COMSOL 6.4 - Acoustic Streaming in a Microchannel Cross Section

Acoustic Streaming in a Microchannel Cross Section

Recent advances in the fabrication of microfluidic systems require handling of live cells and other microparticles. One possibility is to use acoustics and exploit the acoustic radiation force to focus particles and/or separate them based on their acoustical properties. When using acoustics in a microchannel, an acoustic streaming flow will be generated and also affect the particles with a viscous drag force; see Ref. 1.

The particle trajectories are determined by the acoustic radiation force and the viscous drag force. The acoustic radiation force is an effect where momentum is transferred from an acoustic field to particles due to nonlinear terms in the governing equations. This results in a net force acting on the particles — the acoustic radiation force.

Due to the nonlinear terms in the Navier–Stokes equations, harmonic perturbation of the flow will lead to a net time-averaged flow called acoustic streaming. Acoustic streaming is a second-order (nonlinear) acoustic effect. The acoustic streaming influences the viscous drag force on the particles. The trajectory of particles in devices are governed by the balance between the viscous drag force (from the streaming flow) and the acoustic radiation force.

There are losses in the acoustic field that produce a heat source. This can be frictional losses in the viscous boundary layers where acoustic energy is transformed into heat, which can result in heating of a microfluidic system.

The model is of a 2D cross section of a microfluidic channel which can, for example, be used for upconcentrating or separating particles in biological fluid samples. This model is based on pressure acoustics and uses effective boundary conditions (the thermoviscous boundary layer impedance or BLI condition) to include the effects of the viscous boundary layers; see Ref. 2.

Model Definition

The model consists of a rectangular fluid domain actuated by vibrating walls; see Figure 1. It is assumed that the surrounding solid is acoustically hard. The model first computes the acoustic field in a study; then the time-averaged second-order fields, the acoustic streaming flow and acoustic heating in a study; and lastly the particle trajectories in a study.

The acoustic field is modeled with and the is used to account for the damping in the thin viscous boundary layers.

Figure 1: Geometry of the 2D cross section of the microfluidic channel.

The acoustic streaming flow is modeled using the multiphysics couplings and , which from the acoustic field compute and apply the acoustic source terms to the fluid flow interface. The source terms consist of a domain force and a slip velocity on the boundary. The slip velocity includes the contribution from the thin viscous and thermal boundary layers.

The heating from the acoustic field primarily occurs in the viscous boundary layer. The heat generated in the viscous boundary layer is computed analytically (it exists as a predefined variable) and imposed as a boundary layer heat source. To mimic a typical silicon chip with a glass lid, the bottom and the two sides of the rectangle have a constant temperature (due to silicon being a very good heat conductor), while the top is thermally insulating (glass transports heat much less than silicon). Therefore, the temperature will increase at the glass lid, and there will be a temperature gradient across the microfluidic channel.

The particle trajectories are modeled using the interface, which computes the particle trajectories based on the two contributing forces: the acoustic radiation force and the viscous drag force. These two forces depend differently on the size of the particles: for small particles the viscous drag force dominates. whereas the acoustic radiation force dominates for large particles. In most applications, the acoustic radiation force is used to focus particles. This is therefore not possible for particles below a critical particle size, which depends on the material properties of the particle.

Results and Discussion

The system is actuated at the resonance frequency f0 = 1.9652 MHz for the horizontal half-wave resonance. This results in the acoustic field in Figure 2 and an acoustic energy density which is typical for these types of microfluidic devices. The main damping in the system is due to losses in the viscous boundary layers, which therefore determine the Q-factor of the system and the width of the resonance peak. In an actual microfluidic chip there will also be significant damping in the piezoelectric transducer used to actuate the system and in the glue layer. This can widen the resonance peak and lower the resonance frequency.

Figure 2: Acoustic pressure field in the rectangular channel at the horizontal half-wave resonance.

Figure 3: Acoustic streaming flow in the microchannel cross section. The color scale shows amplitude of the fluid flow velocity while the white streaming lines show the direction of the flow field. The streaming consists of the classical four Rayleigh Streaming rolls.

Figure 3 shows the acoustic streaming induced by the acoustic field; the color plot shows the amplitude of the flow field and the white streamlines show the direction of the fluid flow. The streaming flow forms classical Rayleigh streaming rolls, which are typical for boundary-driven streaming. The Rayleigh streaming is induced by the stresses and forces in the viscous boundary layers. In this model, the contributions from the viscous boundary layers are applied as slip velocity and therefore the boundary layers do not have to be numerically resolved.

Figure 4: The temperature field created due to the acoustic heat source in the viscous boundary layers. The white lines represents contour lines of the temperature field.

The heating from the acoustic field induces a temperature gradient across the microchannel. The temperature increase due to the acoustic field is shown in Figure 4. The boundary condition on the acoustic field, , computes the heat generated in the viscous boundary layer in the variable acpr.tvb1.Q_tot. This boundary heat source is used to model the temperature increase in the microfluidic channel. In this example, the acoustic field results in a small temperature increase of the order mK. The temperature increase in an actual microfluidic device will depend on the acoustic field, but also on the thermal properties of the surrounding solid.

Figure 5: The position of particles with radius a = 3 µm at a specific time step. The color of the particles represent the amplitude of their velocity and the lines their trajectory.

The particle trajectory is determined by the acoustic radiation force and the viscous drag force. In Figure 5, the positions of polystyrene particles can be seen at time t = 0.1 s. The color of the particles represents the velocity amplitude and the lines represent the particle trajectory. In Figure 5 the particles have radius a = 3 µm and their trajectories are dominated by the acoustic radiation force focusing the particles in the pressure node. For smaller polystyrene particles with radius a = 0.4 µm, the particle trajectories are shown in Figure 6 at t = 3 s. For the small particles, the viscous drag force dominates; the particles are dragged by the fluid flow, and are not focused in the pressure nodes. The velocities of the small particles are an order of magnitude smaller than those of the large particles.

Figure 6: The position of particles with radius a = 0.4 µm at a time step. The color of the particles represent the amplitude of their velocity and the lines their trajectory.

Notes About the COMSOL Implementation

The implementation is based on pressure acoustics, meaning that the viscous boundary layers are not resolved numerically. Therefore, effective boundary conditions are used to represent the impact of the boundary layers (the condition). This is used for the damping of the acoustic field, slip velocity for the fluid flow, and boundary heat source for the acoustic heating. These are all analytical expressions that are valid when the viscous boundary layer thickness is a lot smaller than the acoustic wavelength and the geometrical length scales; see Ref. 2. Some of the analytical expressions depend on the derivatives at the boundary. Therefore, a thin boundary-layer mesh element is used to improve the accuracy of the normal derivative at the boundary.

References

1. P.B. Muller, R. Barnkob, M.J. Herring Jensen, and H. Bruus, “A numerical study of microparticle acoustophoresis driven by acoustic radiation forces and streaming-induced drag forces,” Lab. Chip., vol. 12, pp. 4617–4627, 2012.

2. J.S. Bach and H. Bruus, “Theory for pressure acoustics with viscous boundary layers and streaming in curved elastic cavities,” J. Acoust. Soc. Am., vol. 144, no. 2, pp. 766–784, 2018.

Acoustics_Module/Nonlinear_Acoustics/acoustic_streaming_microchannel_cross_section

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