Simulating acoustically-actuated flows in complex microchannels using the volume penalization technique

Abstract

We present a volume penalization technique for simulating acoustically-actuated flows in geometrically complex microchannels. Using a perturbation approach, the nonlinear response of an acoustically-actuated compressible Newtonian fluid moving over obstacles or flowing in a geometrically complex domain is segregated into two sub-problems: a harmonic first-order problem and a time-averaged second-order problem, where the latter utilizes forcing terms and boundary conditions arising from the first-order solution. This segregation results in two distinct volume penalized systems of equations. The no-slip boundary condition at the fluid-solid interface is enforced by prescribing a zero structure velocity for the first-order problem, while spatially varying Stokes drift—which depends on the gradient of the first-order solution—is prescribed as the structure velocity for the second-order problem. The harmonic first-order system is solved via MUMPS direct solver, whereas the steady state second-order system is solved iteratively using a novel projection method-based preconditioner. The preconditioned iterative solver for the second-order system is demonstrated to be highly effective and scalable with respect to increasing penalty force and grid resolution, respectively. A novel contour integration technique to evaluate the acoustic radiation force on an immersed object is also proposed. This technique circumvents the use of velocity derivatives within the smeared region. The contour integral is specifically tailored to Cartesian grids. Through a series of test cases featuring representative microfluidic geometries, we demonstrate excellent agreement between the volume penalized and body-fitted grid solutions for the primary first- and second-order fields as well as for the acoustic radiation force that depends on the gradients of these fields. We also identify suitable penalty factors and interfacial smearing widths to accurately capture the first- and second-order solutions. These results provide empirical evidence of the efficacy of the volume penalization method for simulating acoustic streaming problems that have commonly been analyzed using body-fitted methods in the acoustofluidic literature.