We apply the blinking rotlet model to the analysis of stirring in a Stokes flow in rectangular containers. Specifically, we construct the rigorous analytical solution for the two-dimensional bi-harmonic equation in a rectangular domain $|x| \leq a$, $|y| \leq b$ with a rotlet placed at point $(0,c)$. The solution shows that for a certain position of the rotlet $c_0$ which depends on $a$ and $b$, the flow has a stagnation point $(0,-c_0)$ symmetrically placed inside the rectangle. Thus the blinking rotlet model can be constructed for the rectangle in which the rotlet that is off does not disturb the flow. This model seems preferable to the classical blinking vortex flow when discussing chaotic advection by the Stokes flow. When the velocity field is accurately obtained, the detailed study of stirring any passive blob can be done by the adaptive boundary tracking algorithm. Quantitative measurements of stirring are developed and they provide the estimates for the goodness of mixing according to Danckwerts.