Effective Theory

When coupling from an effective theory is used to include the effects of viscous boundary layers. It is a theory that has been developed for modeling acoustics in microfluidic devices, a research area called Acoustofluidics, see Ref. 4, Ref. 6, and Ref. 7. The theory solves the acoustic fields with pressure acoustics and use the Thermoviscous Boundary Layer Impedance boundary condition to include the effects from the viscous boundary layers. It then computes the force contribution from the viscous boundary layers from an analytical expression and impose it as slip velocity on the boundary. This model is valid if the viscous boundary layer thickness is a lot smaller than the acoustic and the characteristic geometry length scales. The benefits of this model is that it is not necessary to numerically resolve the viscous boundary layers.

The acoustic sources are then a slip velocity including both the Stokes slip and the viscous stresses from the boundary layers and a bulk body force containing the acoustic source terms related to the bulk of the domain. The bulk body force can be simplified because the acoustic velocity is given by the pressure field. The second-order equations for the continuity and Navier–Stokes equations become

The acoustic body force faco contains the term that gives rise to traditional Eckart streaming and terms that depend on the gradient of the material parameters (Ref. 6). The gradients in material parameters can either be caused by a solvent (Ref. 5) or a temperature gradient (Ref. 8) and are the source terms for thermoacoustic and baroclinic streaming. The slip velocity vslip is given by the acoustic pressure field p1.

The effective theory is used when coupling from . It has been validated against a full model and experiments in the research area of acoustics in microfluidic devices, see Ref. 4, Ref. 6, and Ref. 8. For more information on the theoretical derivation see Ref. 4 and Ref. 6.