Through a two-dimensional (2D) boundary element method, this paper investigates the asymmetrical deformations of multiple emulsions with different internal structures under rotational shear flows induced by the rotation of the inner cylinder in a two-concentric-cylinder microfluidic system like a rotational viscometer. As the total shear stress (including components and interfacial contribution) of simple emulsions in a shear flow is severely affected by the deformation of the droplets, the deformation parameter D is chosen as the bridge to connect the internal structures of multiple emulsions with their apparent viscosity. Thus, the relations among the internal structures, the deformation of the globule, the torque of the inner cylinder and the apparent viscosity of multiple emulsions are discussed qualitatively in order to disclose the effects of the internal structure on the apparent viscosity. As for the small deformation of multiple emulsions, the inner droplets have both suppressing and enhancing effects on the deformation of the globule, and they will rotate along with the inner circulation which is like a big vortex. Generally speaking, the larger the deformation is, the larger the torque is, which means that the apparent viscosity of multiple emulsions increases along with the increase of the deformation of the globule. As for the large deformation (without breakup) of multiple emulsions, the situations are much more complicated and depend on the complex interaction between the inner droplets and the outermost interface of the globule. (C) 2015 Elsevier Ltd. All rights reserved.