The model sound speed determination for the plane shear fluid flow problem solving by the SPH method

The problem discrete statement by the smoothed particle hydrodynamics method (SPH) include a discretization constants parameters set. Of them particular note is the model sound speed $c_0$, which relates the SPH-particle instantaneous density to the resulting pressure through the equation of state.
The paper describes an approach to the exact determination of the model sound speed required value. It is on the analysis based, how SPH-particle density changes with their relative shift. An example of the continuous medium motion taken the plane shear flow problem; the analysis object is the relative compaction function $\epsilon_{\rho}$ in the SPH-particle. For various smoothing kernels was research the functions of $\epsilon_{\rho}$, that allowed the pulsating nature of the pressures occurrence in particles to establish. Also the neighbors uniform distribution in the smoothing domain was determined, at which shaping the maximum of compaction in the particle.
Through comparison the function $\epsilon_{\rho}$ with the SPH-approximation of motion equation is defined associate the discretization parameter $c_0$ with the smoothing kernel shape and other problem parameters. As a result, an equation is formulated that the necessary and sufficient model sound speed value provides finding. For such equation the expressions of root $c_0$ are given for three different smoothing kernels, that simplified from polynomials to numerical coefficients for the plane shear flow problem parameters.