## 2 Materials and Methods  
### Equations of Motion and Their Solutions  
The schematic diagram of the device for slurry mixing, whipping and shaking, based on the slide-crank mechanism, is shown in Figure 1.  
For numerical calculations of the piston-type crank-slider mechanism shown in Figure 1, the dimensions of the links, engine parameters, and fluid characteristics used are listed in Table 1.  
Figure 1: Device for mixing, whipping and shaking of slurry based on slide-crank mechanism device with indication of parameters of its elements. We accept the following assumptions, simplifications and hypotheses. The density of the liquid in the working volume is considered unchangeable (the density of the process fluid, koumiss and shubat, are close to the density of water). When the liquid passes through the holes of the piston, in the thin applied layer, the flow is assumed to be laminar (\(\rm{Re}=(47.5\div 71.2)<\rm{Re}_{cr}\)), the edge effects are neglected, and the flow in the gap between the cylinder and the piston is assumed to be laminar (\(\rm{Re}=(23.8\div 35.6)<\rm{Re}_{cr}\)) under the conditions of smallness of the radius of the holes and the size of the gap between the piston and the cylinder compared to the radius of the piston. To simplify the equation of motion of the device, we assume the length of the crank is many times less than the length of the connecting rod, and the mass of the engine rotor is many times greater than the mass of the piston. The crank, connecting rod, slider, and piston are considered stiff and perfectly aligned in the absence of motion, with their centers located along a vertical line. The slider is constrained on both sides by guides, ensuring strictly vertical movement. The piston holes are located at the vertices of a regular polygon, with their centers coinciding with the connection point to the slider. The piston's center of mass coincides with its geometric center. The position of the piston holes does not affect the mass center coordinate. At low crankshaft rotation speeds, dynamic balancing of centrifugal forces acting on the crank and connecting rod is not necessarily due to the relatively smallness of these forces. The focus is on the dynamics of technological processes carried out by the piston with the mixing-whipping force in the operating volume, and therefore, compared to the mixing-whipping force, the friction forces in kinematic pairs and in guides of vertical motion of a slide can be neglected, due to their insignificance. Using the Lagrangian method, the differential equations of motion of the mixing and churning device were constructed based on a simple piston crank-slider mechanism for cases of upward and downward movement of the piston (Figure 1). The position of the nonideal system's motor mechanism is determined by the generalized coordinate: the \begin{table}
\begin{tabular}{c l l} \hline \hline
**Parameter** & **Name** & **Value** \\ \hline \(r_{0}\) & crank length & 0.10 m \\ \hline \(l_{cr}\) & connecting rod length & 0.60 m \\ \hline \(M_{cr}\) & connecting rod mass & 1 kg \\ \hline \(m\) & piston mass & 20 kg \\ \hline \(R_{c}\) & cylinder radius & 0.10 m \\ \hline \(R_{1}\) & piston radius & 0.08 m \\ \hline \(R_{2}\) & hole radius & 0.002 m \\ \hline \(n_{2}\) & number of holes & 2 \\ \hline \(h\) & piston height & \(5\times 10^{-3}\) m \\ \hline \(\rho\) & density of fluid & 1038 kg\(\cdot\)m\({}^{-3}\) \\ \hline \(\eta\) & dynamic viscosity of fluid & 1.75 \(\times\) 10\({}^{-3}\) Pa\(\cdot\)s (20 \({}^{\circ}\)C) \\ \hline \(a\) & motor torque & 43 N\(\cdot\)m \\ \hline \(b\) & motor rotor drag & 42.879 N\(\cdot\)m\(\cdot\)s\({}^{-1}\) \\  & force torque & \\ \hline \(J_{mr}\) & motor rotor moment & 60.75 \(\times\) 10\({}^{-3}\) kg\(\cdot\)m\({}^{2}\) \\  & of inertia & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of mechanism links, engine, and fluid. angular coordinate \(\varphi\) of the drive element due to its rotation. Then, the differential equation of Lagrange motion has the following form:  
\[