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Acoustic Force Density Acting on Inhomogeneous Fluids in Acoustic Fields Downloaded from orbit.dtu.dk on: Dec 18, 2017 Acoustic Force Density Acting on Inhomogeneous Fluids in Acoustic Fields Karlsen, Jonas Tobias; Augustsson, Per; Bruus, Henrik Published in: Physical Review Letters Link to article, DOI: 10.1103/PhysRevLett.117.114504 Publication date: 2016 Document Version Publisher's PDF, also known as Version of record Link back to DTU Orbit Citation (APA): Karlsen, J. T., Augustsson, P., & Bruus, H. (2016). Acoustic Force Density Acting on Inhomogeneous Fluids in Acoustic Fields. Physical Review Letters, 117(11), 114504. DOI: 10.1103/PhysRevLett.117.114504 General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. week ending PRL 117, 114504 (2016) PHYSICAL REVIEW LETTERS 9 SEPTEMBER 2016 Acoustic Force Density Acting on Inhomogeneous Fluids in Acoustic Fields † Jonas T. Karlsen,1,* Per Augustsson,2 and Henrik Bruus1, 1Department of Physics, Technical University of Denmark, DTU Physics Building 309, DK-2800 Kongens Lyngby, Denmark 2Department of Biomedical Engineering, Lund University, Ole Römers väg 3, 22363 Lund, Sweden (Received 22 April 2016; revised manuscript received 24 June 2016; published 9 September 2016) We present a theory for the acoustic force density acting on inhomogeneous fluids in acoustic fields on time scales that are slow compared to the acoustic oscillation period. The acoustic force density depends on gradients in the density and compressibility of the fluid. For microfluidic systems, the theory predicts a relocation of the inhomogeneities into stable field-dependent configurations, which are qualitatively different from the horizontally layered configurations due to gravity. Experimental validation is obtained by confocal imaging of aqueous solutions in a glass-silicon microchip. DOI: 10.1103/PhysRevLett.117.114504 The physics of acoustic forces on fluids and suspensions types based on differences in their acoustomechanical has a long and rich history including early work on properties. Not surprisingly, the subtle nonlinear acoustic fundamental phenomena such as acoustic streaming phenomenon of relocation and stabilization of inhomo- [1–4], the acoustic radiation force acting on a particle geneous fluids was discovered in the realm of micro- [5,6] or an interface of two immiscible fluids [7], and fluidics, where typical hydrostatic pressure differences acoustic levitation [8,9]. Driven by applications related to (∼1 Pa) are comparable to, or less than, the acoustic energy particle and droplet handling, the field continues to be densities (1–100 Pa) obtained in typical microchannel active with recent advanced studies of acoustic levitators resonators [38–40]. [10–12], acoustic tweezers and tractor beams [13–15], The main goal of this Letter is to provide a theoretical thermoviscous effects [16–18], and, in general, rapid explanation of this phenomenon. To this end, we extend advances within the field of microscale acoustofluidics acoustic radiation force theory beyond the requirement [19]. In the latter, acoustic radiation forces are used to of immiscible phases, and we present a general theory confine, separate, sort, or probe particles such as micro- for the time-averaged acoustic force density acting on a vesicles [20,21], cells [22–26], bacteria [27,28], and bio- fluid with a continuous spatial variation in density and molecules [29]. Biomedical applications include the early compressibility. The starting point of our treatment is to detection of circulating tumor cells in blood [30,31] and the identify and exploit the separation in time scales between diagnosis of bloodstream infections [32]. the fast time scale of acoustic oscillations and the slow time The theoretical treatment of acoustic forces involves scale of the oscillation-time-averaged fluid motion. We nonlinear models including multiple length and time scales show that gradients in density and compressibility result in [33]. Steady acoustic streaming [34] describes a steady a divergence in the time-averaged momentum-flux-density swirling fluid motion, spawned by fast-time-scale acoustic tensor, which, in contrast to the case of immiscible phases, dissipation either in boundary layers [2] or in the bulk [3]. is generally nonzero everywhere in space. Our theory Similarly, the acoustic radiation force acting on a particle explains the observed relocation and stabilization of inho- [18] or an interface of two immiscible fluids [35,36] is mogeneous fluids. Furthermore, we present an experimen- tal validation of our theory obtained by confocal imaging in due to interactions between the incident and the scattered an acoustofluidic glass-silicon microchip. acoustic waves. This force derives from a divergence in Characteristic time scales.—Consider the sketch in the time-averaged momentum-flux-density tensor, which is Fig. 1 of a long, straight microchannel of cross-sectional nonzero only at the position of the particle or the interface. width W ¼ 375 μm and height H ¼ 150 μm filled with Recently, in microchannel acoustofluidics experiments, ρ ð Þ¼½1 þ ρˆð Þρð0Þ it was discovered that acoustic forces can relocate inho- a fluid of inhomogeneous density 0 r r 0 , κ ð Þ mogeneous aqueous salt solutions and stabilize the result- adiabatic compressibility 0 r , and dynamic viscosity η0ðrÞ. Here, ρˆðrÞ is the relative deviation away from the ing density profiles against hydrostatic pressure gradients ð0Þ [37]. Building on this discovery, isoacoustic focusing was reference density ρ0 . Assuming an acoustic standing half- ω subsequently introduced as an equilibrium cell-handling wave resonance at angular frequencypffiffiffiffiffiffiffiffiffi, the wave number method that overcomes the central issue of cell-size is k ¼ ω=c ¼ π=W, where c ¼ 1= ρ0κ0 is the speed of dependency in acoustophoresis [38]. The method can be sound. In terms of the parameters of the microchannel and considered a microfluidic analog to density gradient of water at ambient conditions, the fast acoustic oscillation centrifugation, achieving spatial separation of different cell time scale t is 0031-9007=16=117(11)=114504(6) 114504-1 © 2016 American Physical Society week ending PRL 117, 114504 (2016) PHYSICAL REVIEW LETTERS 9 SEPTEMBER 2016 ρ 1 1 ∂ρ d ¼ dp ¼ ¼ ρ κ ð Þ 2 ; where 2 ∂ 0 0: 4 dt c dt c p S Here, c is the adiabatic local speed of sound, which depends on the position through the inhomogeneity in κ0 and ρ0. Combining Eqs. (3) and (4) leads to the first-order relation ∂tρ1 þðv1 · ∇Þρ0 ¼ ρ0κ0½∂tp1 þðv1 · ∇Þp0; ð5Þ FIG. 1. Sketch of a long, straight acoustofluidic microchannel where we have discarded terms involving v0, as they are ¼ 40 ¼ 375 μ of length L mm along x, width W m and height negligible for jv0j ≪ c. From the governing equations for H ¼ 150 μm with an imposed half-wave acoustic pressure 2 mass and momentum [41–43] follows j∇p0j ≪ c j∇ρ0j, resonance (sinusoidal curves) inside a glass-silicon chip. A salt and the term involving ∇p0 in Eq. (5) is also negligible. concentration (black, low; white, high) leads to an inhomo- This results in the first-order equations geneous-fluid density ρ0ðrÞ, compressibility κ0ðrÞ, and dynamic viscosity η0ðrÞ. The gravitational acceleration is g ¼ −gez. κ0∂tp1 ¼ −∇ · v1; ð6aÞ ρ0∂tv1 ¼ −∇p1; ð6bÞ 1 1 1 pffiffiffiffiffiffiffiffiffi t ∼ ¼ ¼ W ρ0κ0 ∼ 0.1 μs: ð1Þ ω kc π and the wave equation for the acoustic pressure p1 in an inhomogeneous fluid [42,44], In contrast, the time scales associated with flows 1 1 driven by hydrostatic pressure gradients are much slower. ∂2 ¼ ρ ∇ ∇ ð Þ 2 t p1 0 · ρ p1 : 7 Given the length scale H, the gravitational acceleration c 0 g, and the kinematic viscosity ν0 ¼ η0=ρ0, we estimate pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Note that the curl of Eq. (6b) yields ∇ × ðρ0v1Þ¼0, which ∼ ð ρˆÞ the time scale of inertia tinertia H= g , of viscous implies that acoustics in inhomogeneous fluids should be ∼ 2 ν relaxation trelax H = 0, and of steady shear motion formulated in terms of the mass current potential ϕρ instead ∼ ν ð ρˆÞ tshear 0= Hg , with the latter being obtained by bal- of the usual velocity potential, η ancing the shear stress 0=tshear with the hydrostatic ρ ¼ ∇ϕ ¼ −∂ ϕ ð Þ pressure difference Hρ0gρˆ. Remarkably, in our system 0v1 ρ and p1 t ρ: 8 with ρˆ ≈ 0.1, all time scales are of the order of 10 ms, henceforth denoted as the slow time scale τ, Combining Eqs. (6a) and (8) reveals that the mass current potential ϕρ fulfills the same wave equation as p1. τ ∼ ∼ ∼ ∼ 10 ð Þ The acoustic force density.—The first-order acoustic tinertia trelax tshear ms: 2 fields lead to no netR fluid displacement since the time T average hg1i¼ð1=TÞ 0 g1dt over one oscillation period T Furthermore, for acoustic energy densities Eac of the order of any time-harmonic first-order field g1 is zero. The ρ0gH, the time scale of flows driven by time-averaged acoustic forces is also τ. Hence, we have identified a description of time-averaged effects thus requires the separation of time scales into a fast acoustic time scale t and solution of the time-averaged second-order equations, 5 and the introduction of the time-averaged acoustic a slow time scale τ ∼ 10 t. This separation is sufficient momentum-flux-density tensor hΠi [41], to ensure τ ≫ t in general, even for large variations in parameter values.
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