The motion of long drops in rectangular microchannels at low capillary numbers

We study the motion of a long drop in a rectangular microchannel in the limit the capillary number Ca-> 0 (Ca = μU/σ, where U is the constant drop velocity, μ is the carrier-liquid viscosity, and σ is the interfacial tension). In this limit, the drop has two end caps connected by a long column, which is surrounded by thin films on the microchannel wall and by menisci along the microchannel corners. Integral axial force balances relate the carrier-liquid pressure gradient to the drop-fluid pressure gradient and the contact-line drag, which is the same as that for a long bubble (known) if the viscosity ratio λ « Ca^-1/3 and λ « L, where λ = μ*/μ and μ* is the drop viscosity, and L is the dimensionless drop length. The two pressure gradients also drive unidirectional flows in the drop and in the corner channels along the long middle column. These coupled flows are solved by a finite-element method to yield the pressure gradients for λ = 0 to 100 and various microchannel aspect ratios. We find that in the limit LCa^1/3 -> 0, the contact-line drag dominates and the carrier liquid bypasses the drop through the corner channels alongside the drop. For LCa^1/3 » 1, the contact-line drag is negligible and the corner fluid is stationary. Thus, the drop moves as a leaky piston.