Viscous fingering is an ubiquitous instability that occurs when a less viscous fluid displaces a more viscous fluid in a porous medium. The interface between the two fluids starts to deform, and finger-like patterns emerge and grow. This phenomenon can either increases the mixing in porous media, which is incredibly difficult because of the absence of turbulence that can actively stir the flow or be dramatic to some processes.
The typical example is secondary oil recovery, for which fingering from the injected aqueous solution pushing the more viscous oil in underground reservoirs of porous rocks reduces the sweep efficiency severely.
In nature, chemical compounds commonly mix each other. One of the elemental material in fluid mechanics, air, is predominantly composed of two components: nitrogen ($\text{N}_2$, $\approx 78\%$) and oxygen ($\text{0}_2$, $\approx 21\%$). Mass transfer seems at first sight not something too complicated: a species should move down its concentration gradient. This is known as the first Fick’s law, which can be written for two components as
$$ \mathbf{J}_1=-\mathcal{D}\frac{dc_1}{dz}, ~ \text{and } J_2=-J_1$$