The full linear Navier-Stokes (FLNS) equations are solved for viscous-thermal acoustics, which includes the viscous and thermal effects for modeling small devices. For this case, the FLNS model achieves more accurate simulation result compared to the BLI or LRF model. (Kampinga et al. [[436]])
The following topics related to the full linear Navier-Stokes equation solver are available:
With the assumptions of
linearization,
no mean flow, and
small harmonic perturbation,
the linearized Navier-Stokes equations are derived as follows:
 | (8–244) |
 | (8–245) |
 | (8–246) |
where:
 = stress tensor defined as: |
 = velocity variation |
| T = temperature variation |
| p = pressure variation |
 = dynamic viscosity |
 = second viscosity |
 = unit tensor |
 = volumetric force density |
 = ambient density |
| Cp = heat coefficient at constant pressure per unit mass |
 = heat flow vector defined as: |
 = heat conduction coefficient |
| Q = volumetric heat source |
| p0 = ambient pressure |
| T0 = ambient temperature |
The finite element formulations are obtained by testing the momentum
equation (Equation 8–244), the continuity equation
(Equation 8–245), and the entropy equation
(Equation 8–246) using the Galerkin procedure.
The equations are multiplied by testing function 
, pw and Tw, then integrated
over the volume of the domain with some manipulation to yield the following:
 | (8–247) |
 | (8–248) |
 | (8–249) |
where:
 = volume differential of acoustic domain Ω |
 = surface differential of acoustic domain boundary  |
 = outward normal unit vector on the boundary |
 |
Boundary conditions are applied to the finite element model to solve FLNS equations.
Acoustic boundary conditions may be as follows:
On the boundary location, one acoustic boundary condition—either normal velocity, normal stress, or impedance—should be prescribed.
Thermal boundary conditions may be as follows:
On the boundary location, one thermal boundary condition—either temperature, heat flux, or thermal impedance—should be prescribed.
Viscous boundary conditions may be as follows:
On the boundary location, one viscous boundary condition—either shear velocity, shear force, or viscous impedance—should be prescribed. The tangential total force is equal to the tangential viscous force.
On the FSI interface, the kinetic condition relating to the acting and reaction stress and the kinematic condition are written as:
 | (8–259) |
 | (8–260) |
where:
 = structure stress tensor |
 = fluid stress tensor |
 = outward normal unit vector of structure domain |
 = outward normal unit vector of fluid domain |
 = velocity of fluid |
 = displacement of structure |
There are two boundary layers in the FLNS model, the viscous boundary layer and the thermal boundary layer. The thickness of the viscous boundary layer can be evaluated for two cases.
In a circular pipe, the viscous boundary layer is cast with:
On the top of a flat plate, the viscous boundary layer is given by:
The thermal boundary layer is related to the viscous boundary layer with the Prandtl number:
where:
Pr = Prandtl number