skip to main content


This content will become publicly available on August 1, 2024

Title: Analytical theory for a droplet squeezing through a circular pore in creeping flows under constant pressures

We derived equations and closed-form solutions of transit time for a viscous droplet squeezing through a small circular pore with a finite length at microscale under constant pressures. Our analyses were motivated by the vital processes of biological cells squeezing through small pores in blood vessels and sinusoids and droplets squeezing through pores in microfluidics. First, we derived ordinary differential equations (ODEs) of a droplet squeezing through a circular pore by combining Sampson flow, Poiseuille flow, and Young–Laplace equations and took into account the lubrication layer between the droplet and the pore wall. Second, for droplets wetting the wall with small surface tension, we derived the closed-form solutions of transit time. For droplets with finite surface tension, we solved the original ODEs numerically to predict the transit time. After validations against experiments and finite element simulations, we studied the effects of pressure, viscosity, pore/droplet dimensions, and surface tension on the transit time. We found that the transit time is inversely linearly proportional to pressure when the surface tension is low compared to the critical surface tension for preventing the droplet to pass and becomes nonlinear when it approaches the critical tension. Remarkably, we showed that when a fixed percentage of surface tension to critical tension is applied, the transit time is always inversely linearly proportional to pressure, and the dependence of transit time on surface tension is nonmonotonic. Our results provided a quick way of quantitative calculations of transit time for designing droplet microfluidics and understanding cells passing through constrictions.

 
more » « less
Award ID(s):
1841509 1841473 2210366
NSF-PAR ID:
10490317
Author(s) / Creator(s):
; ; ; ; ; ; ;
Publisher / Repository:
AIP Publishing
Date Published:
Journal Name:
Physics of Fluids
Volume:
35
Issue:
8
ISSN:
1070-6631
Subject(s) / Keyword(s):
["theory, droplets"]
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Soft objects squeezing through small apertures are crucial for many in vivo and in vitro processes. Red blood cell transit time through splenic inter-endothelial slits (IESs) plays a crucial role in blood filtration and disease progression, while droplet velocity through constrictions in microfluidic devices is important for effective manipulation and separation processes. As these transit phenomena are not well understood, we sought to establish analytical and numerical solutions of viscous droplet transit through a rectangular slit. This study extends from our former theory of a circular pore because a rectangular slit is more realistic in many physiological and engineering applications. Here, we derived the ordinary differential equations (ODEs) of a droplet passing through a slit by combining planar Poiseuille flow, the Young–Laplace equations, and modifying them to consider the lubrication layer between the droplet and the slit wall. Compared to the pore case, we used the Roscoe solution instead of the Sampson one to account for the flow entering and exiting a rectangular slit. When the surface tension and lubrication layer were negligible, we derived the closed-form solutions of transit time. When the surface tension and lubrication layer were finite, the ODEs were solved numerically to study the impact of various parameters on the transit time. With our solutions, we identified the impact of prescribed pressure drop, slit dimensions, and droplet parameters such as surface tension, viscosity, and volume on transit time. In addition, we also considered the effect of pressure drop and surface tension near critical values. For this study, critical surface tension for a given pressure drop describes the threshold droplet surface tension that prevents transit, and critical pressure for a given surface tension describes the threshold pressure drop that prevents transit. Our solutions demonstrate that there is a linear relationship between pressure and the reciprocal of the transit time (referred to as inverse transit time), as well as a linear relationship between viscosity and transit time. Additionally, when the droplet size increases with respect to the slit dimensions, there is a corresponding increase in transit time. Most notably, we emphasize the initial antagonistic effect of surface tension which resists droplet passage but at the same time decreases the lubrication layer, thus facilitating passage. Our results provide quantitative calculations for understanding cells passing through slit-like constrictions and designing droplet microfluidic experiments.

     
    more » « less
  2. Inspired by the recent realization of a two-dimensional (2-D) chiral fluid as an active monolayer droplet moving atop a 3-D Stokesian fluid, we formulate mathematically its free-boundary dynamics. The surface droplet is described as a general 2-D linear, incompressible and isotropic fluid, having a viscous shear stress, an active chiral driving stress and a Hall stress allowed by the lack of time-reversal symmetry. The droplet interacts with itself through its driven internal mechanics and by driving flows in the underlying 3-D Stokes phase. We pose the dynamics as the solution to a singular integral–differential equation, over the droplet surface, using the mapping from surface stress to surface velocity for the 3-D Stokes equations. Specializing to the case of axisymmetric droplets, exact representations for the chiral surface flow are given in terms of solutions to a singular integral equation, solved using both analytical and numerical techniques. For a disc-shaped monolayer, we additionally employ a semi-analytical solution that hinges on an orthogonal basis of Bessel functions and allows for efficient computation of the monolayer velocity field, which ranges from a nearly solid-body rotation to a unidirectional edge current, depending on the subphase depth and the Saffman–Delbrück length. Except in the near-wall limit, these solutions have divergent surface shear stresses at droplet boundaries, a signature of systems with codimension-one domains embedded in a 3-D medium. We further investigate the effect of a Hall viscosity, which couples radial and transverse surface velocity components, on the dynamics of a closing cavity. Hall stresses are seen to drive inward radial motion, even in the absence of edge tension. 
    more » « less
  3. The transition from laminar to turbulent flow is of great interest since it is one of the most difficult and unsolved problems in fluids engineering. The transition processes are significantly important because the transition has a huge impact on almost all systems that come in contact with a fluid flow by altering the mixing, transport, and drag properties of fluids even in simple pipe and channel flows. Generally, in most transportation systems, the transition to turbulence causes a significant increase in drag force, energy consumption, and, therefore, operating cost. Thus, understanding the underlying mechanisms of the laminar-to-turbulent transition can be a major benefit in many ways, especially economically. There have been substantial previous studies that focused on testing the stability of laminar flow and finding the critical amplitudes of disturbances necessary to trigger the transition in various wall-bounded systems, including circular pipes and square ducts. However, there is still no fundamental theory of transition to predict the onset of turbulence. In this study, we perform direct numerical simulations (DNS) of the transition flows from laminar to turbulence in a channel flow. Specifically, the effects of different magnitudes of perturbations on the onset of turbulence are investigated. The perturbation magnitudes vary from 0.001 (0.1%) to 0.05 (5%) of a typical turbulent velocity field, and the Reynolds number is from 5,000 to 40,000. Most importantly, the transition behavior in this study was found to be in good agreement with other reported studies performed for fluid flow in pipes and ducts. With the DNS results, a finite amplitude stability curve was obtained. The critical magnitude of perturbation required to cause transition was observed to be inversely proportional to the Reynolds number for the magnitude from 0.01 to 0.05. We also investigated the temporal behavior of the transition process, and it was found that the transition time or the time required to begin the transition process is inversely correlated with the Reynolds number only for the magnitude from 0.02 to 0.05, while different temporal behavior occurs for smaller perturbation magnitudes. In addition to the transition time, the transition dynamics were investigated by observing the time series of wall shear stress. At the onset of transition, the shear stress experiences an overshoot, then decreases toward sustained turbulence. As expected, the average values of the wall shear stress in turbulent flow increase with the Reynolds number. The change in the wall shear stress from laminar to overshoot was, of course, found to increase with the Reynolds number. More interestingly was the observed change in wall shear stress from the overshoot to turbulence. The change in magnitude appears to be almost insensitive to the Reynolds number and the perturbation magnitude. Because the change in wall shear stress is directly proportional to the pumping power, these observations could be extremely useful when determining the required pumping power in certain flow conditions. Furthermore, the stability curve and wall shear stress changes can be considered robust features for future applications, and ultimately interpreted as evidence of progress toward solving the unresolved fluids engineering problem. 
    more » « less
  4. SUMMARY

    Stomata are pores at the leaf surface that enable gas exchange and transpiration. The signaling pathways that regulate the differentiation of stomatal guard cells and the mechanisms of stomatal pore formation have been characterized inArabidopsis thaliana. However, the process by which stomatal complexes develop after pore formation into fully mature complexes is poorly understood. We tracked the morphogenesis of young stomatal complexes over time to establish characteristic geometric milestones along the path of stomatal maturation. Using 3D‐nanoindentation coupled with finite element modeling of young and mature stomata, we found that despite having thicker cell walls than young guard cells, mature guard cells are more energy efficient with respect to stomatal opening, potentially attributable to the increased mechanical anisotropy of their cell walls and smaller changes in turgor pressure between the closed and open states. Comparing geometric changes in young and mature guard cells of wild‐type and cellulose‐deficient plants revealed that although cellulose is required for normal stomatal maturation, mechanical anisotropy appears to be achieved by the collective influence of cellulose and additional wall components. Together, these data elucidate the dynamic geometric and biomechanical mechanisms underlying the development process of stomatal maturation.

     
    more » « less
  5. We experimentally investigate the interfacial instability that emerges when a water droplet is deposited on a bath of glycerol-water solution. Despite the absence of surface tension to stabilize short-wavelength undulations, we observe finite-size fingers that grow and pinch off from the drop. We show that the fingering patterns formed in the experiments resultes from a balance between the outward buoyancy effect and inward Marangoni flow. This induced Marangoni flow inhibits small perturbations and acts as an effective surface tension on the miscible interface of the spreading drop. To characterize the final size and shape of the drop, we perform systematic experiments by varying the drop volume and the glycerol-water volume fraction. In addition, we have developed scaling arguments for the drop’s final radius using key physical forces, and show that the final wavelength is inversely proportional to the Bond number. 
    more » « less