Exact calculation of axisymmetric capillary bridge properties between two unequal-sized spherical particles

Abstract

A calculation method for the meridional profile of axisymmetric bridges between two spheres of different size is introduced in this manuscript. From geometrical data of the capillary bridge, such as the neck radius and boundary conditions (filling and contact angle), the shape of the capillary bridge is calculated analytically as a solution of the Young–Laplace equation. Its free surfaces, of constant mean curvature, may be classified into portions of nodoid, unduloid, and other limit cases. Moreover, other properties of the liquid bridge can be computed analytically, such as the associated capillary force exerted on the solid surfaces, liquid volume, mean curvature, and free surface area.

Keywords

Capillary bridge Young–Laplace equation polydisperse granular media interparticle force

1. Introduction

In granular materials, when a volume of water is introduced between two neighboring particles, a capillary bridge might be formed (Figure 1). The presence of capillary bridges between grains has an important impact on the behavior of unsaturated granular materials, especially in soils. Capillary cohesion, caused by the surface tension, strongly influences the behavior and flow properties of granular materials when water-induced effects are taken into account [1]. Generally, wet granular materials are of primary interest in many fields of engineering science, for example in geomechanics or in wet processing of powders [2 –4]. Thus, many studies have focused on the properties of capillary bridges, where both theoretical [5 –9] and experimental aspects are analyzed [10 –12]. The geometry of capillary bridge profiles corresponds to a solution of the Young–Laplace equation. Nodoid or unduloid shapes are the most frequently encountered profiles in practice (see Gagneux et al. [12]). In some particular limit cases, the meridional profile may be also that of a right circular cylinder, a catenoid, or a portion of sphere [9].

Figure 1.

Pendular regime in wet granular materials.

Many analytical approaches have investigated properties of a capillary bridge between two elastic solids, such as between a sphere and a plane [13] or between two adjacent spheres. However, most of the studies on spherical bodies are restricted to the monodisperse case¹ [7, 9, 14, 15] and primarily focused on the convex meridional profile of the liquid bridge. The coalescence of capillary doublets between touching spheres of the same radius has also been studied theoretically by Gagneux and Millet [16]. However, in most numerical simulations of granular media using discrete element methods, spheres of different diameters, corresponding to a polydisperse granular assembly, are considered. Therefore, it is necessary to analyze all possible cases of capillary bridges between polydisperse particles and to construct a complete information set of the liquid bridge properties (exact shape, volume, free surface area, etc.).

This manuscript is an attempt to provide extended analytical descriptions of the meridional profile of a liquid bridge between two particles of different size, including any wetting angle, for both convex and concave meridians. Moreover, based on the solution of the Young–Laplace equation, analytical formulas for the associated capillary force and liquid volume can be obtained as well. In every case presented in the following sections, an exact parametric description of the meridian is given, and some examples enable the analytical approach presented to be illustrated and verified. Note that some applications to solid–fluid mixture theory [17, 18], or to study the stability of the capillary bridge through the approach developed by Sauer and Duong [19], are conceivable. It should be noted that a recent study with very similar motivations has been conducted numerically by Qiang-Nian et al. [20], using the shooting method for the boundary-value problem (the Runge–Kutta method to integrate a system of ordinary differential equations, associated with the Raphson–Newton iteration method).

2. Calculation of capillary bridge profile characteristics

This manuscript analyses the meridional profile of the capillary bridge between two spheres of different radius. Let us consider two particles with radii $r_{1}$ and $r_{2}$ , $r_{1} < r_{2}$ , separated by a distance $D \geq 0$ , as shown in Figure 2. The origin of the Cartesian coordinates corresponds to the neck of the capillary bridge (where $y^{'} = 0$ , see the appendix). The corresponding half-filling and wetting angles are denoted $δ_{1, 2}$ and $θ_{1, 2}$ , respectively, and $y^{*}$ represents the neck radius of the bridge. Gravity is not taken into account in the following calculations; therefore, the capillary bridge is axisymmetric around the $x$ axis.

Figure 2.

Representation of the capillary bridge between two unequal-sized spherical particles (polydisperse case).

We recall that the associated Young–Laplace equation for axisymmetric profiles is given by

\frac{y^{″}}{{(1 + y^{' 2})}^{3 / 2}} - \frac{1}{y \sqrt{1 + y^{' 2}}} = - \frac{Δ p}{γ} = : H, x \in] x_{c 1}, x_{c 2} [

(1)

where $Δ p$ is the pressure difference between the inside and outside of the capillary bridge and $γ$ denotes the surface tension of the liquid.

Integration of the Young–Laplace equation (equation (1)), or a first integral of the Euler–Lagrange–Beltrami equation via calculus of variations, leads classically to the nonlinear first-order differential equation

1 + y^{' 2} = \frac{4 y^{2}}{H^{2} {(y^{2} - \frac{2 λ}{H})}^{2}} if H \neq 0

(2a)

where $λ$ is a constant, revealing the conservation law

λ = \frac{y}{\sqrt{1 + y^{' 2}}} + \frac{H y^{2}}{2}

(2b)

at any point of the profile. Note that $λ$ is directly linked to the associated capillary force $F^{cap}$ :

F^{cap} = 2 π γ λ = 2 π γ (\frac{y}{\sqrt{1 + y^{' 2}}} + \frac{H y^{2}}{2}) .

(3)

It is important to note that equation (3) is general and valid at any point of the capillary bridge profile (see Gagneux and Millet [9]). This leads to the conventional method of estimation of the capillary force at the neck $y^{*}$ of the bridge, known as the “gorge method” [6, 21]:

F^{cap} = π γ H y^{* 2} + 2 π γ y^{*},

(4)

which is, in fact, a particular case of equation (3).

According to the signs of $H$ and $λ / H$ , equation (2) corresponds to Delaunay roulettes [22]

1 + y^{' 2} = \frac{4 a^{2} y^{2}}{{(y^{2} + ε b^{2})}^{2}}

(5)

with the parameters $a$ and $b$ . The case $ε = 1$ , $a > b > 0$ corresponds to an elliptic roulette (unduloid meridian, Figure 3b), whereas the case $ε = - 1$ , $a > 0$ , $b > 0$ corresponds to a hyperbolic roulette (nodoid meridian, Figure 3a). As shown in Gagneux and Millet [9], the convex meridional profile configuration leads to the pressure difference $Δ p = ε γ / a$ , with $ε = - 1$ for a nodoid and $ε = 1$ for an unduloid.

Figure 3.

Plots of nodoid and unduloid surfaces. (Only half of the surface is shown, to demonstrate the internal structure.)

Thus, the following situations may be encountered:

Convex upper meridian of nodoid with parameters $a = 1 / H$ and $b^{2} = 2 λ / H$ , $H > 0$ , $λ > 0$ ;

Convex upper meridian of unduloid with parameters $a = 1 / | H |$ and $b^{2} = - 2 λ / H$ , $H < 0$ , $λ > 0$ ;

Concave upper meridian of nodoid with parameters $a = 1 / | H |$ and $b^{2} = 2 λ / H$ , $H < 0$ , $λ < 0$ ;

Concave upper meridian of unduloid with parameters $a = 1 / | H |$ and $b^{2} = 2 λ / | H |$ , $H < 0$ , $λ > 0$ .

For a capillary bridge defined by the meridional profile $y (x)$ , the bridge’s volume $V$ is given by:

V = π \int_{x_{c 1}}^{x_{c 2}} y^{2} (x) d x - (V_{c 1} + V_{c 2})

(6)

where

V_{c 1} = \frac{π}{3} {r_{1}}^{3} (1 - \cos δ_{1})^{2} (2 + \cos δ_{1})

and

V_{c 2} = \frac{π}{3} {r_{2}}^{3} (1 - \cos δ_{2})^{2} (2 + \cos δ_{2})

denote the volume of the spherical caps wetted by the liquid on each sphere, with $x_{c 1}$ and $x_{c 2}$ the abscissas of triple points (Figure 2).

In the following sections, each case of the meridional profile of the liquid bridge is calculated and accompanied by numerical examples to confirm the exactitude of the calculation. In this paper, we focus on polydisperse configurations with two spheres of different radii, constituting an extension of the monodisperse case already addressed in Gagneux and Millet [9] and Gagneux et al. [12].

3. Convex upper meridian (negative Gaussian curvature of the free surface)

To identify the property of the meridional profile, we use the line joining the triple points (Figure 4), defined by:

y_{c} (x) = (x - x_{c 1}) \frac{r_{2} \sin δ_{2} - r_{1} \sin δ_{1}}{x_{c 2} - x_{c 1}} + r_{1} \sin δ_{1}

(7)

Figure 4.

Concave upper meridian (generating a convex bridge) or convex upper meridian (generating a concave bridge).

The upper capillary bridge profile $x \mapsto y (x)$ is therefore convex if

y (x) \leq y_{c} (x), \forall x \in [x_{c 1}, x_{c 2}]

(8)

or concave if

y (x) \geq y_{c} (x), \forall x \in [x_{c 1}, x_{c 2}] .

(9)

Intrinsically, we can define the free surface with negative Gaussian curvature (the product of the two principal curvatures) in the case of a convex upper meridian, and positive Gaussian curvature in the case of a concave upper meridian. The sign of the Gauss curvature locally characterizes the surface shape.

In other words, the meridian is considered convex if the curvature turns toward the inside of the liquid bridge, and concave in the contrasting direction of the curvature.

For all upper meridians addressed in this manuscript (concave or convex), the boundary conditions can be determined at triple points (equations (10a), (10b), (10c), and (10d)) and at the neck (equation (10e)) as follows:

{\begin{matrix} y (x_{c 1}) = r_{1} \sin δ_{1} \\ y (x_{c 2}) = r_{2} \sin δ_{2} \\ y^{'} (x_{c 1}) = - \cot (δ_{1} + θ_{1}) \\ y^{'} (x_{c 2}) = \cot (δ_{2} + θ_{2}) \\ y^{'} (0) = 0 \end{matrix}

(10a) (10b) (10c) (10d) (10e)

3.1. Nodoid with convex upper meridian

Result 1. If the observed geometrical data of the capillary bridge are such that

max (r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1}), r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2})) < y^{*} < min (r_{1} \sin δ_{1}, r_{2} \sin δ_{2})

(11)

then the meridian of the capillary bridge is a portion of a nodoid, whose parametric equations are given by:

{\begin{matrix} x (t) = \frac{b^{2}}{a} \int_{0}^{t} \frac{\cos u d u}{(e + \cos u) \sqrt{e^{2} - \cos^{2} u}} \\ y (t) = b \sqrt{\frac{e - \cos t}{e + \cos t}} \end{matrix}

(12)

with $t \in [- τ_{1}, τ_{2}]$ . The bounds $τ_{1}$ , $τ_{2}$ verify the condition $y (τ_{1}) = r_{1} \sin δ_{1}$ , $y (τ_{2}) = r_{2} \sin δ_{2}$ and are given by

\begin{matrix} τ_{1} = \arccos (e \frac{b^{2} - r_{1}^{2} \sin^{2} δ_{1}}{b^{2} + r_{1}^{2} \sin^{2} δ_{1}}) \\ τ_{2} = \arccos (e \frac{b^{2} - r_{2}^{2} \sin^{2} δ_{2}}{b^{2} + r_{2}^{2} \sin^{2} δ_{2}}) \end{matrix}

The eccentricity $e$ is defined as $(\sqrt{a^{2} + b^{2}}) / a$ , where $a = 1 / H > 0$ and $b^{2} = 2 λ / H$ , with $λ > 0$ are given by:

a = \frac{1}{2} \frac{r_{1}^{2} \sin^{2} δ_{1} - y^{* 2}}{y^{*} - r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1})} = \frac{1}{2} \frac{r_{2}^{2} \sin^{2} δ_{2} - y^{* 2}}{y^{*} - r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2})}

(13a)

b^{2} = y^{*} r_{1} \sin δ_{1} \frac{r_{1} \sin δ_{1} - y^{*} \sin δ_{1} \sin (δ_{1} + θ_{1})}{y^{*} - r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1})} = y^{*} r_{2} \sin δ_{2} \frac{r_{2} \sin δ_{2} - y^{*} \sin δ_{2} \sin (δ_{2} + θ_{2})}{y^{*} - r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2})}

(13b)

Moreover, the capillary pressure may be recovered from the measured data as:

\frac{Δ p}{γ} = - \frac{1}{a} < 0, Δ p = - 2 γ \frac{y^{*} - r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1})}{r_{1}^{2} \sin^{2} δ_{1} - y^{* 2}} = - 2 γ \frac{y^{*} - r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2})}{r_{2}^{2} \sin^{2} δ_{2} - y^{* 2}}

(13c)

Finally, the associated liquid volume and free surface area of the capillary bridge can be calculated as:

V = π \frac{b^{4}}{a} \int_{- τ_{1}}^{τ_{2}} \frac{e - \cos t}{e + \cos t} \frac{\cos t d t}{(e + \cos t) \sqrt{e^{2} - \cos^{2} t}} - \frac{π}{3} [r_{1}^{3} {(1 - \cos δ_{1})}^{2} (2 + \cos δ_{1}) + r_{2}^{3} {(1 - \cos δ_{2})}^{2} (2 + \cos δ_{2})]

(13d)

Σ_{2} = 2 π b^{2} \int_{- τ_{1}}^{τ_{2}} \frac{1}{e + \cos t} \sqrt{\frac{e - \cos t}{e + \cos t}} d t

(13e)

Proof. This case corresponds to $H = - Δ p / γ > 0$ and $λ > 0$ . The Young–Laplace equation can be rewritten as

\sqrt{1 + y^{' 2}} = \frac{2 ay}{b^{2} - y^{2}}, 0 < y < b

(14a)

with the parameters

a = \frac{1}{H} > 0, b^{2} = \frac{2 λ}{H} > 0

(14b)

and $ε = - 1$ (see equation (5)), which corresponds to the nodoid case.²□

Combining equation (14a) and the associated boundary conditions at the triple points and the liquid bridge’s neck (equation (10)) yield:

{\begin{matrix} b^{2} - r_{1}^{2} \sin^{2} δ_{1} = 2 a r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1}) \\ b^{2} - r_{2}^{2} \sin^{2} δ_{2} = 2 a r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2}) \\ b^{2} - y^{* 2} = 2 a y^{*} \end{matrix}

(15a) (15b) (15c)

Combining equations (15a) and (15b) leads to the compatibility relation between $(δ_{1}, θ_{1})$ and $(δ_{2}, θ_{2})$ , such that:

r_{1} \sin δ_{1} [2 a \sin (δ_{1} + θ_{1}) + r_{1} \sin δ_{1}] = r_{2} \sin δ_{2} [2 a \sin (δ_{2} + θ_{2}) + r_{2} \sin δ_{2}]

(16)

The parameters $a$ and $b^{2}$ can be obtained as solutions of equation (15):

a = \frac{1}{2} \frac{r_{1}^{2} \sin^{2} δ_{1} - y^{* 2}}{y^{*} - r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1})} = \frac{1}{2} \frac{r_{2}^{2} \sin^{2} δ_{2} - y^{* 2}}{y^{*} - r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2})}

(17a)

b^{2} = y^{*} r_{1} \sin δ_{1} \frac{r_{1} \sin δ_{1} - y^{*} \sin δ_{1} \sin (δ_{1} + θ_{1})}{y^{*} - r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1})} = y^{*} r_{2} \sin δ_{2} \frac{r_{2} \sin δ_{2} - y^{*} \sin δ_{2} \sin (δ_{2} + θ_{2})}{y^{*} - r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2})}

(17b)

provided that the systems (equations (15a) to (15c) and equations (15b) and (15c)) admit a unique solution. This is the case if $y^{*} \neq r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1})$ and $y^{*} \neq r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2})$ . The second condition $y^{*} < min (r_{1} \sin δ_{1}, r_{2} \sin δ_{2})$ comes directly from the condition (equation (8)) of a convex upper meridian.

It is to be noted that $a$ and $b^{2}$ can be also calculated without consulting the neck radius $y^{*}$ , by solving equations (15a) and (15b). We have

a = \frac{1}{2} \frac{- r_{1}^{2} \sin^{2} δ_{1} + r_{2}^{2} \sin^{2} δ_{2}}{r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1}) - r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2})}

(18a)

b^{2} = r_{1} r_{2} \sin δ_{1} \sin δ_{2} \frac{r_{2} \sin δ_{2} \sin (δ_{1} + θ_{1}) - r_{1} \sin δ_{1} \sin (δ_{2} + θ_{2})}{r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1}) - r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2})}

(18b)

Provided the system (equations (15a) and (15b)) admits an unique solution, this is the case if $r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1}) \neq r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2})$ . Naturally, in the case of monodisperse particles, where $r_{1} \sin δ_{1} = r_{2} \sin δ_{2}$ , the system (equations (15a) and (15b)) degenerates and only one condition (for $r_{1} = r_{2} = r$ , $δ_{1} = δ_{2}$ , $θ_{1} = θ_{2}$ ) is necessary, coupled with equation (15c). We recover the resolution already performed by Gagneux and Millet [9] for monodisperse particles.

An important consequence of this remark is that determining the characteristic of capillary bridges from equation (18), although very convenient in some cases, becomes very sensitive when $r_{1} \approx r_{2}$ and the material is the same. In this case, the determinant of the system (equations (15a) and (15b)) approaches zero and the formulas in equation (18) are not recommended.³

Note that the solution presented in equation (18) enables a unique meridional profile to be built from the data set $(r_{1}, r_{2}, δ_{1}, δ_{2}, θ_{1}, θ_{2})$ if and only if conditions $a > 0$ and $b^{2} > 0$ are satisfied, which yields

{\begin{matrix} r_{1} \sin δ_{1} < r_{2} \sin δ_{2} \\ r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1}) > r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2}) \\ r_{1} \sin δ_{1} \sin (δ_{2} + θ_{2}) < r_{2} \sin δ_{2} \sin (δ_{1} + θ_{1}) \end{matrix}

(19a) (19b) (19c)

{\begin{matrix} r_{1} \sin δ_{1} > r_{2} \sin δ_{2} \\ r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1}) < r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2}) \\ r_{1} \sin δ_{1} \sin (δ_{2} + θ_{2}) > r_{2} \sin δ_{2} \sin (δ_{1} + θ_{1}) \end{matrix}

(19d) (19e) (19f)

This approach is not valid in the monodisperse case between two identical spheres, since the condition

r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1}) \neq r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2})

is not satisfied and equation (15) becomes redundant. The condition $r_{1} \sin δ_{1} \neq r_{2} \sin δ_{2}$ is equivalent to $y_{c 1} \neq y_{c 2}$ , meaning that the ordinates of the liquid bridge’s triple points are different.

Figure 5 illustrates an example of determination of the convex upper meridian of a nodoid knowing only the half-filling angle $(δ_{1}, δ_{2})$ and wetting angles ( $θ_{1}, θ_{2}$ ). In the case considered, we have $r_{1} = 8 mm$ , $r_{2} = 10 mm$ , $δ_{1} = 20^{°}$ , $δ_{2} = 17 . 5^{°}$ , and $θ_{1} = θ_{2} = 5^{°}$ . Note that the values considered satisfy the condition $a > 0$ , $b^{2} > 0$ and the compatibility relation (equation (16)).

Figure 5.

Example of upper convex meridian of the capillary bridge with $r_{1} = 8 mm$ , $r_{2} = 10 mm$ , $δ_{1} = 20^{°}$ , $δ_{2} = 17 . 5^{°}$ , $θ_{1} = θ_{2} = 5^{°}$ . Nodoid surface.

Given compatible geometrical data, the following numerical procedure enables the exact parameterization of the upper meridian, a solution of the Young–Laplace equation, to be determined. The values of $(r_{1}, r_{2}, δ_{1}, δ_{2}, θ_{1}, θ_{2})$ considered are obviously compatible and correspond to a solution of the Young–Laplace equation in the nodoid case considered here (they could be observed experimentally for the same given volume $V$ calculated in step 7 of the procedure):

From considered values $(r_{1}, r_{2}, δ_{1}, δ_{2}, θ_{1}, θ_{2})$ , $a$ and $b^{2}$ are calculated using equation (18). Only positive outputs of $a$ and $b^{2}$ are accepted, and satisfy the compatibility condition mentioned in equation (16).

$e = \frac{\sqrt{a^{2} + b^{2}}}{a}, τ_{1} = \arccos (e \frac{b^{2} - r_{1}^{2} \sin^{2} δ_{1}}{b^{2} + r_{1}^{2} \sin^{2} δ_{1}}), τ_{2} = \arccos (e \frac{b^{2} - r_{2}^{2} \sin^{2} δ_{2}}{b^{2} + r_{2}^{2} \sin^{2} δ_{2}})$ are calculated.

The meridional profile is determined using the parametric equations in equation (12), The origin is defined at $x (t = 0)$ and $y (t = 0)$ (as explained in the appendix). The coordinates of triple points ( $x_{c 1}$ , $y_{c 1}$ ), ( $x_{c 2}$ , $y_{c 2}$ ) are calculated as $x_{c 1, 2} = x (τ_{1, 2})$ and $y_{c 1, 2} = y (τ_{1, 2})$ , respectively.

The centers of two spheres $P_{1} (x_{p 1}, 0)$ , $P_{2} (x_{p 2}, 0)$ are located based on the given values, $r_{1}$ , $r_{2}$ , and the calculated results, $x_{c 1}$ , $x_{c 2}$ .

The distance between the two spheres is computed as $D = | x_{p 2} - x_{p 1} | - (r_{1} + r_{2})$ .

The neck radius $y^{*}$ can be defined from the quadratic equation (equation (15c)), which produces $y^{*} = - a \pm \sqrt{a^{2} + b^{2}}$ . Since $y^{*} > 0$ , there is an unique accepted solution $y^{*} = - a + \sqrt{a^{2} + b^{2}}$ . Note that $y^{*}$ automatically satisfies the condition (equation (11)) in the nodoid case being considered.⁴

The associated liquid volume $V$ and the free surface area $Σ_{2}$ can be computed using equations (13d) and (13e).

3.2. Unduloid with convex upper meridian

Result 2. When the observed data are such that

0 < y^{*} < min (r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1}), r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2}))

(20)

then the upper meridian is convex and is a portion of an unduloid whose parameterization is given by:

{\begin{matrix} x (t) = \frac{b^{2}}{a} \int_{0}^{t} \frac{d u}{(1 + e \cos u) \sqrt{1 - e^{2} \cos^{2} u}} \\ y (t) = b \sqrt{\frac{1 - e \cos t}{1 + e \cos t}}, t \in [- τ_{1}, τ_{2}] \end{matrix}

(21)

where

e = \frac{\sqrt{a^{2} - b^{2}}}{a}, a = \frac{1}{| H |}, b^{2} = - \frac{2 λ}{H} .

(22)

and $τ_{1}$ , $τ_{2}$ are defined from the boundary condition at the triple points $y (τ_{1}) = r_{1} \sin δ_{1}$ , $y (τ_{2}) = r_{2} \sin δ_{2}$ . Thus

τ_{1} = \arccos (\frac{1}{e} \frac{b^{2} - r_{1}^{2} \sin^{2} δ_{1}}{b^{2} + r_{1}^{2} \sin^{2} δ_{1}}), τ_{2} = \arccos (\frac{1}{e} \frac{b^{2} - r_{2}^{2} \sin^{2} δ_{2}}{b^{2} + r_{2}^{2} \sin^{2} δ_{2}})

are given by

a = \frac{1}{2} \frac{r_{1}^{2} \sin^{2} δ_{1} - y^{* 2}}{r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1}) - y^{*}} = \frac{1}{2} \frac{r_{2}^{2} \sin^{2} δ_{2} - y^{* 2}}{r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2}) - y^{*}}

(23a)

b^{2} = y^{*} r_{1} \sin δ_{1} \frac{r_{1} \sin δ_{1} - y^{*} \sin δ_{1} \sin (δ_{1} + θ_{1})}{r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1}) - y^{*}} = y^{*} r_{2} \sin δ_{2} \frac{r_{2} \sin δ_{2} - y^{*} \sin δ_{2} \sin (δ_{2} + θ_{2})}{r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2}) - y^{*}}

(23b)

and the pressure difference $Δ p$ may be calculated as:

\frac{Δ p}{γ} = \frac{1}{a} > 0, Δ p = 2 γ \frac{r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1}) - y^{*}}{r_{1}^{2} \sin^{2} δ_{1} - y^{* 2}} = 2 γ \frac{r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2}) - y^{*}}{r_{2}^{2} \sin^{2} δ_{2} - y^{* 2}}

(23c)

The expression of liquid volume and the free surface area stands:

\begin{matrix} V = π \frac{b^{4}}{a} \int_{- τ_{1}}^{τ_{2}} \frac{1 - e \cos t}{1 + e \cos t} \frac{d t}{(1 + e \cos t) \sqrt{1 - e^{2} \cos^{2} t}} \\ - \frac{π}{3} [r_{1}^{3} {(1 - \cos δ_{1})}^{2} (2 + \cos δ_{1}) + r_{2}^{3} {(1 - \cos δ_{2})}^{2} (2 + \cos δ_{2})] \end{matrix}

(23d)

Σ_{2} = 2 π b^{2} \int_{- τ_{1}}^{τ_{2}} \frac{1}{1 + e \cos t} \sqrt{\frac{1 - e \cos t}{1 + e \cos t}} d t

(23e)

Proof. Let us consider the case $H < 0$ and $λ > 0$ . Thus, the Young–Laplace equation can be written as:

\sqrt{1 + y^{' 2}} = \frac{2 ay}{b^{2} + y^{2}}, 0 < y < b .

(24)

Applying the boundary conditions presented in equation (10), equation (24) leads to the system:

{\begin{matrix} b^{2} + r_{1}^{2} \sin^{2} δ_{1} = 2 a r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1}) \\ b^{2} + r_{2}^{2} \sin^{2} δ_{2} = 2 a r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2}) \\ b^{2} - y^{* 2} = 2 a y^{*} \end{matrix}

(25a) (25b) (25c)

Similar to the nodoid case, combining equations (25a) and (25b) leads to the compatibility condition between $(δ_{1}, θ_{1})$ and ( $δ_{2}, θ_{2}$ ):

r_{1} \sin δ_{1} [2 a \sin (δ_{1} + θ_{1}) - r_{1} \sin δ_{1}] = r_{2} \sin δ_{2} [2 a \sin (δ_{2} + θ_{2}) - r_{2} \sin δ_{2}]

(26)

Solution of equations (25a) to (25c) leads to the following expressions of $a$ and $b$ :

a = \frac{1}{2} \frac{r_{1}^{2} \sin^{2} δ_{1} - y^{* 2}}{r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1}) - y^{*}} = \frac{1}{2} \frac{r_{2}^{2} \sin^{2} δ_{2} - y^{* 2}}{r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2}) - y^{*}}

(27a)

b^{2} = y^{*} r_{1} \sin δ_{1} \frac{r_{1} \sin δ_{1} - y^{*} \sin (δ_{1} + θ_{1})}{r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1}) - y^{*}} = y^{*} r_{2} \sin δ_{2} \frac{r_{2} \sin δ_{2} - y^{*} \sin (δ_{2} + θ_{2})}{r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2}) - y^{*}}

(27b)

provided the associated system of equations has an unique solution. This is the case if $y^{*} \neq r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1})$ and $y^{*} \neq r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2})$ . As $y^{*} > 0$ , these conditions lead to:

0 < y^{*} < min (r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1}), r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2}))

(28)

in the case of a convex upper meridian. The condition $a > 0$ , $b^{2} > 0$ , in the case of the convex meridian (unduloid case), leads to the supplementary condition

y^{*} < min (r_{1} \frac{\sin δ_{1}}{\sin (δ_{1} + θ_{1})}, r_{2} \frac{\sin δ_{2}}{\sin (δ_{2} + θ_{2})}),

which is automatically satisfied according to equation (28). Note that the compatibility condition presented in equation (20) is satisfied. □

It is also possible to express $a$ and $b^{2}$ with respect to $δ_{1, 2}$ and $θ_{1, 2}$ only (without involving $y^{*}$ ). The associated expressions result from equations (25a) to (25b):

a = \frac{1}{2} \frac{r_{1}^{2} \sin^{2} δ_{1} - r_{2}^{2} \sin^{2} δ_{2}}{r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1}) - r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2})}

(29a)

b^{2} = r_{1} r_{2} \sin δ_{1} \sin δ_{2} \frac{r_{1} \sin δ_{1} \sin (δ_{2} + θ_{2}) - r_{2} \sin δ_{2} \sin (δ_{1} + θ_{1})}{r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1}) - r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2})}

(29b)

respecting the following conditions:

{\begin{matrix} r_{1} \sin δ_{1} < r_{2} \sin δ_{2} \\ r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1}) < r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2}) \\ r_{1} \sin δ_{1} \sin (δ_{2} + θ_{2}) < r_{2} \sin δ_{2} \sin (δ_{1} + θ_{1}) \end{matrix}

(30a) (30b) (30c)

{\begin{matrix} r_{1} \sin δ_{1} > r_{2} \sin δ_{2} \\ r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1}) > r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2}) \\ r_{1} \sin δ_{1} \sin (δ_{2} + θ_{2}) > r_{2} \sin δ_{2} \sin (δ_{1} + θ_{1}) \end{matrix}

(30d) (30e) (30f)

Figure 6 presents an example of the determination of the convex meridian of an unduloid for the given compatible geometric data: $r_{1} = 7 mm$ , $r_{2} = 10 mm$ , $δ_{1} = 16^{°}$ , $δ_{2} = 7 . 4^{°}$ , $θ_{1} = 16 . 6^{°}$ , $θ_{2} = 26 . 9^{°}$ . Note that for the values considered, the compatibility condition (equation (26)) is again satisfied and corresponds to an unduloid shape according to equation (20). The procedure of resolution is similar to that performed in the nodoid case and is not recalled here, for the sake of simplicity.

Figure 6.

Example of upper convex meridian of the capillary bridge with $r_{1} = 7 mm$ , $r_{2} = 10 mm$ , $δ_{1} = 16^{°}$ , $δ_{2} = 7 . 4^{°}$ , $θ_{1} = 16 . 6^{°}$ , $θ_{2} = 26 . 9^{°}$ . Unduloid surface.

4. Concave upper meridian (positive Gaussian curvature of the free surface)

In this section, concave meridian capillary bridges are investigated. Both of the mean curvatures are positive in this case. Thus, the difference between internal and external pressures, $Δ p$ is positive; therefore, $H = - Δ p / γ < 0 .$

4.1. Nodoid with concave upper meridian

Result 3. When observed geometrical data are such that

max (r_{1} \sin δ_{1}, r_{2} \sin δ_{2}) < y^{*} < min (r_{1} \frac{\sin δ_{1}}{\sin (δ_{1} + θ_{1})}, r_{2} \frac{\sin δ_{2}}{\sin (δ_{2} + θ_{2})})

(31)

the free surface of revolution is a part of a nodoid whose meridian is an arc of a hyperbolic Delaunay roulette.

The associated parameterization of the meridional profile may be written as

{\begin{matrix} x (t) = \frac{b^{2}}{a} \int_{0}^{t} \frac{\cos u d u}{(e + \cos u) \sqrt{e^{2} - \cos^{2} u}} \\ y (t) = b \sqrt{\frac{e - \cos t}{e + \cos t}} \end{matrix}

(32)

with $t \in [π - τ_{1}, π + τ_{2}]$ , where $τ_{1}, τ_{2}$ are defined from the boundary conditions $y (π + τ_{1}) = r_{1} \sin δ_{1}$ and $y (π + τ_{2}) = r_{2} \sin δ_{2}$ . It is important to note that the difference between the concave and convex profiles is the boundary condition value at $τ_{1}$ and $τ_{2}$ . The parameters $a = 1 / | H |$ and $b^{2} = 2 λ / H$ can be calculated as follows:

a = \frac{1}{2} \frac{y^{* 2} - r_{1}^{2} \sin^{2} δ_{1}}{y^{*} - r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1})} = \frac{1}{2} \frac{y^{* 2} - r_{2}^{2} \sin^{2} δ_{2}}{y^{*} - r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2})}

(33a)

b^{2} = y^{*} r_{1} \sin δ_{1} \frac{r_{1} \sin δ_{1} - y^{*} \sin (δ_{1} + θ_{1})}{y^{*} - r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1})} = y^{*} r_{2} \sin δ_{2} \frac{r_{2} \sin δ_{2} - y^{*} \sin (δ_{2} + θ_{2})}{y^{*} - r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2})}

(33b)

\frac{Δ p}{γ} = \frac{1}{a} < 0, Δ p = 2 γ \frac{y^{*} - r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1})}{y^{* 2} - r_{1}^{2} \sin^{2} δ_{1}} = 2 γ \frac{y^{*} - r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2})}{y^{* 2} - r_{2}^{2} \sin^{2} δ_{2}}

(33c)

The associated liquid volume and free surface area are calculated as follows:

V = π \frac{b^{4}}{a} \int_{- τ_{1}}^{τ_{2}} \frac{e - \cos t}{e + \cos t} \frac{\cos t d t}{(e + \cos t) \sqrt{e^{2} - \cos^{2} t}} - \frac{π}{3} [r_{1}^{3} {(1 - \cos δ_{1})}^{2} (2 + \cos δ_{1}) + r_{2}^{3} {(1 - \cos δ_{2})}^{2} (2 + \cos δ_{2})]

(33d)

Σ_{2} = 2 π b^{2} \int_{- τ_{1}}^{τ_{2}} \frac{1}{e + \cos t} \sqrt{\frac{e - \cos t}{e + \cos t}} t

(33e)

Proof. Consider the case $H < 0$ and $λ < 0$ . The associated nonlinear differential equation is written as

\sqrt{1 + y^{' 2}} = \frac{2 ay}{y^{2} - b^{2}}, y > b

(34)

Equation (34) therefore reads

{\begin{matrix} r_{1}^{2} \sin^{2} δ_{1} = b^{2} + 2 a r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1}) \\ r_{2}^{2} \sin^{2} δ_{2} = b^{2} + 2 a r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2}) \\ 2 a y^{*} + b^{2} = y^{* 2} \end{matrix}

(35)

The condition for the unique solution of equation (35) is similar to the convex case. The solution of the equation (35) gives the value of $a$ and $b^{2}$ as:

a = \frac{1}{2} \frac{y^{* 2} - r_{1}^{2} \sin^{2} δ_{1}}{y^{*} - r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1})} = \frac{1}{2} \frac{y^{* 2} - r_{2}^{2} \sin^{2} δ_{2}}{y^{*} - r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2})}

(36a)

b^{2} = y^{*} r_{1} \sin δ_{1} \frac{r_{1} \sin δ_{1} - y^{*} \sin (δ_{1} + θ_{1})}{y^{*} - r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1})} = y^{*} r_{2} \sin δ_{2} \frac{r_{2} \sin δ_{2} - y^{*} \sin (δ_{2} + θ_{2})}{y^{*} - r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2})}

(36b)

In the case of the concave meridian, we have $y^{*} > max (r_{1} \sin δ_{1}, r_{2} \sin δ_{2})$ ; therefore, the conditions $a > 0$ , $b^{2} > 0$ are automatically satisfied. Other results relating to the capillary bridges can be achieved in the same way as in the convex nodoid meridian. □

The solution of $a$ and $b^{2}$ independent of $y^{*}$ is calculated as follows:

a = \frac{1}{2} \frac{r_{1}^{2} \sin^{2} δ_{1} - r_{2}^{2} \sin^{2} δ_{2}}{r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1}) - r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2})}

(37a)

b^{2} = r_{1} r_{2} \sin δ_{1} \sin δ_{2} \frac{r_{2} \sin δ_{2} \sin (δ_{1} + θ_{1}) - r_{1} \sin δ_{1} \sin (δ_{2} + θ_{2})}{r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1}) - r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2})}

(37b)

respecting the following condition:

{\begin{matrix} r_{1} \sin δ_{1} < r_{2} \sin δ_{2} \\ r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1}) < r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2}) \\ r_{1} \sin δ_{1} \sin (δ_{2} + θ_{2}) > r_{2} \sin δ_{2} \sin (δ_{1} + θ_{1}) \end{matrix}

(38a) (38b) (38c)

{\begin{matrix} r_{1} \sin δ_{1} > r_{2} \sin δ_{2} \\ r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1}) > r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2}) \\ r_{1} \sin δ_{1} \sin (δ_{2} + θ_{2}) < r_{2} \sin δ_{2} \sin (δ_{1} + θ_{1}) \end{matrix}

(38d) (38e) (38f)

Figure 7 demonstrates an example of a concave upper meridian of a nodoid corresponding to the given compatible geometrical data $r_{1} = 8 mm$ , $r_{2} = 10 mm$ , $δ_{1} = 22^{°}$ , $δ_{2} = 15^{°}$ , $θ_{1} = 10^{°}$ , $θ_{2} = 10^{°}$ . As mentioned previously, the same procedure of “semi-analytical” resolution of the Young–Laplace equation is performed, except that step $6$ is adapted to the upper concave meridian of the considered case, where we have $y^{*} = a + \sqrt{a^{2} + b^{2}}$ .

Figure 7.

Example of upper concave meridian of the capillary bridge with $r_{1} = 8 mm$ , $r_{2} = 10 mm$ , $δ_{1} = 22^{°}$ , $δ_{2} = 15^{°}$ , $θ_{1} = θ_{2} = 10^{°}$ . Nodoid surface.

4.2. Unduloid with concave upper meridian

Result 4. The concave unduloid meridian case is found when the observed data are such that

y^{*} > max (r_{1} \frac{\sin δ_{1}}{\sin (δ_{1} + θ_{1})}, r_{2} \frac{\sin δ_{2}}{\sin (δ_{2} + θ_{2})})

(39)

The free surface of revolution has the form of an unduloid whose meridional profile is an arc of an elliptic Delaunay roulette. The parametric meridian of the liquid bridge is given by:

{\begin{matrix} x (t) = \frac{b^{2}}{a} \int_{0}^{t} \frac{d u}{(1 + e \cos u) \sqrt{e^{2} - \cos^{2} u}} \\ y (t) = b \sqrt{\frac{1 - e \cos t}{1 + e \cos t}}, t \in [- τ_{1} + π, τ_{2} + π] \end{matrix}

(40)

where $τ_{1}$ , $τ_{2}$ are defined from the conditions $y (π + τ_{1}) = r_{1} \sin δ_{1}$ and $y (π + τ_{2}) = r_{2} \sin δ_{2}$ . The parameters $a = 1 / | H |$ and $b^{2} = 2 λ / | H |$ can be calculated as:

a = \frac{1}{2} \frac{y^{* 2} - r_{1}^{2} \sin^{2} δ_{1}}{y^{*} - r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1})} = \frac{1}{2} \frac{y^{* 2} - r_{2}^{2} \sin^{2} δ_{2}}{y^{*} - r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2})}

(41a)

b^{2} = y^{*} r_{1} \sin δ_{1} \frac{y^{*} \sin (δ_{1} + θ_{1}) - r_{1} \sin δ_{1}}{y^{*} - r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1})} = y^{*} r_{2} \sin δ_{2} \frac{y^{*} \sin (δ_{2} + θ_{2}) - r_{2} \sin δ_{2}}{y^{*} - r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2})}

(41b)

\frac{Δ p}{γ} = \frac{1}{a} > 0, Δ p = 2 γ \frac{y^{*} - r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1})}{y^{* 2} - r_{1}^{2} \sin^{2} δ_{1}} = 2 γ \frac{y^{*} - r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2})}{y^{* 2} - r_{2}^{2} \sin^{2} δ_{2}}

(41c)

The associated liquid volume and free surface area are calculated as follows:

\begin{matrix} V = π \frac{b^{4}}{a} \int_{- τ_{1}}^{τ_{2}} \frac{e - \cos t}{e + \cos t} \frac{\cos t}{(e + \cos t) \sqrt{e^{2} - \cos^{2} t}} \\ - \frac{π}{3} [r_{1}^{3} {(1 - \cos δ_{1})}^{2} (2 + \cos δ_{1}) + r_{2}^{3} {(1 - \cos δ_{2})}^{2} (2 + \cos δ_{2})] \end{matrix}

(41d)

Σ_{2} = 2 π b^{2} \int_{- τ_{1}}^{τ_{2}} \frac{1}{e + \cos t} \sqrt{\frac{e - \cos t}{e + \cos t}} d t

(41e)

Proof. Let us consider the last case, where $H < 0$ and $λ > 0$ . The associated nonlinear differential equation reads

\sqrt{1 + y^{' 2}} = \frac{2 ay}{y^{2} + b^{2}} for y > b,

(42)

with $a > b > 0$ .

Equation (42) becomes a system of equations for $a$ and $b^{2}$ by applying the conditions in equation (10):

{\begin{matrix} b^{2} + r_{1}^{2} \sin^{2} δ_{1} = 2 a r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1}) \\ b^{2} + r_{2}^{2} \sin^{2} δ_{2} = 2 a r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2}) \\ 2 a y^{*} - b^{2} = y^{* 2} \end{matrix}

(43a) (43b) (43c)

Solving the system (equations (43a) to (43c)) provides the value of $a$ and $b^{2}$ as follows:

a = \frac{1}{2} \frac{y^{* 2} - r_{1}^{2} \sin^{2} δ_{1}}{y^{*} - r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1})} = \frac{1}{2} \frac{y^{* 2} - r_{2}^{2} \sin^{2} δ_{2}}{y^{*} - r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2})}

(44a)

b^{2} = y^{*} r_{1} \sin δ_{1} \frac{y^{*} \sin (δ_{1} + θ_{1}) - r_{1} \sin δ_{1}}{y^{*} - r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1})} = y^{*} r_{2} \sin δ_{2} \frac{y^{*} \sin (δ_{2} + θ_{2}) - r_{2} \sin δ_{2}}{y^{*} - r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2})}

(44b)

respecting the compatibility relation determined by equations (43a) and (43b)

r_{1} \sin δ_{1} (r_{1} \sin δ_{1} - 2 a \sin (δ_{1} + θ_{1})) = r_{2} \sin δ_{2} (r_{2} \sin δ_{2} - 2 a \sin (δ_{2} + θ_{2})) .

(45)

In the case of a concave meridian, $y^{*} > max (r_{1} \sin δ_{1}, r_{2} \sin δ_{2})$ ; consequently, the condition $a > 0$ , $b^{2} > 0$ makes

y^{*} > max (r_{1} \frac{\sin δ_{1}}{\sin δ_{1} + θ_{1}}, r_{2} \frac{\sin δ_{2}}{\sin δ_{2} + θ_{2}}) .

The solution of ( $a$ , $b^{2}$ ) apart from $y^{*}$ is presented as follows:

a = \frac{1}{2} \frac{r_{1}^{2} \sin^{2} δ_{1} - r_{2}^{2} \sin^{2} δ_{2}}{r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1}) - r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2})}

(46a)

b^{2} = r_{1} r_{2} \sin δ_{1} \sin δ_{2} \frac{r_{1} \sin δ_{1} \sin (δ_{2} + θ_{2}) - r_{2} \sin δ_{2} \sin (δ_{1} + θ_{1})}{r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1}) - r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2})}

(46b)

respecting the following conditions:

{\begin{matrix} r_{1} \sin δ_{1} < r_{2} \sin δ_{2} \\ r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1}) < r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2}) \\ r_{1} \sin δ_{1} \sin (δ_{2} + θ_{2}) < r_{2} \sin δ_{2} \sin (δ_{1} + θ_{1}) \end{matrix}

(47a) (47b) (47c)

{\begin{matrix} r_{1} \sin δ_{1} > r_{2} \sin δ_{2} \\ r_{1} \sin δ_{1} \sin (δ_{1} + θ_{1}) > r_{2} \sin δ_{2} \sin (δ_{2} + θ_{2}) \\ r_{1} \sin δ_{1} \sin (δ_{2} + θ_{2}) > r_{2} \sin δ_{2} \sin (δ_{1} + θ_{1}) \end{matrix}

(47d) (47e) (47f)

□

Figure 8 demonstrates an example of a concave upper meridional profile of an unduloid liquid bridge for given compatible geometrical data $r_{1} = 2 mm$ , $r_{2} = 9 mm$ , $δ_{1} = 20^{°}$ , $δ_{2} = 10^{°}$ , $θ_{1} = 90^{°}$ , $θ_{2} = 90^{°}$ . The neck radius in this case is calculated as $y^{*} = a + \sqrt{a^{2} - b^{2}}$ .

Figure 8.

Example of upper concave meridian of the capillary bridge with $r_{1} = 2 mm$ , $r_{2} = 9 mm$ , $δ_{1} = 20^{°}$ , $δ_{2} = 10^{°}$ , $θ_{1} = 90^{°}$ , $θ_{2} = 90^{°}$ . Unduloid surface.

4.2.1. Special case where the liquid profile has an inflection point

It is important to note that a special case of unduloid may be encountered in practice, where the meridional profile comprises both convex and concave parts. In other words, the upper meridian (and also the lower one) contains an inflection point $I (x_{I}, y_{I})$ located on the profile, marking the transition between convex and concave meridional profiles ( $y^{″} (x_{I}) = 0$ ). This special (and sensitive) case only occurs for particular values of the geometrical parameters and happens with the parameterization of the upper meridian for the unduloid case. For example, such an unduloid with inflection point is presented in Figure 9 for $r_{1} = 1 mm$ , $r_{2} = 10 mm$ , $δ_{1} = 90^{°}$ , $δ_{2} = 25^{°}$ , $θ_{1} = 0^{°}$ , $θ_{2} = 65^{°}$ .

Figure 9.

Meridional profile with inflection points, unduloid case, with $r_{1} = 1 mm$ , $r_{2} = 10 mm$ , $δ_{1} = 90^{°}$ , $δ_{2} = 25^{°}$ , $θ_{1} = 0^{°}$ , $θ_{2} = 65^{°}$ .

A summary of all the investigated cases is given in Table 1.

Table 1.

Summary of properties of capillary bridges and their meridional profiles.

Criterion	Surface	Meridional profile	Suction, $s$	$H$ , $λ$
Convex meridian $max (r_{1, 2} \sin δ_{1, 2} \sin (δ_{1, 2} + θ_{1, 2}) < y^{*} < min (r_{1, 2} \sin δ_{1, 2})$	Nodoid	Delaunay hyperbolic roulette	$s > 0$	$H > 0$ , $λ > 0$
Convex meridian $0 < y^{*} < min (r_{1, 2} \sin δ_{1, 2} \sin (δ_{1, 2} + θ_{1, 2})$	Unduloid	Delaunay elliptic roulette	$s < 0$	$H < 0$ , $λ > 0$
Concave meridian $max (r_{1, 2} \sin δ_{1, 2}) < y^{*} < min (r_{1, 2} \frac{\sin δ_{1, 2}}{\sin (δ_{1, 2} + θ_{1, 2})})$	Nodoid	Delaunay hyperbolic roulette	$s < 0$	$H < 0$ , $λ < 0$
Concave meridian $y^{*} > max (r_{1, 2} \frac{\sin δ_{1, 2}}{\sin (δ_{1, 2} + θ_{1, 2})})$	Unduloid	Delaunay elliptic roulette	$s < 0$	$H < 0$ , $λ > 0$

5. Conclusion

The properties of capillary bridges between two particles of different sizes are investigated in this manuscript. Two types of upper meridians of capillary bridge profiles (convex and concave) are considered. In each case, the exact geometry of the capillary bridge is determined as a solution of the Young–Laplace equation for given compatible geometrical data. The corresponding capillary force, volume, and free surface area of the liquid bridge are determined explicitly. Some examples of different possible configurations, solutions of the Young–Laplace equation, have been given to validate the correctness of the addressed analytic approach.

The radius of unequal-sized spheres being fixed, we obtain an explicit criterion based on the observation of the contact and wetting angle and the gorge radius, to classify the nature of the surface of revolution and its associated meridian (portion of nodoid or unduloid, with concave or convex meridian). Moreover, a special case of a capillary bridge with a meridian presenting one inflection point is revealed.

In all the cases encountered, illustrative examples are addressed to enforce the analytical approach presented. They correspond to a solution of the Young–Laplace equation obtained for given compatible geometrical data. Of course, they may also be observed experimentally for the same values of geometrical data. A comparison with experimental results obtained for polydisperse particles, following the démarche presented in Gagneux et al. [12] for the monodisperse case, will constitute the aim of a subsequent work.

Footnotes

Appendix

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Notes

References

Sauer

RA.

A contact theory for surface tension driven systems. Math Mech Solids 2014; 21(3): 305–325.

Mitarai

Nori

Wet granular materials. Adv Phys 2006; 55(1–2): 1–45.

Radl

Kalvoda

Glasser

, et al. Mixing characteristics of wet granular matter in a bladed mixer. Powder Technol 2010; 200(3): 171–189.

Schwarze

Gladkyy

Uhlig

, et al. Rheology of weakly wetted granular materials: A comparison of experimental and numerical data. Granular Matter 2013; 15(4): 455–465.

Erle

Dyson

Morrow

NR.

Liquid bridges between cylinders, in a torus, and between spheres. AIChE J 1971; 17(1): 115–121.

Lian

Thornton

Adams

MJ.

A theoretical study of the liquid bridge forces between two rigid spherical bodies. J Colloid Interface Sci 1993; 161(1): 138–147.

Vogel

TI.

Convex, rotationally symmetric liquid bridges between spheres. Pacific J Math 2006; 224: 367–377.

Vogel

TI.

Liquid bridges between balls: The small volume instability. J Math Fluid Mech 2013; 15(2): 397–413.

Gagneux

Millet

Analytic calculation of capillary bridge properties deduced as an inverse problem from experimental data. Transp Porous Media 2014; 105(1): 117–139.

10.

Willett

Adams

Johnson

, et al. Capillary bridges between two spherical bodies. Langmuir 2000; 16(24): 9396–9405.

11.

Rabinovich

Esayanur

Moudgil

BM.

Capillary forces between two spheres with a fixed volume liquid bridge: Theory and experiment. Langmuir 2005; 21(24): 10992–10997.

12.

Gagneux

Millet

Mielniczuk

, et al. Theoretical and experimental study of pendular regime in unsaturated granular media. Eur J Environ Civil Eng 2016; 21(7–8): 840–853.

13.

Rubinstein

Fel

LG.

Theory of pendular rings revisited. arXiv 2012; arXiv:12077096.

14.

Megias-Alguacil

Gauckler

LJ.

Capillary forces between two solid spheres linked by a concave liquid bridge: Regions of existence and forces mapping. AIChE J 2009; 55(5): 1103–1109.

15.

Kruyt

Millet

An analytical theory for the capillary bridge force between spheres. J Fluid Mech 2017; 812: 129–151.

16.

Gagneux

Millet

An analytical framework for evaluating the cohesion effects of coalescence between capillary bridges. Granular Matter 2016; 18(2): 16.

17.

Placidi

Dell’Isola

Ianiro

, et al. Variational formulation of pre-stressed solid–fluid mixture theory, with an application to wave phenomena. Eur J Mech A Solids 2008; 27(4): 582–606.

18.

Dell’Isola

Madeo

Seppecher

Boundary conditions at fluid-permeable interfaces in porous media: A variational approach. Int J Solids Struct 2009; 46(17): 3150–3164.

19.

Sauer

Duong

TX.

On the theoretical foundations of thin solid and liquid shells. Math Mech Solids 2017; 22(3): 343–371.

20.

Qiang-Nian

Jia-Qi

Feng-Xi

Exact solution for capillary bridges properties by shooting method. Z Naturforsch A Phys Sci 2017; 72(4): 315–320.

21.

Gras

Delenne

El Youssoufi

MS.

Study of capillary interaction between two grains: A new experimental device with suction control. Granular Matter 2013; 15(1): 49–56.

22.

Delaunay

Sur la surface de révolution dont la courbure moyenne est constante. J Math Pures Appl 1841; 1841: 309–314.