Analytical Study of Reacting Force of Liquid …lab2/01daygame/naiyou/bunken...Capillarity and...

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International Tribology Conference, Hiroshima 2011 October 30 – November 3, 2011, Hiroshima, Japan -60 -40 -20 0 20 40 60 80 0 10 20 30 40 50 60 70 80 0 20 40 60 80 100 120 k m , N/m θ E , deg. f , Hz 0 0 0 0 -20 -20 -20 -20 -20 -20 -20 0 0 0 0 0 0 20 20 20 20 20 40 40 40 60 θ E , deg. 0 10 20 30 40 50 60 70 80 f, Hz 0 20 40 60 80 100 120 140 Analytical Study of Reacting Force of Liquid Meniscus Bridge (Basic Characteristics by Dynamic Contact Angle Model Considering Capillary Force) Tadahiko Itani * , Kyoko Matsuda, Hiroshige Matsuoka and Shigehisa Fukui Department of Mechanical and Aerospace Engineering, Graduate School of Engineering, Tottori University 4-101 Minami, Koyama, Tottori 680-8552, Japan * Corresponding author: [email protected] 1. Introduction The flying height of a magnetic head has rapidly decreased and contact between the head slider and the disk surface has become unavoidable [1]-[4]. A meniscus bridge of liquid lubricant is formed between the head slider and the magnetic disk. In the present study, the mechanical characteristics of a liquid meniscus bridge confined between vibrating solid surfaces are investigated theoretically considering the dynamical contact angle. 2. Theory 2.1 Dynamical contact angle The dynamical contact angle θ D at the triple line shown in Fig. 1 is considered in the present study. The relationship between the speed of the triple line v c and the dynamical contact angle θ D is given by [5] (1) where γ is the surface tension of the liquid, η is the viscosity, and θ E is the equilibrium (static) contact angle. The dimensionless coefficient l is related to the size of the drop and the size of the liquid molecule, and its numerical value ranges from 15 to 20. (a) Advancing (b) Receding Fig.1 Dynamical contact angle 2.2 Theoretical model A meniscus bridge between a sphere and a plane is shown in Fig. 2. The Reynolds equation based on a thin film assumption is solved in order to analyze the dynamic force. The boundary condition is that the pressure at the meniscus periphery is given by the Laplace pressure. The capillary force along the triple line is also taken into account. Fig. 2 Dynamical contact angle (DCA) model of the liquid meniscus bridge between a sphere and a plane 3. Analytical results: spring constant The spring constant of the liquid meniscus bridge k m is given by the following equation: (3) where C 1 and C 2 are given by (4) (5) The spring constant of the liquid meniscus bridge k m is shown in Fig. 3. The spring constant k m depends not only on θ E but also on the excitation frequency f (=ω /2π). The value of k m can be positive even if θ E is less than 90 degrees. 4. Conclusion The dynamical contact angle was introduced in order to analyze the characteristics of a liquid meniscus bridge between a sphere and a plane. The spring constant of a meniscus bridge was derived by solving the Reynolds equation. The spring constant depends on the excitation frequency and can have positive values under the condition that the contact angle is less than 90 degrees. References [1] Matsuoka, H., Fukui, S. and Kato, T., Tribologists, Vol. 45, No. 10, 2000, pp. 757-768. [2] Matsuda, K., Matsuoka, H. and Fukui, S., Pro. JAST Tribology Conference Fukui, 2010, pp. 93-94. (in Japanese) [3] Inada, K., Kunitomi, S., Matsuoka, H. and Fukui, S., Proc. Mechanical Engineering Congress, 2006, pp. 619-620. (in Japanese) [4] Ishihara, K., Kotakemori, Y., Matsuoka, H. and Fukui, S., Proc. JSME Chugoku-Shikoku Branch Annual Meeting, 2009, pp. 291-292. (in Japanese) [5] Gennes P.-Gi., Brochard-Wyart F., Quere D., Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves, 2003, Springer (a) 3-D view (b) Contour view of (a) Fig. 3 Effects of contact angle and excitation frequency on the spring constant of the meniscus bridge ( ) * * tan cos cos , , 3 c D E D V v V l γ θ θ θ η = = h 0 h(r,t) r 10 z h(r,0) r φ r m (t) r m0 R θ D (t) θ E θ E θ D (t) r 1 (t) δ r 20 r 2 (t) θ Ε θ Ε θ D θ D v c v c θ Ε θ Ε θ D θ D c c 2 1 2 2 10 10 0 2 2 * 2 2 2 0 10 1 3 1 2 cos 3 2 8 1 1 sin tan 1 2 1 2 8 8 E m E E C C r lr h k C C h V r C ω πγ θ ω θ θ =− + + 2 2 1 10 0 0 2 2 2 2 10 10 1 1 sin 2 sin 1 2 4 cos 2 cos 1 2 1 2 8 8 E E E E C C r h h C C R r r C θ θ θ ω θ + + + ( ) { } * 2 1 0 2 cos sin 3 2 tan 2 E E E E V C lh θ θ θ π θ = ( ) ( ) { } 2 0 2 10 2 tan 1 2 tan . E E E h C r π θ θ θ = +

Transcript of Analytical Study of Reacting Force of Liquid …lab2/01daygame/naiyou/bunken...Capillarity and...

Page 1: Analytical Study of Reacting Force of Liquid …lab2/01daygame/naiyou/bunken...Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves, 2003, Springer m (a) 3-D view (b) Contour

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Analytical Study of Reacting Force of Liquid Meniscus Bridge (Basic Characteristics by Dynamic Contact Angle Model Considering Capillary Force)

Tadahiko Itani*, Kyoko Matsuda, Hiroshige Matsuoka and Shigehisa Fukui

Department of Mechanical and Aerospace Engineering, Graduate School of Engineering, Tottori University

4-101 Minami, Koyama, Tottori 680-8552, Japan *Corresponding author: [email protected]

1. Introduction

The flying height of a magnetic head has rapidly decreased and contact between the head slider and the disk surface has become unavoidable [1]-[4]. A meniscus bridge of liquid lubricant is formed between the head slider and the magnetic disk. In the present study, the mechanical characteristics of a liquid meniscus bridge confined between vibrating solid surfaces are investigated theoretically considering the dynamical contact angle. 2. Theory

2.1 Dynamical contact angle The dynamical contact angle θD at the triple line shown in Fig. 1 is considered in the present study. The relationship between the speed of the triple line vc and the dynamical contact angle θD is given by [5]

(1)

where γ is the surface tension of the liquid, η is the viscosity, and θE is the equilibrium (static) contact angle. The dimensionless coefficient l is related to the size of the drop and the size of the liquid molecule, and its numerical value ranges from 15 to 20.

(a) Advancing (b) Receding Fig.1 Dynamical contact angle

2.2 Theoretical model A meniscus bridge between a sphere and a plane is shown in Fig. 2. The Reynolds equation based on a thin film assumption is solved in order to analyze the dynamic force. The boundary condition is that the pressure at the meniscus periphery is given by the Laplace pressure. The capillary force along the triple line is also taken into account.

Fig. 2 Dynamical contact angle (DCA) model of the liquid meniscus bridge between a sphere and a plane

3. Analytical results: spring constant

The spring constant of the liquid meniscus bridge km is given by the following equation:

(3) where C1 and C2 are given by

(4)

(5)

The spring constant of the liquid meniscus bridge km is shown in Fig. 3. The spring constant km depends not only on θE but also on the excitation frequency f (=ω /2π). The value of km can be positive even if θE is less than 90 degrees.

4. Conclusion

The dynamical contact angle was introduced in order to analyze the characteristics of a liquid meniscus bridge between a sphere and a plane. The spring constant of a meniscus bridge was derived by solving the Reynolds equation. The spring constant depends on the excitation frequency and can have positive values under the condition that the contact angle is less than 90 degrees.

References [1] Matsuoka, H., Fukui, S. and Kato, T., Tribologists, Vol. 45, No. 10,

2000, pp. 757-768. [2] Matsuda, K., Matsuoka, H. and Fukui, S., Pro. JAST Tribology

Conference Fukui, 2010, pp. 93-94. (in Japanese) [3] Inada, K., Kunitomi, S., Matsuoka, H. and Fukui, S., Proc.

Mechanical Engineering Congress, 2006, pp. 619-620. (in Japanese)

[4] Ishihara, K., Kotakemori, Y., Matsuoka, H. and Fukui, S., Proc. JSME Chugoku-Shikoku Branch Annual Meeting, 2009, pp. 291-292. (in Japanese)

[5] Gennes P.-Gi., Brochard-Wyart F., Quere D., Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves, 2003, Springer

(a) 3-D view (b) Contour view of (a)

Fig. 3 Effects of contact angle and excitation frequency on the spring constant of the meniscus bridge

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