From Helmholtz's decomposition,

$v=v_{\scriptscriptstyle IR} +v_{\scriptscriptstyle R} $

where $\nabla\times v_{IR} =0$ and $\nabla\cdot v_R=0$

when apply this to the linearized Navier-Stokes equation,

i, imaginary unit

k, reduced frequency

$\gamma$, square root of the Prandtl number

s, shear wave number

$\xi$, viscosity ratio

it splits into two equations, namely,

$iv_{\scriptscriptstyle IR}-{1\over s^2}({4\over 3}+\xi)\nabla^2v_{\scriptscriptstyle IR}=-{1\over k\gamma}\nabla p$

$iv_{\scriptscriptstyle R}-{1\over s^2} \nabla ^2v_{\scriptscriptstyle R}=0$

now, just consider the rotational velocity. Does the following system is over-determined? (3 components of $v_{\scriptscriptstyle R}$, 4 equations)

$\left\{\begin{array}{cols} iv_{\scriptscriptstyle R}-{1\over s^2} \nabla ^2v_{\scriptscriptstyle R}=0 \\ \nabla\cdot v_R=0 \end{array} \right. $