The hydrodynamics of disperse liquid mediaattracts increased attention, first of all, owing to theirnumerous applications in various industries (chemical, medical, pharmaceutical, processing of polymers,etc.). The theoretical analysis of such systems is basedon studying the dynamic behavior of an isolated dropin a flow for determining the strain, orientation, andstability of the drop in various flow modes. To date, anextensive body of experimental data on the mechanisms of deformation and breakup of homogeneousdrops has been accumulated [1–6]. However, thehydrodynamic behavior of composite drops containing one or several internal inclusions (cores) remainslittle studied. At the same time, such objects have adiversity of applications: from increasing the impactresistance of mixed polymer composites to targeteddrug delivery [5, 6]. Therefore, the comprehensiveinvestigation of composite drops is of fundamental andpractical interest.Studying the hydrodynamic behavior of such dropsis complicated by a number of difficulties related to thenecessity of taking into account a large number ofparameters and also the dynamic changes in theshapes of both inner and outer interfaces. In the simplest case, a composite drop consists of a single coreand a shell, which are immersed in a dispersion liquid(Fig. 1). In this case, features of the deformation andshaping of the drop are determined by the relative values of the surface tensions at the interfaces betweenthe components of the medium, their viscosities, andalso the core size as compared to the shell size.It was previously shown that, in a shear flow, a corewith low viscosity and low surface tension takes theshape of an extended dumbbell rotating in the direction of the flow [7]. In this work, we studied the hydrodynamic behavior of a twodimensional compositedrop in which the viscosity of the core is much higherthan that of the shell and also the core–shell interfacial tension significantly increases the surface tensionat the interface between the shell and the dispersionliquid. In such a formulation, the main contribution tothe total strain and orientation of the composite dropIs made by the transformation of the shape of the shell.The deformation and dynamic structuring of this typeare also characteristic of swollen microgels immersedin a thermodynamically incompatible liquid [8–10].Comparison of the hydrodynamic behaviors of composite and homogeneous drops enabled one to understand the effect of flow perturbations caused by thepresence of the viscous core on the deformationbehavior of the composite drop.