*6.1. Shear Wave Decay*

We consider the dynamics in terms of the time of the sinusoidal velocity wave in a square domain. The initial flow velocity in *x* direction is dependent on *y* coordinate and is given by

$$u\_{\mathbf{x}}(x, y, t=0) = \mathcal{U}\_0 \sin(ky), \quad k = \frac{2\pi}{L}, \rho$$

where *L* is the length of the domain equals *N* lattice nodes and *U*<sup>0</sup> = 0.01. The periodic boundary conditions are applied for the present problem. This problem has the following analytical solution

$$u\_{\mathfrak{x}}(\mathfrak{x}, \mathfrak{y}, t) = \mathcal{U}\_0 \sin(ky) e^{-\nu k^2 t}.$$

In the present case, we consider *ν* = 0.001 and *N* = 101, the time step *δt* = 1. We compare the analytical solutions with the velocity profiles obtained by the application of the model (29)–(37) (implemented in the form (42)). The peak velocity time history and the velocity profiles for the different moments of time are plotted, Figure 2. One can see that the simulation results are very similar to the analytical profiles.

It is worth mentioning that it is possible to shorten the model and take *γ* = *λ* = 0, in this case *α* = *β*, and we have only five different collisions. The numerical experiments show that this model becomes unstable for *ν* < 0.1, while the setting (43) allows to model the flow with small viscosity and no instabilities are observed.

**Figure 2.** Shear wave decay. The logarithm of the peak velocity time histories obtained numerically and analytically are presented (**first slide**); velocity profiles at different moments of time (*<sup>t</sup>* <sup>=</sup> <sup>10</sup>5, *<sup>t</sup>* <sup>=</sup> <sup>2</sup> <sup>×</sup> <sup>10</sup>5, *<sup>t</sup>* <sup>=</sup> <sup>3</sup> <sup>×</sup> 105) obtained numerically and analytically are presented (**second slide**), the spatial variable *y* is normalized on the domain length *L*.
