Numerical study of bifurcations in spiral fluid flow with free boundary

The plane stationary flows of viscous incompressible fluid with spiral lines of flow are considered. This class of solutions was discovered by G. Gamel in 1917. Later the solutions of Navier-Stokes equations with spiral lines of flow were explored by V. Pukhnachev, who observed that such solutions could be obtained using the group properties of Navier-Stokes equations and considered as the fluids with free boundaries. In this study, we solve the problem of determining the velocity of fluid flow in a region that is bounded either by a rigid fixed wall and a free boundary or by two free boundaries. The boundaries of the region have the form of logarithmic spirals. The flow rate of fluid through the origin of coordinates is given. The purpose of our work is to determine the relationship between the number of solutions and the parameter values. The parameters are dependent on the Reynolds number and circulation. The bifurcation lines separating the parameter domain into five parts were found analytically. Application of numerical methods makes it possible to find the solutions corresponding to each part. It is shown that multiple (from one to seven) solutions can exist.