The studies in fluid mechanics performed over the last decade at the Lavrent'ev Institute of Hydrodynamics of the Russian Academy of Sciences were to a large degree extensions of the lines of research that were the choice of Academician M. A. Lavrent'ev, the founder of the Siberian Division. The results of the studies performed up to 1987 are presented in [1]. In this paper, we give a brief review of the advancements in the field of hydromechanics at a new stage where, along with the traditional topics that date back to M. A. Lavrent'ev, new promising lines of research have evolved and been investigated. In addition, some results of research at some other Institutes of the Siberian Division of the Russian Academy of Sciences whose lines of investigation are intimately adjacent to those of the Lavrent'ev Institute of Hydrodynamics are also touched upon in this review.Group Analysis of the Equations of Fluid Dynamics.(1) This line of investigation, as applied to the construction of exact solutions of the equations of motion for gases and fluids, was developed in the course of realization of the SUBMODELS program, which was first presented in [2]. This program is based on the fact that many "large" mathematical models that describe physical processes in the form of a system of differential equations E have high symmetry, namely, they admit a fairly wide continuous group G of transformations for the subspace of independent and dependent variables.The object of the SUBMODELS program is to reach the limit of the possibilities involved in such symmetry to find the class of exact solutions of the system E. Although, in the world literature, there are many examples of using symmetry properties for this purpose, the problem of reaching the limit of these properties was first posed in the SUBMODELS program.The idea of the program is based on the fa~ct that any subgroup H C G is a source of exact partial solutions. The search for these solutions reduces to a submodel --the factor system E/H. The latter is simplified compared with E, for example, by reducing the dimension for independent variables. Therefore, if the system E admits the known basic (widest) group G, all possible subgroups of the group G will act as H. Thus, the solution of the purely algebraic problem of compiling the list of all subgroups of the given group G plays an important role in the SUBMODELS program. Actually, it suffices to list the subgroups H C G up to the similarity in G, which is accomplished by the internal automorphisms of the group G. The complete list of unlike subgroups H C G is called the optimal system of subgroups and denoted by OG.The passage to the equations of the submodel E/H consists in establishing additional relations among the invariants of the group H that allows one to determine the desired functions and analyze the compatibility of these relations with the equations E.A detailed description of the SUBMODELS program and the main algorithms of its realization are given in [3]. By virtue of the well-known correspondence between th...
