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In this article, we apply the diffuse-interface model [developed by Shah and Yuan (2011) [21]] for collision and coalescence of two bubbles in a linear shear flow. The governing equations consist of a system of coupled nonlinear partial differential equations for conservation of mass, momentum and phase transport. In the two-phase flow, the diffuse-interface model relaxes certain numerical difficulties for tracking the moving interface. An artificial compressibility based numerical scheme is implemented to study the effects of surface tension on bubbles coalescence and separation. We found the critical value of the surface tension coefficient and observed that lowering the surface tension coefficient from the critical value prevent bubbles to coalesce.