# Theory of Nonlinear Acoustic Forces Acting on Inhomogeneous Fluids

Varun Kumar Rajendran,1 Sujith Jayakumar,1 Mohammed Azharudeen,1 and Karthick
Subramani1<EMAIL_ADDRESS>1 Department of Mechanical Engineering,
Indian Institute of Information Technology, Design and Manufacturing
Kancheepuram, Chennai-600127, India.

###### Abstract

Recently, the phenomena of streaming suppression and relocation of
inhomogeneous miscible fluids under acoustic fields were explained using the
hypothesis on mean Eulerian pressure. In this letter, we show that this
hypothesis is unsound and any assumption on mean Eulerian pressure is
needless. We present a theory of non-linear acoustics for inhomogeneous fluids
from the first principles, which explains streaming suppression and acoustic
relocation in both miscible and immiscible inhomogeneous fluids inside a
microchannel. This theory predicts the relocation of higher impedance fluids
to pressure nodes of the standing wave, which agrees with the recent
experiments.

The acoustic fields imposed on fluids exhibits several interesting nonlinear
acoustic phenomena including acoustic radiation force acting on particle or
interface, and acoustic streaming [1]. This subject has long history,
beginning with early investigations by Faraday [2], Rayleigh [3], King [4],
and Lighthill [5]. Over the last two decades, employing these acoustic forces
in microscale flows has become a rapidly growing research field known as
“Acoustofluidics”. It has far ranging applications in biological [6, 7, 8, 9,
10], medical [11, 12] and chemical sciences [13]. Recently, two interesting
phenomena were observed when the acoustic fields are imposed on inhomogenous
fluid called acoustic relocation [14] and streaming suppression [15]. Karlsen
et al [16] theoretically explained both the above phenomena using acoustic
force density which is divergence of time averaged momentum flux tensor,
$\textbf{\emph{f}}_{ac}=-\nabla\cdot\langle\Pi\rangle=-\nabla\cdot\langle
p_{2}I+\rho_{0}\textbf{\emph{v}}_{1}\textbf{\emph{v}}_{1}\rangle$, where
$\langle p_{2}\rangle$ is time averaged second order mean Eulerian pressure
and
$\nabla\cdot\langle\rho_{0}\textbf{\emph{v}}_{1}\textbf{\emph{v}}_{1}\rangle$
is the time averaged Reynolds stress tensor due to first order acoustic
fields. The mean Eulerian pressure is assumed as $\langle
p_{2}\rangle=(1/2)[\kappa_{0}\langle|p_{1}|^{2}\rangle-\rho_{0}\langle|\textbf{\emph{v}}_{1}|^{2}\rangle]$
and this assumption is claimed as the central hypothesis of their theory [17].
Following are our objections to this assumption: First, like any other
pressure field, the mean Eulerian pressure has to be derived from Navier -
Stokes (N-S) equations. Second, if the assumed $p_{2}$ is second order
Eulerian pressure then the pressure that results from N-S equations after the
substitution of $-\nabla\cdot\langle\Pi\rangle$ lacks clarity. In this letter,
we show that the above hypothesis is unsound and unnecessary.

The main goal of this letter is to develop a theory of nonlinear acoustics for
inhomogenous fluids that explains the phenomena of acoustic relocation and
streaming suppression. Remarkably, our theory predicts that the acoustic
relocation/stabilization of inhomogeneous fluids in a microchannel subjected
to standing acoustic waves is possible only if there exist an impedance
($Z=\rho c$) gradient which agrees well with the recent experiments. We
demonstrate that the amplitude of first order fields are highly dependent on
fluid configuration, thus acoustic energy density ($E_{ac}$) varies
significantly during the process of relocation. Also, we successfully separate
the streaming term and acoustic relocation term from the generalized acoustic
body force, which previously claimed not possible [18]. Furthermore, this
theory also explains the relocation of immiscible fluids under acoustic
fields.

_Physics._ \- The hydrodynamics of inhomogeneous fluids considered in this
study is governed by mass-continuity, momentum and advection-diffusion
equations, [19]

$\partial_{t}\rho+\nabla\cdot\left(\rho\textbf{\emph{v}}\right)=0$ (1a)
$\rho[\partial_{t}\textbf{\emph{v}}+(\textbf{\emph{v}}\cdot\nabla)\textbf{\emph{v}}]=-\nabla
p+\eta\nabla^{2}\textbf{\emph{v}}\\\
+\beta\eta\nabla(\nabla\cdot\textbf{\emph{v}})+\rho\textbf{\emph{g}}$ (1b)
$\partial_{t}\emph{s}+\textbf{\emph{v}}\cdot\nabla\emph{s}=D\nabla^{2}\emph{s}$
(1c) where $\rho$ is the density, v is the velocity, $p$ is the pressure,
$\eta$ is the dynamic viscosity of the fluid, $\xi$ is the bulk viscosity,
$\beta=(\xi/\eta)+(1/3)$, s is the solute concentration and _D_ is
diffusivity. When the fluid is subjected to acoustic waves, following
thermodynamic pressure-density relation in terms of material derivative
$(d/dt)=\partial_{t}+(\textbf{\emph{v}}_{1}\cdot\nabla)$ is also required
[20], $\frac{d\rho}{dt}=\frac{1}{c^{2}}\frac{dp}{dt}$ (1d)

where $c^{2}=(\partial p/\partial\rho)|_{S}$ and $c$ is adiabatic local speed
of sound.

According to perturbation theory, the dependent fields $f$ are decomposed as
[21],

$f=f_{0}(\textbf{\emph{r}},\tau)+f_{1}(\textbf{\emph{r}},\tau,t_{f})+f_{2}(\textbf{\emph{r}},\tau)$
(2)

where $f_{0}$ is zeroth order (background) fields, $f_{1}$ is first order
time-harmonic acoustic fields $f_{1}=f_{a}(\textbf{\emph{r}},\tau)e^{-i\omega
t_{f}}$, actuated at an angular frequency $\omega$ $(\sim 1MHz)$, and $f_{2}$
are second order fields (in general $f_{2}\ll f_{1}$). In microscale flows,
since the hydrostatic pressure $\rho gH(\sim 1Pa)\ll p_{1}(\sim 10^{6}Pa)$,
the variation of pressure and velocity fields due to gravity is accounted in
the second order effects. Thus, in a quiescent fluid, we take the zeroth order
velocity $\textbf{\emph{{v}}}_{0}=0$, and pressure $p_{0}=constant$ ($\nabla
p_{0}=0$).

The first order acoustic fields vary in fast time scale $t_{f}$
($t_{f}\sim\frac{1}{\omega}\sim 0.1\mu s$), whereas the second order
hydrodynamic fields varies in slow time scale $\tau$ $(\tau\gg t_{f})$.
Usually in perturbation theory, $f_{0}(r,\tau)$ is assumed to be constant for
homogeneous fluids. Whereas for inhomogeneous fluids, the variation in
background fields ($\rho_{0}$ and $s_{0}$) with space as well as slow time
scale has to be accounted due to the gravity stratification and second order
acoustic effects. As the first order acoustic fields ($f_{1}$) are sensitive
to inhomogeneous configuration ($\rho_{0}$ and $s_{0}$) which varies with slow
time scale, the amplitude of these acoustic fields are considered to be
function of slow time scale ($f_{a}(\textbf{\emph{r}},\tau)$) unlike
homogeneous fluids, where the amplitude of these fields are only function of
space ($f_{a}(\textbf{\emph{r}})$).

The variation of zeroth order fields in fast time scale ($t_{f}$) can be
neglected. Also, since $D\sim\mathcal{O}(10^{-9})$, the diffusion term is
negligible in fast time scale, due to which composition of any given fluid
particle remains unchanged as it moves. Consequently, the governing equations
up to first-order ($t\sim t_{f}$) reduce to [20]

$\partial_{t}\rho_{1}+\nabla\cdot\left(\rho_{0}\textbf{\emph{{v}}}_{1}\right)=0$
(3a) $\rho_{0}\partial_{t}\textbf{\emph{v}}_{1}=-\nabla
p_{1}+\eta\nabla^{2}\textbf{\emph{v}}_{1}+\beta\eta\nabla(\nabla\cdot\textbf{\emph{v}}_{1})$
(3b) $\partial_{t}s_{1}+\textbf{\emph{v}}_{1}\cdot\nabla\emph{s}_{0}=0$ (3c)
$\partial_{t}\rho_{1}+(\textbf{\emph{v}}_{1}\cdot\nabla)\rho_{0}=(1/c^{2})[\partial_{t}p_{1}]$
(3d)

The first order fields have a harmonic time dependence, thus time average of
these fields are zero. Therefore, first order terms in Eqs.(3) cannot cause
any bulk fluid motion. However, the Navier-Stokes equation is non linear and
the above linearized Eqs.(3) are not exact. Hence, proceeding to solve Eqs.(1)
up to second-order ($t\sim\tau$),

$\left\langle\partial_{t}(\rho_{0}+\rho_{2})\right\rangle+\nabla\cdot\langle\rho_{1}\textbf{\emph{v}}_{1}\rangle+\nabla\cdot\langle\rho_{0}\textbf{\emph{v}}_{2}\rangle=0$
(4a)
$\langle\rho_{1}\partial_{t}\textbf{\emph{v}}_{1}\rangle+\rho_{0}\langle(\textbf{\emph{v}}_{1}\cdot\nabla)\textbf{\emph{v}}_{1}\rangle-\langle(\rho_{0}+\rho_{2})\textbf{\emph{g}}\rangle=-\nabla\langle
p_{2}\rangle\\\
+\eta\nabla^{2}\langle\textbf{\emph{v}}_{2}\rangle+\beta\eta\nabla(\nabla\cdot\langle\textbf{\emph{v}}_{2}\rangle)-\left\langle\rho_{0}\partial_{t}\textbf{\emph{v}}_{2}\right\rangle$
(4b)
$\langle\partial_{t}(s_{0}+s_{2})\rangle+\langle\textbf{\emph{v}}_{2}\cdot\nabla
s_{0}\rangle+\langle\textbf{\emph{v}}_{1}\cdot\nabla
s_{1}\rangle=D\nabla^{2}\langle(s_{0}+s_{2})\rangle$ (4c)
$\langle\partial_{t}(\rho_{0}+\rho_{2})\rangle+\langle(\textbf{\emph{v}}_{1}\cdot\nabla)\rho_{1}\rangle+\langle(\textbf{\emph{v}}_{2}\cdot\nabla)\rho_{0}\rangle=\\\
(1/c^{2})[\langle\partial_{t}p_{2}\rangle+\langle(\textbf{\emph{v}}_{1}\cdot\nabla)p_{1}\rangle]$
(4d)

where $\langle...\rangle$ denotes time average over one oscillation period.
Since the time average of product of two first order fields is nonzero, they
act as a source term for second order fields (slow hydrodynamic time scale).
The first two terms on the left side of Eq.(4b) together comprise the
divergence of Reynolds stress tensor,
$\nabla\cdot\langle\rho_{0}\textbf{\emph{v}}_{1}\textbf{\emph{v}}_{1}\rangle$.

In microscale flows, variation of second order fields with respect to slow
time scale is negligible [1, 21]. From Eqs.(3), $\rho_{1}\ll\rho_{0}$ and
$s_{1}\ll s_{0}$, thus $\rho_{2}\ll\rho_{0}$ and $s_{2}\ll s_{0}$. We can also
neglect $\langle\textbf{\emph{v}}_{1}\cdot\nabla
s_{1}\rangle$,$\langle\rho_{1}\nabla\cdot\textbf{\emph{v}}_{1}\rangle$ since
the first order fields in both these terms are out of phase. Using the above
arguments, combining Eqs.(4a and 4d),

$\langle\rho_{0}\nabla\cdot\textbf{\emph{v}}_{2}\rangle=(1/c^{2})\langle\textbf{\emph{v}}_{1}\cdot\nabla
p_{1}\rangle$ (5)

Substituting the above relation in Eq.(4b) and analysing the order of
magnitude,
$\mathcal{O}(\beta\eta\nabla\frac{1}{\rho_{0}c_{0}^{2}}(\textbf{\emph{v}}_{1}\cdot\nabla)p_{1})\ll\mathcal{O}(\nabla\cdot\langle\rho_{0}\textbf{\emph{v}}_{1}\textbf{\emph{v}}_{1}\rangle)$.
As this term, $(1/c^{2})\langle\textbf{\emph{v}}_{1}\cdot\nabla p_{1}\rangle$
does not contribute in momentum equation, it can be neglected. Hence, Eq.(5)
becomes divergence free and second-order flow is incompressible i.e.,
$\nabla\cdot\langle\textbf{\emph{v}}_{2}\rangle=0$ (Nyborg’s
approximation-[22, 23]). Accounting all the above arguments, the governing
equations reduce to

$\nabla\cdot\langle\textbf{\emph{v}}_{2}\rangle=0$ (6a)
$-\nabla\cdot\langle\rho_{0}\textbf{\emph{v}}_{1}\textbf{\emph{v}}_{1}\rangle+\langle\rho_{0}\textbf{\emph{g}}\rangle-\nabla\langle
p_{2}\rangle+\eta\nabla^{2}\langle\textbf{\emph{v}}_{2}\rangle=0$ (6b)
$\langle\partial_{t}\emph{s}_{0}\rangle+\langle\textbf{\emph{v}}_{2}\cdot\nabla\emph{s}_{0}\rangle=D\nabla^{2}\langle\emph{s}_{0}\rangle$
(6c)

The above Eqs.(6) govern the dynamics of inhomogeneous fluids in microscale
acoustofluidics and can also be derived by another approach, see Supplemental
Material . From Eq.(6b), it is evident that the second-order slow hydrodynamic
flows are created due to the divergence of Reynolds stress tensor consisting
of product of first order fast acoustic fields. Thus, we introduce the body
force due to acoustic fields as

$\textbf{\emph{f}}_{ac}=-\nabla\cdot\langle\rho_{0}\textbf{\emph{v}}_{1}\textbf{\emph{v}}_{1}\rangle$
(7)

It is well-known that the above force is responsible for boundary-driven
Rayleigh streaming and bulk-driven Eckart streaming in homogeneous fluids. In
this letter, we proceed to show the same force is also responsible for
recently observed streaming suppression and acoustic relocation of miscible as
well as immiscible inhomogeneous fluids. A microchannel of width $w=380$ $\mu
m$ and height $h=160$ $\mu m$ containing an inhomogeneous miscible solution is
chosen for study Fig.(1). An acoustic standing half-wave along the width is
imposed in the microchannel by actuating both the side walls at a specific
frequency ${}^{\prime}f^{\prime}$ and wall displacement
${}^{\prime}d^{\prime}$ in x-direction. The first order fields due to this
standing wave are obtained by solving the Eqs.(3) in frequency domain, see
Supplemental Material . These first order fields are then used to obtain the
acoustic body force Eq.(7) to solve the time averaged second order fields
Eqs.(6). At each time step, the first order fields are solved for the updated
inhomogeneous fluid configuration ($\rho_{0},c_{0},\mu_{0}$) from the previous
time step. Hence, both the first order and second order equations are
bidirectionally coupled and solved numerically at all slow time-steps in
COMSOL Multiphysics 5.6.

Figure 1: Sketch of acoustofluidic microchannel with an imposed half-wave
acoustic pressure resonance, containing fluids whose density, speed of sound,
dynamic viscosity and solute concentration are functions of space and time.

_Results._ \- From Fig.(2), acoustic body force $\textbf{\emph{f}}_{ac}$ tends
to relocate high impedance fluid to node and low impedance to anti-nodes
called stable configuration (whereas, any other configuration with impedance
gradient is considered unstable). Once stable configuration is reached, due to
existing impedance gradient, this $\textbf{\emph{f}}_{ac}$ inhibits any fluid
motion due to gravity and suppresses acoustic streaming, that tries to disturb
the stable configuration as seen in Figs.(2 & 3). As time progresses, due to
diffusion, the fluid profile becomes homogeneous, where the same
$\textbf{\emph{f}}_{ac}$ induces boundary-driven Rayleigh streaming. From
these results, it is evident that the acoustic body force
$\textbf{\emph{f}}_{ac}$ is responsible for acoustic relocation and streaming
suppression in inhomogeneous fluids as well as acoustic streaming in
homogeneous fluids. Remarkably, in the process of acoustic relocation and
diffusion as shown in Figs.(2a and 3a), the amplitude of the first order
fields ($p_{a}$ and $v_{a}$) vary significantly (refer Supplemental Material )
as the background $\rho_{0}$ and $c_{0}$ fields change in slow time-scale.
Hence, it is now apparent that assumption of acoustic force density $E_{ac}$
to be constant is incorrect. Surprisingly, it is observed that for the case of
constant impedance Fig.(4), relocation does not occur irrespective of
$\rho_{0}$ and $c_{0}$ configurations, thus any constant impedance
configuration is called neutral configuration. This demonstrates that
impedance gradient is the requisite and governing factor for acoustic
relocation. Along with this, it is also observed from the simulations that,
the sufficient conditions for acoustic relocation include - fluid interface
should not be at node (Supplemental Material ) and the fluid configuration
must be unstable.

Figure 2: (a) Relocation of unstable configuration to stable configuration
with $d=0.261$ nm and $f_{0}=1.96$ MHz. Initially (t=0s), low impedance DI
Water (blue) at the center and high impedance 10% Ficoll PM70 (red) at the
sides. (b) second order velocity $\textbf{\emph{v}}_{2}$. Fluid properties are
taken from [24] Figure 3: (a) Stable configuration with no relocation, low
impedance DI Water (blue) at the sides and high impedance 10% Ficoll PM70
(red) at the center (b) Second order velocity $\textbf{\emph{v}}_{2}$. Here,
$d=0.261$ nm and $f_{0}=1.96$ MHz. Relocation force ($\textbf{\emph{f}}_{rl}$)
stabilizes this configuration against gravity stratification.

In order to explain the above results mathematically we analyse the acoustic
body force,
$\textbf{\emph{f}}_{ac}=-\nabla\cdot\langle\rho_{0}\textbf{\emph{v}}_{1}\textbf{\emph{v}}_{1}\rangle$
in detail

$\textbf{\emph{f}}_{ac}=-\nabla\cdot\langle\rho_{0}\textbf{\emph{v}}_{1}\textbf{\emph{v}}_{1}\rangle=-\rho_{0}\langle\textbf{\emph{v}}_{1}\cdot\nabla\textbf{\emph{v}}_{1}\rangle-\langle\rho_{1}\partial_{t}\textbf{\emph{v}}_{1}\rangle$
(8)

Using the following identity,
$\textbf{\emph{A}}\cdot\nabla\textbf{\emph{A}}=\nabla(\textbf{\emph{A}}^{2}/2)-\textbf{\emph{A}}\times(\nabla\times\textbf{\emph{A}})$
and substituting first-order fields from Eqs.(3). Eq.(8) becomes

$=-\frac{1}{2\rho_{0}}\nabla\langle|\rho_{0}\textbf{\emph{v}}_{1}|^{2}\rangle+\langle\textbf{\emph{v}}_{1}\times\nabla\times(\rho_{0}\textbf{\emph{v}}_{1})\rangle+\frac{1}{2}\kappa_{0}\nabla\langle|p_{1}|^{2}\rangle\\\
-\langle(\kappa_{0}p_{1})(\eta\nabla^{2}\textbf{\emph{v}}_{1}-\beta\eta\nabla(\nabla\cdot\textbf{\emph{v}}_{1}))\rangle$
(9)

For detailed derivation refer Supplemental Material . From the scaling
analysis, last term in Eq.(9) can be neglected, thus reducing to

$=\frac{1}{2}\nabla(\kappa_{0}\langle|p_{1}|^{2}\rangle-\rho_{0}\langle|\textbf{\emph{v}}_{1}|^{2}\rangle)+\\{\langle\textbf{\emph{v}}_{1}\times\nabla\times(\rho_{0}\textbf{\emph{v}}_{1})\rangle\\}\\\
-\frac{1}{2}[\langle|p_{1}|^{2}\rangle\nabla\kappa_{0}+\langle|\textbf{\emph{v}}_{1}|^{2}\rangle\nabla\rho_{0}]$
(10)

Figure 4: For constant impedance fluids any configuration of $\rho_{0}$ and
$c_{0}$ is neutral configuration (a) No relocation due to zero relocation
force ($\textbf{\emph{f}}_{rl}=0$). To clearly show the absence of relocation
force in constant impedance case, gravity is neglected. The presence of
gravity would stratify the fluids. (b) Second order velocity
$\textbf{\emph{v}}_{2}$. Here, $d=2$ nm and $f_{0}=1.96$ MHz.

In the above equation Eq.(10), the first term consists of purely gradient term
(conservative) and curl of that term is zero, thus does not cause relocation
or streaming but only contributes to second order pressure. The third term
within the square bracket is responsible for relocation, since the curl of
this term in general is non-zero. From Eq.(3b), the first order fields in the
inviscid region satisfies the following relation,
$\nabla\times(\rho_{0}\textbf{\emph{v}}_{1})=0$. Thus the term within the
curly brackets is only significant inside the boundary layer ($\delta\sim 1\mu
m$) of first order fields, which is responsible for boundary-driven streaming
in inhomogeneous fluids. The competition between the relocation force (third
term) and streaming force (second term) accounts for streaming suppression as
seen in Figs.(2 and 3). In Eq.(10), we have analytically separated relocation
and streaming causing terms from the general body force term,
$\textbf{\emph{f}}_{ac}$, which was previously claimed not possible [18]. For
homogeneous fluids Eq.(10) reduces to [1, 25]

$\textbf{\emph{f}}_{ac}=-\nabla\cdot\langle\rho_{0}\textbf{\emph{v}}_{1}\textbf{\emph{v}}_{1}\rangle=\frac{1}{2}\left(\kappa_{0}\nabla\langle|p_{1}|^{2}\rangle-\rho_{0}\nabla\langle|\textbf{\emph{v}}_{1}|^{2}\rangle\right)\\\
+\\{\rho_{0}\left\langle\textbf{\emph{v}}_{1}\times(\nabla\times\textbf{\emph{v}}_{1})\right\rangle\\}$
(11)

where the first term is homogeneous second order mean Eulerian pressure and
the second term is responsible for acoustic streaming.

To understand the second order pressure and relocation phenomenon in
inhomogeneous fluids clearly, we study the acoustic body force outside the
boundary layer of first order fields (neglecting streaming term). Thus,
Eq.(10) after time averaging reduces to

$\textbf{\emph{f}}_{ac}=-\nabla\cdot\langle\rho_{0}\textbf{\emph{v}}_{1}\textbf{\emph{v}}_{1}\rangle=\frac{1}{4}\nabla\left(\kappa_{0}\langle|p_{1}|^{2}\rangle-\rho_{0}\langle|\textbf{\emph{v}}_{1}|^{2}\rangle\right)\\\
\left[-\frac{1}{4}\langle|p_{1}|^{2}\rangle\nabla\kappa_{0}-\frac{1}{4}\langle|\textbf{\emph{v}}_{1}|^{2}\rangle\nabla\rho_{0}\right]=\textbf{\emph{f}}_{1}+[\textbf{\emph{f}}_{2}]$
(12)

Considering the case of stable configuration Fig.(3), second order velocity
will be zero as relocation does not take place due to which Eq.(6b) will
reduce to
$-\nabla\cdot\langle\rho_{0}\textbf{\emph{v}}_{1}\textbf{\emph{v}}_{1}\rangle=\nabla\langle
p_{2}\rangle$. Now, it is evident that in inviscid case, both the terms in
Eq.(12) contribute to the mean Eulerian pressure $\langle p_{2}\rangle$.
Whereas, in [16, 17, 18], it is hypothesised that only first term in Eq.(12)
contributes to mean Eulerian pressure, $\langle
p_{2}\rangle=\frac{1}{2}\kappa_{0}\langle|p_{1}|^{2}\rangle-\frac{1}{2}\rho_{0}\langle|\textbf{\emph{v}}_{1}|^{2}\rangle$.
Thus our work clearly demonstrates, the assumption on $p_{2}$ is needless and
can be derived from the governing equations.

In case of constant impedance ($Z_{0}=\rho_{0}c_{0}=constant$) inhomogeneous
fluids, subjected to acoustic standing half wave ($\lambda\approx 2w$), using
the relations $p_{1}=p_{a}sin(k{x})$ and
$v_{1}=\frac{p_{a}}{i\rho_{0}c}cos(k{x})$ where $k=2\pi/\lambda$ is wave
number. The relocation force term $\textbf{\emph{f}}_{2}$ reduces to

$\textbf{\emph{f}}_{2|_{Z=C}}=-\frac{1}{4}|p_{1}|^{2}\nabla\kappa_{0}-\frac{1}{4}|\textbf{\emph{v}}_{1}|^{2}\nabla\rho_{0}=-\nabla\left(\frac{p_{a}^{2}\rho_{0}}{4Z^{2}}\right)$
(13)

Since $\textbf{\emph{f}}_{2}$ reduces to a purely gradient term in case of
constant impedance inhomogeneous fluids, it does not induce relocation or
motion but only contributes towards second order pressure. Thus, we prove that
impedance gradient is the necessary condition for acoustic relocation, which
agrees with experimental results [26, 14].

For the case of variable impedance, the relocation term $f_{2}$ in Eq.(12) is
written as

$\textbf{\emph{f}}_{2}=\frac{p_{a}^{2}\sin^{2}{kx}}{4}\left(\frac{\nabla\rho_{0}}{\rho_{0}^{2}c_{0}^{2}}\right)+\frac{p_{a}^{2}\sin^{2}{kx}}{2}\left(\frac{\nabla
c_{0}}{\rho_{0}c_{0}^{3}}\right)-\\\
\frac{p_{a}^{2}\cos^{2}{kx}}{4}\left(\frac{\nabla\rho_{0}}{\rho_{0}^{2}c_{0}^{2}}\right)$
(14)
$\textbf{\emph{f}}_{2}=-\frac{p_{a}^{2}\cos({2kx})}{4\rho_{0}^{2}c_{0}^{3}}\nabla
Z_{0}+\frac{p_{a}^{2}}{4\rho_{0}c_{0}^{3}}\nabla c_{0}$ (15)

Analogous to Boussinesq approximation, $\frac{\nabla
Z_{0}}{\rho_{0}^{2}c_{0}^{3}}\approx\frac{\nabla
Z_{0}}{\rho_{avg}^{2}c_{avg}^{3}},\frac{\nabla
c_{0}}{\rho_{0}c_{0}^{3}}\approx\frac{\nabla c_{0}}{\rho_{avg}c_{avg}^{3}}$
and substituting Eq.(15) in Eq.(12), separating gradient and non-gradient
terms,

$\textbf{\emph{f}}_{ac}=-\nabla\cdot\langle\rho_{0}\textbf{\emph{v}}_{1}\textbf{\emph{v}}_{1}\rangle=-E_{ac}\cos({2kx})\nabla\hat{Z}_{0}\\\
-\nabla\left(\frac{1}{4}\rho_{0}|\textbf{\emph{v}}_{1}|^{2}-\frac{1}{4}\kappa_{0}|p_{1}|^{2}-E_{ac}\hat{c}_{0}\right)$
(16)

where $E_{ac}=p_{a}^{2}/(4\rho_{avg}c_{avg}^{2})$,
$\hat{c}_{0}=c_{0}/c_{avg}$, $\hat{\rho}_{0}=\rho_{0}/\rho_{avg}$, and
$\hat{Z}_{0}=\hat{\rho}_{0}\hat{c}_{0}$.

It is clear that only first term in Eq.(16) is responsible for acoustic
relocation in inhomogeneous fluids, whereas second term resembles conservative
or purely gradient term, thus induces only pressure. Thus, in addition to
impedance gradient being the necessary factor for relocation, now we write
relocation force in terms of impedance gradient as follows

$\textbf{\emph{f}}_{rl}=-E_{ac}\cos({2kx})\nabla\hat{Z}_{0}$ (17)

The above force term which is a part of generalised force equation Eq.(7), is
responsible for relocation of unstable configuration in Fig.(2) and
maintaining the stable configuration by inhibiting acoustic streaming as well
as gravity stratification in Fig.(3). Whereas, for constant impedance fluids
(Fig.(4)), irrespective of any fluid configuration, relocation force is always
absent ($\textbf{\emph{f}}_{rl}=0$).

In several previous studies [24, 27, 28], including ours [29, 30], Eq.(15) has
been used to study relocation without realising that it contains implicit
gradient or conservative terms which does not contribute to relocation. This
leads to incorrect scaling analysis and thus it is necessary to ignore this
gradient term in such circumstances. Furthermore, Eq.(17) captures all the
aspects of relocation and does not cause scaling issues like Eq.(15). It must
be noted that Eq.(17) is valid only outside the boundary layer of first order
fields, whereas inside the boundary layer, it cannot be reduced from
$\textbf{\emph{f}}_{2}$ of Eq.(12). This is because $p_{1}=p_{a}sin(kx)$
remains same inside and outside the boundary layer ($\delta\sim 1\mu m$), but
$v_{1}\neq\frac{p_{a}}{i\rho_{0}c}cos(kx)$ and $\partial_{y}v_{1}$ is
significant inside the boundary layer due to no slip condition. Thus, in case
of constant impedance fluids, relocation force is absent in the bulk but non-
zero within the boundary layer, which is responsible for the disturbance of
homogeneous streaming as in Fig.(4).

_Immiscible fluids_.- The phenomenon of acoustic relocation of immisicble
fluids due to standing acoustic wave is also governed by Eq.(7). This theory
predicts that for the relocation of immiscible fluids to occur, the applied
acoustic energy density $E_{ac}$ must be greater than the threshold $E_{cr}$
in order to overcome the interfacial tension force. The above prediction is in
agreement with the experimental studies by Hemachandran et al. [26]. However,
in their study relocation is achieved irrespective of the interface location
with respect to node. Also, the frequency employed is much different than the
resonant frequency ($f=c_{avg}/2w$) that corresponds to the standing half-wave
actuated only along the width of microchannel as shown in Fig.(1). On contrary
to their study, when we actuated only side walls, relocation seemed to be
highly dependent on the position of the interface and was absent when the
interface and node coincide. Whereas, when all the walls were actuated in
direction to their surface normal, we achieved relocation, independent of the
interface location, (Fig.(5)) that is in agreement with their experiments
[26]. It is evident that relocation can be achieved through two modes; one is
due to 1D standing wave resulting from the actuation of side walls at 1D
resonant frequency ($f=c_{avg}/2w$) while the other is due to 2D standing wave
resulting due to actuation of all walls at 2D resonant frequency (between
$f=c_{avg}/2w$ and $f=c_{avg}/2h$). Since the relocation in [26] is due to 2D
standing wave, the frequency required ($\sim 2.1-2.4MHz$) for relocation (for
silicone-mineral oil) is different from 1D resonant frequency ($1.66MHz$).

Figure 5: Relocation of immiscible fluids consisting high impedance mineral
oil(red) and low impedance silicone oil(blue) with a surface tension of $1$
$mN/m$ actuated at a frequency of $2.1$ $MHz$ for unstable configurations a)
No relocation for $E_{ac}<E_{cr}$, b) Relocation for $E_{ac}>E_{cr}$ and c)
stable configuration - No relocation for any $E_{ac}$

_Conclusion_ \- We have put forward a theory of nonlinear acoustics that
governs the acoustic phenomena of relocation and streaming suppression in
inhomogeneous miscible and immiscible fluids (including streaming in
homogeneous fluids). This theory also confirms the fact that divergence of
time-averaged Reynolds tensor
$-\nabla\cdot\langle\rho_{0}\textbf{\emph{v}}_{1}\textbf{\emph{v}}_{1}\rangle$
is alone responsible for all the above processes. We showed that first order
fields $p_{1},\textbf{\emph{v}}_{1}$ and energy density $E_{ac}$ vary
significantly during process of acoustic relocation and diffusion.
Importantly, we have proved that impedance gradient is the necessary condition
for relocation. The other conditions for acoustic relocation in 1D and 2D mode
will be addressed in detail using stability analysis for both miscible and
immiscible fluids in upcoming paper. The fundamental understanding from this
study can give new insights for particle (cells/drops/beads) and inhomogeneous
fluid handling in microchannel under acoustic fields.

## References

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