Janus droplet motion in an external flow

We consider a hydrodynamics of a Janus droplet, which consists of two hemispherical domains occupied by different liquids. The simplest problem, a Janus droplet in a uniform at infinity flow, is analyzed. The interfaces are assumed weakly deformable. It is shown, that the velocity field can be represented as a superposition of two fields: for internal surface (i) normal and (ii) parallel to the external flow. In case (i) the flow is axisymmetric; the force imposed on the droplet is found by summation of the series. It is worth noting, that even for equal internal viscosities, the solution for a simple drop [1,2] is not reproduced. Indeed, the internal impermeable interface prohibits a flow of Hadamard-Rybczynski type. Weak deformation of the interfaces is found; it is shown that deformation of the internal surface is larger than that of the drop surface. In case (ii) expansion in Lamb's functions is applied; both the torque and force are found. It is also shown that stable configuration of a torque-free droplet corresponds to case (i) with less viscous fluid on the upstream face.\\[4pt] [1] J. S. Hadamard, Compt. Rend. Acad. Sci. 152, 1735 (1911).\\[0pt] [2] W. Rybczynski, Bull. Acad. Sci. Cracovie (ser. A), 40 (1911).