Mathematical modeling of mudflow dynamics

Despite the on-going efforts of scientists, there are still few scientifically justified mathematical models that give a practical prediction of the origin, dynamics and destructive force of mudflow. Many problems related to the study of mudflows, and especially their dynamics, are not extensively studied due to the complexity of the process. The contributions of Gagoshidze (1949; 1957; 1962; 1970), Natishvili et al. (1976; 1963; 1969), Tevzadze (1971), Beruchashvili et al. (1958; 1969; 1979), Muzaev, Sozanow (1996), Gavardashvili (1986), Fleshman (1978), Vinogradov (1976) towards the study of the hydrology of mudflows deserves attention. In the scientific works of Voinich-Sianozhensky et al. (1984; 1977) and Obgadze (2016; 2019), many different mathematical models have been developed that accurately reflect the dynamics of a mudflow caused by a breaking wave. It should also be noted that many interesting imitation models have been developed by the team of Mikhailov and Chernomorets (1984). In mountainous districts, the first hit of a mudflow is taken on by lattice-type structures offered by Kherkheulidze (1984a; 1984b) that release the flow from fractions of large stones and floating trees. After passing through the lattice-type structures, the mudflow is released from large fractions and turns into a water-mud flow. In order to simulate this flow, a mathematical model based on the baro-viscous fluid model offered by Geniev-Gogoladze (1987; 1985) has been developed, where the averaging formula of Voynich-Sianozhencki is used for the particle density, and for the concentration of the solid phase, the diffusion equation is added to the system dynamics equations. In the given article, for the constructed mathematical model, the exact solution of the one-dimensional flow in the mudflow channel is considered. The problem of stratification of the fluid density under equilibrium conditions is discussed. In the riverbed of the Kurmukhi River, for two-dimensional currents, the problem of flow around the bridge pier with an elliptical cross-section is considered. The Rvachev-Obgadze variation method (1982; 1989a; 1989b) is used to solve the streamlined problem.