  A surfactant-laden droplet of one fluid dispersed in another immiscible fluid
serves as an artificial model system capable of mimicking microbial swimmers.
Either an interfacial chemical reaction or the process of solubilization
generates gradients in interfacial tension resulting in a Marangoni flow. The
resulting fluid flow propels the droplet toward a region of lower interfacial
tension. The advective transport of surfactants sustains the active propulsion
of these droplets. In these systems, the local interfacial tension is affected
by the interfacial reaction kinetics as well as convection and diffusion
induced concentration gradients. The migration of such a surfactant-laden
viscous droplet undergoing an interfacial reaction, suspended in a background
Poiseuille flow is investigated. The focus is specifically on the role of the
surface reaction that generates a non-uniform interfacial coverage of the
surfactant, which in turn dictates the migration velocity of the droplet in the
background flow. Assuming negligible interface deformation and fluid inertia,
the Lorentz reciprocal theorem is used to analytically determine the migration
velocity of the droplet using regular perturbation expansion in terms of the
surface P\'eclet number. We show that the presence of interfacial reaction
affects the magnitude of both stream-wise and cross-stream migration velocity
of the droplet in a background Poiseuille flow. We conclude that the
stream-wise migration velocity is not of sufficient strength to exhibit
positive rheotaxis as observed in recent experimental observations. Additional
effects such as the hydrodynamic interactions with the adjacent wall may be
essential to capture the same.
