# Explanation of the Generalizations of Uncertainty Principle from
Coordinate and Momentum Space Periodicity

Subir Ghosh<EMAIL_ADDRESS>Physics and Applied Mathematics Unit,
Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India

###### Abstract

Generalizations of coordinate $x$-momentum $p_{x}$ Uncertainty Principle, with
$\Delta x$ and $\Delta p_{x}$ dependent terms ($\Delta$ denoting standard
deviation),

$\Delta x\Delta p_{x}\geq i\hbar(1+\alpha\Delta p_{x}^{2}+\beta\Delta x^{2})$

have provided rich dividends as a poor person’s approach towards Quantum
Gravity, because these can introduce coordinate and momentum scales
($\alpha,\beta$ ) that are appealing conceptually. However, these extensions
of Uncertainty Principle are purely phenomenological in nature. Apart from the
inherent ambiguity in their explicit structures, the introduction of
generalized commutations relations compatible with the Uncertainty Principle
has serious drawbacks.

In the present paper we reveal that these generalized Uncertainty Principles
can appear in a perfectly natural way, in canonical quantum mechanics, if one
assumes a periodic nature in coordinate or momentum space, as the case may be.
We bring in to light quite old, (but no so well known), works by Judge and by
Judge and Lewis, that explains in detail how the popularly known structure of
Extended Uncertainty Principle is generated in the case of angle $\phi$
angular momentum $L_{z}$,

$\Delta\phi\Delta L_{z}\geq i\hbar(1+\nu\Delta\phi^{2})$

purely from a consistent implementation of periodic nature of the angle
variable $\phi$, without changing the $\phi,L_{z}$ canonical commutation
relation. We directly apply this formalism to construction to formulate
generalizations in $\Delta x\Delta p_{x}$ Uncertainty Principle. We identify
$\beta$ with an observed length scale relevant in astrophysics context. We
speculate about the $\alpha$ extension.

Introduction: Absence of a fundamental theory of Quantum Gravity has given
rise to phenomenological models of generalized uncertainty relations, that in
a top-down approach, tries to incorporate signatures of Quantum Gravity. Most
notable of such effects is the proposed existence of a minimum length scale,
adhering to String theory ideas. This can be achieved by considering,
(contrary to the Heisenberg Uncertainty Principle (HUP) $\Delta x^{i}\Delta
p_{j}\geq(\hbar/2)\delta^{i}_{j}$), a Generalized Uncertainty Principle (GUP)
of a generic form

$\Delta x^{i}\Delta
p_{j}\geq\frac{\hbar}{2}\delta^{i}_{j}(1+\alpha\frac{2G}{\hbar c^{3}}(\Delta
p_{j})^{2})$ (1)

where $\alpha$ is a dimensionless constant gup . Minimization of (1) for
$\Delta x^{i}$ yields a minimum length of the order of $L_{P}={\sqrt{\hbar
G/c^{3}}}$, the Planck length. However, this is the minimal form and there
exist more elaborate forms of GUP. The non-uniqueness of the GUP structure
stems from the absence of a fundamental theory leading to the GUP in a bottom-
up approach.

The symmetry between $x_{i}$ and $p_{j}$ in the canonical commutation
relations, $[x_{i},p_{j}]=i\hbar\delta_{ij},~{}[x_{i},x_{j}]=[p_{i},p_{j}]=0$
suggests the possibility of a complimentary type of UP, popularly known as
Extended Uncertainty Principle (EUP),

$\Delta x^{i}\Delta
p_{j}\geq\frac{\hbar}{2}\delta^{i}_{j}(1+\beta\Lambda(\Delta x^{i})^{2})$ (2)

where $\beta$ is a dimensionless constant and $\Lambda$ is the Cosmological
Constant eup ; park . This implies a minimum scale for de Sitter momentum of
the order $m_{dS}c=\hbar{\sqrt{\Lambda/3}}$. Finally there is the maximally
extended for GEUP comprising of both $(\Delta x^{i})^{2}$ and $(\Delta
p_{j})^{2}$ in the RHS eup  but we will not go into this in the present work.
In the present paper we will be concerned mostly with EUP (2).

It needs to be stressed that these generalizations of uncertainty relations
are essentially phenomenological in nature. The idea of a minimum length scale
is appealing from different perspectives; high energy scattering of strings
leading to a fundamental length, removing conceptual paradoxes in Black Hole
Physics, … among others. In order to have a deeper insight into the origin of
such non-canonical URs, a well-studied proposal kem  has been to replace the
canonical commutation relations to generalized set of commutations relations,
consistent with GUP, EUP, GEUP. Recall that the canonical (or Heisenberg UP)
is compatible with the canonical commutation relations com . However, in the
present instances, these modified commutators necessary involve operator-
valued terms in the RHS, as for example

$[x^{i},p_{j}~{}]=i\hbar\delta^{i}_{j}(1+\alpha\frac{2G}{\hbar c^{3}}(p)^{2})$
(3)

corresponding to the GUP (1). Apart from the technical problems involved in
quantizing a theory with operator-valued commutation relations, (although a
possible solution is to map non-canonical degrees of freedom to canonical ones
via a Darboux-like transformation), it has been observed recently la  that
such generalizations in UPs can give rise to reference-frame dependent minimum
length scales and modified commutators can induce equivalence principle
violations. To overcome the above mentioned issues and simultaneously generate
the GUP, EUP or EGUP, it has been suggested in la  to replace the usual
position and momentum measurement operators by appropriate Positive Operator
valued Measure, with an inbuilt finite accuracy. The main point of this
construction is that one can still use the canonical variables, commutation
relations and Hamiltonian with Heisenberg and Schrodinger equations unchanged.

The take home message of the above discussion: try to derive the generalized
UPs keeping the canonical structure (of commutators, Hamiltonian, ..) intact.
We resurrect a quite ancient paper by D. Judge j  to show that it is indeed
possible and in the process reveal an oft studied feature of the Universe - it
has a periodicity.

Deriving the Extended Uncertainty Principle: The work of Judge j  deals with
the direct evaluation of uncertainty relation involving $z$-component of
orbital angular momentum, $L_{z}=-i\hbar\partial/\partial\phi$ and the angle
$\phi$. Naively, it should follow the $p_{x},x$ result, that is $\Delta
L_{z}\Delta\phi\geq\hbar/2$ but obviously this is not correct because $\phi$
being an angle, $\Delta\phi\leq\pi/{\sqrt{3}}$, corresponding to a uniform
distribution $f(\phi)$ of $\phi$ as derived below:

$for~{}\pi\geq\phi\geq\\-\pi,~{}f(\phi)=1/(2\pi),(\Delta\phi)^{2}=\int_{-\pi}^{\pi}\phi^{2}f(\phi)d\phi-(\int_{-\pi}^{\pi}\phi
f(\phi)d\phi)^{2}=\pi^{2}/3.$ (4)

Thus as $\Delta L_{z}$ moves towards zero $\Delta\phi$ has to obey the bound
and a contradiction arises. A careful consideration of the fact that wave
functions are periodic in $\phi$, Judge j  was able to derive a modified
uncertainty principle

$\Delta L_{z}\Delta\phi\geq\eta\hbar(1-\frac{3(\Delta\phi)^{2}}{\pi^{2}})$ (5)

where $\eta\approx 0.15$ as derived in j . Notice that it is structurally
identical to the EUP (2). Since the work of Judge j  is not very well-known,
we will repeat some steps of the calculation at the end in an Appendix in our
context of EUP. Here we emphasize the fact that the periodic nature of wave
functions of angle variables is the key factor in deriving the modified UP
(5); the phase space symplectic structure and associated quantum commutators
are strictly canonical.

Substituting $p_{x},x$ in place of $L_{z},\phi$ we derive (See Appendix for
details) the EUPp (the subscript ”p” is to stress that this form is derived
here considering only periodic nature of space)

$\Delta x\Delta p_{x}\geq\frac{\hbar}{2}(1-\frac{12(\Delta
x^{i})^{2}}{L^{2}})$ (6)

where we have assumed the periodicity of wave function $\psi(x=0)=\psi(x=L)$.
With $L=L_{P}/l$ our EUP reduces to

$\Delta x\Delta p_{x}\geq\frac{\hbar}{2}(1-\frac{12l^{2}(\Delta
x^{i})^{2}}{L_{P}^{2}}).$ (7)

Before providing justification for the periodic space assumption let us
quickly see the immediate consequences of this EUPp. In park , generic
phenomenological forms of EUPs,

$\Delta x\Delta p_{x}\geq\frac{\hbar}{2}(1\pm\frac{\beta^{2}(\Delta
x^{i})^{2}}{l^{2}})$ (8)

with $l$ a characteristic length scale and $\beta$ a dimensionless parameter,
have been considered. From park , it is inferred that the positive sign EUP in
(8) implies an AdS space, with a lower bound of $\Delta p_{x}$, whereas the
positive sign EUP in (8) can correspond to a dS space.
(i) The EUPp (6) is an exact and unambiguous result without any arbitrary
parameter since, as we show later, the periodicity parameter $L$ can be
identified with robust observational value. Clearly it is not an outcome of
phenomenological and heuristic arguments.
(ii) The EUPp (6) leads to an absolute maximum value of $\Delta x\leq
L/{\sqrt{12}}$ so that $\Delta p_{x}$ remains positive.
(iii) It is interesting to note that, as pointed out in park , the maximum
value of $\Delta x$ does not have $\hbar$ since it is a consequence of a
periodic feature of classical cosmology.

Justification of the spatial periodicity : Recent astrophysical observations
p1 , (corroborating the earlier work p2 ), have shown that, (contrary to the
expectation of galaxies being randomly scattered in the Universe), mapped in
two or three dimensions, galaxies are clustered on small scales $(\sim
5h^{-1}Mpc,~{}h\approx 0.5-1)$. In fact, more interesting is the finding as
reported in p2  that galaxies tend to remain correlated over a much larger
distance, with a characteristic scale of $\sim 128h^{-1}Mpc$. The excess of
correlation and this apparent periodic distribution of galaxies suggest
possibility of oscillating behaviours in cosmological evolution. This is not
explainable in conventional cosmological models and requires scalar field with
non-minimal couplings. In p3  a model is proposed involving a stable
condensate of a pseudo-Goldstone mode having a periodic distribution of energy
density in radial coordinate.

Conclusion: Conventionally generalizations of $\Delta x,\Delta p_{x}$
uncertainty relation is introduced in a purely ad hoc and phenomenological way
leading to non-uniqueness in its explicit structure and its subsequent
offshoot of non-canonical operator dependent commutation relations yielding
further issues. In this perspective, we fall back to the canonical framework
and following closely early works by Judge j  show that well defined Extended
Uncertainty Relations can be induced by a periodicity in the space
coordinates. We further show that there are observational results showing such
periodicity in astrophysical scenario.

In the present work we used the coordinate representation with
$p_{x}=-i\hbar\partial/\partial x$ and recovered $(\Delta x)^{2}$ corrections
in $\Delta x,\Delta p_{x}$ uncertainty relation. It will be straightforward to
consider the complimentary (Fourier space) scenario where
$x=i\hbar\partial/\partial p_{x}$ and generate the GUP with $(\Delta x)^{2}$
corrections in $\Delta x,\Delta p_{x}$ uncertainty relation.

Appendix : We essentially follow the analysis of Judge j ; j1 ; j2 ,
appropriately modified for our system. We assume the periodicity in wave
function $\psi(x=-L/2)=\psi(x=L/2)$. Define the function

$V(\gamma)=\int_{-L/2}^{L/2}\psi^{*}(x+\gamma)x^{2}\psi(x+\gamma)dx.$ (9)

Note that $\psi^{*}(x)\psi(x)$ is identified with an appropriate distribution
function. The value of $\gamma$, for $L/2\geq\gamma\geq-L/2$, that minimises
$V(\gamma)$ (9) is the mean value of $x$ and $(\Delta x)^{2}$ is the minimum
value of $V(\gamma)$. Note that in general this is different from the
conventional definition of variances

$(\Delta x)^{2}=<(<x-<x>)^{2}>=<x^{2}>-<x>^{2},(\Delta
p_{x})^{2}=<(<p_{x}-<p_{x}>)^{2}>=<p_{x}^{2}>-<p_{x}>^{2}$ (10)

where $<A>$ stands for the expectation value of $A$. Indeed, it is easy to
check that $(\Delta x)^{2}$ computed from (9) and (10) are identical for
$\infty\geq x\geq-\infty$, that is when the domain of $x$ is the real line and
$\psi(\pm\infty)=0$. However for angular variables $\phi$ j ; j1 ; j2  or for
other periodic systems, ( as the model proposed here), the conventional
definition is invalid; for example $\bar{\phi}=\int_{-\pi}^{\pi}d\phi~{}\phi
f(\phi)$ and subsequently, for the same reason
$(\Delta\phi)^{2}=(<x-<x>)^{2}>$, will depend up on the choice of the initial
value of $\phi$. This means, for example, for a shift by $\phi_{0}$,
$\phi=\phi^{\prime}+\phi_{0}$, we will find
$\bar{\phi}\neq\bar{\phi}^{\prime}+\phi_{0}$. The significance of (9) is that
this definition is independent of the choice of the initial line.

This leads us to define, for periodic $\psi(-L/2)=\psi(L/2)$,

$V(\gamma)=\int_{-L/2}^{L/2}\psi^{*}(x+\gamma)x^{2}\psi(x+\gamma)dx$ $(\Delta
p_{x})^{2}=\int_{-L/2}^{L/2}\psi^{*}(x)\alpha^{2}\psi(x)dx,~{}~{}\alpha=-i\hbar\partial/\partial_{x}-<p_{x}>.$
(11)

Note that $(\Delta p_{x})^{2}$ is defined in the conventional way. Let us
construct

$(\Delta
p_{x})^{2}V(\gamma)=<\psi_{\gamma}|\alpha^{2}\psi_{\gamma}><\psi_{\gamma}|x^{2}\psi_{\gamma}>,~{}~{}\psi_{\gamma}(x)=\psi(x+\gamma).$
(12)

In the above both (18,19) are independent of the choice of the initial line.
Hence, exploiting the Schwartz inequality, we have

$(\Delta p_{x})^{2}V(\gamma)\geq|<\psi_{\gamma}|(x\alpha)\psi_{\gamma}>|^{2}.$
(13)

We decompose

$x\alpha=\frac{1}{2}(x\alpha+\alpha x)+\frac{1}{2}(x\alpha-\alpha
x),~{}~{}x\alpha-\alpha x=i\hbar$ (14)

and obtain

$<\psi_{\gamma}|(x\alpha+\alpha
x)\psi_{\gamma}>^{*}=<\psi_{\gamma}|(x\alpha+\alpha
x)\psi_{\gamma}>+2Li\hbar\psi^{*}(L+\gamma)\psi(L+\gamma),$ (15)

$Im~{}<\psi_{\gamma}|(x\alpha+\alpha
x)\psi_{\gamma}>=-L\hbar\psi^{*}(L+\gamma)\psi(L+\gamma).$ (16)

The above steps lead to

$(\Delta
p_{x})^{2}V(\gamma)\geq\frac{\hbar^{2}}{4}(1-L\hbar\psi^{*}(L+\gamma)\psi(L+\gamma))^{2}.$
(17)

Renaming $\alpha\equiv x-<x>,\beta\equiv p_{x}-<p_{x}>$ we have

$(\Delta x)^{2}(\Delta p_{x})^{2}=\int\psi^{*}\alpha^{2}\psi
dx\int\psi^{*}\beta^{2}\psi
dx=\int(\alpha^{*}\psi^{*})(\alpha\psi)dx\int(\beta^{*}\psi^{*})(\beta\psi)dx.$
(18)

Schwartz inequality requires

$(\Delta x)^{2}(\Delta
p_{x})^{2}\geq|\int(\alpha^{*}\psi^{*})(\beta\psi)dx|^{2}=|\int\psi^{*}(\alpha\beta)\psi
dx|^{2}$ (19)

In general we have

$|<f|g>|^{2}\geq(\frac{<f|g>-<g|f>}{2i})^{2}.$ (20)

In the present case with $f\equiv\alpha\psi,g\equiv\beta\psi$ we have

$\displaystyle<f|g>-<g|f>=\int(\psi^{*}(x+\gamma)x(-i\hbar\partial\psi(x+\gamma))dx-\int((+i\hbar\partial\psi^{*}(x+\gamma)x\psi(x+\gamma))dx$
$\displaystyle=-i\hbar\int(\psi^{*}(x+\gamma)x(\partial\psi(x+\gamma))dx-i\hbar\int[\partial\\{\psi^{*}(x+\gamma)x\psi(x+\gamma)\\}-\psi^{*}(x+\gamma)\partial(x\psi(x+\gamma))]dx$
$\displaystyle=-i\hbar\psi^{*}(x+\gamma)x\psi(x+\gamma)|_{0}^{L}+i\hbar\int[-\psi^{*}(x+\gamma)x\partial\psi(x+\gamma)+\psi^{*}(x+\gamma)\partial(x\psi(x+\gamma))]dx$
(21)

This trivially simplifies to

$<f|g>-<g|f>=-i\hbar
L\psi^{*}(L/2+\gamma)\psi(L/2+\gamma)+i\hbar\int\psi^{*}\psi dx$ (22)

for $\psi(L+\gamma)=\psi(\gamma)$, leading to

$<f|g>-<g|f>=i\hbar(1-L\psi^{*}(L+\gamma)\psi(L+\gamma)).$ (23)

$\Delta
p_{x}^{2}V(\gamma)\geq\frac{\hbar^{2}}{4}(1-L\psi^{*}(L+\gamma)\psi(L+\gamma))^{2}$
(24)

The inequality in (24) above applies to all values of $\gamma$. Let us assume
that the initial line is at $\gamma=0$ so that, according to our definition
$V(0)=\Delta x^{2}$.

The following identity will play an important role:

$\displaystyle\frac{dV(y)}{dy}=V^{\prime}(y)=\frac{d}{dy}[\int^{\beta(y)}_{\alpha(y)}g(x,y)dx]=g(\beta(y),y)\beta^{\prime}(y)-g(\alpha(y),y)\alpha^{\prime}(y)$
$\displaystyle+\int^{\beta(y)}_{\alpha(y)}\frac{\partial g(x,y)}{\partial
y}dx.$ (25)

The above yields the relations, where $f(x)\equiv\psi^{*}(x)\psi(x)$,

$V^{\prime}(\gamma)=-2\int^{L/2}_{-L/2}f(x+\gamma)xdx,$ (26)
$V^{\prime\prime}(\gamma)=2[1-Lf(L/2+\gamma)]$ (27)

and the following inequalities,

$0\leq V(\gamma)\leq L^{2}/4,~{}-L\leq
V^{\prime}(\gamma)<L,~{}-\infty<V^{\prime\prime}(\gamma)\leq 2$ (28)

and finally

$\bar{V}=\frac{1}{L}\int^{L/2}_{-L/2}V(\gamma)d\gamma=L^{2}/12.$ (29)

Following the algebraic argument provided in j1 , (that can be directly
applied to the present case of interest), it can be shown that

$\Delta p_{x}\Delta x\geq\frac{\nu\hbar}{2}(1-12\frac{(\Delta
x)^{2}}{L^{2}}),$ (30)

where $nu$ is a numerical constant of $\sim 1$. In fact value of $\nu$ for the
angular momentum - angle uncertainty relation is derived in j ; j1 ; j2  but
we have not included the corresponding analysis since our main concern was to
demonstrate that the EUP structure can be derived naturally from the
assumption of a periodic nature in space in conventional Quantum Mechanics
framework. Thus, we claim to have derived the EUP

$\Delta p_{x}\Delta x\geq\frac{\hbar}{2}(1-12\frac{(\Delta x)^{2}}{L^{2}}).$
(31)

## References

  * (1) M. Maggiore, A Generalized uncertainty principle in quantum gravity, Phys. Lett. B 304, 65 (1993); R. J. Adler and D. I. Santiago, On gravity and the uncertainty principle, Mod. Phys. Lett. A 14, 1371 (1999); F. Scardigli, Generalized uncertainty principle in quantum gravity from micro - black hole Gedanken experiment, Phys. Lett. B 452, 39 (1999).
  * (2) B. Bolen and M. Cavaglia, (Anti-)de Sitter black hole thermodynamics and the generalized uncertainty principle, Gen. Rel. Grav. 37, 1255 (2005); C. Bambi and F. R. Urban, Natural extension of the Generalised Uncertainty Principle, Class. Quant. Grav. 25, 095006 (2008).
  * (3) M. I. Park, The Generalized Uncertainty Principle in (A)dS Space and the Modification of Hawking Temperature from the Minimal Length, Phys. Lett. B 659, 698 (2008);
  * (4) A. Kempf, G. Mangano and R. B. Mann, Hilbert space representation of the minimal length uncertainty relation, Phys. Rev. D 52, 1108-1118 (1995).
  * (5) H. P. Robertson, The Uncertainty Principle, Phys. Rev. 34, 163 (1929); E. Schr¨odinger, About Heisenberg uncertainty relation, Bulg. J. Phys. 26, 193 (1999) [Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. ) 19, 296 (1930)].
  * (6) Matthew J. Lake, Marek Miller, Ray Ganardif and Tomasz Paterekg, Generalised Uncertainty Relations from Finite-Accuracy Measurements; Front. Astron. Space Sci., 10, 1087724 (2023), https://doi.org/10.3389/fspas.2023.1087724 .
  * (7) D. Judge, On the Uncertainty Relation for $L_{z}$ and $\varphi$, Phys. Lett. 5 (1963) 189.
  * (8) D. Judge, Nuovo Cimento 31, 332 (1964)
  * (9) D. Judge and J.T. Lewis, Phys. Lett. 5, 190(1963).
  * (10) G. Busarello, S. Capozziello, R. de Ritis, G. Longo, A. Rifatto, C. Rubano, P. Scudellaro, Apparently periodic Universe, Astronomy and Astrophysics, 283, p. 717 (1994)
  * (11) T. J. Broadhurst, R. S. Ellis, D. C. Koo , A. S. Szalay, Large-scale distribution of galaxies at the Galactic poles, Nature 343, 726–728 (1990)
  * (12) A.A. Anselm, Periodic universe and condensate of pseudo-Goldstone field, Physics Letters B, 260, 39-44 (1991)


pnasresearcharticle Bhosale, Vishwanathan Viscous streaming, an inertial
phenomenon, refers to the steady flows that emerge when a fluid oscillates
around a localized microfeature. There, stresses are concentrated, remodeling
the surrounding flow over short time/length scales. The result is an efficient
flow machinery to mix, transport, trap, or sort at the microscale.
Nonetheless, classically, viscous streaming has been investigated for uniform-
curvature bodies, for which only two distinct flow topologies exist,
constraining its utility and scope. We combine predictive simulations and
versatile experimental techniques to demonstrate how the rational design of
multi-curvature bodies unlocks a vast range of flow topologies, demonstrated
here in compact and tunable microparticle filtration and separation devices.
Overall, our technology transcends previous flow/particle manipulation
limitations, significantly expanding viscous streaming potential. Y.B., T.P.,
and M.G. designed the computational research; G.V. and G.J. designed the
experimental research; Y.B. performed numerical computations and analyzed
data; G.V. performed experimental research and analyzed data; Y.B., G.V.,
T.P., G.J., and M.G. wrote the paper The authors declare no competing
interests. 1Y.B. and G.V. contributed equally to this work. 2To whom
correspondence should be addressed. E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Multi-curvature viscous streaming: flow topology and particle manipulation

Yashraj Bhosale  Mechanical Science and Engineering, University of Illinois at
Urbana-Champaign, Urbana, IL 61801, USA  Giridar Vishwanathan  Mechanical
Science and Engineering, University of Illinois at Urbana-Champaign, Urbana,
IL 61801, USA  Tejaswin Parthasarathy  Mechanical Science and Engineering,
University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA  Gabriel
Juarez  Mechanical Science and Engineering, University of Illinois at Urbana-
Champaign, Urbana, IL 61801, USA  Mattia Gazzola  Mechanical Science and
Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
National Center for Supercomputing Applications, University of Illinois at
Urbana-Champaign, Urbana, IL 61801, USA  Carl R. Woese Institute for Genomic
Biology, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

###### Abstract

Viscous streaming refers to the rectified, steady flows that emerge when a
liquid oscillates around an immersed microfeature, typically a solid body or a
bubble. The ability of such features to locally concentrate stresses produces
strong inertial effects to which both fluid and immersed particles respond
within short length ($\mathcal{O}(100)\leavevmode\nobreak\ \mathrm{\mu}$m) and
time ($\mathcal{O}(10^{-3})$ s) scales, rendering viscous streaming arguably
the most efficient mechanism to exploit inertia at the microscale. Despite
this potential, viscous streaming has been investigated in rather narrow
conditions, mostly making use of bodies of uniform curvature (cylinders,
spheres) for which induced flow topologies are constrained to only two
regimes, classically referred to as single and double layer regimes. This
severely limits the scope of potential applications, and sits in stark
contrast to inertial focusing (the approach that perhaps has most defined the
field of inertial microfluidics), where instead boundaries with multiple
curvatures have long been utilised to magnify inertial effects, expand design
space, and enable novel applications. Here, we demonstrate that a multi-
curvature approach in viscous streaming dramatically extends the range of
accessible flow topologies. We show that numerically predicted, but never
experimentally observed, streaming flows can be physically reproduced,
computationally engineered, and in turn used to enhance particle manipulation,
filtering and separation in compact, robust, tunable and inexpensive devices.
Overall, this study provides an avenue to unlock the utility of viscous
streaming with potential applications in manufacturing, environment, health
and medicine, from particle self-assembly to microplastics removal.

###### keywords:

viscous streaming $|$ particle manipulation $|$ filtration $|$ computational
inertial microfluidics

###### doi:

www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX

This manuscript was compiled on

The recognition that, at the microscale, small but finite inertia can be
employed to manipulate flows and suspended particles, has majorly impacted
microfluidics (1, 2, 3, 4). Inertial effects have been central to the
development of purely passive, hydrodynamic control strategies to align (5),
separate (6), concentrate (7) and mix (8) liquids, particles and chemicals,
delivering high throughputs, operational simplicity, cost effectiveness,
precision and delicacy (biological samples) in a robust, fault-tolerant
fashion. Because of these appealing features, modern inertial microfluidics
has found broad application across disciplines, from engineering (9) and
biology (10) to environment (11), health and medicine (12).

Two main flow phenomena may be leveraged to harness inertia at the microscale:
inertial focusing and viscous streaming. Inertial focusing relies on cross-
streamline particle migration (from the competition between wall lift and
shear gradient forces) and microchannel curvature variations to focus
particles of given properties at key downstream locations, from which they are
collected and processed (1, 3). Due to its success, particularly in clinical
and point-of-care settings (1, 3), inertial focusing has perhaps come to
define the field of inertial microfluidics. Nonetheless, limitations remain:
indeed, inertial focusing operates over relatively large length
($\sim\mathcal{O}(10)$ cm) and time ($\sim\mathcal{O}(1)$ s) scales (2, 3); it
typically relies on highly specialized microchannels, rendering it rather
inflexible to new target particles and applications; it is poorly tunable,
with the flow rate as the sole real-time control parameter, limiting
opportunities for modulation, rapid and localized on/off activations, and
modular integration. Finally, its design process is time consuming and mostly
reliant on trial-and-error, due to the current paucity of reliable, accurate
and predictive simulations (13).

Potential solutions may be instead offered by viscous streaming, a
complementary and perhaps overlooked inertial flow phenomenon. Viscous
streaming refers to the steady, rectified flows that emerge when a fluid
oscillates around a localised microfeature, typically a solid body or a bubble
(14, 15, 16). Such microfeatures have the ability to concentrate stresses (17,
18), thus distort and remodel, even dramatically, the surrounding flow and its
topology (19, 20, 21). The result is a remarkably consistent, controllable and
convenient machinery to shape both flow and particle paths. Indeed, since the
flow reorganises over short length-scales (microfeature size), a spectrum of
high velocity curvatures and gradients is produced, which in turn generate
strong inertial forces on particles (22). This allows for precise, selective
manipulation over compact footprints ($\mathcal{O}(100)\leavevmode\nobreak\
\mathrm{\mu}$m) at the millisecond scale
($\mathcal{O}(10^{-3})\leavevmode\nobreak\ $s) (23, 24), making streaming
arguably the most efficient way to exploit inertia at the microscale.

Despite its potential, viscous streaming has been narrowly explored, with
applications (8, 15, 25) overwhelmingly employing classically understood
bodies of uniform curvature (cylinders, spheres) which admit only two
resulting flow topologies, known as the single and double layer regimes (26,
27). Such a limited reportoire restricts the scope of achievable microfluidic
applications, and sits in stark contrast to inertial focusing, where instead
the importance of boundaries (channel walls) varying in curvature has long
been recognised and leveraged to expand application range and improve
performance (2, 3, 13).

Only recently, computational studies have begun to explore the nexus between
body-curvature manipulation and streaming flow reorganization, revealing a
rich variety of flow topologies and dynamics (19, 20, 21), never
experimentally reported and of potential practical use. Here, we lay out a
technology based on localised microfeature design, directly driven flow
oscillations, and independently controlled frequencies and amplitudes, to
experimentally access such numerically predicted flows. After demonstrating
their existence, we further showcase their utility in computationally
engineered filtering and separation devices that are tunable, versatile and
effective. Overall, this study fundamentally advances our understanding of
viscous streaming and paves the way for its expansion in practical settings,
while contributing to bridging the gap between experimental and computational
inertial microfluidics.

Figure 1:  Streaming flows in lattice arrays. (a) Illustration of the lattice
geometry that consists of alternating cylinders. The repeating unit cell
entails cylinders of two different radii, therefore curvatures $\kappa_{max}$
and $\kappa_{min}$, and a fixed centre-to-centre spacing. The experimental
setup is constituted of a glass-bonded PDMS channel connected to a loudspeaker
that generates oscillatory flows along the horizontal direction. (b, c) Time-
averaged particle pathlines observed in experiments (left) and simulations
(right), for a constant curvature lattice system with
$\kappa_{max}/\kappa_{min}=1$. Two distinct flow topologies, direct
generalization of the well-known (b) single layer (referred to as Phase I
later) and (c) double layer (referred to as Phase II later) regimes are
reported. The flow topologies are four-fold symmetric, and defining saddles
(green circles) and centres (black diamonds) are labeled. Blue represents
clockwise rotation, while orange stands for counter-clockwise rotation. (d)
Time-averaged particle pathlines observed in experiments across multiple unit
cells for a lattice system with $\kappa_{max}/\kappa_{min}=2$.

## Results

### Multiple discrete curvatures – The lattice system

Towards exploring the effects of multiple body-curvatures on viscous
streaming, a simplified setup referred to as the lattice system is considered
(20). The lattice system consists of a 2D array of circular cylinders, with
exactly two distinct curvatures $\kappa_{max}$ and $\kappa_{min}$, arranged in
a checkerboard pattern, and immersed in a background oscillatory flow of
amplitude $A$ and angular frequency $\omega$, as shown in Fig. 1(a). The
center-to-center distance between adjacent cylinders is kept constant at
$6.25/\kappa_{max}$ throughout the study, without loss of generality (20).
This setup allows the injection of multiple, discrete body-curvatures in the
flow system, whose response, at small amplitudes ($A\ll 1/\kappa_{max}$), is
completely specified by only two parameters: the non-dimensional Stokes layer
thickness $\delta_{AC}\kappa_{max}=\kappa_{max}\sqrt{\nu/\omega}$, where $\nu$
is the fluid kinematic viscosity, and the curvature ratio
$\kappa_{max}/\kappa_{min}$. Thus, by keeping $\kappa_{max}$ fixed while
modifying $\kappa_{min}$ and $\omega$, the effects of curvature variations
($\kappa_{max}/\kappa_{min}$) and flow conditions ($\delta_{AC}\kappa_{max}$)
on streaming topology can be studied in a controlled, systematic manner. A
rich phase space composed of hitherto unseen streaming flow topologies was
recently discovered numerically, and underlying bifurcation mechanisms
understood via dynamical systems theory (20). To examine the potential for
microfluidic applications, a realization of the lattice system is attempted in
this study.

#### Experimental realization

To systematically access multi-curvature viscous streaming regimes, an
external driver is used to generate oscillatory flows within a PDMS
microfluidic channel (28), sculpted with a lattice structure at its center. A
loudspeaker is directly connected to a microfluidic tubing, the other end of
which is inserted into the device, as shown in Fig. 1(a). The vibration of the
loudspeaker diaphragm generates a time-varying pressure that produces harmonic
displacement of the liquid within the device, at independently controlled
frequencies and amplitudes.

Three lattice systems are fabricated, consisting of rectangular arrays of
$10\times 8$ cylinders with $r_{min}=\kappa_{max}^{-1}=287\mathrm{\ \mu m}$
and $r_{max}=\kappa_{min}^{-1}=287,\ 333,\ 571\mathrm{\ \mu m}$, resulting in
the curvature ratios $\kappa_{max}/\kappa_{min}\approx 1.0$, $1.2$, and $2.0$.
The investigated range of flow conditions $\delta_{AC}\kappa_{max}$ is
achieved with oscillation frequencies of $50-600$ Hz, and deionized water as
the working liquid. For all cases, the oscillation amplitude ($A\leq 50\
\mu$m) is small compared to the cylinder radii.

Figure 2:  Phase space of streaming flow topologies, identified through the
lattice system. (a) Phase space as a function of $\delta_{AC}\kappa_{max}$ and
$\kappa_{max}/\kappa_{min}$. Black lines indicate transition boundaries
between phases, and black dots indicate the specific point in the phase space
reported here. Streaming flows are classified into distinct phases depending
on their flow topology, characterised by saddles (green circles) and centres
(black diamonds). (b-f) Time-averaged particle pathlines observed in
experiments (left) and simulations (right) for distinct phases, indicated by
roman numerals. Phases in (b,c,d,e) are obtained for
$\kappa_{max}/\kappa_{min}=2$, while (f) employs
$\kappa_{max}/\kappa_{min}=1.2$. Topological markers (saddles and centres) are
identified and indicate good agreement between experiments and simulations.

#### Phase space

We start with the intuitive case of the uniform-curvature lattice
($\kappa_{max}/\kappa_{min}$=1), illustrated in Fig. 1 (b,c). On the left,
observed streaming flow pathlines for a unit cell are shown, while on the
right, corresponding numerical streamlines are juxtaposed, with blue and
orange representing clockwise and counter-clockwise rotations, respectively.
Of particular relevance are the flow critical points, where the velocity is
zero. In two-dimensional incompressible flows, saddles (green circles) and
centers (black diamonds) conveniently offer a sparse yet complete
representation of the flow field, its topology and underlying dynamics.
Further, and importantly, they are of practical use with regards to mixing,
trapping or transport. Specifically, centers are employed to attract and
retain particles (29, 30), while saddles (and connecting streamlines)
partition the flow, enabling particle separation (8, 23) or targeted mixing in
spatially controlled chemistry (8, 25).

We first observe that the cases of Fig. 1 (b) and (c), corresponding to
$\delta_{AC}\kappa_{max}=0.19$ and $\delta_{AC}\kappa_{max}=0.06$, both
exhibit four-fold symmetry, in line with the symmetry of the unit cell. At
relatively large $\delta_{AC}\kappa_{max}$(>0.1), streaming flows result in
single vortices (orange, outlined in red) with distinct centers, neatly
separated by saddles. This topology is an expected, direct generalisation of
the single-layer regime for individual cylinders (26). At smaller
$\delta_{AC}\kappa_{max}$, additional counter-rotating outer vortices (blue,
outlined in red) appear diagonally, squeezing the original vortices into inner
bounded regions adjacent to the cylinders, as shown in Fig. 1c. This is a
generalisation of the double-layer regime for individual cylinders (26). As
can be noticed, experiments accurately replicate numerical solutions.

After establishing our approach in the uniform scenario, lattice systems with
multiple curvatures are explored. As illustrated over six unit cells in Fig.
1(d), a curvature ratio departure from unity ($\kappa_{max}/\kappa_{min}=2$)
breaks four-fold symmetry, leading to a two-fold diagonal symmetry instead.
Evidently, reduced symmetry modifies flow topology, permitting the transport
of material across two of the four vortices. This is a significant
breakthrough from the limitations of the uniform curvature lattice, and
provides an avenue to usefully employ multiple body-curvatures for sculpting,
manipulating and connecting flow regions.

Systematic variation of flow conditions ($\delta_{AC}\kappa_{max}$) and
curvature ratio ($\kappa_{max}/\kappa_{min}\geq$ 1) yield the computationally-
determined phase space of Fig. 2(a), in which seven distinct flow topologies
are identified. Here, we experimentally probe their existence (black dots),
spanning curvature ratio by means of our three lattice channels
($\kappa_{max}/\kappa_{min}=$ 1, 1.2 and 2). For a given lattice, different
flow topologies (phases) can be accessed by modifying
$\delta_{AC}/\kappa_{max}$ via the frequency $\omega$, with low frequencies
corresponding to large $\delta_{AC}\kappa_{max}$, and vice-versa. The case of
$\kappa_{max}/\kappa_{min}=1$ has already been discussed, and we refer to the
topologies of Fig. 1(b) and (c) as Phase I and II, respectively. Of the
remaining five phases, Phases III-VI are achieved with the lattice of
$\kappa_{max}/\kappa_{min}=2$, while Phase VII is achieved with
$\kappa_{max}/\kappa_{min}=1.2$.

Phase III is observed for $\delta_{AC}/\kappa_{max}>0.2$. In this phase, the
inner vortices of the smaller cylinders interact with each other and form a
closed connected bicentric region (marked in orange with two centers and a
saddle), as shown in Fig. 2(b). This bicentric region then separates the inner
vortices (blue) of the larger cylinders.

Phase IV is observed when $0.09<\delta_{AC}\kappa_{max}<0.2$. Here, a single
central vortex (orange with a center) flanked by two saddles (green) is
observed, as shown in Fig. 2(c). This central vortex then separates the inner
vortices (blue) of the larger cylinders.

Phase V is observed when $0.07<\delta_{AC}\kappa_{max}<0.09$. In this case, a
central vortex (orange with a center) identical to the one in Phase IV is
present, as shown in Fig. 2(d). Additionally, new outer vortices (blue with
centers) appear on either side of the central one, in the vicinity of the
smaller cylinders.

Phase VI is seen for $\delta_{AC}\kappa_{max}<0.07$. In this phase, a
bicentric recirculation zone (orange, two centers and a saddle) is observed,
as shown in Fig. 2(e). Additionally, outer vortices (blue with centers) are
seen on either side of this recirculation zone, around the smaller cylinders.

Phase VII is realized for $\delta_{AC}\kappa_{max}\approx 0.11$ and
$\kappa_{max}/\kappa_{min}=1.2$, and shown in Fig. 2(f). Once again, a
bicentric recirculation zone (orange, two centers and a saddle) is seen, but
unlike Phase VI, no outer vortices are present near the smaller cylinders.

These streaming flows, never experimentally observed before, are found to
well-agree with numerical predictions (20), thus establishing their viability
and robustness for potential applications. Indeed, they allow to expose
suspended particles to a sequence of highly varied flow environments, which in
turn can be leveraged (particularly for inertial particles, Stokes number >
0.3) to induce, hasten or modulate cross-streamline migration for manipulation
purposes. In purely oscillatory flows, and for sufficiently long time scales,
all inertial particles are eventually trapped at vortex centers, a fact
previously exploited for secure positioning (31). In the presence of commonly
employed finite transport flows, however, particles’ residence time in the
device is limited and trapping may or may not occur depending on initial
location, and relative strength and topology of the streaming flow. This
implies that appropriate streaming patterns, perhaps enabled by our lattice
system, may be better suited than others to selectively and effectively remove
incoming particles from the mean flow, thus acting as contactless, compact
filters, which we present next.

Figure 3:  Particle filtration with streaming flows in lattice arrays. (a)
Illustration of the lattice array used and the locations examined for particle
concentration. (b) Particle pathlines observed in experiments within the
device, as the particle suspension flows from inlet to outlet. The reduction
in pathlines density, from inlet to outlet, indicates particle trapping within
the device. Example particle trajectories in (c) experiments and (d)
simulations for $\kappa_{max}/\kappa_{min}=2$, $\delta_{AC}\kappa_{max}=0.14$
(Phase IV), and $U_{o}/U_{\infty}=200$. Particle trapping locations are
indicated by red arrows. (e) Filtration efficiency
$\xi=c_{\textrm{out}}/c_{\textrm{in}}$ measured experimentally (dots),
corresponding envelopes (shaded regions, $95$% confidence intervals of a
second-order polynomial fit to the experimental data), and predicted
simulation slopes (dashed lines, with an intercept shift of $\sim 20\%$ along
both axes) as a function of $U_{o}/U_{\infty}$ (oscillations vs. transport
flow velocities) for three different lattice phases (III, IV and V). Particle
fluxes across control surfaces at the inlet ($c_{in}$) and outlet ($c_{out}$)
are evaluated using standard particle tracking velocimetry. The effect of flow
topology (lattice phase) manifests as markedly different filtration
behaviours.

#### Particle filtration

To experimentally examine the operation of the lattice system as a filtration
device, the previously described $\kappa_{max}/\kappa_{min}=2$ lattice channel
is divided into three major sections of observation, labelled as IN, MID and
OUT (Fig. 3a). The first zone, IN, refers to the inlet of the microchannel,
through which a neutrally buoyant suspension of 65 $\mu$m polystyrene
particles in 22% w/w glycerol/water solution is injected at a volumetric flow
rate of 0.1 ml min-1. These inertial particles are tracked over a period of 50
s, and the input particle flux $c_{\textrm{in}}$ across a fixed cross-section
is evaluated (details in figure caption). The second zone, MID, refers to the
core region of the lattice system where significant trapping occurs. Finally,
the third zone, OUT, refers to the outlet of the filter, where the output
particle flux $c_{\textrm{out}}$ is evaluated similarly to the input flux.
Filtration efficiency $\xi=c_{\textrm{out}}/c_{\textrm{in}}$ is then defined
as the relative number of particles flowing through the system, where low
values of $\xi$ indicate high filtration rates, and vice-versa.

Without oscillatory flow, virtually all particles pass through the lattice and
$\xi\sim 100\%$. When instead the filter is activated by turning the
loudspeaker on, complex time-averaged flow topologies emerge from the
interplay between transport flow and streaming, leading to particle retention
within few milliseconds (Fig. 3b). Indeed, as can be appreciated in Fig. 3(c),
inertial particles are extracted from the mean flow, and indefinitely captured
within the streaming vortices (red arrows), a mechanism confirmed in
simulations (Fig. 3d).

Filtering efficiency is then systematically characterised via experiments and
simulations as a function of $U_{o}/U_{\infty}$, which expresses the relative
strengths between streaming ($U_{o}=A\omega$, oscillation velocity) and
transport flow ($U_{\infty}$, free-stream velocity), and
$\delta_{AC}\kappa_{max}$, which determines the streaming flow topology.

Three values of $\delta_{AC}\kappa_{max}$ are chosen, corresponding to Phases
III, IV and V, and for each, $U_{o}/U_{\infty}$ is varied from 50 to 400.
Corresponding filtering efficiencies $\xi$ are plotted in Fig. 3(e), where
dots are experimental measurements, and shaded regions are their envelopes
(caption for details). As expected, for relatively weak streaming
($U_{o}/U_{\infty}<100$), nearly all particles pass through the filter for any
phase considered. When instead the streaming strength is increased, the system
response is dramatically altered. For Phase III (red), filtration rate
improves slowly, remaining relatively ineffective with over $\xi>80\%$ of the
particles flowing through. For Phases IV (green) and V (blue) instead,
filtration rates improve markedly, with about $35\%$ and $10\%$ of the
particles passing through, respectively. When compared with simulation results
(dashed lines), we find qualitative agreement, with the efficiency slopes
correctly captured. Finally, we note that performance can be further improved
by simply adding a few rows of cylinders.

It is worth appreciating that by varying $\delta_{AC}\kappa_{max}$ by a factor
of 2 at $U_{o}/U_{\infty}=200$, an increase in filtration of nearly two orders
of magnitude is numerically predicted and indeed experimentally attained. This
step-change in filtration performance reflects a change in flow topology,
specifically from the less effective Phase III to the more effective Phases IV
and V. We attribute the efficacy of the latter phases to the presence of a
central vortex (orange, Fig. 2 c-d), otherwise absent in Phase III. The mean
flow reconfigures this vortex, which is pushed aside strengthening the
bottom/top vortices near the cylinders, as illustrated in Fig. 3 (d). This, in
turn, facilitates trapping at the indicated locations (red arrows, Fig. 3).

The extraordinary sensitivity afforded by transitioning across streaming
topologies, combined with their ability to almost instantaneously ($<10$ ms)
reconfigure upon frequency $\omega$ variations, can be harnessed to
dynamically control particle distributions in time and space in a purely
hydrodynamic fashion. This makes streaming an adaptive and flexible mechanism,
uniquely complementary to current inertial microfluidic techniques.

Having established the utility of multi-curvature streaming via the case of
discrete curvature configurations, we proceed to demonstrate a distinct
microfluidic application, based on a continuous spectrum of curvatures, at the
single-object level.

Figure 4:  Particle separation using the bullet system. (a-c) Comparison of
simulated and experimental streaming pathlines around a bullet shaped
cylinder, for Phases 1, 2 and 3, respectively. (d) Demonstration of inertial
separation of particles from the wake region at
$\delta_{AC}\kappa_{max}=0.05$. (e) Efficiency of particle separation $\xi$,
defined as the number of particles that flow in the measurement zone, located
in the wake of the bullet, relative to the unperturbed (i.e. no streaming)
reference particle content, plotted against $\delta_{AC}\kappa_{max}$ for
$U_{o}/U_{\infty}=100$. Data points represent experimental results, the shaded
region provides the corresponding envelope ($95$% confidence intervals of a
second-order polynomial fit to the experimental data), and dashed lines show
simulation results.

### Continuous curvature variation – The bullet system

The bullet refers to the object formed by the extrusion of a single convex
shape created by joining a semi-circle of diameter $2/\kappa_{max}$ to a
rounded semi-square of same side-length, as illustrated in Fig. 4 (a). This
particular geometry was rationally designed in-silico to generate streaming
features conducive to enhanced particle transport (19, 20). Here the bullet
system is experimentally realized and subsequently employed for continuous
particle separation, an unattainable application in the case of purely
circular geometries.

#### Experimental realization

We employ the same oscillatory flow generation and channel microfabrication
techniques of the lattice. The bullet is characterized by a major radius of
$1/\kappa_{max}=0.5$ mm, smoothing radius at the corners of 0.1 mm, and
extrusion height of 2.5 mm.

#### Phase space

Analogous to the lattice system, as $\delta_{AC}\kappa_{max}$ is decreased,
distinct flow topologies emerge. We refer to them as Phase 1, 2 and 3, as
illustrated in Figure 4(a-c). We emphasize the remarkable agreement between
computationally designed (top half) and experimentally realised (bottom half)
flow fields.

At higher values of $\delta_{AC}\kappa_{max}$, Phase 1 is observed. In this
phase, a single set of vortices (center, black diamonds), which resemble the
inner vortices observed for a circular cylinder, are seen on the circular
front-end, as shown in Fig. 4(a). On the square rear-end, due to the large
curvature mismatch at the corners, a set of vortices of large size and flow
strength is obtained. In addition to these rear-end vortices, a new set of
outer vortices flank the bullet sides. Decreasing $\delta_{AC}\kappa_{max}$
results in Phase 2. In this phase (Fig. 4b), the original set of vortices at
the circular front-end reconfigures as inner bounded vortices (thickness
defined by green saddles), surrounded by outer counter-rotating vortices,
similar to the double-layer regime of a circular cylinder. Additionally, a
well-defined boundary appears between inner vortices on the circular front-end
and flanking vortices. By further decreasing $\delta_{AC}\kappa_{max}$, the
system transitions to Phase 3 (Fig. 4c), where the square-end vortices become
bounded by the saddle approaching from the far-field. Consequently, flanking
vortices increase in strength and curvature near the rear-end.

The bullet illustrates how rapid computational prototyping enables new and
specialised sets of flow topologies, via microfeature design. We anticipate
that, upon application of a left-to-right transport flow, the flanking
vortices will be effective in laterally deflecting incoming particles.
Additionally, the rear vortices are expected to trap remaining undeflected
particles, rendering the bullet well-suited for continuous separation, which
we present next.

#### Particle separation

To experimentally demonstrate particle separation, a steady transport flow
laden with $65\ \mu$m neutrally buoyant, inertial particles is generated past
the bullet, such that the direction of flow is from the circular-end towards
the square-end. Particle separation is then activated by superimposing
streaming onto the mean flow. This results in the example flow topology of
Fig. 4 (d). As particles are convected around the bullet, they encounter the
flanking vortices, which, as predicted, are observed to deflect particles
aside. Any remaining particle that makes it to the square-end of the bullet is
then readily trapped by the rear vortices. This creates a wake region mostly
free from particles, referred to as the separation zone. We emphasize the
contrast between streaming application and lack thereof, illustrated in Fig.
4(d).

Quantification of particle separation is performed similarly to the lattice
system, where particles are now tracked in a measurement zone (marked in Fig.
4d) before and after streaming is applied. Here, separation efficiency $\xi$
refers to the number of particles that flow in the measurement zone relative
to the unperturbed (i.e. no streaming) reference particle content, and depends
once again on $U_{o}/U_{\infty}$ and $\delta_{AC}\kappa_{max}$. For the fixed
value $U_{o}/U_{\infty}=100$, the efficiency $\xi$ is plotted as a function of
$\delta_{AC}\kappa_{max}$ in Fig. 4(e). As it can be observed, drastically
different behaviors emerge across phases, with Phase 3 being evidently
superior ($<30\%$ of the original particles found in the measurement zone),
relative to the much more modest performance of Phase 1 ($\sim 90\%$ of the
particles still present in the measurement zone). We attribute this difference
to the bounded and stronger rear-end vortices of Phase 3, accompanied by
stronger flanking vortices. Finally, we note the quantitative (within error
margins) agreement with simulations.

## Conclusion

This study fundamentally advances our understanding of viscous streaming
beyond classical settings, through a combination of predictive simulations and
versatile experimental techniques. It illuminates the role of multiple body-
curvatures in establishing and manipulating unconventional flow topologies,
dramatically expanding the intrinsically limited repertoire afforded by
traditional uniform-curvature streaming. It illustrates the utility of
injecting ranges of curvature in the flow, both in a discrete (lattice) and
continuous (bullet) sense, at the level of multiple or individual objects, to
achieve contactless, purely hydrodynamic, and real-time tunable microparticle
filtration and separation. It demonstrates an inexpensive, robust and simply
controllable technological implementation. It establishes the viability and
quantitative accuracy of in-silico design, enabling rapid computational
prototyping. All in all, these innovations provide the tools and understanding
to expand the use of viscous streaming beyond its current niche set of
applications, thus realising its full potential, characterized by vigorous
inertial effects that can be rapidly activated, spatially and temporally
controlled, and modularly composed, synergistically with mainstream inertial
microfluidics.

### Simulation method and numerical implementation

We briefly recap the governing equations and the numerical solution technique.
We consider incompressible viscous flows in a periodic or unbounded domain
$\Sigma$. In this fluid domain, immersed solid bodies perform simple harmonic
oscillations. The bodies are density matched and have support $\Omega$ and
boundary $\partial\Omega$ respectively. The flow can then be described using
the incompressible Navier–Stokes equations Eq. 1

$\boldsymbol{\nabla}\cdot\mbox{\boldmath$u$}=0;\leavevmode\nobreak\
\leavevmode\nobreak\ \frac{\partial\mbox{\boldmath$u$}}{\partial
t}+(\mbox{\boldmath$u$}\cdot\boldsymbol{\nabla})\mbox{\boldmath$u$}=-\frac{\boldsymbol{\nabla}P}{\rho}+\nu\nabla^{2}\mbox{\boldmath$u$},\leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{\boldmath$x$}\in\Sigma\backslash\Omega$ (1)

where $\rho$, $P$, $u$ and $\nu$ are the fluid density, pressure, velocity and
kinematic viscosity, respectively. The dynamics of the fluid–solid system is
coupled via the no-slip boundary condition
$\mbox{\boldmath$u$}=\mbox{\boldmath$u_{s}$}$, where $u_{s}$ is the solid body
velocity. The system of equations is then solved using a velocity–vorticity
formulation with a combination of remeshed vortex methods and Brinkmann
penalization (32). This method has been validated across a range of
flow–structure interaction problems, from flow past bluff bodies to biological
swimming (32, 33, 34, 35, 36, 37). Recently, the accuracy of this method has
been demonstrated in rectified flow contexts as well, capturing steady
streaming responses from arbitrary rigid shapes in 2D and 3D (19, 20, 21).
Lastly, the motion of inertial particles demonstrated in the lattice trapping
and bullet separation sections, is captured using the Maxey-Riley equations
(38).

### Experimental Methods

The channels were fabricated using Polydimethylsiloxane (Sylgard 184™, PDMS,
10:1 resin:crosslinker) molded from a CNC micromachined aluminium mold,
pretreated with silicone mold release spray. The PDMS was cleaned with
isopropanol, perforated, and then bonded to a glass slide after oxygen plasma
treatment for 2 minutes. Polyethylene tubing (PE 200™, ID 1.5 mm, OD 1.9 mm)
was inserted at the inlet and outlet.

A solution of 22% (w/w) glycerol in deionized water with a density and
kinematic viscosity of of $\rho=1.15\ \mathrm{kg/m^{3}}$ and $\nu=1.68\
\mathrm{mm^{2}/s}$, respectively, was used. Polystyrene particles were
suspended and remained neutrally buoyant in the aqueous solution. Particles
with a diameter of $5\ \mathrm{\mu m}$ were used as tracers since the particle
response time $\tau=\rho d^{2}/18\mu$ was equal to $1.6$ $\mu$s, much less
compared to the oscillatory timescales considered in this study. Particles
with a diameter of $65\ \mathrm{\mu m}$ were used for particle manipulation
experiments since the particle response time was equal to $0.4$ ms.

Sinusoidal oscillatory flow of frequencies ranging from $50-600$ Hz was
produced at the outlet by an external oscillatory driver generated by a
loudspeaker diaphragm. Details of the generation and fidelity of oscillatory
flow are reported elsewhere (28). Brightfield microscopy with $2\times$ and
$4\times$ objective lens (depth of field $125\mathrm{\ \mu m}$) were used to
image tracer particles at the channel mid plane. Frame rates between 1-20 fps
were used with an exposure time of $100\ \mathrm{\mu}$s to minimize streaking.

We thank S. Hilgenfeldt for helpful discussions over the course of this work.
The authors acknowledge support by the National Science Foundation under NSF
CAREER Grant No. CBET-1846752 (MG) and by the Blue Waters project (OCI-
0725070, ACI- 1238993), a joint effort of the University of Illinois at
Urbana-Champaign and its National Center for Supercomputing Applications. This
work used the Extreme Science and Engineering Discovery Environment (XSEDE)
(39) Stampede2, supported by National Science Foundation grant no.
ACI-1548562, at the Texas Advanced Computing Center (TACC) through allocation
TG-MCB190004.

## References

  * (1) D Di Carlo, D Irimia, RG Tompkins, M Toner, Continuous inertial focusing, ordering, and separation of particles in microchannels.  Proceedings of the National Academy of Sciences 104, 18892–18897 (2007).
  * (2) D Di Carlo, Inertial microfluidics.  Lab on a Chip 9, 3038–3046 (2009).
  * (3) JM Martel, M Toner, Inertial focusing in microfluidics.  Annual review of biomedical engineering 16, 371–396 (2014).
  * (4) J Zhang, et al., Fundamentals and applications of inertial microfluidics: A review.  Lab on a Chip 16, 10–34 (2016).
  * (5) E Ozkumur, et al., Inertial focusing for tumor antigen–dependent and–independent sorting of rare circulating tumor cells.  Science translational medicine 5, 179ra47–179ra47 (2013).
  * (6) C Wang, SV Jalikop, S Hilgenfeldt, Size-sensitive sorting of microparticles through control of flow geometry.  Applied Physics Letters 99, 034101 (2011).
  * (7) S Ookawara, N Oozeki, K Ogawa, P Löb, V Hessel, Process intensification of particle separation by lift force in arc microchannel with bifurcation.  Chemical Engineering and Processing: Process Intensification 49, 697–703 (2010).
  * (8) BR Lutz, J Chen, DT Schwartz, Microfluidics without microfabrication.  Proceedings of the National Academy of Sciences 100, 4395–4398 (2003).
  * (9) J Seo, MH Lean, A Kole, Membrane-free microfiltration by asymmetric inertial migration.  Applied Physics Letters 91, 033901 (2007).
  * (10) M Masaeli, et al., Continuous inertial focusing and separation of particles by shape.  Physical Review X 2, 031017 (2012).
  * (11) A Schaap, T Rohrlack, Y Bellouard, Lab on a chip technologies for algae detection: a review.  Journal of biophotonics 5, 661–672 (2012).
  * (12) AJ Mach, D Di Carlo, Continuous scalable blood filtration device using inertial microfluidics.  Biotechnology and bioengineering 107, 302–311 (2010).
  * (13) SR Bazaz, et al., Computational inertial microfluidics: a review.  Lab on a Chip 20, 1023–1048 (2020).
  * (14) MS Longuet-Higgins, Viscous streaming from an oscillating spherical bubble.  Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 725–742 (1998).
  * (15) BR Lutz, J Chen, DT Schwartz, Microscopic steady streaming eddies created around short cylinders in a channel: Flow visualization and stokes layer scaling.  Physics of Fluids 17, 023601 (2005).
  * (16) C Wang, SV Jalikop, S Hilgenfeldt, Efficient manipulation of microparticles in bubble streaming flows.  Biomicrofluidics 6, 012801 (2012).
  * (17) P Marmottant, S Hilgenfeldt, Controlled vesicle deformation and lysis by single oscillating bubbles.  Nature 423, 153–156 (2003).
  * (18) P Marmottant, S Hilgenfeldt, A bubble-driven microfluidic transport element for bioengineering.  Proceedings of the National Academy of Sciences 101, 9523–9527 (2004).
  * (19) T Parthasarathy, FK Chan, M Gazzola, Streaming-enhanced flow-mediated transport.  Journal of Fluid Mechanics 878, 647–662 (2019).
  * (20) Y Bhosale, T Parthasarathy, M Gazzola, Shape curvature effects in viscous streaming.  Journal of Fluid Mechanics 898 (2020).
  * (21) FK Chan, Y Bhosale, T Parthasarathy, M Gazzola, Three-dimensional geometry and topology effects in viscous streaming.  arXiv preprint arXiv:2106.08447 XX (2021).
  * (22) S Agarwal, FK Chan, B Rallabandi, M Gazzola, S Hilgenfeldt, An unrecognized inertial force induced by flow curvature in microfluidics.  Proceedings of the National Academy of Sciences 118 (2021).
  * (23) R Thameem, B Rallabandi, S Hilgenfeldt, Particle migration and sorting in microbubble streaming flows.  Biomicrofluidics 10, 014124 (2016).
  * (24) R Thameem, B Rallabandi, S Hilgenfeldt, Fast inertial particle manipulation in oscillating flows.  Physical Review Fluids 2, 052001 (2017).
  * (25) BR Lutz, J Chen, DT Schwartz, Characterizing homogeneous chemistry using well-mixed microeddies.  Analytical chemistry 78, 1606–1612 (2006).
  * (26) J Holtsmark, I Johnsen, T Sikkeland, S Skavlem, Boundary layer flow near a cylindrical obstacle in an oscillating, incompressible fluid.  The journal of the acoustical society of America 26, 26–39 (1954).
  * (27) C Lane, Acoustical streaming in the vicinity of a sphere.  The Journal of the Acoustical Society of America 27, 1082–1086 (1955).
  * (28) G Vishwanathan, G Juarez, Generation and application of sub-kilohertz oscillatory flows in microchannels.  Microfluidics Nanofluidics 24, 1–10 (2020).
  * (29) K Chong, SD Kelly, S Smith, JD Eldredge, Inertial particle trapping in viscous streaming.  Physics of Fluids 25, 033602 (2013).
  * (30) K Chong, SD Kelly, ST Smith, JD Eldredge, Transport of inertial particles by viscous streaming in arrays of oscillating probes.  Physical Review E 93, 013109 (2016).
  * (31) BR Lutz, J Chen, DT Schwartz, Hydrodynamic tweezers: 1. noncontact trapping of single cells using steady streaming microeddies.  Analytical chemistry 78, 5429–5435 (2006).
  * (32) M Gazzola, P Chatelain, WM Van Rees, P Koumoutsakos, Simulations of single and multiple swimmers with non-divergence free deforming geometries.  Journal of Computational Physics 230, 7093–7114 (2011).
  * (33) M Gazzola, C Mimeau, AA Tchieu, P Koumoutsakos, Flow mediated interactions between two cylinders at finite re numbers.  Physics of fluids 24, 043103 (2012).
  * (34) M Gazzola, WM Van Rees, P Koumoutsakos, C-start: optimal start of larval fish.  Journal of Fluid Mechanics 698, 5–18 (2012).
  * (35) M Gazzola, B Hejazialhosseini, P Koumoutsakos, Reinforcement learning and wavelet adapted vortex methods for simulations of self-propelled swimmers.  SIAM Journal on Scientific Computing 36, B622–B639 (2014).
  * (36) M Gazzola, AA Tchieu, D Alexeev, A de Brauer, P Koumoutsakos, Learning to school in the presence of hydrodynamic interactions.  Journal of Fluid Mechanics 789, 726–749 (2016).
  * (37) Y Bhosale, T Parthasarathy, M Gazzola, A remeshed vortex method for mixed rigid/soft body fluid-structure interaction.  Journal of Computational Physics 444, 110577 (2021).
  * (38) MR Maxey, JJ Riley, Equation of motion for a small rigid sphere in a nonuniform flow.  The Physics of Fluids 26, 883–889 (1983).
  * (39) J Towns, et al., Xsede: accelerating scientific discovery.  Computing in science & engineering 16, 62–74 (2014).