Sufficient Energy Estimates of Stability of Unsteady Combined Shear Flows in a Cylindrical Layer

The time evolution of the three-dimensional pattern of initial disturbances imposed on an unsteady flow, which is a combination of one-dimensional $$r\theta $$ - and $$rz$$ -shears of Newtonian viscous fluid in a cylindrical layer infinite in length, is studied. The annular and axial velocities of both cylindrical boundaries, which do not vary in the disturbed motion, are specified. The formulation of the linearized problem in terms of variations in the velocities, the strain rates, the pressure, and the stress deviator is given. To analyze this problem, the method of integral relations is developed. The method makes it possible to obtain sufficient estimates of the development of disturbances in the Hilbert space H2, in particular, Lyapunov stability and asymptotic stability. These estimates include both the kinematic parameters of main flow and harmonics of the annular disturbances and wavenumbers of axial disturbances. For the steady-state main flow in the layer, exponential estimates of stability take place.