Low frequency of a deforming capillary vibration,part 1: Mathematical model

Abstract

The capillary effect has wide applications from biology and textile engineering to nanotechnology especially the micro-electromechanical systems and microfluidics device. The capillary vibration significantly affects its mass transmission. This paper establishes a nonlinear oscillator of a deforming capillary tube. The geometric potential theory is used to estimate the capillary force. The paper reveals that the low-frequency property of capillary vibration plays an important role not only in life but also engineering applications.

Introduction

Capillary phenomenon appears everywhere in our everyday life, and it plays an important role in blood transmission in animals, water transmission in plants, and heat transmission in porous media. Capillary vibration significantly affects the capillary rise or capillary pressure, as a result the mass transfer or heat transfer will be greatly affected. The air permeability^1,2 through a nanofiber membrane^3–6 is mainly affected by the capillary effect, which can be explained by the geometric potential theory,^7–9 implying that any boundary can produce a boundary-induced force. Liu et al. ¹⁰ established a fractional oscillator for ion release from a follow fiber. Lin and Yao ¹¹ gave an experimental verification of the fractional model, and suggested a simple method to estimate its frequency. ¹² All the above capillary vibrations assumed that the capillary tube is small and uniform. However, in most porous media, capillary flows are nonuniform capillaries. For example, wicking efficiency in the small nonuniform capillary between fibers is crucial for the comfort of fabric. It has been reported that the degree of nonuniform decided by yarn twist has a close relationship with the liquid transport efficiency by capillary. ¹³ The alternative frequency between the diverging section and converging section of the capillary greatly affected the wicking velocity. This paper will study a deforming capillary with periodic boundaries and study the low frequency of capillary frequency on the capillary rise.

Lotus-rhizome-node-like deforming capillary

In this paper, we consider a deforming capillary tube with Lotus-rhizome-node-like structure as illustrated in Figure 1. Such a model can be used to describe a nonuniform capillary in porous media, especially the fabrics.

Figure 1.

A deforming capillary tube with Lotus-rhizome-node-like structure.

As a preliminary study, we consider deforming capillary tube as a periodic boundary, which can be written as

$$R=R_0+a\sin \hspace{0.17em}\omega x$$

(1)

where R is the tube radius, $R_0$ is the average radius of the capillary tube, a and $\omega$ are periodic parameters for the tube boundary.

By Newton’s second law, capillary vibration can be written as

$$F=m\ddot{x}$$

(2)

where x is the distance from the equilibrium ( $x_0$ ), m is the total mass in the tube, F is the force arising in capillary force and fluid weight in the tube.

The mass in the tube is

$$m=m_0+A\rho x$$

(3)

where $m_0$ is the total mass at equilibrium, A is the section area of the tube, and $\rho$ is the fluid density.

The capillary force can be calculated by the geometric potential theory^7–9

$$f\propto \frac{1}{r^2}$$

(4)

where f is the capillary force, r is the meniscus radius of the free surface. Generally, the meniscus radius scales with the tube size

$$r\propto R$$

(5)

The capillary force can be written as

$$f=\frac{k}{R^2}$$

(6)

where k is a constant.

The acting force F given in equation (2) can be calculated as

$$F=f-\mathit{mg}=\frac{k}{R^2}-(m_0+A\rho x)g$$

(7)

At equilibrium, the force balance requires

$$\frac{k}{R_0^2}=m_{0}g$$

(8)

Equation (7) can be simplified as

$$F=-2\frac{\mathit{ak}}{R_0^3}\sin \omega x-A\rho \mathit{gx}$$

(9)

We, therefore, obtain the following oscillator for the capillary vibration

$$m\ddot{x}+2\frac{\mathit{ak}}{R_0^3}\sin \omega +A\rho \mathit{gx}=0$$

(10)

or

$$\ddot{x}+\epsilon \sin \omega x+{\omega }_0^{2}x=0$$

(11)

where ${\omega }_0^2=\frac{A\rho g}{m}\approx \frac{\pi R_0^2\rho g}{m}$ , $\epsilon =2\frac{\mathit{ak}}{R_0^3}$ .

Equation (11) can be approximated by the following Duffing equation

$$\ddot{x}+\left(\epsilon \omega +{\omega }_0^2\right)x-\frac{1}{6}\epsilon {\omega }^3{x}^3=0$$

(12)

Homotopy perturbation method

Equation (12) can be effectively solved by the variational iteration method, ¹⁴ the homotopy perturbation method,^15,16 or He’s frequency formulation. ¹⁷

We construct the following homotopy equation

$$\ddot{x}+\left(\epsilon \omega +{\omega }_0^2\right)x-\frac{1}{6}\epsilon {\omega }^{3}p{\omega }^3=0,\hspace{1em}p\in \left[0,1\right]$$

(13)

where p is a homotopy parameter changing from zero to unit.

We assume that the solution can be expanded into the form

$$x=x_0+p{x}_1+p^2{x}_2+\cdots$$

(14)

and the coefficient of the linear term can be expanded as^18–20

$$\epsilon \omega +{\omega }_0^2={\Omega }^2+p{\Omega }_1+p^2{\Omega }_2+\cdots$$

(15)

where $\Omega$ is the frequency of the capillary oscillator. Proceeding the way as the standard homotopy perturbation method, we have

$${\ddot{x}}_0^2+{\Omega }^2{x}_0=0,\hspace{1em}{x}_0\left(0\right)=B,\hspace{1em}{\dot{x}}_1\left(0\right)=0$$

(16)

$${\ddot{x}}_1+{\Omega }^2{x}_1+{\Omega }_1{x}_0-\frac{1}{6}\epsilon {\omega }^3{x}_0^3=0,\hspace{1em}{x}_1\left(0\right)=0,\hspace{1em}{\dot{x}}_1\left(0\right)=0$$

(17)

where we assume that the amplitude is B. The solution of equation (16) is

$$x_0=B\cos \Omega t$$

(18)

In view of equation (18), equation (17) becomes

$${\ddot{x}}_1+{\Omega }^2{x}_1+{\Omega }_1\cos \Omega t-\frac{1}{6}\epsilon {\omega }^3{B}^3{\cos }^3\Omega t=0$$

(19)

By the following identity

$${\cos }^3\Omega t=\frac{3}{4}\cos \Omega t+\frac{1}{4}\cos 3\Omega t$$

(20)

Equation (19) becomes

$${\ddot{x}}_1+{\Omega }^2{x}_1+\left({\Omega }_{1}B-\frac{1}{8}\epsilon {\omega }^3{B}^3\right)\cos \Omega t-\frac{1}{24}\epsilon {\omega }^3{B}^3\cos 3\Omega t=0$$

(21)

No secular term in $x_1$ requires that the coefficient of $\cos \Omega t$ must be zero

$${\Omega }_{1}B-\frac{1}{8}\epsilon {\omega }^3{B}^3=0$$

(22)

From equation (22), we can find the value of ${\Omega }_1$

$${\Omega }_1=\frac{1}{8}\epsilon {\omega }^3{B}^2$$

(23)

If the first approximate is searched for, we set p = 1 and ${\Omega }_i=0(i>1)$ in equation (15)

$$\epsilon \omega +{\omega }_0^2={\Omega }^2+\frac{1}{8}\epsilon {\omega }^3{B}^2$$

(24)

This results in

$${\Omega }^2=\epsilon \omega +{\omega }_0^2-\frac{1}{8}\epsilon {\omega }^3{B}^2$$

(25)

or

$$B=\sqrt{\frac{8({\Omega }^2-\epsilon \omega -{\omega }_0^2)}{\epsilon {\omega }^3}}$$

(26)

where $\Omega$ is the frequency.

Discussion and conclusions

The capillary fluid moves extremely slow, and its vibration near its equilibrium has extremely low frequency, which plays an important role in both life and applications. For a high tree, the water transmission in a hot season is of great importance. The low frequency can effectively guarantee both safe water transmission and extremely low loss of water. In textile engineering, the low frequency of capillary vibration leads to an effective perspiration absorption.

In this paper, we obtain an oscillator for the capillary vibration on the tube with periodic boundary. The frequency of the capillary vibration affects greatly the capillary effect, and this paper elucidates the main factors affecting the frequency, which can be used for optimal design of a porous medium. In the forthcoming papers, we will study the capillary effect numerically and experimentally.