  The process of one fluid pushing another is universally common while
involving complex interfacial instabilities. Particularly, occurring in a
myriad of natural and industrial processes, wavy fingering patterns frequently
emerge when a less viscous fluid pushes another more viscous one, such as water
invading oil, in a porous medium. Such finger-shaped interfaces producing
partial displacement significantly affect the efficiency of numerous
applications, for example, chromatography, printing devices, coating flows,
oil-well cementing, as well as large-scale technologies of groundwater and
enhanced oil recovery (EOR). This classical viscous fingering instability is
notoriously difficult to control because the two fluids' viscosity or mobility
ratio is often fixed and yet the predominant drive of the instability. Although
some strategies have been recently revealed for simple fluids of constant
viscosity, the feasibility of controlling the fundamental viscous fingering
instability for omnipresent complex fluids has not been established. Here, we
demonstrate how to control a common complex fluid (of a power-law fluid with a
yield-stress) using a narrow tapered cell theoretically and experimentally.
