1. Introduction
Capillary rise, the phenomenon where a liquid spontaneously ascends within a narrow space due to intermolecular forces, plays a crucial role in numerous scientific and industrial processes. Studying the capillary rise of liquids in thin tubes is important due to its numerous applications. For example, a porous medium was modeled as a bundle of capillary tubes, De Gennes et al. [
1], Quéré [
2], and Lago and Araujo [
3]. Other applications that essentially require a thorough understanding of the interesting phenomenon include applications in the paper, ink, and textile industries, Delgado [
4].
While extensively studied in simple geometries such as circular tubes, the behavior of capillary rise in more complex geometries, like an eccentric annulus, remains less understood yet equally significant. An eccentric annulus, formed between two cylinders with different radii that are offset from each other, presents unique challenges and opportunities for understanding fluid dynamics.
The study of capillary rise in an eccentric annulus is important for several reasons. First, many natural and engineered systems feature non-concentric cylindrical geometries, where precise fluid management is critical. Applications range from enhanced oil recovery in porous media to microfluidic devices, where controlling fluid distribution in narrow, eccentric spaces can improve efficiency and performance. Furthermore, the behavior of capillary rise in an eccentric annulus can offer insights into the design of advanced materials and systems. For example, in porous media, where the pores can be modeled as bundles of capillary tubes with varying degrees of eccentricity, understanding capillary behavior can inform the development of more effective filtration and absorption materials. In the textile and paper industries, where fibers and sheets can create similar eccentric geometries, predicting fluid movement is essential for optimizing processes like dyeing, printing, and coating. Investigating capillary rise in an eccentric annulus not only fills a critical gap in the fundamental understanding of fluid dynamics but also has significant implications for a wide range of practical applications across various industries.
The mathematical modeling of capillary rise in a tube dates back to Ampt [
5], Lucas [
6], and Washburn [
7]. Since then, the literature has been quite rich on all aspects of this phenomenon. The original celebrated Lucas–Washburn equation ignores gravity and kinetic effects. A model that includes both effects is given by Masoodi et al. [
8], Szekely et al. [
9], and Fries [
10]. The model in abstract symbols reads as
with
being the parameter that hosts the liquid and tube properties, and
is some dimensionless version of the height of the column. An interesting behavior is found near the equilibrium height of the rise. For some range of
, the rise oscillates around its equilibrium destination, while for others, the oscillations are completely dampened, as shown in
Figure 1 (a gif file in
Supplementary Materials file is also attached), Płociniczak and Świtała [
11].
Masoodi et al. [
8] determined that the condition
leads to the onset of oscillations. The findings of this fundamental work were rigorously proven by Płociniczak and Świtała [
11].
Numerous studies have been performed on the flow between two vertical plates (see, for example, [
12,
13,
14,
15,
16]) and on the capillary rise in tubes. The bulk of the research on capillary rise has been related to circular tubes. In this research, we investigate the capillary rise in an eccentric annulus from which a circular tube is a special case. The research is interesting from an application point of view as many instruments involve capillary rise in the annulus of tubes. The annulus is not always, intentionally or unintentionally, perfectly centered. The eccentricity of the annulus can be of significant magnitude. In the study presented herein, we determine the velocity distribution and investigate the effect of the radii ratio and the eccentricity on the transition from monotone to oscillatory rise. To the best of our knowledge, no such work has been carried out.
2. Velocity Profile
Consider an axial flow in an eccentric cylindrical annulus with inner radius
, outer radius
, and center-to-center distance
, as shown in
Figure 2.
The steady developed flow in the annulus is assumed to obey the Poiseuille equation
where
is the velocity in the axial direction
,
is the viscosity, and
is the pressure.
The solution of (1), i.e., the velocity profile, can be obtained using the naturally fit bipolar cylindrical coordinates system
. This system is a three-dimensional orthogonal coordinates system whose projection on a plane perpendicular to the z-axis is composed of two sets of Apollonian circles, as shown in
Figure 2. For more details on the coordinates system, the reader is referred to the work of Moon and Spencer [
17] and Alassar and El-Gebeily [
18]. The Cartesian coordinates system is related to the bipolar system by the relations
with
and
. The inner and the outer cylinders are identified, respectively, by
and
, where
is the focal length given by
The bipolar coordinates system has the scale factors
It is standard procedure, Arfken [
19], to cast Equation (1) in any orthogonal coordinates system using the appropriate scale factors. In bipolar coordinates, Equation (1) takes the form
At the surfaces of the cylinders, no-slip conditions () are considered.
Though inspection, one can notice that
is a particular solution of (5). Considering the symmetry of the problem, we then assume a solution of the form
It is noted here that the velocity has been scaled by
to make it dimensionless. The assumed solution (6) has the same form as that given by Alassar [
20] for the flow in eccentric annuli but with slip. It can be shown, however, that the coefficients in (6) which were calculated numerically by Alassar [
20] due to their coupled nature, decouples in the present no-slip case and can be obtained exactly in closed form. The coefficients are found to be
To obtain (7), one needs, in addition to the standard orthogonality relations of the trigonometric functions, the three special integrals
The average dimensionless velocity,
, can now be calculated as
In summary, the velocity and the average velocity are completely determined if both the radii and the center-to-center distance are known. The center-to-center distance can be replaced by the eccentricity,
,
The focal distance is then
The two numbers representing the two surfaces of the cylinders,
and
, are thus determined.
Figure 3 shows a typical velocity profile (
and
).
3. Capillary Rise
Consider a liquid of density
and viscosity
rising in a vertical capillary channel of cross-sectional area
and perimeter
. The height of the liquid column,
, is governed by Newton’s second law of motion
where
is the surface tension,
is the contact angle,
is the pressure drop in the channel, and
is the gravitational acceleration, Masoodi et al. [
8]. From (9), the pressure drop in a liquid column of height
can be written as
There are several ways to cast (12) into dimensionless form, Fries and Dreyer [
21]. Using
and
, Equation (12) can be rewritten as
where
is the Galileo number,
is the Bond number, and
is the hydraulic radius.
It has been shown that, under consistent boundary conditions (i.e., ) and depending on the value of (i.e., the coefficient of in (14)), the solution of Equation (14) progresses in time either monotonically or oscillates about its equilibrium height. The equilibrium height, known as Jurin’s height, is usually used for the measurement of surface tension. It is obtained from (12) as or the dimensionless value .
According to Płociniczak and Świtała [
11], the solution of (12) remains monotone as long as
or equivalently
It is important to observe that, for a specific liquid and tube material, the condition (15) is purely geometric. We define the equality as the critical condition at which the rise changes from oscillatory to monotone or vice versa.
Figure 4 quantifies the expected effect of the radii ratio and the eccentricity on this critical value. As the eccentricity decreases, the rise becomes more oscillatory. In other words, eccentric annuli tend to dampen oscillations. The oscillations are also reduced when the radii ratio increases as a result of the annulus becoming narrower.
It is noted here that for a circular tube (i.e., as
), we obtain the well-known result
, and the limiting critical values is
. The bottom and top curves in
Figure 4 correspond to, respectively, the concentric (
) and the fully eccentric (
) cases. The critical values for these two limiting cases can be written, in order, as
where
is the second Polygamma function. The condition in (17) is based on the velocity profile given by MacDonald [
22]. For both cases, again,
as
. As
, one can show, perhaps for only theoretical curiosity, that the critical value for the fully eccentric case
, while for the concentric case,
.
Figure 4 is fundamental in determining the dimensions of a channel that supports monotone rise. The following paragraph explains the process.
Given that
and
, one can, for specific values of the radii ratio
and the eccentricity
, find the maximum radius of the outer (equivalently the inner) cylinder that maintains monotone rise. The condition for maintaining monotone then reduces from (15) as
In other words, the value
is the largest outer radius before oscillations appear. As
, condition (18) reduces to
, which is identical to that obtained by Masoodi et al. [
8] for a circular tube.
Figure 5 shows the variation in the maximum value of
required to maintain monotonicity, with the radii ratio and the eccentricity for water at
°C (
,
,
, and
). The corresponding Jurin’s heights are also plotted in
Figure 6. As
,
(
31.26 mm for water at
°C). The calculations can be repeated easily for any other liquid.