Perturbation Theory
In acoustics it is typical to linearize the continuity equation Equation 10-2 and the Navier–Stokes equation Equation 10-3 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
(10-6)
and likewise for all physical fields. In the derivation, it is assumed that the background flow is zero so v0 = 0.
The time-averaged second-order equations consist of the second-order fields and the time-averaged products of the acoustic first-order fields. The products of first-order fields have the form (Ref. 2)
(10-7)
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.
The second-order equation for Equation 10-3 is given as
(10-8)
(10-9)
where Qm is the acoustic source term to the second-order fluid equations. The second-order equation for the Navier–Stokes equations (Equation 10-4) is given as
(10-10)
(10-11)
Here, the acoustic body force appears as the divergence of the momentum flux tensor and the second order stress tensor that depends on the acoustic fields τ11. The stress tensor τ11 is given as
(10-12)
Here, μ1 is the acoustic perturbation for the viscosity due to the acoustic pressure and thermal field. The contribution from τ11 is only included if the check box Include first order viscosity terms is selected. The acoustic perturbation of the viscosity is given as
(10-13)
The last acoustic contribution to the fluid flow equation is the addition of a slip velocity due to the vibrations of the boundary. This occurs because the no-slip boundary condition should be enforced on the vibrating boundary. For a boundary at stationary position s0 and acoustic vibration of s1 the no-slip condition becomes
(10-14)
This gives a slip velocity between the fluid and the wall, known as Stokes slip.
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 10-10 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.