2.1. Flow configuration

We consider a two-dimensional incompressible viscoelastic liquid film flowing down an inclined plane making an angle $\theta$ with the horizontal, as shown in figure 1. The thickness of liquid film is $2d$, and the origin of the Cartesian coordinate system is placed in the middle of the unperturbed liquid film. The inclined plate is forced to oscillate sinusoidally in the streamwise direction ($x$-axis) and wall-normal direction ($\kern 1.5pt y$-axis), with constant frequency $\omega$. The solid line $h(x,t)$ shows the deformed liquid surface. For a given viscoelastic liquid, the density $\rho$ is assumed to be constant. Also, the amplitudes of oscillation in the streamwise and wall-normal directions are $A_x$ and $A_y$. According to the d’Alembert theorem, the governing equations in a non-inertial reference system can be expressed as

(2.1)$$\begin{gather} \partial_x u+\partial_y \upsilon =0, \end{gather}$$

(2.2)$$\begin{gather}\rho(\partial_t u + u\,\partial_x u+\upsilon\,\partial_y u)= \partial_x\tau_{xx}+\partial_y\tau_{xy}+\rho g\sin\theta - \rho A_x \omega^2\sin\omega t, \end{gather}$$

(2.3)$$\begin{gather}\rho(\partial_t \upsilon + u\,\partial_x \upsilon + \upsilon\,\partial_y \upsilon)=\partial_x\tau_{y x}+\partial_y\tau_{y y} -\rho g\cos\theta - \rho{A_y \omega^2}\sin\omega t, \end{gather}$$

where $u$ and $\upsilon$ are velocity components in the streamwise and wall-normal directions, respectively. The inertial forces in two directions are expressed as $\rho A_x \omega ^2\sin \omega t$ and $\rho A_y \omega ^2\sin \omega t$. Here, $g$ is the gravitational acceleration, and $\tau _{ij}$ is the component of the stress tensor, with different expressions for different viscoelastic fluids. Walters’ liquid $B''$ is used in this paper, which represents an approximation to first order in elasticity, i.e. for short or rapidly fading memory fluids, and satisfies the following constitutive equation of state (Beard & Walters Reference Beard and Walters1964; Andersson & Dahl Reference Andersson and Dahl1999):

(2.4)\begin{equation} \tau_{i j} ={-}p\delta_{i j} + 2\mu e_{ij}-2M_0\,\frac{\delta}{\delta t}\,e_{ij}, \end{equation}

where $\mu$ is the limiting viscosity, $e_{ij}=(\partial _{i}u_j + \partial _j u_i)/2$ is the strain rate tensor, $2\mu e_{ij}$ corresponds to the Newtonian stress, $2M_0(\delta /\delta {t})e_{i j}$ corresponds to elastic stress, $M_0$ is the viscoelastic coefficient, $\delta _{i j}$ is the Kronecker symbol, and $p$ is the isotropic pressure. Walters’ liquid $B''$ is chosen because the constitutive equation of this model involves only one non-Newtonian parameter, although this model can describe only shear flows of weakly elastic dilute liquids. The advantages brought by this are that one can easily obtain a deeper insight into the flow behaviour of viscoelastic fluids. Besides, Walters’ liquid $B''$ with limiting viscosity at low shear rates and short memory coefficient is one of the best models to describe the characteristics of a viscoelastic fluid. Nevertheless, it should be noted that Walters’ liquid $B''$ is an approximation to the Oldroyd-B liquid in the limit of small relaxation time. And the flow needs to have the characteristics of low shear rates and weak viscoelasticity when using the constitutive equation. This approximation is reasonable for analysing the flow similar to the boundary layer and liquid film. For typical parameters of Walters’ liquid $B''$ (Walters Reference Walters1960), a mixture of polymethyl methacrylate in pyridine has density $\rho = 0.98\times 10^3\,\mathrm {kg}\,\mathrm {m}^{-3}$, limiting viscosity $\mu = 0.79\,\mathrm {N}\,\mathrm {s}\,\mathrm {m}^{-2}$, and viscoelastic coefficient $M_0 = 0.04\,\mathrm {N}\,\mathrm {s}^{2}\,\mathrm {m}^{-2}$. The upper convected derivative for the strain rate tensor is defined as

(2.5)\begin{equation} \frac{\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}

Figure 1.

Schematic of a viscoelastic fluid on an oscillating inclined plane.

For the boundary conditions, it satisfies no-slip and no-penetration at $y = -d$ since the coordinate system is fixed at the oscillating plate:

(2.6a,b)\begin{equation} u=0,\quad \upsilon=0. \end{equation}

The free surface of the film at $y=d+h(x,t)$ satisfies the normal force conservation, zero tangential force conservation and kinematic boundary condition:

(2.7)$$\begin{gather} p_a + \tau_{i j} n_i n_j =\sigma\,\frac{ \partial_{xx} h } {(1+\partial_x^{2}h)^{3/2}}, \end{gather}$$

(2.8)$$\begin{gather}\tau_{i j} n_j t_i =0, \end{gather}$$

(2.9)$$\begin{gather}\partial_{t} h + u\,\partial_{x} h = \upsilon, \end{gather}$$

where $\sigma$ is the surface tension, $\boldsymbol {n}$ is the unit normal vector, $\boldsymbol {t}$ is the unit tangent vector, and $p_{a}$ is the ambient pressure. It is assumed that there is no external force on the surface of the liquid film, e.g. imposed shear stress, so the tangent stress of the free surface is equal to zero. Then the governing equations and boundary conditions are normalized by introducing $d$ as the length scale, $1/\omega$ as the time scale, $d\omega$ as the velocity scale, and $\rho \omega ^2 d^2$ as the pressure scale. The choice of characteristic time scale can be varied. Considering that the research object of this work is the influence of an oscillating boundary on the flow of a liquid film on an inclined plate, from this point of view, $1/\omega$ is more suitable. If only the influence of the inclined angle on the flow field is discussed, then it is more appropriate to use the liquid film thickness divided by the typical velocity as the time scale. The dimensionless parameters of the problem are as follows: the Reynolds number ${{Re}} =\omega d^2/\nu$ shows the effect of frequency; the Froude number $Fr=\omega d/\sqrt {gd}$ shows the effect of gravity; the Weber number $We=\sigma /\rho \omega ^2 d^3$ shows the effect of surface tension, and the viscoelastic parameter $M=M_0/(\rho d^2)$ shows the effect of viscoelasticity.

For the basic flow, we consider a unidirectional parallel flow, and the unperturbed surface of the viscoelastic liquid film is parallel to the plate at $y = 1$. By simplifying the governing equations and boundary conditions in a reference frame moving with the oscillating inclined plate, the dimensionless equations for the basic flow are obtained:

(2.10a)$$\begin{gather} \partial_{t}U ={-}\partial_{x}P + {{Re}}^{{-}1}\,\partial_{yy}U - M\,\partial_{yyt}U + \frac{\sin\theta}{Fr^{2}} -a_{x} \sin t, \end{gather}$$

(2.10b)$$\begin{gather}0=\partial_{y}P+\frac{\cos\theta}{Fr^{2}}+a_{y}\sin t, \end{gather}$$

with the corresponding boundary conditions

(2.11a)$$\begin{gather} U=0,\quad\text{at}\ y={-}1, \end{gather}$$

(2.11b)$$\begin{gather}\partial_{y}U - M\,Re\,\partial_{yt}U=0,\quad P=P_{a},\quad \text{at}\ y=1, \end{gather}$$

where $U(y,t)$ is the velocity of basic flow, $P=p_a/\rho (\omega d)^2$ is the dimensionless ambient pressure, and $a_x=A_x/d$ and $a_y=A_y/d$ are the oscillation amplitude components in the streamwise and wall-normal directions, respectively. By solving unperturbed parallel flow (2.10)–(2.11), the theoretical solution can be expressed as

(2.12) \begin{gather} U(y,t) = U^s(y) + U^t(y,t) \nonumber\\ = \frac{{{Re}}\sin\theta}{2\,Fr^2}\,(3+2y-y^2) - a_x\,\mathrm{Re} \left[\left\{\frac{\cosh[r (y-1)]}{\cosh(2r)}-1 \right\}\text{e}^{\text{i}t}\right], \end{gather}

(2.13)$$\begin{gather} P(y,t)=P_{a}+\left(\frac{\cos\theta}{F r^{2}}+a_{y}\sin t\right)(1-y), \end{gather}$$

(2.14)$$\begin{gather}V(y,t)=0, \end{gather}$$

with

(2.15)\begin{equation} \left.\begin{gathered} r=\sqrt{\frac{\text{i}\,{{Re}}}{1 - \text{i}M\,{{Re}}}} ={\pm} \varOmega\sqrt{{{Re}}\,S/2}\,(1+\mathrm{i}S),\\ \varOmega=\frac{\sqrt{1+(M\,{{Re}})^2}-M\,{{Re}}}{\sqrt{1+(M\,{{Re}})^2}},\\ S=\sqrt{1+(M\,{{Re}})^2} + M\,{{Re}}, \end{gathered}\right\} \end{equation}

where $\mathrm {Re}[\cdot ]$ represents the real part of a complex function. The basic flow is separated into two parts, namely, the steady part denoted by $U^s(y)$ and the time-dependent flow denoted by $U^t(y,t)$. Note that wall-normal oscillation $a_y$ is involved only in the expression for pressure, which means that the unperturbed velocity is independent of wall-normal oscillation. Besides, the discussed problem degenerates into that of Woods & Lin (Reference Woods and Lin1995) and Lin et al. (Reference Lin, Chen and Woods1996) if the viscoelastic parameter $M$ is zero. Further setting the oscillation amplitudes to zero, the basic flow will degenerate into the stability problem of the Newtonian fluid discussed by Yih (Reference Yih1963).

2.2. Linear stability problem

Considering the linear stability problem, the linear time-dependent Orr–Sommerfeld equations can be derived by introducing the infinitesimal perturbation to the basic flow. By ignoring higher-order small quantities, the perturbed flow variables can be expressed as

(2.16a–d)\begin{equation} u=U+u',\quad \upsilon=\upsilon',\quad p=P+p',\quad h=1+h', \end{equation}

where variables with superscript prime denote perturbation variables. By substituting perturbation variables (2.16a–d) into the governing equations and boundary conditions (2.10)–(2.11), we can obtain the dimensionless perturbation equations. Considering only the onset of instability with respect to two-dimensional disturbance, the continuity will be automatically satisfied by introducing a streamfunction $\psi '(x,y,t)$, such that

(2.17a,b)\begin{equation} u'=\partial_{y}\psi',\quad \upsilon'={-}\partial_{x}\psi'. \end{equation}

In the following stability analysis, we look for the normal mode solution of the form

(2.18)$$\begin{gather} \psi'(x,y,t)={\phi}(y,t)\,\mathrm{e}^{\mathrm{i}k x}, \end{gather}$$

(2.19)$$\begin{gather}h'(x,t)=\eta(t) \,\mathrm{e}^{\mathrm{i}k x}, \end{gather}$$

where $k$ represents the wavelength of infinitesimal perturbation in the streamwise direction. By substituting the normal mode solution into the dimensionless perturbation equations, one can derive the time-dependent Orr–Sommerfeld equation as

(2.20)\begin{equation} (\mathscr{L}+M \mathscr{L}^{2})\,\partial_{t} \phi = \mathscr{L}^{2} \phi/{{Re}}-\mathrm{i} k U(\mathscr{L}+M \mathscr{L}^{2})\phi + \mathrm{i} k(\mathscr{D}^{2} U+M \mathscr{D}^{4} U) \phi, \end{equation}

with boundary conditions

(2.21a)\begin{align} \phi&=\mathscr{D} \phi=0,\quad \text{at}\ y={-}1, \end{align}

(2.21b)\begin{align} M\,{{Re}}\,(\mathscr{L}+2 k^{2})\,\partial_{t} \phi &= (\mathscr{L}+2 k^{2}) \phi -\mathrm{i}kM\,{{Re}}\,[U(\mathscr{L}+2 k^{2})-\mathscr{D}^{2} U] \phi \nonumber\\ &\quad +[\mathscr{D}^{2} U- M\,{{Re}}\,\mathscr{D}^{2}\,\partial_{t} U]\eta,\quad \text{at}\ y=1, \end{align}

(2.21c)\begin{align} \hbox{[}\mathscr{D}+M(\mathscr{L}-2 k^{2})\mathscr{D}]\,\partial_{t} \phi &= (\mathscr{L}-2 k^{2}) \mathscr{D} \phi/{{Re}}-\mathrm{i} k\eta(\cos{\theta}/Fr^2+a_y\sin{t} +k^{2}\,We) \nonumber\\ &\quad -\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,\quad \text{at}\ y=1, \end{align}

(2.21d)\begin{align} \partial_{t} \eta&={-}\mathrm{i} k \phi-\mathrm{i} k U \eta,\quad \text{at}\ y=1, \end{align}

where the linear operators are $\mathscr {L}=\mathscr {D}^2- k^2$ and $\mathscr {D}=\partial _y$. The Floquet system (2.20)–(2.21) governs the linear stability problem. For finite-wavelength instabilities, the differential system should be solved numerically, while long-wavelength solutions can be obtained analytically by an expansion in $k$, and will be discussed in the following.

3.1. Effect of inclined angle ($M = 0$, $We=0.016$, $Fr = 100$)

First, we calculated the distribution of unstable regions on the $(a_x, {{Re}})$-plane at different dip angles when there is no viscoelasticity ($M=0$), as shown in figure 3. The stable and unstable regions are denoted by blue and yellow, respectively. When the inclination angle is small ($\theta <50^{\circ }$), as can be seen from figures 3(a,b), the unstable region presents three families of U-shaped curves (from low-frequency to high-frequency, the first, second and third families in turn), and the flow will be unstable only within a specific ${{Re}}$ range. With the increase of oscillation frequency, the amplitude of oscillation leading to linear instability is also increasing rapidly. Nevertheless, when the inclination angle is in the range $10^{\circ }$–$50^{\circ }$, the instability boundary has hardly changed. When $\theta =60^{\circ }$, a new unstable region appears at the high-frequency oscillation (${{Re}}>45.6$) in figure 3(c), and a very small amplitude of streamwise oscillation in this region can also make the flow unstable. After careful examination, the instability of this area is caused by gravity, which can also explain why the unstable area cannot be found when $\theta$ is small. As the inclination angle continues to increase, such as $\theta =70^{\circ }$ and $80^{\circ }$, the gravity instability region continuously expands to the low frequency and merges with the third family oscillation instability region. With respect to the vertical flat plate (see figure 3f), most of the regions in the $(a_x, {{Re}})$-plane are linearly unstable due to the continuous expansion of the gravity instability region.

Figure 3.

Stability limits in the $(a_x, {{Re}})$-plane for $M = 0$, $We=0.016$, $Fr = 100$, with different inclined angles: (a) $\theta =10^{\circ }$, (b) $\theta =50^{\circ }$, (c) $\theta =60^{\circ }$, (d) $\theta =70^{\circ }$, (e) $\theta =80^{\circ }$, and (f) $\theta =90^{\circ }$. Stable (S) and unstable (U) regions are denoted by blue and yellow bands, respectively.

3.2. Effect of viscoelasticity ($We=0.016$, $Fr = 100$)

Figure 4 shows the effect of viscoelasticity on the stability of the oscillating liquid film. When the tilt angle of the plate is small, U-shaped neutral curves appear in the separated bandwidths of the imposed frequency. Moreover, there is a critical streamwise oscillation amplitude in each unstable frequency range. Disturbance diminishes in the region underneath each curve corresponding to stable modes. As the viscoelasticity increases, the unstable frequency bandwidth corresponding to each family of neutral curves will be narrowed, and the critical streamwise oscillation amplitude will also be reduced, which means that compared with Newtonian fluid, the viscoelastic fluid film is more unstable under the same oscillation parameters. Different from the results for small inclination angle, the neutral curve of liquid film with inclination angle $60^\circ$ has gravity instability region in the high frequency range. The flow in the liquid film is unstable when the external oscillation amplitude is lower than the critical streamwise oscillation amplitude, but all infinitesimal disturbances are stable when the critical oscillation amplitude is exceeded. In other words, streamwise oscillation can restrain the gravity instability in high-frequency oscillation to some extent. For viscoelastic liquid films, however, the results are different. It can be seen that when $M=0.01$, the critical streamwise oscillation amplitude in the gravity instability region is beyond the scope of parameter research, i.e. the suppression effect of streamwise oscillation on the gravity instability of viscoelastic liquid film is not as obvious as that of Newtonian fluid. Actually, the unstable region corresponding to high-frequency oscillation here is formed by the fusion of the oscillation unstable region and the gravity unstable region under the action of viscoelasticity, and this region will expand continuously with the increase of viscoelasticity. For the vertical flat plate, the unstable region occupies most area of the $(a_x, {{Re}})$-plane due to the increasing inclination angle, and the interior of the U-shaped curve is stable at this time. An increase of viscoelasticity makes the neutral curve move in the direction that both $a_x$ and ${{Re}}$ become smaller. In order to study the variation of growth rate with viscoelasticity at different frequencies, we show the functional relationship curve of $\gamma _2$ and ${{Re}}$ under typical parameters (see figure 4d). Considering that the ordinate is logarithmic, only the part with positive growth rate is shown in the figure. It can be observed that the influence of viscoelasticity appears in the following three aspects. The first is that viscoelasticity can increase the growth rate, which is reflected in both low-frequency and high-frequency regions. Second, viscoelasticity makes the stable region or instability move to the low-frequency direction. And third, viscoelasticity may change the unstable region of Newtonian fluid into a stable frequency bandwidth (e.g. when $M=0.015$, $32.2<{{Re}}<40.4$). As a result, for the long-wave instability of an oscillating plate, viscoelasticity is no longer only an unstable effect but also makes the flow more stable under certain parameters.

Figure 4.

The neutral curves for the long-wavelength instability in the ($a_x, {{Re}})$-plane for $We=0.016$, $Fr = 100$, with different $M$ and inclined angles: (a) $\theta =10^{\circ }$, (b) $\theta =60^{\circ }$, and (c) $\theta =90^{\circ }$.(d) The growth rate as functions of ${{Re}}$ for $M=0$, $M=0.01$ and $M=0.015$ with $a_x=6$ in (c).

3.3. Effect of gravity ($\theta =90^{\circ }$, $M = 0.01$, $We=0.016$)

Figure 5 depicts the influence of changing gravity on the stability of a viscoelastic vertical plate liquid film. It can be seen that in the range $10< Fr<200$, the neutral curve also presents a U-shaped curve with stable frequency bandwidth and unstable frequency bandwidth separated from each other, and the stable region is above the neutral curve. With the increase of $Fr$, the critical streamwise oscillation amplitude of the neutral curve decreases, but the frequency range corresponding to the stable region has not changed obviously. When $Fr=500$, the stable regions of the first and second families merge with each other, so that an unstable frequency bandwidth is formed in the range $8.2<{{Re}}<14.9$. If $Fr$ is increased further, then the stable region will further expand to the high-frequency direction, and finally a U-shaped curve with stable frequency bandwidth and unstable frequency bandwidth separated from each other will be formed. The difference is that in the unstable frequency bandwidth, when the streamwise oscillation amplitude exceeds a certain value, the flow will be unstable, i.e. the upper part of the U-shaped curve is the unstable region. It is worth noting that no matter how gravity changes, the flow is always unstable when ${{Re}}<2$. In general, increasing Froude number makes the viscoelastic liquid film more stable under the condition of small streamwise oscillation.

Figure 5.

Stability limits in the ($a_x, {{Re}})$-plane for $\theta =90^{\circ }$, $M = 0.01$, $We=0.016$, with different $Fr$: (a) ${Fr=10}$, (b) $Fr=100$, (c) $Fr=200$, (d) $Fr=500$, (e) $Fr=800$, and (f) $Fr=1000$. Stable (S) and unstable (U) regions are denoted by blue and yellow bands, respectively.

3.4. Effect of surface tension ($\theta =90^{\circ }$, $M = 0.01$, $Fr=100$)

Finally, we discuss the influence of surface tension on the stability of a viscoelastic liquid film. The neutral curves of surface tension with different orders are shown in figure 6. As can be seen, with the increase of $We$, the bottoms of the three U-shaped neutral curves develop in the direction of decreasing $a_x$ and ${{Re}}$, while maintaining a stable frequency bandwidth. Different from before, when $We=0.2$, a convex neutral curve emerges in the lower left corner of the $(a_x, {{Re}})$-plane, as shown by the blue dashed line in figure 6. With the further increase of $We$, the curve will merge with the first family of neutral curves to form a larger stable region, and gradually expand to both low-frequency and high-frequency directions. We conclude, therefore, that increasing the surface tension can make more regions on the $(a_x, {{Re}})$-plane change from unstable regions to stable regions, i.e. the viscoelastic liquid film is more stable to some extent.

Figure 6.

The neutral curves for the long-wavelength instability in the ($a_x, {{Re}})$-plane for $\theta =90^{\circ }$, $M = 0.01$, $Fr=100$, with different $We$ values.

4.1. Wall-normal oscillation results

When the influence of viscoelastic fluid is considered, it is observed that the shape of the neutral curve remains unchanged. The three tongue-shaped harmonic instability regions and the gravity instability region continue to exist, but the critical parameters are altered. When viscoelasticity is incorporated, the unstable boundary for GU shifts to a higher wavenumber. The expansion of the gravity instability region implies that viscoelasticity will destabilize the gravity instability mode. In addition, the right boundaries of resonance unstable regions for SU-1, HU and SU-2 all shift to smaller wavenumbers, and correspond to the changes of critical parameters. Regions SU-1 and SU-2 first appear at $(0.786, 1.16)$ and $(7.12, 2.3)$ in the $(a_y, k)$-plane, and HU is first excited at $(1.8, 3.41)$ for $M=0.01$. The existence of viscoelasticity reduces the critical amplitudes of subharmonic and harmonic instabilities. Therefore, the subharmonic and harmonic instabilities can be generated at lower forcing amplitudes for a wall-normal oscillatory viscoelastic fluid. In order to analyse the influence of viscoelasticity on the four kinds of instability more comprehensively, the variations of temporal growth rate $\gamma _r$ with wavenumber $k$ corresponding to three different wall-normal oscillation amplitudes are also shown in figure 7. The temporal growth profile shows two humps if the amplitude of the oscillating plane is fixed at $a_y=1$; see figure 7(a). The first hump is linked to the gravitational instability and appears in the long-wave regime, while the second hump is associated with the subharmonic instability and appears in the finite-wavelength regime. It can be seen that the maximum growth rates of GU and SU-1 both increase with the increase of viscoelasticity, but the growth rate of the gravity mode is far less than that of subharmonic instability. However, there is no hump corresponding to HU and SU-2, since the forcing amplitude $a_y=1$ is not large enough. For $a_y=5$, along with gravity instability and subharmonic instability, a new hump related to harmonic instability appears. It can be found in figure 7(c) that the peak value increases with the increase of viscoelasticity, and the corresponding wavenumber decreases. Unlike the case $a_y=1$, the corresponding peak value of SU-1 becomes smaller when $a_y=5$. If the wall-normal oscillation amplitude is further increased to $a_y=10$, then the humps of the four instabilities all appear in figure 7(d). Again, with the increase of $M$, the hump corresponding to SU-2 moves in the direction of increasing growth rate and decreasing wavenumber. It is worth mentioning that the maximum values of growth rates for SU-1 and HU diminish with the increasing value of viscoelasticity. Here, combining the above results, an interesting phenomenon is observed that the maximum value of all emerging humps is greater than the growth rate of the existing unstable mode, and viscoelasticity has a destabilizing effect on the emerging modes and makes the existing modes more stable.

Next, we will discuss the influence of the viscoelastic parameter on the stability of higher-frequency oscillating plates. Figure 8 illustrates the neutral curve for the oscillation plate with ${{Re}}=30$. For the case $M=0$, there are still four unstable regions on the neutral curve plane. Compared with the result of low-frequency oscillation, increasing the oscillation frequency increases the gravitational unstable wavenumber range and compresses the neutral curve to lower oscillation amplitude. And the critical amplitudes for the first subharmonic and harmonic instabilities are reduced significantly. Besides, it can be noticed that the region of GU merges with SU-1 in the range $1< a_y<2.16$. When the effect of viscoelasticity is incorporated, the unstable region between gravitational and first subharmonic instabilities increases, e.g. $0.9< a_y<2.36$ for $M=0.01$, and $0.82< a_y< 2.46$ for $M=0.015$. The impact of viscoelasticity on the first subharmonic resonance is manifested in two aspects. With the increase of $M$, the region of SU-1 is gradually squeezed, and the critical oscillation amplitude of SU-1 is almost unchanged, but the corresponding critical wavenumber gradually decreases, as indicated by the triangles in figure 8. The oscillation frequency might be more important than the viscoelastic effect on gravity and first subharmonic instabilities. Nevertheless, the influence of viscoelasticity on HU significantly increases the critical oscillation amplitude and decreases the critical disturbance wavenumber.

Figure 8.

Neutral curves in the $(a_y, k)$-plane for different viscoelasticities $M$. Here, black lines, blue dashed lines and orange dash-dotted lines represent $M=0$, $M=0.01$ and $M=0.015$, respectively. Other parameters are ${{Re}}=30$, $Fr=10$, $We=0.016$, $a_x=0$, $\theta =90^\circ$. The triangles represent the critical amplitude of the resonance wave.

Figure 9(a) shows the variation of growth rate $\gamma _r$ with the wavenumber $k$ for low forcing amplitude $a_y=1$. The growth rates for the gravitational and subharmonic instabilities enhance with the increasing values of viscoelastic parameters at $a_y=1$. The minimum value of the growth rate curve at the depression ($k\approx 0.5$) increases to a value greater than zero, which can explain the phenomenon that the unstable region between GU and SU-1 increases. If the forcing amplitude is increased to a comparatively higher value $a_y=3$, then the hump associated with the first subharmonic resonance becomes weaker, while the harmonic and second subharmonic resonances amplify with the increasing value of $M$ (see figure 9b). Compared with the case of low oscillation frequency (${{Re}}=5$), the maximum growth rate of higher-frequency oscillation when $M<0.013$ corresponds to harmonic resonance instead of hump corresponding to high wavenumber. Further, when the value of the viscoelastic parameter exceeds 0.013, a sharp peak will quickly grow at the hump of SU-2, which will increase with the increase of $M$ and eventually exceed the maximum growth rate of HU. Therefore, for the liquid film with higher-frequency oscillation, the most unstable mode is determined by the liquid viscoelasticity and the forcing amplitude.

Figure 9.

Variation of temporal growth rate $\gamma _r$ with wavelength $k$ for (a) $a_y=1$ and (b) $a_y=3$. Here, the black solid line, green dotted line, blue dashed line, red dash-dot-dotted line, and orange dash-dotted line represent $M=0$, $M=0.005$, $M=0.01$, $M=0.013$ and $M=0.015$, respectively.

4.2. Streamwise oscillation results

Now we will discuss the stability of a viscoelastic oscillating liquid film flowing down an oscillating plate with streamwise oscillation. To investigate the effect of viscoelastic liquid on the stability with different oscillation frequencies ${{Re}}$ under arbitrary wavelength disturbance, we set $Fr=100$, $We=0.016$, $a_x=6$, $\theta =90^\circ$ and $a_y=0$. The neutral curve in the $(k, {{Re}})$-plane is illustrated in figure 10. It is obtained that the neutral curve exhibits three distinct unstable regions specified by the first harmonic resonated unstable region (HU-1), the second harmonic resonated unstable region (HU-2), and the gravitational unstable region (GU) for the given set of parameter values (Lin et al. Reference Lin, Chen and Woods1996). The resonated unstable regions HU-1 and HU-2 are developed owing to the streamwise oscillation of the plane, while the region GU is responsible for the gravitational instability. There are two stable bandwidths among the different unstable regions, and all infinitesimal disturbances will be attenuated within those bandwidths. For the first harmonic resonated unstable region, the neutral curve does not change considerably for $k<0.5$ when the viscoelastic parameter alters. However, with the increase of viscoelasticity, the neutral curve extends to higher wavenumbers for $k>0.5$, and the critical oscillation frequency of HU-1 also increases from 5.71 when $M = 0$, to 6.05 when $M = 0.015$. The viscoelasticity has a significant impact on the region of HU-2, and the critical wavenumber of HU-2 increases rapidly with the increase of $M$. For the gravitational unstable region, it is shown that the increasing $M$ makes the neutral curve gradually show the characteristics of being convex at both ends and concave in the middle, as shown by the blue dashed line in figure 10. Further increasing the viscoelasticity ($M=0.015$), the neutral curve is finally divided into two parts, and a stable bandwidth is formed within the range $32.4<{{Re}}<40.8$. In other words, the increase of viscoelasticity makes the gravity instability region divide into two parts, and the frequency range between the two parts is linearly stable.

Figure 10.

Neutral curves in the $(k, {{Re}})$-plane for different viscoelasticities $M$. Here, black solid lines, blue dashed lines and orange dotted lines represent $M=0$, $M=0.01$ and $M=0.015$, respectively. Other parameters are $Fr=100$, $We=0.016$, $a_x=6$, $\theta =90^\circ$, $a_y=0$. The solid circles show the critical frequencies of the unstable regions.

The variation of temporal growth rate with Reynolds number for different values of $M$ at a specified wavenumber is displayed in figure 11. Two typical wavenumbers are selected to reveal the associated growth rates for the regions HU-1, HU-2 and GU when the viscoelastic parameter varies. For $k=0.05$, when $M=0$, it can be seen in figure 11(a) that the black solid lines are divided into three parts by two stable bandwidths, corresponding to three unstable regions, respectively. Each part of the growth rate curve shows a U-shaped curve with a downward opening. When viscoelasticity is considered, members of the first family of U-shaped curves almost coincide, i.e. viscoelasticity does not affect the growth rate of the low-frequency oscillation plate. The second family of U-shaped curves gradually rises with the increase of $M$, which means that the viscoelasticity promotes the instability of region of HU-2. The third family of curves, representing the gravitational instability region, changes most dramatically when the viscoelasticity alters. The temporal growth rate decreases in the middle of the oscillation frequency of GU, and ultimately vanishes from the growth rate diagram with the increasing value of $M$. As a consequence, the increasing $M$ has a stabilizing effect on the instability in the range of oscillation frequency $32.4<{{Re}}<40.8$. Figure 11(b) shows the variation of the growth rate with oscillation frequency ${{Re}}$ at a higher wavenumber $k = 0.5$. Because gravity instability exists only in the case of small disturbance wavenumber, there are only two families of U-shaped curves in figure 11(b). It can be seen that the first family of curves increases slightly with the increase of viscoelasticity, while the second family of curves increases significantly and is adjoined by the expansion of an unstable frequency range. The above results are fully consistent with the results reported in figure 10.

Figure 11.

(a) Variation of temporal growth rate $\gamma _r$ with oscillation frequency ${{Re}}$ for (a) $k=0.05$ and(b) $k=0.5$. The black solid lines, blue dashed lines and orange dash-dotted lines represent $M=0$, $M=0.01$ and $M=0.015$, respectively.

4.3. Effect of viscoelasticity on shear wave mode

The influence of viscoelasticity in the presence of wall oscillation on the shear wave instabilities is considered in this subsection. When the angle of inclination is sufficiently small, one expects to see the emergence of the shear waves (Lin Reference Lin1967; Chin et al. Reference Chin, Abernath and Bertschy1986; Floryan et al. Reference Floryan, Davis and Kelly1987), which occur at much larger Reynolds numbers. To investigate the effect of viscoelasticity on shear wave mode for an oscillatory liquid film, the parameters of the numerical experiment are set as $\theta =1^\circ$, $Fr=1$, $We=5$, ${{Re}}=293$. Although there exists a critical inclined angle depending on the surface tension, the liquid film will show shear wave instability (Woods & Lin Reference Woods and Lin1996), which is not the main motivation of this subsection. Figure 12(a) displays the variation of growth rate $\gamma _r$ with wavenumber $k$ for the inclined plane without oscillation, i.e. $a_x=0$ and $a_y=0$. The growth rate curve for Newtonian fluid exhibits two unstable modes: shear waves and surface waves. As can be shown, the viscoelasticity significantly affects the shear wave mode while just marginally enhancing the surface wave mode. The most dangerous mode changes from surface wave to shear wave, when the viscoelastic parameter is raised to only 0.0005, which indicates that the viscoelastic effect on the stability of liquid film is very considerable. For the wall-normal oscillating plate, it is seen in figure 12(b) that both the surface waves and shear waves are quickly dominated by the Faraday waves compared with the static plate since the value of the growth rate (black solid line) is far greater than that in figure 12(a). If the viscoelastic parameter $M$ is increased to 0.001, then the Faraday wave will remain unchanged, but the shear mode will be significantly enhanced (black dash-dotted line). For a greater value of forcing amplitude, e.g. the orange line for $a_y=2$, the same result can be observed. It is worth noting that although the appearance of viscoelasticity makes shear waves dominate, increasing the forcing amplitude of oscillation appears to decrease the maximum growth rate of shear waves while maintaining the same value of $M$. This implies that wall-normal oscillation will appropriately suppress shear waves.

Figure 12.

(a) Variation of temporal growth rate $\gamma _r$ with wavelength $k$ in the absence of oscillation. The black solid line, blue dashed line and orange dash-dotted line represent $M=0$, $M=0.0001$ and $M=0.0005$.(b) Variation of temporal growth rate $\gamma _r$ with wavelength $k$. Here, the black solid line and dash-dotted line are $M=0$ and $M=0.001$ when $a_y=1$. The orange lines change $a_y=2$ only with respect to the black lines.

Figure 13(a) shows the effect of streamwise oscillation on the temporal growth rate $\gamma _r$ with $a_y=0$ for $M=0$. When the inclined plate is stationary, there is a shear wave at $k = 0.54$, and the oscillation perpendicular to the wall will amplify the growth rate of this shear wave mode. Interestingly, when there is only streamwise oscillation, the shear wave at this position will be suppressed, but a new hump will appear at $k=0.66$. When $a_x=3.25$, the maximum growth rate of the new unstable mode has the same value as that of surface wave at the same wavenumber. Further increasing the oscillation amplitude to $a_x=3.45$, the new unstable mode is dominant. The newly emerging mode is called the oscillatory mode. What needs special explanation is that the shear mode is determined by the steady-state solution driven by gravity in the basic flow, and the imaginary part of its eigenvalue is not zero. Unlike the shear mode, the eigenvalue of the oscillatory mode is a real number, and the oscillatory mode is caused by streamwise oscillation of the inclined plate. Therefore, the increase of $a_x$ suppresses the shear wave mode but excites the oscillatory mode. This is to be expected physically: as the forcing amplitude increases, the proportion of energy of periodic oscillation part $U^t(y,t)$ in the total energy of basic flow (2.12) increases gradually. If the effect of viscoelasticity is incorporated, then the oscillatory mode starts to enhance, as can be seen in figure 13(b). When the forcing amplitude is increased from that given by the black dashed line in figure 13(b), with the rest of the parameters kept constant, the enhancement of the oscillatory mode becomes very prominent. The maximum amplification rate of the oscillatory mode is actually much larger than the surface waves. Despite this, the growth rate of the surface waves hardly changes whether the amplitude of streamwise oscillation or viscoelastic parameter is increased.

Figure 13.

(a) The effect of streamwise oscillation on the temporal growth rate $\gamma _r$ with $a_y=0$ for $M=0$. The black solid line, blue dots and black dashed line represent $a_x=0$, $a_x=3.25$ and $a_x=3.45$. (b) Variation of temporal growth rate $\gamma _r$ with wavelength $k$ for different $a_x$ and $M$. The blue dots represent $a_x=3.25$ and ${M=0}$. The black dashed line represents $a_x=3.25$ and $M=0.0005$. The orange dash-dotted line represents ${a_x=3.45}$ and $M=0.0005$.