$|\frac{b_{i}}{\varpi^{j}}|r^{|i|}=|\varpi|^{-j}|b_{i}|r^{|i|}\leq|\varpi|^{-j}{r}^{\frac{j|i|}{j+1}}\leq|\varpi|^{-j}{r}^{\frac{jM}{j+1}}$

for $|i|\geq M$. It is thus enough to show

${r}^{\frac{-(M-1)}{j+1}}<\frac{1}{|\varpi|}\quad\text{and}\quad|\varpi|^{-j}{r}^{\frac{jM}{j+1}}\leq
1.$

Rearranging these two inequalities, we obtain the following condition:

$M\geq\frac{j+1}{\log_{|\varpi|}(r)}>M-1,$

which is satisfied for for $M=\lceil\frac{1+j}{\log_{|\varpi|}(r)}\rceil$, as
needed. ∎

######  Theorem 6.2.3.

There is a morphism in $D(\operatorname{Solid})$:

$\alpha_{B^{d,\circ}}\colon\underline{\mathcal{O}_{K}\llbracket
T_{1},\dots,T_{d}\rrbracket}[\varepsilon]\to
R\Gamma(B^{d,\circ}_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}}),$

such that for all $i\geq 0$, $H^{i}(\operatorname{cof}(\alpha_{B^{d,\circ}}))$
is killed by a single power of $p$.

###### Proof.

We apply Theorem 5.6.8 to the tamely semistable cover
$\mathcal{U}=\\{B^{d}_{r_{\ell}}\\}$. The condensed Čech complex
$\check{C}(\mathcal{U},\mathcal{O}_{\operatorname{cond}}^{+})$ is quasi-
isomorphic to
$R\varprojlim_{\ell}H^{0}(B_{r_{\ell}}^{d},\mathcal{O}^{+}_{\operatorname{cond}})$.
By Lemma 6.2.2 and Lemma 6.2.1, the $\mathbb{N}^{2}$-diagram
$H^{0}(B_{r_{\ell}}^{d},\mathcal{O}^{+}_{\operatorname{cond}})/(p^{j})(S)$
satisfies the Mittag-Leffler condition for each profinite set $S$. We then
compute:

$\displaystyle
R\varprojlim_{\ell}H^{0}(B_{r_{\ell}}^{d},\mathcal{O}^{+}_{\operatorname{cond}})$
$\displaystyle\cong
R\varprojlim_{\ell}R\varprojlim_{j}H^{0}(B_{r_{\ell}}^{d},\mathcal{O}^{+}_{\operatorname{cond}})/p^{j}$
$\displaystyle\cong
R\varprojlim_{\ell,j}H^{0}(B_{r_{\ell}}^{d},\mathcal{O}^{+}_{\operatorname{cond}})/p^{j}$
$\displaystyle\cong\varprojlim_{\ell,j}H^{0}(B_{r_{\ell}}^{d},\mathcal{O}^{+}_{\operatorname{cond}})/p^{j}$
$\displaystyle\cong\varprojlim_{\ell}H^{0}(B_{r_{\ell}}^{d},\mathcal{O}^{+}_{\operatorname{cond}})$
$\displaystyle\cong\underline{\mathcal{O}_{K}\llbracket
T_{1},\dots,T_{d}\rrbracket}.$

The first and forth isomorphisms follow from the derived $p$-completeness of
$H^{0}(B_{r_{\ell}}^{d},\mathcal{O}^{+}_{\operatorname{cond}})$. The third
isomorphism is a consequence of the Mittag-Leffler condition. ∎

######  Corollary 6.2.4.

Let $W=W(\overline{\mathbb{F}}_{p})$, let $K=W[1/p]$, and let
$\operatorname{LT}_{K}$ be Lubin–Tate space in height $n$, with
$A=H^{0}(\operatorname{LT}_{K},\mathcal{O}^{+})\cong\mathcal{O}_{K}\llbracket
u_{1},\dots,u_{n-1}\rrbracket$ the Lubin–Tate ring. We have an isomorphism of
graded $K$-vector space objects in $D(\operatorname{Solid})$:

$H^{*}(R\Gamma(\operatorname{LT}_{K,\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+})^{h\mathcal{O}_{D}^{\times}})\otimes_{W}K\cong\Lambda_{K}(x_{1},x_{3},\dots,x_{2n-1})[\varepsilon]\oplus(A^{h\mathcal{O}_{D}^{\times}}\otimes_{W}K)[\varepsilon].$

###### Proof.

$\operatorname{LT}_{K}$ is a rigid-analytic open ball, so Theorem 6.2.3
applies. After taking homotopy fixed points under $\mathcal{O}_{D}^{\times}$,
we find a morphism

$\alpha_{\operatorname{LT}_{K}}^{h\mathcal{O}_{D}^{\times}}\colon
A^{h\mathcal{O}_{D}^{\times}}[\varepsilon]\to
R\Gamma(\operatorname{LT}_{K,\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}})^{h\mathcal{O}_{D}^{\times}}$

By Lemma 6.1.2(2), the cohomology groups of
$\operatorname{cof}(\alpha_{\operatorname{LT}_{K}}^{h\mathcal{O}_{D}^{\times}})\cong\operatorname{cof}(\alpha_{\operatorname{LT}_{K}})^{h\mathcal{O}_{D}^{\times}}$
are all $p$-power torsion. After inverting $p$, we arrive at an isomorphism of
$K$-vector space objects in $D(\operatorname{Solid})$:

$A^{h\mathcal{O}_{D}^{\times}}\otimes_{W}K[\varepsilon]\cong
R\Gamma(\operatorname{LT}_{K,\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}})^{h\mathcal{O}_{D}^{\times}}\otimes_{W}K.$

Now we apply the decomposition $A=W\oplus A^{c}$ from Proposition 2.5.1. By
Lemma 3.8.1 (after extending scalars to $K$), we have
$W^{h\mathcal{O}_{D}^{\times}}\otimes_{W}K\cong K^{h\mathcal{O}_{D}^{\times}}$
is an exterior algebra over $K$. ∎

###  6.3. The pro-étale cohomology of Drinfeld’s symmetric space

Let $\mathcal{H}$ be Drinfeld’s symmetric space of dimension $n-1$. Then
$\mathcal{H}$ admits a semistable formal model $\mathfrak{H}/\mathbb{Z}_{p}$.
For a construction of $\mathfrak{H}$, see [GK05a, §6]. The formal scheme
$\mathfrak{H}/\mathbb{Z}_{p}$ admits an action of
$\operatorname{GL}_{n}(\mathbb{Q}_{p})$ which is compatible with the
isomorphism $\mathfrak{H}_{\mathbb{Q}_{p}}\cong\mathcal{H}$.

Our goal is to control the pro-étale cohomology of $\mathcal{H}$. Key to that
calculation is the following acyclicity result:

######  Theorem 6.3.1.

Let $K/\mathbb{Q}_{p}$ be a local field, and let
$\mathfrak{H}_{\mathcal{O}_{K}}$ be the base change of $\mathfrak{H}$ to
$\operatorname{Spf}\mathcal{O}_{K}$. We have a
$\operatorname{GL}_{n}(\mathbb{Z}_{p})$-equivariant isomorphism in
$D(\operatorname{Solid})$:

$R\Gamma(\mathfrak{H}_{\mathcal{O}_{K}},\mathcal{O}_{\operatorname{cond}})\cong\mathcal{O}_{K}.$

(Here $\mathcal{O}_{K}$ with its $p$-adic topology is considered as a solid
complex.)

######  Remark 6.3.2.

The non-condensed version of the theorem (over $\mathbb{Z}_{p}$) is a special
case of a result of Grosse-Klonne [GK05b]; for the sake of being self-
contained we offer a proof below. The result in [GK05b] also computes the
cohomology of each of the sheaves $\Omega^{j}_{\log}$ on $\mathfrak{H}$ in
terms of certain lattices in Steinberg representations of
$\operatorname{PGL}_{d+1}(\mathbb{Q}_{p})$. This would allow us to control the
$\hat{\mathcal{O}}^{+}$-cohomology of
$\mathcal{H}^{d}_{\mathrm{pro\acute{e}t},C}$. From this we could ultimately
compute the continuous cohomology
$H^{i}_{\operatorname{cts}}(\mathcal{O}_{D}^{1},A)[1/p]$, where
$\mathcal{O}_{D}^{1}\subset\mathcal{O}_{D}^{\times}$ is the subgroup of
elements of reduced norm 1. We do not pursue these computations here.

###### Proof of Theorem 6.3.1.

Let $\kappa$ be the residue field of $K$, and let
$s=\operatorname{Spec}\kappa$. Let $\mathfrak{H}_{s}$ denote the special fiber
of $\mathfrak{H}_{\mathcal{O}_{K}}$. Since
$R\Gamma(\mathfrak{H}_{\mathcal{O}_{K}},\mathcal{O}_{\operatorname{cond}})$ is
$p$-adically complete and its $H^{0}$ is torsion-free (since
$\mathfrak{H}_{\mathcal{O}_{K}}$ is semistable over
$\operatorname{Spf}\mathcal{O}_{K}$, it is also flat), it suffices to show
that $R\Gamma(\mathfrak{H}_{s},\mathcal{O}_{\operatorname{cond}})\cong\kappa$.

We need the following facts about $\mathfrak{H}_{s}$, and in particular its
relation to the Bruhat–Tits building $\operatorname{BT}$ for
$\operatorname{PGL}_{n}(\mathbb{Q}_{p})$ (see [GK05a, §6]):

  1. (1)

$\operatorname{BT}$ is the simplicial complex whose $q$-simplices
$\operatorname{BT}_{q}$ are the homothety classes of flags of
$\mathbb{Z}_{p}$-lattices in $\mathbb{Q}_{p}^{n}$:

$L_{0}\subsetneq L_{1}\subsetneq\cdots\subsetneq L_{q}\subsetneq p^{-1}L_{0}.$

In particular $\operatorname{BT}$ is $(n-1)$-dimensional.

  2. (2)

$\operatorname{BT}$ can be expressed as an increasing union of finite
contractible subcomplexes.

  3. (3)

There is a bijection $\sigma\mapsto Z_{\sigma}$ between the vertices
$\operatorname{BT}_{0}$ of $\operatorname{BT}$ and irreducible components of
$\mathfrak{H}_{s}$.

  4. (4)

A collection of vertices $\sigma$ in $\operatorname{BT}_{0}$ are the vertices
of a simplex in $\operatorname{BT}$ if and only if the $Z_{\sigma}$ have
nontrivial intersection. In that case, the $Z_{\sigma}$ meet transversely.

  5. (5)

For each $q$-simplex $\tau$, let $Z_{\tau}$ be the intersection of
$Z_{\sigma}$ over all vertices $\sigma$ of $\tau$. Then $Z_{\tau}$ is a
rational projective variety. More precisely, it is the result of a sequence of
blow-ups of $(n-1-q)$-dimensional projective space over $\kappa$, performed
along nonsingular subvarieties.

For $\tau\in\operatorname{BT}_{q}$, we let $\tau_{0}$ be its set of vertices,
and we let $U_{\tau}$ be the open subset of
$\cup_{\sigma\in\tau_{0}}Z_{\sigma}$ consisting of points not contained in
$Z_{\sigma^{\prime}}$ for any $\sigma^{\prime}\not\in\tau$. Then the
$U_{\tau}$ are a collection of (generally singular) affine varieties which is
closed under intersection.

The $U_{\tau}$ for $\tau\in\operatorname{BT}_{n-1}$ cover $\mathfrak{H}_{s}$,
so $R\Gamma(\mathfrak{H}_{s},\mathcal{O}_{\operatorname{cond}})$ is computed
by the condensed Čech complex

$\operatorname{lim}_{[q]\in\Delta}\prod_{f\colon[q]\to\operatorname{BT}_{n-1}}\mathcal{O}(U_{f}),$
(6.3.1)

where $U_{f}=\cap_{i\in[q]}U_{f(i)}$. Each $\mathcal{O}(U_{f})$ is taken to be
discrete, and the limit is computed in $D(\operatorname{Solid})$.

For $q>0$ and $\tau\in\operatorname{BT}_{q}$, the affine variety $U_{\tau}$ is
the union of irreducible components $Z_{\sigma}\cap U_{\tau}$ (for $\sigma$ a
vertex of $\tau)$ which meet transversely. Each $Z_{\sigma}\cap U_{\tau}$ is
an affine open in a rational variety $Z_{\sigma}$, and so it is isomorphic to
an open subset of $(n-1)$-dimensional affine space. Therefore there is an
isomorphism of $U_{\tau}$ onto an open subset of $\operatorname{Spec}B$, where

$B=k[x_{1},\dots,x_{n}]/(x_{1}x_{2}\cdots x_{q}),$

which carries $Z_{\sigma}\cap U_{\tau}$ into $\operatorname{Spec}B/(x_{i})$
for the corresponding $1\leq i\leq q$. Thus, the map
$B\to\mathcal{O}(U_{\tau})$ is flat and we have $\mathcal{O}(Z_{\sigma}\cap
U_{\tau})\simeq\mathcal{O}(U_{\tau})/(x_{i})$.

The resolution of $U_{\tau}$ by the disjoint union of the $Z_{\sigma}\cap
U_{\tau}$ gives a quasi-isomorphism:

$\mathcal{O}(U_{\tau})\cong\operatorname{lim}_{[r]\in\Delta}\prod_{\sigma\in\operatorname{BT}_{r}}\mathcal{O}(Z_{\sigma}\cap
U_{\tau}).$ (6.3.2)

This follows from the standard observation that the complex of $B$-modules

$B\to\bigoplus_{1\leq i\leq q}B/(x_{i})\to\bigoplus_{1\leq i<j\leq
q}B/(x_{i},x_{j})\to\cdots$

is exact; base-changing along the flat map $B\to\mathcal{O}(U_{\tau})$ gives
the isomorphism in (6.3.2).

Each $U_{f}$ appearing in (6.3.1) equals $U_{\tau}$ for
$\tau=\cap_{i\in[q]}f(i)$. Substituting (6.3.2) into (6.3.1) and rearranging
the order of limits shows that
$R\Gamma(\mathfrak{H}_{s},\mathcal{O}_{\operatorname{cond}})$ is quasi-
isomorphic to

$\operatorname{lim}_{[r]\in\Delta}\prod_{\sigma\in\operatorname{BT}_{r}}\operatorname{lim}_{[q]\in\Delta}\prod_{f\colon[q]\to\operatorname{BT}_{n-1}}\mathcal{O}(Z_{\sigma}\cap
U_{f}).$ (6.3.3)

For a given $\sigma\in\operatorname{BT}_{q}$, the projective variety
$Z_{\sigma}$ admits a cover by affine varieties $Z_{\sigma}\cap U_{\tau}$,
where $\tau$ runs through $\operatorname{BT}_{n-1}$. Therefore the inner limit
in (6.3.3) computes $R\Gamma(Z_{\sigma},\mathcal{O})$, and we find:

$R\Gamma(\mathfrak{H}_{s},\mathcal{O}_{\operatorname{cond}})\cong\operatorname{lim}_{[r]\in\Delta}\prod_{\sigma\in\operatorname{BT}_{r}}R\Gamma(Z_{\sigma},\mathcal{O})$

We are left with the task of computing the cohomology
$R\Gamma(Z_{\sigma},\mathcal{O})$ of the nonsingular projective variety
$Z_{\sigma}$. Property (5) above states that $Z_{\sigma}$ is constructed by a
sequence of blow-ups of projective space along nonsingular subvarieties. For
any projective space $\mathbf{P}$ over $\kappa$ we have
$R\Gamma(\mathbf{P},\mathcal{O})\cong\kappa$. We now use the general fact
[Hir64, §7, Corollary 2] that if $f\colon X^{\prime}\to X$ is a blow-up of a
nonsingular variety $X$ along a nonsingular subvariety, then
$Rf_{*}\mathcal{O}_{X^{\prime}}\cong\mathcal{O}_{X}$ as complexes, so
$R\Gamma(Z_{\sigma},\mathcal{O})\cong\kappa$ as well.

Therefore

$R\Gamma(\mathfrak{H}_{s},\mathcal{O}_{\operatorname{cond}})\cong\operatorname{lim}_{[r]\in\Delta}\prod_{\sigma\in\operatorname{BT}_{r}}\kappa$

is quasi-isomorphic to the (condensed!) $\kappa$-cohomology of
$\operatorname{BT}$ itself. Since $\operatorname{BT}$ is exhausted by finite
contractible subcomplexes $\operatorname{BT}^{i}\subset\operatorname{BT}$, its
cohomology is $\varprojlim_{i}\kappa\cong\kappa$. ∎

######  Theorem 6.3.3.

There is a $\operatorname{GL}_{n}(\mathbb{Z}_{p})$-equivariant morphism in
$D(\operatorname{Solid})$:

$\alpha_{\mathcal{H}_{K}}\colon\mathcal{O}_{K}[\varepsilon]\to
R\Gamma(\mathcal{H}_{K,\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}})$

such that the $H^{i}(\operatorname{cof}(\alpha))$ are killed by a uniform
power of $p$ for any $i$.

###### Proof.

Combine 5.6.9 with Theorem 6.3.1. ∎

######  Corollary 6.3.4.

We have an isomorphism of graded $K$-vector spaces in
$D(\operatorname{Solid})$:

$H^{*}(R\Gamma(\mathcal{H}_{K,\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+})^{h\operatorname{GL}_{n}(\mathbb{Z}_{p})})\otimes_{\mathcal{O}_{K}}K\cong\Lambda_{K}(y_{1},y_{3},\dots,y_{2n-1})[\varepsilon]$

###### Proof.

After taking homotopy fixed points under
$\operatorname{GL}_{n}(\mathbb{Z}_{p})$ in Theorem 6.3.3, we find a morphism

$\alpha_{\mathcal{H}}^{h\operatorname{GL}_{n}(\mathbb{Z}_{p})}\colon\mathcal{O}_{K}^{h\operatorname{GL}_{n}(\mathbb{Z}_{p})}[\varepsilon]\to
R\Gamma(\mathcal{H}_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}})^{h\operatorname{GL}_{n}(\mathbb{Z}_{p})}.$

By Lemma 6.1.2, the cohomology groups of
$\operatorname{cof}(\alpha_{\mathcal{H}}^{h\operatorname{GL}_{n}(\mathbb{Z}_{p})})\cong\operatorname{cof}(\alpha_{\mathcal{H}})^{h\operatorname{GL}_{n}(\mathbb{Z}_{p})}$
are all $p$-power torsion. After inverting $p$, we arrive at an isomorphism of
$K$-vector space objects in $D(\operatorname{Solid})$:

$K^{h\mathcal{O}_{D}^{\times}}[\varepsilon]\cong
R\Gamma(\mathcal{H}_{K,\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}})^{h\operatorname{GL}_{n}(\mathbb{Z}_{p})}\otimes_{\mathcal{O}_{K}}K$

By Lemma 3.8.1, the continuous cohomology of
$\operatorname{GL}_{n}(\mathbb{Z}_{p})$ is the stated exterior algebra. ∎

###  6.4. Conclusion of the proof

We now complete the proofs of B and of A.

######  Theorem 6.4.1.

For every $i$, the groups
$H^{i}_{\operatorname{cts}}(\mathcal{O}_{D}^{\times},A^{c})$ and
$H^{i}_{\operatorname{cts}}(\mathbb{G}_{n},A^{c})$ are $p$-power torsion.

###### Proof.

We leverage the isomorphism between the two towers, as in Theorem 3.9.1. We
have a diagram of adic spaces over $\operatorname{Spa}K$, where
$K=W(\overline{\mathbb{F}}_{p})[1/p]$:

$\textstyle{\mathcal{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{GL}_{n}(\mathbb{Z}_{p})}$$\scriptstyle{\mathcal{O}_{D}^{\times}}$$\textstyle{\operatorname{LT}_{K}}$$\textstyle{\mathcal{H}_{K}}$

Here $\mathcal{H}_{K}$ is the base change of $\mathcal{H}$ to $K$. The diagram
induces an isomorphism in $D(\operatorname{Solid})$:

$R\Gamma(\operatorname{LT}_{K,\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}})^{h\mathcal{O}_{D}^{\times}}\cong
R\Gamma(\mathcal{H}_{K,\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}})^{h\operatorname{GL}_{n}(\mathbb{Z}_{p})}\cong
R\Gamma(\mathcal{H}_{K,\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}})^{h\operatorname{GL}_{n}(\mathbb{Z}_{p})}.$

After inverting $p$, and combining 6.2.4 and 6.3.4 and ignoring the condensed
structure, we find an isomorphism of graded $K$-vector spaces:

$\Lambda_{K}(x_{1},x_{3},\dots,x_{2n-1})[\varepsilon]\oplus(H^{*}_{\operatorname{cts}}(\mathcal{O}_{D}^{\times},A^{c})\otimes_{W}K)\cong\Lambda_{K}(y_{1},y_{3},\dots,y_{2n-1})[\varepsilon]$

By comparing dimensions degree-wise on either side, we conclude that
$H^{*}_{\operatorname{cts}}(\mathcal{O}_{D}^{\times},A^{c})\otimes_{W}K=0$.

The claim extends from $\mathcal{O}_{D}^{\times}$ to $\mathbb{G}_{n}$, using
the Hochschild–Serre spectral sequence combined with the fact that the
cohomological dimension of
$\mathbb{G}_{n}/\mathcal{O}_{D}^{\times}\cong\hat{\mathbb{Z}}$ is $1$. ∎

###### Proof of A.

This is now immediate from B and Proposition 2.6.2. ∎

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# Shaping potential landscape for organic polariton condensates in double-dye
cavities

Anton D. Putintsev1, Kirsty E. McGhee2, Denis Sannikov1, Anton V.
Zasedatelev1, Julian D. Töpfer1, Till Jessewitsch3, Ullrich Scherf3, David G.
Lidzey2, and Pavlos G. Lagoudakis1 1Hybrid Photonics Laboratory, Skolkovo
Institute of Science and Technology, Territory of Innovation Center Skolkovo,
Bolshoy Boulevard 30, building 1, 121205 Moscow, Russia  2Department of
Physics and Astronomy, University of Sheffield, Hicks Building, Hounsfield
Road, Sheffield S3 7RH, UK  3Macromolecular Chemistry Group and Institute for
Polymer Technology, Bergische Universität Wuppertal, Wuppertal, Germany

###### Abstract

We investigate active spatial control of polariton condensates independently
of the polariton-, gain-inducing excitation profile. This is achieved by
introducing an extra intracavity semiconductor layer, non-resonant to the
cavity mode. Saturation of the optical absorption in the uncoupled layer
enables the ultra-fast modulation of the effective refractive index and,
through excited-state absorption, the polariton dissipation. Utilising these
mechanisms, we demonstrate control over the spatial profile and density of a
polariton condensate at room temperature.

††preprint: APS/123-QED

The ability to shape the potential energy landscape to confine polariton
condensates into geometrically arranged regular or arbitrary arrays underpins
various applications such as topological polaritonics [1, 2, 3, 4], lattice
simulators [5, 6, 7, 8], neural networks [9, 10], and allows for the
investigation of fundamental physics [11, 12]. An essential part in all these
studies is to have tools and techniques to control the polariton energy
landscape that is usually achieved by means of lithographic or optical
methods. The lithographic approach allows for the local control of the
photonic component of the system by a straightforward lithographic modulation
of the cavity length [13], deep etching of micropillars into the structure
[14, 15], metal films deposition on the surface of a grown microcavity
structure [16], or by exploiting natural photonic disorder potentials present
in microcavities due to growth related defects [17, 18]. All-optical methods
to control the energy landscape of polaritons offer higher flexibility in
customizing its shape whilst circumventing inherent disorder. Whether it is a
single gaussian pump or an array of pumps geometrically arranged in multiple-
spot configurations and lattices, forming either ballistically expanding [19]
or trapped polariton condensates [20], one exploits a localized blueshift
effect which at the same time induces additional gain in the system.

Manipulating polariton potentials through additional gain inducing optical
excitation adds a real-valued component to the interaction between polariton
condensates that strongly affects the energy of the condensate. An alternative
approach of utilising local control of polaritons through dissipation via ion
implantation was recently proposed [21] to control the coupling between
nearest neighbours in arrays of polaritons condensates. Furthermore, it would
be desirable to introduce local dissipation in a reversible manner that can
enable both ad-hoc landscape engineering of polariton condensates and real-
time control over the coupling between nearest neighbour condensates without
adding gain to the system. Despite the potential of such an approach, to date,
control of local polariton dissipation has remained illusive both in organic
and inorganic microcavities.

Figure 1:  Top: schematic of a double-dye organic microcavity. Bottom panel
shows the absorbance spectra of the uncoupled BN-PFO dye molecules and the
coupled BODIPY-Br dye molecules (left-axis) and the reflectivity spectrum of
the structure (right-axis). Black vertical dashed lines indicate the spectral
tuning of pump and control excitation beams.

In this Letter, we experimentally demonstrate ultra-fast and reversible
manipulation of the condensate energy and density by shaping both the local
potential and dissipation in the real space at room temperature. We utilize a
double-dye organic microcavity, depicted on the top panel of Fig.1, that
recently was proposed as a new platform for optically controlled polariton
condensate lattices operational at room temperature [22]. The microcavity
includes one layer of strongly coupled BODIPY-Br dye molecules and a blue-
detuned uncoupled layer of the copolymer BN-PFO. We demonstrate polariton
condensation under non-resonant optical excitation and, by simultaneous
excitation of the uncoupled layer, tuning of the condensate energy up to 8
meV. Excited-state absorption of the uncoupled molecular dyes results in
increasing losses for polaritons. In tandem with the energy blueshifts, due to
the decrease of the intracavity effective refractive index at the energy of
the lower polariton mode, excited-state absorption allow us to control the
spatial profile of a polariton condensate.

Polariton density dependent energy blueshifts in semiconductor microcavities
are omnipresent at high excitations and usually corroborate evidence of strong
coupling. However, the mechanism behind blueshifts in crystalline, usually
inorganic, and non-crystalline, usually organic, semiconductors is different.
In the first case, the blueshift is predominantly due to the Coulombic
exchange interaction between Wannier-Mott excitons [23, 24], whereas in the
latter case of Frenkel excitons, it is due to the saturation of molecular
optical transitions. It was recently shown that the blueshifts in molecular-
dye microcavities are not only due to the quenching of the Rabi splitting but
predominantly due to changes of the intracavity refractive index from the
saturation of both coupled and uncoupled molecules [25]. Here, we utilise a
structure design that allows us to decouple these mechanisms.

The structure consists of two layers of different organic materials embedded
within the microcavity. One 285 nm thick-layer of the molecular dye BODIPY-Br
and one 170 nm thick-layer of the conjugated polymer BN-PFO, separated by an
inert polyvinyl alcohol (PVA) spacer layer of 60 nm thickness. The organic
layers are sandwiched in between a bottom SiO2/Nb2O5 and a top SiO2/TiO2
distributed Bragg reflector (DBR), as shown schematically in the top panel of
Fig.1, resulting in quality factor of $\approx 920$; see K. McGhee _et al._
(2023) [22] for a complete description and characterization of the microcavity
structure. The absorption spectra of the BODIPY-Br and BN-PFO layers, and the
reflectivity spectrum of the full cavity are shown on the bottom panel of
Fig.1. Despite the presence of the weakly coupled, blue-detuned to the cavity
mode, BN-PFO absorber in the microcavity, the BODIPY-Br layer couples strongly
to the bare cavity mode exhibiting a Rabi-splitting ($\Omega$) of 103 meV
[22]. The cavity exhibits a small range of available detunings ($\delta$), and
throughout this study the detuning was kept constant at $\approx-$90 meV, with
exciton and photon fractions,
$|X_{e,p}|^{2}=(1\pm\delta/\sqrt{(\delta^{2}+\Omega^{2})})/2$, of $0.17$ and
$0.83$, respectively. These two layers are independently excited using a two
colour optical excitation beam configuration; see dashed black lines in the
bottom panel of Fig.1.

Figure 2:  Polariton photoluminescence (PL) (a) intensity, (b) blueshift of
the PL spectrum in blue squares (left-axis) and the corresponding linewidth in
red dots (right-axis) measured at full width at half maximum (FWHM) vs pump
absorbed fluence (bottom-axis) and pump threshold (top-axis). The vertical
gray shaded line indicates the condensation threshold. Normalized single-shot
dispersion images recorded at $\sim 2\times P_{th}$ in the (c) absence and (d)
presence of the control beam. The white dashed lines indicate the linear
dispersion. The right panel shows the corresponding spectra in black dotted
and red solid respectively. (e) Polariton blueshift vs absorbed fluence of the
control beam. The horizontal solid-gray line indicates the blueshift of the
polariton condensate at $\sim 2\times P_{th}$ in the absence of the control
beam. The dashed-red line is a linear fit to the blueshift dependence. The
black-solid line is the estimated blueshift due to effective refractive index
change at the energy of the polariton condensate due to saturation of the
molecular optical transitions. The difference in the slope between the black-
solid and dashed-red dependencies of the blueshift is attributed to the
excited state absorption.

We examine whether polariton condensation is achievable in the double-dye
microcavity using non-resonant pulsed optical excitation of 2 ps temporal
width tuned at the first Bragg minimum of the reflectivity stop-band; 485 nm.
Figure 2a shows a pump power dependence of polariton photoluminescence (PL) vs
the absorbed pump fluence on the bottom horizontal axis. The onset to non-
linear emission is defined as the crossing of the two black dashed lines in
Fig. 2a, $P_{th}\approx 400$ $\mu$J/cm2, see bottom horizontal axis. At
$P_{th}$, we observe a concomitant collapse of the photoluminescence
linewidth, from 8 meV to 1.8 meV, and a blueshift of the emission energy, see
Fig.2b. Each point on the plots of Figs.2a,b is derived from a dispersion
image obtained under a single excitation pulse, whose fluence is
simultaneously recorded. In Fig.2c we show the single-shot dispersion image of
the lower polariton branch (LPB) at $P_{pump}\sim 2\times P_{th}$. The
emission is blueshifted by $2.5$ meV from the linear dispersion that is
indicated with a white dotted line. Following the formalism of Ref.[25], we
can estimate the two contributions to the blueshift from the saturation of the
molecular optical transitions for this structure (see also Supplementary
Information (SI) Section I). We obtain that the dominant component of the
blueshift, 1.84 meV, is due to the effective refractive index renormalisation,
and 0.66 meV is due to the vacuum Rabi-splitting quenching. The observed
threshold for polariton condensation is similar to that observed previously
for BODIPY-family dye filled microcavities [26, 25, 27, 28], indicating that
the presence of the BN-PFO absorber in the cavity does not affect the dynamics
of polariton condensation under non-resonant excitation at 485 nm.

To explore the effect of saturation of the molecular transitions of the second
layer of molecular dyes on the polariton condensate, we introduce a second
optical excitation “control” beam of 500 fs temporal width, and resonant with
the absorption of BN-PFO at 400 nm, see the bottom panel of Fig.1. We keep the
pump fluence of the 485 nm beam constant at twice the condensation threshold,
$2\times P_{th}$, and perform a fluence dependence of the control beam at zero
time-delay between pump and control pulses. For a control beam fluence of
$\approx 250$ $\mu$J/cm2, BN-PFO absorbs $\approx 180$ $\mu$J/cm2, and BODIPY-
Br absorbs $\approx 2$ $\mu$J/cm2. The latter corresponds to $\approx 2\%$ of
the required absorbed fluence for polariton condensation of BODIPY-Br when
pumped at 400 nm [28], which brings a non-discernible effect to the dynamics
of the condensate. Under these excitation conditions, we perform single-pulse
dispersion imaging, shown in Fig.2d, and observe an additional 5 meV
blueshift, in the presence of the control pulse. The right panel of Fig.2d
shows the PL spectra of the condensate in the presence and absence of the
control beam. Evidently the linewidth of the condensate in the presence of the
control beam is broadened, indicating the onset of dissipation due to excited
state absorption from the uncoupled excitons in BN-PFO. Figure 2e shows the
dependence of the blueshift on the absorbed fluence of the control beam. The
energy shift dependence on the change of effective refractive index is given
by $\Delta E\cong-E_{c}\frac{\Delta n}{n_{eff}}$, where $\Delta E$ is the
energy shift, $E_{c}$ is the bare cavity optical resonance, and $\Delta n$ is
the change of the effective cavity refractive index $n_{eff}$ [25]. For the
density of BN-PFO molecules $n_{0}\approx 1.2\pm 0.2$ g/cm-3, we can estimate
the negative change of the intracavity effective refractive index at the
energy of the polariton condensate using the Kramers-Kronig relation and plot
the corresponding energy shift with a solid black line in Fig.2e, see SI
Section I. The difference between the estimated blueshift and the experimental
observation is attributed to excited state absorption from the uncoupled
molecules that was not previously considered as a contributor to the
mechanisms of blueshifts in organic microcavities [25].

Figure 3:  Pump-control time-delay scan: the pump beam fluence is fixed at
$\sim 2\times P_{th}$ and the control beam absorbed fluence at $\approx 170$
$\mu$J/cm2; positive time-delay corresponds to earlier arrival of the control
pulse. (a) Blueshift dependence on the time-delay, exhibiting a double
exponential decay with two components shown by gray dashed lines. (b)
Polariton PL intensity dependence on the time-delay, exhibiting quenching of
emission with the same characteristic times of the blueshift dependence. The
horizontal-dashed lines indicate the polariton blueshift, (a), and intensity,
(b), at $\sim 2\times P_{th}$ in the absence of the control beam; the vertical
lines indicate the zero-time delay.

We investigate the transient dynamics of the observed blueshifts by scanning
time-delay between the pump and control pulses and recording the dispersion
image of the PL for each excitation pulse. We keep the pump fluence constant
at $2\times P_{th}$, and the absorbed control beam fluence at $\approx 170$
$\mu$J/cm2, see vertical dashed line in Fig.2e. Figure 3a shows the temporal
dependence of the blueshift vs pump-control beam time-delay. The transient
decay of the blueshift can be fitted with a bi-exponential decay. The fast
component has the same characteristic time, $\approx 1.7$ ps, with the rise
dynamics and is dominated by exciton-exciton annihilation in the BN-PFO,
resolution limited here by the temporal width of the pump pulses [29]. The
slow component, $\approx 56$ ps, corresponds to the recombination dynamics of
the excitons in BN-PFO in the absence of non-linear recombination mechanisms.
Figure 3b shows the corresponding temporal dynamics of the condensate PL
intensity. Here, we observe strong quenching of the polariton density at zero
time-delay that follows approximately the same characteristic times of the
blueshift. The depletion of polariton density indicates the onset of strong
polariton dissipation in the presence of a far blue detuned and spatially
separated exciton reservoir. In the absence of intermolecular energy transfer
between the two molecular dyes, the dissipation of polaritons is attributed to
excited state absorption of the BN-PFO excitons, as it was also recently
revealed through transient absorption spectroscopy for BN-PFO [22].

Figure 4:  (a) Schematic of the Gaussian pump and ring-shaped control
excitation beams. The control beam saturates the uncoupled BN-PFO layer at
400nm, and the pump beam non-resonantly injects a polariton condensate in the
strongly coupled BODIPY-Br layer at 485nm. (b) Single-shot real-space emission
image of a polariton condensation at $2\times P_{th}$. (c) Averaged over 100
realizations real-space emission image of the PL from BODIPY-Br layer excited
by $\sim 500$ pJ absorbed energy of the ring-shaped control beam. (d) Single-
shot real-space emission image of the polariton condensate of (b) in the
presence of the ring-shaped control pulse (red-dashed circle) of (c) at zero
time-delay. (e) The dependence of the spatial width (red-markers, left-axis)
of the polariton condensate emission across the vertical blue dashed line of
(b) and the corresponding blueshift (blue-markers, right-axis) from the centre
of the ring of (d) vs the control pulse absorbed energy. (f) The polariton
emission intensity dependence from the center of the ring of (d) vs the
control pulse absorbed energy.

It is apparent that beyond the control over locally induced blueshifts,
excited state absorption induces strong local dissipation, that is both
tuneable and reversible. We utilise spatial modulation of the control beam to
demonstrate ad-hoc localisation of the polariton condensate. We expand the
pump to a $\sim 30$ $\mu$m Gaussian beam at $\approx 2\times P_{th}$ above the
condensation threshold, and use a spatial light modulator to shape the control
beam into a ring of $\sim 20$ $\mu$m in diameter, as shown schematically in
Fig.4a. Figure 4b shows the disorder-limited spatial profile of the polariton
condensate in the absence of the control beam; the top panel shows the PL
intensity cross-section obtained from the horizontal dotted line in Fig.4b.
Figure 4c shows the polariton PL in the presence only of the control beam with
an absorbed energy of $\sim 500$ pJ due to the residual absorption of BODIPY-
Br at 400 nm and re-absorption of BN-PFO emission by BODIPY-Br [22], and the
top panel shows the corresponding horizontal PL intensity cross-section.
Figure 4d shows the respective polariton condensate PL in the presence of both
pump and control beams. By tuning the absorbed fluence of the control beam we
can simultaneously control both the extent of the condensate and its
respective blueshift. Figures 4e,f show the spatial width, blueshift, and
condensate PL intensity vs the control absorbed energy, obtained from a
single-pulse spectrally resolved imaging of the condensate real space PL
across a vertical cross-section annotated with a blue dashed line in Fig.4b.
Across the full range of absorbed control fluences we observe that the locally
induced dissipation out-competes any pump or control beam induced gain.

In conclusion, we investigate active spatial control of polariton blueshifts
and dissipation by introducing an extra intracavity semiconductor layer, non-
resonant to the cavity mode and demonstrate control over the spatial profile
of a polariton condensate at room temperature. Our study realises previously
unexplored physical mechanisms for the controlled preparation and manipulation
of the spatial profile in organic microcavities.

###### Acknowledgements.

This work was supported by the Russian Science Foundation (RSF) grant no.
20-72-10145. K.E.M. and D.G.L. thank the U.K. EPSRC for support via the
Programme Grant ‘Hybrid Polaritonics’ (EP/M025330/1). K.E.M. thanks the EPSRC
for the award of a Doctoral Training Account PhD studentship (EP/R513313/1).

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Supplementary Information

##  I Section I

The total polariton blueshift in our double-layered microcavity can be
decoupled into for independent components (for details see Ref.[25]):

$\displaystyle\Delta E_{LPB}$ $\displaystyle=\Delta
E^{\Omega}_{LPB}+\sum_{i}\Delta E^{c_{i}}_{LPB}+\Delta E^{ESA}_{LPB}=$
$\displaystyle=\frac{\xi_{1}}{2}\frac{s\cdot\hbar\Omega}{\sqrt{1+s^{2}}}+\sum_{i=1}^{2}\frac{\xi_{i}}{2}\frac{E_{X}-|\delta|}{5n_{eff}}F[d_{i}]\cdot\alpha_{i}\cdot\biggl{(}1+\frac{1}{\sqrt{1+s^{2}}}\biggr{)}+\Delta
E^{ESA}_{LPB}$ (S1)

where $\Delta E_{LPB}$ is the total energy shift of LPB that is the sum of
energy shifts due to vacuum Rabi-splitting quenching in strongly coupled
layer, $\Delta E^{\Omega}_{LPB}$, cavity mode renormalization with
contributions to the refractive index modulation from both organic layers,
with $i=1,2$ relating to BODIPY-Br and BN-PFO, respectively, and $\Delta
E^{ESA}_{LPB}$ is the energy shift caused by refractive index modulation due
to the excited state absorption effect in BN-PFO , $\xi$ is the saturation
parameter, $s=\hbar\Omega/|\delta|$ is a dimensionless parameter of strong
coupling, $\alpha$ corresponds to the oscillator strength of the optical
transition, $E_{X}$ is the BODIPY-Br exciton resonance, $\delta=E_{c}-E_{x}$
is the detuning, and $F[d]$ is the Dawson function (integral),
$F[d]=e^{-d^{2}}\int_{0}^{d}e^{x^{2}}dx$.

Lets consider two organic layers separately. In strongly coupled BODIPY-Br
layer in the absence of the control beam the emission is blueshifted by $2.5$
meV from the linear dispersion. Following the formalism of Ref.[25], we can
estimate the two contributions to the blueshift from the saturation of the
molecular optical transitions for this structure. Taking the ratio of two
first terms in the Eq.I we obtain independent of $\xi$ expression:

$\rho=\frac{\Delta E^{c_{1}}_{LPB}}{\Delta
E^{\Omega}_{LPB}}=\frac{(E_{X}-|\delta|)\cdot
F[d_{1}]\cdot\alpha_{1}\cdot(\sqrt{s^{2}+1}+1)}{5n_{ff}\cdot
s\cdot\hbar\Omega_{0}}$ (S2)

Here $E_{X}=2.34672$ eV $(528.33$ nm) is the BODIPY-Br main exciton line,
$\hbar\Omega_{0}=103$ meV is the Rabi-splitting,
$|\delta|=|549.18-528.33|=89.1$ meV is the detuning,
$\alpha_{1}=Abs_{max}\cdot\lambda_{max}/L_{layer}=0.55182\cdot 528.33/285$
[$1\cdot$nm/nm] is the oscillator strength of the optical transition,
$s=\hbar\Omega_{0}/|\delta|=103/89.1$ [meV/meV] is a dimensionless parameter
of strong coupling, $d_{1}=2\sqrt{ln(2)}\cdot\delta/FWHM=2\sqrt{ln(2)}\cdot
89.1/127.181=1.16654$, and $F[1.16654]=0.514261$, $n_{eff}=1.81$. Substituting
these values into S2, we get:

$\rho=2.78684$ (S3)

It allows us to obtain components of the blueshift in BODIPY-Br layer, in the
absence of the control beam, due to the effective refractive index
renormalisation, which is 1.84 meV, and due to the vacuum Rabi-splitting
quenching, which is 0.66 meV.

The uncoupled BN-PFO layer contributes to the blueshift through effective
refractive index change due to two effects: the saturation of the molecular
optical transitions and excited state absorption on top of those saturated
transitions. The first effect is again described by the second term of Eq.I
and scales linearly with the saturation parameter $\xi_{2}$. To estimate its
absolute contribution to the additional blueshift we have to use parameters
relating to BN-PFO layer:
$\alpha_{2}=Abs_{max}\cdot\lambda_{max}/L_{layer}=0.94\cdot 371/170$
[$1\cdot$nm/nm] is the oscillator strength of the optical transition,
$d_{2}=2\sqrt{ln(2)}\cdot\delta/FWHM=2\sqrt{ln(2)}\cdot 1084.3/858=2.10429$
and $F[2.10429]=0.281093$, $n_{eff}=1.81$. Substituting these values into the
following expression:

$\Delta
E^{c_{2}}_{LPB}=\frac{\xi_{2}}{2}\frac{E_{X}-|\delta|}{5n_{eff}}F[d_{2}]\cdot\alpha_{2}\cdot\biggl{(}1+\frac{1}{\sqrt{1+s^{2}}}\biggr{)}$
(S4)

We obtain:

$\Delta E^{c_{2}}_{LPB}\cong 112[meV]\cdot\xi_{2}$ (S5)

Figure S1:  Polariton blueshift vs absorbed fluence of the control beam. The
horizontal solid-gray line indicates the blueshift of the polariton condensate
at $\sim 2\times P_{th}$ in the absence of the control beam. The dashed-red
line is a linear fit to the blueshift dependence. The black-solid line is the
estimated blueshift due to effective refractive index change at the energy of
the polariton condensate due to saturation of the molecular optical
transitions with the error gray-shaded error bar. The difference in the slope
between the black-solid and dashed-red dependencies of the blueshift is
attributed to the excited state absorption.

This expression defines the dependence of the estimated blueshift due to
effective refractive index change at the energy of the polariton condensate on
the saturation of the molecular optical transitions $\xi_{2}$. With the
control pulse absorbed fluence in the BN-PFO layer at 400 nm of $\approx 200$
$\mu$J/cm2, the number of excited molecules is $N\sim 10^{9}$ within a volume
of 53 $\mu$m3 (20 $\mu m$ spot-size and 170nm-thick layer). The estimated
molecular density in BN-PFO layer is $n_{0}\approx 1.2\pm 0.2$ g/cm-3, which,
given the weight of a polymer unit of M${}_{n}=$68300 [22], results in
$N_{0}\approx 5.6\cdot 10^{8}$ available polymer chains within the same 53
$\mu$m3 volume-size. However, the number of chromophores is assumed to be
higher [30], since the chromophores are localized to shorter conjugated
segments (the so-called “effective conjugation length”), and this length is
different for individual polymers. One chain carry multiple chromophores. For
BN-PFO, the binaphthyl units that are randomly incorporated into the chains
are expected to separate oligo/polyfluorene segments (of different length),
and should act as conjugation barriers thus separating individual PFO
chromophores. Thus the number of chromophores can be higher by a factor of 10
(or more). With this in mind, we estimate the number of available absorbing
units $N_{0}\approx 5.6\cdot 10^{10}$ resulting in the saturation parameter
$\xi_{2}\approx N/N_{0}=10^{9}/5.6\cdot 10^{10}\approx 0.018$ at $200$
$\mu$J/cm2 control absorbed fluence. Fig.S1 shows the resulting dependence of
the blueshift due to the saturation of the molecular optical transitions on
the control absorbed fluence, black-solid line. The difference in the slope
between the black-solid and dashed-red dependencies of the blueshift is
$\Delta E^{ESA}_{LPB}$ which is attributed to excited state absorption.


# Theoretical aspects of microscale acoustofluidics

Henrik Bruus  Department of Physics, Technical University of Denmark, DTU
Physics Building 309, DK-2800 Kongens Lyngby, Denmark

(27 February 2017. E-mail<EMAIL_ADDRESS>
###### Abstract

|  Short biography: Henrik Bruus received his PhD degree in physics from the
Niels Bohr Institute, University of Copenhagen in 1990, and then worked as
postdoc at Nordic Institute of Theoretical Physics 1990-92, Yale University
1992-94 and CNRS Grenoble 1994-96. He returned to the Niels Bohr Institute as
associate professor 1997-2001, before moving to the Technical University of
Denmark in 2001. There, he became full professor of lab-chip systems in 2005
and of theoretical physics in 2012. His current research interests comprise
micro/nanofluidics, acoustofluidics, electrokinetics, the physics of on-chip
cell manipulation, the motion of sugar in living plants, and topology-
optimized microflows. He has (co)au-
---|---

thored 134 journal papers on condensed matter physics and microfluidics, 180
conference papers, and 2 monographs, the latest being ”Theoretical
Microfluidics”, Oxford University Press (2008).
Abstract. In this contribution, I summarize some of the recent results within
theory and simulation of microscale acoustofluidic systems that I have
obtained in collaboration with my students and international colleagues. The
main emphasis is on three dynamical effects induced by external ultrasound
fields acting on aqueous solutions and particle suspensions: The acoustic
radiation force acting on suspended micro- and nanoparticles, the acoustic
streaming appearing in the fluid, and the newly discovered acoustic body force
acting on inhomogeneous solutions.

##  I Introduction

Microscale acoustofluidic devices are used increasingly in biology,
environmental and forensic sciences, and clinical diagnostics as reviewed in
the Lab Chip Tutorial Series Bruus  _et al._ (2011), reprinted by the Royal
Society of Chemistry in the book Microscale Acoustofluidics Laurell and
Lenshof (2014).

Following the standard method described in textbooks such as Lighthill
Lighthill (2002), Pierce Pierce (1991), and Landau & Lifshitz Landau and
Lifshitz (1993), we formulate the linear acoustics in fluids and elastic
solids, as well as the non-linear acoustic dynamics in fluids using first-and
second-order perturbation theory in the notation established in the textbook
by Bruus Bruus (2008). The basic fields in both fluids and solids are the
density $\rho$, the stress tensor $\bm{\sigma}$, the internal energy density
$\varepsilon$, and the temperature $T$. These fields are supplemented in the
fluids by the pressure $p$ and the velocity $\bm{v}$, and in the solids by the
displacement $\bm{u}$.

##  II Governing equations

For fluids, advection causes non-linearities, and the governing equations are
formulated in terms of the fluxes of the mass density $\rho$, the momentum
density $\rho\bm{v}$, and the energy density
$\rho\big{(}\varepsilon+\mbox{$\frac{1}{2}$}v^{2}\big{)}$ under the
simplifying assumption of no net body forces and no heat sources,

$\displaystyle\partial_{t}\rho$
$\displaystyle=\bm{\nabla}\\!\cdot\\!\big{[}-\\!\rho\bm{v}\big{]},$ (1a)
$\displaystyle\partial_{t}(\rho\bm{v})$
$\displaystyle=\bm{\nabla}\\!\cdot\\!\big{[}\bm{\sigma}_{\mathrm{f}}-\rho\bm{v}\bm{v}\big{]},$
(1b)
$\displaystyle\partial_{t}\big{(}\rho\varepsilon\\!+\\!\mbox{$\frac{1}{2}$}\rho
v^{2}\big{)}$
$\displaystyle=\bm{\nabla}\\!\cdot\\!\big{[}k_{\mathrm{th}}\bm{\nabla}T+\\!\bm{v}\\!\cdot\\!\bm{\sigma}_{\mathrm{f}}-\\!\rho(\varepsilon+\mbox{$\frac{1}{2}$}v^{2})\bm{v}\big{]},$
(1c)

where the stress tensor $\bm{\sigma}_{\mathrm{f}}$ for fluids is

$\bm{\sigma}_{\mathrm{f}}=-\Big{[}p\\!+\\!\big{(}\mbox{$\frac{2}{3}$}\eta_{0}-\eta_{\mathrm{B}}\big{)}\bm{\nabla}\\!\cdot\\!\bm{v}\Big{]}\bm{1}+\eta_{0}\\!\Big{[}\bm{\nabla}\bm{v}+(\bm{\nabla}\bm{v})^{\mathrm{T}}\>\Big{]}.$
(2)

Here, the material parameters are the shear viscosity $\eta_{0}$, the bulk
viscosity $\eta_{\mathrm{B}}$, and the thermal conductivity $k_{\mathrm{th}}$,
while $\bm{1}$ is the unit tensor and ’T’ means tensor transpose.

For elastic solids with an isobaric thermal expansion coefficient
${\alpha_{p}}$, an isothermal compressibility $\kappa_{T}$, and a specific
heat ratio $\gamma$, the governing equations are linear due to the lack of
advection,

$\displaystyle\rho\partial^{\,2}_{t}\bm{u}$
$\displaystyle=\bm{\nabla}\\!\cdot\\!\bm{\sigma}_{\mathrm{s}}-\frac{{\alpha_{p}}}{\kappa_{T}}\>\bm{\nabla}T,$
(3a) $\displaystyle\partial_{t}T$ $\displaystyle=\gamma
D_{\mathrm{th}}\nabla^{2}T-\frac{\gamma-1}{{\alpha_{p}}}\partial_{t}\bm{\nabla}\\!\cdot\\!\bm{u},$
(3b)

where the stress tensor $\bm{\sigma}_{\mathrm{s}}$ for solids is given in
terms of the longitudinal and transverse sound speeds $c_{\mathrm{L}}$ and
$c_{\mathrm{T}}$, respectively,

$\bm{\sigma}_{\mathrm{s}}=\rho\Big{[}c^{2}_{\mathrm{L}}-2c^{2}_{\mathrm{T}}\Big{]}(\bm{\nabla}\\!\cdot\\!\bm{u})\bm{1}+\rho
c^{2}_{\mathrm{T}}\\!\Big{[}\bm{\nabla}\bm{u}+(\bm{\nabla}\bm{u})^{\mathrm{T}}\>\Big{]}.$
(4)

##  III Acoustic perturbation theory

We consider a system initially at rest, and which has the density $\rho_{0}$
and velocity $\bm{v}=\bm{0}$. We assume that the system is perturbed by
imposing an harmonically oscillating displacement with a small position-
dependent amplitude $\bm{d}$ on parts of the boundary. As a result, harmonic
variations $\rho_{1}$ in the density and $\bm{v}_{1}$ in the velocity are
induced. For a medium with the speed of sound $c_{0}$, the magnitude of the
perturbation can then be characterized by the dimensionless acoustic Mach
number Ma,

$\text{Ma}=\frac{v_{1}}{c_{0}}=\frac{\rho_{1}}{\rho_{0}}\ll 1.$ (5)

Typically in microscale acoustofluidics $\text{Ma}\approx 10^{-4}$, so a
perturbation expansion of any field $f$ in order of Ma makes sense,
$f=f_{0}+f_{1}+f_{2}+\ldots$, where $f_{n}\propto\text{Ma}^{n}$. We expand the
non-linear fluid equations to second order, while the linear solid equations
terminate at first order,

$\begin{array}[]{rll}T&=T_{0}+T_{1}+T_{2},&\quad\bm{u}=\bm{0}+\bm{u}_{1},\\\\[2.84526pt]
\rho&=\rho_{0}+\rho_{1}+\rho_{2},&\quad\bm{v}=\bm{0}+\bm{v}_{1}+\bm{v}_{2},\\\\[4.2679pt]
p&=p_{0}+p_{1}+p_{2}.&\end{array}$ (6)

###  III.1 The first-order equations

Using the complex-valued representation $\mathrm{e}^{-{\mathrm{i}\omega t}}$
of harmonic time dependence with angular frequency $\omega$, the actuation of
the system is modeled by prescribing the motion of its exterior boundaries at
positions $\bm{r}_{0}$,

$\displaystyle\bm{u}_{1}$ $\displaystyle=$
$\displaystyle\bm{d}(\bm{r}_{0})\>\mathrm{e}^{-{\mathrm{i}\omega t}},$
$\displaystyle\;\text{ on solid boundaries},$ (7a) $\displaystyle\bm{v}_{1}$
$\displaystyle=\;$
$\displaystyle-\mathrm{i}\omega\bm{d}(\bm{r}_{0})\>\mathrm{e}^{-{\mathrm{i}\omega
t}},$ $\displaystyle\;\text{ on fluid boundaries}.$ (7b)

All first-order fields thus have the form
$g_{1}=g_{1}(\bm{r})\>\mathrm{e}^{-{\mathrm{i}\omega t}}$, which changes the
time derivative into a simple complex-valued factor,
$\partial_{t}\rightarrow-\mathrm{i}\omega$, and consequently, all first-order
terms contain the harmonic phase-factor $\mathrm{e}^{-{\mathrm{i}\omega t}}$.
This factor is divided out leaving the time-independent, complex-valued
amplitude fields $g_{1}(\bm{r})$.

For fluids in the isentropic case, we have the simplifying relations
$p_{1}=c^{\,2}_{0}\rho_{1}$ and
$T_{1}=\frac{{\alpha_{p}}T_{0}}{c_{\mathrm{p}}\rho_{0}}\>p_{1}$, and the
governing equations for the first-order fields become

$\displaystyle-\mathrm{i}\omega p_{1}$
$\displaystyle=-\rho_{0}c^{\,2}_{0}\>\bm{\nabla}\cdot\bm{v}_{1},$ (8a)
$\displaystyle-\mathrm{i}\omega\rho_{0}\bm{v}_{1}$
$\displaystyle=\bm{\nabla}\cdot\bm{\sigma}_{\mathrm{1f}},$ (8b)

where the first-order stress tensor $\bm{\sigma}_{\mathrm{1f}}$ is Eq. (2)
with all fields substituted by first-order fields. Similarly for the solids,
we obtain the simplified first-order equation of motion,

$-\omega^{2}\rho_{0}\bm{u}_{1}=\bm{\nabla}\cdot\bm{\sigma}_{\mathrm{1s}}.$ (9)

At internal fluid-solid interfaces with the surface normal vector $\bm{n}$, we
demand continuity of the velocity and stress fields,

$\left.\begin{array}[]{rl}\bm{v}_{1}=&-\mathrm{i}\omega\bm{u}_{1}\\\\[2.84526pt]
\bm{\sigma}_{\mathrm{1f}}\\!\cdot\\!\bm{n}=&\bm{\sigma}_{\mathrm{1s}}\\!\cdot\\!\bm{n}\end{array}\\!\\!\right\\}\\!,\text{internal
fluid-solid interfaces}.$ (10)

###  III.2 The time-averaged second-order equations

The basic acoustofluidic dynamics takes place in fluids on a ms-time scale,
much slower than the ultrasonic µs-time scale. It is therefore described by
the time-averaged response over one oscillation period, denoted by angled
brackets $\langle\ldots\rangle$ in the following. The terms of lowest order
contributing to this time average are second-order terms. Moreover, since the
time-average of the time derivative $\big{\langle}\partial_{t}g\big{\rangle}$
of any field $g$ in time-periodic systems vanish,
$\langle\partial_{t}g\rangle=0$, the second-order governing equations of
fluids become

$\displaystyle\bm{\nabla}\cdot\Big{[}\rho_{0}\big{\langle}\bm{v}_{2}\big{\rangle}+\big{\langle}\rho_{1}\bm{v}_{1}\big{\rangle}\Big{]}$
$\displaystyle=0,$ (11a)
$\displaystyle\bm{\nabla}\cdot\Big{[}\big{\langle}\bm{\sigma}_{2}\big{\rangle}-\rho_{0}\big{\langle}\bm{v}_{1}\bm{v}_{1}\big{\rangle}\Big{]}$
$\displaystyle=\bm{0}.$ (11b)

In terms of the momentum flux density tensor

$\bm{\Pi}=\rho\bm{v}\bm{v}-\bm{\sigma},$ (12)

the second-order equation (11b) becomes

$\bm{\nabla}\cdot\big{\langle}\bm{\Pi}_{2}\big{\rangle}=\bm{0}.$ (13)

On the boundary, the velocity is oscillating harmonically, so there
$\big{\langle}\bm{v}\big{\rangle}=\bm{0}$. On the other hand, a boundary point
with the time-averaged position $\bm{r}_{0}$ has the time-dependent position
$\bm{r}=\bm{r}_{0}+\bm{u}_{1}$, so by a second-order Taylor expansion
$\big{\langle}\bm{v}(\bm{r})\big{\rangle}=\big{\langle}\bm{v}(\bm{r}_{0})+\bm{u}_{1}(\bm{r}_{0})\\!\cdot\\!\bm{\nabla}\bm{v}(\bm{r}_{0})\big{\rangle}$.
Thus on the boundary
$\bm{0}=\big{\langle}\bm{v}_{2}\big{\rangle}+\big{\langle}\bm{u}_{1}\cdot\bm{v}_{1}\big{\rangle}$,
and using $\bm{u}_{1}=\frac{\mathrm{i}}{\omega}\bm{v}_{1}$ from Eq. (10), the
boundary condition for $\big{\langle}\bm{v}_{2}\big{\rangle}$ becomes

$\big{\langle}\bm{v}_{2}\big{\rangle}=-\frac{1}{\omega}\big{\langle}(\mathrm{i}\bm{v}_{1})\cdot\bm{\nabla}\bm{v}_{1}\big{\rangle},\text{
on fluid boundaries}.$ (14)

##  IV The radiation force

In a series of papers, we have studied the acoustic radiation force
$\bm{F}^{\mathrm{rad}}$ on a spherical particle of radius $a$ suspended in an
inviscid Bruus (2012), viscous Settnes and Bruus (2012), and thermoviscous
fluid Karlsen and Bruus (2015) with kinematic viscosity $\nu_{0}$ and thermal
diffusivity $D_{\mathrm{th}}$. When exposed to an incident acoustic wave of
angular frequency $\omega$ and wavenumber $k$, a viscous and thermal boundary
layer of width $\delta_{\mathrm{s}}=\sqrt{2\nu_{0}/\omega}$ and
$\delta_{\mathrm{t}}=\sqrt{2D_{\mathrm{th}}/\omega}$, respectively, develop at
the particle surface. In the long-wave limit
$a,\delta_{\mathrm{s}},\delta_{\mathrm{t}}\ll 1/k$, we can express
$\bm{F}^{\mathrm{rad}}$ as the time-averaged momentum flux through any static
surface $\partial\Omega_{0}$ encompassing the particle,

$\bm{F}^{\mathrm{rad}}=-\left\langle\oint_{\partial\Omega_{0}}\\!\\!\bm{\Pi}_{2}\cdot\bm{n}\>\mathrm{d}a\right\rangle=-\int_{\Omega_{0}}\\!\\!\bm{\nabla}\cdot\big{\langle}\bm{\Pi}_{2}\big{\rangle}\>\mathrm{d}V.$
(15)

In spite of Eq. (13), the expression for $\bm{F}^{\mathrm{rad}}$ is not zero
because of delta-function singularities which appear in
$\bm{\nabla}\cdot\big{\langle}\bm{\Pi}_{2}\big{\rangle}$ due to the scattering
of the incident acoustic wave on the particle. When properly evaluating these
singularities we find Settnes and Bruus (2012),

$\displaystyle\bm{F}^{\mathrm{rad}}\\!=-\frac{4}{3}\pi
a^{3}\\!\Big{[}\kappa_{0}\big{\langle}(f_{0}p_{1}^{\mathrm{in}})\\!\bm{\nabla}p_{1}^{\mathrm{in}}\big{\rangle}\big{]}-\frac{3}{2}\rho_{0}\big{\langle}(f_{1}\bm{v}_{1}^{\mathrm{in}})\\!\cdot\\!\bm{\nabla}\bm{v}_{1}^{\mathrm{in}}\big{\rangle}\Big{]},$
(16)

where $p_{1}^{\mathrm{in}}$ and $\bm{v}_{1}^{\mathrm{in}}$ are the first-order
pressure and velocity fields of the incident acoustic wave evaluated at the
particle position, $\kappa_{0}$ is the isentropic compressibility of the
fluid, while $f_{0}$ and $f_{1}$ are the so-called mono- and dipole scattering
coefficients, respectively. The latter are involved functions of the material
parameters as shown in Ref. Karlsen and Bruus (2015). For a fluid particle (a
droplet), $f_{0}$ becomes

$\displaystyle
f_{0}^{\mathrm{fl}}=1-\tilde{\kappa}_{s}+3(\gamma-1)\left(1-\dfrac{\tilde{\alpha}_{p}}{\tilde{\rho}_{0}\>\tilde{c}_{p}}\right)^{2}H(x_{\mathrm{t}},x_{\mathrm{t}}^{\prime}),$
(17a) $\displaystyle
H(x_{\mathrm{t}},x_{\mathrm{t}}^{\prime})=\dfrac{1}{x_{\mathrm{t}}^{2}}\left[\dfrac{1}{1-\mathrm{i}x_{\mathrm{t}}}-\dfrac{1}{\tilde{k}_{\mathrm{th}}}\dfrac{\tan
x_{\mathrm{t}}^{\prime}}{\tan
x_{\mathrm{t}}^{\prime}-x_{\mathrm{t}}^{\prime}}\right]^{-1},$ (17b)

where the tilde represent the droplet-to-medium ratio of the various material
parameters, and where $H(x_{\mathrm{t}},x_{\mathrm{t}}^{\prime})$ is a
function of the particle radius $a$ through the non-dimensionalized thermal
wavenumber $x_{\mathrm{t}}\approx(1+\mathrm{i})a/\delta_{\mathrm{t}}$ for the
fluid medium, and likewise for $x_{\mathrm{t}}^{\prime}$ of the droplet.

Similarly, $f_{1}$ becomes

$\displaystyle f_{1}^{\mathrm{fl}}$
$\displaystyle=\dfrac{2\left(\tilde{\rho_{0}}-1\right)\left(1+F(x_{\mathrm{s}},x_{\mathrm{s}}^{\prime})-G(x_{\mathrm{s}})\right)}{\left(2\tilde{\rho_{0}}+1\right)\big{[}1+F(x_{\mathrm{s}},x_{\mathrm{s}}^{\prime})\big{]}-3G(x_{\mathrm{s}})},$
(18a) $\displaystyle G(x_{\mathrm{s}})$
$\displaystyle=\dfrac{3}{x_{\mathrm{s}}}\left(\dfrac{1}{x_{\mathrm{s}}}-\mathrm{i}\right),$
(18b) $\displaystyle F(x_{\mathrm{s}},x_{\mathrm{s}}^{\prime})$
$\displaystyle=\dfrac{1-\mathrm{i}x_{\mathrm{s}}}{2(1-\tilde{\eta}_{0})+\dfrac{\tilde{\eta}_{0}x_{\mathrm{s}}^{\prime
2}(\tan
x_{\mathrm{s}}^{\prime}-x_{\mathrm{s}}^{\prime})}{(3-x_{\mathrm{s}}^{\prime
2})\tan x_{\mathrm{s}}^{\prime}-3x_{\mathrm{s}}^{\prime}}},$ (18c)

where $x_{\mathrm{s}}=(1+\mathrm{i})a/\delta_{\mathrm{s}}$ is the non-
dimensionalized shear-viscous wavenumber for the fluid medium, and likewise
for $x_{\mathrm{s}}^{\prime}$ of the droplet.

A full experimental verification of our theoretical expression for the
acoustic radiation force has not yet been carried out. However, we have
successfully obtained partial experimental verifications Barnkob  _et al._
(2010); Augustsson  _et al._ (2011); Barnkob  _et al._ (2012a).

##  V Acoustic streaming in homogeneous fluids

In a series papers, we have studied theoretically and numerically the acoustic
streaming in hard-walled acoustofluidic devices Muller  _et al._ (2012, 2013);
Muller and Bruus (2014); Antfolk  _et al._ (2014); Muller and Bruus (2015).
All types of time-averaged acoustic flows are driven by an acoustic body force
$\bm{f}_{\mathrm{ac}}$, which is given in terms of the non-zero divergence of
the acoustic part $\big{\langle}\bm{\Pi}_{\mathrm{ac}}\big{\rangle}$ of the
momentum-flux-density tensor $\big{\langle}\bm{\Pi}_{2}\big{\rangle}$. We
define $\bm{f}_{\mathrm{ac}}$ as Karlsen  _et al._ (2016)

$\bm{f}_{\mathrm{ac}}=-\bm{\nabla}\cdot\big{\langle}\bm{\Pi}_{\mathrm{ac}}\big{\rangle},$
(19a) where $\big{\langle}\bm{\Pi}_{\mathrm{ac}}\big{\rangle}$ is the part of
$\big{\langle}\bm{\Pi}_{2}\big{\rangle}$ that contains all the time-averaged
products of first-order acoustic fields,
$\displaystyle\big{\langle}\bm{\Pi}_{\mathrm{ac}}\big{\rangle}$
$\displaystyle=\big{\langle}p_{11}\big{\rangle}\mathbf{I}+\big{\langle}\rho_{0}\bm{v}_{1}\bm{v}_{1}\big{\rangle},\quad\quad\mathrm{with}$
(19b) $\displaystyle\big{\langle}p_{11}\big{\rangle}$
$\displaystyle=\big{\langle}\frac{1}{2}\kappa_{0}|p_{1}|^{2}-\frac{1}{2}\rho_{0}|\bm{v}_{1}|^{2}\big{\rangle}.$
(19c)

In this expression, $\big{\langle}p_{11}\big{\rangle}$ is a local time-
averaged acoustic pressure. Using the body force $\bm{f}_{\mathrm{ac}}$, we
rewrite Eq. (11b) as an equation for a body-force-driven Stokes flow,

$\bm{\nabla}\cdot\big{\langle}\bm{\sigma}_{2}\big{\rangle}=\bm{f}_{\mathrm{ac}}.\\\
$ (20)

For a planar rigid wall and a homogeneous fluid, we recover the well-known
Rayleigh streaming, with pairs of vortices and anti-vortices each with a
spatial extension of $\frac{1}{4}\lambda\times\frac{1}{4}\lambda$ Muller  _et
al._ (2012). In the same work we added tracer particles and studied the
transition in particle dynamics from being dominated by the radiation force to
be dominated by the Stokes drag force from the acoustic streaming. We
determined the critical radius $a_{0}$ for the cross-over between these two
regimes to be around $1~{}\textrm{\textmu{}m}$ for polystyrene particles in
water at 2 MHz Muller  _et al._ (2012); Barnkob  _et al._ (2012b).

In subsequent studies of rectangular microchannels, we compared the
theoretical predictions of our streaming model with experimental observations
for the standard case of a horizontal standing half-wave acoustic resonance.
In Ref. Barnkob  _et al._ (2012b) we verified experimentally the model for the
cross-over from radiation-force to streaming-drag dominated particle dynamics,
and in Ref. Muller  _et al._ (2013) we found good agreement between
experimental, analytical, and numerical determinations of streaming-dominated
particle paths in all three spatial dimensions. Finally in Ref. Antfolk  _et
al._ (2014), we found good agreement between theory and experiments for the
handling of sub-µm particles in the special case of single-vortex streaming
obtained by phase-specific oscillation of both pairs of opposite walls in a
rectangular channel.

##  VI Acoustic body-force handling of inhomogeneous solutions

Recently, it was discovered experimentally that inhomogeneities in density
$\rho_{0}$, compressibility $\kappa_{0}$, and viscosity $\eta_{0}$ of a fluid,
introduced by a solute concentration $s$, can be relocated and stabilized into
field-dependent configurations by use of acoustic fields Deshmukh  _et al._
(2014); Augustsson  _et al._ (2016). Based on the separation of the fast µs-
time-scale $t$ of acoustics and the slow ms-time-scale $\tau$ of
hydrodynamics, we subsequently developed the following theory Karlsen  _et
al._ (2016) for the acoustic body force responsible for the observed
phenomena.

The density $\rho$ is written as the sum of the slowly evolving hydrodynamic
density $\rho_{0}$ and the quickly oscillating density perturbation
$\rho_{1}$,

$\rho=\rho_{0}(\bm{r},\tau)+\rho_{1}(\bm{r},\tau)\>\mathrm{e}^{-{\mathrm{i}\omega
t}},$ (21)

where $\rho_{1}$ is associated with the acoustic pressure and velocity fields
$p_{1}$ and $\bm{v}_{1}$. Now, the two equations for $p_{1}$ and $\rho_{1}$
cannot be combined into one as in Eq. (8), so inhomogeneous acoustics are
given by three equations,

$\displaystyle-\mathrm{i}\omega\kappa_{0}p_{1}$
$\displaystyle=-\bm{\nabla}\cdot\bm{v}_{1},$ (22a)
$\displaystyle-\mathrm{i}\omega\rho_{0}\kappa_{0}p_{1}$
$\displaystyle=-\mathrm{i}\omega\rho_{1}+\bm{v}_{1}\cdot\bm{\nabla}\rho_{0},$
(22b) $\displaystyle-\mathrm{i}\omega\rho_{0}\bm{v}_{1}$
$\displaystyle=\bm{\nabla}\cdot\bm{\sigma}_{\mathrm{1f}}.$ (22c)

The fluid inhomogeneity is caused by a solute concentration field
$s=s(\bm{r},\tau)$, which is being transported on the slow timescale $\tau$.
The density $\rho_{0}\big{(}s(\bm{r},\tau)\big{)}$, compressibility
$\kappa_{0}\big{(}s(\bm{r},\tau)\big{)}$, and viscosity
$\eta_{0}\big{(}s(\bm{r},\tau)\big{)}$ are all functions of the solute
concentration, and consequently the hydrodynamics is governed by the Stokes
equation with body forces (acoustics and gravity) for the velocity field
$\bm{v}$, the continuity equation for $\rho_{0}$, and the advection-diffusion
equation of the solute $s$,

$\displaystyle\partial_{\tau}(\rho_{0}\bm{v})$
$\displaystyle=\bm{\nabla}\cdot\bm{\sigma}+\bm{f}_{\mathrm{ac}}+\rho_{0}\bm{g},$
(23a) $\displaystyle\partial_{\tau}\rho_{0}$
$\displaystyle=-\bm{\nabla}\cdot\big{(}\rho_{0}\bm{v}\big{)},$ (23b)
$\displaystyle\partial_{\tau}s$
$\displaystyle=-\bm{\nabla}\cdot\big{[}-D\bm{\nabla}s+\bm{v}s\big{]}.$ (23c)

Usual streaming in the homogeneous viscous case Muller  _et al._ (2012, 2013)
is obtained using $s=0$ and $\bm{f}_{\mathrm{ac}}$ of the form

$\bm{f}_{\mathrm{ac}}^{\mathrm{hom}}=-\rho_{0}\bm{\nabla}\cdot\big{\langle}\bm{v}_{1}\bm{v}_{1}\big{\rangle}.$
(24)

The recently discovered slow-time-scale relocation of solute inhomogeneities
into stable field-dependent configurations is obtained in the inhomogeneous
inviscid case, for which $s\neq 0$ and and $\bm{f}_{\mathrm{ac}}$ reduces to
Karlsen  _et al._ (2016); Karlsen and Bruus (2017)

$\bm{f}_{\mathrm{ac}}^{\mathrm{invisc}}=-\frac{1}{4}|p_{1}|^{2}\bm{\nabla}\kappa_{0}(s)-\frac{1}{4}|\bm{v}_{1}|^{2}\bm{\nabla}\rho_{0}(s).$
(25)

In the general inhomogeneous viscous case, which we are currently studying,
the streaming flow co-develop with the solute relocation using
$\bm{f}_{\mathrm{ac}}$ from Eq. (19),

$\bm{f}_{\mathrm{ac}}=-\bm{\nabla}\big{\langle}p_{11}(s)\big{\rangle}-\bm{\nabla}\cdot\big{\langle}\rho_{0}(s)\bm{v}_{1}\bm{v}_{1}\big{\rangle}.$
(26)

##  VII Discussion

Central aspects and selected examples of these recent theoretical developments
in the field of microscale acoustofluidics will be discussed at the Nobel
Symposium on Microfluidics, Sånga-Säby, Sweden, 5 – 8 June 2017.

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