  This paper is an attempt to study the effects of surface topography on the
flow of a droplet (or a bubble) in a low Reynolds number flow regime.
Multiphase flows through a constricted passage find many interesting
applications in chemistry and biology. The main parameters which determine the
flow properties such as flow rate and pressure drop, and govern the complex
multiphase phenomena such as drop coalescence, break-up and snap-off in a
straight channel flow are the viscosity ratio, droplet size and ratio of the
viscous forces to the surface tension forces (denoted by Capillary number). But
in flow through a constricted passage, in addition to the above-mentioned
parameters, various other geometric parameters such as constriction ratio,
length and shape of the constriction, phase angle, and spacing between the
constrictions also start playing an important role. Most of the studies done on
the problem of drop flow through a constricted passage have aimed to understand
the role of physical parameters, with some studies extending their analysis to
understand the variation of one or two geometric parameters. But no study could
be found which explicitly evaluates the role of surface topography. An attempt
has been made to unify the current literature as well as analyze the effect of
the geometric parameters by understanding the physics and mechanisms involved.
The non-dimensional numbers which govern this problem are then identified using
the scaling analysis.
