4.1. Without effects of surfactants and gravity $( {{M_j} = 0,{F^{ - 2}} = 0} )$

When the effects of surfactants and gravity are not considered, (3.16) linear system can be simplified to

(4.1)\begin{equation} \left[ {\begin{array}{cc} {{I_{11}}} & {{I_{12}}}\\ {{I_{31}}} & {{I_{32}}}\end{array}} \right] \left[{\begin{array}{c}{{h_{10}}}\\ {{h_{20}}}\end{array}} \right] = \theta \left[ {\begin{array}{c}{{h_{10}}}\\ {{h_{20}}}\end{array}} \right]. \end{equation}

Solving the equation, we have ${\theta _{1,2}} = \frac {1}{2}( {{I_{11}} + {I_{32}} \pm \sqrt {{{( {{I_{11}} - {I_{32}}} )}^2} + 4{I_{12}}{I_{31}}} } )$. Therefore, whether the flow is stable depends on the sign of $\max ( {{\theta _{1r}},{\theta _{2r}}} )$, where the subscript $r$ represents the real part of the complex number.

First, the phase diagram in the $( {\beta ,n} )$-plane for three typical values of $m$ is shown in figure 2. The stable and unstable regions are denoted by blue and yellow bands. From figure 2(a), it is seen that when $m = 1$, in the $( {\beta ,n} )$-plane, stable and unstable curved stripes appear alternately. As $\beta$ and $n$ increase, the bandwidth of these stripes gradually shrinks. When $0.1 < m < 1$, e.g. $m=0.5$, the phase diagram (see figure 2b) looks similar to that of $m=1$ except for the following two differences. One is some minor fluctuations in the neutral curves (the borders between the stable and unstable regions), the other is that except for the first unstable region from the bottom, the rest of the unstable regions no longer intersect with $n = 0$. For the cases of $0 < m < 0.1$, e.g. $m=0.05$, the phase diagram is shown in figure 2(c). When $0 < \beta < 0.73$, the stable and unstable stripes still appear alternatively, but the bandwidth becomes narrower. When $\beta > 0.73$, the unstable region is absolutely dominated, except for a small stripe close to the $\beta$-axis. Hence, if $\beta$ is high, e.g. $\beta >2$, and $n$ is small enough, e.g. $n<0.01$, the flow stability is enhanced for $m<1$. It is noted that when $m > 1$, the flow is always unstable within the parameter ranges in this study.

Figure 2.

Phase diagram in the $( {\beta ,n} )$-plane for $r = 1$: (a) $m = 1$; (b) $m = 0.5$; (c) $m = 0.05$. Stable and unstable regions are denoted by blue and yellow, respectively.

Next, we would like to explain briefly why the above three values of $m$ are typical and why in the above classification, $m\approx 0.1$ is a critical value. The phase diagram in the $(\beta , \sqrt {m})$-plane for cases $n = 1$ and $r = 1$ is shown in figure 3(a). On the whole, the stable (blue) stripes mainly appear in the upper region where $\sqrt {m} > 0.32$, i.e. approximately $m>0.1$. When $0< m<0.1$ and $\beta >0.8$, the flow is always unstable. The reason for this situation may be that $m$ is too small. When $m$ is small and $\beta$ is large, the lower fluid is sluggish by the plate vibration. Then the interface between the two fluids would vibrate with a small value of $\beta$. Therefore, the effective vibration frequency of the upper fluid is small. In other words, the entire flow system is equivalent to a single layer of fluid being vibrated by a plate at small $\beta$. For the single-layer cases, from our figure 2(a) ($n=0$) or according to Gao & Lu (Reference Gao and Lu2006), we can see that when $\beta$ is small, e.g. $\beta \in ( {0,2.63} )$, the flow is indeed unstable.

Figure 3.

(a) Stability limits in the $( {\beta ,\sqrt m } )$-plane for $n = 1,r = 1$. (b) Stability limits in the $( {\beta ,\sqrt r } )$-plane for $n = 1,m = 0.5$. Stable and unstable regions are denoted by blue and yellow bands, respectively.

Lastly, we would also like to see the density-ratio effect. The phase diagram for cases of $n = 1$, $m = 0.5$ in the $(\beta , \sqrt {r})$-plane is shown in figure 3(b). It is seen that there are also many inclined curved stripes and stable and unstable stripes appear alternately. Comparing figures 3(b) and 2(b), we found that the thickness-ratio effect and the density-ratio effect on the flow stability look similar except that the stripes in figure 3(b) are not so dense.

4.2. Effect of surfactants $( {{M_j} \ne 0,{F^{ - 2}} = 0} )$

In this section we discuss the effect of surfactants. The surfactants include surface surfactant ${M_1}$ and interfacial surfactant ${M_2}$. Here, in all cases, thickness ratio, viscosity ratio and density ratio are fixed to be $n = 1$, $m = 0.5$, $r =0.5$, respectively, but the oscillation frequency $\beta$ is a variable parameter.

First, the neutral stability curves in the $( {\beta ,{M_1}} )$-plane are shown in figure 4(a). These curves divide the plane into three discontinuous unstable regions and a connected stable region, denoted by $U$ and $S$, respectively. For the cases of the clean surface ($M_1=0$), the disturbance mode is unstable in the $\beta$ interval of $\max \{ {{\theta _{1r}}( \beta ),{\theta _{2r}}( \beta )} \} > 0$, and the motion form of the disturbance mode is a standing wave (see Appendix E). It is seen that as ${M_1}$ increases from $0$, the interval of $\beta$ for the unstable region may decrease significantly, e.g. $\beta \in (0,2.63]$ at $M_1=10^{-6}$ while $\beta \in (0,2]$ at $M_1=10^{-4}$, indicating a stabilizing effect of surface surfactant. There are a total of three Floquet exponents. The Floquet exponents corresponding to the neutral curve are two pure imaginary numbers that are conjugate to each other. It means that the first two perturbation modes are travelling waves, and the movement directions of these two modes are opposite. The third Floquet exponent is always negative. For the left-most neutral curve, when $M_1$ continues to increase to a critical value related to $\beta$, the neutral curve becomes a straight line and is independent of $M_1$. Its corresponding Floquet exponent is zero, indicating that the perturbed mode becomes a standing wave again.

Figure 4.

(a) Stability limits in the $( {\beta ,{M_1}} )$-plane at ${M_2} = 0$. (b) Stability limits in the $( {\beta ,{M_2}} )$-plane at ${M_1} = 0$. Stable and unstable regions are denoted by $S$ and $U$, respectively.

Next, the neutral stability curves in the $( {\beta ,{M_2}} )$-plane are shown in figure 4(b). It is seen that these curves divide the plane into alternating unstable and stable regions, which is significantly different from figure 4(a). It seems that the unstable regions do not significantly depend on $M_2$. Therefore, the interfacial surfactant does not play an important role in stabilizing the flow. The possible reason is that interfacial surfactant introduces another disturbance mode (it is defined as the interfacial surfactant mode), but the surface mode, i.e. the deformation of the surface, is dominant. On the other hand, even the interface is slightly contaminated (e.g. ${M_{2}} = {10^{ - 20}}$), the situation may noticeably differ from the clean case (${M_{2}} = 0$). For example, the unstable $\beta$ intervals for the clean and contaminated interfaces are $(4.24, 5.72)$ and $(3.97, 5.72)$, respectively. Another example is that for the clean and contaminated interfaces, $\beta \in (7.44, 8.83)$ and $\beta \in (7.16, 8.83)$, respectively. The unstable intervals $\beta \in ( {4.24,5.72} )$ and $\beta \in ( {7.44,8.83} )$ can be obtained from figure 3(b). In figure 3(b) all cases are $\ {{M_2} = 0}$. When $m=0.5$, the horizontal line of $\sqrt m = \sqrt {0.5} \approx 0.707$ would intersect with the unstable domains, and then the unstable intervals can be read. Similarly, the unstable intervals $( {3.97,5.72} )$ and $( {7.16,8.83} )$ can be seen from figure 4(b) (${{M_2} = {{10}^{ - 20}}}$). Hence, the presence of interfacial surfactants expands the unstable regions. In general, the interfacial surfactant may enhance or suppress flow instability. A similar phenomenon occurs in a steady two-layer flow system (Gao & Lu Reference Gao and Lu2007).

To investigate the behaviour of the growth rates of the four Floquet modes (surface and interface modes, surface and interfacial surfactant modes), we plot the real part of $\theta$, i.e. $\theta _{r}$, as a function of $\beta$ for typical cases, such as (${M_1} = 0$, ${M_2} = 0$), (${M_1} = 10^{-2}$, ${M_2} = 0$) and (${M_1} = 0$, ${M_2} = 10^{-2}$) in figure 5, which are denoted by red, black and blue lines. If the real part sizes of the four eigenvalues of (3.16) are arranged in order, we have four Floquet modes, i.e.  $\textrm {Re} ( {{\theta _1}} ) \ge \textrm {Re} ( {{\theta _2}} ) \ge \textrm {Re} ( {{\theta _3}} ) \ge \textrm {Re} ( {{\theta _4}} )$. It is noticed that the two interface modes and the two surfactant modes are not one-to-one correspondence to the four Floquet modes. We would like to discuss the first mode (${\theta _1}$). From figure 5, it is seen that the black solid line is always below the red dashed line, indicating that the surface surfactant would reduce the maximum perturbation growth rate and, therefore, enhances the flow stability. It is also seen that the blue solid line may be above or below the red dashed line. Hence, the interfacial surfactant may destabilize or stabilize the flow, which depends on $\beta$. In the three insets, the blue solid line is always slightly below the red dashed line, indicating the interfacial surfactant stabilizes the flow mildly. The $\beta$ range in each inset corresponds to that of each unstable region's right-side neutral curve in figure 4(b).

Figure 5.

The real part of $\theta$, i.e. ${\theta _r}$, as a function of $\beta$ for ${M_1} = {10^{ - 2}}, {M_2} = 0$ (black solid lines), ${M_1} = 0,{M_2} = {10^{ - 2}}$ (blue solid lines) and ${M_1} = 0,{M_2} = 0$ (red dashed lines). Insets show the first mode.

For the cases of clean surface and interface ($M_1=0$, $M_2=0$), only the first two Floquet modes are left, which are the surface and interface modes. From figure 5 we can see that the Floquet exponents of surface and interface modes are real, and the interface mode is always stable, so the flow stability depends on the surface mode.

In the presence of surface surfactants ($M_1=10^{-2}$, $M_2=0$), there are only the first three Floquet modes, i.e. the surface mode, the surface surfactant mode and the interface mode. The interface mode is still stable, while the other two disturbance modes may be unstable, e.g. at $\beta \approx 1.1$. In most parameter ranges the three Floquet exponents are all real, corresponding to standing wave modes. However, when $1.04 < \beta < 1.62$, two of the eigenvalues are conjugate complex, which is associated with travelling wave disturbances, one of which propagates to the right with a phase velocity of $O( k )$ and the other one to the left. The maximum growth rate of the disturbance occurs at $\beta \approx 0.64$, where the flow is the most unstable to the long-wave disturbance. For high frequencies, especially $\beta >2$, the interface mode is dominated, and the corresponding eigenvalue tends to be zero (see details in the insets of figure 5). When $\beta >4$, the other two eigenvalues almost no longer depend on $\beta$, and these three eigenvalues are always negative.

In the presence of interfacial surfactants ($M_1=0$, $M_2=10^{-2}$), there are only the first three Floquet modes, i.e. the surface mode, the interface mode and the interfacial surfactant mode. All three disturbance modes may be unstable, e.g. at $\beta \approx 0.4$. In most parameter ranges, the three eigenvalues are all real, corresponding to standing wave modes. However, when $0 < \beta < 1.44$, two of the eigenvalues are conjugate complex numbers, and these two disturbance modes are travelling wave disturbances and have an identical growth rate. The maximum growth rate of disturbance also occurs at $\beta \approx 0.64$. For high frequencies, especially when $\beta >2$, as shown in figure 3, the surface mode is dominated, the corresponding eigenvalue tends to zero. When $\beta >4$, the other two eigenvalues almost no longer depend on $\beta$, and are always negative. Although the surface mode may be unstable, its growth rate is exponentially small and the instability is mild.

4.3. Effects of surfactants and gravity $( {{M_j} \ne 0,{F^{ - 2}} \ne 0} )$

Generally speaking, gravity would always hinder the deformation of the fluid surface and stabilize flows. The flow may be unstable only when the gravity is small enough or the Froude number is large enough. When there is gravity, it is convenient to plot neutral stability curves in the $({\beta ,F} )$-plane (see figure 6). Cases of $M_1 \ne 0$ and $M_2=0$ are shown in figure 4(a–c), while cases of $M_1 = 0$ and $M_2 \ne 0$ are shown in figure 6(d–f). In this section the other key parameters are ${n = 1,m = 0.5,r = 0.5}$. In figure 3(b) the line of $\sqrt {r} = \sqrt {0.5}\approx 0.707$ crosses three unstable regimes when $0 < \beta \le 10$, so there are three families of $U$-shaped neutral curves in figure 6. In each family, the neutral curves are slightly different as $M_1$ or $M_2$ takes different values. The region above each neutral curve is unstable to the long-wavelength disturbances, while the region underneath each curve is stable to long-wavelength disturbances.

Figure 6.

The (a) first, (b) second and (c) third family of the neutral stability curves for the long-wavelength instability in the $( {\beta ,F} )$-plane for different ${M_1}$ (see the labels); the (d) first, (e) second and ( f) third family of the neutral stability curves for the long-wavelength instability in the $( {\beta ,F} )$-plane for different ${M_2}$ (see the labels). The region above (below) each curve is unstable (stable) to long-wavelength disturbances. Results are shown for (a) $M_1 \times 10^2$; (b) $M_1 \times 10^9$; (c) $M_1 \times 10^{15}$; (d) $M_2 \times 10^5$; (e) $M_2 \times 10^{11}$; ( f) $M_2 \times 10^{17}$.

For cases of $M_1 \ne 0$ and $M_2=0$, the neutral curves of the first family are shown in figure 6(a). When ${M_1} > 9.44 \times {10^{ - 2}}$, the neutral curves are identical, which is represented by the thick line. Under the circumstances, the neutral curves are independent on ${M_1}$. If ${M_1}$ slightly decreases, e.g. ${M_1} = 9 \times {10^{ - 2}}$, a dip occurs at the bottom of the thick curve at $\beta \approx 0.7$. This part of the neutral curve depends on ${M_1}$. As $M_1$ decreases gradually, the neutral curves branch off the thick curve and the unstable region expands. As ${M_1} \to 0$, the neutral curve eventually tends to the clean-surface curve (dashed line). Hence, again we see that the surface surfactant can reduce the unstable region and increase the critical Froude number, indicating a stabilizing effect of surface surfactants. For the second family (see figure 6b), the features look similar to those in the first family. However, there are also two significant differences. One is that when ${M_1} > 4.447 \times {10^{ - 9}}$, the neutral curve disappears and the flow tends to be stable. The other is that when $0 < {M_1} < 4.447 \times {10^{ - 9}}$, the neutral curves in figure 6(b) do not overlap, which differ from figure 6(a). They all depend on ${M_1}$, so each entire neutral curve corresponds to the travelling wave mode. For the third family (see figure 6c), its characteristics are identical to that of the second family except for larger critical Froude numbers. A larger critical Froude number is induced by the high oscillation frequency in the third family curves because the Stokes layer caused by the oscillation is only confined to the vicinity of the wall, and the instability caused by the oscillation is very weak. Therefore, only a small amount of gravity is required to suppress the flow instability.

For cases of $M_1 =0$ and $M_2\ne 0$, the first family of the neutral stability curves in figure 6(d) is basically similar to figure 6(a) and the unstable region is very limited. For the second family (see figure 6e), the $\beta$ range of the second family is relatively higher and the family's basic characteristics are similar to those of the first family except that the left side of the curves is always constant and independent of ${M_2}$. The characteristics of the third family (see figure 6f) are similar to those of the second family except for a larger critical Froude number.

4.4. Instability mechanism

The instability of oscillatory film flow can be explained straightforwardly through the effects of the disturbance flow field. First, the disturbance velocity causes the volume flow rate unevenly distributed in time and space, which is related to the disturbance streamfunction. The volume flow rates of the upper and lower film are

(4.2a)\begin{align} {Q_{v1}}\left( {x,t} \right) &= \int_{{\eta '_2}}^{n + {\eta '_1}} {{u_1}\left( {x,y,t} \right){\textrm{d} y}} + \int_{ - 1}^{{\eta '_2}} {{u_2}\left( {x,y,t} \right){\textrm{d} y}} = \int_0^n {{U_1}\left( {y,t} \right){\textrm{d} y}} \nonumber\\ &\quad + \int_{ - 1}^0 {{U_2}\left( {y,t} \right){\textrm{d} y}} + {U_1}\left( {n,t} \right){\eta '_1} + {\psi '_1}\left( {x,n,t} \right) + O( {{\varepsilon ^2}} ), \end{align}

(4.2b)\begin{align} {Q_{v2}}\left( {x,t} \right) &= \int_{{\eta '_2}}^{n + {\eta '_1}} {{u_1}\left( {x,y,t} \right){\textrm{d} y}} = \int_0^n {{U_1}\left( {y,t} \right){\textrm{d} y}} \nonumber\\ &\quad + [ {{U_1}\left( {n,t} \right){\eta '_1} + {\psi '_1}\left( {x,n,t} \right)} ] - [ {{U_1}\left( {0,t} \right){\eta '_2} + {\psi '_1}\left( {x,0,t} \right)} ] + O( {{\varepsilon ^2}} ), \end{align}

respectively. Furthermore, the two disturbed volume flow rates can be defined as

(4.3a)\begin{gather} {Q '_{v1}}\left( {x,t} \right) = {U_1}\left( {n,t} \right){\eta '_1} + {\psi '_1}\left( {x,n,t} \right), \end{gather}

(4.3b)\begin{gather}{Q '_{v2}}\left( {x,t} \right) = \left[ {{U_1}\left( {n,t} \right){\eta '_1} + {\psi '_1}\left( {x,n,t} \right)} \right] - \left[ {{U_1}\left( {0,t} \right){\eta '_2} + {\psi '_1}\left( {x,0,t} \right)} \right], \end{gather}

respectively. In order to satisfy local conservation of volume, the presence of disturbance of volume flow rate induces spatial and temporal changes of the surface and interface, e.g. increase of surface and interface. Second, the disturbance velocity would rearrange the velocity distribution and cause local concentration or dilution of surfactants. Therefore, we introduced the expressions of surface surfactant flux and interfacial surfactant flux

(4.4a)\begin{align} {Q_{s1}}\left( {x,t} \right) &= {u_1}\left( {x,n + {\eta '_1},t} \right){\varGamma _1}\left( {x,t} \right)\nonumber\\ &= {U_1}\left( {n,t} \right) + {U_1}\left( {n,t} \right){\varGamma '_1}\left( {x,t} \right) + \frac{{\partial {\psi '_1}}}{{\partial y}}\left( {x,n,t} \right) + O( {{\varepsilon ^2}} ), \end{align}

(4.4b)\begin{align} {Q_{s2}}\left( {x,t} \right) &= {u_1}\left( {x,n + {\eta '_1},t} \right){\varGamma _1}\left( {x,t} \right) - {u_2}\left( {x,{\eta '_2},t} \right){\varGamma _2}\left( {x,t} \right)\nonumber\\ &= {U_1}\left( {n,t} \right) - {U_2}\left( {0,t} \right) + \left[ {{U_1}\left( {n,t} \right){\varGamma '_1}\left( {x,t} \right) + \frac{{\partial {\psi '_1}}}{{\partial y}}\left( {x,n,t} \right)} \right]\nonumber\\ & \quad - \left[ {{U_2}\left( {0,t} \right){\varGamma '_2}\left( {x,t} \right) + \frac{{\partial {\psi '_2}}}{{\partial y}}\left( {x,0,t} \right) + \frac{{\partial {U_2}}}{{\partial y}}\left( {0,t} \right){\eta '_2}} \right] + O( {{\varepsilon ^2}} ). \end{align}

Furthermore, the two disturbed surfactant fluxes can be defined as

(4.5a)\begin{align} {Q '_{s1}}\left( {x,t} \right) &= {U_1}\left( {n,t} \right){\varGamma '_1}\left( {x,t} \right) + \frac{{\partial {\psi '_1}}}{{\partial y}}\left( {x,n,t} \right), \end{align}

(4.5b)\begin{align} {Q'_{s2}}\left( {x,t} \right) &= \left[ {{U_1}\left( {n,t} \right){\varGamma '_1}\left( {x,t} \right) + \frac{{\partial {\psi '_1}}}{{\partial y}}\left( {x,n,t} \right)} \right] \nonumber\\ &\quad - \left[ {{U_2}\left( {0,t} \right){\varGamma '_2}\left( {x,t} \right) + \frac{{\partial {\psi '_2}}}{{\partial y}}\left( {x,0,t} \right) + \frac{{\partial {U_2}}}{{\partial y}}\left( {0,t} \right){\eta '_2}} \right]. \end{align}

Flow instability is reflected in the exponential growth of the amplitude of the interface (including surface and interface) waves and the amplitude of the surfactant concentration waves. The former depends on the phase difference ${\varDelta _{{Q'_{vj}} - {\eta '_j}}}$ between the disturbance volume flow rates and the interface waves, and the latter depends on the phase difference ${\varDelta _{{Q'_{sj}} - {\varGamma '_j}}}$ between the surfactant fluxes and the surfactant concentration waves, where superscript $'$ denotes the corresponding order of $O( \varepsilon )$. According to Gao & Lu (Reference Gao and Lu2006), if ${\varDelta _{{Q'_{vj}} - {\eta '_j}}}$ and ${\varDelta _{{Q'_{sj}} - {\varGamma '_j}}}$ are defined on the interval $( { - {\rm \pi},{\rm \pi} } ]$, the situation when ${\varDelta _{{Q'_{vj}} - {\eta '_j}}}$ and ${\varDelta _{{Q'_{sj}} - {\varGamma '_j}}}$ take negative values is shown in figure 7. Consider the control body between the two vertical lines in figure 7($a$), the volume flow rate out of the control body $Q_{v1}^{'OUT}$ is greater than the volume flow rate into the control body $Q_{v1}^{'IN}$, the peak position moves downward, so the amplitude of the surface wave would attenuate. Similarly, we can explain the change of the surface surfactant concentration wave, as shown in figure 7($b$). Consider a control body between the two vertical lines, since the mass of surfactant out of the control body $Q_{s1}^{'OUT}$ is greater than the mass of surfactant into the control body $Q_{s1}^{'IN}$, the amplitude of the surfactant concentration wave would attenuate. Therefore, the surfactant tends to be evenly distributed. Similarly, we can explain the change of the interface wave and the interfacial surfactant concentration wave, but the control body here is the upper film, as shown in figure 7($c$,$d$). In summary, when ${\varDelta _{{Q'_{vj}} - {\eta '_j}}}$ and ${\varDelta _{{Q'_{sj}} - {\varGamma '_j}}}$ are negative, the flow is stable to the long-wave disturbance; otherwise, it is unstable.

Figure 7.

Sketch for the instability mechanism: ($a$,$c$) ${\varDelta _{{Q'_{vj}} - {\eta '_j}}} < 0$ suppress the interfacial deflection; ($b$,$d$) ${\varDelta _{{Q'_{sj}} - {\varGamma '_j}}} < 0$ smoothes the surfactant distribution. The large arrows represent the downward motion of the crest. The disturb surfactant concentration ${\varGamma '_j}$ is denoted by the width of the grey area and the large arrows represent the decrease of the surfactant concentration.

Since we only care about the mean growth of the flow, here we focus on the steady part of the disturbance, and the instantaneous contribution of the unsteady part is zero in the time-average sense (the integral of the unsteady part of the disturbance in a period is zero). Based on (4.3a) and (4.5a), the contributions of the steady part of the disturbances are described as (3.15a) and (3.15b), respectively. Based on (4.3b) and (4.5b), from (3.15) we have

(4.6a)\begin{align} &{\left( {{U_1}{h_{11}} + {\phi _{11}}} \right)^{\left( S \right)}} - {\left( {{U_2}{h_{21}} + {\phi _{21}}} \right)^{\left( S \right)}}\notag\\ &\quad ={-} \textrm{i}R{F^{ - 2}}\left[ {\left( {\frac{n}{2} + {n^2} + \frac{{{n^3}}}{{3~m}}} \right)r{h_{10}} - \frac{n}{2}\left( {r - 1} \right){h_{20}}} \right]\nonumber\\ &\qquad - \textrm{i}R\left[ {\left( {nm + \frac{{{n^2}}}{2}} \right){M_1}{\xi _{10}} + n{M_2}{\xi _{20}}} \right] + \textrm{i}R\left[ {\left( {{I_{11}} - {I_{31}}} \right){h_{10}} + \left( {{I_{12}} - {I_{32}}} \right){h_{20}}} \right], \end{align}

(4.6b)\begin{align} &{\left( {{U_1}{\xi _{11}} + \frac{{\partial {\phi _{11}}}}{{\partial y}}} \right)^{\left( S \right)}} - {\left( {{U_2}{\xi _{21}} + \frac{{\partial {\phi _{21}}}}{{\partial y}} + \frac{{\partial {U_2}}}{{\partial y}}{h_{21}}} \right)^{\left( S \right)}}\nonumber\\ &\quad ={-} \textrm{i}R{F^{ - 2}}\frac{{{n^2}}}{{2~m}}r{h_{10}} - \textrm{i}Rn{M_1}{\xi _{10}} + \textrm{i}R\left[ {\left( {{I_{21}} - {I_{41}}} \right){h_{10}} + \left( {{I_{22}} - {I_{42}}} \right){h_{20}}} \right]. \end{align}

The complex amplitudes of the steady part of the disturbance are represented by (3.15a), (3.15b), (4.6a) and (4.6b), three terms of which represent the contributions of gravity, Marangoni stress and fluid inertia, respectively. From (3.15a), it can be seen that the phase difference between the disturbed volume flow rate caused by gravity and the surface wave is $- {{\rm \pi} / 2}$ (see Appendix F). Similarly, in (4.6a) the phase difference between the disturbed volume flow rate caused by gravity and the interface wave is also $- {{\rm \pi} / 2}$. Therefore, gravity always enhances stability. For the surface wave, it can be seen from (3.15a) that the contribution of the inertial effect depends on the signs of ${I_{11}}$, when ${I_{11}} > 0$ $( { < 0} )$, the phase difference between the disturbed volume flow rate caused by inertia and the surface wave is ${{\rm \pi} / 2}( { - {{\rm \pi} / 2}} )$. Similarly, for the interface wave, it is seen from (4.6a) that when ${I_{12}} - {I_{32}} > 0$ $( { < 0} )$, the phase difference between the disturbed volume flow rate caused by inertia and the interface wave is ${{\rm \pi} / 2}( { - {{\rm \pi} / 2}} )$. For phase difference ${{\rm \pi} / 2}$ and $- {{\rm \pi} / 2}$, the effect of inertia would weaken and enhance the flow stability, respectively. For the surface surfactant, it can be seen from (3.15b) that the phase difference between the disturbed surface surfactant flux and the surface wave caused by Marangoni stress is $- {{\rm \pi} / 2}$, so the surface surfactant always enhances the flow stability. For the interfacial surfactant, (4.6b) shows that the disturbed interfacial surfactant flux is not directly affected by the interfacial surfactant. Hence, the interfacial surfactant may enhance or suppress flow instability. This is in contrast to the surface surfactant, which always enhances flow stability.