Bifurcation Dynamics of a Particle-Encapsulating Droplet in Shear Flow

Abstract

To understand the behavior of composite fluid particles such as nucleated cells and double emulsions in flow, we study a finite-size particle encapsulated in a deforming droplet under shear flow as a model system. In addition to its concentric particle-droplet configuration, we numerically explore other eccentric and time-periodic equilibrium solutions, which emerge spontaneously via supercritical pitchfork and Hopf bifurcations. We present the loci of these solutions around the codimension-two point. We adopt a dynamic system approach to model and characterize the coupled behavior of the two bifurcations. By exploring the flow fields and hydrodynamic forces in detail, we identify the role of hydrodynamic particle-droplet interaction which gives rise to these bifurcations. DOI: 10.1103/PhysRevLett.119.064502 Droplets, capsules, and vesicles in flow often exhibit interestingly rich dynamics even in the linear shear flow [1][2][3][4][5][6][7][8][9]. Despite substantial work on the dynamics of these soft systems enclosing homogeneous fluids, limited effort has been directed to studying their behavior when they include an internal structure. However, such a configuration is common in nature and engineering applications: cells like leukocytes and megakaryocytes contain a nucleus of up to 50%-80% of themselves in volume [10]; double emulsions playing an important role in chemical and pharmaceutical engineering are featured with a coreshell geometry [11][12][13]; droplet-based encapsulation for high-throughput biological assays utilizes droplets as microchambers to compartment cells for analysis at the single-cell level, where the cell size can be comparable to the droplet size in certain applications [14][15][16].These fluid particles are characterized by complex hydrodynamic interactions between the internal structures and the external interface. Few works conducted for nucleated model cells in shear [17][18][19][20] all assumed their compound structures to be concentric, preserving the rotational symmetry of order 2 (C2) about the y axis and reflection symmetry about the y ¼ 0 shear plane [see Fig. 1(a)]. The symmetries do hold for a single shear-driven deformable particle that attains a steady ellipsoidal shape undergoing tank-treading motion [21][22][23]]. Yet, they are not guaranteed in the presence of an internal structure.In this Letter we focus on the stability of the concentricity of composite fluid particles. By considering a droplet encaging a spherical particle as a model system, we formulate the following questions: will the composite structures remain concentric? How does the dynamics depend on interfacial tension and particle size? What is the role of the hydrodynamic interaction?We begin our discussion by presenting 3D hydrodynamic simulations of a compound particle droplet subjected to unbounded shear U ∞ ¼ G · x, in the creeping flow regime, where the only nonzero component G xz ¼ _ γ represents the shear rate [ Fig. 1(a)]. The incompressible Stokes equations are solved by a boundary integral meth...

Impinging planar jets: hysteretic behaviour and origin of the self-sustained oscillations

“…The predicted normal form describes the nonlinear interactions between global modes A (steady) and B (oscillating) and reduces the full dynamics to a low-dimensional model, as typical of WNL formulations. For codimensions larger than one, as in the present case, which displays a codimension-2 point, the normal form often predicts successfully the system behaviour (Crawford & Knobloch 1991;Meliga et al 2009a;Zhu & Gallaire 2017). Indeed, a quantitative comparison of our WNL results against DNS, in terms of oscillation frequency and mode amplitudes, confirms the validity of the WNL analysis and, in particular, the existence of a narrow region of hysteresis for AR < AR C 2 and Re BA < Re < Re SHB .…”

“…For codimensions larger than one, as in the present case, which displays a codimension-2 point, the normal form often predicts successfully the system behaviour (Crawford & Knobloch 1991; Meliga et al. 2009 a ; Zhu & Gallaire 2017). Indeed, a quantitative comparison of our WNL results against DNS, in terms of oscillation frequency and mode amplitudes, confirms the validity of the WNL analysis and, in particular, the existence of a narrow region of hysteresis for and .…”

Creeping motion of a solid particle inside a spherical elastic cavity

“…In addition, a combination of multipole expansion and Faxén's theorem has been used by Zia and collaborators [41,42], providing the elements of the grand mobility tensor of finitesized particles moving inside a rigid spherical cavity. Additional works addressed the low-Reynolds-number locomotion inside a viscous drop [43][44][45], or the dynamics of a particleencapsulating droplet in flow [46,47]. arXiv:1802.00353v2 [physics.flu-dyn] 14 Aug 2018 Despite enormous studies on particle motion inside a rigid cavity or a viscous drop, to the best of our knowledge, no works have been yet conducted to investigate particle motion inside a deformable elastic cavity.…”

Deformation and breakup of a compound droplet in three-dimensional oscillatory shear flow

“…Tiribocchi et al (2020) used a lattice Boltzmann and finite difference hybrid method to investigate the steady states of a non-coalescing multi-core compound droplet under simple shear flow, and found that for a two-core compound droplet, the inner droplets periodically rotate around the mass center of the outer one, exhibiting an oscillatory steady state. Zhu and Gallaire (2017) numerically studied the dynamics of a particleencapsulated droplet under simple shear and discovered two new bifurcation scenarios that the particle executes spanwise migration and/or in-plane orbiting motion. However, the imposed external flows in the above studies are all limited to the time-independent cases.…”