On instability of a viscous jet with bi-quadratic velocity profile (Q2767633)

Using the method of small perturbations in linear formulation, the author studies the hydrodynamic stability of a circular cylindrical jet of viscous fluid. The influence of surrounding medium is neglected. More precisely, the distribution of the velocity in axial direction is taken in the form \(W(r) =-\varepsilon r^4 +2\varepsilon r^2 +1\) which satisfies conditions \(W'(0)=W'(1)=0\) both at the axis and at the free surface (\(\varepsilon\) is small parameter). The author studies stability of this flow under axisymmetric perturbations of a special kind. After linearization, the system of Navier-Stokes equations is reduced to a fourth-order differential equation. The equation is solved by Frobenius technique (the method of power series). The analysis of solution shows that viscosity can play a destabilizing role for some parameters of velocity profile.