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.
Computing the viscous effect in early-time drop impact dynamics
The impact of a liquid drop on a solid surface involves many intertwined physical effects, and is influenced by drop velocity, surface tension, ambient pressure and liquid viscosity, among others. Experiments by Kolinski et al. ( Phys. Rev. Lett. , vol. 112, no. 13, 2014 b , p. 134501) show that the liquid–air interface begins to deviate away from the solid surface even before contact. They found that the lift-off of the interface starts at a critical time that scales with the square root of the kinematic viscosity of the liquid. To understand this, we study the approach of a liquid drop towards a solid surface in the presence of an intervening gas layer. We take a numerical approach to solve the Navier–Stokes equations for the liquid, coupled to the compressible lubrication equations for the gas, in two dimensions. With this approach, we recover the experimentally captured early time effect of liquid viscosity on the drop impact, but our results show that lift-off time and liquid kinematic viscosity have a more complex dependence than the square-root scaling relationship. We also predict the effect of interfacial tension at the liquid–gas interface on the drop impact, showing that it mediates the lift-off behaviour.
more »
« less
- Award ID(s):
- 1753203
- PAR ID:
- 10430264
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 945
- ISSN:
- 0022-1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
more » « less
-
In this work, we use the immersed boundary method with four extensions to simulate a moving liquid–gas interface on a solid surface. We first define a moving contact line model and implements a static-dynamic friction condition at the immersed solid boundary. The dynamic contact angle is endogenous instead of prescribed, and the solid boundary can be non-stationary with respect to time. Second, we simulate both a surface tension force and a Young's force with one general equation that does not involve estimating local curvature. In the third extension, we splice liquid–gas interfaces to handle topological changes, such as the coalescence and separation of liquid droplets or gas bubbles. Finally, we re-sample liquid–gas interface markers to ensure a near-uniform distribution without exerting artificial forces. We demonstrate empirical convergence of our methods on non-trivial examples and apply them to several benchmark cases, including a slipping droplet on a wall and a rising bubble.more » « less
-
McCartney, J.S. ; Tomac, I. (Ed.)Immiscible multiphase flow in porous media is largely affected by interfacial properties, manifested in contact angle and surface tension. The gas-liquid surface tension can be significantly altered by suspended particles at the interface. Particle-laden interfaces have unique properties, for example, a lower surface tension of interfaces laden with surfactants or nanoparticles. This study investigates the impacts of a motile bacterium Escherichia coli ( E. coli , strain ATCC 9637) on the air-water surface tension. Methods of the maximum bubble pressure, the du Noüy ring, and the pendant droplet are used to measure the surface tension of the motile-bacteria-laden interfaces. Measured surface tension remains independent to the E. coli concentration when using the maximum bubble pressure method, decreases with increased E. coli concentration in the du Noüy ring method, and presents time-dependent changes by the pendant drop method. The analyses show that the discrepancies may come from the different convection-diffusion processes of E. coli in the flow among various testing methods.more » « less
-
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
-
In this paper, we numerically investigate drop impact on a micro-well substrate to understand the phenomena of non-wettability. The simulation is carried out by solving three-dimensional incompressible Navier–Stokes equations using a density projection method and an adaptive grid refinement algorithm. A very sharp interface reconstruction algorithm, known as the moment-of-fluid method, is utilized to identify the multi-materials and multi-phases present in the computation domain. Our simulations predicted that a micro-well with a deep cavity can significantly reduce a solid–liquid contact in the event of drop impact. The results from the drop impact on the micro-well substrate are compared with results from drop impact on a flat substrate. Significant differences are observed between these two cases in terms of wetted area, spreading ratio, and kinetic energy. Our simulation shows that under the same conditions, a drop is more apt to jump from a micro-well substrate than from a flat surface, resulting in smaller wetted area and shorter contact time. Based on the simulation results, we draw a drop jumping region map. The micro-well substrate has a larger region than the flat surface substrate. Finally, we present a comparative analysis between a flat substrate and a substrate constructed with a dense array of micro-wells and, therefore, show that the array of micro-wells outperforms the smooth substrate with regard to non-wettability and drop wicking capability.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/jfm.2022.445
