ABSTRACT
For very small liquid droplets, the gravitational effect may be negligibly small compared to the interfacial tension, so the term ∆p/γ on the left side of Eq. (7) can be considered constant along the whole interface. Eq. (7) can be then integrated to:
(8)
where C1 (=∆p/γ) and C2 are the system constants, which are restricted to the following boundary conditions
(9)
Combination of Eqs. (8) and (9) gives:
(10)
Eq. (10), together with the relationship:
(11)
allows now the gradient of the drop profile, dy/dx, to be expressed as a function of x, which, after substitution for in Eq. (11) with the expression of Eq. (10) followed by expansion and simplification as described in detail by Carroll [19], can be written as:
(12)
where a is defined as:
(13)
With the following transformation: x2=h2(1−k2 sin2 φ),
(14)
where
(15)
Eq. (12) can be rewritten as
(16)
which can further be integrated to give y=±[a·r·F(φ, k)+h·E(φ, k)],
(17)
where F(φ, k) and E(φ, k) are the Legendre standard incomplete elliptic integrals of the first and second kind, respectively, and are defined as
(18)
(19)
Equations (12)–(15) and (17) give the expression of the drop profile in a drop-on-fiber system where the gravitational effect is negligibly small compared to the interfacial tension. To make it easier for the later generalization, it is useful to transfer these equations into their corresponding dimensionless forms by dividing all the used dimensional variables by the radius of the fiber rf involved in the system:
(13a) X2=H2(1−k2 sin2 φ),
(14a) k2=1−a2/H2,
(15a) Y=±[a·F(φ, k)+H·E(φ, k)],
(17a)
where X, Y, and H are the dimensionless counterparts of x, y, and h, respectively (in this text we use capital and lowercase letters to describe the dimensionless and dimensional variables, respectively). As can be seen from these relations, a drop profile, as represented by a collection of (X, Y)-coordinates, may be expressed as a function of H and θ and can be obtained through the numeric evaluation of the elliptic integrals F and E in Eq. (14a) and (16a). The Legendre’s standard elliptic integrals F and E can be transformed to the Carlson’s elliptic integrals [14, 15] of the first and the second kind, RF and RD, which are more suitable for the numerical evaluation [16, 17]:
F(φ, k)=sin φ·RF(cos2 φ, 1−k2 sin2 φ, 1), (20)
(21)
B. Diverse Characteristic Drop Parameters
(a) Drop Length. Combining Eq. (17a) with Eqs. (20) and (21), one obtains the dimensionless length L of a droplet with maximum height H and contact angle θ:
(22)
The right side in Eq. (22) is a function of H and θ only [as denoted by f(H, θ)], which means that the contact angle θ in a drop-on-fiber system can be unambiguously determined from its length L and height H.
