Note on Capillary Rise in Elliptic and Semielliptic Tubes

. In this note, the dimensions of elliptic and semielliptic tubes in which the capillary rise starts oscillating about its Jurin ’ s height are determined. Comparisons with the capillary rise in the classical circular and in semicircular tubes are made.


Introduction
The literature on the phenomenon of capillary rise in tubes is quite rich due to numerous applications such the ink industry and flow through porous media.Modeling the dynamics of the rise dates back to the work of Lucas [1] and Washburn [2].Neglecting gravity and kinetic effects, the famous Lucas-Washburn equation uses a momentum balance to describe the change of the meniscus height with time.The neglected forces were later added to build up the well-known equation described here.
The capillary rise in a tube can be seen as a transition from an initial to a final state.The final column height is known as equilibrium or Jurin's height.In case of a monotonous rise, the viscous contribution leads to a smooth transition to the equilibrium height without any overshoots.This can be compared to the theory of linearly damped oscillators (underdamped, critically damped, and overdamped, based on the damping ratio).The initial state of the oscillator, such as a pendulum or a spring-mass system, is out of equilibrium.The oscillator reaches equilibrium in different ways, depending on the damping of the system.The monotonous rise of a liquid in a capillary is an overdamped case.The dimensions of a tube under which the rise remains monotone have been determined for circular geometry, Masoodi et al. [3].In this note, we analyze tubes of elliptic and semielliptic cross sections from which the circular and semicircular tubes are special cases.
The height, hðtÞ, of a liquid of density ρ and viscosity μ, rises in a vertical capillary tube of cross-sectional area A and perimeter C according to where γ is the surface tension, θ is the contact angle, ΔP is the pressure drop in the tube, and g is the acceleration due to gravity, Masoodi et al. [3].It is noted here that the celebrated equation (1) neglects the effect of the displaced fluid.
The effect of the displaced fluid which is not considered here is discussed by Shan et al. [4].Using Hagen-Poiseuille law, with the rate of rise inside the tube being the same as the average velocity of the liquid, the pressure drop can be written as where R = 2A/C is the hydraulic radius, L is the length of the tube, Q is the volume flow rate, and ψ is a dimensionless function that depends on the geometry of the crosssectional area of the tube.The function ψ, for a tube of any cross-sectional area, can be extracted from Hagen-Poiseuille flow solution if recast in the form of equation (2).For a circular tube, ψ = 1/8.
Equation ( 1) can now be rewritten as The left side of equation ( 3) is the capillary pressure, P c = 2γ cos θ/R.The equation is subject to the initial conditions hð0Þ = 0 and dh/dtð0Þ = 0.It is noticed here that for large t, h ⟶ h e = ð1/ρ gÞð2γ cos θ/RÞ = P c /ρ g, where h e is the equilibrium position known as Jurin's height.
There are several dimensionless scaling methods for equation (3) , Fries and Dreyer [5].We choose the method of Masoodi et al. [3] to write equation (3) in dimensionless form as where The dimensionless Bond and Ohnesorge numbers are defined, respectively, as Bo = ρ g R 2 /γ and Oh = μ/ ffiffiffiffiffiffiffiffiffi ffi ρ R γ p .It can be shown, Płociniczak and Świtała [6], that the condition ω = 1/4 is the boundary between a monotone and oscillatory rise.If ω 1/4, the rise of the liquid starts to oscillate about its equilibrium position (Jurin's height).

Elliptic Tubes
Solving the Hagen-Poiseuille flow equation in an elliptic tube, Duan and Muzychka [7], with major and minor axes of 2a and 2b, respectively, and recasting the solution in the form of equation ( 2), one finds that ψ = ðEð1 − ε 2 ÞÞ 2 /π 2 ð1 , where ε = b/a is the axis ratio, and Eð⋅Þ is the complete elliptic integral of the second kind.It follows that the rise of the liquid in an elliptic tube is monotone if ω = ðBo/OhÞ 2 ψ 2 /2 cos θ < 1/4, or equivalently Figure 1(a) shows the supremum of ð1/ ffiffiffiffiffiffiffiffiffiffi cos θ p ÞðBo/OhÞ for which the rise is monotone.It is noted here that the limiting values for a circular tube (i.e., as ε ⟶ 1) are ψ = 1/8 and ð1/ ffiffiffiffiffiffiffiffiffiffi cos θ p ÞðBo/OhÞ < 8/ ffiffi ffi 2 p .Given the hydraulic radius for an elliptic tube R = π b/2 Eð1 − ε 2 Þ and the expressions for Bond and Ohnesorge numbers, one can, for a specific value of the axis ratio, find the supremum length of the minor (or equivalently the major) axis of the tube that maintains a monotone rise.The condition for maintaining a monotone rise can be shown to follow The limiting value for a circular tube, therefore, is b < 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi γμ 2 cos θ/g 2 ρ 3 5 p . This limiting value is identically obtained by Masoodi et al. [3]. Figure 2(a) shows the variation of the maximum value b with the aspect ratio for water at 20 °C (ρ = 998:23kg m −3 , μ = 1:002 × 10 −3 Pa s, γ = 72:86 × 10 −3 N m −1 , and θ = 0 ∘ ).The corresponding equilibrium (Jurin's) heights (h e = 2γ cos θ/ρgR) are also plotted. 2 Journal of Applied Mathematics

Semielliptic Tubes
The solution of the Hagen-Poiseuille flow equation in a semielliptic tube obtained by Alassar and Abushoshah [8], recast in the form of equation ( 2), gives the geometry func- It follows that the rise of the liquid in a semielliptic tube is monotone if Once again, given R = πb/ð2 + 2Eð1 − ε 2 ÞÞ and the expressions for Bond and Ohnesorge numbers, then for a specific value of the axis ratio, the condition for maintaining a monotone rise in a semielliptic tube can be shown to be The limiting value for a semicircular tube (again as ε . Figure 2 (b) shows these maximum values for water at 20 °C along with the corresponding Jurin's heights.

Conclusion
The critical lengths of the major and minor axes of elliptic and semielliptic tubes in which a transition from a monotone to oscillatory capillary rise takes place have been determined (conditions ( 6) and ( 8)).The first condition ( 6) is shown to reduce to the well-known result of a circular tube.Condition (8), in the limit as the axis ratio approaches unity, provides the critical dimensions for a semicircular tube.Surface tension ψ: Geometry function ω: Dimensionless parameter defined in equation ( 4) ε: Axis ratio ε = b/a.

Nomenclature 2a :Figure 2 :
Figure 2: Maximum b to maintain monotone rise for water at 20 °C and the corresponding equilibrium height.