On the stability of inhomogeneous fluids under acoustic fields

A two-dimensional fluid domain subjected to a standing acoustic half-wave in the X-direction, with two fluids separated by a sharp vertical interface as shown in figure 1.a-c is considered for the stability analysis. Before beginning the analysis, it is necessary to understand the equilibrium of the system in the absence of interfacial tension. In a completely enclosed domain, a fluid initially at rest ($\textbf{\emph{V}}=0$) will remain at rest (or equilibrium) if the body force can be completely absorbed in pressure, $\textbf{\emph{f}}_{rl}=\bm{\nabla}P$ from (1b) and (4). By taking the curl of the above relation, the condition for equilibrium is given as $\bm{\nabla}\times\textbf{\emph{f}}_{rl}=0$. Thus,

$$-\frac{E_{ac}}{z_{avg}}\left[\frac{\partial}{\partial x}\left(\cos(2k_{w}x)\frac{\partial z}{\partial y}\right)-\frac{\partial}{\partial y}\left(\cos(2k_{w}x)\frac{\partial z}{\partial x}\right)\right]=0.$$

(5)

It is clear from (5) that the given fluid configuration will be in an equilibrium state, only if the direction of the acoustic standing wave is normal to the fluid-fluid interface (the direction of the acoustic standing wave is parallel to the direction of the impedance gradient) as shown in figure 1($a$-$c$). Since $\bm{\nabla}\times\textbf{\emph{f}}_{rl}\neq 0$ for the configuration shown in figure 1$(d)$, it is not in equilibrium and tends to relocate to the stable configuration without any perturbations. The stability nature of the equilibrium configurations is analysed by imposing infinitesimal perturbations on the interface. Now we proceed to show that in the absence of interfacial tension, the configuration shown in figure 1$(a)$ is in unstable equilibrium (perturbations grow), figure 1$(b)$ is in stable equilibrium (perturbations decay), and figure 1($c$) is in neutral equilibrium (perturbations neither grow nor decay).

The effect of viscosity is neglected in the stability analysis, as it governs only the timescale of the phenomenon and does not contribute to the stability criterion. Although the physical properties are non-uniform in an inhomogeneous system, the fluid particles considered in the flow field have constant density $\rho$, speed of sound $c$, and impedance $Z$. Thus, the material derivative of all properties is zero, which includes the incompressibility condition $(D\rho/Dt=\partial\rho/\partial t+\textbf{\emph{V}}\bm{\cdot\nabla}\rho=0)$. By combining the incompressibility condition with (1a) and neglecting the viscosity, the governing equations (1) reduce to

	$$\displaystyle\frac{\partial U}{\partial x}+\frac{\partial V}{\partial y}=0,$$		(6a)
	$$\displaystyle\rho\frac{DU}{Dt}=-\frac{\partial P}{\partial x}-\frac{E_{ac}\cos({2k_{w}x})}{z_{avg}}\frac{\partial Z}{\partial x},$$		(6b)
	$$\displaystyle\rho\frac{DV}{Dt}=-\frac{\partial P}{\partial y}-\frac{E_{ac}\cos({2k_{w}x})}{z_{avg}}\frac{\partial Z}{\partial y},$$		(6c)
where $U,V$ are the X-component and Y-component of the velocity field V. Since the body force term is a function of impedance, the below impedance relation is required for the closure.
	$$\frac{DZ}{Dt}=\frac{\partial Z}{\partial t}+U\frac{\partial Z}{\partial x}+V\frac{\partial Z}{\partial y}=0.$$		(6d)

Now, the flow fields are decomposed into an unperturbed zeroth-order stationary state and infinitesimal perturbations as $U=u_{0}+\delta u,\ V=v_{0}+\delta v,\ P=p_{0}+\delta p,\ \rho=\rho_{0}+\delta\rho$ and $Z=z_{0}+\delta z$. In this study, the variation of acoustic impedance is considered only in the X-direction (figure 1($a$-$c$)), $z_{0}=z_{0}(x)$. At the stationary state $(u_{0}=v_{0}=0)$, the unperturbed zeroth-order equations become, $\frac{\partial p_{0}}{\partial x}=-\frac{E_{ac}\cos({2k_{w}x})}{z_{avg}}\frac{\partial z_{0}}{\partial x}$ from 6b, $\frac{\partial p_{0}}{\partial x}=0$ from 6c and $\frac{\partial z_{0}}{\partial t}=0$ from 6d. Using the above zeroth order relations and neglecting the second-order terms in (6), the first-order perturbation equations governing the stability becomes

	$$\displaystyle\frac{\partial\delta u}{\partial x}+\frac{\partial\delta v}{\partial y}=0,$$		(7a)
	$$\displaystyle\rho_{0}\frac{\partial\delta u}{\partial t}=-\frac{\partial\delta p}{\partial x}-\frac{E_{ac}\cos({2k_{w}x})}{z_{avg}}\frac{\partial\delta z}{\partial x},$$		(7b)
	$$\displaystyle\rho_{0}\frac{\partial\delta v}{\partial t}=-\frac{\partial\delta p}{\partial y}-\frac{E_{ac}\cos({2k_{w}x})}{z_{avg}}\frac{\partial\delta z}{\partial y},$$		(7c)
	$$\displaystyle\frac{\partial(\delta z)}{\partial t}=-\delta u\frac{\partial z_{0}}{\partial x}.$$		(7d)

Analysing the disturbances into normal modes, the amplitude of the disturbances $\delta u,\ \delta v,\ \delta\rho,\ \delta p,$ and $\delta z$ takes the following form

$$A(x,y,t)=A(x)exp(ik_{y}y+nt),$$

(8)

where $k_{y}$ is the wavenumber considered along the Y-direction. Applying the above amplitude relations in the form (8) in (7),

	$$\displaystyle\frac{\partial\delta u}{\partial x}+ik_{y}\delta v=0,$$		(9a)
	$$\displaystyle\rho_{0}n\delta u=-\frac{\partial\delta p}{\partial x}-\frac{E_{ac}\cos{\left(2k_{w}x\right)}}{z_{avg}}\frac{\partial\delta z}{\partial x},$$		(9b)
	$$\displaystyle\rho_{0}n\delta v=-ik_{y}\delta p-ik_{y}\frac{E_{ac}\cos{\left(2k_{w}x\right)}}{z_{avg}}\delta z,$$		(9c)
	$$\displaystyle n\delta z=-\delta u\frac{\partial z_{0}}{\partial x}.$$		(9d)

The partial notation is dropped since the only derivatives in (9) are with respect to the $x$ coordinate. Multiplying by $ik_{y}$ throughout (9c) and combining with (9a) and (9d), we obtain,

$$\delta p=-\rho_{0}\frac{n}{{k_{y}}^{2}}\frac{d\delta u}{dx}+E_{ac}\frac{\cos(2k_{w}x)}{z_{avg}}\frac{\delta u}{n}\frac{dz_{0}}{dx}.$$

(10)

substituting (9d) and (10) in (9b) results in,

$$\frac{d}{dx}\left(\rho_{0}\frac{d\delta u}{dx}\right)-\rho_{0}k_{y}^{2}\delta u=-E_{ac}\frac{2k_{w}\delta u}{z_{avg}}\frac{k_{y}^{2}}{n^{2}}\frac{dz_{0}}{dx}\sin(2k_{w}x).$$

(11)

Considering two uniform fluids of different impedance $Z_{A}$ and $Z_{B}$ separated by interfaces positioned at $x_{s}$,

	$$z_{0}=z_{A}+(z_{B}-z_{A})H(x-x_{s}),$$		(12a)
	$$\frac{dz_{0}}{dx}=(z_{B}-z_{A})\delta(x-x_{s}),$$		(12b)

where $H(x-x_{s})$ is the Heaviside step function at $x=x_{s}$ and $\delta(x-x_{s})$ is the Dirac’s $\delta$-function at $x=x_{s}$. Substituting (12b) in (11),

$$\frac{d}{dx}\left(\rho_{0}\frac{d\delta u}{dx}\right)-\rho_{0}k_{y}^{2}\delta u=-E_{ac}\frac{2k_{w}\delta u}{z_{avg}}\frac{k_{y}^{2}}{n^{2}}\sin(2k_{w}x)(z_{B}-z_{A})\delta(x-x_{s}).$$

(13)

Equation (13) is the governing differential equation for the stability of inhomogeneous fluids (without interfacial tension). For a uniform region on either side of the interface(s) where there are no discontinuities in the impedance, the governing equation (13) reduces to

$$\frac{d^{2}\delta u}{dx^{2}}-k_{y}^{2}\delta u=0.$$

(14)

The solution of (14) is of the form $\delta u=C_{1}e^{k_{y}(x-x_{s})}+C_{2}e^{-k_{y}(x-x_{s})}$ where $C_{1},C_{2}$ are constants. Since $\delta u$ must vanish at the boundaries, we can write the solution as,

	$$\delta u_{B}=Ce^{k_{y}(x-x_{s})}\quad\quad(x<x_{s}),$$		(15a)
	$$\delta u_{A}=Ce^{-k_{y}(x-x_{s})}\quad\quad(x>x_{s}),$$		(15b)

where the constant $C$ in (15) is chosen to ensure continuity in velocity across the interfaces. For the solution at the interface $(x=x_{s})$, we integrate (13) along infinitesimal distance $(dx\approx 0)$, the second term in the left-hand side of the equation is zero and the remaining terms are,

$$\Delta\left(\rho_{0}\frac{d\delta u_{s}}{dx}\right)=-E_{ac}\frac{2k_{w}\delta u_{s}}{z_{avg}}\frac{k_{y}^{2}}{n^{2}}(z_{B}-z_{A})\int{\left(\sin(2k_{w}x)\delta(x-x_{s})\right)dx},$$

(16)

where $\delta u_{s}$ is the value of $\delta u$ at $x=x_{s}$. Using, (15) and the Dirac delta identity $\int f(x)\delta(x-a)dx=f(a)$ to solve for eigenvalue $n$ in (16).

$$\rho_{A}(-k_{y}\delta u_{s})-\rho_{B}(k_{y}\delta u_{s})=-E_{ac}\frac{2k_{w}\delta u_{s}}{z_{avg}}\frac{k_{y}^{2}}{n^{2}}(z_{B}-z_{A})\sin(2k_{w}x_{s}),$$

(17)

Rearranging (17), the dispersion relation $n$ for the stability problem becomes

$$n=\sqrt{\frac{k_{y}}{\rho_{A}+\rho_{B}}\phi E_{ac}(z_{B}-z_{A})\sin(2k_{w}x_{s})}.$$

(18)

where $\phi=2k_{w}/z_{avg}$. The dispersion relation (18) establishes the acoustic stability criterion when inhomogeneous fluids (without interfacial tension) in a microchannel are subjected to a standing acoustic wave. If the eigenvalue $n$ is imaginary in (18), then the configuration is in a stable equilibrium and the configuration is in an unstable equilibrium when the eigenvalue $n$ is real. For a standing acoustic half-wave, in (18), the values of $\frac{k_{y}}{\rho_{B}+\rho_{A}},\phi$ and $E_{ac}$ are always positive. Thus, the sign of $z_{B}-z_{A}$ (initial configuration of the fluids) and $\sin(2k_{w}x_{s})$ (relative location of the interface with respect to the standing acoustic wave) decide the nature of the eigenvalue in (18). $z_{B}-z_{A}$ is positive when high-impedance fluid is present to the right of the interface, and negative when high-impedance fluid is present to the left of the interface. $\sin(2k_{w}x_{s})$ has a negative value to the left of the node ($x_{s}$ is negative), a positive value to the right of the node ($x_{s}$ is positive) and zero when the interface coincides with the node (centre of the microchannel) or anti-node (sides of the microchannel) $(x_{s}=0)$.

As per the above arguments, the inhomogeneous system in figure 1($a$) is in an unstable equilibrium as eigenvalue $n$ is real, and the system in figure 1($b$) is in a stable equilibrium as eigenvalue $n$ is imaginary. It can be concluded from the above discussion and figures 2($a$-$i$) and 2($a$-$ii$) that, a system is said to be acoustically stable (unstable) if the initial configuration of the fluids is in such a way that the low (high) impedance fluid is present at the anti-node(s) and the high (low) impedance fluid is present at the node(s). This conclusion is consistent with the demonstration of acoustic relocation of fluids within a microchannel by Deshmukh et al. (2014). For the case where the interface coincides with the node, $\sin(2k_{w}x_{s})=0$. Thus, the system is in a neutral equilibrium ($n=0$) as shown in figures 1($c$) and 2($a$-$iii$). The above analysis can be easily extended to an inhomogeneous system consisting of multiple interfaces. In this case, the eigenvalues evaluated at the fluid interfaces govern the nature of the system. Figure 2($b$) shows the stability of two interface systems that are widely used in acoustofluidic applications. It can be seen from figure 2($b$-$i$) that when high impedance fluid is at the sides (anti-nodes), the eigenvalue at both the interfaces $(IF_{1}$ and $IF_{2})$ is real and hence the system is in unstable equilibrium. The system is in stable equilibrium in figure 2($b$-$ii$), as the eigenvalue at both interfaces is imaginary.

Proceeding to solve for immiscible fluids, the effect of surface tension must be accounted for. The discontinuity in impedance occurring in the interfaces ($x_{s}$) is modelled by including the interfacial tension effects in the X momentum equation (9b) as, (Chandrasekhar, 1961)

$$\rho_{0}n\delta u=-\frac{\partial\delta p}{\partial x}-\frac{E_{ac}\cos{\left(2k_{w}x\right)}}{z_{avg}}\frac{\partial\delta z}{\partial x}-k_{y}^{2}\sum_{s}T\delta x_{s}\delta(x-x_{s}).$$

(19)

where $T$ is the interfacial tension and $\delta x_{s}$ denotes the perturbation of the interfaces and $\frac{d}{dt}\delta x_{s}=\delta u_{s}\implies\delta x_{s}=\frac{\delta u_{s}}{n}$. The governing differential equation for stability between inhomogeneous fluids with interfacial tension is obtained similar to the case without interfacial tension, as in the previous $\S$ 2.1,

$$\frac{d}{dx}\left(\rho_{0}\frac{d\delta u}{dx}\right)-\rho_{0}k_{y}^{2}\delta u=-E_{ac}\frac{2k_{w}\delta u}{z_{avg}}\frac{k_{y}^{2}}{n^{2}}\sin(2k_{w}x)(z_{B}-z_{A})\delta(x-x_{s})\\ +\frac{k_{y}^{2}}{n^{2}}\sum_{s}{k_{y}^{2}\left(T\delta u_{s}\right)\delta(x-x_{s})},$$

(20)

Integrating (20) across an infinitesimal distance ($dx\approx 0$) and solving for the dispersion relation $n$,

$$n=\sqrt{\frac{k_{y}}{\rho_{1}+\rho_{2}}\left(\phi E_{ac}(z_{B}-z_{A})\sin(2k_{w}x_{s})-k_{y}^{2}T\right)}.$$

(21)

Equation (21) establishes the acoustic stability criterion when fluids with interfacial tension are subjected to a standing acoustic wave. It can be seen from (21) that the interfacial tension $(T)$ and wavenumber of the perturbation $(k_{y})$ play a role in the stability of immiscible fluids.

In the presence of interfacial tension ($T>0$), the fluid system shown in figure 1($b$) is always stable, as the negative sign of $(z_{B}-z_{A})\sin(2k_{w}x_{s})$ results in an imaginary eigenvalue $n$ in (21). Whereas, for the fluid system shown in figure 1($a$), the sign of $(z_{B}-z_{A})\sin(2k_{w}x_{s})$ is positive in (21). Thus, the system is conditionally stable, and the stability is determined by the relative magnitudes of $\phi E_{ac}(z_{B}-z_{A})\sin(2k_{w}x_{s})$ and $k_{y}^{2}T$. The fluid system (figure 1($a$) becomes unstable ($n$ is real) if the acoustic force density $F_{rl}$ ($\phi E_{ac}(z_{B}-z_{A})\sin(2k_{w}x_{s})$) dominates (or is greater than) the interfacial force density $F_{int}$ ($k_{y}^{2}T$) and becomes stable ($n$ is imaginary) if the interfacial force density dominates the acoustic force density. For the case where the interface coincides with the node ($\sin(2k_{w}x_{s})=0$ and eigenvalue $n$ is imaginary) and the system is in a stable equilibrium, as shown in figures 1($c$) and 2($a$-$iii$).

Now, for conditionally stable configuration, we proceed to find the minimum energy density required to relocate the fluid systems in figures 1($a$) and 2($b$-$i$) with interfacial tension ($T>0$). Since the interface height, $h$ is finite, this leads to the quantization of the possible modes $k_{y}=k_{h_{n}}=n\pi/h$. The minimum (critical) acoustic energy density $(E_{cr})$ required to relocate the fluid system is decided by the first conceivable mode, $k_{h_{1}}=k_{h}=\pi/h$ and the critical acoustic energy density is obtained by limiting the eigenvalue $n$ to zero in (21). Thus,

$$E_{cr}=\frac{k_{h}^{2}Tz_{avg}}{sin(2k_{w}x_{s})2k_{w}(z_{B}-z_{A})}.$$

(22)

If the applied energy density $E_{ac}$ is less than the critical energy density $E_{cr}$ ($E_{ac}<E_{cr}$), the interfacial tension succeeds in stabilizing a potentially unstable configuration. The same system becomes unstable and eventually relocates to a stable configuration when $E_{ac}>E_{cr}$. The above discussions on the equilibrium nature of different inhomogeneous fluid configurations (with and without interfacial tension) are clearly summarised in figure 2.

For the numerical simulations shown in 3, we employ equation (4) as body force and assumed $E_{ac}$ to be constant throughout the relocation process. The boundary condition for the analysis is no slip at the walls and the pressure is constrained at a point (bottom left corner of the channel). In the absence of interfacial tension $(T=0\ mN/m)$, it is observed that for any $E_{ac}>0$, the HLH configuration undergoes relocation to a stable LHL configuration as in figure 3($a$) (the simulation is shown for $E_{ac}$ $=80$ $J/m^{3}$). In this case, the magnitude of $E_{ac}$ only influences the timescale of the relocation process by competing with the viscosity. While, in the presence of interfacial tension, $T=1\ mN/m$, the HLH fluid configuration remained stable for all energy densities below $88\ J/m^{3}$, and relocation is observed for all energy densities above $89\ J/m^{3}$. These simulations are in close agreement with the critical acoustic energy density $E_{cr}=88.78\ J/m^{3}$ predicted by (22) for mineral-silicon oil combination. Simulation results of other fluid combinations shown in figure 5 also agree with (22). When the applied $E_{ac}$ is just above $E_{cr}$, the fluids take a much longer time to relocate. Thus for convenience, the simulation is shown for $E_{ac}=120\ J/m^{3}$ in figure 3($a$).

For LHL configuration with and without interfacial tension ($T\geq 0$), for any $E_{ac}>0$, the relocation of fluid is not observed, and the system remained stable as shown in figure 3($b$) (the simulation is shown for $E_{ac}$ $=120$ $J/m^{3}$). In the HL configuration, the node of the standing acoustic half-wave coincides with the fluid-fluid interface. Here for fluids with interfacial tension, relocation is not observed for any $E_{ac}>0$, and the fluid system remained stable (figure 3($c$)). Whereas, for fluids without interfacial tension, the HL configuration is observed to be in neutral equilibrium (figure 3($c$)). These simulation results of unstable, stable, and neutral equilibrium of inhomogeneous fluids (figure 3) are in agreement with the stability criteria (from (18) and (21)) that we established theoretically in $\S$ 2.

Thus far, in the theoretical stability analysis ($\S$ 2) as well as in the numerical simulations ($\S$ 3.1), a simplified equation $\textbf{\emph{f}}_{rl}$ (4) is employed as a body force with the assumption of constant $E_{ac}$ (the amplitudes of first-order fields $p_{a}$ and $v_{a}$ do not vary during relocation). In this section, the generalized acoustic body force $\textbf{\emph{f}}_{ac}$ (2) is employed and the first-order fields required to calculate the above $\textbf{\emph{f}}_{ac}$ are obtained from the wave equations (frequency domain - see appendix A) by actuating the channel walls at a frequency $\nu$ with a wall displacement $d_{0}$. There are two reasons for using generalized acoustic body force $\textbf{\emph{f}}_{ac}$: 1. To show the relocation predicted by $\textbf{\emph{f}}_{rl}$ and $\textbf{\emph{f}}_{ac}$ is approximately the same. When we use much simpler $\textbf{\emph{f}}_{rl}$ instead of the complex $\textbf{\emph{f}}_{ac}$, the first-order field equations are not required to be solved which will significantly reduce the computation time for simulation of relocation of inhomogeneous fluids. 2. To explain the previous microchannel experiments in immiscible fluid relocation (Hemachandran et al., 2019).

For one-directional (1-D) standing half-wave simulations, the sidewalls are actuated in phase at a displacement $d_{0}$ at a frequency $\nu$. In laminar flow equations, the boundary conditions used are no-slip at all walls, and the pressure is constrained at a point (bottom left corner of the channel). To disregard the effect of streaming, the first-order acoustic fields (see Appendix A) are allowed to slip in the frequency domain.

Figure 6($a$), shows the HLH configuration subjected to 1-D standing half-wave by actuating sidewalls at a displacement $d_{0}$ of $0.21$ $nm$ and a frequency $\nu$ of $1.73\ MHz$. In this case, it is observed that the resulting pressure amplitude $P_{a}$ of $0.67\ MPa$ ($E_{ac}$ = $85.58\ J/m^{3}$), could not relocate the fluids in the HLH configuration and thus remains stable. Whereas, when the displacement is increased to $0.22$ $nm$, the resulting pressure amplitude of $P_{a}=0.69\ MPa$ ($E_{ac}$ = $86.22\ J/m^{3}$), could relocate the HLH configuration to a stable equilibrium as shown in figure 6($b$). From the above discussion, the critical acoustic energy density is found to be $E_{cr}$ = $85.9\ \pm 0.32$ $J/m^{3}$. This value of $E_{cr}$ obtained through generalized body force $\textbf{\emph{f}}_{ac}$ is in close agreement (deviation of $3.24\%$) with the simplified relocation force $f_{rl}$ employed to derive stability criterion ($\S$ 2).

In the case of a 1-D standing half-wave, when the interface of the fluid coincides with the pressure node ($x_{s}=0$), for any $E_{ac}$, relocation is not observed using both $\textbf{\emph{f}}_{ac}$ and $\textbf{\emph{f}}_{rl}$ (figures 3($c$) and 4($c$) as predicted by the stability criteria (21). However, Hemachandran et al. (2019) through experiments demonstrated the relocation of fluids irrespective of the location of the vertical interface $x_{s}$. In their experiments, the frequency employed ($2.1$ $MHz$) is far from the 1-D resonant half-wave frequency ($\nu$ = $1.6$ $MHz$ $\approx$ $c_{avg}/2w$). In our previous work (Rajendran et al., 2022), we have shown that the above relocation is due to standing two-directional (2-D) acoustic wave (frequency $f$ $=$ $2.1\ MHz$ between $c_{avg}/2w$ and $c_{avg}/2h$) as shown in figure 4($d$). From figure 4($d$) it is clear that the pressure node is not vertical but inclined with respect to the fluid interface owing to the 2-D actuation (all four walls are actuated at $d_{0}$). The above 2-D relocation can be clearly explained by the fact that if the fluid interface and node are not perpendicular to each other, then $\bm{\nabla}\times\textbf{\emph{f}}_{rl}$ $\neq 0$. This implies when a sufficient energy density is applied, the fluid system 4($d$) will not be in equilibrium and relocation begins without imposing any perturbations unlike the other relocation discussed in this work.

When the 1-D acoustic standing wave is imposed on fluids with interfacial tension, the configurations (figures 1($b$), 2($b$-$i$), 3($a$) having high impedance fluid at the anti-node and the low impedance fluid at the node, become conditionally stable. From (21), it is evident that the stability of the above inhomogeneous fluid configurations is governed by the ratio of $F_{rl}$ and $F_{int}$, which is called as acoustic Bond number ($Bo_{a}$), given by

$$Bo_{a}=\frac{F_{rl}}{F_{int}}=\frac{\phi E_{ac}\Delta Z\sin\left(2k_{w}x_{s}\right)}{k_{h}^{2}T}$$

(23)

The $Bo_{a}$ that separates the stable and unstable region is called critical acoustic Bond number $Bo_{a,cr}$. From (21)

$$Bo_{a,cr}=1$$

(24)

For $Bo_{a}>Bo_{a,cr}$, the above configurations become unstable (relocation occurs), and for $Bo_{a}<Bo_{a,cr}$ the configurations remain stable. Figure 5 shows the simulation results of different immiscible fluid combinations. The relocation and non-relocation regimes predicted by the simulations are in line with (24). It must also be noted that the fluids with higher interfacial tension require a higher acoustic force for relocation.

The height of the channel $h$ plays a critical role in the stability of immiscible fluids. For a given $E_{ac}$, the increase in $h$ weakens the stabilizing effect of interfacial tension force, as analysed theoretically in $\S$ 2. From (22), it can be inferred that the critical acoustic energy density is inversely proportional to the square of the channel height $(E_{cr}\propto 1/h^{2})$. In figure 6($a$), for a microchannel of height $h=80\ \mu m$ consisting of mineral-silicone oil with interfacial tension $T=1\ mN/m$, the fluid system is stable as the applied $E_{ac}\ (120\ J/m^{3}$) is lower than the critical energy density ($E_{cr}=384\ J/m^{3}$). Whereas for $h=160\ \mu m$ and keeping the remaining parameters same, fluid relocation is observed as applied $E_{ac}\ (120\ J/m^{3})$ is higher than the critical energy density $(E_{cr}=88.78\ J/m^{3})$.

The above discussion on the effect of channel height on acoustic relocation has high relevance in practical applications. To relocate fluids with high interfacial tension of $\mathcal{O}(10^{1}-10^{2})$ $mN/m$, in commonly used acoustofluidic channels of height ranging from $100\ \mu m$ to $200\ \mu m$, the required $E_{ac}$ becomes $\approx\mathcal{O}(10^{4})\ J/m^{3}$, which is very high compared to the $E_{ac}$ employed in typical acoustofluidic experiments ($\mathcal{O}(10^{2}-10^{3})$ $J/m^{3}$). The equation (22) tells that the above problem can be solved by increasing the channel height as $E_{ac}\propto 1/h^{2}$. Hence, the depth (height) of the channel is a crucial aspect to be considered during the fabrication of an acoustofluidic microchannel for handling high interfacial tension fluids.