In this paper, Part 7 of the thematic tutorial series “Acoustofluidics – exploiting ultrasonic standing waves, forces and acoustic streaming in microfluidic systems for cell and particle manipulation ”, we present the theory of the acoustic radiation force; a second-order, time-averaged effect responsible for the acoustophoretic motion of suspended, micrometre-sized particles in an ultrasound field.

We present a numerical study of the transient acoustophoretic motion of microparticles suspended in a liquid-filled microchannel and driven by the acoustic forces arising from an imposed standing ultrasound wave: the acoustic radiation force from the scattering of sound waves on the particles and the Stokes drag force from the induced acoustic streaming flow. These forces are calculated numerically in two steps. First, the thermoacoustic equations are solved to first order in the imposed ultrasound field taking into account the micrometer-thin but crucial thermoviscous boundary layer near the rigid walls. Second, the products of the resulting first-order fields are used as source terms in the time-averaged second-order equations, from which the net acoustic forces acting on the particles are determined. The resulting acoustophoretic particle velocities are quantified for experimentally relevant parameters using a numerical particle-tracking scheme. The model shows the transition in the acoustophoretic particle motion from being dominated by streaming-induced drag to being dominated by radiation forces as a function of particle size, channel geometry, and material properties.

We calculate the acoustic radiation force from an ultrasound wave on a compressible, spherical particle suspended in a viscous fluid. Using Prandtl-Schlichting boundary-layer theory, we include the kinematic viscosity of the solvent and derive an analytical expression for the resulting radiation force, which is valid for any particle radius and boundary-layer thickness provided that both of these length scales are much smaller than the wavelength of the ultrasound wave (millimeters in water at megahertz frequencies). The acoustophoretic response of suspended microparticles is predicted and analyzed using parameter values typically employed in microchannel acoustophoresis.

In the second part of the thematic tutorial series “Acoustofluidics – exploiting ultrasonic standing waves forces and acoustic streaming in microfluidic systems for cell and particle manipulation”, we develop the perturbation theory of the acoustic field in fluids and apply the result in a study of acoustic resonance modes in microfluidic channels.

We consider pressure-driven, steady-state Poiseuille flow in straight channels with various cross-sectional shapes: elliptic, rectangular, triangular, and harmonic-perturbed circles. A given shape is characterized by its perimeter $\mathcal{P}$ and area $\mathcal{A}$ which are combined into the dimensionless compactness number $\mathcal{C}={\mathcal{P}}^2∕\mathcal{A}$, while the hydraulic resistance is characterized by the well-known dimensionless geometrical correction factor $\alpha$. We find that $\alpha$ depends linearly on $\mathcal{C}$, which points out $\mathcal{C}$ as a single dimensionless measure characterizing flow properties as well as the strength and effectiveness of surface-related phenomena central to lab-on-a-chip applications. This measure also provides a simple way to evaluate the hydraulic resistance for the various shapes.