Effect of the small forced oscillations on the stability of the steady motion of fluid (Q794837)

The author investigates the effect of small forced oscillations on the steady state motion of a viscous incompressible fluid. From the usual set of fluid dynamic equations, he looks for solution to the velocity of the fluid in the form \(\vec v=\vec v_ 0+\epsilon \vec v_ 1(t)+\vec u(t),\) where \(\vec v_ 0\) is the steady flow velocity, \(\epsilon \vec v_ 1(t)\) is the periodic component, representing external perturbation and \(\epsilon\) is a small parameter, while \(\vec u(t)\) is the time-dependent component of the velocity without the influence of the external force, satisfying the linearized hydrodynamic equation (*) \(d\vec u/dt=A_ 0\vec u+\nu K\vec u+\epsilon F(t)\vec u,\) where the operator \(A_ 0\) satisfies \(A_ 0\vec u=-P[(\vec u\cdot \nabla)\vec v_ 0+(\vec v_ 0\cdot \nabla)\vec u],\) and P is the orthogonal projection operator of the vector space \(L_ 2\) onto space H obtained from the closure property of the finite set which is defined in the three-dimensional region \(\Omega\) in which the viscous incompressible fluid moves. In (*), \(K\vec u=P\Delta \vec u\) and \(\nu\) is the eigenfrequency. Using perturbation and complex variable techniques, the author obtains an expression for the second order harmonic frequency \(\nu_ 2\) of the flow, while \(\nu_ 1=0\). The merit of this paper lies in the operator methodology in finding \(\nu_ 2\) via the stated process. [This is an abridged version, the complete review is available on demand.]