2.1. Governing equations of the oscillatory flow

In this context, Martínez-Calvo et al. (Reference Martínez-Calvo, Sevilla, Peng and Stone2020) analysed the inertialess startup flow, following on from the steady problem solved by Christov et al. (Reference Christov, Cognet, Shidhore and Stone2018), introducing scalings that balance all velocity components in the conservation of mass equation. Ramos-Arzola & Bautista (Reference Ramos-Arzola and Bautista2021) also used these scalings for a non-Newtonian version of the problem. Here, unlike these prior works, we adopt the scaling of the 3-D problem introduced by Boyko & Christov (Reference Boyko and Christov2023), which leads to a leading-order problem that can be spanwise-averaged. Although both approaches are valid within the assumed order of approximation for a shallow and wide channel, only the latter approach allows us to make analytical progress in the oscillatory flow problem. Specifically, we scale both cross-sectional velocities, $v_x$ and $v_y$ , so that they are smaller than the axial one, $v_z$ , by a factor of $\epsilon \stackrel {\text{def}}{=} h_0/\ell$ , where $\epsilon \ll 1$ for a slender channel.

Figure 1. Schematic of the 3-D deformable shallow and slender rectangular microchannel geometry of initial (undeformed) height $h_0$ , axial length $\ell$ and transverse width $w$ , such that $\ell \gg w \gg h_0$ . The top wall (darker colour) is an elastic plate structure of thickness $b$ that can deform from $y=h_0$ to $y=h(x,z,t)$ , where $h(x,z,t)-h_0=u_y(x,z,t)$ is the vertical displacement of the fluid–solid interface. The top wall is clamped (no displacement) along the planes $x=\pm w/2$ (and $0\leqslant z\leqslant \ell$ ), while taking the outlet pressure as gauge, $p|_{z=\ell }=0$ , ensures no deformation along the plane $z=\ell$ (and $-w/2 \leqslant x \leqslant +w/2$ ). An oscillatory inlet pressure, $p|_{z=0} = p_{\textit{in}}(t)$ of amplitude $p_0$ and angular frequency $\omega$ , drives the flow.

With all this in mind, we introduce the dimensionless variables for the problem:

(2.1) \begin{align} X = \frac {x}{w},\quad Y = \frac {y}{h_0},\quad Z = \frac {z}{\ell },\quad T = \frac {t}{\omega ^{-1}},\quad V_X = \frac {v_x}{\epsilon v_c},\quad V_Y = \frac {v_y}{\epsilon v_c},\nonumber \\ V_Z = \frac {v_z}{v_c}, \quad P = \frac {p}{p_0}, \quad U_X = \frac {u_x}{u_c},\quad U_Y = \frac {u_y}{u_c},\quad U_Z = \frac {u_z}{u_c}, \quad H = \frac {h}{h_0}. \end{align}

Here, $p_0$ and $\omega$ are the amplitude and angular frequency, respectively, of the inlet pressure oscillations driving the flow. The scales $v_c$ and $u_c$ for the axial velocity and wall displacement, respectively, are determined below. In terms of the dimensionless variables from (2.1), the governing partial differential equations (PDEs) of leading-order-in- $\epsilon$ unsteady flow in a 3-D channel (i.e. a slender channel under the lubrication approximation for which $\epsilon \ll 1$ (Leal Reference Leal2007)) are

(2.2a) \begin{align} \delta \frac {\partial V_X}{\partial X} + \frac {\partial V_Y}{\partial Y} + \frac {\partial V_Z}{\partial Z} &= 0, \end{align}

(2.2b) \begin{align} 0 &= - \frac {\partial P}{\partial X}, \end{align}

(2.2c) \begin{align} 0 &= - \frac {\partial P}{\partial Y}, \end{align}

(2.2d) \begin{align} {\textit{Wo}}^2 \left [ \frac {\partial V_Z}{\partial T} + \frac {\beta }{\gamma } \left ( \delta V_X \frac {\partial V_Z}{\partial X} + V_Y \frac {\partial V_Z}{\partial Y} + V_Z \frac {\partial V_Z}{\partial Z} \right) \right ] &= - \frac {\partial P}{\partial Z} + \delta ^2 \frac {\partial ^2 V_Z}{\partial X^2} + \frac {\partial ^2 V_Z}{\partial Y^2}, \end{align}

where we have chosen the axial velocity scale as $v_c = h_0^2 p_0/(\ell \mu _f)$ to balance viscous and pressure forces in (2.2d ). In scaling the problem this way, several key dimensionless numbers emerge:

(2.3a) \begin{align} \delta &\stackrel {\text{def}}{=} \frac {h_0}{w}, \end{align}

(2.3b) \begin{align} {\textit{Wo}}^2 &\stackrel {\text{def}}{=} \frac {h_0^2/(\mu _f/\rho _f)}{\omega ^{-1}} = \frac {\text{transverse momentum diffusion time scale}}{\text{oscillation time scale}}, \end{align}

(2.3c) \begin{align} \beta &\stackrel {\text{def}}{=} \frac {u_c}{h_0} = \frac {\text{top wall displacement scale}}{\text{initial channel height}}, \end{align}

(2.3d) \begin{align} \gamma &\stackrel {\text{def}}{=} \frac {u_c/(\epsilon v_c)}{\omega ^{-1}} = \frac {\text{elastoviscous time scale}}{\text{oscillation time scale}}. \\[9pt] \nonumber \end{align}

Here, $\delta$ is the channel’s cross-sectional (inverse) aspect ratio ( $\delta \ll 1$ for wide channels); $\beta$ is the compliance number (Christov et al. Reference Christov, Cognet, Shidhore and Stone2018), where wall displacement scale $u_c$ is introduced below upon specifying the elasticity model; $Wo $ is the Womersley number (Womersley Reference Womersley1955a ); and $\gamma$ is the elastoviscous number (Elbaz & Gat Reference Elbaz and Gat2014; Zhang & Rallabandi Reference Zhang and Rallabandi2024), where the elastoviscous time scale emerges from comparing the vertical displacement scale $u_c$ (set by elasticity) with the vertical velocity scale $\epsilon \textit {v}_c$ (set by the balance of viscous and pressure forces). For a stiff top channel wall (weakly compliant channel), we would expect $\beta \ll 1$ . On the other hand, we do not impose any restrictions on $\gamma$ or ${\textit{Wo}}$ a priori. Although it may be tempting to rewrite $\beta /\gamma$ in (2.2d ) as a Reynolds number (Pande et al. Reference Pande, Wang and Christov2023), it will become clear below why that is not a good idea (Zhang & Rallabandi Reference Zhang and Rallabandi2024).

Now, assuming a wide (shallow) channel, $\delta \ll 1$ , so that we can neglect all terms at $O(\delta)$ and higher in (2.2), we have

(2.4a) \begin{align} \frac {\partial V_Y}{\partial Y} + \frac {\partial V_Z}{\partial Z} &= 0, \end{align}

(2.4b) \begin{align} 0 &= - \frac {\partial P}{\partial X}, \end{align}

(2.4c) \begin{align} 0 &= - \frac {\partial P}{\partial Y}, \end{align}

(2.4d) \begin{align} {\textit{Wo}}^2 \left [ \frac {\partial V_Z}{\partial T} + \frac {\beta }{\gamma } \left ( V_Y \frac {\partial V_Z}{\partial Y} + V_Z \frac {\partial V_Z}{\partial Z} \right) \right ] &= - \frac {\partial P}{\partial Z} + \frac {\partial ^2 V_Z}{\partial Y^2}. \end{align}

As usual, from (2.4b , c ), we immediately conclude that $P=P(Z,T)$ only.

The flow obeys no-slip and no-penetration conditions along the bottom wall of the channel:

(2.5a,b) \begin{align} V_Z|_{Y=0} = 0,\quad V_Y|_{Y=0} = 0. \end{align}

Along the deformable (thus, moving) top wall, $y=h(x,z,t)$ , a kinematic condition, $\boldsymbol{v} = D\boldsymbol{u}/Dt$ , applies. In dimensionless component form, the kinematic condition is

(2.6a) \begin{align} V_X|_{Y=H} &= \left.\left (\gamma \frac {\partial U_X}{\partial T} + \frac {u_c}{w} V_X\frac {\partial U_X}{\partial X} + \frac {u_c}{h_0} V_Y\frac {\partial U_X}{\partial Y} + 0\right)\right |_{Y=H}, \\[-12pt] \nonumber \end{align}

(2.6b) \begin{align} V_Y|_{Y=H} &= \left.\left (\gamma \frac {\partial U_Y}{\partial T} + \frac {u_c}{w} V_X \frac {\partial U_Y}{\partial X} + \frac {u_c}{h_0} V_Y \frac {\partial U_Y}{\partial Y} + 0\right)\right |_{Y=H}, \\[-12pt] \nonumber \end{align}

(2.6c) \begin{align} V_Z|_{Y=H} &= O(\epsilon), \\[9pt] \nonumber \end{align}

having used the third condition to simplify the first two. Further simplification can only be obtained upon specifying the governing equations of the wall deformation (in § 2.2). Since we restrict ourselves to pressure-driven flows (as described in more detail in § 3), the inlet and outlet pressures are known:

(2.7a,b) \begin{align} P|_{Z=0} = P_{\textit{in}}(T),\quad P|_{Z=1} = 0. \end{align}

2.2. Governing equations of the top wall deformation

Assuming a thin structure, we employ plate theory. The central tenet of plate theory is that the structure is thin and thus the vertical displacement $u_y$ (in the thin direction) does not vary with $y$ (Reddy Reference Reddy2007). Thus, the bottom surface of the top wall (i.e. the fluid–solid interface) is located at $h_0+u_y$ upon deformation of the wall (see figure 1).

Anand, Muchandimath & Christov (Reference Anand, Muchandimath and Christov2020) provide a full derivation and discussion of the equations of motion of slender and shallow thick plates obeying the theory due to Reissner (Reference Reissner1945) and Mindlin (Reference Mindlin1951), which governs the flexural deformation of isotropic, elastic plates deduced from the 3-D equations of elasticity when the plate’s thickness is non-trivial – a so-called first-order shear deformation theory (Reddy Reference Reddy2007; Challamel & Elishakoff Reference Challamel and Elishakoff2019). Here, we summarise the key details. The in-plane plane displacements are given by $u_x = u_x^0(x,z) + \bar {y} \phi _x(x,z)$ and $u_z = u_z^0(x,z) + \bar {y} \phi _z(x,z)$ , where $\phi _x$ and $\phi _z$ are rotations of the normal vector to the plate’s midsurface about the $x$ and $z$ axes, respectively, and $\bar {y}$ is measured from the midsurface, i.e. $\bar {y} = y - h_0 - b/2$ . The PDEs for $u_x^0(x,z)$ and $u_z^0(x,z)$ decouple from those for $\phi _y(x,z)$ , $\phi _z(x,z)$ and $u_y(x,z)$ . In the absence of external in-plane loads, these PDEs are homogeneous. Thus, from the clamped boundary conditions (BCs) (as in figure 1), it follows that $u_x^0 = u_z^0 \equiv 0$ . Consequently, the plate problem reduces to determining the vertical displacement, $u_y$ , and the rotations of the normal, $\phi _x$ and $\phi _z$ . A balancing argument shows that the appropriate scales for $\phi _x$ and $\phi _z$ are $u_c/w$ and $u_c/\ell$ , respectively.

For a slender and shallow plate, neglecting terms of $O(\delta)$ and $O(\epsilon)$ , $\varPhi _Z$ drops out of the governing plate equations, which now feature only $U_Y$ and $\varPhi _X$ (Anand et al. Reference Anand, Muchandimath and Christov2020). Our starting point is these equations, at the leading order in $\delta$ and $\epsilon$ , namely

(2.8a) \begin{align} {St}_s\frac {\partial ^2 U_Y}{\partial T^2} &= \frac {1}{720}\frac {\partial ^3 \varPhi _X}{\partial X^3} + P(Z,T), \end{align}

(2.8b) \begin{align} 0 &= \varPhi _X + \frac {\partial U_Y}{\partial X} - \frac {\mathscr{T}}{60} \frac {\partial ^2\varPhi _X}{\partial X^2}, \\[9pt] \nonumber \end{align}

where $\mathscr{T} \stackrel {\text{def}}{=} 10(b/w)^2/(1-\nu _s)$ is a dimensionless parameter quantifying the shallowness of the plate. Having balanced transverse bending with the pressure load from the flow in (2.8a ), $u_c = w^4 p_0 / (720 D_b)$ , where $D_b = E b^3/[12(1-\nu _s^2)]$ is the flexural rigidity of a plate in pure bending (Reddy Reference Reddy2007), $E$ is Young’s modulus and $\nu _s$ is Poisson’s ratio. The factor of $1/720$ included in $u_c$ might not be obvious now; it is included to eliminate all numerical prefactors in (4.6) below.

Similar to Martínez-Calvo et al. (Reference Martínez-Calvo, Sevilla, Peng and Stone2020), we have retained the displacement’s inertia in (2.8a ), but neglected rotary inertia in (2.8b ) as is standard in the literature (see e.g. Zienkiewicz, Taylor & Zhu (Reference Zienkiewicz, Taylor and Zhu2013), Chap. 13). The inertia of the plate is thus quantified by the solid’s Strouhal number:

(2.9) \begin{equation} {St}_s \stackrel {\text{def}}{=} \frac {\rho _s b u_c/p_0}{\omega ^{-2}} = \frac {(\text{solid's time scale for response to loading})^2}{(\text{oscillation time scale})^2}, \end{equation}

where $\rho _s$ is the density of the elastic wall material.

Generalising the steady results of Shidhore & Christov (Reference Shidhore and Christov2018), (2.8) describe transverse bending of the wall, i.e. the deformation in any $(X,Y)$ plane for fixed $Z$ , subject to the normal load imposed by the hydrodynamic pressure $P(Z,T)$ . The axial tension and bending are neglected as they scale with $\epsilon \ll 1$ (Anand et al. Reference Anand, Muchandimath and Christov2020). The corresponding clamped BCs supplementing (2.8) are

(2.10a,b) \begin{align} U_Y |_{X=\pm 1/2} = 0,\quad \varPhi _X |_{X=\pm 1/2} = 0. \end{align}

Observe that if we let $\mathscr{T}\to 0$ (i.e. $b/w\to 0$ ), then $\varPhi _X = - \partial U_Y/\partial X$ from (2.8b ), and substituting this relation into (2.8a ) and (2.10b ) reduces these to the corresponding equation of Kirchhoff–Love (thin-plate) theory (see e.g. Howell, Kozyreff & Ockendon Reference Howell, Kozyreff and Ockendon2009).

Finally, based on this discussion, we have deduced that $U_X = O(\delta)$ and $U_Z = O(\epsilon)$ . Thus, the in-plane displacements are negligible compared with the vertical (out-of-plane) displacement $U_Y$ at leading order, which significantly simplifies the kinematic condition (2.6).

2.3. Coupling flow to deformation

Since the in-plane displacements $U_X$ and $U_Z$ are negligible, the dimensionless kinematic condition (2.6) reduces to three independent BCs. The $X$ and $Z$ components, (2.6a ) and (2.6c ), are essentially no-slip BCs: $V_X|_{Y=H}=V_Z|_{Y=H}=0$ . While the $Y$ component (2.6b ) dictates that the fluid’s vertical velocity matches that of the wall:

(2.11) \begin{equation} V_Y|_{Y=H} = \frac {\gamma }{\beta } \frac {\partial H}{\partial T}, \end{equation}

where $H(X,Z) = 1 + \beta U_Y(X,Z)$ is the dimensionless height of the fluid channel. We recognise that $\gamma /\beta$ can also be interpreted as a fluidic Strouhal number (Ramachandra Rao Reference Ramachandra Rao1983; Ward & Whittaker Reference Ward and Whittaker2019; Inamdar, Wang & Christov Reference Inamdar, Wang and Christov2020; Pande et al. Reference Pande, Wang and Christov2023). Equation (2.11) is key to the two-way coupling of the flow and deformation.

Next, we seek to manipulate the kinematic condition (2.11) into one involving the flow rate. To this end, we define the flow rate $Q$ by integrating over the cross-sectional area in an $(X,Y)$ plane:

(2.12) \begin{equation} Q = \int _{-1/2}^{+1/2} \int _0^H V_Z \,\textrm{d} Y \textrm{d} X. \end{equation}

Then, a lengthy but straightforward calculation using the Leibniz rule to swap the order of $Y$ integration and $Z$ differentiation shows that the kinematic BC (2.11) can be combined with (2.12) to re-express the conservation of mass (2.4a ) as the continuity equation:

(2.13) \begin{equation} \frac {\partial Q}{\partial Z} + \frac {\gamma }{\beta } \frac {\partial }{\partial T} \int _{-1/2}^{+1/2} \big [ 1 + \beta U_Y(X,Z,T) \big ] \,\textrm{d} X = 0. \end{equation}

2.4. Summary of the governing equations and dimensionless numbers

In summary, (2.4d ), (2.8) and (2.13) are the governing equations of the oscillatory flow in a 3-D shallow and slender channel with a plate-like deformable top wall in pure bending.

The key dimensionless groups of the problem are summarised in table 1. The experimental system, discussed next in § 3, was designed to achieve an oscillatory flow with ${\textit{Wo}},\gamma =O(1)$ in the weakly compliant regime $\beta$ small (but not negligible) in a slender $\epsilon \ll 1$ and shallow $\delta \ll 1$ channel, which are the key approximations made in the theoretical analysis. These considerations lead to the parameter values/ranges given in table 1. Additionally, we learn from table 1 that the channels constructed have negligible wall inertia, ${St}_s\ll 1$ , allowing us to consider the plate deformation as quasi-static.

Table 1. Key dimensionless numbers of the 3-D elastoinertial rectification problem, based on the characteristic displacement scale $u_c$ for a plate and a characteristic axial velocity scale $v_c$ under lubrication theory. Typical values/ranges are based on the experimental set-up (§ 3). Negligible numbers are taken as zero in the analysis (i.e. the theory is ‘at leading order’ in these parameters), while small quantities are taken into account; perturbatively in the case of $\beta$ .

3.1. Oscillatory flow generation and shaping

Our experimental set-up is shown in figure 2. Previous experimental investigations of oscillatory flow generation in microchannels (Pielhop, Klaas & Schröder Reference Pielhop, Klaas and Schröder2015; Raj et al. Reference Raj, Dasgupta and Chakraborty2019; Vishwanathan & Juarez Reference Vishwanathan and Juarez2020; Dörner et al. Reference Dörner, Schröder and Klaas2021; Levenstein et al. Reference Levenstein2022; Vishwanathan & Juarez Reference Vishwanathan and Juarez2022, Reference Vishwanathan and Juarez2023) involved synchronising mechanical vibrations directly to the fluid itself in a microfluidic channel. Notably, the way that Vishwanathan & Juarez (Reference Vishwanathan and Juarez2020, Reference Vishwanathan and Juarez2022, Reference Vishwanathan and Juarez2023) achieved this is by attaching an inlet tube to the diaphragm of a speaker, such that the fluid within the channel was driven by the oscillations initiated from the inlet tube, ultimately creating an oscillatory flow. Inspired by this approach, we customised our oscillatory flow generation module as follows. We utilised a function generator (GH-CJDS66, Koolertron) connected with a speaker (DR-US200275, Drok) to ensure a robust signal input. We further introduced a PDMS (Sylgard 184, Dow Corning) liquid chamber positioned towards the speaker, with its membrane linked to the speaker diaphragm via a rigid, 3-D-printed connector to facilitate efficient mechanical vibration transmission. The chamber was first fabricated using 3-D-printed moulding techniques, and the membrane (with thickness of the order of 0.5 mm) sealing the liquid chamber was then bonded by inverting the chamber onto a liquid layer of PDMS mixture, which was subsequently cured at $90\,^\circ {\textrm {C}}$ for one hour to ensure solidification and secure adhesion. All PDMS components were fabricated with a 10:1 (w/w) ratio of silicone elastomer base to curing agent.

Figure 2. Experimental system with oscillatory flow in a 3-D deformable rectangular microchannel. (a) Set-up schematic. The entire interior space of the system is completely filled with the fluid prior to the experiments. To initiate the flow, an analogue sinusoidal signal generated by the function generator is transmitted into the speaker, enabling its diaphragm to vibrate. The deformable membrane of the liquid chamber (linked to the speaker diaphragm via a rigid, 3-D-printed connector shown in dark blue) transmits these vibrations, causing the oscillation of the fluid within both the chamber and the following channel. (b) Microchannel configuration. The channel features five pressure ports connecting to the data acquisition system (pressure transducer and PC). The five ports (each of width $w_p$ ) are evenly spaced with an axis-to-axis interval $\ell _p$ in the flow direction. The microchannel section between the first port and the outlet is covered with a deformable PDMS film of length $\ell$ at the top, and the section of the channel ahead of the first port is covered by a rigid glass slide, with its front edge precisely aligned to the centre of the first port.

In each experiment, this chamber was filled with the working fluid to generate sufficient pressure amplitude. Once the analogue sinusoidal signal from the function generator was transmitted into the speaker, which enabled its diaphragm to vibrate, the connected deformable membrane of the chamber thereby vibrated, causing the fluid oscillations inside the chamber and transmitting them to the microchannel. Before each experiment, we carefully eliminated any entrapment of air within the entire interior space, such as air pockets in all pressure ports and tubing, which is found to be important for obtaining reliable pressure amplitudes by the pressure transducer. In addition, the microchannel outlet was submerged in a liquid reservoir filled with the working fluid slightly above the level of the microchannel to maintain a constant hydrostatic pressure at the outlet, which is approximately atmospheric pressure given the small height.

3.2. Fabrication of the microchannels with deformable top walls

To fabricate the rectangular microchannel with deformable top walls, we follow the same procedure as Chun et al. (Reference Chun, Boyko, Christov and Feng2024), where a 3-D-printing technique (Mars Resin 3D Printer, ELEGOO, USA) was employed to manufacture the reverse mould with the designated channel dimensions as listed in table 2. A mixing ratio of 10:1 (w/w) between the silicone elastomer base and the curing agent was used to prepare the PDMS elastomer. The PDMS mixture was subsequently poured into the 3-D-printed mould and degassed under vacuum for 1 h to fully remove entrapped air bubbles. The mixture was then cured in an oven at $90\,^\circ {\textrm {C}}$ for 24 h. Upon curing, the PDMS block as the channel substrate was carefully detached from the mould, followed by the fluid inlet and pressure measurement ports being precisely punctured by a 2 mm disposable biopsy punch. To probe the time-varying pressure distribution along the flow direction, the five ports (spaced $\ell _p$ apart; see figure 2 b) were designed to connect to a pressure transducer for pressure measurements along the channel. All ports (with width $w_p$ at the branching point where they intersect with the main channel) were located to the side of the channel (figure 2 b) for convenient experimental operation. The lateral cross-sectional view of the entire port is shown in the inset of figure 2(b). Additionally, a smooth geometric transition following an elliptic arc was employed at the microchannel inlet to minimise any secondary flows created at sharp corners.

Table 2. Dimensions of the microchannel and elastic properties of the deformable walls.

Two thin PDMS films were used as the deformable top walls. One PDMS film with a thickness of $0.43 \pm 0.01\,{\textrm {mm}}$ was fabricated in the laboratory following the procedure of Chun et al. (Reference Chun, Boyko, Christov and Feng2024). The other PDMS film (GASKET-UT-200PK) with a thickness of $0.2 \pm 0.005\,{\textrm {mm}}$ was purchased from SIMPore Inc., USA. The films and microchannel substrate were treated with a 4.5 MHz hand-held corona treater (BD-20AC, Electro-Technic Products, USA) for 30 s, and then we brought the film and the channel substrate together into conformal contact for bonding. We note that this bonding approach for oscillatory flows was proven to be robust, as confirmed by the reproducibility of our experiments. We further attached a rigid glass slide on top of the thin PDMS film, other than the section between the first port and the outlet, to confine the deformability solely to this section. We confirmed all dimensions of the channel and the thin PDMS film by microscope visualisation (table 2). We also measured the Young’s modulus $E$ and Poisson’s ratio $\nu _s$ of the thin PDMS film using dynamic mechanical analysis (Q800 DMA, TA instruments, USA) at a room temperature of $20\,^\circ {\textrm {C}}$ . For the PDMS films that we fabricated in the laboratory, the measurements of the Young’s moduli were performed after 24 h to ensure the material properties of the PDMS films were fully stabilised. The density $\rho _s$ from the datasheet is reported.

To validate our experimental system for oscillatory flows, we also fabricated a rigid microchannel, which is the counterpart of the deformable microchannels with the same dimensions and configuration, except that the top wall was replaced by a large PDMS block. Using the same fabrication procedures as for the channel substrate, the PDMS block was fabricated to be as thick as the channel substrate (thickness $\approx 1\,{\textrm {cm}}$ ), ensuring that its rigidity prevents any deformation under the typical imposed hydrodynamic pressure (amplitude $\approx 0.1\,{\textrm {kPa}}$ ) used for the experiments with the thin-film top walls. Subsequently, the large PDMS block was attached to its dedicated channel substrate by undergoing the same surface treatment as for the thin-film top walls described above.

3.3. Working fluids and pressure measurements

Two different viscous Newtonian fluids, deionised water and an aqueous solution of 50 wt% glycerine, were used to tune the Womersley number ${\textit{Wo}}$ and the elastoviscous number $\gamma$ . The working fluids’ densities and viscosities are reported in table 3, which were measured right before each experiment. The viscosity of the working fluids was quantified by flow sweep tests performed using a stress-controlled rheometer (DHR-3, TA Instruments, USA).

Table 3. Physical properties of the working fluids.

For the pressure measurement, the gauge pressure at each pressure port was recorded by a pressure transducer (PX409-10WGUSBH, OMEGA, USA) wired to the PC with control software (Digital Transducer Application, OMEGA, USA), using a sampling rate of $1000\,{\textrm {Hz}}$ for all cases, which is much larger than the frequency of the pressure signal in our experiments (typically set between $1$ and $16\,{\textrm {Hz}}$ ). Prior to each experiment, the pressure transducer was calibrated using a column of water. Then, the entire interior space of the system, including the tubing connected to the pressure transducer, was completely filled with the working fluid. Thereafter, the top opening of the PDMS chamber (used to fill it with fluid) was kept closed, and the channel outlet was completely submerged under the free surface of the fluid in the reservoir. The five pressure ports were also blocked by the tubing connected to the pressure transducer and four plugs to maintain a hermetically sealed liquid environment.

We measured the pressure distribution of both the primary flow (derived in § 4.1) and the secondary streaming flow (derived in § 4.2). For the primary flow, we synchronised the pressure’s time variation in each port with one pressure transducer as follows. At the onset of each experiment, the function generator was configured to produce a sinusoidal waveform with a predetermined frequency and voltage. As the speaker diaphragm continuously oscillated under the excitation of the function generator, pressure variations over time were recorded. For a given pressure input, characterised by an amplitude and frequency, we first recorded data at the first port, $p_1(t)=p(z_1,t)$ , as shown in figure 2(b) (hereafter referred to as ‘port 1’) for a sufficiently long duration. Next, to synchronise the $p_i(t)$ data of port $i$ (where $i = 2$ , 3, 4 or 5) onto the time axis of port 1’s data under the same pressure input, we implemented a staged recording method for port $i$ . First, we began the data collection on port 1. Second, we transferred the pressure sampling tube from port 1 to port $i$ . Third, we continued data collection at port $i$ before concluding the session. This entire process was completed within the total duration of port 1’s data acquisition in the first session. To align the pressure datasets, we applied a temporal adjustment by shifting the whole $p_i(t)$ curve of the second session along the time axis until its initial time slot measuring port 1 precisely overlapped with port 1’s reference data from the first session, eliminating any phase discrepancy. This approach ensured that the data recorded for each port were accurately synchronised with the time axis established by port 1’s dataset. The same procedure was systematically repeated for all pressure measurement ports, ultimately yielding a fully synchronised pressure dataset across all five ports for a given pressure input.

On the other hand, to measure the weak streaming pressure in the secondary flow for each pressure port, we first obtained a baseline pressure value with no flow (the speaker off), then recorded the pressure signal with the flow (the speaker on). The difference between the cycle-averaged value of the pressure signal with the flow and the baseline pressure reflects the streaming effect therein. We used these pressure differences to further calculate the cycle-averaged streaming pressure, which, in this case, was the sole measurement in which we are ultimately interested, meaning the synchronisation between ports was not critical.

4.1. Primary flow: $O(1)$ solution

Substituting the expansion (4.1) into the momentum equations (2.4b ) and (2.4c ), we find that the primary pressure $P_0$ does not vary with $X$ or $Y$ . Then, at $O(1)$ , the $Z$ -momentum equation (2.4d ) and the no-slip BCs from (2.5a ) and (2.6c ) become

(4.2) \begin{equation} \left. \begin{array}{l} \displaystyle {\textit{Wo}}^2 \frac {\partial V_{Z,0}}{\partial T} = - \frac {\textrm{d} P_0}{\textrm{d} Z} + \frac {\partial ^2 V_{Z,0}}{\partial Y^2}, \\[16pt] \displaystyle V_{Z,0}|_{Y=0} = V_{Z,0}|_{Y=1} = 0. \end{array} \right \} \end{equation}

We expect the primary flow to be harmonic; thus, we introduce phasors: $V_{Z,0}(Y,Z,T)=\textrm {Re}[V_{Z,0,a}(Y,Z)\textrm{e}^{\textrm{i} T}]$ and $P_{0}(Z,T)=\textrm {Re}[P_{0,a}(Z)\textrm{e}^{\textrm{i} T}]$ . Substituting the phasors into (4.2), the solution for the axial velocity phasor’s amplitude is easily found (see e.g. Pande et al. Reference Pande, Wang and Christov2023) to be

(4.3) \begin{equation} V_{Z,0,a}(Y,Z) = \frac {1}{\textrm{i}{{\textit{Wo}}}^2}\left [1-\frac {\cos \left (\textrm{i}^{3/2} (1-2Y){{\textit{Wo}}}/2\right)}{\cos \left (\textrm{i}^{3/2} {{\textit{Wo}}}/2 \right)}\right ] \left ( -\frac {\textrm{d} P_{0,a}}{\textrm{d} Z} \right). \end{equation}

Now, the flow rate phasor’s amplitude is evaluated by substituting (4.3) into (2.12) to yield

(4.4) \begin{equation} \left. \begin{array}{l} \displaystyle Q_{0,a} = \int _{-1/2}^{+1/2} \int _{0}^{1} V_{Z,0,a} \,\textrm{d} Y \textrm{d} X = \mathfrak{f}({\textit{Wo}}) \left (-\frac {\textrm{d} P_{0,a}}{\textrm{d} Z}\right),\\[16pt] \displaystyle \mathfrak{f}({\textit{Wo}}) \stackrel {\text{def}}{=} \frac {1}{\textrm{i} {\textit{Wo}}^2}\left [1 - \frac {1}{\textrm{i}^{3/2}{{\textit{Wo}}}/2}\tan \left (\textrm{i}^{3/2}{{\textit{Wo}}}/2\right)\right ]. \end{array} \right \} \end{equation}

Next, we determine the top wall displacement. Having verified that ${St}_s\ll 1$ , we neglect the wall’s inertia. Then, the solution of (2.8) subject to the BCs (2.10) (see e.g. Shidhore & Christov Reference Shidhore and Christov2018) is

(4.5) \begin{equation} U_Y(X,Z,T) = 30 \left (\frac {1}{4} - X^2\right) \left [ \left (\frac {1}{4} - X^2\right) + \frac {\mathscr{T}}{5} \right ] P(Z,T). \end{equation}

Substituting (4.5) into (2.13), performing the $X$ integration and then substituting the perturbation expansions for $Q$ and $P$ in terms of phasors, the continuity equation becomes

(4.6) \begin{equation} \frac {\textrm{d} Q_{0,a}}{\textrm{d} Z} + \left (1 + \mathscr{T}\right) \gamma \textrm{i} P_{0,a}(Z) = 0. \end{equation}

Finally, substituting the flow rate–pressure gradient relation (4.4) into the continuity equation (4.6), and taking into account the pressure BCs (2.7), we arrive at a boundary-value problem (BVP) for the primary pressure amplitude:

(4.7) \begin{equation} \left. \begin{array}{l} \displaystyle \mathfrak{f}({\textit{Wo}}) \frac {\textrm{d}^2 P_{0,a}}{\textrm{d} Z^2} = \textrm{i} \left (1 + \mathscr{T}\right) \gamma P_{0,a}(Z), \\[16pt] \displaystyle P_{0,a}(0) = 1,\quad P_{0,a}(1) = 0. \end{array} \right \} \end{equation}

Here, according to the experimental system’s set-up, we have taken $P_{\textit{in}}(T) = \textrm {Re}[\textrm{e}^{\textrm{i} T}] + O(\beta)$ , having imposed the amplitude of the pressure oscillations as the characteristic pressure scale via (2.1). We have left open the possibility of $O(\beta)$ corrections to the oscillatory pressure BC, which may arise from how the oscillatory flow was generated in the experiments. We return to this issue in § 4.2.

The solution to the BVP (4.7) is easily found to be

(4.8a,b) \begin{align} P_{0,a}(Z) = \frac {\sinh \big (\kappa (1-Z)\big)}{\sinh \kappa },\quad \kappa = \kappa ({\textit{Wo}},\gamma,\mathscr{T}) \stackrel {\text{def}}{=} \sqrt {\frac {\textrm{i}\left (1 + \mathscr{T}\right)\gamma }{\mathfrak{f}({\textit{Wo}})}}, \end{align}

which has the same form as the corresponding solution in an axisymmetric deformable tube (Ramachandra Rao Reference Ramachandra Rao1983; Dragon & Grotberg Reference Dragon and Grotberg1991; Zhang & Rallabandi Reference Zhang and Rallabandi2024). This solution is illustrated in figure 3. We observe that $\kappa /\sqrt {(1 + \mathscr{T})\gamma }$ is solely a function of ${\textit{Wo}}$ with asymptotics of $\sim \sqrt {3\textrm{i}} (2 + \textrm{i}{\textit{Wo}}^2/10)$ as ${\textit{Wo}}\to 0$ and $\sim \sqrt {\textrm{i}} (1 + \sqrt {\textrm{i}} {\textit{Wo}})$ as ${\textit{Wo}}\to \infty$ . The large- ${\textit{Wo}}$ asymptotics show a much faster ( $\sim {\textit{Wo}}$ ) growth for the 3-D rectangular channel compared with the 3-D axisymmetric tube ( $\sim \sqrt {{\textit{Wo}}}$ ) in Zhang & Rallabandi (Reference Zhang and Rallabandi2024).

Figure 3. (a) Dependence of the reduced complex ‘wavenumber’ $\kappa /\sqrt {(1+\mathscr{T})\gamma } = \sqrt {\textrm{i}/\mathfrak{f}({\textit{Wo}})}$ on ${\textit{Wo}}$ and its asymptotic behaviours (dashed curves labelled with arrows). (b) Shape of the primary pressure amplitude’s axial distribution, $\textrm {Re}[P_{0,a}(Z)]$ from (4.8a ), for ${\textit{Wo}} = 1$ (solid) and ${\textit{Wo}} = 3$ (dashed) and across a range of $\gamma$ values.

Observe that as the elastoviscous adjustment of the wall becomes instantaneous compared with the oscillation time scale ( $\gamma \to 0$ ), i.e. the channel is effectively rigid, then $\kappa \to 0$ and $P_{0,a}(Z) \to 1 - Z$ from (4.8), as expected.

4.2. Secondary flow: $O(\beta)$ solution

To define the secondary (streaming) flow problem, we must determine the axial velocity at the deformable wall in terms of known quantities. Following Boyko, Stone & Christov (Reference Boyko, Stone and Christov2022), we proceed by domain perturbation (Lebovitz Reference Lebovitz1982; Leal Reference Leal2007). To this end, recall that $H=1+\beta U_Y = 1+\beta U_{Y,0}+O(\beta ^2)$ , and expand the axial velocity at the wall in a Taylor series:

(4.9) \begin{equation} V_Z|_{Y=H} = V_{Z,0} |_{Y=1} + \left. \beta U_{Y,0} \frac {\partial V_{Z,0}}{\partial Y}\right |_{Y=1} + \beta V_{Z,1}|_{Y=1} + O(\beta ^2). \end{equation}

Enforcing the no-slip BC $V_Z|_{Y=H}=0$ due to (2.6c ), (4.9) requires that

(4.10a,b) \begin{align} V_{Z,0}|_{Y=1} = 0,\quad V_{Z,1}|_{Y=1} = - \left. U_{Y,0} \frac {\partial V_{Z,0}}{\partial Y}\right |_{Y=1}. \end{align}

Observe that the flow-induced deformation of the channel leads to an effective slip velocity $V_{Z,1}|_{Y=1}$ in (4.10b ) along the original location of the wall (Anand & Christov Reference Anand and Christov2020; Zhang & Rallabandi Reference Zhang and Rallabandi2024).

Since we are interested only in the streaming (or rectified) flow component, we now apply the cycle-averaging operator $\langle \, \boldsymbol{\cdot }\, \rangle \stackrel {\text{def}}{=} (1/2\unicode {x03C0}) \int _{0}^{2\unicode {x03C0}} (\, \boldsymbol{\cdot }\,) \,\textrm{d} T$ to (2.4d ) and (2.13) at $O(\beta)$ to obtain

(4.11a) \begin{align} \frac {{\textit{Wo}}^2}{\gamma } \left \langle V_{Y,0} \frac {\partial V_{Z,0}}{\partial Y} + V_{Z,0} \frac {\partial V_{Z,0}}{\partial Z} \right \rangle &= - \frac {\textrm{d} \langle P_1\rangle }{\textrm{d} Z} + \frac {\partial ^2 \langle V_{Z,1} \rangle }{\partial Y^2}, \end{align}

(4.11b) \begin{align} \frac {\partial \langle Q_1 \rangle }{\partial Z} &= 0. \end{align}

Observe that the left-hand side of (4.11a ) is independent of $X$ , and so is $\langle P_1 \rangle$ (due to the cycle-averaged (2.4b ) at $O(\beta)$ ). Thus, we can $X$ -average the $O(\beta)$ problem’s governing equation (4.11):

(4.12a) \begin{align} \frac {{\textit{Wo}}^2}{\gamma } \left \langle V_{Y,0} \frac {\partial V_{Z,0}}{\partial Y} + V_{Z,0} \frac {\partial V_{Z,0}}{\partial Z} \right \rangle &= - \frac {\textrm{d} \langle P_1\rangle }{\textrm{d} Z} + \frac {\partial ^2 \langle \!\langle V_{Z,1} \rangle \!\rangle }{\partial Y^2}, \end{align}

(4.12b) \begin{align} \frac {\partial \langle \!\langle Q_1 \rangle \!\rangle }{\partial Z} &= 0, \end{align}

where $\langle \!\langle \,\boldsymbol{\cdot }\, \rangle \!\rangle \stackrel {\text{def}}{=} (1/2\unicode {x03C0}) \int _{-1/2}^{+1/2} \int _{0}^{2\unicode {x03C0}} (\, \boldsymbol{\cdot }\,) \,\text{d} T\text{d} X$ denotes the simultaneous $T$ and $X$ averaging. The $T$ averages involving $O(1)$ phasors $\mathcal{A}=\text {Re}{[}\mathcal{A}_a \text{e}^{\text{i} T}{]}$ and $\mathcal{B}=\text {Re}{[}\mathcal{B}_a \text{e}^{\text{i} T}{]}$ are calculated by the standard rule $\langle \mathcal{A} \mathcal{B} \rangle = ({1}/{2}) \text {Re}{[}\mathcal{A}_a^* \mathcal{B}_a{]} = ({1}/{2}) \text {Re}{[}\mathcal{A}_a \mathcal{B}_a^*{]}$ , where a star superscript denotes complex conjugate.

Four conditions are required to simultaneously and uniquely determine $\langle \!\langle V_{Z,1}\rangle \!\rangle$ , $\langle \!\langle Q_1 \rangle \!\rangle$ and $\langle P_1\rangle$ from (4.12). The suitable BCs now correspond to no slip at $Y=0$ (from averaging (2.5a )) and effective slip at $Y=1$ (from averaging (4.10b )):

(4.13a,b) \begin{align} \langle \!\langle V_{Z,1} \rangle \!\rangle |_{Y=0} = 0,\quad \langle \!\langle V_{Z,1} \rangle \!\rangle |_{Y=1} = - \left \langle \!\left \langle \left. U_{Y,0} \frac {\partial V_{Z,0}}{\partial Y}\right |_{Y=1} \right \rangle \!\right \rangle. \end{align}

The conditions in the experiment are such that the membrane in the liquid-filled chamber used for oscillatory flow generation (recall figure 2) does not allow any net flow through the system. Then, according to (4.12b ), $\langle \!\langle Q_1 \rangle \!\rangle = \text{const.}$ , and this constant must be zero throughout. The outlet is open to the gauge pressure per (2.7). Thus, the remaining BCs are

(4.14a,b) \begin{align} \langle \!\langle Q_1 \rangle \!\rangle |_{Z=0} = 0,\quad \langle P_1 \rangle |_{Z=1} = 0. \end{align}

Using (4.5), the slip velocity becomes

(4.15) \begin{equation} \left \langle \!\left \langle \left. U_{Y,0} \frac {\partial V_{Z,0}}{\partial Y}\right |_{Y=1} \right \rangle \!\right \rangle = \left \langle \int _{-1/2}^{+1/2} U_{Y,0} \,\textrm{d} X \left. \frac {\partial V_{Z,0}}{\partial Y}\right |_{Y=1} \right \rangle = \left \langle \left. (1+\mathscr{T}) P_0 \frac {\partial V_{Z,0}}{\partial Y}\right |_{Y=1} \right \rangle. \end{equation}

It is convenient to now rewrite $\langle \!\langle V_{Z,1}\rangle \!\rangle$ in a way to eliminate the pressure gradient on the right-hand side of (4.12a ) and simultaneously satisfy the slip BC (4.13b ). Specifically, let

(4.16) \begin{equation} \langle \!\langle V_{Z,1} \rangle \!\rangle (Y,Z) = - \frac {1}{2}\frac {\textrm{d} \langle P_1\rangle }{\textrm{d} Z} Y(1-Y) - Y (1+\mathscr{T}) \left.\left \langle P_0 \frac {\partial V_{Z,0}}{\partial Y} \right |_{Y=1}\right \rangle + \widetilde {\langle \!\langle V_{Z,1} \rangle \!\rangle }(Y,Z). \end{equation}

Then, substituting (4.16) into (4.12a ), we obtain a new BVP for $\widetilde {\langle \!\langle V_{Z,1} \rangle \!\rangle }$ , subject to homogeneous BCs:

(4.17) \begin{equation} \left. \begin{array}{l} \displaystyle \frac {{\textit{Wo}}^2}{\gamma } \left \langle V_{Y,0} \frac {\partial V_{Z,0}}{\partial Y} + V_{Z,0} \frac {\partial V_{Z,0}}{\partial Z} \right \rangle = \frac {\partial ^2 \widetilde {\langle \!\langle V_{Z,1} \rangle \!\rangle }}{\partial Y^2},\\[16pt] \displaystyle \widetilde {\langle \!\langle V_{Z,1} \rangle \!\rangle }|_{Y=0} = \widetilde {\langle \!\langle V_{Z,1} \rangle \!\rangle }|_{Y=1} = 0. \end{array} \right \} \end{equation}

An analytical solution appears unlikely since the left-hand side of the ordinary differential equation in (4.17) is a complicated, complex-valued function of $Y$ and $Z$ , but some useful approximations are discussed in Appendix A. Instead, we solve the linear two-point BVP (4.17) numerically using Matlab’s bvp4c subroutine, which implements a finite-difference solver with residual-based error control (Kierzenka & Shampine Reference Kierzenka and Shampine2001), with absolute and relative tolerance of $10^{-6}$ .

Finally, from (4.12b ), (4.14a ) and (4.16), we conclude that

(4.18) \begin{equation} \langle \!\langle Q_1 \rangle \!\rangle = \int _0^1 \langle \!\langle V_{Z,1} \rangle \!\rangle \,\textrm{d} Y = - \frac {1}{12}\frac {\textrm{d} \langle P_1\rangle }{\textrm{d} Z} + \mathfrak{Q}(Z) = 0, \end{equation}

where, for convenience, we have let

(4.19) \begin{equation} \mathfrak{Q}(Z) \stackrel {\text{def}}{=} \underbrace {- \frac {1}{2} (1+\mathscr{T}) \left \langle \left. P_0 \frac {\partial V_{Z,0}}{\partial Y} \right |_{Y=1}\right \rangle }_{\text{from effective slip}} + \underbrace {\int _0^1 \widetilde {\langle \!\langle V_{Z,1} \rangle \!\rangle } \, \textrm{d} Y}_{\text{from advective inertia}}. \end{equation}

Equation (4.18) is a first-order differential equation for the streaming pressure $\langle P_1 \rangle$ subject to the outlet BC (4.14b ), which is easily solved to obtain

(4.20) \begin{equation} \langle P_1 \rangle (Z) = - 12 \int _Z^1 \mathfrak{Q}(\tilde {Z})\, \textrm{d} \tilde {Z}. \end{equation}

In the experiment, the membrane in the liquid-filled chamber used for oscillatory flow generation imposes a weak, $O(\beta)$ , non-zero mean pressure at the inlet, which is consistent with (4.20). To plot $\langle P_1 \rangle (Z)$ , we evaluate $\mathfrak{Q}(Z)$ from (4.19) wherein the integral over $Y$ is computed numerically using the trapezoidal rule via Matlab’s trapz with $\Delta Y=0.0101$ from the numerical solution for $\widetilde {\langle \!\langle V_{Z,1} \rangle \!\rangle }$ of BVP (4.17). Then, the indefinite integral in (4.20) is evaluated numerically using Matlab’s cumtrapz using $\Delta Z=0.0204$ .

5.1. Validation in a rigid channel

To validate the experimental system, we first assessed its performance using a rigid channel ( $\gamma =0$ ) with deionised water as the working fluid. This validation is accomplished by comparing the experimental pressure data for the rigid channel with the $\gamma \to 0$ limit of the theory, which gives $P(Z,T) = \textrm {Re}[(1-Z)\textrm{e}^{\textrm{i} T}]$ from (4.8). We considered three validation cases with different input frequencies (corresponding to ${\textit{Wo}} = 2.5$ , $3.32$ and $3.96$ ). The case of ${\textit{Wo}} = 2.5$ is shown as an example in figure 4. We observe good agreement between the theory and experiments for the axial distribution of $P$ and its variation over time in figure 4(b). The same holds for the other two validation experiments (not shown).

Note that the experimental pressure data time series (figure 4 a) does not show any phase difference between the pressure signals at the different axial positions, which is to be contrasted with the results for the deformable channel below (§ 5.2). The signals’ amplitudes exhibit a trend of linear attenuation with $z$ , characterised by a constant multiplicative relationship between the values at different $z$ . Correspondingly, in figure 4(b), these observations are reflected in the linear variation of $P$ with $Z$ at every $T$ over half an oscillation cycle.

Figure 5. Experimental measurements of the evolution of the pressure over time at different axial positions (pressure port locations) for the deformable channel for smaller and larger compliance numbers (left-hand column versus right-hand column) and smaller and larger Womersley numbers (top row versus bottom row). Specifically, (a) ${\textit{Wo}}=0.537$ ( $\gamma =0.109$ ), (b) ${\textit{Wo}}=0.537$ ( $\gamma =0.913$ ), (c) ${\textit{Wo}}=2.15$ ( $\gamma =1.745$ ) and (d) ${\textit{Wo}}=2.15$ ( $\gamma =14.6$ ).

5.2. Comparison of primary pressure oscillations in deformable channels

Next, we turn to the experiments in the deformable channels, based on a 50 wt% glycerine solution as the working fluid. In this subsection, we discuss the primary (purely oscillatory) pressure (§ 4.1) in the compliant channel. In figure 5, we show the experimentally measured pressure evolution at each of the different pressure ports for four pairs of values of the Womersley and compliance numbers – a low value ${\textit{Wo}}=0.537$ (slow flow oscillation) and a high value ${\textit{Wo}}=2.15$ (fast flow oscillation), as well as two different orders of magnitude of $\beta$ , namely $\simeq 10^{-2}$ and $\simeq 10^{-1}$ . To change the compliance number $\beta$ , the thickness of the deformable top wall was varied in the experiments (recall table 2). In figure 5, the mean pressure has been subtracted to set the outlet pressure as the gauge and to make the signal purely oscillatory.

Comparing the pressure time series in figure 5 (e.g. figure 5 b) with time series in the rigid channel in figure 4(a), we observe a distinct phase difference developing between the time series collected at different $z$ (i.e. at different pressure ports). Furthermore, the decrease in the signals’ amplitudes is not proportional in the deformable channel, unlike in the rigid channel. These differences are expected to arise from the nonlinear two-way coupling of the flow and deformation (increasing $\beta$ ).

Figure 6. Comparison of the dimensionless axial pressure distribution in a deformable channel between experiment (symbols) and theory (solid curves), i.e. $P_0(Z,T) = \textrm {Re}[P_{0,a}(Z)\textrm{e}^{\textrm{i} T}]$ based on (4.8), with (a) ${\textit{Wo}}=0.537$ ( $\beta =0.0208$ ), (b) ${\textit{Wo}}=0.537$ ( $\beta =0.164$ ), (c) ${\textit{Wo}}=1.42$ ( $\beta =0.02$ ), (d) ${\textit{Wo}}=1.42$ ( $\beta =0.125$ ), (e) ${\textit{Wo}}=2.15$ ( $\beta =0.0167$ ) and (f) ${\textit{Wo}}=2.15$ ( $\beta =0.104$ ). The evolution of the pressure distribution is shown over a full cycle (thus $T=2\unicode {x03C0}$ overlaps $T=0$ ).

To clearly demonstrate the nonlinear coupling, in figure 6, we compare the dimensionless primary pressure distribution $P_0(Z,T) = \textrm {Re}[P_{0,a}(Z)\textrm{e}^{\textrm{i} T}]$ from (4.8) predicted by the theory with the experimental data. We neither construct $P_1(Z,T)$ , which is not straightforward (Zhang & Rallabandi Reference Zhang and Rallabandi2024), nor neglect its $O(\beta)$ contribution, as we have ensured a one-to-one comparison of primary theoretical and experimental pressures by removing the mean of the experimental time series. Notably, unlike the steady (Christov et al. Reference Christov, Cognet, Shidhore and Stone2018) and startup (Martínez-Calvo et al. Reference Martínez-Calvo, Sevilla, Peng and Stone2020) problems, the key dimensionless groups influencing the pressure distribution in the deformable channel are the Womersley number ${\textit{Wo}}$ and the elastoviscous number $\gamma$ , not the compliance number $\beta$ .

The results in figure 6 show a good agreement between the theory of the primary pressure oscillations $P_0(Z,T)$ and the experimental measurements over a wide range of ${\textit{Wo}}$ and $\gamma$ . We observe that the agreement between theory and experiment is better for $0.1\lt \gamma \lt 1$ than for $\gamma \gt 1$ . This deviation can be attributed to the fact that for larger values of $\gamma \gt 1$ , the combined effect of compliance and oscillations is strong, and the time it takes the top wall to adjust to the flow oscillations becomes much longer than the oscillation time scale, localising the majority of the pressure variation near the channel’s inlet. This rapid, localised variation is more challenging to capture using equally spaced pressure ports in the experiments. Nevertheless, the overall agreement between theory and experiment on the trend of $P_0(Z,T)$ and how it changes with ${\textit{Wo}}$ and $\gamma$ is good, thus not only validating the predicted nonlinear pressure distribution (4.8) but also providing the first experimental demonstration of the strong coupling between flow oscillations and wall deformations, even in weakly compliant channels ( $\beta \ll 1$ ).

5.3. Comparison of streaming pressure profiles in deformable channels

Next, we turn to the elastoinertial rectification phenomenon, namely the theoretical prediction that $\langle P \rangle /\beta \equiv \langle P_1 \rangle \ne 0$ due to the nonlinear coupling of the flow’s inertia with the wall deformation (§ 4.2). All data shown in this section are based on deionised water as the working fluid. To this end, in figure 7, we compare the streaming pressure $\langle P_1 \rangle (Z)$ calculated numerically from (4.20) (as described above) with the corresponding quantity extracted from the experiments.

Figure 7. (a) Comparison of streaming pressure (cycle-averaged pressure) distribution, $\langle P \rangle /\beta = \langle P_1 \rangle$ , between the experimental data (symbols), the theoretical prediction based on numerically evaluating (4.20) (solid curves) and the closed-form approximation (A7) (dashed curves, overlapping the solid curves) in a deformable channel with $\beta \approx 0.17$ and $\gamma =0.15$ ( ${\textit{Wo}} = 1.25$ ), $\gamma =0.6$ ( ${\textit{Wo}} = 2.5$ ) and $\gamma =1.048$ ( ${\textit{Wo}} = 3.13$ ) achieved by changing the input frequency in the same channel. (b) Ad hoc modification of the theory by multiplying the effective slip term in (4.19) by $0.4$ to demonstrate the sensitivity of the streaming pressure to the effective slip contribution.

This comparison is more challenging than the previous one in § 5.2 because we are now dealing with small quantities that are $O(\beta)$ . Consequently, the error bars on the experimental data in figure 7 are much larger as the pressure values being measured push the experimental system to its sensitivity limit. Nevertheless, in figure 7(a), we see a reasonable agreement between experiment and theory regarding the trend of the streaming pressure distribution along the channel. Interestingly, $\langle P_1\rangle$ is less sensitive to $\gamma$ under the present flow conditions, and both the theory curves and experimental data cluster together.

The largest disagreement is at $Z=0$ , at the first pressure port, which may be expected as this is the location in the experimental system that is least likely to satisfy all the assumptions of the theory. Specifically, as shown in figure 2, there is a rigid section attached to the inlet of the deformable channel in the experiments, which constrains the displacement along the inlet plane. However, this possibility is not accounted for in the theory – the leading-order equation (2.8) for the displacement does not allow for BCs to be imposed in $Z$ and thus cannot capture this localised clamping at the inlet, which previous work showed is confined to a ‘boundary layer’ of thickness $O(\sqrt {w/\ell })$ (Wang & Christov Reference Wang and Christov2021). While the inlet conditions on the displacement have no discernible effect on $P_0(Z)$ , as demonstrated in § 5.2 by the excellent agreement between theory and experiments at $Z=0$ , these conditions may affect the weaker effects being investigated at $O(\beta)$ .

To test the hypothesis that the inlet conditions may affect the agreement there, we turn to (4.19) and (4.20), from which we observe that the streaming pressure is generated by a competition between effective wall slip (at the location of the undeformed wall) and advective inertia. Specifically, the effective slip is a direct function of the displacement, per (4.15). Therefore, we expect this $O(\beta)$ quantity to be possibly strongly affected by the inlet restrictions in the experiments. Specifically, if the displacement in the experiments near the inlet is constrained, or otherwise reduced, then this term might be overestimated by the theory. To test this hypothesis, we check the sensitivity of the $\langle P_1\rangle (Z)$ profile to the magnitude of the effective slip term. We find that even approximately halving this term can account for much of the disagreement between theory and experiment, especially as $Z\to 0$ , as shown by the ad hoc modification in figure 7(b).

Though the ad hoc modification improves the agreement to quite an extent for the lowest frequency (lowest ${\textit{Wo}}$ and $\gamma$ values), it still exhibits disagreement for the larger frequencies (larger ${\textit{Wo}}$ and $\gamma$ values). In fact, fully suppressing the effective slip term brings the streaming pressures at the higher frequency even closer to the experimental data points. To further investigate the frequency-dependent nature of this agreement/disagreement, we would need to better assess how well our theory models the effective slip term in (4.19). The present theory evaluates this term using the result that $U_{Y,0} \propto P_0$ , per (4.15), and the shear $\partial V_{Z,0}/\partial Y$ based on (4.3). We would need a more advanced experimental set-up, which measures the velocity field and the deformation independently, to evaluate the accuracy of each of these results. Alternatively, measuring just the slip velocity at $Y=1$ experimentally, say using particle image velocimetry, and comparing it with (4.10b ) could be another independent check. These directions will be pursued in forthcoming investigations.

Despite the limitations and challenges of these measurements, the experimental data appear to capture the key effects of ${\textit{Wo}}$ and $\gamma$ on the streaming pressure profile, including the non-monotonic behaviour with respect to $\gamma$ (in particular for $Z\gt 0.5$ ), though admittedly, the error bars on the experimental measurements for different $\gamma$ overlap.