Acoustics in fluids can produce many interesting phenomena because of the **nonlinear Navier─Stokes equation**. The field studying this is called acoustofluidics. In recent years, the manipulation of cells and particles with **acoustofluidics** has been of keen interest for **medical and diagnostic applications**. The method is a gentle and label-free method of manipulation that can quickly sort and handle cells based on characteristics of the cells.

In this blog post, we will briefly touch upon some of the phenomena in acoustofluidics as well as how this can be calculated in COMSOL^{®}.

## Microscale Acoustofluidics

Because the governing equations are the nonlinear Navier─Stokes equation, finding solutions can be a complicated affair.

However, the acoustic effects can be expressed as a **harmonic perturbation** to a quiescent fluid solution. By expanding the solution into a power series expansion, the first harmonic terms can be identified,

which leads to pressure acoustics,

Now by solving **Cauchy’s momentum equation** and Maxwell’s first equation,

the electromechanical piezoelectric elements can be calculated. Implementing the correct boundary conditions at the piezoelectric electrode interface, and having free stress and zero free charges at the interfaces,

gives the last necessary conditions for calculating the electromechanical part of the system. By calculating the **effective boundary conditions** on the pressure acoustics, as done in [2], in the orthogonal direction compared to the surface, it is possible to calculate all the first-order fields in the system. The pressure acoustic boundary condition between the solid and liquid surface interface, where ζ is the orthogonal coordinate, is

The precise process used when implementing it in the weak form will be reserved for another blog post. Otherwise please click on this link.

## Acoustofluidics for cell or particle handling

Acoustofluidics is a popular research topic for the manipulation of cells or particles, in particular in combination with microfluidics. The reason is twofold. The sizes of human cell diameters in the blood range from **red blood cells** (**RBC**), of roughly 5 µm, up to roughly 30 µm for **circulating tumor cells** (**CTC**). This size, therefore, sets some constraints on the system. Ideally, the wavelength of the system should be at least 10 times that of the cells and, because of the methods used to manipulate them, the frequency should also be as large as possible.

The two methods for manipulating cells with acoustofluidics are typically either the acoustic radiation force or the acoustic streaming. The **acoustic radiation force** arises from the nonlinear interaction of a wave with the scattered wave from the cell or particle. The details deriving the radiation force can be found in this article [3]. The radiation force can be calculated from the first-order **pressure and velocity fields in the fluid** and the relative **compressibility and density** of the cell or particle compared to the fluid,

typically have coefficients such that they focus into the pressure nodes of the standing pressure field. Another effect the acoustics have on the system is **acoustic streaming**. This is the **time-averaged second-order effect** that generates **flow rolls in the fluid**. The fluid also affects the particles with a **drag force**,

which is proportional to the velocity difference between the fluid and the particle, and proportional to the radius to the first power. Since the acoustic radiation force is proportional to the radius of the cell or particle cubed, then there is a crossover in size where the particle motion is either dominated by the acoustic streaming (small particles) or the acoustic radiation force (larger particles). Typically, the crossover is a diameter of around 2 µm.

## Acoustofluidics in microfluidic glass devices for cell manipulations

Many devices have been made with the intention of sorting, rinsing, or concentrating cells[4-6]. Here is one such example designed with multiple purposes in mind. First, let us just consider the fluidics of the device without the acoustics. Having two inlets and two outlets it is possible to only enter particles in one of the inlets. The flow profile can be calculated in COMSOL® by solving for the Navier─Stokes equation and defining the inlet boundary condition.

Because of the low **Reynolds number**, the system is very laminar, and therefore the fluid combines, flows along the device, and separates with minimal **mixing**. This means that cells that enter one of the inlets will enter the corresponding outlet. This can be illustrated by tracking cells as they move along the device. In this case, green particles are used to portray the cells and are mixed into the flow in the outer inlet, and water is inserted evenly into the two inlets. The cells flow along the devices and into the outlet. Since the system is highly **laminar**, the fluids do not mix and the cells that enter the outer inlet exit the outer outlet as well.

However, when the acoustics is activated then it is possible to move the cells from the outer flow into the middle flow of the channel. Just to be clear, it is not the fluid that changes place but the cells that migrate from the outer flow into the inner flow.

As can be seen in the illustration this means that cells that entered the outer inlet, and would otherwise have exited the outer outlet, now exit the inner outlet when the acoustics is turned *on*. This can have many uses:

1. The **cell outlet can be chosen and controlled like a switch**, without changing other flow conditions.

2. Since the cells have exchanged fluid, the process can be used to clean the cells, by potentially having two different fluids in the inlets, one dirty with the cells and one clean without cells. The **cells at the exit have then been moved to the clean fluid** before they exit the device.

3. The fluid that now does not contain the cell can also have its uses. One such example is blood with blood cells and blood plasma. By removing the blood cells from the blood plasma, the **clean blood plasma** can be studied alone. There are of course other ways to achieve these three examples, but this method has its advantages and disadvantages. One of the main ones is the low volume needed for this process, and that it can be done in a flow-through setup.

# Summary

Here we gave a very brief introduction to acoustofluidics as well as the governing equations that are necessary to understand the system. This included the Navier─Stokes equation, pressure acoustics, piezoelectric electromechanics, and the acoustic radiation force. With these equations it is possible to do some investigations into acoustofluidic devices that are not dominated by acoustic streaming which we did not cover here. With that, we show an example of such a device calculated in COMSOL® and how this can be used to manipulate and sort cells in a system and what uses that could have.

[1] Wei, Qiu, T. Baasch and Laurell, T., Enhancement of Acoustic Energy Density in Bulk-Wave-Acoustophoresis Devices Using Side Actuation. Phys. Rev. Applied, 4: 044043. (2022)

[2] J. S. Bach and H. Bruus, Bulk-driven acoustic streaming at resonance in closed microcavities. Phys. Rev. E 100, 023104 (2019).

[3] M. Settnes and H. Bruus, Theoretical analysis of viscous corrections to the acoustic radiation force on cells in microchannel acoustophoresis. In J. Landers, A. Herr, D. Juncker, N. Pamme, and J. Bienvenue (eds.), Proc. 15th MicroTAS, 2 – 6 October 2011, Seattle (WA), USA, 160–162 (CBMS) (2011).

[4] A. G. Steckel and H. Bruus, Numerical study of bulk acoustofluidic devices driven by thin-film transducers and whole-system resonance modes. The Journal of the Acoustical Society of America 150(1), 634–645 (2021)

[5] Olm, F., Lim, H.C., Schallmoser, K., Strunk, D., Laurell, T. and Scheding, S., Acoustophoresis Enables the Label-Free Separation of Functionally Different Subsets of Cultured Bone Marrow Stromal Cells. Cytometry, 99: 476-487. (2021)

[6] Havers, M., Broman, A., Lenshof, A. et al. Advancement and obstacles in microfluidics-based isolation of extracellular vesicles. Anal Bioanal Chem, 1618-2650 (2022).