3.1. Scaling and identification of the dominant effects

Assume the working fluid is incompressible and Newtonian, with a density of $\rho _f$ and dynamic viscosity of $\mu$. With the displacement of the fluid–solid interface denoted as $u_y(x,z,t)$, the deformed channel height can be written as $h(x,z,t) = h_0 +u_y(x,z,t)$. Then, the deformed configuration of the fluid domain is $\{ (x,y,z) | -w/2\leq x \leq +w/2,\ 0\leq y\leq h(x,z,t),\ 0\leq z\leq \ell \}$. Further, we assume that $h(x,z,t)\ll w\ll \ell$, i.e. the slenderness and shallowness assumptions on the conduit hold true even after its deformation. The former assumption is important because it allows us to use $h_0$ as the scale for $y$.

Under these assumptions, the governing equations are the unsteady incompressible Navier–Stokes equations, which take the form

(3.1a)\begin{gather} \underbrace{\frac{\partial v_x}{\partial x}}_{O(1)} + \underbrace{\frac{\partial v_y}{\partial y}}_{O(1)} + \underbrace{\frac{\partial v_z}{\partial z}}_{O(1)} = 0,\end{gather}

(3.1b)\begin{gather}\underbrace{\frac{\partial v_x}{\partial t}}_{O (\epsilon^{2}\delta^{{-}2}\widehat{Re})} \!+\! \underbrace{v_x\frac{\partial v_x}{\partial x}}_{O (\epsilon^{2}\delta^{{-}2}\widehat{Re})} \!+\! \underbrace{v_y\frac{\partial v_x}{\partial y}}_{O (\epsilon^{2}\delta^{{-}2}\widehat{Re})} \!+\! \underbrace{v_z\frac{\partial v_x}{\partial z}}_{O (\epsilon^{2}\delta^{{-}2}\widehat{Re})} ={-}\underbrace{\frac{1}{\rho_f}\frac{\partial p}{\partial x}}_{O(1)} \!+\! \underbrace{\frac{\mu}{\rho_f} \frac{\partial^{2} v_x}{\partial x^{2}}}_{O(\epsilon^{2})} + \underbrace{\frac{\mu}{\rho_f}\frac{\partial^{2} v_x}{\partial y^{2}}}_{O(\epsilon^{2}\delta^{{-}2})} + \underbrace{\frac{\mu}{\rho_f}\frac{\partial^{2} v_x}{\partial z^{2}}}_{O(\epsilon^{4}\delta^{{-}2})}, \end{gather}

(3.1c)\begin{gather}\underbrace{\frac{\partial v_y}{\partial t}}_{O (\epsilon^{2}\widehat{Re})} + \underbrace{v_x\frac{\partial v_y}{\partial x}}_{O(\epsilon^{2}\widehat{Re})} + \underbrace{v_y\frac{\partial v_y}{\partial y}}_{O (\epsilon^{2}\widehat{Re})} + \underbrace{v_z\frac{\partial v_y}{\partial z}}_{O(\epsilon^{2}\widehat{Re})} ={-}\underbrace{\frac{1}{\rho_f}\frac{\partial p}{\partial y}}_{O(1)} + \underbrace{\frac{\mu}{\rho_f} \frac{\partial^{2} v_y}{\partial x^{2}}}_{O(\epsilon^{2} \delta^{2})} + \underbrace{\frac{\mu}{\rho_f}\frac{\partial^{2} v_y}{\partial y^{2}}}_{O(\epsilon^{2})} + \underbrace{\frac{\mu}{\rho_f}\frac{\partial^{2} v_y}{\partial z^{2}}}_{O(\epsilon^{4})}, \end{gather}

(3.1d)\begin{gather}\underbrace{\frac{\partial v_z}{\partial t}}_{O (\widehat{Re})} + \underbrace{v_x\frac{\partial v_z}{\partial x}}_{O(\widehat{Re})} + \underbrace{v_y \frac{\partial v_z}{\partial y}}_{O(\widehat{Re})} + \underbrace{v_z\frac{\partial v_z}{\partial z}}_{O (\widehat{Re})} ={-}\underbrace{\frac{1}{\rho_f} \frac{\partial p}{\partial z}}_{O(1)} + \underbrace{\frac{\mu}{\rho_f}\frac{\partial^{2} v_z}{\partial x^{2}}}_{O(\delta^{2})} + \underbrace{\frac{\mu}{\rho_f} \frac{\partial^{2} v_z}{\partial y^{2}}}_{O(1)} + \underbrace{\frac{\mu}{\rho_f}\frac{\partial^{2} v_z}{\partial z^{2}}}_{O(\epsilon^{2})}, \end{gather}

with the order of magnitude of each term listed underneath, based on the scales from table 2.

Table 2. The scales for the variables in the incompressible Navier–Stokes equations (3.1).

In table 2, $\mathcal {V}_c$ is the characteristic velocity scale. Specifically, to ensure the conservation of mass of (3.1a), $\epsilon \mathcal {V}_c/\delta$, $\epsilon \mathcal {V}_c$ and $\mathcal {V}_c$ are chosen to be the characteristic scales for the velocity components $v_x$, $v_y$ and $v_z$, respectively. Also, as is standard for low-Reynolds-number flow, to achieve a balance between the pressure and the viscous stresses in (3.1d), the characteristic pressure scale, $\mathcal {P}_c$, and $\mathcal {V}_c$ are related by $\mathcal {P}_c=\mu \mathcal {V}_c \ell /h_0^{2}$. If the volumetric flow rate, $q$, at the inlet is fixed, we can choose $\mathcal {V}_c = q/(wh_0)$, then $\mathcal {P}_c = \mu q \ell /(wh_0^{3})$. However, if the pressure drop, ${\rm \Delta} p = p|_{z=0}- p|_{z=\ell }$, is prescribed, $\mathcal {P}_c = {\rm \Delta} p$ and, accordingly, $\mathcal {V}_c = {\rm \Delta} p h_0^{2}/(\mu \ell )$. The Reynolds number is defined as $Re=\rho _f\mathcal {V}_c h_0/{\mu }$. However, owing to the shallowness and slenderness of the fluid domain ($\epsilon \ll \delta \ll 1$), the inertial effects in the $z$-momentum equation (3.1d) are more dominant than in the other two momentum equations. In this scaling, $\widehat {Re}=\epsilon Re$ emerges as the sole prefactor of the inertial terms in (3.1d) (rather than $Re$), thus we say that the reduced Reynolds number $\widehat {Re}$ is more suitable for quantifying the inertia of this flow. Finally, $\mathcal {T}_f$ is taken to be the characteristic time scale for axial advection (the dominant flow direction): $\mathcal {T}_f=\ell /\mathcal {V}_c$.

3.2. Reduction: lubrication approximation

Recall that we are interested in flow in a shallow and slender microchannel ($h_0\ll w\ll \ell$) such that $\epsilon =h_0/\ell \ll \delta =h_0/w \ll 1$. Based on the discussion above, it is clear that the dominant balance of terms occurs in the $z$-momentum equation (3.1d). Only the pressure terms are left in (3.1b) and (3.1c), indicating that, at the leading order in $\epsilon$ and $\delta$, the hydrodynamic pressure $p$ is only a function of the streamwise location $z$, as in the classic lubrication approximation (White Reference White2006; Panton Reference Panton2013). More importantly, this argument is true even at finite Reynolds number, i.e. $\widehat {Re}= O(1)$, which is typical of the microfluidic experiments of Verma & Kumaran (Reference Verma and Kumaran2013) that we compare with. Specifically, with $\widehat {Re}= O(1)$, the dominant balance in the flow-wise momentum equation (3.1d) occurs between fluid inertia, the pressure gradient and viscous forces. The same balance was employed by Inamdar, Wang & Christov (Reference Inamdar, Wang and Christov2020) to derive a 1-D FSI model from the 2-D Navies–Stokes equations (but under different assumptions on the solid mechanics problem).

Examining further the right-hand side of (3.1d), the balance of forces at the leading order indicates that $\partial p/\partial z \sim \partial \tau _{yz}/\partial y$ because the shear stress is $\tau _{yz} \sim \mu \partial v_z/\partial y$. Introducing $\mathcal {S}_c$ as the characteristic scale for $\tau _{yz}$ and substituting the other scales from table 2, this balance suggests that $\mathcal {P}_c/\ell = \mathcal {S}_c/h_0$, leading to $\mathcal {S}_c=(h_0/\ell) \mathcal {P}_c=\epsilon \mathcal {P}_c$. For $\epsilon \ll 1$, we conclude that $\tau _{yz}\ll p$. Hence, at the leading order in $\epsilon$ and $\delta$, $p(z)$ is the only flow-induced load exerted on the fluid–solid interface.

4.1. Scaling and identification of the dominant effects

For the solid mechanics problem, it is more convenient to use the $\hat {o}_{\hat {x} \hat {y} \hat {z}}$ coordinate system, where we denote the displacement of the fluid–solid interface by $u_{\hat {y}}$, as shown in figure 1(a). We consider the case in which the maximum of $\hat {u}_y$ is small compared with the smallest dimension of the solid, so that the small-strain theory of linear elasticity is applicable. Specifically, if the wall is ‘thick’, meaning $w \lesssim d \ll \ell$, we require that $\hat {u}_y \ll w$. However, if the wall is ‘thin’, meaning $d\lesssim w \ll \ell$, we require that $\hat {u}_y \ll d$ (Wang & Christov Reference Wang and Christov2021).

The following discussion proceeds along the lines of Wang & Christov (Reference Wang and Christov2019). However, here, we provide a more general derivation for the reader's convenience. First, using the scales from table 3, the balance between the Cauchy stresses and the solid inertia within the wall, neglecting any body forces, is

(4.1a)\begin{gather} \underbrace{\rho_s\frac{\partial^{2} u_{\hat{x}}}{\partial t^{2}}}_{O(\rho_s \mathcal{U}_{c,x}/\mathcal{T}_f^{2}) } + \underbrace{\frac{\partial \sigma_{\hat{x} \hat{x}}}{\partial \hat{x}}}_{O({\mathcal{D}_{\hat{x} \hat{x}}}/{w})} + \underbrace{\frac{\partial \sigma_{\hat{x} \hat{y}}}{\partial \hat{y}}}_{O({\mathcal{D}_{\hat{x} \hat{y}}}/{d})} + \underbrace{\frac{\partial \sigma_{\hat{x} \hat{z}}}{\partial \hat{z}}}_{O({\mathcal{D}_{\hat{x} \hat{z}}}/{\ell})} = 0, \end{gather}

(4.1b)\begin{gather}\underbrace{\rho_s\frac{\partial^{2} u_{\hat{y}}}{\partial t^{2}}}_{O(\rho_s\mathcal{U}_c/\mathcal{T}_f^{2})} + \underbrace{\frac{\partial \sigma_{\hat{x} \hat{y}}}{\partial \hat{x}}}_{O({\mathcal{D}_{\hat{x} \hat{y}}}/{w})} + \underbrace{\frac{\partial \sigma_{\hat{y} \hat{y}}}{\partial \hat{y}}}_{O(\mathcal{P}_c/{d})} + \underbrace{\frac{\partial \sigma_{\hat{y} \hat{z}}}{\partial \hat{z}}}_{O(\epsilon\mathcal{P}_c/\ell)} = 0, \end{gather}

(4.1c)\begin{gather}\underbrace{\rho_s\frac{\partial^{2} u_{\hat{z}}}{\partial t^{2}}}_{O(\rho_s \mathcal{U}_{c,z}/\mathcal{T}_f^{2})} + \underbrace{\frac{\partial \sigma_{\hat{x} \hat{z}}}{\partial \hat{x}}}_{O({\mathcal{D}_{\hat{x} \hat{z}}}/{w})} + \underbrace{\frac{\partial \sigma_{\hat{y} \hat{z}}}{\partial \hat{y}}}_{O(\epsilon\mathcal{P}_c/{d})} + \underbrace{\frac{\partial \sigma_{\hat{z} \hat{z}}}{\partial \hat{z}}}_{O({\mathcal{D}_{\hat{z} \hat{z}}}/{\ell})} = 0. \end{gather}

Here, $\sigma _{\hat {x} \hat {x}}$, $\sigma _{\hat {x} \hat {y}}$, $\sigma _{\hat {x} \hat {z}}$, $\sigma _{\hat {y} \hat {y}}$, $\sigma _{\hat {y} \hat {z}}$ and $\sigma _{\hat {z} \hat {z}}$ are the six independent components of the Cauchy stress in the solid. The order of magnitude of each term is listed underneath, based on the scales from table 3.

Table 3. The scales for the variables in the linear elastodynamics equations (4.1).

In table 3, $\mathcal {U}_{c,x}$, $\mathcal {U}_c$ and $\mathcal {U}_{c,z}$ are the characteristic scales for $u_{\hat {x}}$, $u_{\hat {y}}$ and $u_{\hat {z}}$, respectively. We immediately assume that $\mathcal {U}_{c,x}\ll \mathcal {U}_c$ and $\mathcal {U}_{c,z}\ll \mathcal {U}_c$, meaning that the wall is primarily bulging upwards, as in experiments. This assumption has previously been quantitatively validated against experiments (Christov et al. Reference Christov, Cognet, Shidhore and Stone2018; Shidhore & Christov Reference Shidhore and Christov2018; Wang & Christov Reference Wang and Christov2019). Then the most prominent inertial term in the solid is in (4.1b). Note that the time scale for (4.1) is still the fluid's axial advection time scale, $\mathcal {T}_f$, in order to ensure the coupling between the solid and the fluid mechanics problems. Note that this choice of time scale is different from the so-called ‘viscous–elastic’ one used in related works (Elbaz & Gat Reference Elbaz and Gat2014; Martínez-Calvo et al. Reference Martínez-Calvo, Sevilla, Peng and Stone2020). In the latter papers, the characteristic (common) time scale $\mathcal {T}_c$ was chosen based on the kinematic boundary condition at the fluid–solid interface, i.e. $\partial \bar {u}_y/\partial t = v_y$, leading to a fluid time scale of $\mathcal {T}_c=\bar {\mathcal {U}}_c/(\epsilon \mathcal {V}_c) = (\bar {\mathcal {U}}_c/h_0)(\ell /\mathcal {V}_c)= \beta \mathcal {T}_f$. However, since $\beta=\bar{\mathcal{U}}_c/h_0$ is typically at $O(1)$ in our work (discussed in § 4.4), these two different choices of the fluid time scale, $\mathcal {T}_c$ and $\mathcal {T}_f$, do not differ significantly. To further elucidate the time scales involved, the magnitude of the inertial term in (4.1b) can be written as $\rho _s\mathcal {U}_c/\mathcal {T}_f^{2}=\rho _s\mathcal {U}_c/\mathcal {T}_s^{2} \times (\mathcal {T}_s/\mathcal {T}_f)^{2}$, where we have explicitly introduced the solid time scale, $\mathcal {T}_s$. Note that when the soft solid's density is similar to the fluid's density (see example values in table 4 below), the elastic wave speed in the solid is $\sqrt {E/\rho _s}\gg \mathcal {V}_c$. Therefore, the development of the solid deformation is expected to be much faster than the flow, i.e. $\mathcal {T}_s \ll \mathcal {T}_f$, which makes the solid's inertia a weak effect. We will quantify the solid's weak inertia in § 4.4, but for now it can be neglected to find the deformation profile (bulging of the fluid–solid interface due to the hydrodynamic pressure).

Table 4. The dimensional and dimensionless parameters of the 1-D FSI model.

To this end, let us consider the balance of the Cauchy stresses in (4.1). Due to the traction balance at the fluid–solid interface, it can be inferred that $\mathcal {D}_{\hat {y}\hat {y}} = \mathcal {P}_c$ and $\mathcal {D}_{\hat {y} \hat {z}} = \epsilon \mathcal {P}_c$, as tabulated in table 3. For convenience, we introduce $\gamma = d/w$ as the spanwise aspect ratio of the solid wall. Then, a balance in (4.1b) can only occur between the second and the third terms, yielding $D_{\hat {x} \hat {y}} = \mathcal {P}_c/\gamma$. At the same time, the balance of the three terms in (4.1c) gives $\mathcal {D}_{\hat {x} \hat {z}} = \epsilon \mathcal {P}_c w/d = \epsilon \mathcal {P}_c/\gamma$ and $\mathcal {D}_{\hat {z} \hat {z}} = \epsilon \mathcal {P}_c\ell /d = \delta \mathcal {P}_c/\gamma$. Finally, from (4.1a), the only remaining possibility is that the second term balances the third term, indicating $\mathcal {D}_{\hat {x} \hat {x}}=w\mathcal {D}_{\hat {x} \hat {y}}/d = \mathcal {P}_c/\gamma ^{2}$.

So far, we have only required that the elastic solid is slender, i.e. $d\ll \ell$, which is equivalent to $\gamma \ll \delta /\epsilon$, and covers a large range of wall thicknesses (recall that $\gamma =d/w$, $\epsilon =h_0/\ell$ and $\delta =h_0/w$). However, it is also expected that $\gamma \gg \epsilon$, such that $\mathcal {D}_{\hat {x} \hat {z}}$ is a small quantity, excluding the case of an extremely thin wall. In fact, recalling that the application of linear elasticity requires that $\hat {u}_y\ll d$, any prominent deformation in a thin-walled microchannel is likely beyond the scope of the linear elastic theory.

Therefore, with $\epsilon \ll \gamma \ll \delta /\epsilon$ as well as $\epsilon \ll \delta \ll 1$, we conclude that $\sigma _{\hat {x} \hat {z}}$ and $\sigma _{\hat {y} \hat {z}}$ are negligible in comparison with the other stress components. Depending on the wall thickness, the relative magnitude among the remaining four stress components can change. For example, if $\gamma ^{2} \gg 1$, we can further neglect $\sigma _{\hat {x} \hat {x}}$ as in Wang & Christov (Reference Wang and Christov2019). Nevertheless, no matter how $d$ varies, the dominant balances in (4.1a) and (4.1b) occur in the cross-sectional $(\hat {x},\hat {y})$ plane, which reduces the original 3-D elasticity problem to a 2-D plane-strain problem. Since we showed in § 3 that $p$ is a function of $z$ only at the leading order (in $\epsilon$), the deformation of the $(\hat {x},\hat {y})$ cross-sections at different $z$-locations (recalling $z=\hat {z}$) decouple from each other. At each cross-section, the deformation is then determined by the local hydrodynamic pressure $p(z,t)$. Therefore, generally, we can express the displacement of the fluid–solid interface at the leading order (in $\epsilon$) as

(4.2)\begin{equation} u_y(x,z,t) = u_{\hat{y}}(\hat{x}, \hat{z}, t)= \mathfrak{f}(x)p(z,t), \end{equation}

with $\mathfrak {f}(x)$ being the spanwise deformation profile. The separation-of-variables form of (4.2) suggests that the cross-sectional deformation profiles at different $z$-locations are, in a sense, self-similar. The displacement is fully determined by the local pressure, showing that the fluid–solid interface behaves like a Winkler foundation (Winkler Reference Winkler1867; Dillard et al. Reference Dillard, Mukherjee, Karnal, Batra and Frechette2018), with a variable stiffness represented by $1/\mathfrak {f}(x)$. Importantly, this Winkler-foundation-like mechanism is not an assumption here, but rather it is a consequence of the slenderness of the top wall. Also, note that the assumption of $\mathcal {T}_s\ll \mathcal {T}_f$ has been applied here, meaning that the solid promptly responds to pressure changes in the flow.

It is also worth mentioning that if the top wall is thin with $\epsilon \ll \gamma \lesssim 1$, the elasticity problem is usually taken to be a plane stress problem, and a 1-D engineering model is usually available for the displacement out of plane (i.e. $u_y$ here), such as the Kirchhoff–Love (Love Reference Love1888; Timoshenko & Woinowsky-Krieger Reference Timoshenko and Woinowsky-Krieger1959) and Reissner–Mindlin (Reissner Reference Reissner1945; Mindlin Reference Mindlin1951) plate theories. However, this fact does not fundamentally contradict with our plane-strain reduction because the decoupling of the cross-sections remains true (Christov et al. Reference Christov, Cognet, Shidhore and Stone2018; Shidhore & Christov Reference Shidhore and Christov2018; Anand, Muchandimath & Christov Reference Anand, Muchandimath and Christov2020) due to the separation of scales, $w\ll \ell$.

Moreover, the discussion above is only based on the balance of Cauchy stresses, and does not involve the boundary conditions either on the sides (i.e. at $x=\pm w/2$) or at the upper surface of the wall (i.e. at $\hat {y}=d$ or $y=h_0 +d$). The decoupling of the cross-sectional deformation is just a consequence of the wall slenderness. However, the boundary conditions do have an important influence on the displacement field in the solid, which gives rise to different forms of $\mathfrak {f}(x)$ in (4.2) (Wang & Christov Reference Wang and Christov2021).

4.2. Reduction: introducing the width-averaged (effective) height

Since we seek a 1-D model dependent on $z$ only, the $x$-dependence can be eliminated by averaging the displacement of the fluid–solid interface over $x$:

(4.3)\begin{equation} \bar{u}_y(z,t) = \frac{1}{w}\int_{{-}w/2}^{{+}w/2} u_y(x,z,t)\, \mathrm{d}\kern0.06em x = \underbrace{\left[ \frac{1}{w}\int_{{-}w/2}^{{+}w/2} \mathfrak{f}(x)\, \mathrm{d}\kern0.06em x\right]}_{1/k} p(z,t), \end{equation}

having used (4.2). Then, the width-averaged height $\bar {h}$ of the channel is

(4.4)\begin{equation} \bar{h}(z,t) = \frac{1}{w}\int_{{-}w/2}^{{+}w/2} h_0 + u_y(x,z,t)\, \mathrm{d}\kern0.06em x = h_0 + \frac{1}{k}p(z,t). \end{equation}

An important quantity in the above equations is the proportionality constant, $k$, which represents the effective stiffness of the interface and further highlights the Winkler-foundation-like mechanism of deformation of the fluid–solid interface.

Equation (4.4) simplifies the FSI problem in two aspects. First, the deformation of the interface is further reduced from 2-D to 1-D. Second, the flow in the channel is reduced from 3-D to 2-D, and this reduction does not change any key statements made before because we can still appeal to the lubrication approximation for 2-D flows due to $h_0\ll \ell$. In other words, if we start from the 2-D incompressible Navier–Stokes equations (simply neglecting $v_x$ and $x$-dependent terms), we still find the lubrication approximation is applicable up to $\widehat {Re} = O(1)$, as in § 3.2.

In the following discussion, we will take $\bar {h}$ as the effective height of the deformed channel and consider the reduced system sketched in figure 1(b). Note that the 2-D configuration in figure 1(b) is not a priori assumed but, rather, it is derived from the 3-D configuration in figure 1(a) based on averaging the deformation across $x$, via (4.3) (or (4.4)). This approach is in contrast to the earlier work of, e.g. Skotheim & Mahadevan (Reference Skotheim and Mahadevan2004), who assumed a 2-D configuration of the elastic solid in the $(y,z)$ plane from the outset. Taking $\bar {h}$ as the effective deformed height was first suggested by Gervais et al. (Reference Gervais, El-Ali, Günther and Jensen2006). As shown by Wang & Christov (Reference Wang and Christov2021), the error introduced by this averaging approach is controllable.

4.3. Extension: introducing weak deformation effects

So far, the discussion in the previous subsections was based on the leading-order (in $\epsilon$) theory, which does not take into account the possible restrictions imposed at the inlet and the outlet (i.e. at $z=0$ and $z=\ell$). As shown in figure 1(b), the movement of the fluid–solid interface at both ends is often physically restricted. In this sense, we can think of the solid mechanics problem as being essentially a boundary layer problem. While the Winkler-foundation-like mechanism is dominant outside any ‘boundary layers’, with (4.3) being the (outer) solution there, another mechanism plays a role within thin (boundary) layers near $z=0,\ell$, each potentially admitting inner solutions that regularize the problem and allow the enforcement of end constraints.

Since the bulging of the top wall unavoidably introduces stretching along $z$ in the solid, a simple extension of (4.3) can be achieved by introducing weak constant tension into the formulation (Wang & Christov Reference Wang and Christov2021). First, for the Winkler-foundation-like mechanism to be dominant, the tension needs to be ‘weak’. Second, the tension force can be taken to be constant because its variation along $z$ is proportional to the fluid's tangential traction at the interface, which is a small quantity per the lubrication approximation, as shown in § 3 (see also Hewitt, Balmforth & De Bruyn Reference Hewitt, Balmforth and De Bruyn2015). The ‘regularized’ governing equation for the width-averaged fluid–solid interface's displacement $\bar {u}_y$ is then written as

(4.5) \begin{equation} \underbrace{\rho_s b^{{\star}} \frac{\partial^{2} \bar{u}_y}{\partial t^{2}}}_{\text{inertia}} + \underbrace{k\bar{u}_y}_{\text{stiffness}} - \underbrace{\chi_t\frac{\partial^{2} \bar{u}_y}{\partial z^{2}}}_{\text{tension}} = \underbrace{p(z,t)}_{\text{load}}, \end{equation}

where $\rho _s$ denotes the solid density, $b^{\star }$ represents the effective thickness of the interface (discussed in § A.1), which is introduced so that the first term can represent the bulk inertial effects of the solid. Recalling that $k$ is the effective stiffness introduced from (4.3), the second term represents the dominant Winkler-foundation-like effect, while $\chi _t$ is introduced to model the tension per unit width (discussed in § A.2). In the case of $\chi _t$ being constant, the tension can be written in terms of the transverse displacement $\bar {u}_y$ (see e.g. Howell, Kozyreff & Ockendon (Reference Howell, Kozyreff and Ockendon2009), § 4.3). The weakness of the solid inertia and the tension may not be obvious from (4.5), although the scaling analysis in § 4.1 anticipates it.

Equation (4.5) is essentially the equation of motion of a Kramer-type surface, which has been used extensively in the study of high-$Re$ (boundary layer) flows over compliant coatings. The goal of the latter studies is to understand how to delay the laminar–turbulence transition (Gad-el Hak Reference Gad-el Hak2002). However, microchannel flows cannot reach such high $Re$ values. More surprisingly, the application of (4.5) in modelling soft microchannels leads to different conclusions from the compliant coating studies. Compliance of the channel wall can actually promote (instead of delay) the laminar–turbulence transition in pressure-driven flow thanks to the FSI-induced instabilities. This effect can be successfully exploited for micromixing.

4.4. Non-dimensionalization: introducing the FSI parameter

We can now make the 1-D interface motion equation (4.5) dimensionless to better illustrate the relative magnitude of the different terms (effects). Capital letters are used to denote the dimensionless variables.

The first step is to determine the characteristic scale, $\bar {\mathcal {U}}_c$, for the fluid–solid interface displacement. The dominant deformation effect in (4.3), suggests that we should take $\bar {\mathcal {U}}_c = \mathcal {P}_c/k$, recalling that the scale for $p$ is $\mathcal {P}_c$. Then, the dimensionless version of (4.3) is simply

(4.6)\begin{equation} \bar{U}_Y(Z,T) = P(Z,T). \end{equation}

Still using $h_0$ to scale $\bar {h}$, the dimensionless effective channel height (see (4.4)) becomes

(4.7)\begin{equation} \bar{H}(Z,T) = 1 + \frac{\bar{\mathcal{U}}_c}{h_0}\bar{U}_Y(Z,T) = 1 + \beta \bar{U}_Y(Z,T). \end{equation}

Here, we have introduced another dimensionless parameter, $\beta =\bar {\mathcal {U}}_c/h_0=\mathcal {P}_c/(kh_0)$. It is clear from (4.7) that $\beta$ translates the interface displacement into the deformation of the fluid domain, capturing the ‘strength’ of the fluid–solid coupling. Thus, $\beta$ is the ‘FSI parameter’ of our model.

The dependence of $\beta$ on the system properties comes through $\mathcal {P}_c$ and $k$. While $\mathcal {P}_c$ is determined by the flow conditions (i.e. the viscosity of the fluid, the flow rate and the geometry of the undeformed channel), $k$ is determined by the material properties, the geometry and the boundary conditions on the compliant wall. To explicitly show this, we write

(4.8)\begin{equation} \frac{1}{k} = \frac{1}{w}\int_{{-}w/2}^{{+}w/2} \mathfrak{f}(x)\, \mathrm{d}\kern0.06em x= \int_{{-}1/2}^{{+}1/2} \mathfrak{f}(wX)\, \mathrm{d}\kern0.06em X = \frac{\xi}{\bar{E}}\underbrace{\int_{{-}1/2}^{{+}1/2} F(X)\, \mathrm{d}\kern0.06em X}_{\mathcal{I}_1}= \frac{\xi\mathcal{I}_1}{\bar{E}}. \end{equation}

The definition of $k$ from (4.3) is used in the first step. The second step is making the integral dimensionless. In the third step, the assumption of a linearly elastic solid has been invoked with $k\propto \bar {E}$ and $\mathfrak {f}(x)\propto 1/\bar {E}$. Note that $\bar {E} = E/(1-\nu _s^{2})$ here, which means that $k$ is well defined as $\nu _s\to 1/2^{-}$, because of the plane-strain reduction from 3-D to 2-D, with $E$ being Young's modulus and $\nu _s$ being the Poisson ratio, respectively. Then, $F(X)$ is introduced as the dimensionless self-similar deformation profile, and $\xi$ is the resulting prefactor after $x$ is scaled by $w$. In the last step, $\mathcal {I}_1=\int _{-1/2}^{+1/2} F(X)\, \mathrm {d}\kern0.06em X$ was introduced to simplify the expression. While the effect of the material properties of the solid wall are captured by $\bar {E}$, the influence of the wall geometry and the boundary conditions are taken into account by both $\xi$ and $\mathcal {I}_1$. As mentioned in § 4.1, $\mathfrak {f}(x)$ takes different forms in different situations, thus giving different expressions for $\xi$ and $\mathcal {I}_1$. For example, for the thick-walled microchannel considered by Wang & Christov (Reference Wang and Christov2019), $\xi =w$ and $\mathcal {I}_1\approx 0.542754$. Meanwhile, for the microchannels with thick-plate-like top walls considered by Shidhore & Christov (Reference Shidhore and Christov2018), $\xi =w^{4}/(2d^{3})$ and $\mathcal {I}_1={1}/{30}+{(d/w)^{2}}/[{3\kappa (1-\nu _s)}]$, with $\kappa$ being a ‘shear correction factor’ (typically, $\kappa =1$). Nevertheless, the key point is that both $\xi$ and $\mathcal {I}_1$ can be obtained a priori, by solving the corresponding elasticity problem, as analytical expressions.

Using the scales from tables 2 and 3, the dimensionless version of (4.5) is

(4.9)\begin{equation} \theta_I \frac{\partial^{2} \bar{U}_Y}{\partial T^{2}} + \bar{U}_Y -\theta_t\frac{\partial^{2} \bar{U}_Y}{\partial Z^{2}} = P, \end{equation}

where $\theta _I = \rho _s b^{\star }\mathcal {\bar {U}}_c/(\mathcal {T}_f^{2} \mathcal {P}_c)$ and $\theta _t = \chi _t \bar {\mathcal {U}}_c/(\ell ^{2}\mathcal {P}_c)$ are introduced above as the inertial coefficient and the tension coefficient, respectively. We expect that $\theta _I\ll 1$ and $\theta _t\ll 1$ because both the solid inertia and the tension effect are weak. Then, the leading-order solution (as $\theta _I,\theta _t\to 0$) of (4.9) is (4.6), as required by the above discussion. Even though $\theta _I\ll 1$ and $\theta _t\ll 1$, it is important to properly model these weak effects because they have significant influence on the global instability of the system (as we will see in § 7.1).

First, let us justify $\theta _I\ll 1$. Recalling that $\mathcal {P}_c=\mu \mathcal {V}_c/(\epsilon h_0)$, $\widehat {Re}=\epsilon \rho _f\mathcal {V}_c h_0/\mu$, and $\beta = \bar {\mathcal {U}}_c/h_0$, $\theta _I$ can be written as

(4.10)\begin{equation} \theta_I = \frac{\rho_s b^{{\star}}\mathcal{\bar{U}}_c}{\mathcal{T}_f^{2} \mathcal{P}_c}= \epsilon\frac{\rho_f\mathcal{V}_ch_0}{\mu}\frac{\mathcal{\bar{U}}_c}{{h_0}}\frac{b^{{\star}}h_0}{\ell^{2}}\frac{\rho_s}{\rho_f} = \epsilon \widehat{Re}\beta\frac{b^{{\star}} }{\ell }\frac{\rho_s}{\rho_f}. \end{equation}

Since $\epsilon \ll 1$, $b^{\star } \leq d\ll \ell$, $\widehat {Re}= O(1)$, $\beta$ is typically $O(1)$ and $\rho _s\simeq \rho _f$ in the microchannel setting (because polydimethylsiloxane (known as PDMS) has a similar density to water), we have justified $\theta _I\ll 1$. Note that time in (4.5) is scaled by $\mathcal {T}_f$, as before, to ensure the fluid–solid coupling. Thus, $\theta _I\ll 1$ corresponds to $\mathcal {T}_s^{2}\ll \mathcal {T}_f^{2}$, meaning that the solid responds to pressure changes in the flow promptly, as discussed in § 4.1. It is also helpful to note that, using (4.8), we can also write

(4.11)\begin{equation} \theta_I = \mathcal{I}_1\frac{\rho_s b^{{\star}} \xi}{\mathcal{T}_f^{2} \bar{E}}, \end{equation}

which indicates that, for a more rigid solid (i.e. with the increase of $\bar {E}$), the solid deformation develops much faster than the flow.

Next, using (4.8) again, the dimensionless tension coefficient can be written as

(4.12)\begin{equation} \theta_t = \frac{\chi_t \bar{\mathcal{U}}_c}{\ell^{2}\mathcal{P}_c} =\mathcal{I}_1\frac{\chi_t \xi}{\bar{E}\ell^{2}}. \end{equation}

If $\chi _t$ is deformation-induced (thus time-dependent), substituting equation (A3) into (4.12), we obtain

(4.13)\begin{align} \theta_t(T) &=\mathcal{I}_1\frac{\xi}{\bar{E}\ell^{2}} \frac{E b^{{\star}}}{\ell} \int_0^{\ell} \frac{1}{2}\left(\frac{\partial \bar{u}_y}{\partial z}\right)^{2}\, \mathrm{d} z \nonumber\\ &= (1-\nu_s^{2})\mathcal{I}_1 \frac{\xi b^{{\star}}\bar{\mathcal{U}}_c^{2} }{\ell^{4}} \int_0^{1} \frac{1}{2} \left(\frac{\partial \bar{U}_Y}{\partial Z}\right)^{2} \, \mathrm{d} Z \nonumber\\ &= \tilde{\theta}_t \int_0^{1} \frac{1}{2} \left(\frac{\partial \bar{H}}{\partial Z}\right)^{2} \, \mathrm{d} Z, \end{align}

where

(4.14)\begin{equation} \tilde{\theta}_t = \frac{1}{\beta^{2}}\times (1-\nu_s^{2})\mathcal{I}_1 \frac{\xi b^{{\star}}\bar{\mathcal{U}}_c^{2} }{\ell^{4}} =(1-\nu_s^{2})\mathcal{I}_1 \epsilon^{2}\frac{\xi b^{{\star}}}{\ell^{2}}. \end{equation}

Note that we have used (4.7) in the last step of the manipulations in (4.13). Also, note that $\tilde {\theta }_t$ is not related to the flow conditions, and $\tilde {\theta }_t \ll 1$ for microchannels. Within linear elasticity, the integral in (4.13) is $O(1)$, thus $\theta _t\ll 1$ as well.

Finally, the key dimensional and dimensionless parameters are summarized in table 4. Typical values for microfluidic systems are given to justify the bigness/smallness assumptions made.

5.1. Exemplar cases and preview of results

In the remainder of this paper, we address the steady-state features, the dynamic response and also the linear stability of the non-flat steady state of the proposed 1-D FSI model. To explore these issues, we have chosen exemplar cases with typical dimensional and dimensionless values given in tables 4 and 5. The values for the geometrical and material properties are taken and/or modified from Gervais et al. (Reference Gervais, El-Ali, Günther and Jensen2006). The long and shallow microchannels are assumed fabricated via soft lithography, with a thick top wall. The leading-order steady response of such microchannels has been solved by Wang & Christov (Reference Wang and Christov2019), according to which $\mathcal {I}_1 = 0.542754$ and $\xi =w$ for calculating $\beta$ in table 5.

Table 5. The dimensional and dimensionless parameters for the exemplar cases considered.

Similar to the experiments of Verma & Kumaran (Reference Verma and Kumaran2013), cases C1 to C3 and C4 to C6 summarized in table 5 are each based on a single microchannel, operated under different flow conditions. As catalogued in table 6, the steady and dynamic responses of the new 1-D FSI model match several experimental observations qualitatively, which indicates that the proposed model can provide unique insights into this unstable FSI problem. However, we cannot perform direct quantitative comparisons between our 1-D model and the experiments of Verma & Kumaran (Reference Verma and Kumaran2013), for the following reasons. First, the experiments’ soft wall was compressed upon a rigid outer surface, unlike our model wherein we have a soft wall that bulges outwards in an unconstrained manner (being stress-free on its outer surface). Further, in the experiments, the wall thickness was comparable to the channel's width, with two sidewalls made rigid. Consequently, the deformation field within the compliant wall in the microchannels fabricated by Verma & Kumaran (Reference Verma and Kumaran2013) is described by a different leading-order theory of the flow-induced deformation than the theories considered herein (recall § 4). At this time, it is not clear whether an exact solution (along the lines of Wang & Christov (Reference Wang and Christov2019)) could be obtained for the deformation in the configuration fabricated by Verma & Kumaran (Reference Verma and Kumaran2013). The main difference would be in the definition of $\beta$. Nevertheless, since the experiments did consider long and shallow microchannels with a slender deformable wall, the assumptions made in §§ 3 and 4 apply. Therefore, the FSI physics in these experiments are expected to be captured by the theoretical framework proposed herein. Indeed, as discussed by Verma & Kumaran (Reference Verma and Kumaran2013), the FSI-induced instabilities are generic, thus are not expected to be an ‘accidental’ phenomenon occurring only in some specific experimental devices. It follows that our qualitative comparisons below are meaningful and useful for validating the proposed 1-D FSI model.

Table 6. Qualitative comparison between the experimental observations of Verma & Kumaran (Reference Verma and Kumaran2013) and the predictions of the proposed global 1-D FSI model.

7.1. Eigenspectra of the exemplar cases

Here, it is illustrative to plot dimensional quantities to show how a corresponding physical system would behave. To this end, we write the dimensional frequency as

(7.7)\begin{equation} f_g = \frac{\omega_G}{2{\rm \pi} \mathcal{T}_f} = \frac{\omega_G q}{2{\rm \pi} \ell w h_0}. \end{equation}

Note that $f_g\in \mathbb {C}$, such that $\textrm{Re} (f_g)$ is the oscillatory frequency of the corresponding eigenmode, i.e. $[\tilde {H}, \tilde {Q}]^{T}$ in our formulation, while $\textrm{Im}(f_g)$ is the eigenmode's growth/decay time constant.

On the other hand, since (4.9) essentially represents a mechanical oscillator, the dimensionless and dimensional natural frequency of the oscillator, denoted as $F_N$ and $f_n$, respectively, can be approximated as

(7.8a,b)\begin{equation} F_N \approx \frac{1}{2{\rm \pi} \sqrt{\theta_I}}, \quad f_n \approx \frac{F_N}{\mathcal{T}_f} = \frac{F_N q}{\ell w h_0}. \end{equation}

Unlike $f_g$, both $F_N,f_n\in \mathbb {R}$. Also note that, (7.8a,b) do not take the tension effect into account, but since tension is weak, we believe (7.8a,b) provide a good approximation to the natural frequency of the system.

In figure 4, we show the calculated eigenspectra for the six exemplar cases from table 5. First, observe that the eigenspectra are symmetric about the imaginary axis in the complex plane. This symmetry is a consequence of the formulation of the generalized eigenvalue problem (see (7.4)), as the matrix $\boldsymbol{\mathsf{A}}$ on the left-hand side is purely real, while the matrix $\mathrm {i}\boldsymbol{\mathsf{B}}$ on the right-hand side is purely imaginary. This symmetry is also a feature of the eigenvalue analyses in Inamdar et al. (Reference Inamdar, Wang and Christov2020) and Wang & Christov (Reference Wang and Christov2020).

Figure 4. Eigenspectra of the exemplar cases from table 5: (a) $q=1500\ \mathrm {\mu }$l min$^{-1}$ (C1 and C4); (b) $q=6000\ \mathrm {\mu }$l min$^{-1}$ (C2 and C5); (c) $q=9000\ \mathrm {\mu }$l min$^{-1}$ (C3 and C6). The symbols represent the discrete eigenvalues for $E=1$ MPa and $E=2$ MPa, respectively. The red horizontal line mark the position of real axis, i.e. $\textrm{Im}(f_g)=0$. The dash–dotted lines mark the natural frequencies calculated by (7.8b). The insets in each panel have the same vertical axes.

Second, the 1-D FSI system transitions from stability (figure 4a) to instability (figures 4b and 4c) as the flow rate is increased. The fluid inertia becomes more prominent as the flow rate is increased, indicated by the magnitude of $\widehat {Re}$, which is associated with the nonlinear terms on the left-hand side of (5.3b). For the six exemplar cases from table 5, we observe that the linear instability typically occurs for $\widehat {Re}\simeq 1$, or equivalently $Re\simeq 100$, which is close to the values reported for the microchannel experiments showing instability (Verma & Kumaran Reference Verma and Kumaran2013).

Third, it is observed that the unstable regions in figures 4(b) and 4(c) are close to the natural frequency, indicating that the instabilities are related to the resonance of the wall. Due to the weak solid inertia, the natural frequency calculated from (7.8b), as well as $\textrm{Re} (f_g)$ in the unstable region, is as high as $\simeq 10^{4}$ Hz. This fact also matches the local linear stability analysis of Verma & Kumaran (Reference Verma and Kumaran2013), who reported that the least stable mode oscillates at a frequency of the order of $10^{4}$ Hz. Moreover, one of the common features of the eigenspectra in figure 4 is that, away from the unstable region $\textrm{Im}(f_g)\to 0^{-}$ as $|\textrm{Re} (f_g)|$ becomes large. This observation indicates that, apart from those modes that grow for the given flow conditions, the stable modes that oscillate with higher frequency decay slowly, which highlights the computational stiffness of the 1-D FSI system.

Lastly, the compliance of the wall does influence the shape of the eigenspectra. Since the stiffer wall has a larger natural frequency, its unstable region is located at higher frequencies than its softer counterpart. Note that in figure 4, the two zoom-in insets in each panel have the same range for their vertical axes. Then it can be shown that the more compliant wall has the larger growth rate (or the smaller decay rate in the linearly stable case) for the least stable mode, which could be related to the experimental observations that the softer microchannel is more prone to instabilities (Verma & Kumaran Reference Verma and Kumaran2013).

Since both $\theta _I$ and $\theta _t$ are small quantities in the solid mechanics (4.9), it is informative to show a comparison of the linear stability between cases either with $\theta _I=0$ or $\theta _t=0$. We consider three different such cases: (i) $\theta _I\ll 1$ and $\theta _t\ll 1$; (ii) $\theta _I = 0$ and $\theta _t\ll 1$; (iii) $\theta _I=0$ and $\theta _t=0$. The base states of both cases (i) and (ii) are the same, governed by (6.1). However, in case (iii), the system cannot satisfy $\bar {H}(0)=0$, thus the corresponding base state should be taken from (6.4a,b). As shown in figure 5(a), we observe that even though the solid inertia and tension are weak in the system, the inclusion of these weak effects changes the eigenspectrum fundamentally. The non-flat base state from (6.4a,b) is shown to be linearly stable. With only $\theta _t\ne 0$, case (ii) can predict linear stability only while case (i) with both $\theta _I,\theta _t\ne 0$ is linearly unstable. Furthermore, as shown in both cases (ii) and (iii), for $\theta _I=0$, the eigenmodes oscillate with higher frequencies. The reason is that, in this case, the natural frequency $F_N \to \infty$ as $\theta _I\to 0$.

Figure 5. (a) Comparison of the eigenspectra of the case with $\theta _I\ll 1$ and $\theta _t\ll 1$ (+), the case with $\theta _t\ll 1$ but $\theta _I=0$ ($\bullet$, blue), and the case with $\theta _I=0$ and $\theta _t=0$ ($\blacktriangle$, pink). The dimensionless parameters are taken as per case C3 in table 5. (b) Contour plot of $\textrm{Im}(\omega _G)$ of the least stable mode as a function of $\theta _I$ and $\theta _t$, with $\widehat {Re}$ and $\beta$ taken from case C3 in table 5. The dashed line marks $\theta _I = \theta _t$.

To further investigate the effect of $\theta _I$ and $\theta _t$, we calculated the growth/decay rate of the least stable mode by taking different combinations of $\theta _I$ and $\theta _t$, fixing $\widehat {Re}$ and $\beta$ as the values corresponding to case C3. As shown in figure 5(b), for both $\theta _I$ and $\theta _t$ across five orders, linear instabilities are only observed when $\theta _I$ is at least one order larger than $\theta _t$. Since the dimensionless phase speed of the transverse waves along the fluid–solid interface is $\sqrt {\theta _t/\theta _I}$, then the linear instability occurs when the transverse waves propagate (much) slower than the flow.

7.2. Eigenfunctions of the exemplar cases

Each eigenvalue is associated with an eigenfunction pair, i.e. $[\tilde {H}, \tilde {Q}]^{T}$ via (7.4). For the eigenvalues with larger $|\textrm{Re} (\omega _G)|$, the corresponding eigenfunctions are more oscillatory in space. For example, for the purely decay mode with $\textrm{Re} (\omega _G)=0$, the corresponding eigenfunctions are found to be purely real and non-oscillatory, as shown in figure 6. However, for the least stable modes of the linearly unstable cases from table 5, as shown in figure 7, with $|\textrm{Re}(\omega _G)|\gg 1$, the corresponding eigenfunctions are highly oscillatory in space. The corresponding eigenvalues for the eigenfunctions in figures 6 and 7 are tabulated in table 7.

Figure 6. The eigenfunctions of the pure decay eigenmodes of the six exemplar cases from table 5. The eigenfunctions have been scaled such that $\max |\tilde {Q}|=1$.

Figure 7. The eigenfunctions of the least stable modes for the linearly unstable cases, i.e. C2, C3, C5 and C6 in table 5. The eigenfunctions have been scaled such that $\max |\tilde {Q}|=1$.

Table 7. The dimensionless eigenvalues, $\omega _G$, for the pure decay modes and for the least stable modes of the exemplar cases from table 5.

We do not tabulate the least stable modes of the linearly stable cases in table 7 because it is hard to pick out the least stable mode due to the limitation of the numerical method. In that case, we observe that the least stable mode is always the farthest eigenvalue away from the imaginary axis, if computed with the Chebyshev pseudospectral method using different number of Gauss–Lobatto points $N$. In principle, there are an infinite number of eigenvalues in the 1-D FSI system, resolving all of which would require an infinite number of Gauss–Lobatto points. Unfortunately, $N$ cannot be arbitrarily large because matrices in (7.4) will become ill-conditioned.

Let us take a closer look at the eigenfunctions of the least stable modes in figure 7. For all the cases, both $\tilde {H}$ and $\tilde {Q}$ exhibit large oscillations near the channel outlet ($Z=1$), which echoes the experimental observation that the instabilities were always first observed near the outlet in the converging section of the microchannel (Verma & Kumaran Reference Verma and Kumaran2013). Furthermore, for a larger growth rate, the difference in oscillations between the outlet and the inlet is more prominent (comparing C2 and C3, or C5 and C6).

The wavy forms of the eigenfunctions in figure 7 further inspire us to conduct a Fourier transform in space for each case. We have used SciPy's fft routine with a Blackman window. The results are summarized in figure 8 and the abscissa represents the reciprocal of the dimensionless wavelength, denoted by $\varLambda =\lambda /\ell$. Here, $\lambda$ is the dimensional wavelength. Interestingly, there are always two peaks for all the cases. The major peak is located at $1/\varLambda \simeq 100$, meaning the dominant wavelength is of the order of the channel height (recall $\ell /h_0\approx 333$). This observation could be related to the results of the local linear stability analysis of Verma & Kumaran (Reference Verma and Kumaran2013), who found that the most unstable modes have a wavelength comparable to the channel height.

Figure 8. The spatial Fourier transform of the eigenfunctions from figure 7. Case C3 is scaled up by a factor of $10$, while case C6 is scaled up a factor of $4$.

8.1. Evolution from a perturbed inflated state

First, to validate the linear stability results from § 7, at $T=0$, we perturb the steady-state solution of the 1-D FSI model using the eigenfunction of a specific mode. Then, $\textrm{Im}(\omega _G)$ indicates the growth/decay rate of the perturbation, while $2{\rm \pi} /\textrm{Re}(\omega _G)$ is the period of the perturbation's oscillations.

The first example is the linearly stable case C1 ($q=1500$ $\mathrm {\mu }$l min$^{-1}$) perturbed from the steady state with the eigenfunctions of the pure decay mode tabulated in table 7. The shapes of the corresponding eigenfunctions are shown in figure 6. The decay rate of the perturbation to the steady flow agrees well with the corresponding $\textrm{Im}(\omega _G)$, as shown in figure 9(a). No oscillations are observed in the evolution because $\textrm{Re}(\omega _G)=0$ in this case.

Figure 9. Time history of the difference between the instantaneous outlet flow rate and the steady one, i.e. $|Q(1,T)-Q_0(1)|$ (solid, left-hand axes, also recall that $Q_0(Z)\equiv 1$ for the chosen boundary conditions), and the axially average deformed height, $\langle H\rangle (T) = \int _0^{1} \bar {H}(Z,T)\, \mathrm {d} Z$ (dashed, right-hand axes). (a) The steady state is perturbed using the eigenfunctions of the pure decay mode of case C1. (b) The steady state is perturbed using the eigenfunctions of the most unstable mode of case C3. The dot–dashed trendlines represent the growth/decay of perturbations, based on the imaginary part of the eigenvalues from table 7. In panel (b), the computed period of oscillation is marked. The value in parenthesis is from linear stability analysis, i.e. $2{\rm \pi} /\textrm{Re}(\omega _G)$.

The second example corresponds to the linearly unstable case C3. At $T=0$, the steady state is perturbed by the eigenfunction of the corresponding most unstable mode. The shapes of the eigenfunctions are shown in figure 7, while the corresponding eigenvalues are given in table 7. Both the simulated growth rate and the oscillation period agree well with the linear stability analysis, as shown in figure 9(b). Here, note that the tension coefficient $\theta _t$ in (4.9) is estimated from the instantaneous wall deformation, thus it is time dependent. Meanwhile $\theta _t$ is fixed to be the steady-state value for the purposes of the linear stability analysis. The good agreement between the numerical simulation and the linear stability analysis indicates that neglecting the time dependence of $\theta _t$ for the linear stability analysis is valid. The actual simulations for each case are conducted for a longer time window. Interestingly, although the oscillations of the system are sustained, no saturated periodic state emerges during the course of the simulations. The simulations for cases C3 and C5 are available in Appendix B.

8.2. Evolution from a flat initial state

Starting the simulations with an undeformed channel initial condition (as in (5.6b)) would be more realistic of how a microfluidic device might be operated. With (5.6a,b) as the initial conditions, cases C1 and C4 are linearly stable and reach the steady state without detectable oscillations. The evolution of the representative quantities for case C4 are shown in figure SM.2 in the supplementary material and movies, wherein a video of the evolution under case C1 is also provided. In this subsection, we focus on the two linearly unstable cases C3 and C6. All of the results shown below have been verified by time step refinement (see the supplementary material and movies and figure SM.3 therein, for example).

First, the evolution of the outlet flow rate and the inlet pressure for cases C3 and C6 are shown in figure 10. In both cases, the outlet flow rate first decreases and then increases, reaching a value close to the imposed flow rate at late times. Meanwhile, the inlet pressure increases to a value slightly below the steady-state inlet pressure. Small-amplitude oscillations are observed in the evolution of both quantities. More importantly, the oscillations become magnified at later times in the simulation, which suggests that these unstable cases will not reach the steady state. It can be shown (by running longer-time simulations) the that these oscillations are not unbounded. Nevertheless, similar to the cases discussed in § 8.1, no saturated periodic state appears to emerge during the time window of the simulations shown in this subsection.

Figure 10. Time histories of the outlet flow rate and the inlet pressure for (a) case C3 and (b) case C6, respectively, with (5.6a,b) being the initial conditions. The dot–dashed lines mark the flow rate at steady state, while the dashed lines mark the inlet pressure at the steady state.

It is more enlightening to contrast the two simulations shown in figure 10. For these two cases, all system parameters are the same, except that Young's modulus for case C3 is half of that for case C6. With a more compliant wall, the instabilities under case C3 develop more quickly. Specifically, more ‘violent’ oscillations are observed in case C3 for the outlet flow rate and the inlet pressure than in case C6. These oscillation amplitudes could be, qualitatively, representative of the observations in dye-stream experiments. In other words, dye break up could be expected when more violent oscillations occur in softer channels, while the dye steam may just oscillate (without breaking up) if the channel is less compliant (thus, the oscillations in the flow rate and pressure are milder). Indeed, it was observed in the experiments that the dye breaks-up at lower $Re$ in softer channels (Verma & Kumaran Reference Verma and Kumaran2013).

Next, let us take a closer look at the evolution of the fluid–solid interface. The example in figure 11 corresponds to case C3 in figure 10(a), while the evolution under case C6 is qualitatively similar. Initially, for $0\leq t\leq 1\ {\rm ms}$ as shown in figure 11(a), the interface bulges near the channel inlet first because the pressure is relatively high there. At the same time, transverse waves are shed and propagate from the inlet to the outlet until they are reflected at the downstream boundary of the domain. There is a dramatic increase in the total volume of fluid in the channel at this stage. Thereafter, the deformation of the wall stops growing, but the transverse waves still propagate back and forth along the fluid–solid interface. Compared with figure 11(a), the transverse waves have smaller wavelength and amplitudes. Furthermore, the interface shape at this stage is close to the steady-state shape, as seen in figure 11(b). The deviation of the interface's dynamic deformation from the steady one is plotted for $4.0\ {\rm ms} \leq t\leq 6.5\ {\rm ms}$ in figure 11(c), where the wave propagation can be clearly observed. Interestingly, after a while, the oscillations near the channel's outlet continue to grow and become larger than the oscillations anywhere else along the channel, which explains why the variations of the outlet flow rate appear more prominent compared with those of the inlet pressure in figure 10. This observation is also corroborated by the experimental observation that the instabilities always initiate near the channel's outlet.

Figure 11. Evolution of the shape of the fluid–solid interface from the flat initial condition (5.6b) (movie available in the supplementary material and movies). (a) Shape of the interface for $0 \leq t \leq 1$ ms. (b) Comparison of the interface shape at $t=3.5$ ms with the steady state. (c) Space–time plot of the difference between the instantaneous interface shape $\bar {u}_y$ and the steady state $\bar {u}_y^{s}$, i.e. $\bar {u}_y - \bar {u}_y^{s}$, for $4.0\ {\rm ms}\leq t \leq 6.5\ {\rm ms}$.

To better emphasize the difference in the wall motions near the inlet versus near the outlet, the vertical velocities of the fluid–solid interface at $z=0.9\ell$ (near the outlet) and $z=0.1\ell$ (near the inlet) of case C3 are plotted in figure 12 (see the supplementary material and movies for the details of reconstructing the velocities). The three stages discussed in figure 11 can also be identified from the time histories of the vertical velocities shown in figure 12. At early times, since the wall bulges first near the channel inlet, the vertical velocity at $z=0.1\ell$ is larger. The motion at $z=0.9\ell$ starts after the transverse waves reach the channel outlet. In the intermediate stage, during which the channel volume does not change significantly, the oscillations at both positions remain relatively small, until the motion near the outlet becomes amplified and leads to a striking difference in the oscillatory amplitudes at the two positions.

Figure 12. (a) Time histories of the vertical velocity of the fluid–solid interface of case C3 at $z=0.9\ell$ (near the outlet) and $z=0.1\ell$ (near the inlet), respectively. (b) Fourier transform of the corresponding time histories from (a). The dotted line marks the natural frequency calculated from (7.8b).

Figure 12(b) shows the corresponding time histories in the frequency (Fourier) domain. We observe that the peak is near the natural frequency of the wall (predicted by (7.8b)), indicating a resonant phenomenon. Due to the FSI, which gives rise to the transverse waves along the fluid–solid interface, the pressure oscillations exhibit a frequency close to the natural frequency of the wall as well, causing a feedback. Note that, the resonances are self-excited as no oscillatory components are introduced in the initial conditions (5.6a,b). Further, the oscillations are self-sustained as they do not die out during the entire simulation time window. Consequently, the demonstrated FSI-induced instabilities could be an effective and inexpensive way of enhancing mixing at the microscale.