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The axisymmetric unsteady two-phase flow problem is explored. The flow domain is defined by two co-axial circular cylinders and is axial symmetric. The Dirichlet-type boundary condition is used on the inner cylindrical surface, whereas the Robin-type boundary condition is used on the outer cylindrical surface. The velocities are computed analytically using a new form of the Weber transform that is suited for these boundary conditions. The effect of the slip parameter on velocities is investigated using numerical simulations and graphical representations. The studied problem is new in the literature because there do not exist any analytical studies regarding the problems with boundary conditions of Dirichlet type on the inner cylinder (the no-slip on the wall) and boundary conditions of Robin type on the outer cylinder (the mixture slipping on the wall). A new integral transform of Weber type has been employed to determine analytical solutions for such problems, together with the Laplace transform. The studied problem could generate analytical solutions for more two-phase flow problems in annular domains since the translational motions of the inner cylinder and the outer cylinder are given by arbitrary functions of the time t.