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In this paper, we theoretically investigate the migration of a surfactant covered droplet in a Poiseuille flow by including the surface viscosities of the droplet. We employ a regular perturbation expansion for low surface Péclet numbers and solve the problem up to a second-order approximation. We represent the drop surface as a two-dimensional homogeneous fluid using the Bousinessq–Scriven law and employ Lamb's general solution to represent the velocity fields inside and outside the droplet. We obtain an expression for the cross-stream migration velocity of the droplet, where the surface viscosities are captured by the Bousinessq numbers for surface shear and surface dilatation. We elucidate the influence of the surface viscosities on the migration characteristics of the droplet and the surfactant redistribution on the droplet surface. Our study sheds light on the importance of including the droplet surface viscosities to accurately predict the migration characteristics of the droplet. 

In this work, we theoretically investigate the motion of an arbitrarily shaped particle in a linear density stratified fluid with weak stratification and negligible inertia. We calculate the hydrodynamic force and torque experienced by the particle using the method of matched asymptotic expansions. We analyse our results for two classes of particles (non-skew and skew) depending on whether the particle possesses a centre of hydrodynamic stress. For both classes, we derive general expressions for the modified resistance tensors in the presence of stratification. We demonstrate the application of our results by considering some specific examples of particles settling in a direction parallel to the density gradient by considering both the limits of high ( $$Pe\gg 1$$ ) and low ( $$Pe\ll 1$$ ) Péclet numbers. We find that presence of stratification causes a slender body to rotate and settle along the broader side due to the contribution of the hydrostatic torque. Our work sheds light on the impact of stratification on the transport of arbitrarily shaped particles in density stratified environments in low-Reynolds-number regimes. 

In this work, we theoretically investigate the swimming velocity of a Taylor swimming sheet immersed in a linearly density-stratified fluid. We use a regular perturbation expansion approach to estimate the swimming velocity up to second order in wave amplitude. We divide our analysis into two regimes of low ( $$\ll O(1)$$ ) and finite Reynolds numbers. We use our solution to understand the effect of stratification on the swimming behaviour of organisms. We find that stratification significantly influences motility characteristics of the swimmer such as the swimming speed, hydrodynamic power expenditure, swimming efficiency and the induced mixing, quantified by mixing efficiency and diapycnal eddy diffusivity. We explore this dependence in detail for both low and finite Reynolds number and elucidate the fundamental insights obtained. We expect our work to shed some light on the importance of stratification in the locomotion of organisms living in density-stratified aquatic environments.