The results of experimental studies of the conditions of loss of stability of the shape of a single dispersed-phase inclusion (droplet and bubble) during its motion in a viscous fluid at low Reynolds numbers are presented. It is shown that in the conditions considered the deformation of an initially spherical inclusion occurs due to the development of the Rayleigh-Taylor instability, as a critical value of the Bond number is attained. It is found that the onset of deformation of the phase interface and the instability mechanism depend strongly on the particle motion regime. A range of critical Reynolds numbers, corresponding to the boundaries of the regions of the Rayleigh-Taylor and Kelvin-Helmholtz instabilities, is determined.An interest in the physics of phenomena occurring in disperse media is associated with their decisive role in a number of applications related to two-phase flows in power plants, the formation of atmospheric precipitations, heat transfer in boiling, cavitation, underwater acoustics, atomization of liquids, flotation, and other engineering processes. In these applications, important factors are the size and the shape of droplets/bubbles, the regimes of their motion, and the mechanisms of instability leading to deformation and fragmentation of the dispersed phase. The processes of dispersed-phase motion, the modes of particle deformation and fragmentation, and the instability mechanisms are some of the classic problems of hydrodynamics, which attracted the attention of researchers from different generations for a long time. To date, a number of fundamental theoretical solutions has been obtained and many results of experimental and numerical studies of these processes have been accumulated. Fairly detailed reviews of both classic and contemporary results of studies of deformation and fragmentation of dispersed inclusions are given in [1][2][3][4][5][6][7][8][9][10].Two instability mechanisms in droplet motion in a flow, namely, the Kelvin-Helmholtz and Rayleigh-Taylor instabilities, are known. In accordance with these mechanisms, the deformation and fragmentation of the dispersed phase occur at certain critical values of the Weber and Bond numbers [11]. In most of wellknown studies, the main criterion responsible for the deformation and fragmentation of a droplet in a flow is the Weber number (the development of the Kelvin-Helmholtz instability). From the Hadamard solution [12] for a spherical dispersed inclusion moving in a viscous fluid at low Reynolds numbers, it follows that the difference of the normal stresses on the particle surface is constant and does not tend to deform it. Based on this, G.K. Batchelor [13] come to the following conclusion. If the viscosities and densities of the dispersed phase and the carrier medium are such that at low Reynolds numbers the inertial forces can be neglected, then there are no limitations on the size of a spherical inclusion of the dispersed phase.The results of theoretical studies [9,11,14] showed that at a critical value of the Bond number Bo cr...