Analysis of capillary flows in non-uniform cross-sectional capillaries

Auto-ejection of liquid from a nozzle

Auto-ejection of liquid is an important process in engineering applications, and is also very complicated since it involves interface moving, deforming, and jet breaking up. In this work, a theoretical velocity of meniscus at nozzle exit is first derived, which can be used to analyze the critical condition for auto-ejection of liquid. Then a consistent and conservative axisymmetric lattice Boltzmann (LB) method is proposed to study the auto-ejection process of liquid jet from a nozzle. We test the LB model by conducting some simulations, and find that the numerical results agree well with the theoretical and experimental data. We further consider the effects of contraction ratio, length ratio, contact angle, and nozzle structure on the auto-ejection, and observe some distinct phenomena during the ejection process, including the deformation of meniscus, capillary necking, and droplet pinch off. Finally, the results reported in the present work may play an instructive role on the design of droplet ejectors and the understanding of jetting dynamics in microgravity environment.

Pouring of Grains onto Liquid Surfaces: Dispersion or Lump Formation?

This study considers the consequences of adding grains to an air-liquid interface from a funnel. Depending on the grain contact angle and liquid surface tension, the interface is found to support a single or multiple layers of grains, forming a granular stack. By continuing to add grains, the stacks grow until either the lower grains disperse in the liquid, or the complete stack breaks free from the surface and sinks as a dry powder lump. Herein, the effects of grain contact angle, density, and size on these processes are studied experimentally, and a theoretical analysis is given. The maximum number of grains contained in a floating stack and its critical depth are observed to increase as the grain size decreases. The maximum number of grains scales with the bond number (Bo) as Bo^-1.82 when stack detachment is observed and with an exponent  -2.0 when grains disperse into the liquid. As a result of these different scaling exponents, a critical bond number above which grains wet and disperse can be identified. Favorable conditions for dispersion are achieved with larger grains and, to a lesser extent, by lower surface tension and contact angle. The critical bond number separating grain dispersion from lump formation increases with an increasing grain contact angle, thus providing a physical justification for increasing grain size with common processes such as granulation or agglomeration. Conversely, a quantitative framework to interpret the limitations in dispersing small grains is proposed, justifying the need for low contact angle or liquids with low surface tensions, both favored by the use of surfactants. The present findings have identified conditions under which lump formation occurs, and hence how these undesired phenomena can be avoided in applications requiring the efficient dispersion of grains across a liquid interface.

Bio-inspired liquid transport via elastocapillary interaction of a thin membrane with a liquid meniscus

We report bio-inspired (from a hummingbird's tongue) liquid transport via elastocapillary interaction of a thin membrane with a liquid meniscus. A soft wedge-thin rectangular membrane forming a wedge with a rigid substrate and a flat thin rectangular membrane undergo large deformation while interacting with liquid menisci. The membrane deformation leads to the formation of confinement which in turn results in elastocapillary flow along the membrane length. A simple theoretical model based on the Euler Bernoulli law is used to predict the membrane deformation profiles, which compare well with that obtained from experiments. In the wedge case, the membrane surface and liquid are selected such that the Concus-Finn criterion is not satisfied to contrast the present case of elastocapillary flow from the typical corner flow reported in the literature. The meniscus location versus time studies indicated that the flow exhibits the typical Washburn regime with , except for a sudden increase in velocity at the end of the membrane length. The effects of membrane thickness and width, liquids and substrates were studied to determine the expression for the modified Washburn constant W_m in both the wedge and flat membranes. It was found that gravity plays a role for Bo > 0.94 and for Bo = 1.9, the effect of inclination angle on the flow was studied. The elastocapillary flow with thin membranes could open up an opportunity for a new area, namely "membrane microfluidics" or "lab on a membrane", for diagnostics and other applications.

Modeling the abnormally slow infiltration rate in mesoporous films

The rate of infiltration of fluids into bimodal porous systems was modeled to decrease as the ratio of the pore size to pore neck size was increased.

Capillary Imbibition into Converging Tubes: Beating Washburn's Law and the Optimal Imbibition of Liquids

We consider the problem of capillary imbibition into an axisymmetric tube for which the tube radius decreases in the direction of increasing imbibition. For tubes with constant radius, imbibition is described by Washburn's law (referred to here as the BCLW law to recognize the contributions of Bell, Cameron, and Lucas that predate Washburn). We show that imbibition into tubes with a power-law relationship between the radius and axial position generally occurs more quickly than imbibition into a constant-radius tube. By a suitable choice of the shape exponent, it is possible to decrease the time taken for the liquid to imbibe from one position to another by a factor of 2 compared to the BCLW law. We then show that a further small decrease in the imbibition time may be obtained by using a tube consisting of a cylinder joined to a cone of 3 times the cylinder length. For a given inlet radius, this composite shape attains the minimum imbibition time possible. We confirm our theoretical results with experiments on the tips of micropipettes and discuss the possible significance of these results for the control of liquid motion in microfluidic devices.

Dynamics of a capillary invasion in a closed-end capillary

The position of fluid invasion in an open capillary increases as the square root of time and ceases when the capillary and hydrostatic forces are balanced, when viscous and inertia terms are negligible. Although this fluid invasion into open-end capillaries has been well described, detailed studies of fluid invasion in closed-end capillaries have not been explored thoroughly. Thus, we demonstrated, both theoretically and experimentally, a fluid invasion in closed-end capillaries, where the movement of the meniscus and the invasion velocity are accompanied by adiabatic gas compression inside the capillary. Theoretically, we found the fluid oscillations during invasion at short time scales by solving the one-dimensional momentum balance. This oscillatory motion is evaluated to determine which physical forces dominate the different conditions, and is further described by a damped driven harmonic oscillator model. However, this oscillating motion is not observed in the experiments. This inconsistency is due to the following: first, a continuous decrease in the radius of the curvature caused by decreasing the invasion velocity and increasing pressure inside the closed-end capillary, and second, the shear stress increase in the short time scale by the plug like velocity profile within the entrance length. The viscous term of modified momentum equation can be written as K(8μl/rc(2))(dl/dt) by using the multiplying factor K, which represents the increase of shear stress. The K is 7.3, 5.1, and 4.8 while capillary aspect ratio χc is 740, 1008, and 1244, respectively.

Geometry-induced asymmetric capillary flow

When capillary flow occurs in a uniform porous medium, the depth of penetration is known to increase as the square root of time. However, we demonstrate in this study that the depth of penetration in multi-section porous layers with variation in width and height against the flow time is modified from this diffusive-like response, and liquids can pass through porous systems more readily in one direction than the other. We show here in a model and an experiment that the flow time for a negative gradient of cross-sectional widths is smaller than that for a positive gradient at the given total height of porous layers. The effect of width and height of local layers on capillary flow is quantitatively analyzed, and optimal parameters are obtained to facilitate the fastest flow.

Optimal design of porous structures for the fastest liquid absorption

Porous materials engineered for rapid liquid absorption are useful in many applications, including oil recovery, spacecraft life-support systems, moisture management fabrics, medical wound dressings, and microfluidic devices. Dynamic absorption in capillary tubes and porous media is driven by the capillary pressure, which is inversely proportional to the pore size. On the other hand, the permeability of porous materials scales with the square of the pore size. The dynamic competition between these two superimposed mechanisms for liquid absorption through a heterogeneous porous structure may lead to an overall minimum absorption time. In this work, we explore liquid absorption in two different heterogeneous porous structures [three-dimensional (3D) circular tubes and porous layers], which are composed of two sections with variations in radius/porosity and height. The absorption time to fill the voids of porous constructs is expressed as a function of radius/porosity and height of local sections, and the absorption process does not follow the classic Washburn's law. Under given height and void volume, these two-section structures with a negative gradient of radius/porosity against the absorption direction are shown to have faster absorption rates than control samples with uniform radius/porosity. In particular, optimal structural parameters, including radius/porosity and height, are found that account for the minimum absorption time. The liquid absorption in the optimized porous structure is up to 38% faster than in a control sample. The results obtained can be used a priori for the design of porous structures with excellent liquid management property in various fields.

Analytical Model of Capillary Suction Velocity of a Pipe with Multiple Sizes

Absorption of water in concrete is often described by simple linear water uptake vs. square-root-of-time law. This model based on the classical capillary model derived from the Hagen-Poiseuille’ flow which travel on a single pipe dimension. However, in cement-like material concrete, the pore structure is complex, including for example, variable pore radius and complicated topological changes. In this work, an analytical velocity model of Hagen Poiseuille’s flow on a pipe with multiply sizes is constructed. This model takes the pipe geometry as described by two sets of parameters α_i and β_i;i=1,2…,N into account. In addition, a numerical study of other flow models using Navier Stokes and Stokes models has been conducted. The analytical model is in good agreement with the numerical models at a series of different pipe geometry sizes.