines: black dashed lines, orange solid lines, blue solid lines, and red dotted lines, each representing different viscoelastic parameters. The lines exhibit various trends, including sharp increases and decreases, indicating different stability behaviors. Panel B: Another line graph displays the real part of a variable (θr/Re^2) on the vertical axis as a function of beta (β) on the horizontal axis. The graph includes the same line styles as Panel A. An inset within this panel zooms in on the interval from 10.5 to 12.5 for beta, showing detailed behavior around this range. The main graph and the inset together illustrate how the real part of the variable changes with beta, highlighting specific trends and interactions among the different lines.

The cases with gravity, which correspond to terrestrial experiments and industrial thin-film applications, represent the most common situation. When the influence of gravity is considered, in the $\textrm{Re}-\beta$ plane, the neutral curves corresponding to long-wave instability present U-shaped curves with separated bandwidths of frequency, where the separated bandwidths refer to stable regions lying between long-wave instability intervals. Underneath the curve and outside the bandwidth range, all infinitesimal long-wave disturbances are stable. The physical origin of these U-shaped neutral curves is the stabilising action of gravity on the horizontal film. In the presence of gravity, the oscillatory forcing must be sufficiently strong to overcome this stabilising effect, so that instability appears only above a critical $\textrm{Re}$ . This gives rise to a U-shaped curve in the $\textrm{Re}-\beta$ plane, with the minimum indicating the frequency in which the oscillatory motion is most effective in destabilising the film. Wang et al. (Reference Wang, Du, Xiao and Zhao2023) studied the viscoelastic liquid film containing surfactants, and Jia et al. (Reference Jia, Xiao, Wang, Zhao and Xue2026) also studied the Newtonian liquid film containing surfactants under the quasi-periodic oscillation. Here, we take both viscoelasticity and the quasi-periodic oscillation into account under the influence of surfactant effects. Figure 8( $a$ ) displays the influence of different viscoelastic parameters on the neutral curves with $\chi =1$ , $\omega =\sqrt {37}$ , $\gamma =1$ and $M_{s}=0.01$ when $M=0$ , $M=0.001$ , $M=0.0015$ and $M=0.002$ . In comparison with the results for single-frequency oscillations studied by Wang et al. (Reference Wang, Du, Xiao and Zhao2023), the presence of quasi-periodic oscillations leads to the emergence of two local minima in the lower segment of the first branch of neutral stability curves, which correspond to the two primary frequencies, $\omega _{1}$ and $\omega _{2}$ , of the quasi-periodic oscillations. By contrast, each of the remaining branches of the neutral curve exhibits only a single local minimum, suggesting that the effect of the quasi-periodic oscillations is confined to the first branch. For low-frequency oscillation ( $\beta \lt 2.09$ ), the neutral curves corresponding to different viscoelastic parameters nearly coincide, which means that viscoelasticity hardly affects the stability of low-frequency oscillation liquid film. Within the range of $4.56\lt \beta \lt 8.50$ , both the critical Reynolds number and the imposed oscillation frequency progressively decrease with the increase of viscoelasticity. However, in the high-frequency region ( $\beta \gt 9.21$ ), the results exhibit a distinctly different trend with increasing viscoelastic parameters. Firstly, for the Newtonian fluid ( $M=0$ ), the unstable region enclosed by the U-shaped branches within the high-frequency region contracts monotonically with increasing imposed oscillation frequency $\beta$ . When the viscoelastic parameter $M$ increases to $0.001$ , the unstable region no longer contracts monotonically but instead occupies a larger parameter space. This indicates that viscoelasticity now significantly destabilises the liquid film under high-frequency oscillations, which is markedly different from the results obtained under low-frequency oscillations. As $M$ increases further to $0.0015$ , the range of the two U-shaped curves in the high-frequency region continues to expand. However, within the range $10.17\lt \beta \lt 14.30$ , a stable region occupying the majority of the high-frequency region appears, implying that under this viscoelastic parameter the viscoelasticity plays a stabilising role in the high-frequency region. Finally, when $M$ increases to $0.002$ , the high-frequency region is entirely dominated by a single U-shaped curve, indicating that the influence of viscoelasticity once again turns into a more significantly destabilising effect. To clarify the above observations, figure 8( $b$ ) presents the real part of the growth rate as functions of $\beta$ in the interval $9\lt \beta \lt 15$ with fixed $\textrm{Re}=10^{5}$ , under identical parameters as in figure 8( $a$ ). Although (3.24) admits two roots under this condition, only one root is within the selected parameter range. All the growth rate curves exhibit a local minimum, and this minimum follows a trend of first decreasing and then increasing as the viscoelastic parameter $M$ increases. When $M=0.001$ , the minimum value is relatively small. As $M$ increases to $0.0015$ , the minimum value significantly decreases, and the corresponding oscillation frequency increases to $\beta =13.43$ , which causes the growth rate to be negative across the interval $10.17\lt \beta \lt 14.30$ , corresponding to the stable region observed in this frequency range in figure 8( $a$ ). When $M$ continues to increase to $0.002$ , the minimum value increases significantly and the corresponding oscillation frequency decreases to $\beta =9.75$ , resulting in an unstable flow throughout the entirely high-frequency region. In summary, in the presence of gravity, viscoelasticity has little effect on the stability of low-frequency oscillation liquid film. However, it exhibits a dual regulation effect on the stability of high-frequency oscillation liquid film, initially destabilising, then stabilising and finally destabilising again the liquid film as $M$ increases.

Figure 9.

The neutral curves in the $\textrm{Re}-\beta$ plane for different frequency ratios with $\chi =1$ , $\gamma =1$ , $M=0.002$ and $M_{s}=0.01$ . ( $a$ ) The black solid lines, orange dashed lines and blue solid lines represent $\omega =\sqrt {37}$ , $\omega =\sqrt {7}$ and $\omega =\sqrt {2}$ , respectively. ( $b$ ) The black solid lines, orange dashed lines and blue solid lines represent $\omega =1/\sqrt {37}$ , $\omega =1/\sqrt {7}$ and $\omega =1/\sqrt {2}$ , respectively.

Figure 9 depicts the effect of different frequency ratios on the neutral curves with $\chi =1$ , $\gamma =1$ , $M=0.002$ and $M_{s}=0.01$ . Results for $\omega \gt 1$ are shown in figure 9( $a$ ), while those for $\omega \lt 1$ appear in figure 9( $b$ ). When $\omega \gt 1$ , for the neutral curves corresponding to different frequency ratios $\omega$ , the bandwidths of the first branches in the interval $0\lt \beta \lt 2.11$ remain nearly invariant, indicating a weak sensitivity of the instability structure to the frequency ratios in the low-frequency range for $\omega \gt 1$ . As $\omega$ decreases from $\sqrt {37}$ to $\sqrt {2}$ , i.e. as $\omega _{1}$ and $\omega _{2}$ become increasingly close, the two minima on the first branch of the unstable frequency band progressively coalesce into a single minimum. This merging behaviour reflects the modulation of resonance interactions under increasingly similar excitation frequencies. However, in the high-frequency range $\beta \gt 10.28$ , the neutral curves for different frequency ratios exhibit significant differences. When $\omega =\sqrt {37}$ , the high-frequency region is almost entirely dominated by U-shaped unstable frequency bands. As the frequency ratio decreases, this U-shaped unstable frequency band gradually separates, and the range of the unstable region contracts, indicating that when $\omega \gt 1$ , the stability in the high-frequency range is progressively enhanced as the frequency ratio decreases. However, the results reported by Jia et al. (Reference Jia, Xiao, Wang, Zhao and Xue2026) for Newtonian fluids indicate that, in the absence of viscoelasticity, stability in the high-frequency region remains unaffected by variations in the frequency ratio $\omega$ . In figure 9( $b$ ), as the frequency ratio $\omega$ increases, the bandwidths of the first branches in the neutral curves rapidly decrease rapidly, suggesting that, when $\omega \lt 1$ , increasing the frequency ratios significantly stabilises the liquid film in the low-frequency region. Meanwhile, on the neutral stability curve corresponding to $\omega =1/\sqrt {37}$ , it still exhibits two discernible minima. In contrast to the results for Newtonian fluids in Jia et al. (Reference Jia, Xiao, Wang, Zhao and Xue2026), in the high-frequency region $\beta \gt 25.31$ , when viscoelasticity is present, the neutral stability curves for different frequency ratios are all stacked together, while no such overlap occurs in the neutral stability curves in the high-frequency region under Newtonian fluid conditions. This demonstrates that the instability structure in viscoelastic fluids is only weakly sensitive to the frequency ratios when $\omega \lt 1$ . In conclusion, under viscoelastic effects, as opposed to Newtonian fluids, different frequency ratios exert completely distinct effects upon the neutral curves in the high-frequency region, indicating that, in high-frequency region, viscoelasticity has an obvious modulation effect on the influences of frequency ratio $\omega$ . These results indicate that the effect of the frequency ratio $\omega$ is not independent of viscoelasticity. Rather, the quasi-periodic oscillation introduces two time scales, and the elastic memory of the fluid modifies how the film responds to these time scales. This coupling is responsible for the much stronger frequency-ratio dependence observed here than in the corresponding Newtonian fluids, especially in the high-frequency region.

Figure 10.

The neutral curves in the $\textrm{Re}-\beta$ plane for different amplitude ratios with $\chi =1$ and the black solid lines, orange solid lines and blue dashed lines represent $\gamma =0.1$ , $\gamma =1$ and $\gamma =1.5$ , respectively. Panels show ( $a$ ) $\omega =\sqrt {37}$ , $M=0.002$ and $M_{s}=0.01$ , ( $b$ ) $\omega =1/\sqrt {37}$ , $M=0.001$ and $M_{s}=10^{-6}$ .

Figure 10. Long description

Panel A: A line graph shows neutral curves in the plane for different amplitude ratios with beta ranging from 0 to 16 and Re ranging from 1 to 10^6. The black solid lines, orange solid lines, and blue dashed lines represent different amplitude ratios. Panel B: Another line graph shows neutral curves in the plane for different amplitude ratios with beta ranging from 0 to 32 and Re ranging from 1 to 10^6. The black solid lines, orange solid lines, and blue dashed lines represent different amplitude ratios.

The neutral curves in the $\textrm{Re}-\beta$ plane for different amplitude ratios with two distinct combinations of parameters when $\gamma =0.1$ , $\gamma =1$ and $\gamma =1.5$ are illustrated in figure 10. Figure 10( $a$ ) illustrates the influence of the amplitude ratio $\gamma$ at a large frequency ratio $\omega =\sqrt {37}$ . Under these conditions, the bandwidths of the neutral curves for different amplitude ratios $\gamma$ remain almost unchanged. However, within the range $7.24\lt \beta \lt 8.82$ , an increase in the amplitude ratio $\gamma$ results in a slight rise in the corresponding imposed oscillation frequency $\beta$ . Under weak quasi-periodic forcing with $\gamma =0.1$ , the first branch in the interval $0\lt \beta \lt 2.11$ exhibits only a single minimum. As $\gamma$ increases, a second minimum appears on the first branch and rapidly decreases. This behaviour is attributed to the increased amplitude of the second primary frequency $\omega _{2}$ in the quasi-periodic forcing. In the high-frequency region $\beta \gt 14.03$ , although the bandwidth of the unstable frequency band remains unchanged with varying amplitude ratio $\gamma$ , the critical Reynolds number continuously decreases as $\gamma$ increases, leading to a significant expansion of the unstable region. This indicates that, at a large frequency ratio $\omega =\sqrt {37}$ , increasing the amplitude ratio significantly destabilises the film flow in the high-frequency region. However, the results for a Newtonian fluid (Jia et al. Reference Jia, Xiao, Wang, Zhao and Xue2026) demonstrate that, under identical parameter conditions, the neutral curves in the high-frequency region remain unchanged for different amplitude ratios. This suggests that, under this current circumstance, the influences of the amplitude ratio on high-frequency stability are primarily attributable to viscoelastic effects. Figure 10( $b$ ) presents the effect of amplitude ratio $\gamma$ under a small frequency ratio $\omega =1/\sqrt {37}$ . In the low-frequency region $0\lt \beta \lt 6.43$ , when $\gamma =0.1$ , two separate U-shaped unstable frequency bands appear. As the amplitude ratio $\gamma$ increases, these two U-shaped unstable frequency bands merge into a single continuous unstable region. Subsequent analysis reveals that, in the low-frequency region, this phenomenon reflects the suppression of the finite-wavelength instability as $\gamma$ increases, facilitating a transition toward purely long-wave-driven instability. Within the interval $10.34\lt \beta \lt 26.11$ , the widths of the three U-shaped unstable frequency bands all expand significantly with increasing $\gamma$ , demonstrating that instability in this region is significantly enhanced. Finally, in the region $\beta \gt 29.82$ , the neutral curves corresponding to different $\gamma$ are all stacked together, indicating that high-frequency instability is only weakly influenced by the amplitude ratio $\gamma$ at a small $\omega =1/\sqrt {37}$ . Nevertheless, this phenomenon of overlap does not occur in Newtonian fluids with identical parameters. In summary, in high-frequency region, viscoelasticity also has a significant modulation effect on the influences of amplitude ratio gamma $\gamma$ .