5.1. Effect of viscoelastic properties on fluid perturbations

A S-BAW induces perturbations in the viscoelastic fluid present in the microchannel. We analyse the variation of the first-order fields $(\kern0.7pt \tilde {p}_1, \tilde {{v}}_1)$ along the channel width ($\kern0.7pt y$-direction) (refer to figure 1a) and depth ($z$-direction) in the absence of particles by numerically solving (2.11)–(2.13), as detailed in § 2.3. These perturbed fields act as incoming fields for the particle, hence we name them as $p_{in}$ and $v_{in}$. Here, $v_{in}$ indicates the component of perturbed velocity field along the $y$-direction. The perturbed fields are characterized using dimensionless first-order fluid pressure, $p_{in}^* = \tilde {p}_{in}/(\rho _0 c_0 A \omega )$, and dimensionless first-order fluid velocity, $v_1^* = \tilde {v}_{in} / (A \omega )$ (Vargas et al. Reference Vargas, Campos-Silva, Méndez, Arcos and Bautista2022), where $A$ is the amplitude of vibration of the channel, and $\rho _0$ and $c_0$ are respectively the density and speed of sound of the viscoelastic fluid. The variations of the incoming fields are shown in figure 4. Here, $y^* = y/(W/2)$ and $z^* = z/(D/2)$. We compare the field fluctuations in a viscoelastic fluid for $De=0.1, 1.0, 10, 300$ and $10^5$ with those in a Newtonian fluid ($De = 0$), for a fixed total viscosity ($\mu ^* = 1001$, with $\mu _p^* = 1000$ and $\mu _s^* = 1$), density ($\rho ^* = 1$) and speed of sound ($c^* = 1$), thereby decoupling the elastic effects from the viscous effects.

Figure 4. Numerical results of dimensionless incoming pressure $(p_{in}^*)$ and velocity $(v_{in}^*)$ fields in Newtonian ($De=0$) and viscoelastic fluids ($De=0.1, 1, 10, 300, 10^5$) at $\mu _p^*=1000$, $\mu _s^*=1$, $\rho ^*=1$ and $c^*=1$. Variation of (a) $p_{in}^*$ along $y^*$ (along width or S-BAW direction, at $z^*=1$), (b) $v_{in}^*$ along $y^*$ ($\kern0.7pt y^*=0$ and 2 as the channel walls, $y^*=1$ as the centre), (c) $p_{in}^*$ along $z^*$ (along depth or perpendicular to S-BAW, at $y^*=0.5$) and (d) $v_{in}^*$ along $z^*$ (symmetric about the channel centre $z^*=1$, $z^*=0$ is the channel wall perpendicular to S-BAW direction), refer to figure 1. (e) Theoretical and numerical variation of dimensionless boundary layer thickness (${\delta }_{v e}^*$), viscous shear wavelength (${\lambda }_{ve}^*$), $\alpha$ (${\lambda }_{ve}^*/{\delta }_{v e}^*$) and $E_{ac}^*$ with $De$.

We depict the distribution of $p_{in}^*$ along the channel width $y^*$ in figure 4(a), where its magnitude varies from a maximum at the channel walls ($\kern0.7pt y^* = 0$ or $y^* = 2$) to zero at the channel centre ($\kern0.7pt y^* = 1$). A zero value of $p_{in}^*$ indicates a pressure node, signifying the absence of the primary ARF. In contrast, $v_{in}^*$ attains a maximum at the centre and decreases to zero at the walls, suggesting a velocity node at the walls, as shown in figure 4(b). The fluid field variation along $z^*$ is symmetric about the channel centre $z^* = 1$. A decrease in $p_{in}^*$ from the wall $(z^* = 0)$ to the centre $(z^* = 1)$ is seen in figure 4(c), while figure 4(d) depicts oscillations in $v_{in}^*$ near the wall that stabilize at the centre. These fluctuations can be attributed to viscous shear disturbances and acoustic boundary layer formation resulting from the no-slip condition at the walls and acoustic perturbations (Hamilton, Ilinskii & Zabolotskaya Reference Hamilton, Ilinskii and Zabolotskaya2003; Baasch & Dual Reference Baasch and Dual2018). Specifically, the ABLs formed at the top and bottom walls of the microchannel (shown in figure 1a) affect the pressure and velocity distribution along the $z$-direction. A change in nature of shear waves near particles with viscoelastic parameters has been reported by Doinikov et al. (Reference Doinikov, Fankhauser and Dual2021a). However, in their study, the discussion is limited to the effect of boundary layer formation on a particle in a viscoelastic fluid from the perspective of acoustic streaming. Here, we study the effects of viscoelasticity on the incoming waves and unravel the effects of boundary layer formation at the channel walls on the perturbations.

Here, the viscous shear disturbances propagate from the walls to the bulk, in the $z$-direction, i.e. perpendicular to the S-BAW direction, influenced by the fluid properties. The characteristics of viscous shear waves with $De$ are shown in figure 4(e), where the viscous shear wave is characterized using the dimensionless boundary layer thickness (${\delta }_{ve}^*=\delta _{ve}/a$) and the dimensionless viscous shear wavelength (${\lambda }_{ve}^*=\lambda _{ve}/a$). The parameter ${\delta }_{ve}^*$ indicates the penetration distance of viscous disturbances, while ${\lambda }_{ve}^*$ provides an idea about the wavelength of shear waves or the number of oscillations in the vicinity of the channel wall, as obtained from (2.20) and (2.21). In the absence of elasticity ($De \approx 0$) or in low elasticity fluids (at very low values of $De =0.1$), the shear waves attenuate easily due to higher viscous effects, and oscillations are overdamped (shown in figure 4d), indicated by a smaller ${\delta }_{ve}^*$ and larger ${\lambda }_{ve}^*$ (${\lambda }_{ve}^* > {\delta }_{ve}^*$) in figure 4(e). Here, $\varsigma =\rho _0 \omega /(2\mu _{bf} \mu ^*)$ and $\chi =0$ (2.23a,b). From (2.13), (2.20) and (2.21), the dimensionless complex viscosity $\mu _c^* (=\mu _c/\mu _{bf})$, $\delta _{ve}^*$ and $\lambda _{ve}^*$ reduce to

(5.1a–c)\begin{equation} \mu_c^*|_{De \rightarrow 0}= \mu_s^*+\mu_p^*, \quad \delta_{ve}^*|_{De \rightarrow 0}=\frac{1}{a}\sqrt{\frac{2 \mu_{bf}(\mu_s^*+\mu_p^*)}{\rho_0 \omega}} \quad \textrm{and} \quad \lambda_{ve}^*|_{De \rightarrow 0}=2 {\rm \pi}\delta_{ve}^*. \end{equation}

As $De$ increases to intermediate values ($De=1.0$ and $10$) or in moderate elasticity fluids, the amplitude, frequency and propagation distance of viscous shear waves increase due to the transition from a viscous to an elastic state. This transition is shown by the variation of $v_{in}^*$ over a larger distance from the channel wall in figure 4(d). Further, as depicted in figure 4(e), there is an increase in ${\delta }_{ve}^*$ and a decrease in ${\lambda }_{ve}^*$, with ${\delta }_{ve}^*$ being greater than ${\lambda }_{ve}^*$. Here, $\mu _c^*$, $\varsigma$ and $\chi$ reduce to

(5.2a–c) \begin{gather}\begin{gathered} \mu_c^*|_{De}= \mu_s^*+\frac{\mu_p^*}{1+iDe}, \nonumber\\ \varsigma|_{De}=\frac{\rho_0 \omega[\mu^*+\mu_s^* De^2]}{2 \mu_{bf} [\mu^{*2}+\mu_s^{*2} De^2]} \quad \textrm{and} \quad \chi|_{De}=\frac{\rho_0 \omega[\mu^* -\mu_s^*]De}{2 \mu_{bf} [\mu^{*2}+\mu_s^{*2}De^2]}. \end{gathered}\end{gather}

By substituting the above values of $\varsigma$ and $\chi$ into (2.20) and (2.21), we obtain the variation of $\delta _{ve}^*$ and $\lambda _{ve}^*$ for intermediate $De$ values.

At a higher $De$ or for high elasticity fluids, disturbances attenuate faster with further increases in $De$ ($De = 300$ and $10^5$) due to a decrease in the contribution of $\mu _p^*$ to $\mu _c^*$, as indicated by a decrease in ${\delta }_{ve}^*$ in figure 4(d), which stabilizes at higher $De$ values. At very high $De$ ($=10^5$), the polymer contribution to the complex viscosity approaches zero, restricting the propagation of viscous shear waves to a smaller distance from the wall, indicated by variations in $v_{in}^*$ very close to the wall in figure 4(d) and negligible variation of ${\delta }_{ve}^*$ and ${\lambda }_{ve}^*$ with $De$ in figure 4(e), where ${\lambda }_{ve}^*>\delta _{ve}^*$. Here, $\varsigma =\rho _0 \omega /(2\mu _{bf} \mu _s^*)$ and $\chi =0$. The values of $\mu _c^*$, $\delta _{ve}^*$ and $\lambda _{ve}^*$ reduce to

(5.3a–c)\begin{equation} \mu_c^*|_{De \rightarrow \infty}= \mu_s^*, \quad \delta_{ve}^*|_{De \rightarrow \infty}=\frac{1}{a}\sqrt{\frac{2 \mu_{bf}\mu_s^*}{\rho_0 \omega}} \quad \textrm{and} \quad \lambda_{ve}^*|_{De \rightarrow \infty}=2 {\rm \pi}\delta_{ve}^*. \end{equation}

The variation in amplitude of $v_{in}^*$ along $y^*$ and $z^*$ shows a non-monotonic pattern with increasing $De$, as shown in figures 4(b) and 4(d). At low $De$ $(De \leq 1)$, higher momentum transfer to the fluid occurs with increasing $De$ or elasticity, causing an increase in $v_{in}^*$. Beyond $De=1$, due to a significant reduction in effective viscosity, $v_{in}^*$ decreases. Our results in figures 4(a) and 4(c) illustrate an increase in pressure amplitude of incoming waves from $De = 0$ to $De = 300$ at a fixed $\mu ^*$, followed by a small decrease and then negligible variation with further increases in $De$. However, the rate of increase in $p_{in}^*$ with $De$ is minimal for fluids with low elasticity ($De \ll 1$), more pronounced for fluids with moderate elasticity (intermediate $De$) and negligible for fluids with high elasticity $(De=10^5)$. Interestingly, in narrow microchannels where the acoustic boundary layer thickness $\delta _{ve}$ is comparable to the channel depth $D$ but much smaller than the channel width $W$ and acoustic wavelength $\lambda _0$ ($\delta _{ve} \sim D \ll W \sim \lambda _0$), Vargas et al. (Reference Vargas, Campos-Silva, Méndez, Arcos and Bautista2022) reported a continuous increase in pressure with relaxation time, with pressure being proportional to $De^{1/2}$. However, this relationship does not hold for commonly used microchannels in acoustofluidics, where $\delta _{ve}$ is smaller and both $D$ and $W$ are comparable to $\lambda _0$ ($\delta _{ve} < D \sim W \sim \lambda _0$). Thus our findings highlight a remarkable differences in the behaviour of pressure waves in commonly used microchannels as compared with narrow microchannels, highlighting the unique attributes of the present study.

Generally, a fluid undergoes relaxation when exposed to external perturbations, and it depends on the relaxation time scale $\tau$. In a viscous fluid, $\tau \sim 0$, and in an elastic body, $\tau \sim \infty$. Viscoelastic fluids have a relaxation time between the two extremes (Bird, Armstrong & Hassager Reference Bird, Armstrong and Hassager1977; Joseph Reference Joseph2013), and $De$ represents the fluid relaxation time $(\tau )$ with respect to the time period of acoustic perturbation $(t_f=1/\omega )$. For viscoelastic fluids with $De \ll 1$ ($\tau \ll t_f$), stress generated by the perturbation relaxes faster than the actuation period, causing a low pressure amplitude. At high $De$ ($\tau \gg t_f$), stress generated by the perturbation relaxes very slowly, showing higher incoming pressure amplitude compared with a Newtonian fluid ($De = 0$). Therefore, viscoelastic fluids exhibit a distinct behaviour compared with Newtonian fluids due to the variations in fluid's elasticity, ${\delta }_{ve}^*$ and ${\lambda }_{ve}^*$, as shown in figure 4(a–d).

We introduce a parameter $\alpha = {\lambda }_{ve}^* / {\delta }_{ve}^*= {\lambda }_{ve} / {\delta }_{ve}$, shown in figure 4(e), to study the relative variation of the length scales related to the viscous shear wave. The viscous shear wave parameter $\alpha$ attains a value of approximately $2 {\rm \pi}$ for low elasticity ($De \ll 1$) and high elasticity fluids $De$ ($De \gg 1$). For moderate elasticity fluids (moderate $De$ values), $\alpha$ decreases up to $De \approx 10$, beyond which it increases. This trend can be attributed to the variations in ${\delta }_{ve}^*$ and ${\lambda }_{ve}^*$, providing deeper insights into the viscoelasticity of fluids. The variations in the incoming fields directly indicate changes in the AED of the fluid. We characterize AED using the dimensionless AED ($E_{ac}^*$), defined as the ratio of $E_{ac}$ for a viscoelastic fluid to that of the base Newtonian fluid, $E_{ac}^* = {E_{ac}^{ve}}/{E_{ac}^{bf}}$, as shown in figure 4(e). Interestingly, at a fixed value of $\mu ^*$, we observe that, with increasing elasticity or $De$ (keeping $t_f$ fixed), $E_{ac}^*$ increases, mirroring the variation in amplitude of the incoming pressure wave, as shown in figure 4(a). The rate of increase, however, varies across the different ranges of $De$. For low elasticity fluids ($De\ll 1$), the change in $E_{ac}^*$ is negligible. Conversely, moderate elasticity fluids (moderate $De$) show a significant rise in $E_{ac}^*$ with $De$, corresponding to a marked increase in incoming pressure wave amplitude in figure 4(a). Further increase in $De$ results in a negligible change in the incoming pressure field, leading to a nearly fixed value of $E_{ac}^*$ in highly elastic fluids ($De \gg 1$). Viscous effects dominate in low elasticity fluids, while elasticity effects surpass viscous effects in high elasticity fluids. Higher viscous effects indicate an energy dissipation state, whereas higher elasticity effects indicates an energy storage state. Thus, with increasing $De$, a viscoelastic fluid transitions from an energy dissipation state (relaxation mode or relaxing state of AED) to an energy storage state (frozen mode or freezing state of AED) where energy dissipation is minimized (Bird et al. Reference Bird, Armstrong and Hassager1977; Joseph Reference Joseph2013).

In the next section, we investigate the transition of AED with $De$ at different $\mu _p^*$ and $\mu _s^*$, that affects the ARF and particle migration. Our study unveils novel insights into the energy states of viscoelastic fluids under varying conditions, contributing significantly to the understanding of fluid behaviour in microfluidic environments exposed to the acoustic waves.

5.2. Effect of viscoelastic properties on acoustic energy density

Acoustic waves can transport energy without a net fluid transfer (Lighthill Reference Lighthill1978). Viscous attenuation alters the acoustic field and AED in viscous fluids (Allegra & Hawley Reference Allegra and Hawley1972). However, the impact of viscoelastic properties on AED in micro-confinements is not well studied. Therefore, we numerically analyse AED transition in a microchannel due to variations in the viscoelastic parameters $\mu _p^*$, $\mu _s^*$ and $De$ (see §§ 2.1 and 3). Using the theoretical formulations in § 2.2 for $\delta _{ve}$, $\lambda _{ve}$ and $\alpha (=\lambda _{ve}/\delta _{ve})$, we relate variations in dimensionless AED $(E_{ac}^* = {E_{ac}^{ve}}/{E_{ac}^{bf}})$ to the characteristics of viscous shear waves.

We depict the variation of $E_{ac}^*$ with $\mu _p^*$ in figure 5(a) for three different cases: (i) small $De$ ($De \leq ~1$, low elasticity fluids), (ii) intermediate $De$ ($1 < De \leq 100$, moderate elasticity fluids) and (iii) high $De$ ($De>100$, high elasticity fluids), with $\mu _s^* = 1$, $\rho ^* = 1$ and $c^* = 1$ fixed in all cases. For low elasticity fluids ($De \leq 1$), viscous effects dominate, with ${\delta }_{ve}^* < {\lambda }_{ve}^*$ and $\alpha > 1$. The variation of ${\delta }_{ve}^*$ and ${\lambda }_{ve}^*$ with $\mu _p^*$ is shown in figure 5(b-i) using solid and dashed lines, respectively. The corresponding variations in $\alpha$ are illustrated in figures 5(c) and 5(d). According to (5.1a–c) and figure 5(b-i), increasing $\mu _p^*$ amplifies $\delta _{ve}^*$ and viscous dissipation, reducing incoming fields and consequently the AED, as shown in figure 5(a-i). The decay is more rapid at smaller $\mu _p^*$ and slower at higher $\mu _p^*$, due to a higher rate of change in ${\delta }_{ve}^*$ and $\alpha$ at smaller $\mu _p^*$, which stabilizes at a higher $\mu _p^*$ (see figures 5a-i, 5b-i and 5c,d). Thus, viscoelastic fluids with $De \ll 1$ and $\tau \ll t_f$ exhibit energy dissipation at a faster rate similar to viscous fluids, depicting the relaxation mode. Interestingly, $E_{ac}^*$ increases with $De$ for all $\mu _p^*$, because an increase in $\tau$ decreases $\alpha$, leading to an increase in the incoming fields and energy storage, as mentioned in § 5.1. For moderate elasticity fluids ($1 < De \leq ~100$), elastic effects start to dominate over viscous effects, with $\delta _{ve}^* > \lambda _{ve}^*$ and $\alpha < 1$ for $De > 3$ (see figures 5b-ii and 5c,d). In this range, $E_{ac}^*$ decreases with $\mu _p^*$, but the rate of decrease slows down due to enhanced elastic effects, indicating a reduced contribution of polymer viscosity to viscous dissipation (see (5.2a–c)). As $De$ increases, the incoming pressure $p_{in}^*$ rises (figure 4a), and $E_{ac}^*$ approaches 1 for $De = 100$, suggesting a transition to a freezing state of AED (frozen mode) at all $\mu _p^*$.

Figure 5. (a) Numerical variation of dimensionless acoustic energy density ($E_{ac}^*$) and corresponding (b) theoretical variation of $\delta _{ve}^*$ and $\lambda _{ve}^*$ with $\mu _p^*$ at different $De$: (i) $De \leq 1$, (ii) $1 < De \leq 100$, (iii) $De>100$. (c) Contour plot of $\alpha$ with $De$ and $\mu _p^*$. (d) Variation of $\alpha$ with $\mu _p^*$ at different $De$. Here, we consider $\mu _s^*=1$, $\rho ^*=1$ and $c^*=1$.

For high elasticity fluids ($De>100$), elastic effects dominate over viscous effects, with $\delta _{ve}^* > \lambda _{ve}^*$ and $\alpha > 1$ (see figures 5b-iii and 5c,d). Here, $E_{ac}^*$ becomes nearly independent of $\mu _p^*$ and $De$ (see (5.3a–c)), as shown in figure 5(a-iii). Viscoelastic fluids with $De \gg 1$ and $\tau \gg t_f$ indicate energy storage, representing a freezing state of energy or frozen mode (Bird et al. Reference Bird, Armstrong and Hassager1977; Joseph Reference Joseph2013). The frozen mode shows a state of viscoelastic fluid where the AED is independent of $De$ and $\mu _p^*$. From figure 5(a-i–a-iii), we conclude that viscoelastic fluids exposed to ultrasound transition from a relaxing to a freezing state with increasing $De$, regardless of $\mu _p^*$. This suggests that viscoelastic fluids have a higher AED compared with a Newtonian fluid for a fixed total viscosity. The increase in $p_{in}^*$ with $De$ in figure 4 supports this finding. At lower $De$, the effects of $\mu _p^*$ (see (5.1a–c)) is predominant, while at higher $De$, acoustic energy dissipation is determined by $\mu _s^*$ (see (5.3a–c)).

To study the influence of solvent viscosity on AED of low, moderate and high elasticity fluids, the variation of $E_{ac}^*$ with $\mu _s^*$ at different $De$ and $\mu _p^*$ is presented in figure 6(a). For $\mu _p^* = 1$, figure 6(a-i) shows a decrease in $E_{ac}^*$ with $\mu _s^*$, which is due to the fact that an increase in $\mu _s^*$ leads to the growth of ${\delta }_{ve}^*$, causing enhanced viscous dissipation and a decrease in $p_{in}^*$. Interestingly, for $\mu _p^* = 1$, $E_{ac}^*$ appears to be independent of $De$ at high $\mu _s^*$, where variation of $\alpha$ with $De$ is negligible (see figures 6b-i and 6d-i). The inset in figure 6(a-i) shows $E_{ac}^*$ increases with $De$ for low $\mu _s^*$, accompanied by a small variation in $\alpha$ (see figure 6d-i). The variation of $E_{ac}^*$ with $\mu _s^*$ and $De$ for $\mu _p^* = 10^3$ is presented in figure 6(a-ii), showing a decrease in $E_{ac}^*$ with $\mu _s^*$ similar to $\mu _p^* = 1$. Notably, $E_{ac}^*$ varies significantly with $De$ at smaller $\mu _s^*$, attributed to the variation of $\delta _{ve}^*$ and $\lambda _{ve}^*$ (see figure 6b-ii) and the change in $\alpha$ from $<1$ to $2{\rm \pi}$ (see figure 6c). At higher $\mu _s^*$, AED follows the same trend as that at lower $\mu _s^*$ with increasing $De$, but due to a higher viscous dissipation, the freezing state $E_{ac}^*$ reduces to a smaller value. In summary, viscoelastic fluids transits from a relaxing to a freezing state in AED for all $\mu _s^*$. In the freezing state, $E_{ac}^*$ decreases with increasing $\mu _s^*$, with the rate of decrease dependent on $\mu _p^*$ and $De$. Interestingly, increasing $\mu _p^*$ (or $\mu ^*$) does not always result in a decrease in the pressure amplitude – at higher $De$, $\mu _p^*$ has negligible effect on pressure field and AED. This contrasts with the study by Vargas et al. (Reference Vargas, Campos-Silva, Méndez, Arcos and Bautista2022) that reported a continuous decrease in pressure wave amplitude with viscosity varying as $\mu ^{-3/5}$ in narrow microchannels. The difference likely arises because they considered $\delta _{ve}^*$ only as a function of viscosity and neglected higher-order viscous effects by assuming $2D/\lambda _0 \ll 1$, which is applicable to narrow channels. In our study, $D \sim \lambda _0$ and therefore we consider the effects of both viscosity and elasticity in $\delta _{ve}$ and $\lambda _{ve}$ (see (2.20) and (2.21)).

Figure 6. (a) Numerical variation of dimensionless AED ($E_{ac}^*$) and corresponding (b) theoretical variation of $\delta _{ve}^*$ and $\lambda _{ve}^*$ with $\mu _s^*$ at different $De$ $(0, 0.1, 1, 10, 10^3, 10^7)$ for (i) $\mu _p^*=1$ and (ii) $\mu _p^*=10^3$. (c) Variation of $\alpha$ with $\mu _p^*$ at different $De$. (d) Contour variation of $\alpha$ with $\mu _s^*$ and $De$ for (i) $\mu _p^*=1$ and (ii) $\mu _p^*=10^3$. We consider $\rho ^*=1$ and $c^*=1$.

In the following section, we discuss how variations in the acoustic energy transition affect the ARF on a particle in a viscoelastic fluid.

5.3. Effect of acoustic energy transitions on acoustic radiation force

The ARF can induce particle migration in a fluid exposed to ultrasound (Barnkob et al. Reference Barnkob, Augustsson, Laurell and Bruus2010; Bruus Reference Bruus2012b). The effects of boundary layer formation over particles suspended in unbounded fluids of different viscoelastic properties on the ARF are discussed in Doinikov et al. (Reference Doinikov, Fankhauser and Dual2021b). However, most acoustomicrofluidic devices consider microchannels of dimensions of the order of the acoustic wavelength. Here, apart from boundary layer formation over the particles, the boundary layer formation on the channel walls and dissipation in viscoelastic fluid also become significant, influencing ARF and particle migration, which is missing in the literature. We focus on the dynamics of particle migration in microchannels and for $a \ll \lambda _0$. We consider the influence of viscoelastic fluid properties on ARF, characterized by AED (§ 5.2) and ACF. In contrast to Doinikov et al. (Reference Doinikov, Fankhauser and Dual2021a,Reference Doinikov, Fankhauser and Dualb), we successfully decouple the effects of $E_{ac}^{ve}$ and $\varPhi$ on ARF theoretically, as described in (2.34) in § 2.3. Our approach considers the ABL formation on both the microchannel walls and the particles, reflected in AED and ACF, respectively. From (2.34), we find that ARF is directly proportional to AED and ACF. Therefore, the ABL formation on the channel walls and the consequent variation in AED with $\mu _p^*$, $\mu _p^*$ and $De$ (given in § 5.2) significantly impacts ARF and particle migration. Further, we observe that increasing the fluid elasticity enhances the AED and consequently the ARF due to boundary layer formation over the channel walls. However, we do not observe any change in the sign of the force as compared with Doinikov et al. (Reference Doinikov, Fankhauser and Dual2021b), even with the influence of the channel boundary layers, likely due to the high driving frequency used in our system (order of MHz).

To further differentiate the effect of viscoelasticity on AED from that on ACF, we analyse the relative variation in ACF for a particle in fluids with and without viscoelasticity, as detailed in supplemental material § 4.1. This variation is expressed as $\varOmega = \{\varPhi ({\kappa }^*_P, {\rho }^*_P, {\delta }_{ve}^*, {\lambda }_{ve}^*) - \varPhi ({\kappa }^*_P, {\rho }^*_P, 0, 0)\} / \varPhi ({\kappa }^*_P, {\rho }^*_P, 0, 0)$ in percentages. We find that the values of $\varOmega$ vary by up to $0.3\,\%, 3.75\,\%, 2.6\,\%, 11.8\,\%, 27\,\%$ and $39\,\%$ for polystyrene (PS), MESC2.10 cells, melamine resin, polymethacrylate, silica and Pyrex particles, respectively, in viscoelastic fluid with $\mu _s^* = 1, \mu _p^* = 10, De = 200, \rho ^* = 1.005$ and ${c^* = 1}$. To study the effect of viscoelastic parameters and AED transition on particle migration dynamics inside a microchannel, we focused on PS microparticles for which the ACF in a viscoelastic fluid, compared with that in the base Newtonian fluid, is minimal (only $0.3\,\%$), as given in supplemental material § 4.1. Therefore, the dimensionless acoustic radiation force $F_{rad}^*$ ($F_{rad}^* = F_{rad}^{ve} / F_{rad}^{bf}$) follows the same trend as the variation of $E_{ac}^*$ with the viscoelastic properties of the fluid, as detailed in supplemental material § 4.2.

5.4. Effect of viscoelastic properties on particle migration

A particle suspended in a fluid with a positive ACF experiences ARF towards acoustic pressure node (Barnkob et al. Reference Barnkob, Augustsson, Laurell and Bruus2010; Bruus Reference Bruus2012b). Here, through appropriate channel design and actuation, a pressure node is formed at the channel centre, as shown in figure 4(a). We investigate how changes in the AED and acoustic force field, resulting from the viscoelastic parameters $\mu _p^*, \mu _s^*$ and $D e$, affect the particle migration dynamics. For the first time, we conduct experiments to study particle migration dynamics in a viscoelastic fluids within a microchannel exposed to ultrasound and validate the theoretical expressions given by (2.40) and (2.42) in § 2.4. We use PS beads suspended in PEO and PVP-based viscoelastic fluids, offering a positive ACF as obtained from (2.35).

We present the trajectories and velocities of PS particles during their migration, with different initial locations ($\kern0.7pt y_0^*=y_0/(W/2)$), $y_0^*=0.125,0.25$ and 0.50, in PEO $0.4\ \mathrm {MDa}$ solutions at two different concentrations, $C=2.5\,\%$ and $3.5\,\%$, in figure 7. For PEO $0.4\ \mathrm {MDa}$ solution with $C=2.5\,\%$, the fluid properties are: $\mu ^*=75\ (\mu _s^*=1, \mu _p^*=74), De=7.4 \times 10^4, \rho ^*= 1.025, c^*=1.011, \xi \approx 1$. Similarly, the properties for $C=3.5\,\%$ are: $\mu ^*=203\ (\mu _s^*=1, \mu _p^*=202), De=25.5 \times 10^4, \rho ^*=1.045, c^*=1.015, \xi \approx 1$ (Zimm Reference Zimm1956; Ebagninin, Benchabane & Bekkour Reference Ebagninin, Benchabane and Bekkour2009; Faroughi et al. Reference Faroughi, Fernandes, Miguel Nóbrega and McKinley2020). We normalize the instantaneous locations of the particles with the half-width of the channel, $y^*=y /(W / 2)$. We define an acoustic time scale $(t_{a c}=D \lambda _0 \mu / a^2 E_{a c} \varPhi )$ based on the solvent (DI water) to non-dimensionalize the migration time as $t^*=t / t_{a c}$. The local velocity of the particles is normalized using a reference velocity $(V_{r e f}=(W / 2) / t_{a c})$, the dimensionless particle velocity $V_P^*=V_P / V_{r e f}$. The particle diameter is $15 \ \mathrm {\mu }\mathrm {m}$ and therefore blockage ratio becomes $\beta =a /(W / 2)=0.0375$. The transducer is operated at a driving voltage $U_{p p}=13.2\ \mathrm {V}$, where $U_{p p}^*=U_{p p}/U_{p p}^{min}=1.14$.

Figure 7. (a) Experimental images showing the time history of the transverse position of a particle released from $y_0^*=0.125$ in a PEO 0.4 MDa solution with $C=2.5\,\%$. The particle is marked by a red dashed circle. Comparison of theoretical and experimental variation of (i) dimensionless particle trajectory (transverse location vs time of migration) and (ii) dimensionless particle velocity along $y^*$ in PEO 0.4 MDa at two polymer concentrations: (b) $C=2.5\,\%$: $\mu ^*=75$ ($\mu _s^*=1, \mu _p^*=74$), $De=7.4 \times 10^4$ and (c) $C=3.5\,\%$: $\mu ^*=203$ ($\mu _s^*=1, \mu _p^*=202$), $De=25.5 \times 10^4$. The particle is released from initial locations $y_0^*=0.125, 0.25$ and $0.5$. The dimensionless central migration time is represented as $T^*$.

We initially study the effect of varying ARF across the channel width (S-BAW or $y$-direction) on particle migration in a viscoelastic fluid. Since the ARF acts only along the $y$ direction, particle translation is one-dimensional in the $x$–$y$ plane, with negligible drift in the $x$ direction. The time-lapsed experimental images showing migration of a 15 $\mathrm {\mu }$m PS particle with $y_0^*=0.125$ in PEO 0.4 MDa solution of $C=2.5\,\%$ is shown in figure 7(a). The corresponding time evolution of particle position $(y^*)$ and particle migration velocity ($V_P^*$) are shown in figures 7(b-i) and 7(b-ii), respectively. Under an acoustic field, the particles migrate from their initial positions to a final equilibrium position with a varying velocity, and the final equilibrium position is represented by $y^*=1 \pm 0.05$ (the central pressure nodal region is indicated as a band in figure 7a). The value of $V_P^*$ shows a non-monotonic variation during its migration, where $V_P^*$ is found to be small closer to the channel centre and wall, but becomes maximum closed to $y^* \approx 0.5$, shown in figure 7(b-ii). We compare the experimental data with the predictions from the theoretical expressions given by (2.40) and (2.42) in § 2.4, which shows a close agreement, with $V_P \propto \sin {(2 k_y y)}$. The maximum velocity seen at $y^* \approx 0.5$ is due to the higher acoustic radiation force, which becomes zero at the walls and the channel centre, here, $F_{rad} \propto E_{ac}^{ve} \sin {(2 k_y y)}$, as given by (2.40).

To further investigate the effect of initial locations of a particle on acoustic migration characteristics in viscoelastic fluids, we consider particles at $y_0^*=0.125,0.25$ and $0.5$. For PEO fluid with $C=2.5\,\%$, $\mu ^*=75$ and $De=7.4\times 10^4$, the particle trajectories change with the initial location and the migration time to reach the centre of the channel decreases with increase in the $y_0^*$, illustrated in figure 7(b-i). It is interesting to observe that the migration velocities across $y^*$ are found to follow a fixed curve, irrespective of the values of $y_0^*$, illustrated in figure 7(b-ii), agreeing with our theoretical prediction (2.42).

Viscoelastic fluids show a coupled variation of viscous and elastic properties with increasing polymer concentration (Ebagninin et al. Reference Ebagninin, Benchabane and Bekkour2009; Joseph Reference Joseph2013). We study the coupled variation of $\mu ^*$ (or $\mu _p^*$, where $\mu _s^*=1$) and $De$ on acoustic particle migration in figures 7(b-i,ii) and 7(c-i,ii). At a fixed $y_0^*$, a coupled increase in both $\mu ^*$ and $De$ slows down particle migration due to the combined increase in drag force and decrease in acoustic energy density, shown in figure 7(c-i,ii). The value of $V_P^*$ is found to be smaller at a higher $\mu ^*$ and $De$. However, the effect of initial location remains same as figure 7(b-ii). A quantitative comparison between theoretical predictions and experimental data shows a close agreement in terms of the particle trajectory and velocity variations within $10\,\%$.

Now we discuss the individual effects of $\mu ^*$ and $De$ and the corresponding AED transition on acoustic particle migration, theoretically and experimentally, as shown in figure 8. We consider PVP $0.36\ \mathrm {MDa}\ (C=0.1\,\%, \mu ^*=1.9, \mu _s^*=1, \mu _p^*=0.9, De=91)$, PEO $0.4 \ \mathrm {MDa}$ ($C=0.2\,\%, \mu ^*= 1.9, \mu _s^*=1, \mu _p^*=0.9, De=178$) and Newtonian fluid (DI water $+0.3\,\%$ Glycerol, $\mu ^*= 1.9, {De}=0$) all having a fixed $\rho ^*(\rho ^* \approx 1.007)$ and $c^*(c^* \approx 1.003)$, keeping $\mu ^*$ fixed and only varying $De$, as shown in figure 8. Here, we find that an increase in the elasticity or $De$ of the fluid enhances the particle migration, shown by a reduced central migration time $(T^*=t^*|_{y^* \approx 1})$ and a higher $V_P^*$ in figures 8(a) and 8(b), respectively. Therefore, as compared with the Newtonian fluid $(De=0)$, particles migrate faster in PEO 0.4MDa fluid with $De=178$. This is mainly because of the acoustic energy transition with $De$, as discussed in § 5.2. Due to the viscous nature, the Newtonian fluid exhibits higher viscous dissipation and remains in the relaxation mode, giving rise to a low AED in the fluid. However, in high elasticity fluids, the elastic nature helps to reduce the viscous dissipation and store the energy, offering high AED and consequently higher ARF. Here, we observe a strong agreement between our experimental data and the predictions from our theory and simulations, all of which suggest that the AED transition is attributed to the fluid elasticity giving rise to a faster particle migration in viscoelastic fluids than in Newtonian fluid.

Figure 8. Comparison of theoretical and experimental variation of dimensionless (a) trajectory ($\kern0.7pt y^*$ vs $t^*$) and (b) velocity ($V_P^*$ vs $y^*$) of particle in PEO 0.4 MDa, PVP 0.36 MDa and Newtonian (DI water+0.3$\,\%$ glycerol) fluid of constant viscosity $\mu ^*=1.9$. The initial location of particle is considered as $y_0^*=0.125$. Each data point corresponds to a value averaged over repeated experiments.

Next, we discuss the effect of fluid elasticity and the corresponding AED transition on the particle migration dynamics at different $\mu _p^*$ and $\mu _s^*$, presented in figure 9. We express the theoretical variation of particle migration time in terms of dimensionless central migration time $T^*$, which is defined as the ratio of the time taken by a particle to reach the channel centre (nodal plane) $(t_c = t|_{y^* \approx 1})$ to the acoustic time scale $(t_{a c}), T^*=t_c / t_{a c}$. We illustrate the variation of $T^*$ with $\mu _p^*$ in figure 9(a) for three different scenarios: (a) low elasticity fluids ($De\leq 1$), (b) moderate elasticity fluids ($1< De\leq 100$) and (c) high elasticity fluids ($De\geq 200$), with $\mu _s^*=1, \rho ^*=1$ and $c^*=1$.

Figure 9. Theoretical variation of (a) dimensionless particle central migration time $(T^*)$ with $\mu _p^*$ at different $De$: (i) $De \leq 1$, (ii) $1 < De \leq 100$, (iii) $De>100$ for $\mu _s^*=1$, and (b) $T^*$ with $\mu _s^*$ at different $De$ for (i) $\mu _p^*=1$ and (ii) $\mu _p^*=10^3$. Here, $\beta =0.0375, U_{pp}^*=1.14, \rho _P^*=1.05$ and $y_0^*=0.125$. (c) Experimental variation of $T^*$ with blockage ratio ($\beta$, at $U_{pp}^*=1.14$) and dimensionless peak-to-peak voltage ($U_{pp}^*$, at $\beta =0.0375$) for PEO 1 MDa solution with $C=1.05\,\%$ ($\mu ^*=45.1$, $De=5.7 \times 10^4$), where $y_0^*=0.125$. The theoretical variation of $T^*$ with $\beta$ from (2.40) is shown by a solid line, where $T^* \propto \beta ^{-2}$. For $T^*$ with $U_{pp}^*$, the experimental fit becomes $T^* \propto U_{pp}^{*-2}$.

For low elasticity fluids ($De \leq 1$), viscous effects dominate and fluids are in the relaxation state. Therefore, an increase in $\mu _p^*$ amplifies the viscous dissipation, and consequently the AED and ARF decrease, as shown in figure 5(a-i). Moreover, with increase in $\mu _p^*$ drag forces increases. The combined effects of these variations gives rise to a significant increase in $T^*$ with $\mu _p^*$ in figure 9(a-i). However, an increase in $De$ leads to an increase in the AED in the fluid, which causes a decrease in the $T^*$ as well as the rate of increase in $T^*$ with $\mu _p^*$. The influence of $De$ on $T^*$ becomes more pronounced at a higher $\mu _p^*$. For moderate elasticity fluids ($1 < De \leq 100$), $T^*$ increases with $\mu _p^*$ at a moderate rate in figure 9(a-ii). Here, the elastic effects start dominating viscous effects, fluid starts to transition from the relaxation mode to the frozen mode, as shown in figures 4(e), 5(a-ii) and 5(b-ii). The increased $E_{ac}^*$ in moderate elasticity fluids compared with low elasticity fluids causes a higher ARF, which reduces $T^*$ as well as rate of increase in $T^*$ with $\mu _p^*$ in figure 9(a-ii). A further increase in $De$ leads to an increase in AED, causing a further decrease in $T^*$ as well as the rate of increase in $T^*$ with $\mu _p^*$, as illustrated in figure 9(a-ii). For high elasticity fluids ($De>100$), $T^*$ slowly increases with $\mu _p^*$ and is nearly independent of $De$, as shown in figure 9(a-iii). Here, elastic effects suppress viscous effects, and fluid attains the frozen state, as shown in figures 4(e), 5(a-iii) and 5(b-iii). Therefore, $E_{ac}^*$ and thus the ARF attain their maximum values and becomes independent of $De$ and $\mu _p^*$. This phenomenon reduces $T^*$ and rate of increase in $T^*$ with $\mu _p^*$ in figure 9(a-iii). In summary, the higher elasticity fluids show shorter central migration time as compared with low elasticity fluids in all range of $\mu _p^*$ due to transition of AED from a relaxation mode to a frozen mode.

We depict the variation of $T^*$ with $\mu _s^*$ at different $De$ and $\mu _p^*$ in figures 9(b-i) and 9(b-ii). For $\mu _p^*=1, \ T^*$ consistently increases with $\mu _s^*$ and is independent of $De$, as shown in figure 9(b-i). This can be attributed to the viscous dissipation and consequent decrease in $E_{a c}^*$ with $\mu _s^*$ (refer to figure 6a-i). For $\mu _p^*=10^3, T^*$ increases with $\mu _s^*$ but decreases with $De$ for smaller $De$ and becomes independent of ${De}$ at higher ${De} ({De}=10^3 \ \textrm {and} \ 10^7)$, as illustrated in figure 9(b-ii). Here, the fluid response transition from the relaxation mode to the frozen mode, causing a decrease in $T^*$ with increasing $D e$ for low elasticity and moderate elasticity fluids. However, in high elasticity fluids, the $E_{a c}^*$ attains a maximum value and then becomes independent of $De$, as similar to figure 6(a). In summary, an augmentation in the solvent viscosity of a viscoelastic fluid decelerates particle migration and concurrently suppresses elastic effects, particularly at lower polymer viscosity range.

In the above analysis, we consider a fixed particle size or blockage ratio $(\beta =a /(W / 2)=0.0375)$ and dimensionless driving voltage $U_{p p}^*(U_{p p}^*=U_{p p} / U_{p p}^{\min }=1.14$), which indicate the acoustic power input. We perform theoretical and experimental investigations on the effects of $\beta$ and $U_{p p}^*$ on particle migration in a viscoelastic fluid in figure 9(c). Here, $\beta$ is varied experimentally by varying the particle size between 5 and $25 \ \mathrm {\mu }\mathrm {m}$. The variation of $T^*$ with $\beta$ and $U_{p p}^*$ is depicted in figure 9(c) for PEO 1 MDa solution with $C=1.05\,\%$, with $\mu ^*=45.1, \mu _p^*=44.1, \mu _s^*=1, De=5.7 \times 10^4, \rho ^*=1.017$ and $c^*=1.005$. We find that the central migration time decreases with an increase in the blockage ratio or particle size. This decrease is primarily attributed to the rise in the ARF, evident from (2.42), where $T^* \propto \beta ^{-2}$, similar to the Newtonian case (Barnkob et al. Reference Barnkob, Augustsson, Laurell and Bruus2010). We find that particles migrate more rapidly towards the node at higher powers and lower $\mu ^*$ owing to the increase in AED and ARF. A power-law fit to the experimental data shows $T^* \propto U_{p p}^{*-2}$, agreeing with the Newtonian case (Barnkob et al. Reference Barnkob, Augustsson, Laurell and Bruus2010). The dependency of $T^*$ with $\beta$ and $U_{p p}^*$ is also studied with two other PEO $1\ \mathrm {MDa}$ solutions: $C=0.74\,\%$ with $\mu ^*=21.3, De= 3.7 \times 10^4$; and $C=1.05\,\%$ with $\mu ^*=45.1, De=5.7 \times 10^4 ; C=1.4\,\%$ with $\mu ^*=69.5, De=1.1 \times 10^5$, Here, the rate of change of $T^*$ increases/decreases depending upon $\mu ^*$ and $De$ of the fluid, as presented in supplemental material § 4.3.

In summary, particle migration in viscoelastic fluids within a microchannel is significantly influenced by the fluid properties and acoustic boundary layer formation. From a biological standpoint, fluids such as blood, plasma, mucus, saliva, sputum, synovial fluid, protein solution and hydrogel have relaxation times ranging from $10^{-5}$ to $10^{-2}$ s (Thurston Reference Thurston1972; Rainer & Ribitsch Reference Rainer and Ribitsch1985; Stokes & Davies Reference Stokes and Davies2007; Pan et al. Reference Pan, Filobelo, Pham, Galkin, Uzunova and Vekilov2009; Li et al. Reference Li, Ren, Huang, Yao, Cuento, McCoy, Cameron, Levine and Huang2016; Chaudhuri et al. Reference Chaudhuri, Cooper-White, Janmey, Mooney and Shenoy2020), thus categorized as moderate and high elasticity fluids. Our study shows that changes in fluid elasticity or $De$ causes the AED transition from the dissipation state to the storage state. This transition increases the ARF and accelerates particle migration. At a fixed input power, viscoelastic fluid in a microchannel acts as an energy dissipation device at low $De$ (low elasticity) and as an energy storage device at higher $De$ (high elasticity). Therefore, our study suggests that the Newtonian fluid assumption may not be valid for bio-fluids under all operating conditions. Moreover, by carefully controlling $De$, it is possible to manipulate the particle migration dynamics, at a fixed power. The outcome of the present study may be crucial for designing effective microfluidic systems for bio-fluids.