2.1. Flow configuration

We consider a two-dimensional incompressible viscoelastic liquid film with insoluble surfactant in the horizontal oscillation plane, as shown in figure 1. The density and limiting kinematic viscosity of the fluid are $\rho$ and $\nu$. The oscillation plane is infinite in the $x^*$-direction with velocity $U_{0}\cos \omega t^*$, where $U_{0}$ and $\omega$ are the amplitude and frequency of the oscillation, respectively. The superscript ‘$*$’ denotes the dimensional variables. The origin is located at the free surface and the upper surface of the film is described by $y^* = \eta ^*(x^*, t^*)$. The non-Newtonian liquid film is covered by a monolayer of insoluble surfactant with concentration $\varGamma ^*(x^*, t^*)$. Here, the velocity components in the streamwise and vertical directions are represented by $u^*_x = u^*$ and $u^*_y = \upsilon ^*$, respectively. When the liquid film is at rest, the thickness of the film is $d$, as shown by the dash–dotted line in figure 1.

Figure 1.

Schematic geometry of the physical system under consideration. A thin viscoelastic film is confined between its deformable free surface $\eta ^*(x^*, t^*)$ and a horizontal oscillation plane. The dash–dotted line corresponds to the undeformed surface in the quiescent base state.

The governing equations are given by

(2.1a)\begin{gather} \partial_{i} u^*_{i} = 0, \end{gather}

(2.1b)\begin{gather}\rho(\partial_{t^*} u^*_{i}+u^*_{j} \partial_{j} u^*_{i}) = \rho g_{i}+\partial_{j} \tau^*_{ij}, \end{gather}

where $\tau ^*_{i j}$ is stress tensor and the subscript $i, j$ are summed over the $x^*$- and $y^*$-directions. In this paper, Walter's liquid $B''$ is considered as the rheological model, which is a first-order elastic approximation of Newtonian behaviour and possesses a rapidly fading memory. The constitutive equation of liquid $B''$ is (Beard & Walters Reference Beard and Walters1964; Andersson & Dahl Reference Andersson and Dahl1999)

(2.2)\begin{equation} \tau^*_{i j} ={-}p^*\delta_{i j} + 2\rho\nu 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, $2\rho \nu e^*_{ij}$ is the Newtonian stress while $2M_0(\delta /\delta {t^*})e^*_{i j}$ corresponds to elastic stress, $M_0$ is the viscoelastic coefficient. The corotational derivative of the strain rate tensor is defined as (Shrestha Reference Shrestha1970; Samanta Reference Samanta2017)

(2.3)\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}

The corresponding boundary conditions are:

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

   (2.4a,b)\begin{equation} u^*=U_0\cos\omega t^*, \quad \upsilon^*=0; \end{equation}

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

   (2.5)\begin{equation} \partial_{t^*}\eta^* = \upsilon^* - u^* \partial_{x^*}\eta^*; \end{equation}

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

   (2.6a)\begin{gather} \tau^*_{i j} n_i n_j = \gamma^* \frac{ \partial_{{x^*}{x^*}} \eta^* } {(1+\partial_{x^*}^{2} \eta^*)^{3/2} }, \end{gather}
   (2.6b)\begin{gather}\tau^*_{i j} n_j t_i ={-} \frac{ \partial_{x^*} \gamma^* } {(1+\partial_{x^*}^{2} \eta^*)^{{-}1/2} }, \end{gather}
   where $\gamma ^*$ is the surface tension, ${\boldsymbol {n}}$ is the unit normal vector and ${\boldsymbol {t}}$ is the unit tangent vector.

Note that the tangential stress at the free surface is not equal to zero, which is different from that in the work of Samanta (Reference Samanta2017). Owing to the existence of surfactant, the dynamic condition requires a balance between the hydrodynamic traction, the surface tension and the Marangoni traction. The right-hand term of (2.6b) is the Marangoni stress term. The surface tension will depend on the concentration of the local surfactant. For the two-dimensional stability problem, the surfactant concentration $\varGamma ^*(x^*, t^*)$ is controlled by the following transport equation (Halpern & Frenkel Reference Halpern and Frenkel2003):

(2.7)\begin{equation} \partial_{t^*}(H \varGamma^*)+\partial_{x^*}(H \varGamma^* u^*) = D_{s} \partial_{x^*} (H^{{-}1} \partial_{x^*} \varGamma^* ), \end{equation}

where $H=\sqrt {1+\eta _{x^*}^{*2}}$. Here $D_s$ is the surface molecular diffusivity of the surfactant, which can usually be ignored. Considering that the disturbance at the free surface is infinitesimal, the relation between the surface tension $\gamma ^*$ and the surfactant concentration $\varGamma ^*(x^*, t^*)$ can be approximated as

(2.8)\begin{equation} \gamma^* = \gamma_0 -E( \varGamma^*-\varGamma_{0} ), \end{equation}

where $E$ is the surface elasticity and is a constant, $\varGamma _0$ is the basic value of the surfactant concentration, corresponding to a uniform surface tension $\gamma _0$.

To normalize the governing equation and boundary conditions, the following characteristic scales are introduced. The mean thickness of liquid film $d$ is selected as the characteristic length scale, $\nu /d$ as the velocity scale, $1/\omega$ as the time scale, and $\rho \nu ^2/d^2$ as the pressure scale. The surfactant concentration and surface tension are normalized by $\varGamma _{0}$ and $\gamma _0$, respectively.

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

(2.9a–f)\begin{gather} Re=\frac{U_0d}{\nu}, \quad \chi=\frac{gd^3}{2\nu^2}, \quad Ca=\frac{\rho\nu^2}{\gamma_0d},\quad M=\frac{M_0}{\rho d^2},\quad \beta=\sqrt{\frac{\omega d^2}{2\nu}},\quad M_s=\frac{E\varGamma_0}{\rho U_0^2 d}, \end{gather}

where $Re$ is the Reynolds number, the Galileo number $\chi$ is the ratio of gravity force to viscous force, the Capillary number $Ca$ shows the effect of the surface tension, the viscoelastic parameter $M$ represents the effect of the viscoelasticity, the Womersley number $\beta$ is the ratio of the mean thickness of the film to the thickness of the Stokes layer, $M_s$ represents the influence of surface surfactant.

For the basic flow, the surface is flat, i.e. $y=\eta =0$, and we assume that the surface pressure is zero. The surfactant concentration $\varGamma$ and surface tension $\gamma$ are normalized to unity. By the method of separation of variables, we obtain the unsteady flow solution on the oscillating plane

(2.10a,b)\begin{equation} U(y,t)={\rm Re}\left[\dfrac{Re\cosh[\varOmega(1+\mathrm{i}S)y]\mathrm{e}^{\mathrm{i}t}}{\cosh[\varOmega(1+\mathrm{i}S)]}\right],\quad V(y,t)=0, \end{equation}

with

(2.11a,b)\begin{equation} S=(1+4M^2\beta^4)^{1/2}+2M\beta^2, \quad \varOmega=\beta \sqrt{ \frac{ (1+4M^2\beta^4)^{1/2} -2M\beta^2}{1+4M^2\beta^4} }. \end{equation}

Here ${\rm Re}[\cdot ]$ represents the real part of that complex function. The pressure is induced by gravity

(2.12)\begin{equation} P(y)={-}2\chi y,\quad -1\leq y\leq 0. \end{equation}

Note that (2.9a–f) is identical to that of Samanta (Reference Samanta2017), which means the surfactant has no effect on the basic flow. This is to be expected physically, as the curvature of the free surface for the basic flow is zero, the effect of surface tension on the basic flow disappears. Besides, when the viscoelastic parameter $M=0$, (2.9a–f) will degenerate into that of Or (Reference Or1997).

2.2. Linear stability problem

For the linear stability problem, it is assumed that the disturbances are infinitesimal to the basic state (2.9a–f) and (2.12). The non-dimensional variables are

(2.13a–e)\begin{equation} u=U+\tilde{u},\quad \upsilon=\tilde{\upsilon}, \quad p=P+\tilde{p}, \quad \varGamma=1+\tilde{\varGamma}, \quad \gamma=1+\tilde{\gamma}. \end{equation}

Considering that the problem we are dealing with is two-dimensional, the continuity in (2.1a) is automatically satisfied by introducing a stream function $\tilde {\psi }(x,y)$, such that

(2.14a,b)\begin{equation} \tilde{u}=\partial_{y}\tilde{\psi}, \quad \tilde{\upsilon}={-}\partial_{x}\tilde{\psi}. \end{equation}

In the following stability analysis, general infinitesimal disturbances for modal analysis are decomposed into the following form:

(2.15a–c)\begin{equation} \tilde{\psi}(x,y,t)={\phi}(y,t)\mathrm{e}^{\mathrm{i}k x}, \quad \tilde{\eta}(x,t)=h(t)\mathrm{e}^{\mathrm{i}k x}, \quad \tilde{\varGamma}=\xi(t)\mathrm{e}^{\mathrm{i}k x}, \end{equation}

where $k \in \mathbb {R}$ is the wavenumber in the streamwise direction. Note that the pressure $\tilde {p}$ can be eliminated by combining the Navier–Stokes equations, and hence it does not appear in the linearized system. The time-dependent perturbation equation governing the stability of viscoelastic liquid film is

(2.16)\begin{equation} 2 \beta^{2}(\mathscr{L}+M \mathscr{L}^{2}) \partial_{t} \phi=\mathscr{L}^{2} \phi-\mathrm{i} k[U(\mathscr{L}+M \mathscr{L}^{2})-(\mathscr{D}^{2} U+M \mathscr{D}^{4} U)] \phi, \end{equation}

with differential operators $\mathscr {D}=\partial _{y}$ and $\mathscr {L}=\mathscr {D}^2-k^2$. The boundary conditions on the oscillation wall are given by

(2.17)\begin{equation} \phi=\mathscr{D} \phi=0, \quad \text{at } y={-}1. \end{equation}

The linearized kinematic boundary condition and transport equation for surfactant are expressed by

(2.18a)\begin{gather} 2 \beta^{2} \partial_{t} h={-}\mathrm{i} k \phi-\mathrm{i} k U h, \quad \text{at } y=0, \end{gather}

(2.18b)\begin{gather}2 \beta^{2} \partial_{t} \xi={-}\mathrm{i} k \partial_y\phi - \mathrm{i} k U \xi, \quad \text{at } y=0. \end{gather}

The linearized conditions for the normal and tangential stresses at $y=0$ are

(2.19a)\begin{align} 2 \beta^{2}[\mathscr{D}+M(\mathscr{L}-2 k^{2}) \mathscr{D}] \partial_{t} \phi&={-}\mathrm{i} k[U\{\mathscr{D}+M(\mathscr{L}-2 k^{2}) \mathscr{D}\}.\nonumber\\ &\quad +M(\mathscr{D}^{2} U \mathscr{D}-\mathscr{D}^{3} U)] \phi \nonumber\\ &\quad +(\mathscr{L}-2 k^{2}) \mathscr{D} \phi-2 \mathrm{i} k\left(\chi+\frac{k^{2}}{2 Ca}\right) h, \end{align}

(2.19b)\begin{align} 2 \beta^{2} M(\mathscr{L}+2 k^{2}) \partial_{t} \phi&=(\mathscr{L}+2 k^{2}) \phi- \mathrm{i}kM [U(\mathscr{L}+2 k^{2})-\mathscr{D}^{2} U] \phi \nonumber\\ &\quad +[\mathscr{D}^{2} U-2 \beta^{2} M \mathscr{D}^{2} \partial_{t} U]h + M_s Re^2 \mathrm{i}k\xi. \end{align}

The time-dependent Orr–Sommerfeld equation (2.16) subject to conditions (2.17)–(2.19) forms a Floquet system. For finite-wavelength instabilities, the Floquet system has to be solved numerically, while the long-wavelength instability can be analytically obtained by a series expansion in $k$, and we will discuss the long-wavelength solutions below.

3.1. Absence of gravity $\chi =0$

In some microgravity environments, the absence of gravity often occurs, so it is necessary to consider the stability of oscillating liquid film flow in this situation. We can discuss the influence of viscoelastic $M$ and surfactant $M_s$ on neutral stability conditions by directly analysing (3.27). For the long-wavelength instability ($k\rightarrow 0$), the neutral condition becomes

(3.28)\begin{equation} \max( 4I_1 - 2I_2, I_1 - \textit{Re}^2 M_s ) = 0. \end{equation}

It is worth noting that the expression of (3.28) used here is consistent with Gao & Lu (Reference Gao and Lu2006) in form, but the influence of viscoelasticity is introduced into the expressions $I_1$ and $I_2$ in this paper. Figure 2 displays the influence of viscoelasticity on the $(M_s-\beta )$ neutral curve under given partial parameters. For Newtonian fluid ($M = 0$), the characteristics of the neutral curves (black solid lines) are consistent with those found by Gao & Lu (Reference Gao and Lu2006), i.e. for a given $M_s$, with the increase of $\beta$, the flow will alternately show stable and unstable properties in the corresponding frequency interval. And the neutral curve will become a vertical straight line (curve to straight line) with the increase of $M_s$, accompanied by the narrowing of the unstable interval. The curve part is mainly controlled by $I_1-\textit {Re}^2M_s$ corresponding to the conjugate complex roots (travelling waves), while the vertical straight lines are controlled by $2I_1-I_2$ corresponding to the standing waves. For different viscoelastic parameters, there is no qualitative change in these stability characteristics. Nevertheless, with the increase of viscoelasticity, for low frequency oscillation ($\beta <6$), the unstable region will be narrowed and move towards the direction of smaller $\beta$. In addition, increasing viscoelasticity will also cause the shrinkage position to be raised. For the higher frequency oscillation ($\beta >6$), there is no obvious characteristic of the change of the unstable interval in the presence of viscoelasticity.

Figure 2.

The neutral curves in the $M_s\unicode{x2013}\beta$ plane for different viscoelastic parameters when $\chi =0, Ca=0.05, \textit {Re}=200$. Here, the black solid, blue dashed and orange dash–dotted lines stand for $M = 0$, $M = 0.005$ and $M = 0.01$, respectively.

In order to explore more clearly the influence of viscoelasticity on different Floquet modes in the presence of surfactants, two groups of typical values of $M_s$ and $M$ are selected. Figure 3 shows the variation of the real part of the growth rate of two Floquet modes with the oscillation frequency $\beta$. For comparison, the growth curves of Newtonian fluid ($M=0$) and clean surface ($M_s=0$) are also shown in black solid lines. 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. For a larger $M_s$ (see figure 3a), with the increase of viscoelasticity, the growth rate corresponding to the most unstable mode also increases, and the maximum growth rate within the given parameter range does not exceed that of the clean surface. This means that although viscoelasticity promotes the growth of disturbance, the effect of destabilization is not as obvious as the effect of stabilization from surfactant. The increase of viscoelasticity has no obvious effect on the oscillation frequency corresponding to the maximum growth rate ($\beta =1.1$), but for high frequency ($\beta >4$), the position of the local maximum of the growth rate curve will move in the direction of decreasing frequency (see the inset). Floquet exponents are real numbers in most frequency ranges, which is shown as two branches of the growth rate curve, and the characteristic mode is a standing wave. In a small frequency range, such as $1.94<\beta <2.62$ when $M=0.01$, the two branches merge into a curve. The Floquet exponent is a pair of conjugate complex numbers, and the positive and negative imaginary parts represent the opposite propagation directions of travelling-wave modes. In figure 3(b), a smaller surfactant parameter ($M_s = 10^{-4}$) is introduced, and it is found that when the viscoelasticity is large enough, the flow is still more unstable than that of Newtonian fluid with a clean surface, despite the presence of surfactant. Therefore, when surfactant and viscoelasticity exist at the same time, the stability of oscillating liquid film depends on their competitive relationship.

Figure 3.

The real part of $\theta /\textit {Re}^2$ as functions of $\beta$ when $\textit {Re}=200$ for (a) $M_s=0.01$ and (b) $M_s=10^{-4}$ with $M=0$ (black dashed lines), $M=0.005$ (orange solid lines) and $M=0.01$ (blue dash–dotted lines). The black thin lines represent Newtonian fluids without surfactants.

3.2. Presence of gravity $\chi =1$

When the influence of gravity is considered, the neutral curves corresponding to long-wave instability present the U-shaped curves with separated bandwidth of frequency. Underneath the curve and outside the bandwidth range, all infinitesimal long-wave disturbances are stable. Dandapat & Gupta (Reference Dandapat and Gupta1975) and Samanta (Reference Samanta2017) studied the viscoelastic liquid film on the clean surface, and Gao & Lu (Reference Gao and Lu2006) also studied the Newtonian liquid film containing surfactants. Here, we consider both viscoelastic and surfactant effects. Figure 4(a) shows the influence of different viscoelastic parameters on the neutral curve when $M_s=0.01$ is fixed. For low-frequency oscillation ($\beta < 3$), the neutral curves corresponding to different viscoelastic parameters are all stacked together, which means that viscoelasticity hardly affects the stability of low-frequency oscillation liquid film. When $4<\beta <6$, with the increase of viscoelasticity, the critical Reynolds number and the imposed oscillation frequency gradually decrease. The interesting result is that, high frequency oscillation ($\beta >10$) significantly enhances the instability of viscoelastic liquid film, which is different from low-frequency oscillation. The effects of different surfactants on the stability of viscoelastic liquid film are shown in figure 4(b). The neutral curves are plotted in the $\textit {Re}\unicode{x2013}\beta$ plane parameterized by the value of $M_s$ with fixed $M=0.01$. The results show that with the increase of $M_s$, the unstable frequency bandwidth decreases gradually and the corresponding Reynolds number increases. Similar to the results in the absence of gravity, the surfactant will still weaken the instability of viscoelastic liquid film and make the flow more stable in the presence of gravity. It is worth mentioning that the scattered points ($M_s = 0.01$) and thin lines ($M_s = 0.001$) of the second and third family neutral curves ($4<\beta <8$) in figure 4(b) are coincident. This is because with the increase of $M_s$, the dominant term of (3.27) changes from $I_1-\textit {Re}^2M_s$ to $2I_1-I_2$, and $M_s$ no longer influences the neutral curve.

Figure 4.

The neutral curves in the $\textit {Re}\unicode{x2013}\beta$ plane for different viscoelastic parameters and surfactants with $\chi =1$. (a) Specified surfactant $M_s=0.01$, the blue solid lines, black dashed lines and orange solid lines represent $M=0$, $M=0.005$ and $M=0.01$, respectively. (b) Specified viscoelastic parameter $M=0.01$, the blue solid lines, black dashed lines, orange solid lines and open circles represent $M_s=0$, $M_s=10^{-4}$, $M_s=10^{-3}$ and $M_s=10^{-2}$, respectively.

5.1. Clean surface in the absence/presence of gravity

The influences of viscoelasticity in the presence of gravity are considered by Samanta (Reference Samanta2017). However, the results of the absence of gravity are not involved in his work. Before considering the combined effects of viscoelasticity and surfactant, let us briefly analyse the stability of clean-surface viscoelasticity liquid film without gravity. Figure 6 shows the Reynolds number and its corresponding wavenumber versus imposed frequency for two viscoelastic parameters ($M=0.005$ and 0.01). As described in the results of long-wavelength stability, the disturbances are unstable over alternating frequency intervals, and the boundaries of unstable regions are indicated by vertical dash–dotted lines in figures 6(a) and 6(c). Different from the long-wavelength results, when the Reynolds number exceeds the critical value, finite-wavelength instability occurs in the region where the long-wavelength disturbance is stable. If $\beta$ continues to increase to the next long-wavelength instability interval, although the finite-wavelength instability still exists (see the blue dotted lines), the critical Reynolds number $\textit {Re}_c$ is not determined by the finite-wavelength disturbance due to the long-wave mode being still unstable when $\textit {Re}\rightarrow 0$. These phenomena are similar to those found by Gao & Lu (Reference Gao and Lu2008). In addition, the neutral curve in the presence of gravity ($\chi = 1$) is also given in figure 6(a), in which the U-shaped curve represents the result of the long-wavelength instability. It can be seen that with the increase of $\beta$, the flow instability is first dominated by the long-wave mode. When $\beta$ increases to 2.4925, the critical Reynolds number corresponding to the long-wave mode is equal to that of the finite-wavelength, and then the oscillation frequency range is dominated by the finite-wavelength mode until the neutral curve intersects the second U-shaped neutral curve. Thus, in the presence of gravity, $\textit {Re}_c$ is a continuous function of the imposed frequency, although the long-wavelength mode and the finite-wavelength mode alternate with increasing frequency, which is quite different from the results in the absence of gravity. Two points should be noted in figures 6(a) and 6(c). On the one hand, the critical Reynolds number corresponding to a finite-wavelength seems to be insensitive to gravity except for near $\beta =8$, which can be seen from the coincidence of the neutral curves $\chi = 0$ (blue solid lines) and $\chi = 1$ (open circles). In fact, for $\beta =3$ in figure 6(a), the critical Reynolds number corresponding to $\chi = 1$ is 58.95, while that corresponding to $\chi = 0$ is 59.77, with a difference of only $1.37\,\%$. On the other hand, the oscillation frequencies corresponding to the bifurcation points in the presence of gravity (orange solid circle) and the absence of gravity (open diamond) are also affected slightly by gravity. For example, $\beta$ values corresponding to the presence and absence of gravity at the first bifurcation point are 2.4925 and 2.5564, respectively. Considering that gravity has little influence on the boundary of the stability region and the critical Reynolds number, the results with or without gravity will not be distinguished in the following analysis unless otherwise specified.

Figure 6.

Stability characteristics in the absence of surfactants with $Ca=0.05$ for different viscoelastic parameters $M=0.005$ (a,b) and $M=0.01$ (c,d). Panels (a,c) represent the stability boundaries in the ($\beta, \textit {Re}$)-plane and panels (b,d) represent variation of the corresponding wavenumber $k$ with $\beta$. The U-shaped orange solid lines and open circles represent long- and finite-wavelength instability for $\chi =1$. The blue solid lines represent finite-wavelength instability for $\chi =0$. The vertical dash–dotted lines represent the boundaries of unstable regions for $\chi =0$.

For the larger viscoelasticity ($M=0.01$), as shown in figure 6(c), $\textit {Re}$ increases with the increase of $\beta$ for the low-frequency oscillation ($2.4970 < \beta < 3.86$), while in the high-frequency region ($8.014 < \beta < 11.22$), $\textit {Re}$ shows a decreasing trend, i.e. Reynolds number does not always monotonically increase with the imposed frequency. In addition, when the flow instability changes from long-wavelength mode to finite-wavelength mode, the critical wavenumber does not necessarily change continuously, as shown in figure 6(d). The discontinuity of wavenumber is explained as follows. For $\chi =0$, the first branch point ($\text {P}_1$) is located at $\beta _1=2.4968$, and we plot the neutral curves for the slightly higher and lower values of $\beta _1$ in figure 7(a). It can be seen that for $\beta =2.4970$ (orange solid line), the critical wavenumber corresponding to the local minimum of the neutral curve is 0.2, while for $\beta =2.4967$ (blue solid line), the neutral curve monotonically increases from the coordinate origin $(k, Re)=(0, 0)$. Therefore, the critical wavenumber can be continuously transitioned from long-wavelength instability to finite-wavelength instability. For the second branch point ($\text {P}_2$), $\beta _2=5.1305$, the neutral curve for slightly higher frequency has a local minimum at $k=0.54$ (see figure 7b), which makes the critical wavenumber discontinuity near $\beta _2$. The characteristics of the first two bifurcation points are qualitatively consistent with the results obtained by Gao & Lu (Reference Gao and Lu2008) in the study of oscillating Newtonian fluid films with surfactants. The neutral curve near that third branch point ($\text {P}_3, \beta _3=7.9909$) is different from the first two branch points. It first separates a new branch at $\text {P}_3$ with a rapid increase in Reynolds number (see blue solid line in figure 6c). Then, in a small range of $\beta$, it intersects with another monotonically decreasing neutral curve at $\text {P}_4$. It can be seen from figure 6(d), in the $k\unicode{x2013}\beta$ plane, the critical wavenumber increases rapidly from zero at first, after that, it appears discontinuous with the increase of $\beta$, and continuously changes at a higher wavenumber. Note that the parameters used here are consistent with figure 5 in Samanta (Reference Samanta2017). However, there is no stable bandwidth in the present work for both $\chi =0$ and $\chi =1$. On the contrary, a neutral curve with finite-wavelength is found. In order to more comprehensively analyse the different structural characteristics of critical parameter changes near $\beta _3$, the neutral curves corresponding to different $\beta$ on the $\textit {Re}\unicode{x2013}k$ plane are plotted in figure 7(c). It can be seen that when the frequency is less than $\beta _3$, the neutral curve increases monotonically over the range of $0 < k < 0.5$, which may explain why the critical wavenumbers increase from zero. Nevertheless, when the long-wavelength instability is eliminated, the neutral curves corresponding to $\beta = 7.991$ contain two local minima which correspond to a small wavenumber of 0.08 and a larger wavenumber of 4.2, respectively. With the increase of $\beta$, the Reynolds number corresponding to the extreme point of the smaller wavenumber increases rapidly while that of the larger wavenumber decreases slowly. Until $\beta$ increases to 8.014, the two local lowest points have the same Reynolds number $\textit {Re}=138.6$ corresponding to the point $\text {P}_4$, and then the global minimum of the neutral curve is controlled by the part of large wavenumber. This is why the critical Reynolds number is continuous in figure 6(c) and the corresponding wavenumber jumps from 0.32 to 4.2. It should be noted that the Floquet exponents corresponding to the above critical parameters are all zero, i.e. the characteristic mode always exists as a standing wave.

Figure 7.

The neutral curves in the neighbourhood of branch point when $M=0.01$, $M_s=0$, $Ca=0.05$ and $\chi =0$: (a) first branch point $\text {P}_1$; (b) second branch point $\text {P}_2$; (c) third branch point $\text {P}_3$.

5.2. Effect of surfactants in the absence/presence of gravity

In this section, we will focus on the combined effect of viscoelasticity and surfactants on the stability of oscillating liquid films. The typical parameter $M_s=0.001$ is selected for analysis. Similar to the previous discussion, the presence or absence of gravity has an extremely limited effect on finite-wavelength stability, as shown in figure 8. Due to the effect of the surfactant on the long-wavelength instability, the bandwidth of each instability interval is shorter than that in the absence of the surfactant, which means that the oscillation frequency range occupied by the finite-wavelength instability will be wider, which can be observed by comparing figure 6(a) with figure 8(a). There are three significant differences in the neutral curve for finite-wavelength instability when considering surfactants. First, in the low-frequency region, with the increase of $\beta$, the unstable mode changes from the long-wavelength mode to the finite-wavelength mode. Careful examination of the Floquet exponent reveals that $\mu$ corresponding to the finite-wavelength mode is a pair of conjugate complex numbers, which means that the disturbance is a travelling wave. Continue increasing $\beta$ to 3.5, where there is a bifurcation point (P) to the right of which corresponds to the higher wavenumber, as shown in figure 8(b). Different from the travelling-wave mode on the left-hand side of the bifurcation point, the mode corresponding to the high wavenumber is a standing-wave. Second, in the intermediate frequency region ($4.8<\beta <6.9$), the finite-wavelength neutral curve shows a discontinuity at $\beta =6.7$, which is caused by the standing-wave mode competition between two different wavelengths. This case has been analysed in detail in the work of Or (Reference Or1997) and Gao & Lu (Reference Gao and Lu2008), and will not be described here. Third, for the high-frequency region ($\beta >7.44$) in figure 8(a), compared with the Newtonian fluid, the critical Reynolds number in the presence of viscoelasticity is no longer monotonically increasing, while the critical wavenumber corresponding to the finite-wavelength mode is gradually increasing with $\beta$. More importantly, as shown in figure 8(c), for higher viscoelastic parameters ($M=0.01$), the viscoelasticity causes the instability of the oscillating liquid film to be determined by the standing-wave mode of finite wavelength over a relatively large frequency range ($4.6<\beta <11.4$). The above results again imply that viscoelasticity has a significant effect on the stability of the high-frequency oscillation liquid film.

Figure 8.

Stability characteristics in the presence of surfactants with $M_s=0.001$ and $Ca=0.05$ for different viscoelastic parameters $M=0.005$ (a,b) and $M=0.01$ (c,d). Panels (a,c) represent the stability boundaries in the ($\beta, \textit {Re}$)-plane and panels (b,d) represent variation of the corresponding wavenumber $k$ with $\beta$. The meanings of curves and symbols are consistent with figure 6.

In order to study the stability characteristics of surfactants to viscoelastic liquid film under finite-wavelength disturbance more clearly, we calculated the variation of critical Reynolds number and critical wavenumber with oscillation frequency for more values of $M_s$, as shown in figure 9. The left and right endpoints of each curve represent the boundary between long-wave instability and finite-wavelength instability. Figures 9(a) and 9(b) show the effects of surfactants on the critical Reynolds number and critical wavenumber in the low-frequency range, respectively. In order to better distinguish the differences between different curves, in figure 9(a), the same method is adopted as in Gao & Lu (Reference Gao and Lu2008) to deal with the ordinate. It can be seen that with the increase of $M_s$, the range of finite-wavelength instability widens, which is discussed in the long-wavelength stability analysis considering gravity (see figure 4b). Different surfactant concentrations will lead to the competition between travelling-wave mode and standing-wave mode. For example, when $M_s=0$, the unstable mode is a standing wave, but when the surfactant concentration is very small ($0 < M_s \leq 10^{-4}$), the unstable mode becomes a travelling wave. And with the increase of $M_s$, the critical Reynolds number increases, which means that surfactants can make the flow more stable. Further increasing the influence of surfactant ($M_s = 10^{-3}$ or $10^{-2}$), the critical curve becomes two branches. The left-hand branch represents the travelling-wave mode with a smaller critical wavenumber, while the right-hand branch represents the standing-wave mode and has a large critical wavenumber. With the increase of $M_s$, the intersection of the two branches will move in the direction of decreasing oscillation frequency. Nevertheless, the flow in the presence of surfactant with $M_s \leq 10^{-2}$ is always more stable than that in the absence of surfactant. When $M_s$ is large enough, e.g. $M_s=10^{-1}$, the unstable modes in the whole low-frequency oscillation range are standing waves, which is consistent with the clean-surface case. In addition, the curve of $M_s = 0$ and the curve of $M_s = 10^{-1}$ intersect, which means that whether the flow is stabilized or destabilized by the surfactant depends on the oscillation frequency of the plate. The effects of surfactants on critical Reynolds number and critical wavenumber in higher frequency range are shown in figures 9(c) and 9(d). For the clean-surface case, the unstable mode is still a standing wave, and the curve is discontinuous in the range of $6.76<\beta <7.92$, which is dominated by long-wavelength instability. When the concentration of surfactant is increased, $M_s = 10^{-5}$ or $10^{-4}$, the unstable mode will change from a travelling wave to a standing wave with the increase of $\beta$, and the oscillation frequencies at the mode change points are $\beta =8.4$ and 5.4, respectively, which are marked by arrows in figure 9(d). When $M_s$ is further increased, $M_s=10^{-3}$, the unstable mode reverts to the standing-wave type, and the discontinuity in the unstable curve is caused by the competition of different standing-wave modes. It is worth mentioning that surfactants can promote or inhibit flow stability at both lower and higher oscillation frequencies.

Figure 9.

Effects of surfactants on the finite-wavelength instability characteristics of the flow in different frequency range with $\chi =1$, $Ca=0.05$ and $M=0.01$. Variation of (a) $\text {e}^{-\beta }\textit {Re}$ and (b) the corresponding critical wavenumber $k$ with $\beta$ in low-frequency. Variation of (c) $\textit {Re}$ and (d) the corresponding critical wavenumber $k$ with $\beta$ in higher frequency.