A two-dimensional boundary integral method is developed to investigate the rheology of the multiple-emulsion globules with orderly structures up to n layers and up to m(i) droplets in the i-th layer in microchannels of various geometries. Particularly, the globule rheology of three type of multiple emulsions (ME), (i.e., concentric ME with multiple layers, double emulsions containing multiple inner droplets and ME with asymmetric internal structures), under a modest extensional flow in a cross-slot is studied numerically. The displacement, deformation and collision of inner droplets will generate resistance of different strengths to inner circulations in the shell of double emulsions, which will cause the difference of the outer shear subjected by the globules with various internal structures and vary their viscosities. Slow movement of the inner droplets might induce the coalescence among them, and induce the merging of the inner and outer interfaces, which might be useful in their controlled coalescence for internal chemical reactions and in their controlled release for drug delivery. (c) 2013 Elsevier Ltd. All rights reserved.