Subscriber access provided by UNIV OF UTAH
Article
Dynamics of a capillary invasion in a closed-end capillary Hosub Lim, Anubhav Tripathi, and Jinkee Lee Langmuir, Just Accepted Manuscript • DOI: 10.1021/la501927c • Publication Date (Web): 01 Jul 2014 Downloaded from http://pubs.acs.org on July 7, 2014
Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.
Langmuir is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.
Page 1 of 25
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
Dynamics of a capillary invasion in a closed-end capillary
Hosub Lim1, Anubhav Tripathi2 and Jinkee Lee1† 1
Multiscale Fluid Mechanics Laboratory, School of Mechanical Engineering, Sungkyunkwan University, Suwon, 440-746, Rep. of Korea
2
Biochemical Engineering Laboratory, School of Engineering, Brown University, Providence, RI 02912, USA
†
Corresponding author: Jinkee Lee, tel:+82-31-299-4845, email:
[email protected] 1 ACS Paragon Plus Environment
Langmuir
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 2 of 25
Abstract 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 in order 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 close-ended 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 dl by using the rc2 dt
multiplying factor K, which represents the increase of shear stress. The K is 7.3, 5.1 and 4.8 while capillary aspect ratio ߯ is 740, 1008 and 1244, respectively.
Keywords: Capillary invasion, Surface tension, Radius of curvature, Contact angle, Mathematical modeling
2 ACS Paragon Plus Environment
Page 3 of 25
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
Introduction The fluid invasion in open-ended capillary tubes was first studied by Bell & Cameron 1, and detailed results are presented by Lucas 2 and Washburn 3. They investigated fluid invasion in the capillaries of constant cross section with negligible gravity and inertia. By balancing the viscous and capillary forces, the fluid invasion was shown to satisfy l(t ) = rcσ cosθ 2µ t where l(t ) represents the position of the invasion of meniscus with time t, rc is the radius of the capillary, µ is the viscosity of the fluid, σ is the surface tension of the fluid, and θ is the contact angle at the fluid/glass interface. This result is often referred to as the Lucas-Washburn law. Numerous analytical, numerical and experimental studies have also been performed in order to study fluid invasion in open-ended capillaries under different physical conditions. Hoffmann
4
described the effect of the flow on the apparent gas-liquid interfacial shape during
their movement. Jiang et al.
5
revealed the dynamic correlation using the Capillary number.
Ichikawa and Satoda 6 provided the interfacial dynamics of advancing and receding the contact angle inconsistency, and Sibold et al.
7
and Chebbi
8
showed the contact angle changes in an
inertial regime through an experiment and numerical simulation with an asymptotic analysis. Xue et al.
9
wrote about the hydrostatic effect of a spontaneous rise in order to describe the
relationship between the liquid front height and time. Lorenceau et al.
10
conducted an
experiment of oscillating motion in the capillary by enforcing to partially be immersed in the initial boundary condition. The results convey the oscillation in a long time stage and a liquid finger developed in a very short time stage. Fries and Dreyer
11
investigated short time inertial
regime for their transition as well as developed a dimensionless analysis method. Quere
12
described the inertial regime of a capillary rise, where the results are calculated for the height
3 ACS Paragon Plus Environment
Langmuir
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 4 of 25
and velocity of the liquid. It shows that the early stage rise is found to be linear and that oscillation is possible in order to generate a low viscosity fluid. Subsequently, Zumed et al.
13
conducted similar experiments with different solutions including a surfactant. Not only these various physical conditions, but geometrical or material dependency were investigated as well such as non-uniform capillary cross-section effects 15
14
and porous media fluid imbibition effects
. Very recently, the early regime of capillary rising has been investigated. Siddhartha et al.
found that the inviscid regime and their time scale
16
gravity or viscosity dominate a capillary
rising 17. Although there were numbers of researches on fluid invasion in open-end capillaries, only a few researches conducted fluid invasion in closed-ended capillaries to the best of our knowledge. The fluid movement in a closed capillary is important for applying various electro mechanical devices, which are used to measure fluid pressures via gas compression applications could be possible. While Deutsch
19
18
, or other
described the invasion of fluids without
considering the effects of inertia and gravity, Radiom et al.
20
studied capillary invasion by
ignoring the inertial and viscous contributions. Fazio and Iacono
21
performed a numerical
investigation in order to understand the capillary invasion phenomenon. There studies provided less insights into the effects of viscous, inertia and surface tension on capillary invasion for their dynamics. In this manuscript, we performed a detailed study on the fluid rise in closed-end capillaries (figure 1), where the rise is accompanied by adiabatic gas compression. The theoretical analysis is developed in the following section. Subsequently, the theoretical predictions are compared with the experimental result and then the theory is modified. Lastly, we conclude with a discussion of potential applications for this work.
4 ACS Paragon Plus Environment
Page 5 of 25
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
Theoretical development Fluid invasion in a closed-end capillary can be described using a momentum balance of the rising fluid within the capillary. Consider a vertical capillary, as shown in figure 1. By balancing the inertial, viscous, capillary, hydrostatic and entrapped air pressure, the differential equation describing the liquid rise l(t ) inside the capillary tube can be expressed as
−ρ
d dl 8µl dl 2σ cosθ − − ρg (h − l ) − (Patm − Pg ) , l = dt dt rc2 dt rc
(1)
where the last term arises due to the air compression at the closed-end of the capillary. Here, h is the depth at which the capillary is immersed in the liquid of density ρ. In this one dimensional momentum balance equation, the liquid contact angle θ with capillary is assumed to be constant. The gas pressure Pg (t ) inside the capillary can be evaluated as Patm (l c (l c − l(t ))) , γ
assuming none of the gas is dissolving in liquid during the fluid invasion, where Pg (t = 0) = Patm , l c is the capillary tube length, and γ = 1.4 for an adiabatic compression of air inside the
capillary. . Using dimensionless parameters L =
l σt l h ,H = , T = , χ c = c , equation (1) is rc rc µ rc rc
transformed to χ γ 1 d dL dL c − 2 cos θ − Bo (H − L ) − β 1 − L + 8L χ c − L Oh 2 dT dT dT
where Bo =
=0
(2)
ρgrc2 µ P r is the Bond number, Oh = is the Ohnesorge number and β = atm c is σ σ ρrcσ
the ratio of atmospheric to capillary pressure. In a closed-end capillary system, the air pressure is
5 ACS Paragon Plus Environment
Langmuir
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 6 of 25
not effective on the meniscus’ shape because the meniscus is the closed curve of the circumference of the inner diameter of the tube. The values for dimensionless parameters were set as Bo =0.1, 0.01, Oh =0.02, 0.002, β =450, H =0 and χ c = 200, 500, 103, 104, 105 to compare the dimensionless capillary rise L (T ) of various conditions. These values were chosen according to the dimensional and dimensionless quantities of typical parameters provided in table 1. Figure 2A presents the capillary rise L (T ) as a function of time T for different capillary tube heights χ c when the tube inner surface is perfectly wettable, θ ≅ 0 . As shown in figure 2A, the Washburn rule balances the viscous and capillary forces in order to obtain L(T ) = T 2 . For open and closed capillaries, we examine the physical aspects of the fluid invasion in terms of three slope regimes, when Bo = 0.01 , Oh = 0.02 and β = 450 . At a small T , the second order inertial term is dominant; moreover, by balancing the inertial and capillary forces, equation (2) results in L = 2OhT for T 6.3 . For large T ( T >> 10 5 ), the inertial effects are negligible, such that χ c Bo
the liquid ceases to rise and the slope dL dT becomes zero. In the case of an open capillary, 6 ACS Paragon Plus Environment
Page 7 of 25
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
the final height of the liquid rise approaches L(T → ∞) ~ 2 cosθ Bo + H , where this final height is governed by the capillary and hydrostatic force. For closed capillaries, the invasion ceases not by the gravity force, but by the air compression, and the final height of the liquid decreases to L(T → ∞ ) ~
βγ 2 cos θ + HBo
