Beads, bubbles and drops in microchannels: stability of centred position and equilibrium velocity

Abstract

Understanding and predicting the dynamics of dispersed micro-objects in microfluidics is crucial in numerous natural, industrial and technological situations. In this paper, we experimentally characterized the equilibrium velocity $V$ and lateral position $\varepsilon$ of various dispersed micro-objects, such as beads, bubbles and drops, in a cylindrical microchannel over an unprecedentedly wide range of parameters. By varying the dimensionless object size ( $d \in [0.1; 1]$ ), the viscosity ratio ( $\lambda \in [10^{-2}; \infty [$ ), the density ratio ( $\varphi \in [10^{-3}; 2]$ ), the Reynolds number ( $Re \in [10^{-2}; 10^2]$ ) and the capillary number ( $Ca \in [10^{-3}; 0.3]$ ), we offer an exhaustive parametric study exploring various dynamics from the non-deformable viscous regime to the deformable inertial regime, thus enabling us to highlight the sole and combined roles of inertia and capillary effects on lateral migration. Experiments are compared and agree well with a steady three-dimensional Navier–Stokes model for incompressible two-phase fluids, including the effects of inertia and possible interfacial deformations. This model enables us to propose a correlation for the object velocity $V$ as functions of $d$ , $\varepsilon$ and $\lambda$ , obtained in the ${Re}={Ca}=0$ limit, but valid for a larger range of values of ${Re}$ and ${Ca}$ delimited by the validity of the linear regime. Next, we present stability maps for the centred position showing that non-deformable objects dominated by inertial effects are only stable if large enough, typically for $d \gtrsim 0.7$ , whereas deformable objects dominated by capillary effects can be stable for much smaller sizes, provided the viscosity ratio is outside the range $0.7 \lesssim \lambda \lesssim 10$ , in which deformability also plays a destabilizing effect, as for inertia.

Flow in fetoplacental-like microvessels in vitro enhances perfusion, barrier function, and matrix stability

“…Note that velocities and shear stress predictions are obtained using 2D segmented projections and do not account for the exact spatial complexity of 3D vessels nor the wall effects or size-dependent effects of beads. It was reported previously that the presence of common additives in the media ( 59 ), tracer beads size, and their properties ( 60 ), such as deformability, contribute to elastic no-slip boundaries at the fluid interfaces and influence the velocities.…”

Flow in fetoplacental microvessels in vitro enhances perfusion, barrier function, and matrix stability

“…Note that velocities and shear stress predictions are obtained using 2D segmented projections and do not account for the exact spatial complexity of 3D vessels, nor the wall effects or size-dependent effects of beads. It was reported previously that the presence of common additives in the media [64], tracer beads size, and their properties [65], such as deformability, contribute to elastic no-slip boundaries at the fluid interfaces influence the velocities.…”

Influence of immersed particles on the stability of the liquid–liquid interface in a two-layer channel flow

“…After using the divergence theorem and rearranging terms, as well as considering that the normal vector to the generatrix is , we obtain where the surface integral at the generatrix has been written, by virtue of the divergence theorem (B1 b ), as . For more details of this procedure, readers are referred to Rivero-Rodriguez & Scheid (2018), Ruiz-Martín et al (2022 b ) and Cappello et al (2023). The right-hand side of (2.10 b ) can be written at using the divergence theorem (B1 b ) to give where is the rotation tensor, and and are the vectors tangential to the unperturbed and perturbed interface , respectively. Finally, using (B8) and (B9), and considering that they are valid for any arbitrary , (2.10 b ) can be rewritten as Similarly, the kinematic condition (2.9) can be written at the unperturbed interface by integrating (2.1) for liquid 2 on the same generated volume, yielding …”

“…After using the divergence theorem and rearranging terms, as well as considering that the normal vector to the generatrix is , we obtain where the surface integral at the generatrix has been written, by virtue of the divergence theorem (B1 b ), as . For more details of this procedure, readers are referred to Rivero-Rodriguez & Scheid (2018), Ruiz-Martín et al (2022 b ) and Cappello et al (2023).…”