4.1.1. Creeping flow

A typical spectrum of the VR in the creeping flow is shown in figure 3(a). The figure shows the existence of a downstream travelling unstable mode ($\omega _r>0$). From the analysis of Smith & Davis (Reference Smith and Davis1982), the NR case is linearly stable in the creeping flow. However, from figure 3(a), there is an unstable mode for a viscoelastic liquid layer, hereafter termed the ‘liquid elastic mode’. This clearly shows that the elasticity of the liquid leads to the predicted instability. Another discrete mode in figure 3(a) is $\omega =k$, which remains stable for the parameters considered here. Thus, its variation with the other parameters will not be explored in the following discussion. Using the analytical solutions (3.3) and (3.4), the shooting code employing the vpasolve MATLAB function gives $\omega = 0.433075 + 0.023960i$, in excellent agreement with the eigenvalue obtained using the pseudo-spectral method thereby validating the latter. This further affirms the genuine nature of the liquid elastic mode.

Figure 3.

Panel (a) shows the spectrum of the flow at $Re=0$, $k=0.5$, $T=1$, $n=1$, $Ma=0$, $\beta =0$ and $W=0.3$ for the VR case obtained using the pseudo-spectral method. A downstream travelling unstable mode ($\omega = 0.433075 + 0.023960i$) exists in the spectrum since $\omega _i>0$. Such an unstable mode is absent for the Newtonian liquid layer sheared by the air (Smith & Davis Reference Smith and Davis1982), thereby showing that the liquid elasticity is responsible for the destabilisation of this mode. Besides the unstable mode, a neutrally stable mode with $\omega =k$ is also present. The overlap of the unstable mode for $N=50$ and $N=75$ confirms the genuine nature of the mode. The continuous spectrum line at $\omega _i=-3.3333$ is made up of multiple stable modes. The number of modes in the continuous spectrum depends on the number of collocation points utilised in obtaining the spectrum. The same unstable mode has been also predicted by the shooting code. Panel (b) illustrates the movement of the continuous spectrum due to variation in $W$. For $W=0.3$ and $W=0.1$, the continuous spectrum is located at $\omega _i\sim -3.3333$ and $\omega _i \sim -10$, respectively, which confirms the rule for the continuous spectrum as $\omega _i=-1/W$.

The continuous spectrum in figure 3(a) corresponds to $\omega _i \sim -3.3333$, which is an equivalent of the high-frequency Gordotsov–Leonov line for the plane Couette flow reported in the literature (Renardy & Renardy Reference Renardy and Renardy1986; Joshi & Shankar Reference Joshi and Shankar2019). It must be noted that the plane Couette flow of a viscoelastic liquid exhibits a high-frequency Gordotsov–Leonov line at $\omega _i \sim -1/(2W)$ (Joshi & Shankar Reference Joshi and Shankar2019). However, in the present case from figure 3(b), we observe that the continuous spectrum line obeys the rule $\omega _i \sim -1/W$. Therefore, a viscoelastic liquid layer sheared by the air not only introduces a new mode of instability but also affects the continuous spectrum compared with the plane Couette flow of a viscoelastic liquid.

To further understand the new mode of instability, the variation of $\omega _i$ with $k$ for select values of $W,n,\beta$ and $T$ is shown in figure 4. For $T=0$, Smith & Davis (Reference Smith and Davis1982) predicted that, for the Newtonian mode to become unstable, $Re>34.2$, thus $Re_c=34.2$, thereby showing that a minimum $Re$ is needed for the existence of the Newtonian mode. However, from figure 4, the flow is unstable at an arbitrary $W$, implying an absence of the minimum $W$ needed for the instability to exist. Thus, a viscoelastic liquid layer sheared by the air and flowing past a rigid solid is unconditionally unstable. However, from figure 4(a), the parameter $W$ does affect the growth rate of the perturbations. Also, at low $k$, the growth rate shows characteristic scaling $\omega _i \sim Wk^{2}$. According to the analysis of Smith & Davis (Reference Smith and Davis1982), $Re_c$ for a Newtonian liquid rapidly increases with an increasing $T$. However, from figure 4(b), the unstable mode still exists, albeit with a lesser growth rate and at low values of $k$.

Figure 4.

Variation in the growth rate $\omega _i$ with the disturbance wavenumber $k$ in the creeping-flow limit at $Ma=0$. Panel (a) shows that the liquid layer is unconditionally unstable since a minimum $W$ is unnecessary for the instability to exist. The Weissenberg number only affects the growth rate of the disturbances. The curve for $W=0.1$ is incomplete due to the elimination of the negative values of $\omega _i$ to use a log scale. Panel (b) shows the effect of the surface tension on the dispersion curves. An increasing surface tension suppresses the instability at moderate and high wavenumbers but fails to stabilise at low wavenumbers, thereby showing that the flow remains unconditionally unstable. Panel (c) illustrates the destabilising effect of the shear-thinning nature of the fluid. Panel (d) shows the decrease in the growth rate of the perturbations as the contribution of the solvent viscosity signified by $\beta$ is increased. Parameters are: (a) $n=1$, $\beta =0$ and $T=0$; (b) $n=1$, $\beta =0$ and $W=0.1$; (c) $W=0.1$, $\beta =0$ and $T=1$; (d) $n=1$, $T=1$ and $W=0.1$.

The effect of the shear thinning on the growth rate is illustrated in figure 4(c). A decreasing power-law index signifies increasing shear thinning. From figure 4(c), as the shear-thinning nature of the fluid increases the growth rate of the liquid elastic mode increases. The increasing contribution of the solvent to the solution, i.e. $\beta$ leads to a decrease in the growth rate of the liquid elastic mode, as shown in figure 4(d). This is to be expected since the NR case is linearly stable in the creeping-flow limit and increasing $\beta$ implies an increase in the Newtonian solvent contribution.

The mechanism of the liquid elastic mode can be understood as follows. For simplicity, consider the case of a UCM fluid sheared by air which corresponds to $n=1$ and $\beta =0$. For the existence of an instability, a term is necessary which can lead to an energy transfer from the base-state quantities to the perturbation quantities. From (2.10) for $n=1$ and $\beta =0$, there are several terms including the convective terms $\textrm {i}kWy\tilde {\tau }_{xx}$, $\textrm {i}kWy\tilde {\tau }_{xy}$ and $\textrm {i}kWy\tilde {\tau }_{yy}$ and other interaction terms $-4\textrm {i}kW^{2} \tilde {v}_x$, $-2\textrm {i}kW^{2} \tilde {v}_y$, $-2WD\tilde {v}_x$, $-2W\tilde {\tau }_{xy}$ in the constitutive equations that satisfy this criterion. However, similar terms are also present for the plane Couette flow of UCM liquid yet it is linearly stable in the creeping-flow limit (Gorodtsov & Leonov Reference Gorodtsov and Leonov1967; Renardy & Renardy Reference Renardy and Renardy1986; Joshi & Shankar Reference Joshi and Shankar2019). This indicates that the terms responsible for the instability exhibited by VR may not be the terms present in the linearised constitutive equation. The other terms that represent the interaction between the base-state and perturbed state quantities are in the boundary conditions at $y=1$, i.e. $(-2\textrm {i}k\tilde h)$, which is also present for the Newtonian liquid, and $(-2\textrm {i}kW\tilde h)$, which is only present for the viscoelastic liquid. Since the NR case is stable in the creeping flow, the instability predicted in the present study could be due to the term $-2\textrm {i}kW\tilde h$.

To verify the above hypothesis, eigenvalues are obtained with/without the term $(-2\textrm {i}kW\tilde h)$ in the boundary condition (2.12d), as shown in table 3. Table 3 shows that the above hypothesis is indeed correct and the term responsible for the existence of an unstable liquid elastic mode is $(-2\textrm {i}kW\tilde h)$. The term $(-2\textrm {i}kW\tilde h)$ in the basic form is $(\textrm {i}k (\bar \tau _{yy}-\bar \tau _{xx}) \tilde h)$, where the quantity $(\bar \tau _{yy}-\bar \tau _{xx})$ is the first normal stress difference across the air–liquid interface in the base state. Thus, the term $(-2\textrm {i}kW\tilde h)$ originates from the first normal stress difference, a feature unique to a viscoelastic liquid, thereby showing that the liquid elastic mode is driven by the first normal stress difference exhibited by the viscoelastic liquid. The plots of the normalised perturbations for the liquid elastic mode are shown in figure 5. Since the energy for the perturbations is supplied by the term $(-2\textrm {i}kW\tilde h)$ at the air–liquid interface, the perturbations exhibit the highest variation at $y=1$.

Figure 5.

The normalised perturbations, for the liquid at $W=0.3$, $\beta =0$, $n=1$, $T=1$, $k=0.5$, $Ma=0$ and $Re=0$ for the eigenvalue $\omega =0.433075 + 0.023960$. Here, $v'_x=Re[ \tilde {v}_x \,\textrm {e}^{\textrm {i}kx} ]$ and $v'_y=Re[ \tilde {v}'_y \,\textrm {e}^{\textrm {i}kx} ]$. For convenience, the axes have been normalised to the interval $[0,1]$. The length of the domain in the $x$-direction is equal to a wavelength ($2{\rm \pi} /k$) of the perturbations. The plots shows that the perturbations exhibit maximum variation near the air–liquid interface, thereby indicating the destabilisation caused by the term $(-2\textrm {i}kW\tilde h)$. The perturbations vanish at $y=0$ as needed by the no-slip and impermeability conditions at $y=0$; (a) $v'_x$ and (b) $v'_y$.

Table 3.

The most unstable eigenvalue ($\omega =\omega _r +\textrm {i} \omega _i$) for the VR case with/without the $(-2\textrm {i}kW\tilde h)$ term in the tangential stress boundary condition (2.12d) at $n=1$, $\beta =0$, $Ma=0$ and $T=1$. The presence of the stable eigenvalues in the second column of the table shows that the term responsible for the predicted instability is $(-2\textrm {i}kW\tilde h)$. The term $(-2\textrm {i}kW\tilde h)$ originates from the first normal stress difference across the air–liquid interface. Thus, the liquid elastic mode is triggered by the first normal stress difference exhibited by a viscoelastic liquid.

4.1.2. A simple model for liquid elastic mode

The above discussion shows that the driving force for the liquid elastic mode is the first normal stress difference exhibited by a viscoelastic solid across the air–liquid interface. In the following analysis, we derive a simple model which will incorporate the basic driving force for the liquid elastic mode based on the long-wave analysis. Thus, we scale $x=X/\epsilon$ and $t=\tau /\epsilon$ and expand the fluid quantities as

(4.1)$$\begin{gather} v_x = \bar v_x + \epsilon v'_x +\cdots, \end{gather}$$

(4.2)$$\begin{gather}v_y = \epsilon v'_y +\cdots, \end{gather}$$

(4.3)$$\begin{gather}p = p'+\cdots, \end{gather}$$

where $\epsilon \ll 1$. Further, we assume creeping flow ($Re=0$) and UCM fluid in the absence of the surfactant which implies $n=1,\beta =0$ and $Ma=0$. The above expansion closely follows the linear stability analysis derivation discussed in § 2. Next, we substitute the above expansions in (2.5). At $O(1)$, recalling $\bar p=0$ for the present problem, we obtain the base-state quantities specified in (2.8) and the air–liquid interface is flat. At $O(\epsilon )$ we obtain the perturbed state equations which can be further simplified due to the long-wave approximation

(4.4)$$\begin{gather} \frac{\partial^{2} v'_x}{\partial y^{2}}-\frac{\partial p'}{\partial X}=0, \end{gather}$$

(4.5)$$\begin{gather}\frac{\partial p'}{\partial y}=0. \end{gather}$$

The boundary conditions at $O(\epsilon )$ are

(4.6)$$\begin{gather} \text{at} \ y=0,\quad v'_x=0, \end{gather}$$

(4.7)$$\begin{gather}\text{at} \ y=1+\epsilon h(X,\tau),\quad \frac{\partial v'_x}{\partial y} - \underline{2\hat W \frac{\partial h}{\partial X}}=0; \quad p'={-}\hat T \frac{\partial^{2} h}{\partial X^{2}}, \end{gather}$$

where $\hat W=\epsilon W$ and $\hat T=\epsilon ^{3} T$ and the underlined term is due to the first normal stress difference exhibited by the viscoelastic fluid. Thus, $v'_x$ at $O(\epsilon )$ is

(4.8)\begin{equation} v'_x = \frac{1}{2} y \left[ 4\hat W \frac{\partial h}{\partial X} -\hat T (y-2) \frac{\partial^{3} h}{\partial X^{3}} \right]. \end{equation}

The kinematic boundary condition at $O(\epsilon )$ is

(4.9)\begin{equation} \frac{\partial h}{\partial t} + \frac{\partial h}{\partial X} + \frac{\partial}{\partial X} \int_0^{1} v'_x \,\textrm{d} y=0. \end{equation}

Substituting (4.8) in the above equation yields

(4.10)\begin{equation} \frac{\partial h}{\partial \tau} + \frac{\partial h}{\partial X} + \frac{\partial}{\partial X} \left[ \frac{1}{3} \left(3\hat W\frac{\partial h}{\partial X} +\hat T \frac{\partial^{3} h}{\partial X^{3}} \right) \right]=0. \end{equation}

Rescaling the above equation yields

(4.11)\begin{equation} \frac{\partial h}{\partial t} + \frac{\partial h}{\partial x} + \frac{\partial}{\partial x} \left[ \frac{1}{3} \left(3 W\frac{\partial h}{\partial x} + T \frac{\partial^{3} h}{\partial x^{3}} \right) \right]=0, \end{equation}

where $h$ is still a small deviation of the air–liquid interface from the base-state equation $y=1$. Equation (4.11) is strictly applicable in the limit of small perturbations.

To carry out the linear stability analysis, we substitute $h=\tilde h \,\textrm {e}^{\textrm {i}(kx-\omega t)}$ and simplify to yield

(4.12)\begin{equation} \omega = k - \tfrac{1}{3} \textrm{i} k^{4} T + \textrm{i} k^{2} W. \end{equation}

A comparison between the dispersion curves predicted by using the simple model and the numerical approach is shown in figure 6. The above model correctly predicts $\omega _i \sim Wk^{2}$ at low $k$, as shown in figure 4(a), thereby capturing the importance of the first normal stress difference exhibited by the viscoelastic fluid at the air–liquid interface. Also, the simple model succeeds in capturing the stabilisation due to the increasing air–liquid interfacial tension shown in figure 4(b). However, for $k>0.1$, both the models disagree, which clearly points out the stabilising influence of the other terms at high $k$, a feature missing in the simple model.

Figure 6.

A comparison of the growth rate predicted by the proposed simple model and numerical prediction for the liquid elastic mode at $n=1$, $W=0.1$, $T=0.1$ and $\beta =0$ in the creeping-flow limit. Panel (a) shows the dispersion curves predicted by both methods for the whole range of $k$. Panel (b) illustrates the excellent agreement between two methods for $k<0.1$. (a) Complete curves and (b) magnified low $k$ region.

It must be noted that, here, we have assumed a planar flow geometry due to the small thickness of the saliva and mucus layers which removes the curvature effects through the air–liquid interfacial tension. From the analysis of Zhou et al. (Reference Zhou, Peng, Zhang and Zhuge2014), the surface tension plays a destabilising role on the long-wave ($k<1$) instabilities for a viscoelastic film coating a tube and flowing under the action of gravity and imposed shear at the free surface. Thus, given the presence of the long-wave instabilities in the present study if we consider a flow coating a deformable tube and sheared by the airflow then the surface tension may result in the growth rate enhancement of the liquid elastic mode.

4.1.3. Non-zero $Re$

The effect of an increasing inertia, i.e. $Re$, on the growth rate of the perturbations is shown in figure 7. For the NR case, Smith & Davis (Reference Smith and Davis1982) predicted a destabilising effect of increasing inertia on the Newtonian mode. However, from figure 7, an increasing inertia has a stabilising effect on the liquid elastic mode. The stabilising effect of increasing inertia further affirms the purely elastic nature of the liquid elastic mode.

Figure 7.

The effect of variation in $Re$ on the growth rate of the perturbations for the liquid elastic mode at $n=1$, $\beta =0$, $Ma=0$, $T=1$ and $W=0.1$. The figure shows the stabilising effect of an increasing inertia on the elastic mode.

4.2.1. Creeping flow

For the ND case, there are three eigenvalues in the creeping-flow limit, as shown in table 4. The unstable mode is a downstream travelling mode while the remaining two stable modes are upstream travelling. As per the stability analysis of Smith & Davis (Reference Smith and Davis1982), the NR case is linearly stable in the creeping-flow limit. Thus, the instability predicted here is a pure consequence of the elasticity of the deformable layer. Henceforth the mode of instability introduced by the elastic nature of the solid will be referred to as the ‘solid elastic mode.’

Table 4.

Typical eigenspectrum in the creeping-flow limit at $k=0.8$, $H=10$, $\varGamma =0.5$ and $T=1$ for the ND case by using the analytical solution. The other two modes always remain stable. The unstable mode predicted by the pseudo-spectral method is $0.607433 + 0.052304i$, which shows an excellent agreement with the eigenvalue predicted by the analytical solution thereby validating the former.

To further understand the role of the deformable layer in introducing the solid elastic mode, dispersion curves are plotted for select values of $\varGamma$ and $T$ in figure 8. A plane Couette flow of a Newtonian liquid past a deformable layer in the creeping-flow limit exhibits a finite-wave instability referred to as the ‘viscous instability’ in the literature (Kumaran et al. Reference Kumaran, Fredrickson and Pincus1994; Kumaran & Muralikrishnan Reference Kumaran and Muralikrishnan2000; Gkanis & Kumar Reference Gkanis and Kumar2003; Patne & Shankar Reference Patne and Shankar2017; Patne et al. Reference Patne, Giribabu and Shankar2017; Joshi & Shankar Reference Joshi and Shankar2019). For the existence of the viscous instability, a minimum value of $\varGamma$ is necessary, thus a finite critical $\varGamma$, i.e. $\varGamma _c$ exists. However, the dispersion curves of figure 8(a) illustrate that, for an arbitrary $\varGamma$, the flow is unstable, thereby ruling out the existence of $\varGamma _c$. Therefore, a Newtonian liquid sheared past a neo-Hookean solid is unconditionally unstable. The effect of the air–liquid interfacial tension $T$ on the solid elastic mode is shown in figure 8(b). An increasing $T$ has a stabilising effect on the high-wavenumber part of the solid elastic mode but fails to stabilise the low-wavenumber part of the solid elastic mode. From figure 8(b), the growth rate increases with an increasing $H$. However, for $H>10$ the growth rate is relatively insensitive to variation in $H$.

Figure 8.

Variation in the growth rate $\omega _i$ with the disturbance wavenumber $k$ in the creeping-flow limit for the ND case. Panel (a) shows the dispersion plots for $Re=0$, $Ma=0$ and $T=0$. The plots show that, similar to the viscoelastic liquid layer sheared past a rigid solid, the Newtonian liquid layer sheared past a deformable layer is unconditionally unstable since a minimum $\varGamma$ is not needed for the instability to exist. The parameter $\varGamma$ only affects the growth rate of the disturbances. Panel (b) shows the effect of the air–liquid interfacial tension on the dispersion curves for $\varGamma =0.5$. An increasing surface tension suppresses the instability at moderate and high wavenumbers but fails to stabilise at low wavenumbers, thereby showing that the flow remains unconditionally unstable. Panel (c) shows the effect of variation in the dimensionless thickness of the deformable solid on the growth rate of the solid elastic mode; (a) $H=10$ and $T=0$, (b) $H=10$ and $\varGamma =0.5$ and (c) $T=1$ and $\varGamma =0.5$.

The mechanism of the solid elastic mode can be understood as follows. Kumaran et al. (Reference Kumaran, Fredrickson and Pincus1994), Kumaran & Muralikrishnan (Reference Kumaran and Muralikrishnan2000) and Gkanis & Kumar (Reference Gkanis and Kumar2003) showed that the viscous instability is triggered due to the shear work done by the flowing liquid on the deformable layer. This shear work is transferred via the tangential velocity continuity interface condition (2.12g) at the liquid–deformable-solid interface at $y=0$. The particular term responsible for the transfer of the shear work in (2.12g) is $(\tilde {u}_y)$, which in basic form is $(D\bar v_x \tilde {u}_y)$. However, in the present study for the ND case, another term $(-2\textrm {i}k\tilde h)$ at $y=1$ in the normal stress continuity boundary condition (2.12e) is equally important, as shown in table 5. In basic form, the term $(-2\textrm {i}k\tilde h)$ can be written as $(-2\textrm {i}k \bar \tau _{xy} \tilde h)$. But at $y=1$, in the base state, the shear stress in the liquid is equal to the shear stress exerted by the air thus, in dimensionless terms $\bar \tau _{xy} = \tau _a R/(\mu V_m)$, modifying the term to $(-2\textrm {i}k \tau _a R/(\mu V_m) \tilde h)$. Therefore, the energy for the solid elastic mode is provided by the shear stress exerted by the air on the liquid. To conclude, the energy for the destabilisation of the solid elastic mode is provided by the shear stresses exerted by the air at the air–liquid interface and by the liquid at the liquid–deformable-solid interface.

Table 5.

The most unstable eigenvalue for the ND case with/without $(-2\textrm {i}k\tilde h)$ term in the normal stress boundary condition (2.12e) at $H=10$, $Ma=0$ and $T=1$. From the second column of the table the solid elastic mode is stable in the absence of $(-2\textrm {i}k\tilde h)$, thereby showing the importance of the same in triggering the solid elastic mode. The shear stress exerted at the air–liquid interface is responsible for the term $(-2\textrm {i}k\tilde h)$. A similar vital role is also played by the term $(\tilde {u}_y)$ in the tangential velocity continuity condition (2.12g) at the liquid–deformable-solid interface ($y=0$). Therefore, the solid elastic mode is combined effect of the shear stresses exerted by the air at the air–liquid interface and by the liquid at the liquid–deformable-solid interface.

The plots of the normalised velocity perturbations $v'_x$ and $v'_y$ in the liquid and displacement perturbations $u'_x$ and $u'_y$ in the deformable solid are shown in figure 9. Since the destabilisation mechanism of the solid elastic mode is the shear stresses exerted by the air at the air–liquid interface and by the liquid at the liquid–deformable-solid interface, the velocity perturbations exhibit maximum variation near both the air–liquid and liquid–deformable-solid interfaces. However, the displacement perturbations exhibit highest variation near the liquid–deformable-solid interface due to the shear stresses exerted by the liquid on the deformable solid while they vanish at $y=-H$, which corresponds to $y=0$ in figures 9(c) and 9(d), as required by the boundary condition (2.12j).

Figure 9.

The normalised perturbations for the liquid and deformable-solid layer for the ND case at $\varGamma =0.5$, $T=1$, $k=0.8$, $H=10$, $Ma=0$ and $Re=0$ for the solid elastic mode $\omega =0.433075 + 0.023960$. Here, $u'_x=Re[ \tilde {u}_x \,\textrm {e}^{\textrm {i}kx} ]$ and $u'_y=Re[ \tilde {u}'_y \,\textrm {e}^{\textrm {i}kx} ]$. For convenience, the axes have been normalised to the interval $[0,1]$. The length of the domain in the $x$-direction is equal to a wavelength ($2{\rm \pi} /k$) of the perturbations. The velocity perturbations (panels a and b) show maximum variation both near the air–liquid ($y=1$) and liquid–deformable-solid ($y=0$) interface, thereby indicating the destabilisation caused by the terms $(-2\textrm {i}k\tilde h)$ and $(\tilde {u}_y)$. For the solid displacement perturbations (panels c and d), the perturbations exhibit maximum at the liquid–deformable-solid ($y=1$) interface and vanish at the rigid-solid–deformable-solid ($y=0$) interface; (a) $v'_x$, (b) $v'_y$, (c) $u'_x$ and (d) $u'_y$.

It must be noted that, similar to the first normal stress difference exhibited by the viscoelastic fluid at the air–liquid interface, the neo-Hookean solid also exhibits a first normal stress difference across the liquid–deformable-solid interface. As discussed in § 4.1, the first normal stress difference exhibited by the viscoelastic fluid across the air–liquid interface gives rise to the liquid elastic mode. Similarly, Gkanis & Kumar (Reference Gkanis and Kumar2003) predicted that the first normal stress difference exhibited by the neo-Hookean solid also exhibits a new mode of instability, termed the ‘short-wave instability’, for $\varGamma >1$. However, for the processes under consideration here, $\varGamma <1$, thereby ruling out the presence of the short-wave instability and thus the role of the first normal stress difference exhibited by the neo-Hookean solid at the liquid–deformable-solid interface.

4.2.2. Non-zero $Re$

The effect of the inertia on the solid elastic mode is shown in figure 10. Similar to the liquid elastic mode, increasing inertia has a stabilising effect on the solid elastic mode, thereby confirming its purely elastic nature. The viscous mode of instability predicted by Kumaran et al. (Reference Kumaran, Fredrickson and Pincus1994), Kumaran & Muralikrishnan (Reference Kumaran and Muralikrishnan2000) and Chokshi & Kumaran (Reference Chokshi and Kumaran2007) for the plane Couette flow past a deformable solid in the creeping-flow limit can be destabilised by increasing inertia. For $Re>10$, the viscous mode of instability exhibits a characteristic scaling $\varGamma _c \sim Re^{-1}$ (Chokshi & Kumaran Reference Chokshi and Kumaran2008a; Giribabu & Shankar Reference Giribabu and Shankar2017; Tanmay et al. Reference Tanmay, Patne and Shankar2018). However, from figure 10, the inertia has a stabilising effect on the solid elastic mode, which further sets apart the viscous mode of instability from the solid elastic mode predicted here. The stabilising effect of the increasing inertia also implies the purely elastic nature of the solid elastic mode similar to the liquid elastic mode.

Figure 10.

The effect of variation in $Re$ on the growth rate of the solid elastic mode at $H=10$, $T=1$, $Ma=0,$ and $\varGamma =0.5$. The figure shows the stabilising effect of an increasing inertia on the solid elastic mode.

4.3.1. Creeping flow

From §§ 4.1 and 4.2, the liquid and solid elastic modes are unconditionally unstable. For the VD case, the elastic characteristic of the solid and liquid can either lead to two different unstable modes representing the elasticity of the liquid and solid, or the two modes could reinforce each other, i.e. undergo a resonance to result in a more destructive mode of instability. Figure 11 shows that, for the VD case, the resonance pathway is preferred, thereby leading to a single mode resulting from the resonance of the liquid and solid elastic modes thus the name ‘resonance mode.’ For the resonance mode, both the elastic modes reinforce each other, which leads to a higher growth rate. The resonance between both modes for the whole range of $k$ and select values of the other parameters is shown in figure 12.

Figure 11.

The solid elastic mode ($\varGamma =0.2,W=0$), liquid elastic mode ($\varGamma =0,W=0.2$) and resonance mode ($\varGamma =0.2,W=0.2$) at $n=1$, $\beta =0$, $Re=0$, $k=0.5$, $Ma=0$, $T=1$ and $H=10$ obtained using the pseudo-spectral method. Here, the resonance mode refers to the unstable mode for the VD case. The spectrum shows that the resonance mode exhibits a higher growth rate compared with the solid and liquid elastic modes as a result of the resonance between the solid and liquid elastic modes, thus the name resonance mode. To obtain the liquid elastic mode, $\varGamma =0$ and for the solid elastic mode $W=0$.

Figure 12.

Variation in the growth rate $\omega _i$ with the disturbance wavenumber $k$ in the creeping-flow limit for the VD case at $Ma=0$. The figure illustrates an increase in growth rate due to the resonance between the liquid and solid elastic modes. Panel (a) illustrates the growth rate increase as $W$ increases, thereby showing the proof of the resonance between the liquid and solid elastic modes over whole $k$. A similar effect is also exhibited if $W$ is fixed and $\varGamma$ is increased. The stabilising effect of the air–liquid interfacial tension on the resonance mode is illustrated in panel (b). The destabilising effect of increasing $H$ on the resonance mode is shown in panel (c). Panel (d) illustrates destabilising effect of the shear thinning on the resonance mode. The stabilising effect of increasing $\beta$ is shown in panel (e). The figure illustrates that the resonance mode exhibits features of both the liquid and solid elastic modes, thereby affirming its origin: (a) $H=10$, $\varGamma =0.2$, $\beta =0$, $n=1$ and $T=1$; (b) $H=10$, $W=0.2$, $\beta =0$, $n=1$ and $\varGamma =0.2$; (c) $\varGamma =0.2$, $\beta =0$, $n=1$, $W=0.2$ and $T=1$; (d) $T=1$, $W=0.2$, $\beta =0$, $H=10$ and $\varGamma =0.2$; (e) $T=1$, $W=0.2$, $H=10$, $n=1$ and $\varGamma =0.2$.

The resonance mode has combined characteristics of both the modes, thus it is unconditionally unstable. Additionally, the growth rate of the resonance mode will increase with increasing $\varGamma ,W$ and $H$ or decreasing $n$ while its growth rate will decrease with increasing $T$ and $\beta$, as shown in figure 12. As $\beta \rightarrow 1$, the dispersion curve approaches the corresponding curve for the purely solid elastic mode explored in § 4.2. The physical mechanism responsible for the resonance mode is a combination of the mechanisms responsible for the destabilisation of the liquid and solid elastic modes. Thus, the first normal stress difference exhibited by the viscoelastic liquid across the air–liquid interface and the shear stresses exerted by the air at the air–liquid interface and by the liquid on the deformable solid at the liquid–deformable-solid interface are responsible for the existence of the resonance mode.

In addition to the first normal stress difference across the air–liquid interface, there is also the first normal stress difference across the liquid–deformable-solid interface due to the viscoelastic nature of the liquid. From § 4.1, the first normal stress difference across the air–liquid interface leads to the liquid elastic mode. However, the first normal stress difference across the liquid–deformable-solid interface due to the viscoelastic fluid does not introduce a new mode of instability, instead, it only affects the growth rate of the unstable modes and thus does not play a major role in determining the stability of the flow.

Please note that in the present study we have neglected the existence of the PCL, which may lead to some modifications in the predictions obtained in the present study regarding the solid elastic mode due to its proximity to the deformable muscle layers. However, it will not affect the liquid elastic mode which originates at the air–liquid interface. Additionally, the solid elastic mode exists even for a Newtonian fluid layer subjected to airflow induced shear, thus the solid elastic mode will exist even in the presence of the PCL. Thus, the PCL layer can affect the growth rate predicted for the solid elastic and resonance modes but may not affect the liquid elastic mode.

4.3.2. Non-zero $Re$

Since the resonance mode is a result of the resonance between the liquid and solid elastic modes, the inertia is expected to lead to the stabilisation. The effect of increasing inertia on the resonance mode as illustrated in figure 13 is indeed stabilising, which further affirms its purely elastic nature.