We numerically study the dynamics of a three-dimensional contractile fluid droplet in the bulk and under confinement. We show that varying activity leads to a variety of shapes and motile regimes whose motion is driven by an interplay between spontaneous flows and elasticity. In the bulk the droplet self-propels unidirectionally, acquiring either an almost spherical shape at intermediate activity or a peanut-like geometry for larger values. Under confinement, the droplet exhibits a previously unreported oscillating dynamics characterized by periodic hits against opposite walls of a microchannel while moving forward. These results could be of interest for the study of artificial microswimmers and their biological analogs, such as living cells.
Introduction. -Active droplets are a bio-inspired class of soft material where self-propulsion stems from an energy supply located within the droplets or in their surroundings [1][2][3][4]. A particular sub-class is represented by active gel droplets where the driving force is generated by a hierarchically-assembled active liquid crystal of biological origin, such as suspensions of microtubules and kinesin or actomyosin solutions [5][6][7]. The former are an example of extensile materials, which pull the fluid inwards along the equator and emit it axially, while the latter are contractile where the fluid dynamics is reversed. These systems are of particular interest for studying the mechanics of cell locomotion [8][9][10] and for the design of artificial microswimmers, potentially useful in material science [11,12].
While a large body of theoretical studies have been dedicated to modeling the physics of active gel droplets in two [13][14][15][16][17][18][19] and three dimensional setups [8][9][10][20][21][22][23], much less is known about the physics under confinement [24][25][26], especially in three dimensions [27]. In this work, we move a step further and investigate the hydrodynamics of a contractile fluid droplet confined within a three dimensional microchannel, a geometry that could potentially reproduce the cell migration in a realistic (or physiological) environment. Following previous works [24,25], we use a continuum approach incorporating hydrodynam-ics, where the advection-relaxation equations govern the evolution of a phase field, accounting for active material concentration, and a vector field describing its orientation, while the Navier-Stokes equations control the global velocity. Simulations in bulk, while confirming previous shapes and dynamic regimes [8], unveil a novel motile state where a peanut-like droplet hosting an integer topological defect (of charge 1 or -1) self-propels. Under mild confinement, at sufficiently high activity a highly stretched droplet is found to exhibit a periodic motion characterized by with repeated bumps against opposite walls. This dynamics is suppressed under a high confinement regime.
Methods. -The hydrodynamic model used in this work follows those presented, for example, in [20,[24][25][26]. We consider a fluid mixture consisting of an active gel droplet immersed in a passive Newtonian fluid of constant density ρ. The droplet phase is described by a scalar field ϕ(r, t), positive in the droplet and zero outside, while the ordering properties of the active material contained in the droplet are captured by a polar liquid crystal vector p(r, t), representing a coarse-grained average of the orientation of the internal constituents (e.g an actin filament). Finally, the global fluid velocity (of droplet and solvent) is tracked by a vector field v(r, t).
The dynamics of the phase field ϕ is governed by an p-1 arXiv:2602.16408v1 [cond-mat.soft] 18 Feb 2026 advection-relaxation equation
where M is the mobility and µ = δF δϕ is the chemical potential, being F the free energy (see later in the text).
The equation of the polarization field p reads
where s = χD-Ω, with D = (W+W T )/2 and Ω = (W-W T )/2 being the symmetric and antisymmetric parts of the velocity gradient tensor W αβ = ∂ β v α (Greek indices denote Cartesian components). Also, the parameter χ is a shape factor, positive for rod-like particles and negative for disk-like ones, Γ is the rotational viscosity and h = δF δp is the molecular field governing the relaxation of the liquid crystal.
Fluid density ρ and velocity v obey the continuity and Navier Stokes equations which, in the limit of incompressible fluid, are
where P is the isotropic fluid pressure. On the right hand side of Eq.( 4)
, where ζ is the activity (positive for extensile materials and negative for contractile ones) and d is the dimension of the system. Moreover, σ passive αβ is the sum of three contributions, the dissipative σ viscous
where η is the viscosity, the elastic
, where κ is the elastic constant of the liquid crystal and the interfacial
, where f is the free energy density.
The free energy F used to compute the thermodynamic forces (chemical potential, molecular field and stress tensor) appearing in Eqs.( 1)-( 2)-(4) can be written as an expansion up to the fourth order of ϕ and p capturing the bulk properties of the active gel plus gradient terms describing interfacial features of the fluid mixture and local distortions of the liquid crystal [28,29]. It is given by
where a and k are positive constants controlling surface tension σ = 8ak 9 and interface thickness ξ = 2k a . The first term of Eq.( 5) allows for the coexistence of two minima, ϕ = ϕ 0 inside the droplet and ϕ = 0 outside, while the second one penalizes the formation of fluid interfaces. The other terms represent the liquid crystal contribution, where the first two, multiplied by the constant α, describe the bulk properties (with ϕ cr = ϕ 0 /2 critical
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