In acoustics it is typical to linearize the continuity equation Equation 10-1 and the Navier–Stokes equation
Equation 10-2 to derive a set of governing equations for the acoustic field. The linearization of the governing equations is the first-order equations of a perturbation scheme. Acoustic streaming is the time-averaged second-order equations of the perturbations scheme. Thus, a field can be described by the background zeroth-order field, first-order acoustic field and time-averaged second-order field. The pressure and velocity fields are decomposed as
The acoustic source terms in the second-order contributions appear in both the continuity equation as a mass source term Qm, in Navier–Stokes equations as an acoustic body force
faco, and on the boundary condition for the velocity field as a slip-velocity
vslip.
where Qm is the acoustic source term to the second-order fluid equations. The second-order equation for the Navier–Stokes equations (
Equation 10-3) is given as
When coupling from Thermoviscous Acoustics, Frequency Domain, the contributions to the fluid flow equations is the mass source term
Qm, the acoustic body force
faco, and the Stokes slip. The fluid flow described in
Equation 10-9 is a
Creeping Flow since the inertial term is neglected. It is possible to use separation of timescales instead of perturbation theory to derive the governing equations, in this case the inertial terms are present, so it is mathematical sound to couple to the
Laminar Flow interface, see
Ref. 7.