Topological transitions in core-annular flow of oil and water are described. Drops of one liquid are suspended in the second liquid and transported through a pipeline under a pressure gradient. We examine the merging of an array of drops in this two-fluid flow. Larger drops are found to merge and the flow undergoes a topological transition to core-annular flow. In core-annular flow of two liquids, one liquid lies at the core and the other surrounds it. For the case of large oil bubbles in water, a steady core-annular flow with bamboo waves on the interface is achieved. For the case of large water bubbles in oil, the ensuing core-annular flow has continually breaking waves at the interface.