Upper Estimates of the Perturbation Growth under TriaxialSpreading–Drain in Infinite Viscous Space

The time evolution of a system of small perturbations imposed on a triaxial homogeneous spreading–drain in an infinite three-dimensional space of a Newtonian incompressible fluid is investigated. In the first part of the paper, it is assumed that the main motion is stationary and the velocity field is defined by only two constants. In this case, the linearized problem for the velocity and pressure perturbations is reduced to a spectral problem in which the real part of the spectral parameter is related to the nature of the exponential decay or growth of the initial perturbations. Based on the method of integral relations for quadratic functionals, an upper bound for this parameter is obtained. In the second part of the paper, a more general case of unsteady triaxial spreading–drain is considered. For the perturbation growth, we derive an upper integral bound that includes a time function completely determined by the velocity field of the main fluid flow.