Preferential flow penetration in a network of identical channels

It is intuitive to expect a fluid to evenly penetrate two identical daughter channels supplied by the same source when these channels experience the same back pressure. In this work, we show that this might not be the case. We consider the displacement of one fluid by another in a network composed of two identical channels that linearly vary in radius. When surface tension is important, this simple shape variation can cause the fluid to primarily enter only one of the two branches, i.e., we find that the even or uniform fluid penetration into the network is not always stable. Displacement flows in small channels exhibit an interplay between viscous forces and capillary forces. We analyze the dynamics of our system, which features a hydrodynamic instability that is governed by the viscosities of the fluids, their wetting properties, the flow penetration speed, and the gradient characterizing the channel shape.